62
FOR AID USE ONLY AGENCY FOR INTERNATIONAL DEVELOPMENT WASHINGTON. D. C. 20523 BIBLIOGRAPHIC INPUT SHEET A. PRIMARY I. SUBJECT Agriculture API0-0000-G518 CLASSI- E e.SECONDARY FICATION Water resources and management--Colombia 2.TITLE AND SIBTITLE A hydrid computer model of the hydrologic syster within the Atlantico 3 area of Colombia,South America 3. AUTHOR(S) Riley,J.P.; Israelsen,E.K. 5. NUMBER OF PAGES 6. ARC NUMBER 4. DOCUMENT DATE 1971 71p. ARC 7. REFERENCE ORGANIZATION NAME AND ADDRESS Utah State 8. SUPPLEMENTARY NOTES (SponsoringOrganiation,Publiahras, Avaliability) 9. ABSTRACT 11. PRICE OF DOCUMENT 10. CONTROL NUMBER PN-RAA- 0 3 13. PROJECT NUMBER 12. DESCRIPTORS Atlantico 3 Project 14. CONTRACT NUMBER Colombia Farm crops CSD-2167 Res. i1omput. programs 15 TYPE OF DOCUMENT AID 890-1 (4"74)

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Page 1: FOR AID USE ONLY

FOR AID USE ONLYAGENCY FOR INTERNATIONAL DEVELOPMENT

WASHINGTON D C 20523

BIBLIOGRAPHIC INPUT SHEET A PRIMARY

I SUBJECT Agriculture API0-0000-G518 CLASSI- E

eSECONDARYFICATION Water resources and management--Colombia

2TITLE AND SIBTITLE

A hydrid computer model of the hydrologic syster within the Atlantico 3 area

of ColombiaSouth America 3 AUTHOR(S)

RileyJP IsraelsenEK

5 NUMBER OF PAGES 6 ARC NUMBER4 DOCUMENT DATE

1971 71p ARC

7 REFERENCE ORGANIZATION NAME AND ADDRESS

Utah State

8 SUPPLEMENTARY NOTES (SponsoringOrganiationPubliahras Avaliability)

9 ABSTRACT

11 PRICE OF DOCUMENT10 CONTROL NUMBER

PN-RAA- 0 3 13 PROJECT NUMBER12 DESCRIPTORS

Atlantico 3 Project 14 CONTRACT NUMBERColombia

Farm crops CSD-2167 Resi1omput programs 15 TYPE OF DOCUMENT

AID 890-1 (474)

A HYBRID COMPUTER MODEL OF THE HYDROLOGIC SYSTEM WITHIN THE ATLANTICO 3 AREA

OF COLOMBIA SOUTH AMERICA

Prepared by

3 Paul Riley Eugene K Israelsen

UtahWater Research Laboratory Utah State University

Logan Utah

June 1971

TABLE OF CONTENTS

introduction

Page

The Initial Model Model ImprovementModel Calibration

151

Management Studies

Suggested Data Collection Program

Plan of Future Work

5

8

10

Research Utilization

Appeidices 22

LIST OF FIGURES

Figure Page

1 Grid system for the study area Atlantico 3 Colombia - 13

2 Land surface topography of the Atlantico 3 area Colombia 14

3 Groundwater levels after 6 months without drainage 15

4 Groundwater levels after drainage

12 months without 16

5 Groundwater levels after 12 months Drainage rate = 10 cmmonth 17

6 Groundwater levels after 24 months Drainage rate = 10 cmmonth 18

7 Groundwater levels after 36 months Drainage rate = 10 cmmonth 19

8 Groundwater levels after 48 months Drainage rate = 10 cmmonth 20

9 Groundwater levels after 60 months Drainage rate = 10 cmmonth 21

ii

A Progress Report on Work Accomplished in Computer Simulation Under Project WG-69 for the Period January 1 to June 30 1971

Introduction

The initial Model

Computer simulation under this project was initiated in January

1970 with the development of an initial hydrologic model of the Atlantico

3 area in northern Colombia The model was based on a time increment

of one month and considered a space grid of 2 000 meters A descripshy

tion of the work accomplished during January 1 to February 28 1970

is attached as Appendix A

Model Improvement

A summary of progress during the period March 1 to December

31 1970 is attached as Appendix B Itwas stated in the progress reshy

port for March I toDecember 311970 (Appendix B) that efforts were

made during this period to improve the initial simulation model develshy

oped by Morris et al (1970) (Appendix A) by emphasizing the followshy

ing areas of study and by testingth6evisedmodel for proper operashy

tion

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology

and unsaturated groundwater flow In this respect it was

considered that the mnodel should be capable of reflecting

topographic influences upon groundwater levels

5- Capability for considering different soil permeability coshy

efficients at each grid point

6 Addition of the salinity dimension to the model in accordshy

ance with previous work at Utah State University

7 Improvement of the model using hydrologic data which ICo

become available since the completion of the initial study

8 Perform continuing sensitivity studies to establish priorshy

ities and resolution needs for data collection programs

In connection with the preceding list the following is a brief

description of the progress that was made on the project during the

period March]1 to December 31 1970

1 The initial model approximated the area under considerashy

tion by a rectangle with its four edges as boundaries

This approximation caused difficulty in properly defining

the boundary conditions at various times The revised

model as described in Appendix B considers all possishy

bleboundary irregularities and therefore handles areas

of any shape Be this revision of the model Item 1 has

been accomplished

2 Because of the increase in the memory capacity of the

computer and thedecrease in required memory space

due to the revised solution method for the partial differ-

ential equations which described the groundwater fluctushy

3

ations a significant increase in the grid density was made

possible The grid increment in the revised model is 625

meters (Figuire 1) compared to the-Z000meters of the inishy

tial model Tle total number of the grid points within the

area is now 849 For each of these grid points the effecshy

tive percolatipn to (or withdrawal from ) the groundwater

during each tine increment was simulated by the surface

component of the model This computed quantity at each

grid point was then fed into the groundwater component of

the modelto simulate the groundwater table fluctuations

The Dirichlet type boundary condition for the groundwater

model was properly defined on the basis of the available

data The input data for the surface model were precipishy

tation temperature soil type and the corresponding crop

pattern in terms of crop coefficients and irrigation reshy

quirements soil moisture holding capacity initial soil

moisture and swamp storage crop densities and a toposhy

graphic parameter The inputs to the groundwater model

include the initial water table levels water table levels

along the boundaries at different times and the transmisshy

sivity And specific storage of the aquifer The model was

availshycalibrated over a period where reliable data were

able to identify the model parameters- Items 2 and 3 of

the preceding list were thus fulfilled

3 To represent the location variations of the groundwater

table due to topographic influences as specified in Item 4

a topographicparameter which characterize the drainage

or collection of surface water was introduced in the reshy

vised model For the Atlantico 3 area the value for this

parameter at each grid point was determined from a toposhy

graphic map (Figure 2)

4 There was not yet sufficient data available within the

Atlantico 3 area to properly define variations in the soil

permeability The assumption of a homogineous soil

was therefore retained in the revised model However

the model contains sufficient resolution to characterize

these variations and when -permeability data become

available at different locations in the area the model

can be revised in this regard

5 Item 6 also has not yet been accomplished primarily beshy

cause of the lack of water quality data Techniques have

already been developed at USU for adding the water qualishy

ty dimensions to hydrologic simulation models and this

vill be done for the Atlantico 3 modef when the necess ary

vater quality data become available

6 In accordance with Item 7 all relevant data that have beshy

come available since the completion of the initial model

halve been incorporated into the operation of the revised

model

7 The sensitivity studies referred tomyItem 8 were conducted

by observing the model responses of both the surface and

groundwater systems to various parameters such as

phreatophyte density agricultural crop pattern irrigation

supply and soil moisture holding capacity These analyses

suggested several areas of additional data needs within the

system and these needs will be discussed in a subseqient

part of this report

Model Calibration

The revised model was calibrated by using data taken during

1969 While meteorologic data wereavailable for the three years

of 1967 1968 and 1969 adequate information on groundwater levels

could be obtained for only 1969 Although the calibration of a monthshy

ly model over a period of only one year leaves room for question it shy

is considered that the relative magnitudes of the various parameters

associated with the model have been established In addition conshy

siderable insight into operation of the prototype system has been

provided As more data become available for subsequent years the

calibration of Lhe model will be improved

Management Studies

Based on the soil land classification and precipitation data

for the study area croppatterns and the correspnding crop coef-

ficients and irrigation rates wete assumed as shown by Table 1

Table 1 Crop-pattern crop-coefficients and irrigation for different soils

Soil Group Item Crop Jan

Crop-pattern weighted crop-coefficient and irrigation rate Feb Mar Apr May Jun Jul Aug SeptI Oct Nov Dec

1 Crop pattern Ci trus -Peanuts Maize

Crop coeff Irr rate

J65 112

-75 112

55 90

60 45

45 60

60 60

75 60

60 60

60 45

60 60

60 60

50 60

2 Crop pattern

Crop coeff Irr rate

Cotton Sorghum

70 112

50 90

20 0

20 0

30 45

60 60

90 60

60 60

40 60

65 60

90 90

90 112

3 Crop pattern Grasses - -

4

Crop coeff Irr rate

_Crop-coeff Irr rate

Bare Soil

80 90

10 0

80 90

10 0

80 90

10 0

80 75

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 75

10 0

80 90

10 0

-Inmmonth irrigation efficiency = 06

7

According to available information existing densities of the native

secshyphreatophytes vary from about 50 percent in the south-eastern

tion of the arep to approximately 20 percent in the-north-western -part

To investigate the responses of the groundwater table to areduction

in the area of phreatophytes and to the application of irrigation water

to cultivated crops the model was operated under the following

assumptions

1 Half of the native phreatophytes were assumed to be reshy

placed by the cultivated crops shown in Table 1

2 No sub-surface drainage was established

3 The available precipitation and evaporation data for the

period of )967 through 1969 were assumed to be represhy

sentative for the area

Figures 3 and 4 show the simulated groundwater surface within

area at the end of 6 and 12 months after the assumed developmentthe

outlined above These figures suggest that the groundwater table

would build up quickly to the root zone unless a suitable drainage

system were installed to remove excess waler from the area

To estimate the rate of drainage required to prevent the buildshy

up of the groundwater table to undesirable levels several drainage

rates were assumed in simulacing the groundwater table movement

The assumption of a uniform drainage rate of 10 cm per month over

the entire area results in the groundwater contour maps shown in

Figures 5 through 9 It is noted that although the groundwater table

+ (Z []

wbpthe tt

Thus m o e~ s l

at suit-able depth thip~gh~uV t e

pf

rA o (V

With particulart4efe once to the A6400

collection

1 ientyiz cm

program in ISgosted t

PrecipiaJ onlnoVillllt

athuedI4amp J

at

t~~Ve Atlantico 3 arl

utb Itle depets tr O thtjit

and that poabeD

+total of ai -0 Fi t p t

titt

rntltesg e dta a

mtow

i

I-1

--

o Al

+ +Iti~UgU mto4ih

714

and~tht1i~ JRiIuas14-11 Tl

Ah

11

cedure This is a time-consuming and costly process

Therefore as a part of this study a self-optimizing scheme

has been developed and soon will be incorporated in the simshy

ulation model for automatic identification of these paramshy

eters In this way it will be possible to efficiently apply

the model to any prototype area for which sufficient verifishy

cation-data are available

3 As previously discussed tothis point it has been necessary

to either assume or rather grossly approximate many data

used in the model of the Atlantico 3 area As additional

data for this area become available they will be used to furshy

ther improve and test the model

Research Utilization

Although the present study is directed specifically to the reshy

3arch needs for the Atlantico 3 area the simulation model developed

entirely general and can be applied to different geographic areas

addition the philosophy and techniques used in the analysis can

e applied equally well to many problems of similar nature

Presentations based primarily on the initial model were made

t the IV Latin American Congress on Hydraulics Mexico City Aushy

ust 1970 at the 6th American Water Resource Conference Las Vegas

[evada November 1970 and at an International Symposium on Groundshy

iater held at Pale rmoo Sicily inDecember 1970 The paper Upon

hich these Presentations were based is included as Appendix A

A description of the revised model and its applications is now

)eing prepared as a paper to be submitted to an appropriate technical

journal This model was also briefly described in a presentation to

he participants of the seminar on Water Resources Planning which

vas held at Utah State University in June 1971

13

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 2: FOR AID USE ONLY

A HYBRID COMPUTER MODEL OF THE HYDROLOGIC SYSTEM WITHIN THE ATLANTICO 3 AREA

OF COLOMBIA SOUTH AMERICA

Prepared by

3 Paul Riley Eugene K Israelsen

UtahWater Research Laboratory Utah State University

Logan Utah

June 1971

TABLE OF CONTENTS

introduction

Page

The Initial Model Model ImprovementModel Calibration

151

Management Studies

Suggested Data Collection Program

Plan of Future Work

5

8

10

Research Utilization

Appeidices 22

LIST OF FIGURES

Figure Page

1 Grid system for the study area Atlantico 3 Colombia - 13

2 Land surface topography of the Atlantico 3 area Colombia 14

3 Groundwater levels after 6 months without drainage 15

4 Groundwater levels after drainage

12 months without 16

5 Groundwater levels after 12 months Drainage rate = 10 cmmonth 17

6 Groundwater levels after 24 months Drainage rate = 10 cmmonth 18

7 Groundwater levels after 36 months Drainage rate = 10 cmmonth 19

8 Groundwater levels after 48 months Drainage rate = 10 cmmonth 20

9 Groundwater levels after 60 months Drainage rate = 10 cmmonth 21

ii

A Progress Report on Work Accomplished in Computer Simulation Under Project WG-69 for the Period January 1 to June 30 1971

Introduction

The initial Model

Computer simulation under this project was initiated in January

1970 with the development of an initial hydrologic model of the Atlantico

3 area in northern Colombia The model was based on a time increment

of one month and considered a space grid of 2 000 meters A descripshy

tion of the work accomplished during January 1 to February 28 1970

is attached as Appendix A

Model Improvement

A summary of progress during the period March 1 to December

31 1970 is attached as Appendix B Itwas stated in the progress reshy

port for March I toDecember 311970 (Appendix B) that efforts were

made during this period to improve the initial simulation model develshy

oped by Morris et al (1970) (Appendix A) by emphasizing the followshy

ing areas of study and by testingth6evisedmodel for proper operashy

tion

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology

and unsaturated groundwater flow In this respect it was

considered that the mnodel should be capable of reflecting

topographic influences upon groundwater levels

5- Capability for considering different soil permeability coshy

efficients at each grid point

6 Addition of the salinity dimension to the model in accordshy

ance with previous work at Utah State University

7 Improvement of the model using hydrologic data which ICo

become available since the completion of the initial study

8 Perform continuing sensitivity studies to establish priorshy

ities and resolution needs for data collection programs

In connection with the preceding list the following is a brief

description of the progress that was made on the project during the

period March]1 to December 31 1970

1 The initial model approximated the area under considerashy

tion by a rectangle with its four edges as boundaries

This approximation caused difficulty in properly defining

the boundary conditions at various times The revised

model as described in Appendix B considers all possishy

bleboundary irregularities and therefore handles areas

of any shape Be this revision of the model Item 1 has

been accomplished

2 Because of the increase in the memory capacity of the

computer and thedecrease in required memory space

due to the revised solution method for the partial differ-

ential equations which described the groundwater fluctushy

3

ations a significant increase in the grid density was made

possible The grid increment in the revised model is 625

meters (Figuire 1) compared to the-Z000meters of the inishy

tial model Tle total number of the grid points within the

area is now 849 For each of these grid points the effecshy

tive percolatipn to (or withdrawal from ) the groundwater

during each tine increment was simulated by the surface

component of the model This computed quantity at each

grid point was then fed into the groundwater component of

the modelto simulate the groundwater table fluctuations

The Dirichlet type boundary condition for the groundwater

model was properly defined on the basis of the available

data The input data for the surface model were precipishy

tation temperature soil type and the corresponding crop

pattern in terms of crop coefficients and irrigation reshy

quirements soil moisture holding capacity initial soil

moisture and swamp storage crop densities and a toposhy

graphic parameter The inputs to the groundwater model

include the initial water table levels water table levels

along the boundaries at different times and the transmisshy

sivity And specific storage of the aquifer The model was

availshycalibrated over a period where reliable data were

able to identify the model parameters- Items 2 and 3 of

the preceding list were thus fulfilled

3 To represent the location variations of the groundwater

table due to topographic influences as specified in Item 4

a topographicparameter which characterize the drainage

or collection of surface water was introduced in the reshy

vised model For the Atlantico 3 area the value for this

parameter at each grid point was determined from a toposhy

graphic map (Figure 2)

4 There was not yet sufficient data available within the

Atlantico 3 area to properly define variations in the soil

permeability The assumption of a homogineous soil

was therefore retained in the revised model However

the model contains sufficient resolution to characterize

these variations and when -permeability data become

available at different locations in the area the model

can be revised in this regard

5 Item 6 also has not yet been accomplished primarily beshy

cause of the lack of water quality data Techniques have

already been developed at USU for adding the water qualishy

ty dimensions to hydrologic simulation models and this

vill be done for the Atlantico 3 modef when the necess ary

vater quality data become available

6 In accordance with Item 7 all relevant data that have beshy

come available since the completion of the initial model

halve been incorporated into the operation of the revised

model

7 The sensitivity studies referred tomyItem 8 were conducted

by observing the model responses of both the surface and

groundwater systems to various parameters such as

phreatophyte density agricultural crop pattern irrigation

supply and soil moisture holding capacity These analyses

suggested several areas of additional data needs within the

system and these needs will be discussed in a subseqient

part of this report

Model Calibration

The revised model was calibrated by using data taken during

1969 While meteorologic data wereavailable for the three years

of 1967 1968 and 1969 adequate information on groundwater levels

could be obtained for only 1969 Although the calibration of a monthshy

ly model over a period of only one year leaves room for question it shy

is considered that the relative magnitudes of the various parameters

associated with the model have been established In addition conshy

siderable insight into operation of the prototype system has been

provided As more data become available for subsequent years the

calibration of Lhe model will be improved

Management Studies

Based on the soil land classification and precipitation data

for the study area croppatterns and the correspnding crop coef-

ficients and irrigation rates wete assumed as shown by Table 1

Table 1 Crop-pattern crop-coefficients and irrigation for different soils

Soil Group Item Crop Jan

Crop-pattern weighted crop-coefficient and irrigation rate Feb Mar Apr May Jun Jul Aug SeptI Oct Nov Dec

1 Crop pattern Ci trus -Peanuts Maize

Crop coeff Irr rate

J65 112

-75 112

55 90

60 45

45 60

60 60

75 60

60 60

60 45

60 60

60 60

50 60

2 Crop pattern

Crop coeff Irr rate

Cotton Sorghum

70 112

50 90

20 0

20 0

30 45

60 60

90 60

60 60

40 60

65 60

90 90

90 112

3 Crop pattern Grasses - -

4

Crop coeff Irr rate

_Crop-coeff Irr rate

Bare Soil

80 90

10 0

80 90

10 0

80 90

10 0

80 75

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 75

10 0

80 90

10 0

-Inmmonth irrigation efficiency = 06

7

According to available information existing densities of the native

secshyphreatophytes vary from about 50 percent in the south-eastern

tion of the arep to approximately 20 percent in the-north-western -part

To investigate the responses of the groundwater table to areduction

in the area of phreatophytes and to the application of irrigation water

to cultivated crops the model was operated under the following

assumptions

1 Half of the native phreatophytes were assumed to be reshy

placed by the cultivated crops shown in Table 1

2 No sub-surface drainage was established

3 The available precipitation and evaporation data for the

period of )967 through 1969 were assumed to be represhy

sentative for the area

Figures 3 and 4 show the simulated groundwater surface within

area at the end of 6 and 12 months after the assumed developmentthe

outlined above These figures suggest that the groundwater table

would build up quickly to the root zone unless a suitable drainage

system were installed to remove excess waler from the area

To estimate the rate of drainage required to prevent the buildshy

up of the groundwater table to undesirable levels several drainage

rates were assumed in simulacing the groundwater table movement

The assumption of a uniform drainage rate of 10 cm per month over

the entire area results in the groundwater contour maps shown in

Figures 5 through 9 It is noted that although the groundwater table

+ (Z []

wbpthe tt

Thus m o e~ s l

at suit-able depth thip~gh~uV t e

pf

rA o (V

With particulart4efe once to the A6400

collection

1 ientyiz cm

program in ISgosted t

PrecipiaJ onlnoVillllt

athuedI4amp J

at

t~~Ve Atlantico 3 arl

utb Itle depets tr O thtjit

and that poabeD

+total of ai -0 Fi t p t

titt

rntltesg e dta a

mtow

i

I-1

--

o Al

+ +Iti~UgU mto4ih

714

and~tht1i~ JRiIuas14-11 Tl

Ah

11

cedure This is a time-consuming and costly process

Therefore as a part of this study a self-optimizing scheme

has been developed and soon will be incorporated in the simshy

ulation model for automatic identification of these paramshy

eters In this way it will be possible to efficiently apply

the model to any prototype area for which sufficient verifishy

cation-data are available

3 As previously discussed tothis point it has been necessary

to either assume or rather grossly approximate many data

used in the model of the Atlantico 3 area As additional

data for this area become available they will be used to furshy

ther improve and test the model

Research Utilization

Although the present study is directed specifically to the reshy

3arch needs for the Atlantico 3 area the simulation model developed

entirely general and can be applied to different geographic areas

addition the philosophy and techniques used in the analysis can

e applied equally well to many problems of similar nature

Presentations based primarily on the initial model were made

t the IV Latin American Congress on Hydraulics Mexico City Aushy

ust 1970 at the 6th American Water Resource Conference Las Vegas

[evada November 1970 and at an International Symposium on Groundshy

iater held at Pale rmoo Sicily inDecember 1970 The paper Upon

hich these Presentations were based is included as Appendix A

A description of the revised model and its applications is now

)eing prepared as a paper to be submitted to an appropriate technical

journal This model was also briefly described in a presentation to

he participants of the seminar on Water Resources Planning which

vas held at Utah State University in June 1971

13

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

i~ 7 I

T -II -r-

-shy

44~~~

use n 1rtptoi~tw~ist 4 4 P

WY94

W

LL

VAshy

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 3: FOR AID USE ONLY

TABLE OF CONTENTS

introduction

Page

The Initial Model Model ImprovementModel Calibration

151

Management Studies

Suggested Data Collection Program

Plan of Future Work

5

8

10

Research Utilization

Appeidices 22

LIST OF FIGURES

Figure Page

1 Grid system for the study area Atlantico 3 Colombia - 13

2 Land surface topography of the Atlantico 3 area Colombia 14

3 Groundwater levels after 6 months without drainage 15

4 Groundwater levels after drainage

12 months without 16

5 Groundwater levels after 12 months Drainage rate = 10 cmmonth 17

6 Groundwater levels after 24 months Drainage rate = 10 cmmonth 18

7 Groundwater levels after 36 months Drainage rate = 10 cmmonth 19

8 Groundwater levels after 48 months Drainage rate = 10 cmmonth 20

9 Groundwater levels after 60 months Drainage rate = 10 cmmonth 21

ii

A Progress Report on Work Accomplished in Computer Simulation Under Project WG-69 for the Period January 1 to June 30 1971

Introduction

The initial Model

Computer simulation under this project was initiated in January

1970 with the development of an initial hydrologic model of the Atlantico

3 area in northern Colombia The model was based on a time increment

of one month and considered a space grid of 2 000 meters A descripshy

tion of the work accomplished during January 1 to February 28 1970

is attached as Appendix A

Model Improvement

A summary of progress during the period March 1 to December

31 1970 is attached as Appendix B Itwas stated in the progress reshy

port for March I toDecember 311970 (Appendix B) that efforts were

made during this period to improve the initial simulation model develshy

oped by Morris et al (1970) (Appendix A) by emphasizing the followshy

ing areas of study and by testingth6evisedmodel for proper operashy

tion

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology

and unsaturated groundwater flow In this respect it was

considered that the mnodel should be capable of reflecting

topographic influences upon groundwater levels

5- Capability for considering different soil permeability coshy

efficients at each grid point

6 Addition of the salinity dimension to the model in accordshy

ance with previous work at Utah State University

7 Improvement of the model using hydrologic data which ICo

become available since the completion of the initial study

8 Perform continuing sensitivity studies to establish priorshy

ities and resolution needs for data collection programs

In connection with the preceding list the following is a brief

description of the progress that was made on the project during the

period March]1 to December 31 1970

1 The initial model approximated the area under considerashy

tion by a rectangle with its four edges as boundaries

This approximation caused difficulty in properly defining

the boundary conditions at various times The revised

model as described in Appendix B considers all possishy

bleboundary irregularities and therefore handles areas

of any shape Be this revision of the model Item 1 has

been accomplished

2 Because of the increase in the memory capacity of the

computer and thedecrease in required memory space

due to the revised solution method for the partial differ-

ential equations which described the groundwater fluctushy

3

ations a significant increase in the grid density was made

possible The grid increment in the revised model is 625

meters (Figuire 1) compared to the-Z000meters of the inishy

tial model Tle total number of the grid points within the

area is now 849 For each of these grid points the effecshy

tive percolatipn to (or withdrawal from ) the groundwater

during each tine increment was simulated by the surface

component of the model This computed quantity at each

grid point was then fed into the groundwater component of

the modelto simulate the groundwater table fluctuations

The Dirichlet type boundary condition for the groundwater

model was properly defined on the basis of the available

data The input data for the surface model were precipishy

tation temperature soil type and the corresponding crop

pattern in terms of crop coefficients and irrigation reshy

quirements soil moisture holding capacity initial soil

moisture and swamp storage crop densities and a toposhy

graphic parameter The inputs to the groundwater model

include the initial water table levels water table levels

along the boundaries at different times and the transmisshy

sivity And specific storage of the aquifer The model was

availshycalibrated over a period where reliable data were

able to identify the model parameters- Items 2 and 3 of

the preceding list were thus fulfilled

3 To represent the location variations of the groundwater

table due to topographic influences as specified in Item 4

a topographicparameter which characterize the drainage

or collection of surface water was introduced in the reshy

vised model For the Atlantico 3 area the value for this

parameter at each grid point was determined from a toposhy

graphic map (Figure 2)

4 There was not yet sufficient data available within the

Atlantico 3 area to properly define variations in the soil

permeability The assumption of a homogineous soil

was therefore retained in the revised model However

the model contains sufficient resolution to characterize

these variations and when -permeability data become

available at different locations in the area the model

can be revised in this regard

5 Item 6 also has not yet been accomplished primarily beshy

cause of the lack of water quality data Techniques have

already been developed at USU for adding the water qualishy

ty dimensions to hydrologic simulation models and this

vill be done for the Atlantico 3 modef when the necess ary

vater quality data become available

6 In accordance with Item 7 all relevant data that have beshy

come available since the completion of the initial model

halve been incorporated into the operation of the revised

model

7 The sensitivity studies referred tomyItem 8 were conducted

by observing the model responses of both the surface and

groundwater systems to various parameters such as

phreatophyte density agricultural crop pattern irrigation

supply and soil moisture holding capacity These analyses

suggested several areas of additional data needs within the

system and these needs will be discussed in a subseqient

part of this report

Model Calibration

The revised model was calibrated by using data taken during

1969 While meteorologic data wereavailable for the three years

of 1967 1968 and 1969 adequate information on groundwater levels

could be obtained for only 1969 Although the calibration of a monthshy

ly model over a period of only one year leaves room for question it shy

is considered that the relative magnitudes of the various parameters

associated with the model have been established In addition conshy

siderable insight into operation of the prototype system has been

provided As more data become available for subsequent years the

calibration of Lhe model will be improved

Management Studies

Based on the soil land classification and precipitation data

for the study area croppatterns and the correspnding crop coef-

ficients and irrigation rates wete assumed as shown by Table 1

Table 1 Crop-pattern crop-coefficients and irrigation for different soils

Soil Group Item Crop Jan

Crop-pattern weighted crop-coefficient and irrigation rate Feb Mar Apr May Jun Jul Aug SeptI Oct Nov Dec

1 Crop pattern Ci trus -Peanuts Maize

Crop coeff Irr rate

J65 112

-75 112

55 90

60 45

45 60

60 60

75 60

60 60

60 45

60 60

60 60

50 60

2 Crop pattern

Crop coeff Irr rate

Cotton Sorghum

70 112

50 90

20 0

20 0

30 45

60 60

90 60

60 60

40 60

65 60

90 90

90 112

3 Crop pattern Grasses - -

4

Crop coeff Irr rate

_Crop-coeff Irr rate

Bare Soil

80 90

10 0

80 90

10 0

80 90

10 0

80 75

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 75

10 0

80 90

10 0

-Inmmonth irrigation efficiency = 06

7

According to available information existing densities of the native

secshyphreatophytes vary from about 50 percent in the south-eastern

tion of the arep to approximately 20 percent in the-north-western -part

To investigate the responses of the groundwater table to areduction

in the area of phreatophytes and to the application of irrigation water

to cultivated crops the model was operated under the following

assumptions

1 Half of the native phreatophytes were assumed to be reshy

placed by the cultivated crops shown in Table 1

2 No sub-surface drainage was established

3 The available precipitation and evaporation data for the

period of )967 through 1969 were assumed to be represhy

sentative for the area

Figures 3 and 4 show the simulated groundwater surface within

area at the end of 6 and 12 months after the assumed developmentthe

outlined above These figures suggest that the groundwater table

would build up quickly to the root zone unless a suitable drainage

system were installed to remove excess waler from the area

To estimate the rate of drainage required to prevent the buildshy

up of the groundwater table to undesirable levels several drainage

rates were assumed in simulacing the groundwater table movement

The assumption of a uniform drainage rate of 10 cm per month over

the entire area results in the groundwater contour maps shown in

Figures 5 through 9 It is noted that although the groundwater table

+ (Z []

wbpthe tt

Thus m o e~ s l

at suit-able depth thip~gh~uV t e

pf

rA o (V

With particulart4efe once to the A6400

collection

1 ientyiz cm

program in ISgosted t

PrecipiaJ onlnoVillllt

athuedI4amp J

at

t~~Ve Atlantico 3 arl

utb Itle depets tr O thtjit

and that poabeD

+total of ai -0 Fi t p t

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11

cedure This is a time-consuming and costly process

Therefore as a part of this study a self-optimizing scheme

has been developed and soon will be incorporated in the simshy

ulation model for automatic identification of these paramshy

eters In this way it will be possible to efficiently apply

the model to any prototype area for which sufficient verifishy

cation-data are available

3 As previously discussed tothis point it has been necessary

to either assume or rather grossly approximate many data

used in the model of the Atlantico 3 area As additional

data for this area become available they will be used to furshy

ther improve and test the model

Research Utilization

Although the present study is directed specifically to the reshy

3arch needs for the Atlantico 3 area the simulation model developed

entirely general and can be applied to different geographic areas

addition the philosophy and techniques used in the analysis can

e applied equally well to many problems of similar nature

Presentations based primarily on the initial model were made

t the IV Latin American Congress on Hydraulics Mexico City Aushy

ust 1970 at the 6th American Water Resource Conference Las Vegas

[evada November 1970 and at an International Symposium on Groundshy

iater held at Pale rmoo Sicily inDecember 1970 The paper Upon

hich these Presentations were based is included as Appendix A

A description of the revised model and its applications is now

)eing prepared as a paper to be submitted to an appropriate technical

journal This model was also briefly described in a presentation to

he participants of the seminar on Water Resources Planning which

vas held at Utah State University in June 1971

13

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

i~ 7 I

T -II -r-

-shy

44~~~

use n 1rtptoi~tw~ist 4 4 P

WY94

W

LL

VAshy

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 4: FOR AID USE ONLY

LIST OF FIGURES

Figure Page

1 Grid system for the study area Atlantico 3 Colombia - 13

2 Land surface topography of the Atlantico 3 area Colombia 14

3 Groundwater levels after 6 months without drainage 15

4 Groundwater levels after drainage

12 months without 16

5 Groundwater levels after 12 months Drainage rate = 10 cmmonth 17

6 Groundwater levels after 24 months Drainage rate = 10 cmmonth 18

7 Groundwater levels after 36 months Drainage rate = 10 cmmonth 19

8 Groundwater levels after 48 months Drainage rate = 10 cmmonth 20

9 Groundwater levels after 60 months Drainage rate = 10 cmmonth 21

ii

A Progress Report on Work Accomplished in Computer Simulation Under Project WG-69 for the Period January 1 to June 30 1971

Introduction

The initial Model

Computer simulation under this project was initiated in January

1970 with the development of an initial hydrologic model of the Atlantico

3 area in northern Colombia The model was based on a time increment

of one month and considered a space grid of 2 000 meters A descripshy

tion of the work accomplished during January 1 to February 28 1970

is attached as Appendix A

Model Improvement

A summary of progress during the period March 1 to December

31 1970 is attached as Appendix B Itwas stated in the progress reshy

port for March I toDecember 311970 (Appendix B) that efforts were

made during this period to improve the initial simulation model develshy

oped by Morris et al (1970) (Appendix A) by emphasizing the followshy

ing areas of study and by testingth6evisedmodel for proper operashy

tion

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology

and unsaturated groundwater flow In this respect it was

considered that the mnodel should be capable of reflecting

topographic influences upon groundwater levels

5- Capability for considering different soil permeability coshy

efficients at each grid point

6 Addition of the salinity dimension to the model in accordshy

ance with previous work at Utah State University

7 Improvement of the model using hydrologic data which ICo

become available since the completion of the initial study

8 Perform continuing sensitivity studies to establish priorshy

ities and resolution needs for data collection programs

In connection with the preceding list the following is a brief

description of the progress that was made on the project during the

period March]1 to December 31 1970

1 The initial model approximated the area under considerashy

tion by a rectangle with its four edges as boundaries

This approximation caused difficulty in properly defining

the boundary conditions at various times The revised

model as described in Appendix B considers all possishy

bleboundary irregularities and therefore handles areas

of any shape Be this revision of the model Item 1 has

been accomplished

2 Because of the increase in the memory capacity of the

computer and thedecrease in required memory space

due to the revised solution method for the partial differ-

ential equations which described the groundwater fluctushy

3

ations a significant increase in the grid density was made

possible The grid increment in the revised model is 625

meters (Figuire 1) compared to the-Z000meters of the inishy

tial model Tle total number of the grid points within the

area is now 849 For each of these grid points the effecshy

tive percolatipn to (or withdrawal from ) the groundwater

during each tine increment was simulated by the surface

component of the model This computed quantity at each

grid point was then fed into the groundwater component of

the modelto simulate the groundwater table fluctuations

The Dirichlet type boundary condition for the groundwater

model was properly defined on the basis of the available

data The input data for the surface model were precipishy

tation temperature soil type and the corresponding crop

pattern in terms of crop coefficients and irrigation reshy

quirements soil moisture holding capacity initial soil

moisture and swamp storage crop densities and a toposhy

graphic parameter The inputs to the groundwater model

include the initial water table levels water table levels

along the boundaries at different times and the transmisshy

sivity And specific storage of the aquifer The model was

availshycalibrated over a period where reliable data were

able to identify the model parameters- Items 2 and 3 of

the preceding list were thus fulfilled

3 To represent the location variations of the groundwater

table due to topographic influences as specified in Item 4

a topographicparameter which characterize the drainage

or collection of surface water was introduced in the reshy

vised model For the Atlantico 3 area the value for this

parameter at each grid point was determined from a toposhy

graphic map (Figure 2)

4 There was not yet sufficient data available within the

Atlantico 3 area to properly define variations in the soil

permeability The assumption of a homogineous soil

was therefore retained in the revised model However

the model contains sufficient resolution to characterize

these variations and when -permeability data become

available at different locations in the area the model

can be revised in this regard

5 Item 6 also has not yet been accomplished primarily beshy

cause of the lack of water quality data Techniques have

already been developed at USU for adding the water qualishy

ty dimensions to hydrologic simulation models and this

vill be done for the Atlantico 3 modef when the necess ary

vater quality data become available

6 In accordance with Item 7 all relevant data that have beshy

come available since the completion of the initial model

halve been incorporated into the operation of the revised

model

7 The sensitivity studies referred tomyItem 8 were conducted

by observing the model responses of both the surface and

groundwater systems to various parameters such as

phreatophyte density agricultural crop pattern irrigation

supply and soil moisture holding capacity These analyses

suggested several areas of additional data needs within the

system and these needs will be discussed in a subseqient

part of this report

Model Calibration

The revised model was calibrated by using data taken during

1969 While meteorologic data wereavailable for the three years

of 1967 1968 and 1969 adequate information on groundwater levels

could be obtained for only 1969 Although the calibration of a monthshy

ly model over a period of only one year leaves room for question it shy

is considered that the relative magnitudes of the various parameters

associated with the model have been established In addition conshy

siderable insight into operation of the prototype system has been

provided As more data become available for subsequent years the

calibration of Lhe model will be improved

Management Studies

Based on the soil land classification and precipitation data

for the study area croppatterns and the correspnding crop coef-

ficients and irrigation rates wete assumed as shown by Table 1

Table 1 Crop-pattern crop-coefficients and irrigation for different soils

Soil Group Item Crop Jan

Crop-pattern weighted crop-coefficient and irrigation rate Feb Mar Apr May Jun Jul Aug SeptI Oct Nov Dec

1 Crop pattern Ci trus -Peanuts Maize

Crop coeff Irr rate

J65 112

-75 112

55 90

60 45

45 60

60 60

75 60

60 60

60 45

60 60

60 60

50 60

2 Crop pattern

Crop coeff Irr rate

Cotton Sorghum

70 112

50 90

20 0

20 0

30 45

60 60

90 60

60 60

40 60

65 60

90 90

90 112

3 Crop pattern Grasses - -

4

Crop coeff Irr rate

_Crop-coeff Irr rate

Bare Soil

80 90

10 0

80 90

10 0

80 90

10 0

80 75

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 75

10 0

80 90

10 0

-Inmmonth irrigation efficiency = 06

7

According to available information existing densities of the native

secshyphreatophytes vary from about 50 percent in the south-eastern

tion of the arep to approximately 20 percent in the-north-western -part

To investigate the responses of the groundwater table to areduction

in the area of phreatophytes and to the application of irrigation water

to cultivated crops the model was operated under the following

assumptions

1 Half of the native phreatophytes were assumed to be reshy

placed by the cultivated crops shown in Table 1

2 No sub-surface drainage was established

3 The available precipitation and evaporation data for the

period of )967 through 1969 were assumed to be represhy

sentative for the area

Figures 3 and 4 show the simulated groundwater surface within

area at the end of 6 and 12 months after the assumed developmentthe

outlined above These figures suggest that the groundwater table

would build up quickly to the root zone unless a suitable drainage

system were installed to remove excess waler from the area

To estimate the rate of drainage required to prevent the buildshy

up of the groundwater table to undesirable levels several drainage

rates were assumed in simulacing the groundwater table movement

The assumption of a uniform drainage rate of 10 cm per month over

the entire area results in the groundwater contour maps shown in

Figures 5 through 9 It is noted that although the groundwater table

+ (Z []

wbpthe tt

Thus m o e~ s l

at suit-able depth thip~gh~uV t e

pf

rA o (V

With particulart4efe once to the A6400

collection

1 ientyiz cm

program in ISgosted t

PrecipiaJ onlnoVillllt

athuedI4amp J

at

t~~Ve Atlantico 3 arl

utb Itle depets tr O thtjit

and that poabeD

+total of ai -0 Fi t p t

titt

rntltesg e dta a

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i

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o Al

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11

cedure This is a time-consuming and costly process

Therefore as a part of this study a self-optimizing scheme

has been developed and soon will be incorporated in the simshy

ulation model for automatic identification of these paramshy

eters In this way it will be possible to efficiently apply

the model to any prototype area for which sufficient verifishy

cation-data are available

3 As previously discussed tothis point it has been necessary

to either assume or rather grossly approximate many data

used in the model of the Atlantico 3 area As additional

data for this area become available they will be used to furshy

ther improve and test the model

Research Utilization

Although the present study is directed specifically to the reshy

3arch needs for the Atlantico 3 area the simulation model developed

entirely general and can be applied to different geographic areas

addition the philosophy and techniques used in the analysis can

e applied equally well to many problems of similar nature

Presentations based primarily on the initial model were made

t the IV Latin American Congress on Hydraulics Mexico City Aushy

ust 1970 at the 6th American Water Resource Conference Las Vegas

[evada November 1970 and at an International Symposium on Groundshy

iater held at Pale rmoo Sicily inDecember 1970 The paper Upon

hich these Presentations were based is included as Appendix A

A description of the revised model and its applications is now

)eing prepared as a paper to be submitted to an appropriate technical

journal This model was also briefly described in a presentation to

he participants of the seminar on Water Resources Planning which

vas held at Utah State University in June 1971

13

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 5: FOR AID USE ONLY

A Progress Report on Work Accomplished in Computer Simulation Under Project WG-69 for the Period January 1 to June 30 1971

Introduction

The initial Model

Computer simulation under this project was initiated in January

1970 with the development of an initial hydrologic model of the Atlantico

3 area in northern Colombia The model was based on a time increment

of one month and considered a space grid of 2 000 meters A descripshy

tion of the work accomplished during January 1 to February 28 1970

is attached as Appendix A

Model Improvement

A summary of progress during the period March 1 to December

31 1970 is attached as Appendix B Itwas stated in the progress reshy

port for March I toDecember 311970 (Appendix B) that efforts were

made during this period to improve the initial simulation model develshy

oped by Morris et al (1970) (Appendix A) by emphasizing the followshy

ing areas of study and by testingth6evisedmodel for proper operashy

tion

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology

and unsaturated groundwater flow In this respect it was

considered that the mnodel should be capable of reflecting

topographic influences upon groundwater levels

5- Capability for considering different soil permeability coshy

efficients at each grid point

6 Addition of the salinity dimension to the model in accordshy

ance with previous work at Utah State University

7 Improvement of the model using hydrologic data which ICo

become available since the completion of the initial study

8 Perform continuing sensitivity studies to establish priorshy

ities and resolution needs for data collection programs

In connection with the preceding list the following is a brief

description of the progress that was made on the project during the

period March]1 to December 31 1970

1 The initial model approximated the area under considerashy

tion by a rectangle with its four edges as boundaries

This approximation caused difficulty in properly defining

the boundary conditions at various times The revised

model as described in Appendix B considers all possishy

bleboundary irregularities and therefore handles areas

of any shape Be this revision of the model Item 1 has

been accomplished

2 Because of the increase in the memory capacity of the

computer and thedecrease in required memory space

due to the revised solution method for the partial differ-

ential equations which described the groundwater fluctushy

3

ations a significant increase in the grid density was made

possible The grid increment in the revised model is 625

meters (Figuire 1) compared to the-Z000meters of the inishy

tial model Tle total number of the grid points within the

area is now 849 For each of these grid points the effecshy

tive percolatipn to (or withdrawal from ) the groundwater

during each tine increment was simulated by the surface

component of the model This computed quantity at each

grid point was then fed into the groundwater component of

the modelto simulate the groundwater table fluctuations

The Dirichlet type boundary condition for the groundwater

model was properly defined on the basis of the available

data The input data for the surface model were precipishy

tation temperature soil type and the corresponding crop

pattern in terms of crop coefficients and irrigation reshy

quirements soil moisture holding capacity initial soil

moisture and swamp storage crop densities and a toposhy

graphic parameter The inputs to the groundwater model

include the initial water table levels water table levels

along the boundaries at different times and the transmisshy

sivity And specific storage of the aquifer The model was

availshycalibrated over a period where reliable data were

able to identify the model parameters- Items 2 and 3 of

the preceding list were thus fulfilled

3 To represent the location variations of the groundwater

table due to topographic influences as specified in Item 4

a topographicparameter which characterize the drainage

or collection of surface water was introduced in the reshy

vised model For the Atlantico 3 area the value for this

parameter at each grid point was determined from a toposhy

graphic map (Figure 2)

4 There was not yet sufficient data available within the

Atlantico 3 area to properly define variations in the soil

permeability The assumption of a homogineous soil

was therefore retained in the revised model However

the model contains sufficient resolution to characterize

these variations and when -permeability data become

available at different locations in the area the model

can be revised in this regard

5 Item 6 also has not yet been accomplished primarily beshy

cause of the lack of water quality data Techniques have

already been developed at USU for adding the water qualishy

ty dimensions to hydrologic simulation models and this

vill be done for the Atlantico 3 modef when the necess ary

vater quality data become available

6 In accordance with Item 7 all relevant data that have beshy

come available since the completion of the initial model

halve been incorporated into the operation of the revised

model

7 The sensitivity studies referred tomyItem 8 were conducted

by observing the model responses of both the surface and

groundwater systems to various parameters such as

phreatophyte density agricultural crop pattern irrigation

supply and soil moisture holding capacity These analyses

suggested several areas of additional data needs within the

system and these needs will be discussed in a subseqient

part of this report

Model Calibration

The revised model was calibrated by using data taken during

1969 While meteorologic data wereavailable for the three years

of 1967 1968 and 1969 adequate information on groundwater levels

could be obtained for only 1969 Although the calibration of a monthshy

ly model over a period of only one year leaves room for question it shy

is considered that the relative magnitudes of the various parameters

associated with the model have been established In addition conshy

siderable insight into operation of the prototype system has been

provided As more data become available for subsequent years the

calibration of Lhe model will be improved

Management Studies

Based on the soil land classification and precipitation data

for the study area croppatterns and the correspnding crop coef-

ficients and irrigation rates wete assumed as shown by Table 1

Table 1 Crop-pattern crop-coefficients and irrigation for different soils

Soil Group Item Crop Jan

Crop-pattern weighted crop-coefficient and irrigation rate Feb Mar Apr May Jun Jul Aug SeptI Oct Nov Dec

1 Crop pattern Ci trus -Peanuts Maize

Crop coeff Irr rate

J65 112

-75 112

55 90

60 45

45 60

60 60

75 60

60 60

60 45

60 60

60 60

50 60

2 Crop pattern

Crop coeff Irr rate

Cotton Sorghum

70 112

50 90

20 0

20 0

30 45

60 60

90 60

60 60

40 60

65 60

90 90

90 112

3 Crop pattern Grasses - -

4

Crop coeff Irr rate

_Crop-coeff Irr rate

Bare Soil

80 90

10 0

80 90

10 0

80 90

10 0

80 75

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 75

10 0

80 90

10 0

-Inmmonth irrigation efficiency = 06

7

According to available information existing densities of the native

secshyphreatophytes vary from about 50 percent in the south-eastern

tion of the arep to approximately 20 percent in the-north-western -part

To investigate the responses of the groundwater table to areduction

in the area of phreatophytes and to the application of irrigation water

to cultivated crops the model was operated under the following

assumptions

1 Half of the native phreatophytes were assumed to be reshy

placed by the cultivated crops shown in Table 1

2 No sub-surface drainage was established

3 The available precipitation and evaporation data for the

period of )967 through 1969 were assumed to be represhy

sentative for the area

Figures 3 and 4 show the simulated groundwater surface within

area at the end of 6 and 12 months after the assumed developmentthe

outlined above These figures suggest that the groundwater table

would build up quickly to the root zone unless a suitable drainage

system were installed to remove excess waler from the area

To estimate the rate of drainage required to prevent the buildshy

up of the groundwater table to undesirable levels several drainage

rates were assumed in simulacing the groundwater table movement

The assumption of a uniform drainage rate of 10 cm per month over

the entire area results in the groundwater contour maps shown in

Figures 5 through 9 It is noted that although the groundwater table

+ (Z []

wbpthe tt

Thus m o e~ s l

at suit-able depth thip~gh~uV t e

pf

rA o (V

With particulart4efe once to the A6400

collection

1 ientyiz cm

program in ISgosted t

PrecipiaJ onlnoVillllt

athuedI4amp J

at

t~~Ve Atlantico 3 arl

utb Itle depets tr O thtjit

and that poabeD

+total of ai -0 Fi t p t

titt

rntltesg e dta a

mtow

i

I-1

--

o Al

+ +Iti~UgU mto4ih

714

and~tht1i~ JRiIuas14-11 Tl

Ah

11

cedure This is a time-consuming and costly process

Therefore as a part of this study a self-optimizing scheme

has been developed and soon will be incorporated in the simshy

ulation model for automatic identification of these paramshy

eters In this way it will be possible to efficiently apply

the model to any prototype area for which sufficient verifishy

cation-data are available

3 As previously discussed tothis point it has been necessary

to either assume or rather grossly approximate many data

used in the model of the Atlantico 3 area As additional

data for this area become available they will be used to furshy

ther improve and test the model

Research Utilization

Although the present study is directed specifically to the reshy

3arch needs for the Atlantico 3 area the simulation model developed

entirely general and can be applied to different geographic areas

addition the philosophy and techniques used in the analysis can

e applied equally well to many problems of similar nature

Presentations based primarily on the initial model were made

t the IV Latin American Congress on Hydraulics Mexico City Aushy

ust 1970 at the 6th American Water Resource Conference Las Vegas

[evada November 1970 and at an International Symposium on Groundshy

iater held at Pale rmoo Sicily inDecember 1970 The paper Upon

hich these Presentations were based is included as Appendix A

A description of the revised model and its applications is now

)eing prepared as a paper to be submitted to an appropriate technical

journal This model was also briefly described in a presentation to

he participants of the seminar on Water Resources Planning which

vas held at Utah State University in June 1971

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

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use n 1rtptoi~tw~ist 4 4 P

WY94

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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77 777

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we ~~77 4 - -~ 3$ 7

1

244Th 4 4 ~ ttL-144

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271

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 6: FOR AID USE ONLY

considered that the mnodel should be capable of reflecting

topographic influences upon groundwater levels

5- Capability for considering different soil permeability coshy

efficients at each grid point

6 Addition of the salinity dimension to the model in accordshy

ance with previous work at Utah State University

7 Improvement of the model using hydrologic data which ICo

become available since the completion of the initial study

8 Perform continuing sensitivity studies to establish priorshy

ities and resolution needs for data collection programs

In connection with the preceding list the following is a brief

description of the progress that was made on the project during the

period March]1 to December 31 1970

1 The initial model approximated the area under considerashy

tion by a rectangle with its four edges as boundaries

This approximation caused difficulty in properly defining

the boundary conditions at various times The revised

model as described in Appendix B considers all possishy

bleboundary irregularities and therefore handles areas

of any shape Be this revision of the model Item 1 has

been accomplished

2 Because of the increase in the memory capacity of the

computer and thedecrease in required memory space

due to the revised solution method for the partial differ-

ential equations which described the groundwater fluctushy

3

ations a significant increase in the grid density was made

possible The grid increment in the revised model is 625

meters (Figuire 1) compared to the-Z000meters of the inishy

tial model Tle total number of the grid points within the

area is now 849 For each of these grid points the effecshy

tive percolatipn to (or withdrawal from ) the groundwater

during each tine increment was simulated by the surface

component of the model This computed quantity at each

grid point was then fed into the groundwater component of

the modelto simulate the groundwater table fluctuations

The Dirichlet type boundary condition for the groundwater

model was properly defined on the basis of the available

data The input data for the surface model were precipishy

tation temperature soil type and the corresponding crop

pattern in terms of crop coefficients and irrigation reshy

quirements soil moisture holding capacity initial soil

moisture and swamp storage crop densities and a toposhy

graphic parameter The inputs to the groundwater model

include the initial water table levels water table levels

along the boundaries at different times and the transmisshy

sivity And specific storage of the aquifer The model was

availshycalibrated over a period where reliable data were

able to identify the model parameters- Items 2 and 3 of

the preceding list were thus fulfilled

3 To represent the location variations of the groundwater

table due to topographic influences as specified in Item 4

a topographicparameter which characterize the drainage

or collection of surface water was introduced in the reshy

vised model For the Atlantico 3 area the value for this

parameter at each grid point was determined from a toposhy

graphic map (Figure 2)

4 There was not yet sufficient data available within the

Atlantico 3 area to properly define variations in the soil

permeability The assumption of a homogineous soil

was therefore retained in the revised model However

the model contains sufficient resolution to characterize

these variations and when -permeability data become

available at different locations in the area the model

can be revised in this regard

5 Item 6 also has not yet been accomplished primarily beshy

cause of the lack of water quality data Techniques have

already been developed at USU for adding the water qualishy

ty dimensions to hydrologic simulation models and this

vill be done for the Atlantico 3 modef when the necess ary

vater quality data become available

6 In accordance with Item 7 all relevant data that have beshy

come available since the completion of the initial model

halve been incorporated into the operation of the revised

model

7 The sensitivity studies referred tomyItem 8 were conducted

by observing the model responses of both the surface and

groundwater systems to various parameters such as

phreatophyte density agricultural crop pattern irrigation

supply and soil moisture holding capacity These analyses

suggested several areas of additional data needs within the

system and these needs will be discussed in a subseqient

part of this report

Model Calibration

The revised model was calibrated by using data taken during

1969 While meteorologic data wereavailable for the three years

of 1967 1968 and 1969 adequate information on groundwater levels

could be obtained for only 1969 Although the calibration of a monthshy

ly model over a period of only one year leaves room for question it shy

is considered that the relative magnitudes of the various parameters

associated with the model have been established In addition conshy

siderable insight into operation of the prototype system has been

provided As more data become available for subsequent years the

calibration of Lhe model will be improved

Management Studies

Based on the soil land classification and precipitation data

for the study area croppatterns and the correspnding crop coef-

ficients and irrigation rates wete assumed as shown by Table 1

Table 1 Crop-pattern crop-coefficients and irrigation for different soils

Soil Group Item Crop Jan

Crop-pattern weighted crop-coefficient and irrigation rate Feb Mar Apr May Jun Jul Aug SeptI Oct Nov Dec

1 Crop pattern Ci trus -Peanuts Maize

Crop coeff Irr rate

J65 112

-75 112

55 90

60 45

45 60

60 60

75 60

60 60

60 45

60 60

60 60

50 60

2 Crop pattern

Crop coeff Irr rate

Cotton Sorghum

70 112

50 90

20 0

20 0

30 45

60 60

90 60

60 60

40 60

65 60

90 90

90 112

3 Crop pattern Grasses - -

4

Crop coeff Irr rate

_Crop-coeff Irr rate

Bare Soil

80 90

10 0

80 90

10 0

80 90

10 0

80 75

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 75

10 0

80 90

10 0

-Inmmonth irrigation efficiency = 06

7

According to available information existing densities of the native

secshyphreatophytes vary from about 50 percent in the south-eastern

tion of the arep to approximately 20 percent in the-north-western -part

To investigate the responses of the groundwater table to areduction

in the area of phreatophytes and to the application of irrigation water

to cultivated crops the model was operated under the following

assumptions

1 Half of the native phreatophytes were assumed to be reshy

placed by the cultivated crops shown in Table 1

2 No sub-surface drainage was established

3 The available precipitation and evaporation data for the

period of )967 through 1969 were assumed to be represhy

sentative for the area

Figures 3 and 4 show the simulated groundwater surface within

area at the end of 6 and 12 months after the assumed developmentthe

outlined above These figures suggest that the groundwater table

would build up quickly to the root zone unless a suitable drainage

system were installed to remove excess waler from the area

To estimate the rate of drainage required to prevent the buildshy

up of the groundwater table to undesirable levels several drainage

rates were assumed in simulacing the groundwater table movement

The assumption of a uniform drainage rate of 10 cm per month over

the entire area results in the groundwater contour maps shown in

Figures 5 through 9 It is noted that although the groundwater table

+ (Z []

wbpthe tt

Thus m o e~ s l

at suit-able depth thip~gh~uV t e

pf

rA o (V

With particulart4efe once to the A6400

collection

1 ientyiz cm

program in ISgosted t

PrecipiaJ onlnoVillllt

athuedI4amp J

at

t~~Ve Atlantico 3 arl

utb Itle depets tr O thtjit

and that poabeD

+total of ai -0 Fi t p t

titt

rntltesg e dta a

mtow

i

I-1

--

o Al

+ +Iti~UgU mto4ih

714

and~tht1i~ JRiIuas14-11 Tl

Ah

11

cedure This is a time-consuming and costly process

Therefore as a part of this study a self-optimizing scheme

has been developed and soon will be incorporated in the simshy

ulation model for automatic identification of these paramshy

eters In this way it will be possible to efficiently apply

the model to any prototype area for which sufficient verifishy

cation-data are available

3 As previously discussed tothis point it has been necessary

to either assume or rather grossly approximate many data

used in the model of the Atlantico 3 area As additional

data for this area become available they will be used to furshy

ther improve and test the model

Research Utilization

Although the present study is directed specifically to the reshy

3arch needs for the Atlantico 3 area the simulation model developed

entirely general and can be applied to different geographic areas

addition the philosophy and techniques used in the analysis can

e applied equally well to many problems of similar nature

Presentations based primarily on the initial model were made

t the IV Latin American Congress on Hydraulics Mexico City Aushy

ust 1970 at the 6th American Water Resource Conference Las Vegas

[evada November 1970 and at an International Symposium on Groundshy

iater held at Pale rmoo Sicily inDecember 1970 The paper Upon

hich these Presentations were based is included as Appendix A

A description of the revised model and its applications is now

)eing prepared as a paper to be submitted to an appropriate technical

journal This model was also briefly described in a presentation to

he participants of the seminar on Water Resources Planning which

vas held at Utah State University in June 1971

13

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

i~ 7 I

T -II -r-

-shy

44~~~

use n 1rtptoi~tw~ist 4 4 P

WY94

W

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VAshy

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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7)

we ~~77 4 - -~ 3$ 7

1

244Th 4 4 ~ ttL-144

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~fl~KHYBRID COMPUTER $R~1~ m

271

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7727 ~

16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 7: FOR AID USE ONLY

3

ations a significant increase in the grid density was made

possible The grid increment in the revised model is 625

meters (Figuire 1) compared to the-Z000meters of the inishy

tial model Tle total number of the grid points within the

area is now 849 For each of these grid points the effecshy

tive percolatipn to (or withdrawal from ) the groundwater

during each tine increment was simulated by the surface

component of the model This computed quantity at each

grid point was then fed into the groundwater component of

the modelto simulate the groundwater table fluctuations

The Dirichlet type boundary condition for the groundwater

model was properly defined on the basis of the available

data The input data for the surface model were precipishy

tation temperature soil type and the corresponding crop

pattern in terms of crop coefficients and irrigation reshy

quirements soil moisture holding capacity initial soil

moisture and swamp storage crop densities and a toposhy

graphic parameter The inputs to the groundwater model

include the initial water table levels water table levels

along the boundaries at different times and the transmisshy

sivity And specific storage of the aquifer The model was

availshycalibrated over a period where reliable data were

able to identify the model parameters- Items 2 and 3 of

the preceding list were thus fulfilled

3 To represent the location variations of the groundwater

table due to topographic influences as specified in Item 4

a topographicparameter which characterize the drainage

or collection of surface water was introduced in the reshy

vised model For the Atlantico 3 area the value for this

parameter at each grid point was determined from a toposhy

graphic map (Figure 2)

4 There was not yet sufficient data available within the

Atlantico 3 area to properly define variations in the soil

permeability The assumption of a homogineous soil

was therefore retained in the revised model However

the model contains sufficient resolution to characterize

these variations and when -permeability data become

available at different locations in the area the model

can be revised in this regard

5 Item 6 also has not yet been accomplished primarily beshy

cause of the lack of water quality data Techniques have

already been developed at USU for adding the water qualishy

ty dimensions to hydrologic simulation models and this

vill be done for the Atlantico 3 modef when the necess ary

vater quality data become available

6 In accordance with Item 7 all relevant data that have beshy

come available since the completion of the initial model

halve been incorporated into the operation of the revised

model

7 The sensitivity studies referred tomyItem 8 were conducted

by observing the model responses of both the surface and

groundwater systems to various parameters such as

phreatophyte density agricultural crop pattern irrigation

supply and soil moisture holding capacity These analyses

suggested several areas of additional data needs within the

system and these needs will be discussed in a subseqient

part of this report

Model Calibration

The revised model was calibrated by using data taken during

1969 While meteorologic data wereavailable for the three years

of 1967 1968 and 1969 adequate information on groundwater levels

could be obtained for only 1969 Although the calibration of a monthshy

ly model over a period of only one year leaves room for question it shy

is considered that the relative magnitudes of the various parameters

associated with the model have been established In addition conshy

siderable insight into operation of the prototype system has been

provided As more data become available for subsequent years the

calibration of Lhe model will be improved

Management Studies

Based on the soil land classification and precipitation data

for the study area croppatterns and the correspnding crop coef-

ficients and irrigation rates wete assumed as shown by Table 1

Table 1 Crop-pattern crop-coefficients and irrigation for different soils

Soil Group Item Crop Jan

Crop-pattern weighted crop-coefficient and irrigation rate Feb Mar Apr May Jun Jul Aug SeptI Oct Nov Dec

1 Crop pattern Ci trus -Peanuts Maize

Crop coeff Irr rate

J65 112

-75 112

55 90

60 45

45 60

60 60

75 60

60 60

60 45

60 60

60 60

50 60

2 Crop pattern

Crop coeff Irr rate

Cotton Sorghum

70 112

50 90

20 0

20 0

30 45

60 60

90 60

60 60

40 60

65 60

90 90

90 112

3 Crop pattern Grasses - -

4

Crop coeff Irr rate

_Crop-coeff Irr rate

Bare Soil

80 90

10 0

80 90

10 0

80 90

10 0

80 75

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 75

10 0

80 90

10 0

-Inmmonth irrigation efficiency = 06

7

According to available information existing densities of the native

secshyphreatophytes vary from about 50 percent in the south-eastern

tion of the arep to approximately 20 percent in the-north-western -part

To investigate the responses of the groundwater table to areduction

in the area of phreatophytes and to the application of irrigation water

to cultivated crops the model was operated under the following

assumptions

1 Half of the native phreatophytes were assumed to be reshy

placed by the cultivated crops shown in Table 1

2 No sub-surface drainage was established

3 The available precipitation and evaporation data for the

period of )967 through 1969 were assumed to be represhy

sentative for the area

Figures 3 and 4 show the simulated groundwater surface within

area at the end of 6 and 12 months after the assumed developmentthe

outlined above These figures suggest that the groundwater table

would build up quickly to the root zone unless a suitable drainage

system were installed to remove excess waler from the area

To estimate the rate of drainage required to prevent the buildshy

up of the groundwater table to undesirable levels several drainage

rates were assumed in simulacing the groundwater table movement

The assumption of a uniform drainage rate of 10 cm per month over

the entire area results in the groundwater contour maps shown in

Figures 5 through 9 It is noted that although the groundwater table

+ (Z []

wbpthe tt

Thus m o e~ s l

at suit-able depth thip~gh~uV t e

pf

rA o (V

With particulart4efe once to the A6400

collection

1 ientyiz cm

program in ISgosted t

PrecipiaJ onlnoVillllt

athuedI4amp J

at

t~~Ve Atlantico 3 arl

utb Itle depets tr O thtjit

and that poabeD

+total of ai -0 Fi t p t

titt

rntltesg e dta a

mtow

i

I-1

--

o Al

+ +Iti~UgU mto4ih

714

and~tht1i~ JRiIuas14-11 Tl

Ah

11

cedure This is a time-consuming and costly process

Therefore as a part of this study a self-optimizing scheme

has been developed and soon will be incorporated in the simshy

ulation model for automatic identification of these paramshy

eters In this way it will be possible to efficiently apply

the model to any prototype area for which sufficient verifishy

cation-data are available

3 As previously discussed tothis point it has been necessary

to either assume or rather grossly approximate many data

used in the model of the Atlantico 3 area As additional

data for this area become available they will be used to furshy

ther improve and test the model

Research Utilization

Although the present study is directed specifically to the reshy

3arch needs for the Atlantico 3 area the simulation model developed

entirely general and can be applied to different geographic areas

addition the philosophy and techniques used in the analysis can

e applied equally well to many problems of similar nature

Presentations based primarily on the initial model were made

t the IV Latin American Congress on Hydraulics Mexico City Aushy

ust 1970 at the 6th American Water Resource Conference Las Vegas

[evada November 1970 and at an International Symposium on Groundshy

iater held at Pale rmoo Sicily inDecember 1970 The paper Upon

hich these Presentations were based is included as Appendix A

A description of the revised model and its applications is now

)eing prepared as a paper to be submitted to an appropriate technical

journal This model was also briefly described in a presentation to

he participants of the seminar on Water Resources Planning which

vas held at Utah State University in June 1971

13

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

i~ 7 I

T -II -r-

-shy

44~~~

use n 1rtptoi~tw~ist 4 4 P

WY94

W

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VAshy

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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1

244Th 4 4 ~ ttL-144

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271

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 8: FOR AID USE ONLY

3 To represent the location variations of the groundwater

table due to topographic influences as specified in Item 4

a topographicparameter which characterize the drainage

or collection of surface water was introduced in the reshy

vised model For the Atlantico 3 area the value for this

parameter at each grid point was determined from a toposhy

graphic map (Figure 2)

4 There was not yet sufficient data available within the

Atlantico 3 area to properly define variations in the soil

permeability The assumption of a homogineous soil

was therefore retained in the revised model However

the model contains sufficient resolution to characterize

these variations and when -permeability data become

available at different locations in the area the model

can be revised in this regard

5 Item 6 also has not yet been accomplished primarily beshy

cause of the lack of water quality data Techniques have

already been developed at USU for adding the water qualishy

ty dimensions to hydrologic simulation models and this

vill be done for the Atlantico 3 modef when the necess ary

vater quality data become available

6 In accordance with Item 7 all relevant data that have beshy

come available since the completion of the initial model

halve been incorporated into the operation of the revised

model

7 The sensitivity studies referred tomyItem 8 were conducted

by observing the model responses of both the surface and

groundwater systems to various parameters such as

phreatophyte density agricultural crop pattern irrigation

supply and soil moisture holding capacity These analyses

suggested several areas of additional data needs within the

system and these needs will be discussed in a subseqient

part of this report

Model Calibration

The revised model was calibrated by using data taken during

1969 While meteorologic data wereavailable for the three years

of 1967 1968 and 1969 adequate information on groundwater levels

could be obtained for only 1969 Although the calibration of a monthshy

ly model over a period of only one year leaves room for question it shy

is considered that the relative magnitudes of the various parameters

associated with the model have been established In addition conshy

siderable insight into operation of the prototype system has been

provided As more data become available for subsequent years the

calibration of Lhe model will be improved

Management Studies

Based on the soil land classification and precipitation data

for the study area croppatterns and the correspnding crop coef-

ficients and irrigation rates wete assumed as shown by Table 1

Table 1 Crop-pattern crop-coefficients and irrigation for different soils

Soil Group Item Crop Jan

Crop-pattern weighted crop-coefficient and irrigation rate Feb Mar Apr May Jun Jul Aug SeptI Oct Nov Dec

1 Crop pattern Ci trus -Peanuts Maize

Crop coeff Irr rate

J65 112

-75 112

55 90

60 45

45 60

60 60

75 60

60 60

60 45

60 60

60 60

50 60

2 Crop pattern

Crop coeff Irr rate

Cotton Sorghum

70 112

50 90

20 0

20 0

30 45

60 60

90 60

60 60

40 60

65 60

90 90

90 112

3 Crop pattern Grasses - -

4

Crop coeff Irr rate

_Crop-coeff Irr rate

Bare Soil

80 90

10 0

80 90

10 0

80 90

10 0

80 75

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 75

10 0

80 90

10 0

-Inmmonth irrigation efficiency = 06

7

According to available information existing densities of the native

secshyphreatophytes vary from about 50 percent in the south-eastern

tion of the arep to approximately 20 percent in the-north-western -part

To investigate the responses of the groundwater table to areduction

in the area of phreatophytes and to the application of irrigation water

to cultivated crops the model was operated under the following

assumptions

1 Half of the native phreatophytes were assumed to be reshy

placed by the cultivated crops shown in Table 1

2 No sub-surface drainage was established

3 The available precipitation and evaporation data for the

period of )967 through 1969 were assumed to be represhy

sentative for the area

Figures 3 and 4 show the simulated groundwater surface within

area at the end of 6 and 12 months after the assumed developmentthe

outlined above These figures suggest that the groundwater table

would build up quickly to the root zone unless a suitable drainage

system were installed to remove excess waler from the area

To estimate the rate of drainage required to prevent the buildshy

up of the groundwater table to undesirable levels several drainage

rates were assumed in simulacing the groundwater table movement

The assumption of a uniform drainage rate of 10 cm per month over

the entire area results in the groundwater contour maps shown in

Figures 5 through 9 It is noted that although the groundwater table

+ (Z []

wbpthe tt

Thus m o e~ s l

at suit-able depth thip~gh~uV t e

pf

rA o (V

With particulart4efe once to the A6400

collection

1 ientyiz cm

program in ISgosted t

PrecipiaJ onlnoVillllt

athuedI4amp J

at

t~~Ve Atlantico 3 arl

utb Itle depets tr O thtjit

and that poabeD

+total of ai -0 Fi t p t

titt

rntltesg e dta a

mtow

i

I-1

--

o Al

+ +Iti~UgU mto4ih

714

and~tht1i~ JRiIuas14-11 Tl

Ah

11

cedure This is a time-consuming and costly process

Therefore as a part of this study a self-optimizing scheme

has been developed and soon will be incorporated in the simshy

ulation model for automatic identification of these paramshy

eters In this way it will be possible to efficiently apply

the model to any prototype area for which sufficient verifishy

cation-data are available

3 As previously discussed tothis point it has been necessary

to either assume or rather grossly approximate many data

used in the model of the Atlantico 3 area As additional

data for this area become available they will be used to furshy

ther improve and test the model

Research Utilization

Although the present study is directed specifically to the reshy

3arch needs for the Atlantico 3 area the simulation model developed

entirely general and can be applied to different geographic areas

addition the philosophy and techniques used in the analysis can

e applied equally well to many problems of similar nature

Presentations based primarily on the initial model were made

t the IV Latin American Congress on Hydraulics Mexico City Aushy

ust 1970 at the 6th American Water Resource Conference Las Vegas

[evada November 1970 and at an International Symposium on Groundshy

iater held at Pale rmoo Sicily inDecember 1970 The paper Upon

hich these Presentations were based is included as Appendix A

A description of the revised model and its applications is now

)eing prepared as a paper to be submitted to an appropriate technical

journal This model was also briefly described in a presentation to

he participants of the seminar on Water Resources Planning which

vas held at Utah State University in June 1971

13

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

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T -II -r-

-shy

44~~~

use n 1rtptoi~tw~ist 4 4 P

WY94

W

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VAshy

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

~4t

ii-gt r 777~ ~

77 777

~ 715 7 gtCN~JY44~7

3~I- t~ 77 -4777777

z)7~77~t77777 777777 ) 1A ~~4~ti77 c4 2-~ I 7

-~ ~ NI-shy

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mZ274~7 N

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77 S- --4r~ amp~7~C~

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W4rlt2~PK 2 ~ -~k4t~Ntxflt

- 2 -

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~ 777 7741a47

7 x- ~W AI47

77 ~777T 7-1-7-- i2777744 7777A 73 j7 J~X1~VP~4 77

7~74 - ~ r 2 n

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7)

we ~~77 4 - -~ 3$ 7

1

244Th 4 4 ~ ttL-144

~4 c~JJ~ t U -

~fl~KHYBRID COMPUTER $R~1~ m

271

-7 417 77777 77 s 1

44 44 ~ - 27A-~~ ~ 7

NJ 7 ~shy

(177lt N744t ~

~

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Lamp~~5t ~2fl6

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--~-17747~~~t ~

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0

777777-5rfT77rY2clr~27fl~1~LY1~r7

7 I 3NL1 ~ Cl

47 (777tgt 7t77t~7J777t4v~7ttc - s7t$~-7w2A3t~~4 - -

77 - 1(~7~V7 7P~~2fl~ ~tiSi 7lt 7777 ~-4 77W7~

~

74

273 7

14~ 72if rb

7~

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7 A7f7L7~7~7$

7 777

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~

74~Agt77N~7747Y7777

r20F 7 4A~7 ~ 0~r- 77

7 s77t7 4c~t 7 Il rCl44 j$r~x~77 777 ~K 17~7 ~

I 7 771 77723 ~

lt

7 7~7 ~f

~77 7 7 V ~ 2 7

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7 7

7727 ~

16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 9: FOR AID USE ONLY

7 The sensitivity studies referred tomyItem 8 were conducted

by observing the model responses of both the surface and

groundwater systems to various parameters such as

phreatophyte density agricultural crop pattern irrigation

supply and soil moisture holding capacity These analyses

suggested several areas of additional data needs within the

system and these needs will be discussed in a subseqient

part of this report

Model Calibration

The revised model was calibrated by using data taken during

1969 While meteorologic data wereavailable for the three years

of 1967 1968 and 1969 adequate information on groundwater levels

could be obtained for only 1969 Although the calibration of a monthshy

ly model over a period of only one year leaves room for question it shy

is considered that the relative magnitudes of the various parameters

associated with the model have been established In addition conshy

siderable insight into operation of the prototype system has been

provided As more data become available for subsequent years the

calibration of Lhe model will be improved

Management Studies

Based on the soil land classification and precipitation data

for the study area croppatterns and the correspnding crop coef-

ficients and irrigation rates wete assumed as shown by Table 1

Table 1 Crop-pattern crop-coefficients and irrigation for different soils

Soil Group Item Crop Jan

Crop-pattern weighted crop-coefficient and irrigation rate Feb Mar Apr May Jun Jul Aug SeptI Oct Nov Dec

1 Crop pattern Ci trus -Peanuts Maize

Crop coeff Irr rate

J65 112

-75 112

55 90

60 45

45 60

60 60

75 60

60 60

60 45

60 60

60 60

50 60

2 Crop pattern

Crop coeff Irr rate

Cotton Sorghum

70 112

50 90

20 0

20 0

30 45

60 60

90 60

60 60

40 60

65 60

90 90

90 112

3 Crop pattern Grasses - -

4

Crop coeff Irr rate

_Crop-coeff Irr rate

Bare Soil

80 90

10 0

80 90

10 0

80 90

10 0

80 75

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 75

10 0

80 90

10 0

-Inmmonth irrigation efficiency = 06

7

According to available information existing densities of the native

secshyphreatophytes vary from about 50 percent in the south-eastern

tion of the arep to approximately 20 percent in the-north-western -part

To investigate the responses of the groundwater table to areduction

in the area of phreatophytes and to the application of irrigation water

to cultivated crops the model was operated under the following

assumptions

1 Half of the native phreatophytes were assumed to be reshy

placed by the cultivated crops shown in Table 1

2 No sub-surface drainage was established

3 The available precipitation and evaporation data for the

period of )967 through 1969 were assumed to be represhy

sentative for the area

Figures 3 and 4 show the simulated groundwater surface within

area at the end of 6 and 12 months after the assumed developmentthe

outlined above These figures suggest that the groundwater table

would build up quickly to the root zone unless a suitable drainage

system were installed to remove excess waler from the area

To estimate the rate of drainage required to prevent the buildshy

up of the groundwater table to undesirable levels several drainage

rates were assumed in simulacing the groundwater table movement

The assumption of a uniform drainage rate of 10 cm per month over

the entire area results in the groundwater contour maps shown in

Figures 5 through 9 It is noted that although the groundwater table

+ (Z []

wbpthe tt

Thus m o e~ s l

at suit-able depth thip~gh~uV t e

pf

rA o (V

With particulart4efe once to the A6400

collection

1 ientyiz cm

program in ISgosted t

PrecipiaJ onlnoVillllt

athuedI4amp J

at

t~~Ve Atlantico 3 arl

utb Itle depets tr O thtjit

and that poabeD

+total of ai -0 Fi t p t

titt

rntltesg e dta a

mtow

i

I-1

--

o Al

+ +Iti~UgU mto4ih

714

and~tht1i~ JRiIuas14-11 Tl

Ah

11

cedure This is a time-consuming and costly process

Therefore as a part of this study a self-optimizing scheme

has been developed and soon will be incorporated in the simshy

ulation model for automatic identification of these paramshy

eters In this way it will be possible to efficiently apply

the model to any prototype area for which sufficient verifishy

cation-data are available

3 As previously discussed tothis point it has been necessary

to either assume or rather grossly approximate many data

used in the model of the Atlantico 3 area As additional

data for this area become available they will be used to furshy

ther improve and test the model

Research Utilization

Although the present study is directed specifically to the reshy

3arch needs for the Atlantico 3 area the simulation model developed

entirely general and can be applied to different geographic areas

addition the philosophy and techniques used in the analysis can

e applied equally well to many problems of similar nature

Presentations based primarily on the initial model were made

t the IV Latin American Congress on Hydraulics Mexico City Aushy

ust 1970 at the 6th American Water Resource Conference Las Vegas

[evada November 1970 and at an International Symposium on Groundshy

iater held at Pale rmoo Sicily inDecember 1970 The paper Upon

hich these Presentations were based is included as Appendix A

A description of the revised model and its applications is now

)eing prepared as a paper to be submitted to an appropriate technical

journal This model was also briefly described in a presentation to

he participants of the seminar on Water Resources Planning which

vas held at Utah State University in June 1971

13

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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77 777

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z)7~77~t77777 777777 ) 1A ~~4~ti77 c4 2-~ I 7

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7)

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1

244Th 4 4 ~ ttL-144

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271

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 10: FOR AID USE ONLY

Table 1 Crop-pattern crop-coefficients and irrigation for different soils

Soil Group Item Crop Jan

Crop-pattern weighted crop-coefficient and irrigation rate Feb Mar Apr May Jun Jul Aug SeptI Oct Nov Dec

1 Crop pattern Ci trus -Peanuts Maize

Crop coeff Irr rate

J65 112

-75 112

55 90

60 45

45 60

60 60

75 60

60 60

60 45

60 60

60 60

50 60

2 Crop pattern

Crop coeff Irr rate

Cotton Sorghum

70 112

50 90

20 0

20 0

30 45

60 60

90 60

60 60

40 60

65 60

90 90

90 112

3 Crop pattern Grasses - -

4

Crop coeff Irr rate

_Crop-coeff Irr rate

Bare Soil

80 90

10 0

80 90

10 0

80 90

10 0

80 75

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 60

10 0

80 75

10 0

80 90

10 0

-Inmmonth irrigation efficiency = 06

7

According to available information existing densities of the native

secshyphreatophytes vary from about 50 percent in the south-eastern

tion of the arep to approximately 20 percent in the-north-western -part

To investigate the responses of the groundwater table to areduction

in the area of phreatophytes and to the application of irrigation water

to cultivated crops the model was operated under the following

assumptions

1 Half of the native phreatophytes were assumed to be reshy

placed by the cultivated crops shown in Table 1

2 No sub-surface drainage was established

3 The available precipitation and evaporation data for the

period of )967 through 1969 were assumed to be represhy

sentative for the area

Figures 3 and 4 show the simulated groundwater surface within

area at the end of 6 and 12 months after the assumed developmentthe

outlined above These figures suggest that the groundwater table

would build up quickly to the root zone unless a suitable drainage

system were installed to remove excess waler from the area

To estimate the rate of drainage required to prevent the buildshy

up of the groundwater table to undesirable levels several drainage

rates were assumed in simulacing the groundwater table movement

The assumption of a uniform drainage rate of 10 cm per month over

the entire area results in the groundwater contour maps shown in

Figures 5 through 9 It is noted that although the groundwater table

+ (Z []

wbpthe tt

Thus m o e~ s l

at suit-able depth thip~gh~uV t e

pf

rA o (V

With particulart4efe once to the A6400

collection

1 ientyiz cm

program in ISgosted t

PrecipiaJ onlnoVillllt

athuedI4amp J

at

t~~Ve Atlantico 3 arl

utb Itle depets tr O thtjit

and that poabeD

+total of ai -0 Fi t p t

titt

rntltesg e dta a

mtow

i

I-1

--

o Al

+ +Iti~UgU mto4ih

714

and~tht1i~ JRiIuas14-11 Tl

Ah

11

cedure This is a time-consuming and costly process

Therefore as a part of this study a self-optimizing scheme

has been developed and soon will be incorporated in the simshy

ulation model for automatic identification of these paramshy

eters In this way it will be possible to efficiently apply

the model to any prototype area for which sufficient verifishy

cation-data are available

3 As previously discussed tothis point it has been necessary

to either assume or rather grossly approximate many data

used in the model of the Atlantico 3 area As additional

data for this area become available they will be used to furshy

ther improve and test the model

Research Utilization

Although the present study is directed specifically to the reshy

3arch needs for the Atlantico 3 area the simulation model developed

entirely general and can be applied to different geographic areas

addition the philosophy and techniques used in the analysis can

e applied equally well to many problems of similar nature

Presentations based primarily on the initial model were made

t the IV Latin American Congress on Hydraulics Mexico City Aushy

ust 1970 at the 6th American Water Resource Conference Las Vegas

[evada November 1970 and at an International Symposium on Groundshy

iater held at Pale rmoo Sicily inDecember 1970 The paper Upon

hich these Presentations were based is included as Appendix A

A description of the revised model and its applications is now

)eing prepared as a paper to be submitted to an appropriate technical

journal This model was also briefly described in a presentation to

he participants of the seminar on Water Resources Planning which

vas held at Utah State University in June 1971

13

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

i~ 7 I

T -II -r-

-shy

44~~~

use n 1rtptoi~tw~ist 4 4 P

WY94

W

LL

VAshy

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

~4t

ii-gt r 777~ ~

77 777

~ 715 7 gtCN~JY44~7

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z)7~77~t77777 777777 ) 1A ~~4~ti77 c4 2-~ I 7

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1

244Th 4 4 ~ ttL-144

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271

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 11: FOR AID USE ONLY

7

According to available information existing densities of the native

secshyphreatophytes vary from about 50 percent in the south-eastern

tion of the arep to approximately 20 percent in the-north-western -part

To investigate the responses of the groundwater table to areduction

in the area of phreatophytes and to the application of irrigation water

to cultivated crops the model was operated under the following

assumptions

1 Half of the native phreatophytes were assumed to be reshy

placed by the cultivated crops shown in Table 1

2 No sub-surface drainage was established

3 The available precipitation and evaporation data for the

period of )967 through 1969 were assumed to be represhy

sentative for the area

Figures 3 and 4 show the simulated groundwater surface within

area at the end of 6 and 12 months after the assumed developmentthe

outlined above These figures suggest that the groundwater table

would build up quickly to the root zone unless a suitable drainage

system were installed to remove excess waler from the area

To estimate the rate of drainage required to prevent the buildshy

up of the groundwater table to undesirable levels several drainage

rates were assumed in simulacing the groundwater table movement

The assumption of a uniform drainage rate of 10 cm per month over

the entire area results in the groundwater contour maps shown in

Figures 5 through 9 It is noted that although the groundwater table

+ (Z []

wbpthe tt

Thus m o e~ s l

at suit-able depth thip~gh~uV t e

pf

rA o (V

With particulart4efe once to the A6400

collection

1 ientyiz cm

program in ISgosted t

PrecipiaJ onlnoVillllt

athuedI4amp J

at

t~~Ve Atlantico 3 arl

utb Itle depets tr O thtjit

and that poabeD

+total of ai -0 Fi t p t

titt

rntltesg e dta a

mtow

i

I-1

--

o Al

+ +Iti~UgU mto4ih

714

and~tht1i~ JRiIuas14-11 Tl

Ah

11

cedure This is a time-consuming and costly process

Therefore as a part of this study a self-optimizing scheme

has been developed and soon will be incorporated in the simshy

ulation model for automatic identification of these paramshy

eters In this way it will be possible to efficiently apply

the model to any prototype area for which sufficient verifishy

cation-data are available

3 As previously discussed tothis point it has been necessary

to either assume or rather grossly approximate many data

used in the model of the Atlantico 3 area As additional

data for this area become available they will be used to furshy

ther improve and test the model

Research Utilization

Although the present study is directed specifically to the reshy

3arch needs for the Atlantico 3 area the simulation model developed

entirely general and can be applied to different geographic areas

addition the philosophy and techniques used in the analysis can

e applied equally well to many problems of similar nature

Presentations based primarily on the initial model were made

t the IV Latin American Congress on Hydraulics Mexico City Aushy

ust 1970 at the 6th American Water Resource Conference Las Vegas

[evada November 1970 and at an International Symposium on Groundshy

iater held at Pale rmoo Sicily inDecember 1970 The paper Upon

hich these Presentations were based is included as Appendix A

A description of the revised model and its applications is now

)eing prepared as a paper to be submitted to an appropriate technical

journal This model was also briefly described in a presentation to

he participants of the seminar on Water Resources Planning which

vas held at Utah State University in June 1971

13

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

i~ 7 I

T -II -r-

-shy

44~~~

use n 1rtptoi~tw~ist 4 4 P

WY94

W

LL

VAshy

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 12: FOR AID USE ONLY

+ (Z []

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at suit-able depth thip~gh~uV t e

pf

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With particulart4efe once to the A6400

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program in ISgosted t

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utb Itle depets tr O thtjit

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11

cedure This is a time-consuming and costly process

Therefore as a part of this study a self-optimizing scheme

has been developed and soon will be incorporated in the simshy

ulation model for automatic identification of these paramshy

eters In this way it will be possible to efficiently apply

the model to any prototype area for which sufficient verifishy

cation-data are available

3 As previously discussed tothis point it has been necessary

to either assume or rather grossly approximate many data

used in the model of the Atlantico 3 area As additional

data for this area become available they will be used to furshy

ther improve and test the model

Research Utilization

Although the present study is directed specifically to the reshy

3arch needs for the Atlantico 3 area the simulation model developed

entirely general and can be applied to different geographic areas

addition the philosophy and techniques used in the analysis can

e applied equally well to many problems of similar nature

Presentations based primarily on the initial model were made

t the IV Latin American Congress on Hydraulics Mexico City Aushy

ust 1970 at the 6th American Water Resource Conference Las Vegas

[evada November 1970 and at an International Symposium on Groundshy

iater held at Pale rmoo Sicily inDecember 1970 The paper Upon

hich these Presentations were based is included as Appendix A

A description of the revised model and its applications is now

)eing prepared as a paper to be submitted to an appropriate technical

journal This model was also briefly described in a presentation to

he participants of the seminar on Water Resources Planning which

vas held at Utah State University in June 1971

13

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 13: FOR AID USE ONLY

11

cedure This is a time-consuming and costly process

Therefore as a part of this study a self-optimizing scheme

has been developed and soon will be incorporated in the simshy

ulation model for automatic identification of these paramshy

eters In this way it will be possible to efficiently apply

the model to any prototype area for which sufficient verifishy

cation-data are available

3 As previously discussed tothis point it has been necessary

to either assume or rather grossly approximate many data

used in the model of the Atlantico 3 area As additional

data for this area become available they will be used to furshy

ther improve and test the model

Research Utilization

Although the present study is directed specifically to the reshy

3arch needs for the Atlantico 3 area the simulation model developed

entirely general and can be applied to different geographic areas

addition the philosophy and techniques used in the analysis can

e applied equally well to many problems of similar nature

Presentations based primarily on the initial model were made

t the IV Latin American Congress on Hydraulics Mexico City Aushy

ust 1970 at the 6th American Water Resource Conference Las Vegas

[evada November 1970 and at an International Symposium on Groundshy

iater held at Pale rmoo Sicily inDecember 1970 The paper Upon

hich these Presentations were based is included as Appendix A

A description of the revised model and its applications is now

)eing prepared as a paper to be submitted to an appropriate technical

journal This model was also briefly described in a presentation to

he participants of the seminar on Water Resources Planning which

vas held at Utah State University in June 1971

13

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 14: FOR AID USE ONLY

A description of the revised model and its applications is now

)eing prepared as a paper to be submitted to an appropriate technical

journal This model was also briefly described in a presentation to

he participants of the seminar on Water Resources Planning which

vas held at Utah State University in June 1971

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

i~ 7 I

T -II -r-

-shy

44~~~

use n 1rtptoi~tw~ist 4 4 P

WY94

W

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VAshy

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 15: FOR AID USE ONLY

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

i~ 7 I

T -II -r-

-shy

44~~~

use n 1rtptoi~tw~ist 4 4 P

WY94

W

LL

VAshy

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 16: FOR AID USE ONLY

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

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use n 1rtptoi~tw~ist 4 4 P

WY94

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 17: FOR AID USE ONLY

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

i~ 7 I

T -II -r-

-shy

44~~~

use n 1rtptoi~tw~ist 4 4 P

WY94

W

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VAshy

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 19: FOR AID USE ONLY

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

i~ 7 I

T -II -r-

-shy

44~~~

use n 1rtptoi~tw~ist 4 4 P

WY94

W

LL

VAshy

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 20: FOR AID USE ONLY

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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7727 ~

16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 21: FOR AID USE ONLY

19

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

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use n 1rtptoi~tw~ist 4 4 P

WY94

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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271

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 23: FOR AID USE ONLY

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

i~ 7 I

T -II -r-

-shy

44~~~

use n 1rtptoi~tw~ist 4 4 P

WY94

W

LL

VAshy

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 24: FOR AID USE ONLY

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COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 25: FOR AID USE ONLY

COMBINED SURFACE WATER-GROUNDWATER ANALYSIS

OF HYDROLOGICAL SYSTEMS WITH THE AID I

OF THE HYBRID COMPUTER

Introduction

Thecontinuously increasing demands on our limited water resources

have necessitated usingmodern computing techniques to make effective use

The advent of the hybrid computer has made possibleof these resources

systems and the continuousresourcethe rapid solution of complex water

display of these solutions for verification or optimization studies For

water resource management purposes it is necessary to analyze the combined

surface water-groundwater system rather than carrying out separate analyses

for each system

under conditions of irrigated agriculture there existsFor instance

crop growth is inhibited The propera groundwater level abovewhich

management of groundwater systems for agriculture and other purposes requires

an understanding of the factors that control the water levels in these

aquifers including the net input or output to groundwater from the continuous

A hybridhydrologic processes that occur in the surface water system

computer model enables a rapid appraisal of these factors and provides a

levels under various management alternativesmeans of predicting future water

Historically the surface water supplies inmost areas have been

developed first and the groundwater resource has been-considered only when

the surface supply has proved inadequate to meet the demand There is now

Groundwater system - considered as all water within saturated zone

Surface water system -unsaturated zone and hydraulic and hydrologic

processes at ground level

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

i~ 7 I

T -II -r-

-shy

44~~~

use n 1rtptoi~tw~ist 4 4 P

WY94

W

LL

VAshy

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

~4t

ii-gt r 777~ ~

77 777

~ 715 7 gtCN~JY44~7

3~I- t~ 77 -4777777

z)7~77~t77777 777777 ) 1A ~~4~ti77 c4 2-~ I 7

-~ ~ NI-shy

c ~XT~LY 7 4~3C~7r2i~d

1 7 7~ I744~lt7

7 4

~r7S -

~72~ r~ir~nr 7 ~ t77

-

~ tj N ~ - shy1

mZ274~7 N

24rv-vamp $ ~1amp7t- 7 V 7~~~t~Ztk7shy7 77 - 7 77A1

77 S- --4r~ amp~7~C~

shy

2~ ~vA t 7

W4rlt2~PK 2 ~ -~k4t~Ntxflt

- 2 -

~C 1

~ 777 7741a47

7 x- ~W AI47

77 ~777T 7-1-7-- i2777744 7777A 73 j7 J~X1~VP~4 77

7~74 - ~ r 2 n

7 ~ 7 4 t 4 c1r1r774 7~ 77777777 Sr vr~d - ~ ~

7)

we ~~77 4 - -~ 3$ 7

1

244Th 4 4 ~ ttL-144

~4 c~JJ~ t U -

~fl~KHYBRID COMPUTER $R~1~ m

271

-7 417 77777 77 s 1

44 44 ~ - 27A-~~ ~ 7

NJ 7 ~shy

(177lt N744t ~

~

7r 77 -C7 2)~Lf

4 771) shy ~

Lamp~~5t ~2fl6

-t~4 wr~t4~ 7777 7st~Ct44y7 ~ 7 7 t7 f4 7 7 71

--~-17747~~~t ~

~77

7 71 ~

~ ~- h~4tt7 4 ~3~524~

-

1 -7

- 7

--4

0

777777-5rfT77rY2clr~27fl~1~LY1~r7

7 I 3NL1 ~ Cl

47 (777tgt 7t77t~7J777t4v~7ttc - s7t$~-7w2A3t~~4 - -

77 - 1(~7~V7 7P~~2fl~ ~tiSi 7lt 7777 ~-4 77W7~

~

74

273 7

14~ 72if rb

7~

~ sr~fl77~

7 A7f7L7~7~7$

7 777

~ ~ kampi 7

~

74~Agt77N~7747Y7777

r20F 7 4A~7 ~ 0~r- 77

7 s77t7 4c~t 7 Il rCl44 j$r~x~77 777 ~K 17~7 ~

I 7 771 77723 ~

lt

7 7~7 ~f

~77 7 7 V ~ 2 7

7k~ 7J7~ 7 7

7 -~~

77 tj~ ampt7 44t lY7N77t ~

7 7

7727 ~

16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 26: FOR AID USE ONLY

2

growing recognition that groundwater resources have many inherent advantages

particularly for storage purposes However the efficient utilization of

the groundwater resources of an area usually requires that both surface

and groundwater supplies be considered as one integrated system

Objecti ve

The general objective of the present study is to investigate the

fluctuations of the groundwater levels in the study area (see Figure 1)

under various conditions of land use Substitution of the native phreatoshy

phyte vegetation by agricultural crops reduces extraction from groundwater

supplies Groundwater levels are also influenced by irrigation of agriculshy

tural crops The computer simulation study discussed herein was therefore

proposed to provide estimates of attenuation rates and equilibrium levels

of the groundwater under various management alternatives such as areal

variations of native vegetation and crop patterns and varying irrigation

application rates

Study Area

The project required the simulation of the groundwater levels in

a region near the coast of north western Colombia South America The

boundary and groundwater conditions for the 300 square kilometer area

(approximate) are shown by Figure 1 For purposes of spatial definition

a rectangular grid wassuperimposed on the area as shown by Figure 1

The land ismainlylow-lying with little variation in elevation and there

are no major surface streams Vegetative cover is currently largely native

but the area has been designated for extensive agricultural development

The groundwater basin beneath this area is recharged by inflows from

the river canal reservoir and mountins to the north and by deep percolation

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 27: FOR AID USE ONLY

3

R Magdalena

Vari able boundary values at all boundary nodes

y

Variable input to ground water at all internal nodes

A A

AyA

-1 -- 0AX Ax =Ay =2000meters Mountai ns A

Guajaro Reservoir

- 0 1 2 3 4 5 6

1000 m ----- z Section A-A

Water table level

Figure 1 Plan and section of the study area

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

i~ 7 I

T -II -r-

-shy

44~~~

use n 1rtptoi~tw~ist 4 4 P

WY94

W

LL

VAshy

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

~4t

ii-gt r 777~ ~

77 777

~ 715 7 gtCN~JY44~7

3~I- t~ 77 -4777777

z)7~77~t77777 777777 ) 1A ~~4~ti77 c4 2-~ I 7

-~ ~ NI-shy

c ~XT~LY 7 4~3C~7r2i~d

1 7 7~ I744~lt7

7 4

~r7S -

~72~ r~ir~nr 7 ~ t77

-

~ tj N ~ - shy1

mZ274~7 N

24rv-vamp $ ~1amp7t- 7 V 7~~~t~Ztk7shy7 77 - 7 77A1

77 S- --4r~ amp~7~C~

shy

2~ ~vA t 7

W4rlt2~PK 2 ~ -~k4t~Ntxflt

- 2 -

~C 1

~ 777 7741a47

7 x- ~W AI47

77 ~777T 7-1-7-- i2777744 7777A 73 j7 J~X1~VP~4 77

7~74 - ~ r 2 n

7 ~ 7 4 t 4 c1r1r774 7~ 77777777 Sr vr~d - ~ ~

7)

we ~~77 4 - -~ 3$ 7

1

244Th 4 4 ~ ttL-144

~4 c~JJ~ t U -

~fl~KHYBRID COMPUTER $R~1~ m

271

-7 417 77777 77 s 1

44 44 ~ - 27A-~~ ~ 7

NJ 7 ~shy

(177lt N744t ~

~

7r 77 -C7 2)~Lf

4 771) shy ~

Lamp~~5t ~2fl6

-t~4 wr~t4~ 7777 7st~Ct44y7 ~ 7 7 t7 f4 7 7 71

--~-17747~~~t ~

~77

7 71 ~

~ ~- h~4tt7 4 ~3~524~

-

1 -7

- 7

--4

0

777777-5rfT77rY2clr~27fl~1~LY1~r7

7 I 3NL1 ~ Cl

47 (777tgt 7t77t~7J777t4v~7ttc - s7t$~-7w2A3t~~4 - -

77 - 1(~7~V7 7P~~2fl~ ~tiSi 7lt 7777 ~-4 77W7~

~

74

273 7

14~ 72if rb

7~

~ sr~fl77~

7 A7f7L7~7~7$

7 777

~ ~ kampi 7

~

74~Agt77N~7747Y7777

r20F 7 4A~7 ~ 0~r- 77

7 s77t7 4c~t 7 Il rCl44 j$r~x~77 777 ~K 17~7 ~

I 7 771 77723 ~

lt

7 7~7 ~f

~77 7 7 V ~ 2 7

7k~ 7J7~ 7 7

7 -~~

77 tj~ ampt7 44t lY7N77t ~

7 7

7727 ~

16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 28: FOR AID USE ONLY

4

from the land surface during the wet season when precipitation rates exceed

evapotranspiration The depth to groundwater as shown on Section A-A

(plotted from observations during January 1969) varies between one meter

at the edge to 10 meters at the center Superimposed on this general

groundwater pattern are a number of localized areas of high and low water

levels which indicate localized recharge from swamps or evapotranspiration

by native phreatophytes Extractions from the groundwater basin occur as

transpiration by deep rooted phreatophytic vegetation These losses maintain

groundwater levels at approximately 10 meters beneath the land surface at

the center of the area Thus unless a drainage system is provided the

substitution of large areas of native vegetation by relatively shallowshy

rooted agricultural crops likely will eventually produce undesirably high

water table levels The problem is further compounded because irrigation

of agricultural crops is necessary in this region and the unused irrigation

waters deep percolating to the saturated zone will accelerate the rise of

water table levels

Theoreti cal Considerations

Surface Water System For the particular area under consideration

no surface outflow from the area occurs Therefore all of the water input

to the area either is lost by evaporation or enters the unsaturated groundshy

water regime through infiltration A portion of the water in the unsaturated

zone is abstracted by the process of evapotranspiration The remainder moves

downward by deep percolation to the saturated groundwater regime

There are numerous methods available to estimate the rate of evaposhy

transpiration These methods have found application to particular problems

but are not generally applicable for all purposes For the problem under

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 29: FOR AID USE ONLY

5

study the following formula is conslidered apPlicable (Christiansen and

Hargreaves 1969)

Etp = KEv )

in which Etp = estimated potential evapotranspiration

Ev = pan evaporation and

K = an experimentally determined crop coefficient which is dependent

upon crop species and stage of growth

The actual evapotranspiration isusually less than the potential

evapotranspiration when soil moisture is limited Many approaches have been

proposed by different investigators to relate the actual evapotranspiration

and the potential evapotranspiration For the problem under study the linear

relationship introduced by Thornthwaite and Mather (1955) isassumed applicable

The actual evapotranspiration thus can be estimated as follows

Et = Etp when Ms gt Mes (2)

E = Et- M s when M lt M (3)t es s es

Evapotranspiration losses maybe derived from either above or below

a water table (or both) depending upon the type of vegetation soil moisture

content and depth to the groundwatertable For the present study the

assumpti on was made that the cul ti vated crops draw water from only the

unsaturated soil and that the deep-rooted native plants are phreatophytic

innature and derive water from both above and below the groundwater table

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

i~ 7 I

T -II -r-

-shy

44~~~

use n 1rtptoi~tw~ist 4 4 P

WY94

W

LL

VAshy

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 30: FOR AID USE ONLY

6

Groundwater system The following discussion briefly describes the

development of the mathematical equations used in this study to express the

movement of water within the saturated zone A section through the aquifer

in the study area is shown byFigure 2

North boundary of study area South boundary of study area

Mountains

Canal del Dique

water table -

hi Datum for Eq 9 hi

I Saturated Zoneh

________Pervious

igr 8 e--Impervious

Figure 2 Section through the aquifer in the study area

Consider a three dimensional element of the aquifer as shown by

Figure 3 The various symbols indicated in Figures 2 and 3 are defirled

+ Ias follows

h i(q+dq) Y oh

X h (q + dq)

Figure 3 An elemental volume from the aquifer in the studyarea

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

i~ 7 I

T -II -r-

-shy

44~~~

use n 1rtptoi~tw~ist 4 4 P

WY94

W

LL

VAshy

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

~4t

ii-gt r 777~ ~

77 777

~ 715 7 gtCN~JY44~7

3~I- t~ 77 -4777777

z)7~77~t77777 777777 ) 1A ~~4~ti77 c4 2-~ I 7

-~ ~ NI-shy

c ~XT~LY 7 4~3C~7r2i~d

1 7 7~ I744~lt7

7 4

~r7S -

~72~ r~ir~nr 7 ~ t77

-

~ tj N ~ - shy1

mZ274~7 N

24rv-vamp $ ~1amp7t- 7 V 7~~~t~Ztk7shy7 77 - 7 77A1

77 S- --4r~ amp~7~C~

shy

2~ ~vA t 7

W4rlt2~PK 2 ~ -~k4t~Ntxflt

- 2 -

~C 1

~ 777 7741a47

7 x- ~W AI47

77 ~777T 7-1-7-- i2777744 7777A 73 j7 J~X1~VP~4 77

7~74 - ~ r 2 n

7 ~ 7 4 t 4 c1r1r774 7~ 77777777 Sr vr~d - ~ ~

7)

we ~~77 4 - -~ 3$ 7

1

244Th 4 4 ~ ttL-144

~4 c~JJ~ t U -

~fl~KHYBRID COMPUTER $R~1~ m

271

-7 417 77777 77 s 1

44 44 ~ - 27A-~~ ~ 7

NJ 7 ~shy

(177lt N744t ~

~

7r 77 -C7 2)~Lf

4 771) shy ~

Lamp~~5t ~2fl6

-t~4 wr~t4~ 7777 7st~Ct44y7 ~ 7 7 t7 f4 7 7 71

--~-17747~~~t ~

~77

7 71 ~

~ ~- h~4tt7 4 ~3~524~

-

1 -7

- 7

--4

0

777777-5rfT77rY2clr~27fl~1~LY1~r7

7 I 3NL1 ~ Cl

47 (777tgt 7t77t~7J777t4v~7ttc - s7t$~-7w2A3t~~4 - -

77 - 1(~7~V7 7P~~2fl~ ~tiSi 7lt 7777 ~-4 77W7~

~

74

273 7

14~ 72if rb

7~

~ sr~fl77~

7 A7f7L7~7~7$

7 777

~ ~ kampi 7

~

74~Agt77N~7747Y7777

r20F 7 4A~7 ~ 0~r- 77

7 s77t7 4c~t 7 Il rCl44 j$r~x~77 777 ~K 17~7 ~

I 7 771 77723 ~

lt

7 7~7 ~f

~77 7 7 V ~ 2 7

7k~ 7J7~ 7 7

7 -~~

77 tj~ ampt7 44t lY7N77t ~

7 7

7727 ~

16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 31: FOR AID USE ONLY

7

qx =the flow in the x direction

qy =the flow in the y direction

h = the head of water at any point in the aquiferabove the

impermeable layer

hb the boundary value of h

- I = the input to (+) oroutput (-) from the surface water

The following assumptions are made inthe derivation of the groundwater

flow equation

1 Isotropic unconfined aquifer

2Homogeneous porous media

3 Flow lines horizontal

4 Uniform velocity over depth of flow proportional to the slope of

the groundwater surface (Darcys Law)

5 Compressibility effects neglected

6 Effective porosltye = storage coefficientS

From the principle of continuity for an incremental time period 6t

qx6t + qy6t plusmn I6x6y6t = (q + 6q)x6t + (q + 6q)y6t + e6h6x6y

aqx + + I = e h (4)axay axay

From the Darcy equation

ah a X - (h) (5 q k(hay) -h and - I axk (5) w oe 2aitX 2

where k is t -ecoefficient of~permeability

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 32: FOR AID USE ONLY

B

Similarly

(6)- a2(h2) 6ly aq~~= - k

axay 2 ay2 _

Substituting Equations (5) and (6)in Equation (4)yields

32(h2) + a2(h2) 21 - 2e Dh = S (7) k ka t T at3X2 ay2

where T = kh is the transmissivity of the aquifer

Expanding Equation (7) gives

ph 2a h12 plusmn21 2e ah

2ha~ ~ 2 +2 +2 _ k = k at (8)ay2 Bay

ax2

Neglectinh)2 and fahi2 x 2 2y =h)Neglecting ax| and Y1 and substituting - x

2h aa2h ah = h - - and - in Equation (8) gives2 2 at atay ay

a2h a2 h I e ah S )h (k9-)2 Tt ay Tax2

where h is the height~of the water table above a particular datum situated

a distance h0 above the impermeable layer

Equation (7)is the complete equation in that no terms are neglected

in its derivation and Equation (9)is its linearized version Errors due

to neglecting the terms j and -h only become appreciable for large

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

i~ 7 I

T -II -r-

-shy

44~~~

use n 1rtptoi~tw~ist 4 4 P

WY94

W

LL

VAshy

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 33: FOR AID USE ONLY

9

water surface slopes which are not typical of the groundwater levels in

the study area Measuring water table fluctuations from a fixed height

ho above the impermeable layer improves computing accuracy in that the

full dynamic range of the analog componentin the computer is utilized

Hybrid computer Implementation of Model

A schematic flow diagram of the surface water-groundwater system is shown

by Figure 4 and each component of this system will be briefly discussed

The spatial unit adopted for the model was 000 meters as shown by Figure 1

A one month time increment was used All data input to the model were

averaged values on the basis of the space and time scales adopted Data

are input to the model through the digital component of the hybrid computer

The input data are precipitation temperatureUnsaturated Regime

pan evaporation crop densities crop coefficients soil moisture holding

capacity initial soil moisture content and irrigation rates Digital

computations are made to determine the amount of water applied to the soil

surface the extraction from groundwater storage and the initial soil

analogmoisture content and this information is then transferred to the

component The processes of evapotranspiration and percolation are simulated

by the analog component and transferred back to the digital device as shown

in Figure 5 Typical computer output for the model of the unsaturated regime

is shown by Table 1

Saturated Regime The computation method used to model the groundshy

water system is an iterative adaptation of the usual all-analog method

commonly employed insolving the diffusion equation This technique allows

sharing of the analog equipment required for each spatial division andthe

thus essentially replaces the need for large quantities of analog computing

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

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use n 1rtptoi~tw~ist 4 4 P

WY94

W

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VAshy

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

~4t

ii-gt r 777~ ~

77 777

~ 715 7 gtCN~JY44~7

3~I- t~ 77 -4777777

z)7~77~t77777 777777 ) 1A ~~4~ti77 c4 2-~ I 7

-~ ~ NI-shy

c ~XT~LY 7 4~3C~7r2i~d

1 7 7~ I744~lt7

7 4

~r7S -

~72~ r~ir~nr 7 ~ t77

-

~ tj N ~ - shy1

mZ274~7 N

24rv-vamp $ ~1amp7t- 7 V 7~~~t~Ztk7shy7 77 - 7 77A1

77 S- --4r~ amp~7~C~

shy

2~ ~vA t 7

W4rlt2~PK 2 ~ -~k4t~Ntxflt

- 2 -

~C 1

~ 777 7741a47

7 x- ~W AI47

77 ~777T 7-1-7-- i2777744 7777A 73 j7 J~X1~VP~4 77

7~74 - ~ r 2 n

7 ~ 7 4 t 4 c1r1r774 7~ 77777777 Sr vr~d - ~ ~

7)

we ~~77 4 - -~ 3$ 7

1

244Th 4 4 ~ ttL-144

~4 c~JJ~ t U -

~fl~KHYBRID COMPUTER $R~1~ m

271

-7 417 77777 77 s 1

44 44 ~ - 27A-~~ ~ 7

NJ 7 ~shy

(177lt N744t ~

~

7r 77 -C7 2)~Lf

4 771) shy ~

Lamp~~5t ~2fl6

-t~4 wr~t4~ 7777 7st~Ct44y7 ~ 7 7 t7 f4 7 7 71

--~-17747~~~t ~

~77

7 71 ~

~ ~- h~4tt7 4 ~3~524~

-

1 -7

- 7

--4

0

777777-5rfT77rY2clr~27fl~1~LY1~r7

7 I 3NL1 ~ Cl

47 (777tgt 7t77t~7J777t4v~7ttc - s7t$~-7w2A3t~~4 - -

77 - 1(~7~V7 7P~~2fl~ ~tiSi 7lt 7777 ~-4 77W7~

~

74

273 7

14~ 72if rb

7~

~ sr~fl77~

7 A7f7L7~7~7$

7 777

~ ~ kampi 7

~

74~Agt77N~7747Y7777

r20F 7 4A~7 ~ 0~r- 77

7 s77t7 4c~t 7 Il rCl44 j$r~x~77 777 ~K 17~7 ~

I 7 771 77723 ~

lt

7 7~7 ~f

~77 7 7 V ~ 2 7

7k~ 7J7~ 7 7

7 -~~

77 tj~ ampt7 44t lY7N77t ~

7 7

7727 ~

16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 34: FOR AID USE ONLY

10

pr

gs Pr yes

Qirr - It+Qs lt I I

no tss S rI =+ Q +Q FE

r irr stPga

I MsE 1

y e siDP 0 lt

SQIg gt1 -9 t 2

Figure 4 Schematic diagram of the surface water-groundwater system for Atlantico 3 Project

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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271

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 35: FOR AID USE ONLY

Extraction from GW storage by native plants

0A AiD deep percolatio

S 2

IR

DA

Surface Input

( Ms

A+

DA

----

AID0ID

0

Initial Soil moisture

SS)

- e _

Soil Moisture

Et of the cultivated Et of the R1

crops culfivated crop

AD Analog to Digital

DA Digital to Analog

Fig 5 Analog circuit for surface water system

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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271

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 36: FOR AID USE ONLY

T1I L

o I 4_ -

i0PT 30 FO 1

1 28 11i- -

204 shy

0 J61 i

1 263 167 10 6 O _~

2 019 176 20 8l O I)-S j 77 4 91 199 20 9 6 153 155 10 75 Goshy

13 173 20 0 -734 9 125 185 20 80 7n

S 10 144 169 20 75 0c 1183 Ii 2 0 0

PT 31 FNES- 240 FIC 120 CO-P

RIES Available soi l moistre SU

i FIC - Initial soil 1stIAW c L

OP Densty of-rati Ovetst L

PPT Nonthly i-0 i 4mi

EYP MnthlypoR m

cm Coeffic4n4mis fo1 COP oVfit tI

Ar ftn~it A -

444Tfllri

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

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WY94

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 37: FOR AID USE ONLY

15

hi1jn KLDJjl

NY Ax

Figure 7 Diagram showing location of terms in Equation(12) on grid network

Integrating Equation (12) gives

7+jn h-ln hij+lnT r 4 +h +h hijn plusmn hn( 2 jx) j

(13) The magnitude and time scaled version of equaton (13) can 2be implementwd

on the analog computer as shown in Figure 8 Note that only one ntegrator

is required With the aid of the digital computer this integrator can be

moved along each node in turn with the appropriate values of h_

etc being provided from digital storage

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

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WY94

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 38: FOR AID USE ONLY

16

(i amp etc T S(Ax)2 -

- Initial Groundwater Level Values (t=O)

h

DAM IO

ADCl

Im T 4()m T (ampX)

Tm() Inputs from Surface DAM Digital to Analog Multiplier Water System ADC Analog to Digital ConverterDAM 2

Q Potentiometer

Figure 8 Scaled analog circuit for the solution of Equation (13) on the hybrid computer

Integration at each node is carried out for a specific time period

of for example one year and the values of h corresponding to each

time increment (one month) within the specified time period are stored by

the digital computer (see Figure 9) The error e between successive h

versus t curves at each node is tested by the digital computer and a solution

is obtained when Ee2 becomes less than a specified tolerance

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

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WY94

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 39: FOR AID USE ONLY

17

h e

1st run

2nd run 7 t

Boundary Nodes

-

Internal

Nodes

Figure 9 Diagram showing integration procedure

Model Verification

Lack of adequate data on rainfall evapotranspiration rooting depths

areal distribution and type of vegetation and aquifer properties meant

The model willthat some gross assumptions had to be made at this stage

Groundwater contourbe continually refined as furtherdata become available

maps prepared from levels taken from about 500 boreholes over a period of

two yearswere available for the area

The effects of the aquifer permeability Kand storage coefficient

Swere studied by varying one of these parameters at a time for an idealized

aquifer with constant boundary conditions (water table level at 100 meters)

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

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WY94

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

~4t

ii-gt r 777~ ~

77 777

~ 715 7 gtCN~JY44~7

3~I- t~ 77 -4777777

z)7~77~t77777 777777 ) 1A ~~4~ti77 c4 2-~ I 7

-~ ~ NI-shy

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1 7 7~ I744~lt7

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mZ274~7 N

24rv-vamp $ ~1amp7t- 7 V 7~~~t~Ztk7shy7 77 - 7 77A1

77 S- --4r~ amp~7~C~

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2~ ~vA t 7

W4rlt2~PK 2 ~ -~k4t~Ntxflt

- 2 -

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~ 777 7741a47

7 x- ~W AI47

77 ~777T 7-1-7-- i2777744 7777A 73 j7 J~X1~VP~4 77

7~74 - ~ r 2 n

7 ~ 7 4 t 4 c1r1r774 7~ 77777777 Sr vr~d - ~ ~

7)

we ~~77 4 - -~ 3$ 7

1

244Th 4 4 ~ ttL-144

~4 c~JJ~ t U -

~fl~KHYBRID COMPUTER $R~1~ m

271

-7 417 77777 77 s 1

44 44 ~ - 27A-~~ ~ 7

NJ 7 ~shy

(177lt N744t ~

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777777-5rfT77rY2clr~27fl~1~LY1~r7

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47 (777tgt 7t77t~7J777t4v~7ttc - s7t$~-7w2A3t~~4 - -

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7 s77t7 4c~t 7 Il rCl44 j$r~x~77 777 ~K 17~7 ~

I 7 771 77723 ~

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7 7~7 ~f

~77 7 7 V ~ 2 7

7k~ 7J7~ 7 7

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7 7

7727 ~

16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 40: FOR AID USE ONLY

18

and constant initial conditions of-the same value The aquifer levels (see

Figures 10 and 11) were plotted for a uniform net withdrawal from the groundshy

water basin Iof 01 meters per month at each node Figures 10 and 11

indicate that the parameter K determines the shape of the groundwater profile

while S determines the level of the water in the aquifer (for a given I)and

has a rather minor inFluence on shape

1000

I = -01 mmonthnode I = - 01 mmonthnode S = 01 K = 100 mmonth K(mmonth) S

1000 g50 500 020=

-

t 40000 120 016

60 100 -0 014

20 012 01 900

4J

008 850 __ ____

0 1 2 3 0 1 2

Grid Point No Grid Point No

Figure 10 Diagram showing effect Figure 11 Diagram showing effect of varying K on water levels of varying S on water levels inidealized aquifer after 1 in idealized aquifer after 1 year year

1000

950

900

850 3

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

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use n 1rtptoi~tw~ist 4 4 P

WY94

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

~4t

ii-gt r 777~ ~

77 777

~ 715 7 gtCN~JY44~7

3~I- t~ 77 -4777777

z)7~77~t77777 777777 ) 1A ~~4~ti77 c4 2-~ I 7

-~ ~ NI-shy

c ~XT~LY 7 4~3C~7r2i~d

1 7 7~ I744~lt7

7 4

~r7S -

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-

~ tj N ~ - shy1

mZ274~7 N

24rv-vamp $ ~1amp7t- 7 V 7~~~t~Ztk7shy7 77 - 7 77A1

77 S- --4r~ amp~7~C~

shy

2~ ~vA t 7

W4rlt2~PK 2 ~ -~k4t~Ntxflt

- 2 -

~C 1

~ 777 7741a47

7 x- ~W AI47

77 ~777T 7-1-7-- i2777744 7777A 73 j7 J~X1~VP~4 77

7~74 - ~ r 2 n

7 ~ 7 4 t 4 c1r1r774 7~ 77777777 Sr vr~d - ~ ~

7)

we ~~77 4 - -~ 3$ 7

1

244Th 4 4 ~ ttL-144

~4 c~JJ~ t U -

~fl~KHYBRID COMPUTER $R~1~ m

271

-7 417 77777 77 s 1

44 44 ~ - 27A-~~ ~ 7

NJ 7 ~shy

(177lt N744t ~

~

7r 77 -C7 2)~Lf

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Lamp~~5t ~2fl6

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--~-17747~~~t ~

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--4

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777777-5rfT77rY2clr~27fl~1~LY1~r7

7 I 3NL1 ~ Cl

47 (777tgt 7t77t~7J777t4v~7ttc - s7t$~-7w2A3t~~4 - -

77 - 1(~7~V7 7P~~2fl~ ~tiSi 7lt 7777 ~-4 77W7~

~

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7 A7f7L7~7~7$

7 777

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~

74~Agt77N~7747Y7777

r20F 7 4A~7 ~ 0~r- 77

7 s77t7 4c~t 7 Il rCl44 j$r~x~77 777 ~K 17~7 ~

I 7 771 77723 ~

lt

7 7~7 ~f

~77 7 7 V ~ 2 7

7k~ 7J7~ 7 7

7 -~~

77 tj~ ampt7 44t lY7N77t ~

7 7

7727 ~

16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 41: FOR AID USE ONLY

19

The water table profile foran aquifer permeability of 200 meters per

month corresponded closely with the observed profile in the existing aquifer

The value of the storage coefficient required to give water levels in close

as theseagreement with those in the aquifer was more difficult to determine

value ofS equal to 01 gave reasonablelevels also depend on I However a

values and subsequent studies using the model were carried out using this

value

The above values for the aquifer parameters K and S were tested by

study of the growth and shape of the groundwater mounds and depressionsa

For example a mound with a base width of approximately 4000 meters grew to

a height of 35 meters above the level of the surrounding aquifer during a

simulation period of one year The simulation of the mound in the idealized

carried out by setting I = + 007 meters per month at the centralaquifer was

zero value for I at all other nodes The results arenode and assuming a

shown graphically by Figure 12 and demonstrate once again that the assumptions

of K = 200 meters per month and S = 01 are reasonable The choice of I in

this case was based on the fact that approximately 80 percent of the available

annual rainfall reached the groundwater table at this point

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 42: FOR AID USE ONLY

20

I = 007 mmonth

~i S =01 K = 100

1050

K-K300

E 1000

01 2 3 Grid Point No = 007 mmonth

gt K 200 mmonth

1050 9-S 4 = 008

4JS=O02

1000 _ --

0 1 2 3

Grid Point No - Observed groundwater levels

Figure 12 Effect of varying K and S for an input to groundwater of + 007 mmonth at central node only

The values of K = 200 meters per month and S = 01 were further

tested by a simulation study of the entire aquifer for the year 1969

Groundwater records were available for this period A comparison between

observed water table levels and those simulated under conditions ofnative

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 43: FOR AID USE ONLY

21

vegetation are shown in Table 2 and Figure 13 Close agreement was achieved

between recorded and simulated water table levels and the model was therefore

considered to be verified at this stage of study

Management Studies

The verified model was used to provide estimates of the attenuation

rates and equilibrium levels of the water table under various cropping and

irrigation practices Table 3 presents an assumed crop pattern weighted

crop coefficients and assumed irrigation rates for the various soil groups

within the study area Agricultural crop distribution within the area was

thus based on the soil group occurring at each grid point shown by Figure 1

Native vegetation density was taken as being that proportion of the total

area occupied by native vegetation For example under a density of native

vegetation equal to 02 one fifth of the total area represented by each grid

Point (four square kilometers) was assumed to be occupied by native vegetation

The remainder of the area represented by a particular grid point was assumed

to be occupied by the distribution of agricultural crops corresponding to

the soil type at that grid point (Table 3) Thus on the basis of soil type

combinations of native vegetation and cultivated crop cover were developed

for the entire area

Computed equilibrium water table elevations inmeters at each grid

point under four conditions of vegetative cover and irrigation are shown by

Table 2 Corresponding water tableprofiles for Sections A-C and B-C (see

the sketch accompanying Table 2) are shownby Figure 13

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 44: FOR AID USE ONLY

Table 2 Groundwater levels for December 1969

ICanaldel Dique

+ + + + + +A + + + + +

B + ~C+ + + + + + + + + + + + + + + + + + + + +

+ + + + + + + + + + +

I Boundary of study area Groundwater levels tabulated for these points

Sketch showing grid point locations within the study area

Observed

976 1014 1015 1017 1005 997 963 1011 962 960 962 995 975 973 989 959 979 957 997 973 970 980 1006 958 961 962 973 946 976 983 956 965 974 1005 995 962 959 956 953 957 971 970 964 972 1005 995 991 968 965 957 968 980 967 970 970

Simulated - Native vegetation DDP = 025 K = 200 mmonth S = 01

1000 998 1001 1003 997 993 989 990 988 984 986 1002 985 981 990 976 971 968 972 970 969 976 1009 984 968 965 961 959 959 963 962 963 969 1014 988 966 959 955 954 956 960 963 967 975 1019 992 971 961 954 956 962 970 975 989 194

Simulated - Partly cultivated and irrigated DDP = 02 K = 200 mmonth S = 01

999 997 999 1000 995 991 988 989 986 982 985 1002 983 977 975 971 967 966 971 968 967 975 1007 983 967 960 957 954 954 960 958 961 967 1013 986 965 957 950 948 951 957 958 963 972 1019 991 968 959 950 952 959 976 972 985 991

Simulated - Partly cultivated and irrigated DDP = 01 K = 200 mmonth S = 01

1006 1005 1003 1003 1004 1001 998 998 995 986 991 1006 992 986 985 983 980 978 976 978 976 979

966 966 968 966 9751015 988 971 970 970 967 1021 994 969 961 962 961 963 967 969 969 981 1021 993 975 962 959 962 968 975 980 993 999

Simulated - Partly cultivated and irrigated DDP = 00 K = 200 mmonth S = 01

1013 1013 1006 1007 1013 1012 1008 1007 1004 990 997 1010 1008 996 996 996 993 989 982 989 985 983 1023 993 975 980 983 980 978 972 978 971 984 1029 1003 972 965 973 974 975 978 980 974 990 1022 996 981 966 968 978 978 985 990 1002 1007

= DDP = native vegetation density For uncultivated areas DDP 025

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

T 1TT tult~Te1nt J

S~ y Z

1

i~ 7 I

T -II -r-

-shy

44~~~

use n 1rtptoi~tw~ist 4 4 P

WY94

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 45: FOR AID USE ONLY

Table 3 Crop-pattern crop-coefficients and irrigation for different soils

Soil Crop-pattern weighted crop-coefficient and irrigation rate Group Item Crop Jan Feb Mar Apr May Jun IJul Aug Sept Oct- Nov Dec

123 Crop pattern Citrus Peanuts

Maize

Crop coeff 65 75 55 60 45 60 75 60 60 60 60 50 Irr rate2 100 100 100 50 50 50 50 50 50 50 50 100

4 Crop pattern Cotton Sorghum

Crop coeff 70 50 20 20 30 60 90 60 40 65 90 90 Irr rate 2 100 100 0 0 50 50 50 50 50 50 50 100

56 Crop pattern Grasses - - -

Crop coeff80 80 i 80 80 80 80 80 80 80 80 80 8C Irr rate2 100 100 100 50 50 50 50 -50 50 50 50 100

78 Crop coeff Bare Soil 10 10 10 10 10 10 10 10 l0 10 10 10 Irr rate2 0 -0 0 0 0 0 0 0 0 0 0 0

1See Appendix 1

In mmonth

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

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use n 1rtptoi~tw~ist 4 4 P

WY94

W

LL

VAshy

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

~4t

ii-gt r 777~ ~

77 777

~ 715 7 gtCN~JY44~7

3~I- t~ 77 -4777777

z)7~77~t77777 777777 ) 1A ~~4~ti77 c4 2-~ I 7

-~ ~ NI-shy

c ~XT~LY 7 4~3C~7r2i~d

1 7 7~ I744~lt7

7 4

~r7S -

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-

~ tj N ~ - shy1

mZ274~7 N

24rv-vamp $ ~1amp7t- 7 V 7~~~t~Ztk7shy7 77 - 7 77A1

77 S- --4r~ amp~7~C~

shy

2~ ~vA t 7

W4rlt2~PK 2 ~ -~k4t~Ntxflt

- 2 -

~C 1

~ 777 7741a47

7 x- ~W AI47

77 ~777T 7-1-7-- i2777744 7777A 73 j7 J~X1~VP~4 77

7~74 - ~ r 2 n

7 ~ 7 4 t 4 c1r1r774 7~ 77777777 Sr vr~d - ~ ~

7)

we ~~77 4 - -~ 3$ 7

1

244Th 4 4 ~ ttL-144

~4 c~JJ~ t U -

~fl~KHYBRID COMPUTER $R~1~ m

271

-7 417 77777 77 s 1

44 44 ~ - 27A-~~ ~ 7

NJ 7 ~shy

(177lt N744t ~

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--~-17747~~~t ~

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777777-5rfT77rY2clr~27fl~1~LY1~r7

7 I 3NL1 ~ Cl

47 (777tgt 7t77t~7J777t4v~7ttc - s7t$~-7w2A3t~~4 - -

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7 A7f7L7~7~7$

7 777

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~

74~Agt77N~7747Y7777

r20F 7 4A~7 ~ 0~r- 77

7 s77t7 4c~t 7 Il rCl44 j$r~x~77 777 ~K 17~7 ~

I 7 771 77723 ~

lt

7 7~7 ~f

~77 7 7 V ~ 2 7

7k~ 7J7~ 7 7

7 -~~

77 tj~ ampt7 44t lY7N77t ~

7 7

7727 ~

16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 46: FOR AID USE ONLY

C

24

1050

1000 Simulated (DDP 00)

Simulated (DDP = 01)

Simulated (native vegetation 950 S DDP = 025)

V= 00 11 22 33 Simulated (DOP = 02) Grid Point No

Section A-C

1050 Simulated (DDP 00)

Simulated (DDP =01)

d 1000 Simulated (native vegetation)

Simulated (DDP = 02)

950 -- -

Secti on B-C

Observed water table levels

Fig 13 Observed and simulated water tablelevels for December 1969

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

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use n 1rtptoi~tw~ist 4 4 P

WY94

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VAshy

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

~4t

ii-gt r 777~ ~

77 777

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3~I- t~ 77 -4777777

z)7~77~t77777 777777 ) 1A ~~4~ti77 c4 2-~ I 7

-~ ~ NI-shy

c ~XT~LY 7 4~3C~7r2i~d

1 7 7~ I744~lt7

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77 S- --4r~ amp~7~C~

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W4rlt2~PK 2 ~ -~k4t~Ntxflt

- 2 -

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7 x- ~W AI47

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we ~~77 4 - -~ 3$ 7

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244Th 4 4 ~ ttL-144

~4 c~JJ~ t U -

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271

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7 I 3NL1 ~ Cl

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7 7

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 47: FOR AID USE ONLY

25

Discussions and Conclusions

The work reported herein has demonstrated the utility of the hybria

computer for detailed simulation of highly complex and dynamic water resource

systems The hybrid which combines the ddvantage of both the analog and

digital computers is particularly applicable to problems involving differshy

ential equations and where interpretation of results and problem insight

are facilitated by the man in the loop configuration and graphical display

of output Inaddition for the type of iterative routines that are characshy

teristic of simulation problems the hybrid computer shows considerable economies

over the all digital approach (Chubb 1970)

Inthis study sensitivity enalyses with the simulation model provided

considerable insight into the unctioning of the prototype system In addition

the model yielded useful estimates of the effects of various management

alternatives on water table levels within the study area

Further work is now in progress to develop a refined model of the

unsaturated portion of the aquifer to include variable permeability at each

node and to generalize the digital program so that a prototype boundary of

any shape may be specified Eventually the model will be expanded to include

the economic dimensions so that optimal solutions may be found in terms

of particular economic objective functions Even at the present exploratory

stage the model has proved useful in determining the type and accuracy of

data required to define the system and in establishing guide lines for

future development

- ~ ~ ~ lJ ~ ~T ~ ~ ~ V 4

74

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S~ y Z

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WY94

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 48: FOR AID USE ONLY

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A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

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7 7~7 ~f

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7727 ~

16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

Page 49: FOR AID USE ONLY

A PROGRESS REPORT ON WORK ACCOMPLISHED IN COMPUTER SIMULATION UNDER

PROJECT WG-69 DURING THE PERIOD MARCH 1 TO DECEMBER 31 1970

J P Riley

INTRODUCTION

During the initial phaseof the computer simulation study of the

Atlantico 3 area of Colombia a model was developed to simulate groundshy

water levels as functions of precipitation crop-pattern density of the

native phreatophyte and irrigation This work was performed during the

period January 1 to April 30 1970 and is described in the attached papshy

er by Morris et al (1970) Because of time and data limitationsthe

following simplifying assumptions were incorporated in the initial model

of Morris et al

(1) The area was approximated by a rectangular grid system with

regular boundaries

(2) A grid spacing of two km was assumed This assumption was

necessary partly because of thd limitation of memory space

in the computer

(3) The influences of topographic variations upon groundwater

levels due to swamps and waterways were neglected

Even though the initial model was very grosssensitivity studies

provided considerable insight into the operation of the prototype sysshy

tem and indicated that system definition could be considerably improved

by obtaining additional field data As a result of thi initial study

it was recommended that the following data be obtained on a monthly

basis tor a period of three toj four years

1 The distribution and density of native plants

2 Agricultural cropping patterns including spatial and time

distribution

3 Plant root distribution patterns (both native and agricuiltural)

4 Irrigation system layout and monthly diversions for each irrigashy

tion canal

5 Major drainages and the amount of drainage for each month (list

individually for each drainage canal)

6 Monthly precipitation pan evaporation and monthly mean temperashy

ture for all of the stations inside and nearby the study area

7 Depths of the aquifer

8- Soil moisture holding characteristics

9 Mean monthly water levels for RMagdalena and Canal del Dique

10 Aquifer permeabilities (saturated) at various locations and depths

Ifavailable the following data are required for a detailed study of the

hydrology and hydraulic processes of the area

1 Daily data for items (4) (5) and (6) above

2 Hydraulic conductivity as a function of soil moisture

3 Capillary potential as a function of soil moisture

Items (2)and (3)above will need to be determined experimentally

It was decided that concurrent with the data collection program

efforts would be continued to improve the computer simulation model

These efforts would emphasize the following areas of study

1 Capability for simulating a boundary of any irregular shape

2 Capability for considering variable boundary conditions and

variable inputs at each grid point

3 An increased grid density of perhaps 12 km

4 An increased resolution with respect to surface hydrology and

In this respect itwas consideredunsaturated groundwater flow

that the model should be capable of reflecting topographic influshy

ences upon qroundwater levels

5 Capability for considering different soil permeability coefshy

ficients at each grid point

6 Addition of the salinity dimension to the model in accordance

with previous work at Utah State University

7 Improvement of the model using hydrologic data which has become

available sine the completion of the initial study

8 Perform continuing sensitivity studies to establish priorities

and resolution needs for data collection programs

The following is a brief description of progress that is being made

It is emphasized thatin accordance with theabove listed eight points

although this study is being directed specifically to the Atlantico 3

area the model is entirely general and its application isnot inany

way limited to a particular geographic area

Surface Model

The previous model was based on the assumption that all of the water

entering the area by precipitation and surface runoff either is lost by

evapotranspiration or infiltrates the soil The effects of chanqes in surshy

face storage quantities (swamp) on the local variations of the groundwater

table were thus neglected To overcome this deficiency a topoqraphic pashy

rameter which indicates thedrainage or collection of surface water was

introduced in therevised model Inaddition a rectangular qrid spacing

of 0625 km was adopted rather than the 20 km spacing used in thfe initial

model The simulated deeo percolation or withdrawal at each grid point

represents the input or output of the groundwater model

A copy of the computer program for the surface model isgiven in

Appendix 1 Sample output of this program is given by Appendix 3

Groundwater Model

As was discussed in the paper by Morris et al groundwater level fluctuations ina homogeneous unconfined aquifer can be described by the

following equation

92h + 2h I = Eah x + + T T at

inwhich

h is the height of groundwater surface above the impervious datum

x and y are the space coordinates

I is the net vertical input per unit area to the groundwater

c is the effective porosity (or specific field)

T is the transmissivity of the aquifer and

t is time

Equation (1) is a linear partial differential equation of the parabolic

type

The numerical solution of parabolic partial differential equations

can be accomplished either by explicit or implicit methods An implicit

difference schemeis usually desirable because of its unconditional stashy

bility and high accuracy However application of the implicit method to

a two-dimensional unsteady flow problem as described by Equation (1)leads

to difference equations which involve five unknowns per equation and the

simplified version of the Gaussion elimination method for the special trishy

diagonal system of a one-dimensional problem is no longer applicable A

method which has the stability advantages of implicit procedures and yet

5

retains a system of equations with a tridiagonal coefficient matrix thus

allowing a straight forward solution is the alternating direction method

Like alldiscussed by Peaceman and Rachford (1955) and Douglas (1961)

difference methods the procedure approximates the partial differential

equations and boundary conditions of the problem by equivalent differences

except that finite difference operators are applied twice for each time

step The difference equation for the first half-time step is implicit

only in one direction and that for the second half-time step is implicit

only in the other direction Indifference form Equation I can be written

as follows n n+l

jl 1 = T [62 hi + 62 hij + U) (na)

In+lh n+l -h +l T (bAt2 T[ x ijh + 6y2 ii shyi3 ij = [62 hi +tl (2b)

inwhich the Ss denote second central difference operators Written out

in full and rearranged with Ax = Ay these equations become

- hi~ + (I TX )hi- - hi~~Tx TX 2e i-lsj i e ~

TA h0 + (IL) hn+ TA + Al o+1 (3a)

2 j-I C ij 2c ij+l 2c i1

TX l l TX -2 hn+lhTx h+] 2e hij-l1 + ( - i 24 lj+l

nlT ij + (1I) h +- + T(3b) w h 3 2 hi+lj 2 Ai3

inwhich 2 = AA)

Incorporating boundary conditions with irregular boundaries as

shown inFigure 1(a) through 2(d) Equation (3a) becomes

FXY

AX sPxA rAx P I IBJ IB+lj_ R ID-lj ID i

-1B 1 IB+Ij p qAx ID-lj ID ---- 4 SAX A pax1 tx -

AX Ijl - - 1~jl [N

(a) (b) (c) (d)

Fiqure 1 Irregular Boundaries

TX T T hh+TX n(C1) hIBj - e(l+p) IB+lj = cp(l+p) P q(l+q) hQ +

(l- ) hnB + T h+ At In l

E(l+q) TBj+l +2 IBJ

for i = IBand boundaries (a)and (b)respectively

Tx + 2T hij _ i rThi-l~ + ( TxT F2TA hh+l J injC

(l-f) h n + TA n +t n+l

+l ) ii cJ+l 2c ij

for IB lt i lt ID

T A TX h T x n T) n OT(l hID 1j + (I+ 7-) =r h+h h r h + Sl hcI h + r h -r(l+r)R IDi

Tx hn At n+1

e(1+s) IDj+l + 26 IDj

for i = IDand boundaries (c)and (d)respectively

Similarly Equation (3b) becomes

7

(1+T) n+1 Tn T x T T = +-) iB iJB+1 S(l+s) hs + cr~iW+r hR + (1-T) hiJB+

CSi sJ c T x~s I AtB~+linSTs

T A h-lJB +A tB C(l+r) 2c 138

for j = JB and boundary (c)

hn+l (1+11-1) T k W1 xn+l + T x(1I JB c--+q) iJB+l = hA +q(l+ h + ( T h +Ep(1+p) hp ( eP hiJB +

T A h h+loB iJB- re+ At n+1

for j JB and boundary (a)TA n~ TX) hn+l TX hn+l

+ i~j1(I ij i~j+1 I his j + (I-1_ hi

jh9+1~l+I hh (4b+ TT

Shi+lj + r ij

for JB lt j lt JDTx hn+1 T D c(+)h I T(I+S) iD-1 (I+) hn+lEMShi~io1 - TS(I+S)A n+1 + Ter(l+r)X hR + T +hiJD = hs Tx(1)hiJD

Tx h +At tn+l (Tr) i-1JD + c iJD

for j = JD and boundary (d)

TA hn+l + (1+ T) hn+l T n+l + TAx + (l+q) iJD-1 + q i3D - q(1+q) h0 cp(+p) p

0 Tx TA + At n+l hJ D C(+P) hi+lJD + T2 iJD

forj = JD and boundary (b)

This scheme requires less memory space and comnuting timethan the

implicit scheme used indue initial study (Morris et al 1970) Thus

for given-levels of core storage and solution time model resolution can

be increased A computer proqram has been written to solveEquation (4a)

and (4b) and this program is containedin Appendix 2 The program is

now being tested and it isexpectedthat output will be obtained in

early February 1971

APPENDIX I

YBRID COMPUTER PROGRAM FOR THE

SUR ACE AND UNSATURATED FLOW REGIMES

SIMULATE NET PERCOLATION 12)LDIMENSION C(4pl2)O(4p2)CDP(12)CG(12)PPT(D312)PEV(33 tBG(32)LND (32) pEDP(12) REAL MICMCSoMESMS

INITIATE CONFIGURATION CALL OSHfIN (IERR5e) CALL QSC(1IERR)

I PAUSE 0001 READ(69g) AICtACSAES

99 FORMAT(3F53) READ AND CHECK BASIC DATA t CRFREAD(6 111) LMNSLINCLNHYiNYRLYRORPPTRTNFMNFMXDAMTA

4 2 )I11 FORMATCI63I52F422FS532F51F

RAuAMTA1 WRITE(6211) RPPTIRTNoFMNoFMXRApCRF

fit FORMAT(7H RPPT upF42p3Xo5HRTN vpF423Xp5HFMN vpF533Xv5HFX NPF

1533XdHRA xF523Xp5HCRF upF42) DO 2 IIm1NSL READ(6pl12) (C(IIKK) rKKvlr 1 2 )

2 WRITE(6212) IIr(C(IIoKK)KKml12) 112 FORMAT(12F52) 112 FORMATU3H C(tIto4HtKK)12FS2)

00 3 IIvIONSLREAD(6p 113) (G(IloKK) PKKglp 1E)

3 WRITEM6e213) IIC(llIKK)OKKxlpl2)

113 FORMAT(12F53) 3 FORMATC3H Q(I14HKK)pl2F63) READ(6114) (CDPCKK)pKKWPI12)

14 FORMAT (12F52) WRITE(6214) (COP(KK)tKKXI12) 14 FORMAT(8H CDPCKK)12F62)

REAO(6e 115) (CGCKK) oKKwGI 12)

115 FORMAT(12F2) WRITE(62153 (CGCKK)KKu 12)

115 FORMAT(8H CG(KK) 012F62) DO 4 JJI1NCLSDO 4 I(mlpNYR

4 READ(6116) (PPT(JJKpKK)iKKU12) 116 FORMAT(12F53)

00 5 JJuINCL

t DO 5 KalNYR S REAO(6116) (PEVCJJrKIK)pKKUI12) IZDENTIFY BOUNDARIES 00 6 JclfM

6 READ(6117) LBG(J)tLND(J) 17 FORMAT (213)

REPEAT CALCULATIONS FOR EACH GRID POINT D0 20 J1tM LBuLBG (J) LDwLND (J) DO 20 IBLBLD WRITE (6216)

MCS MES TPG CARY)I J LS LC LH OOP MIC6 FORMAT(50H READ(6018) LSLCLHDDPMICMCSMESTPGCARY

R FORMAT (313s 4F53rF41F5o3) IF SSW C ON ASSUME NO DATA OCT 023440 J 8 IF(TPGL-1) CARYvo5000 MICvAIC

U MCSvACS MESmAES

8 IF(CARYGT0) MICNMCS WRITE(6218) IJPLSeLCLHDDPMICuMCSMESETPGtCARY

218 FORMAT(5I3p4F63pF5S1F6v3) IF SSW B ON PRINT TITLE OCT 023500 J 7 WRITE (6v219)

219 FORMATC74H YEAR MN PPT PEV 0 TSP ETP ET EVG M 1 DP EDP CARY) SET POT VALUES AND SET UP INITIAL CONDITION

7 CALL QWPR(4HP0I0tMICIERR) CALL OWPR(4HPOIIwMCSIIERR) CALL QWPR(4HP012jMESpIERR) CALL QSIC(IERR)

REPEAT CALCULATIONS FOR EACH MONTH 00 20 KvINYR LYRuLYRO+K-1 DO 19 KKIlp12 PTOoPPT(LCKKK) F(CARYLT1) PT02PTO OCLSKK) IFCPTQGTFMN) PTQOPTOC1-RTNTPG) TSPvPTQ+CARY IF(LHEQI) TSP=TSP+RPPTCRFRAPPT(LCKoKK) IF(TSPLTFMX) GO TO 10 CARYxTSP-FMX TSPuFMX GO TO 1

10 CARYsO 11 ETPaC (1-DDP)C(LSKK) DDP(CDP(KK)-CG(KK)))PEV(LCKKK)

AAxETP(I0MrES)

EVGDDPCG (KK)PEV(LCpKpKK)

TRANSFER DATA FROM DIGITAL TO ANALOG CALL 0WJDAR(TSPfoofIERR) CALL QWJDAR(AAp0IpIERR)

12 CALL ORLBB(ITESToIERR) IF(ITESTEQ1200) GO TO 12

13 CALL ORLBB(TTESTIERN) IF(ITESTNE200) GO TO 13 CALL nSOP(IERR)

14 CALL ORLBB(ITESTIERR) IF(ITESTEO200) GO TO 14 CALL QSH(IERR) TRANSFER DATA BACK TO DIGITAL CALL ORBADR(DP01IERR) CALL QRBADRCETptp1IERR) CALL QRBAVR(MS21IERR) EDP (KV) vDP-EVG ETmET10 IF SSW B ON PRINT DATA OCT 023500 J mip

WRITE(6220) LYRKKpPPT(LCKKK)PEV(LCKKK)Q(LSKK)FTSPETPEYI IEVGvMSDPEDP(KK)vCARY

120 FORMAT(I5I3p1IF63) 1 CONTINUE

IF SSW A ON PUNCH DATA OCT 023600 J 20 WRITE(5o221) (EDP(KK)KKoI12)

221 FORMAT(12FP63 20 CONTINUE

STOP END

~4t

ii-gt r 777~ ~

77 777

~ 715 7 gtCN~JY44~7

3~I- t~ 77 -4777777

z)7~77~t77777 777777 ) 1A ~~4~ti77 c4 2-~ I 7

-~ ~ NI-shy

c ~XT~LY 7 4~3C~7r2i~d

1 7 7~ I744~lt7

7 4

~r7S -

~72~ r~ir~nr 7 ~ t77

-

~ tj N ~ - shy1

mZ274~7 N

24rv-vamp $ ~1amp7t- 7 V 7~~~t~Ztk7shy7 77 - 7 77A1

77 S- --4r~ amp~7~C~

shy

2~ ~vA t 7

W4rlt2~PK 2 ~ -~k4t~Ntxflt

- 2 -

~C 1

~ 777 7741a47

7 x- ~W AI47

77 ~777T 7-1-7-- i2777744 7777A 73 j7 J~X1~VP~4 77

7~74 - ~ r 2 n

7 ~ 7 4 t 4 c1r1r774 7~ 77777777 Sr vr~d - ~ ~

7)

we ~~77 4 - -~ 3$ 7

1

244Th 4 4 ~ ttL-144

~4 c~JJ~ t U -

~fl~KHYBRID COMPUTER $R~1~ m

271

-7 417 77777 77 s 1

44 44 ~ - 27A-~~ ~ 7

NJ 7 ~shy

(177lt N744t ~

~

7r 77 -C7 2)~Lf

4 771) shy ~

Lamp~~5t ~2fl6

-t~4 wr~t4~ 7777 7st~Ct44y7 ~ 7 7 t7 f4 7 7 71

--~-17747~~~t ~

~77

7 71 ~

~ ~- h~4tt7 4 ~3~524~

-

1 -7

- 7

--4

0

777777-5rfT77rY2clr~27fl~1~LY1~r7

7 I 3NL1 ~ Cl

47 (777tgt 7t77t~7J777t4v~7ttc - s7t$~-7w2A3t~~4 - -

77 - 1(~7~V7 7P~~2fl~ ~tiSi 7lt 7777 ~-4 77W7~

~

74

273 7

14~ 72if rb

7~

~ sr~fl77~

7 A7f7L7~7~7$

7 777

~ ~ kampi 7

~

74~Agt77N~7747Y7777

r20F 7 4A~7 ~ 0~r- 77

7 s77t7 4c~t 7 Il rCl44 j$r~x~77 777 ~K 17~7 ~

I 7 771 77723 ~

lt

7 7~7 ~f

~77 7 7 V ~ 2 7

7k~ 7J7~ 7 7

7 -~~

77 tj~ ampt7 44t lY7N77t ~

7 7

7727 ~

16 CONTINUE

SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS

17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19

18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))

19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21

20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR

21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I

2FIT GO TO 2f5

HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I

2FTT GO TO 26

a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT

GO TO 26 24 IF(MOBNE4) GO TO 25

HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2

D(J)uTLAMH$NI(EPS8P(I)(1+SP()))TLAMHPS(EPSRP(I)(l+RP(I

2FIT 25 I4ONlwHJCI2)

HPSu (HI (1J)+H0I (1 J) )2

IF(POSHEQo1) HPSuHoCI-IJ) D(J)uTLAMHON1(EPSQPCI) (14QPCI)))4+TLAMHPSCEPSPP(I) (1 PP(I

1)) )+(-TLAMCEPSPP CI)))HO(IoJ)TLAMCEPS (1PPCI))) HOCI4J) 2FIT

26 CONTINUE CALL TRIDCJBpJDApBCDoH) WRITE(61203) IpKK (H(IfJ)pJm1DMPI)

203 FORMAT(2I58F823C(10X8F82)) DO 27 JnJBtJD

27 HO(XIJ)EH(IPJ)

28 CONTINUE DO 59 JvIMHOI (IFwJ) Hl CIF J)

59 H I(2J)uHI(2J) DO 60 IxIBID HOJ CI j 1)NHJ (I o 1)

60 HOJ (I o2)HJCI p2) 29 CONTINUE 30 CONTINUE

STOP END SOLVING A SYSTEM OF LINEAR SIMULTAN-OUS EQUATIONS HAVING A TRIDIAGONAL COEFFICIENT MATRIX SUBROUTINE TRID(IFvLpApBCDH) DIMENSION ACI) B(1) C(1) D(I) H(I) PBETA(40) GAMMA(40)

BETA (IF) uB (IF) GAMMA (IF) vD (IF)BETA (IF) IFPIxIF I DO 1 IuIFP1L BETA(I)uB(I)-A(I) C(I)BETA(I-1)

1 GAMMA (I)z(DCI)A(I)GAMMA(I-1))BETA(I)H(L)GAMMA (L) LASTxL-IF DO P KnlPLAST IuL-K

2 HCI)GAMMA(I)-CCI)H(I+1)BETACI) RETURN END

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