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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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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
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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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
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
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
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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
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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
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
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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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1
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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
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
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T -II -r-
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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
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1
244Th 4 4 ~ ttL-144
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271
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-
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777777-5rfT77rY2clr~27fl~1~LY1~r7
7 I 3NL1 ~ Cl
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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
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
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T -II -r-
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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
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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~
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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
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7)
we ~~77 4 - -~ 3$ 7
1
244Th 4 4 ~ ttL-144
~4 c~JJ~ t U -
~fl~KHYBRID COMPUTER $R~1~ m
271
-7 417 77777 77 s 1
44 44 ~ - 27A-~~ ~ 7
NJ 7 ~shy
(177lt N744t ~
~
7r 77 -C7 2)~Lf
4 771) shy ~
Lamp~~5t ~2fl6
-t~4 wr~t4~ 7777 7st~Ct44y7 ~ 7 7 t7 f4 7 7 71
--~-17747~~~t ~
~77
7 71 ~
~ ~- h~4tt7 4 ~3~524~
-
1 -7
- 7
--4
0
777777-5rfT77rY2clr~27fl~1~LY1~r7
7 I 3NL1 ~ Cl
47 (777tgt 7t77t~7J777t4v~7ttc - s7t$~-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
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-
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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
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3~I- t~ 77 -4777777
z)7~77~t77777 777777 ) 1A ~~4~ti77 c4 2-~ I 7
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7)
we ~~77 4 - -~ 3$ 7
1
244Th 4 4 ~ ttL-144
~4 c~JJ~ t U -
~fl~KHYBRID COMPUTER $R~1~ m
271
-7 417 77777 77 s 1
44 44 ~ - 27A-~~ ~ 7
NJ 7 ~shy
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~
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7 I 3NL1 ~ Cl
47 (777tgt 7t77t~7J777t4v~7ttc - s7t$~-7w2A3t~~4 - -
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7 777
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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
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
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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
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-
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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 $R~1~ m
271
-7 417 77777 77 s 1
44 44 ~ - 27A-~~ ~ 7
NJ 7 ~shy
(177lt N744t ~
~
7r 77 -C7 2)~Lf
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Lamp~~5t ~2fl6
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--~-17747~~~t ~
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-
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777777-5rfT77rY2clr~27fl~1~LY1~r7
7 I 3NL1 ~ Cl
47 (777tgt 7t77t~7J777t4v~7ttc - s7t$~-7w2A3t~~4 - -
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~
74
273 7
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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
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
~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
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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
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
+ (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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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
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
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
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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-
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44~~~
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
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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
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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
15
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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
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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
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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
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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
17
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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
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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
19
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20
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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
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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
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use n 1rtptoi~tw~ist 4 4 P
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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
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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
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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
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
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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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
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
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
~4t
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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
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
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
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
~4t
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77 777
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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
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
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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
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
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
~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 $R~1~ m
271
-7 417 77777 77 s 1
44 44 ~ - 27A-~~ ~ 7
NJ 7 ~shy
(177lt N744t ~
~
7r 77 -C7 2)~Lf
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Lamp~~5t ~2fl6
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--~-17747~~~t ~
~77
7 71 ~
~ ~- h~4tt7 4 ~3~524~
-
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- 7
--4
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777777-5rfT77rY2clr~27fl~1~LY1~r7
7 I 3NL1 ~ Cl
47 (777tgt 7t77t~7J777t4v~7ttc - s7t$~-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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~
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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
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
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
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
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
~4t
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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
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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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
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
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
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77 777
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7)
we ~~77 4 - -~ 3$ 7
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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
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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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
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1 7 7~ I744~lt7
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-
~ tj N ~ - shy1
mZ274~7 N
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77 S- --4r~ amp~7~C~
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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 $R~1~ m
271
-7 417 77777 77 s 1
44 44 ~ - 27A-~~ ~ 7
NJ 7 ~shy
(177lt N744t ~
~
7r 77 -C7 2)~Lf
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--~-17747~~~t ~
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-
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777777-5rfT77rY2clr~27fl~1~LY1~r7
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7 A7f7L7~7~7$
7 777
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~
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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
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
~4t
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77 777
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we ~~77 4 - -~ 3$ 7
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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
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
~4t
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77 777
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3~I- t~ 77 -4777777
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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
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
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1 7 7~ I744~lt7
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mZ274~7 N
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77 S- --4r~ amp~7~C~
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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
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~fl~KHYBRID COMPUTER $R~1~ m
271
-7 417 77777 77 s 1
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16 CONTINUE
SECONDHALF-TIME STEP IMPLICIT IN Y-DIRECTIONv EXPLICIT IN X-DIRECTION SET COEFFICIENT ARREYS DO 28 IGIPL RDSHxRP (I) POSHvPP (I) MBBEJIBB(I) MDBmJDB(I) JBnJBG (I) jOiJND (I) JBPuJB+1 JDt~uJO-1 00 17 JnJBPpJDM A(J)PTLAM (2EPS) BCJ)ul 4TLAMEPS
17 C(J~uA(J) IF(MBBNEo3) GO TO 18 B(JB)31 4TLAM(EPSSP (I)) CCJB) m-TLAM (EPS(1+SP(l))) GO TO 19
18 BCJB)u14TLAi4(EPSQP(I)) C(JB)niTLAM (EPS (1QP(I)))
19 1FCMDBNE4) GO TO 20 A(JD) -TLAM (EPS(1+SPCI))) B(JO)m TLAM(EP8SP(I)) GO TO 21
20 A(JD)umTLAM(EPS(14DP(I))) B(JO)a1+TLAm (EPSOP (I)) COMPUTE RIGHT-HAND SIDE VECTOR
21 DO 26 JnJBJD DUMYnFI (IJ) FITaOTDUMY (2EPS) IF(JGTJB) GO TO 23 IF(MBBNE3) GO TO 22 ISNIDHJ (11) HRSiz(HI (2J)+HOI (2bvJ))2o IF(RDSHiO1) HRSuHP C141J) D(J)UTLAMHSNI(EPSSP(I)(1+SP(I)))+TLAMHRSEPSRP(IJ1+RP(I
2FIT GO TO 2f5
HPSu (HI (1J)+H01 (1J))2o IFCPDSHEOI) HPSmHcOCIU1J) D(J)PTLAMHON1CEPSQP(I)(1+QP(I)))+TLAMHPS(EPSPPC(I)(l+PP(I
2FTT GO TO 26
a3 IF(JGEJD) GO TO 24 DCJ)TLAMHO(Im1J)(20EPS)+(1mTLAMEPS)HO(IUJ)+ITLAMH0(I1IFJ) 1(2EPS)+FIT
GO TO 26 24 IF(MOBNE4) GO TO 25
HSN~aHJ (I2) HR~u (HI (2J)HoI(2J))2
D(J)uTLAMH$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
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
~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~
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2~ ~vA t 7
W4rlt2~PK 2 ~ -~k4t~Ntxflt
- 2 -
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~ 777 7741a47
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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
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~fl~KHYBRID COMPUTER $R~1~ m
271
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44 44 ~ - 27A-~~ ~ 7
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--~-17747~~~t ~
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777777-5rfT77rY2clr~27fl~1~LY1~r7
7 I 3NL1 ~ Cl
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~
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r20F 7 4A~7 ~ 0~r- 77
7 s77t7 4c~t 7 Il rCl44 j$r~x~77 777 ~K 17~7 ~
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lt
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~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
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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S~ y Z
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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
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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
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
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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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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
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