I

' :

APPLI ED

Ven Te ChcurDo/'d R.lrrlo'riment

LorryW. Mqys

= t -it'nli

--=-5*- -

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CONTENT:

a

i l

f

PrefaceFart I Hydrologic Processes

1 Introduction1.1 Hydrologic Cycle1.2 Systems Concept1.3 Hydrologic System Model1.4 Hydrologic Model Classification1.5 The Development of Hydrology

2 Hydrologic Processes2.1 Reynolds Transport Theorem2.2 Continuity Equations2.3 Discrete Time Continuity2.4 Momentum Equations2.5 Open Channel Flow2.6 Forous Medium Flow2.7 Energy Balanc'e2.8 TiansportProcesses

3 Atmospheric Water3.1 Atmospheric Circulation3.2 Water Vapor3.3 Precipitation3.4 Rainfall3.5 Evaporation3.6 Evapotranspiration

4 Subsurface Water

I

) t '

2l.,

2t2\33(4t.1 .

5-

5-5(6i7 i8(o l

99

(

vi ABour rHE AUTHoRS

B . S . ( 1 9 7 0 ) a n d M . S . ( | g 7 | ) d e g r e e s f r o m t h e U n i v e r s i t y o f M i s s o u r i a t R o l l a 'a f te rwh ichheserved in theU.s .Armysta t ionedat theLawrenceL ivermoreLabora tor ' v inCa l i fo rn ia .Dr .Mayshas 'beenveryac t ive in researchandteach.ing u, ,tt" University of Texas in the areas of hydrology' hydraulics' and water

i"!ou.." systems analysis. In addition he has served as a consultant in these areas

to various go\rernmenr ug"n"*, and industries including the u.s.-Army corps of

Engineers, the Attorney"G"n"tul'' Office of Texas' the United Nations' NATO'

;#w;; ;;1, fiJihe Government of Taiwan. He is a registered engineer

in seven states and has been active in committees with the Americau society of

Civil Engineers and other professional organizations'

CONTENTf

*j 1

€

Part 1PrefaceHydrologic Processes

1 Introduction1.1 Hydrologic Cycle1.2 Systems Concept1.3 Hydrologic System Model1.4 Hydrologic Model Classification1.5 The Development of Hydrology

2 Hydrologic Processes2.1 Reynolds Thansport Theorem2.2 Continuity Equations2.3 Discrete Time Continuity2.4 Momentum Equations2.5 Open Channel Flow2.6 Forous Medium Flow2.7 Energy Balancb2.8 TfansportProcesses

3 Atmospheric Water3.1 Atmospheric Circulation3.2 Water Vapor3.3 Precipitation3.4 Rainfall3.5 Evaporation3.6 Evapotranspiration

4 Subsurface Water

) t ,

2l

z.2rL -

33(41.

4 .

5-

5-5(6r7 i8(9 l

99

(

viii APPLIED HYDRoLoGY

4.3 Green-AmPt Method4.4 Fonding time

5 Surface Water5.1 Sources of Streamflow

5.2 Streamflow HYdrograPh

5.3 Excess Rainfali and Direct Runoff

5.4 Abstractions usi4g Infiltration Equations

5.5 SCS Method fir Abstractions

5.6 Flow DePth and VelocitY

5.'7 Travel Time5.8 Stream Networks

6 Hydrologic Measurement6.1 Hydrologic Measurement Sequence

6.2 Measurement of Atmospheric Water

6.3 Measurement of Surface Water

6.4 Measurement of Subsurface Water

6.5 Hydroiogic Measurement Systems

6.6 Measurement of Physiographic Characteristics

1 1 0t1'7

127

127

135140147155164r66

r75

L'�l6t'l9184r92192198

hrt 2 Hydrologic AnalYsis

77 . 17 .2t . J'7.4

7 .5-1.6

7 . 7'7.8

Unit HydrographGeneral Hydrologic SYstem Model

Response Functions of Linear Systems

The Unit HydrograPhUnit HydrograPh Derivation

Unit HydrograPh APPlication

Unit Hydrograph by Matrix Calculation

Synthetic Unit HYdrograPhUnit Hydrographs for Different Rainfall Durations

201

2022042132t62r8221LZ3

230

a A a

24224525225'7260

z'�t2z t t

280282287

8 Lumped Flow Routing8.1 Lumped SYstem Routing

8.2 Level Frol Routing8.3 Runge-Kutta Method

8.4 Hydrologic River Routing

8.5 Linear Reservoir Model

9 Distributed Flow Routing9. I Saint-Venant Equations

9.2 Classification of Distributed Routing Models

9.3 Wave Motiong.4 Analytical Solution of the Kinematic Wave

/\

9.5 Finite-Difference Approximations9.6 Numerical Solution of the Kinematic Wave9.7 Muskingum-Cunge Method

10 Dynamic Wave Routing10.1 Dynamic Stage-Discharge Relationships10.2 Implicit Dynamic Wave Model10.3 Finite Difference Equations10.4 Finite Difference Solution10.5 DWOPER Model10.6 Flood Routing in Meandering Rivers10.'l Dam-Break Flood Routine

11 Hydrologic Statistics1.I Probabilistic Tleatment of Hydrologic DataLZ Frequency and Probability Functions1.3 Statistical Parameters1.4 Fitting a Probability Distribution1.5 Probability Distributions for Hydrologic Variables

CONTENTS IX

290294302

310

3 1 1I 3 r4

i. i 316f 320

325326330

Frequency AnalysisReturn hriodExtreme Value DistributionsFrequency Analysis Using Frequency FactorsProbability PlottingWater Resources Council MethodReliability of Analysis

hrt 3 Hydrologic Design

t21 2 . lt 2 .2

I z . +

12.512 .6

350

350354359363371

380

380385389394398405

L3 Hydrologic Design13.l Hydrologic Design Scale13.2 Selection of the Design Level13.3 First Order Analysis of Uncertainty13.4 Composite Risk Analysis13.5 Risk Analysis of Safety Margins and Safety Factors

t4 Design Stormsl4.l Design Precipitation Depth14.2 Intensity-Duration-Frequency Relationships14.3 Design Hyetographs from Storm Event Analysis14.4 Design Precipitation Hyetographs from IDF Relationships14.5 Estimated Limiting Storms14.6 Calculation of Probable Maximum kecipitation

4r6

4164204)1

433436

414

444454459465470475

APPLIED HYDROLOGY

CHAPTER1T

!i . f

INTRODUCTION

Water is the most abundant substance on earth, the principal constituent of all

living things, and a major force constantly shaping the surface of the earth. It is

also a key factor in air-conditioning the earth for human existence and in influenc-

ing the progress of civilization. Hydrology, which treats all phases of the earth's

water, is a subject of great importance for people and their environment' Practical

applications of hydrology are found in such tasks as the design and operation of

hydraulic structures, water supply, wastewater treatment and disposal, irrigation,

drainage, hydropower generation, flood control, navigation, erosion and sediment

control, salinity control, pollution abatement, recreational use of water' and fish

and wildlife protection. The role of applied hydrology is to help analyze the

problems involved in these tasks and to provide guidance for the pianning and

management of water resources.The hydrosciences deal with the waters of the earth: their distribution and

circulation, their physical and chemical properties, and their interaction with the

environment, including interaction with living things and, in particular, human

beings. Hydrology may be considered to encompass all the hydrosciences, or

defined more strictly as the study of the hydrologic cycle, that is, the endless

circulation of water between the earth and its atmosphere. Hydrologic knowledge

is applied to the use and control of water resources on the land areas of the earth;

ocean waters are the domain of ocean engineering and the marine Sciences.

Changes in the distribution, circulation, or temperature of the earth's waters

can have far-reaching effects; the ice ages, for instance, were a manifestation of

such effects. Changes may be caused by human activities. Feople till the soil'

irrigate crops, fertilize land, clear forests, pump groundwater, build dams, dump

wastes into rivers and lakes. and do manv other constructive or destructive things

2 ,cppLlro HYDRot-ocY

1.1 HYDRGLOGIC CYCLEWater on earth exists in a space called the hydrosphere which extends about

15 km up into rhe atmosphere and about I km down into the iithosphere, the

crust of the earth. Water circulates in the hydrosphere through the maze of paths

constituting the hydrologic cycle.The hydrologic cycle is the central focus of hydrology. The cycle has no

beginning or end, and its qrany processes occur continuously. As Shown schemat-

icatty ln-fig. Ll. l, watei evaporates from the oceans and the land surface to

become part of the atmosphere; water vapor is transported and lifted in the atmo-

sphere until it condenses and precipitctres on the land or the oceans; precipitated

water may be i4tercepted by vegetation, become overland fow over the ground

surface. irtfiLtrtttt: into the ground. flow through the soil as subsurfat:e fow, anrJ

discharge into streams as surface runoff. Much of the intercepted water and sur-

face runoff returns to the atmosphere through evaporation' The infiltrated water

may percoiate cieeper te recharge groundwater, later emerging in springs or seep-

ing into streams to form surface runoff, and finally flowing out to the sea or

evaporating into the atmosphere as the hydrologic cycle continues'

Estimating the total amount of water on the earth and in the various processes

of the hydroiogic cycle has been a topic of scientific exploration since the second

half of the nineteenth century. However, quantitative data are scarce, particularly

over the oceans, and so the amounts of water in the various components of the

globai hydrologic cycle are still not known precisely'

Table 1.1.1 lists estimated quantities of water in various forms on the earth'

About 96.5 percent of all the earth's water is in the oceans. If the earth were a

uniform sphere. this quantity would be sufficient to cover it to a depth of about

2.6 km (f.O m;). Of the remainder, 1.7 percent is in the polar ice, 1.7 percent in

groundwater and only 0.1 percent in the surface and atmospheric water systems.

ihe atmospheric water system, the driving force of surface water hydrology,

contains onty tZ,lOO km3 of water, or less than one part in 100,000 of all the

earth's water.of the earrh's fresh water, about two-thirds is polar ice and most of the

remainder is groundwater going down to a depth of 200 to 600 m. Most ground-

water is saline below this depth. Only 0.006 percent of fresh water is contained

in rivers. Biological water, fixed in the tissues of plants and animals, makes up

about 0.003 percent of all fresh water, equivalent to half the volume contained

in rivers.Although the water content of the surface and atmospheric water systems iS

relatively small at an1' given moment, immense quantities of water annually pass

through them. The global annual water balance is shown in Table 1. 1.2; Fig. 1. I ' I

shows the ma.jor components in units relative to an annual iand precipitation

volume of 100. lt can be seen that evaporation from the land surface consumes

6i percent of this precipitation, the remaining 39 percent forming runoff to the

o..inr, mostl-v as surt'ace water. Evaporation from the oceans contributes nearly^^ -- -^ . - i r ^r - r . - - -n<nhar in mniqrrr re An:r lvs is of the f low and storaqe of water in

q

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e :

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t !l i * v| l1 1 ' v 'l l

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h*TRoDUCTION 54 eppi-tro HYDROLocY

TABLE T. I .1Estimated world water quantities

TABLE 1.1.2Global annual water balance

Item Volume(km3)

Ocean LandArea(106 km:)

Percent of Percent of

total water fresh water

Oceans

Groundwater

Fresh

Sal ine

Soil Morsture

Polar rce

Other ice and st i t , i i

Lakes

Fresh

Sal ine

N{arshes

Rivers

Biological n atei'

Atmospheric t'atcr

Total water

Fresh water

1 6 1 . 3

l3lr8

I 34 .8

82.0

1 6 . 0

0. - l

t . 2

0 . 8

2 .7

148.8

5 1 0 . 0

5 1 0 . 0

5 t0 .0

1 4 8 . 8

1,338,000,000

r0,530,000

12,870,000

16.500

24,023,500

340,600

9l.000

85,400

I I ,470

2,120

I , 1 2 0

12,900

1,385,984,610

3s,029,210

96.s

0 .16

0.93

0.0012

t . T

0.025

0.007

0.006

0.0008

0.0002

0.0001

0.001

100

2.5

0 .26

0.03

0.006

0.003

0.04

t00

Area (km2)

Precipitation

Evaporation

Runoff to oceanRiversGroundwaterTotal runoff

(km3/yr)(mm/yr)

(iniyr)

(km3/y0(mrn/yr)

(in/y0

(km3/y0(km3/y0(km3/yg(mm/yr)

(in/yr)

361,300,000

458,00012'70

50

505,0001400

: ) f

r 48,800,000

1 r9,0008003 1

12,000484

t 9

44,7002200

47,0003 r 6

t 2

i'

{

Table from world \!arer Balance and water Resources of the Earth, copyright, UNESCO' 1978

Example 1.1.1, Estimate lhe residence time of global atmospheric moisture.

solution. The residence time T,- is the average duration for a water molecule to

p a s s t h r o u g h a s u b s y s t e m o f t h e h y d r o i o g i c c y c l e ' I t i s c a l c u l a t e d b y d i v i d i n g t h evolume of water S in storage by the flow rate Q'

sT - -t r -

Q

( 1 . 1 . 1 )

Table from World Water Balance and Water Resources of the Earth, Copyright.UNESCO. 1978

constant, the distribution of this water is continually changing on continents, inregions, and within locai drainage basins.

The hydrology of a region is determined by its weather patrerns and byphysical factors such as topography, geology and vegetation. Also. as civiliza-tion progresses, human activities gradually encroach on the natural water envi-ronment, altering the dynamic equilibrium of the hydrologic cycle and initiatingnew processes and events. For example, it has been theorized that because ofihe burning of fossil fuels, the amount of carbon dioxide in the atmosphere isincreasing. This could result in a warming of the earth and have far-reachingeffects on global hydrology.

I.2 SYSTEMS CONCEPT

Hydrologic phenomena are extremely complex, and may never be fully under-stood. However, in the absence of perfect knowledge, they may be representedin a simplified way by means of the systems concept. A system is a set ofconnected parts that form a whole. The hydrologic cycle may be treated as asystem whose components are precipitation, evaporation, runoff, and other phasesof the hydrologic cycle. These components can be grouped into subsystems ofthe overall cycle; to analyze the total system, the simpler subsystems can betreated separately and the results combined according to the interactions betweenthe subsystems.

In Fig. 1.2.1, the global hydrologic cycle is represented as a system. Thedashed lines divide it into three subsytems: the atmospheric water system contain-ing the processes ofprecipitation, evaporation, interception, and transpiration; thesttrface water system containing the processes of overland flow, surface runoff.

The volume of atmospheric moisture (Table 1.1.1) is 12,900 km3 Th: f low rate of

moisture from the atmosphere as precipitat ion (Table 1.1.2) is 458,000 + 119'000

: 577.000 kmi/yr, ,o th" ou..ug" residence time for moisture in the atmosphere

is j', : I2,gO0l5'7"7,000 : 0.022 yr = 8'2 days' The very short residence time

f o r m o i s t u r e i n t h e a t m o s p h e r e i s o n e r e a s o n w h y w e a t h e r c a n n o t b e f o r e c a s taccurateiy more than a few days ahead. Residence times for other components of

the hydrologic cycle are similarly computed. These values are averages of quantities

that may exhibit considerable spatial variat ion'

Although the conce pt of the hydrologic cycle is simple, the phenomenon is enor-

mo u s r ic o * p I "1 : I I lT:i ::"^; t: I : ::: i::l "::,'fl :i'];:', ::: ix'n::'iffffi i

(

6 apprtEo HYDRoLilc\

the subsurface tt{iter ,)stem containing the processes of infiltration, groundwater

recharge, subsurface flow ancl groundwater flow. Subsurface flow takes piace in

the soil near the iand surface; groundwater flow occurs deeper in the soil or rock

strata.For most practical problems, only a few processes of the hydrologic cycle

are considered at a time" and then only considering a small portion of the earth's

surface. A more restricted system definition than the global hydrologic system is

appropriate for such treatment; and is developed from a concept of the controi

volume. In fluid mechaniA, the application of the basic principles of mass,

momentum, and energy to a fluid flow system is accompiished by using a control

voiume. a reference frame drawn in three dimensions through which the fluid

FIGURE 1.2.IR l n ^ L - d i r o r r m r e e i . L p n t r t i o n n f t h e C l o h t l h v d r o l o g i c s v s t e m

(

'ucroN 7

flows. The control volume provides the framework for applying the laws ofconservation of mass and energy and Newton's second law to obtain practicalequations of motion. In developing these equations, it is not necessary to knowthe precise flow pattern inside the control volume. What must be known are theproperties of the fluid flow at the control surface, the boundary of the controivolume. The fluid inside the control volume is treated as a mass, whic\ maybe represented as being concentrated at one point in space when considering theaction of external forces such as gravity. {

By analogy, a hydrologic system is defined as a structure or volume inspace, surrounded by a boundary, that accepts water and other inputs, operateson them internally , and produces them as outputs (Fig. 1 .2.2) . The structure (for

surface or subsurface flow) or volume in space (for atmospheric moisture flow) is

the totality of the flow paths through which the water nlay pass as throughptn fromthe point it enters the system to the point it leaves. The boundary is a continuoussurface defined in three dimensions enclosing the volume or structure. A *-orkingmedium enters the system as input, interacts with the structure and other media,and leaves as output. Physical, chemical, and biological processes operate on theworking media within the system; the most common working media involved inhydrologic analysis are water, air, and heat energy.

The procedure of developing working equations and models of hydrologicphenomena is similar to that in fluid mechanics. In hydrology, however, there isgenerally a greater degree of approximation in applying physical laws because thesystems are larger and more complex, and may involve several working media.Also, most hydrologic systems are inherently random because their major inputis precipitation, a highly variable and unpredictable phenomenon. Consequently,statistical analysis plays a large role in hydrologic anaiysis.

Example 1.2,1. Represent the storm rainfall-runoff process on a watershed as ahydrologic system.

Solution. A watershed is the area of land draining into a stream at a given location.The watershed divide is a line dividing land whose drainage flows toward the givenstream from land whose drainage flows away from that stream. The system boundaryis drawn around the watershed by projecting the watershed divide vertically upwardsand downwards to horizontal planes at the top and bottom (Fig. 1.2.3). Rainfallis the input, distributed in space over the upper plane; streamflow is the output,concentrated in space at the watershed outlet. Evaporation and subsurface flow couldalso be considered as outputs, but they are small compared with streamflow duringa storrn. The structure of the system is the set of flow paths over or through the soiland includes the tributary streams which eventually merge to become.sffeamflow atthe watershed outlet.

FIGURE 1.2.2Schematic representation of system operation

!

-c

5

Inl ' i l tration

8 ...ppt-Iro HYDRoLocY

PreciPitation / (t)

i l l- - - -

- , - - ] - - .

( ' ' r * - li - : : : - * - - ' l

,"'.. J2---.=)<-[ - \l\

=-\----- |

i :srreamflowo(t)i -**--

- ---- l( )- - . . * . *

* - - - / '

FIGURE I.2.3The n'atershed as a hydrologic system'

If the sur{ace and soil of a watershed are examined in great detail, the num-

ber of possibie flow paths becomes enorrnous' Along any path' the shape' slope'

and boundary roughness may be changing continuously from place to place and

these factors may also uury in time as the soil becomes wet' Also' precipitation

varies randoml-Vin space und titn". Because of these great complications' it is not

possible to desiribe iome hydrologic processes with exact physical laws. By using

it-," ,yrt".n concept, effort is directed to the construction of a model relating inputs

and outputs rather than to the extremely difficult task of exact representation of

the system details, which may not be significant from a practical point of view

o. rl]uy not be known. Neveriheless, knowledge of the physical system helps in

developing a good model and verifying its accuracy'

1.3 HYDROLOGIC SYSTEM MODEL

The objective of hydrologic system analysis is to study the system operation

and predict its output. A-hydrologic system model is an approximation of t}te

actual system; its inputs and outputs are measurable hydrologic variables and its

structure is a set of equations linkiqg the inputs and outputs' central to the model

StruCture is th,: concept of a system transformation'

Let the rnput and output be expressed as functions of time' 1(t) and O(D

respectively, foi r belonging to ttre time range 7 under consideration' The system

p..forlrr, a,transformation of the input into the output represented by

t l.- Syrrem boundaryWetershed I ----.--.--.- ,Watershed surface I

Q @ : O I ( ' ( 1 . 3 . 1 )

which is pailed the transformation equation of the system' The symbol C) is a

tansfer function between the input and the output. If this relationship can be ex-

f..soO Ly an aigebraic equation, then O is an algebraic operator' For example' if

rNTRoDUcrroN 9

Q G ) : C I ( t )

where C is a constant, then the transfer function is the operator

Q( t )I (t)

If the transformation is described by a differential equation, then the qansfer

function serves as a differential operator. For example, a linear reservlir has its

storage S related to its outflow O by t

S : k Q ( 1 . 3 . 4 )

where ft is a constant having the dimensions of time. By continuity, the time rate

of change of storage dSldt is equal to the difference between the input and the

output

* : , r D _ � e , , ,at

Eliminating S between the two equations and rearranging,

o# * e( t ) : I ( t )

o : Q U ) : 1I(t) 1 + kD

where D is the differential operator dldt. If the transformation equation has been

determined and can be solved, it yields the output as a function of the input.

Equation (1 .3.7) describes a linear system if k is a constant. If /c is a function

of the input 1or the output 0 then (1.3.7) describes a nonlinear system which is

much more difficult to solve.

I.4 HYDROLOGIC MODEL CLASSIFICATION

Hydrologic models may be divided into two categories: physical models and

aistract models. Physical models include scaie models which represent the system

on a reduced scale, such as a hydraulic model of a dam spillway; and analog

models, which use another physical system having properties similar to those of

the prototype. For example, the Hele-Shaw model is an analog model that uses

the movement of a viscous fluid between two closely spaced parallel plates to

model seepage in an aquifer or embankment.Abstiact models represent the system in mathematical form. The system

operation is described by a set of equations linking the input and the output

variables. These variablei may be functions of space and time, and they may also

be probabilistic or randomvariables which do not have a fixed value at a particular

( r . 3 .2 )

(1 .3 .3 )

( 1 . 3 . 5 )

( 1 . 3 . 6 )

( r . 3 . 7 )

i

t

10 APPLIEDHYDR0i - ( )c lY

point in space and time but instead are described by probabii ity distributions. For

erample, tomorow's rainfall at a particular location cannot be forecast exactly

but the probabiiity that there will be some tain can be estimated. The most general

representation ol such variables is a random field' a region of space and time

within which the value of a variable at each point is defined by a probability

distribution (Vanmarcke, 1983). For example, the precipitation intensity in a

thunderstorm varies rapidly in time, and from one location to another, and cannot

be predicted accurately, so it is reasonable to represent it by a random field'

Trying to develop a lnodel with random variables that depend on all three

space dimensions and time is a formidable task, and for most practical purposes

it is necessary to simplify the model by negiecting some sources of variation'

Hydrologic models may be classified by the ways in which this simplification is

accomplished. Three basic decisions to be made for a model are: Wiil the model

variables be randorn or not? Wiil they vary or be uniform in space? Wili they

vary or be constant in time? The model may be located in a "tree" according to

these choices, as sho$'n in Fig. 1.4. 1.A determinisric model does not consider randomness; a given input always

produces the same output. A stochastic model has outputs that are at least par-

iially random. One might say that deterministic models make forecasts while

stochastic models make predictions. Although ali hydrologic phenomena involve

some randomness. the resulting variability in the output may be quite small when

compared to the variability resulting from known factors. In such cases, a deter-

ministic modei is appropriate. If the random variation is large, a stochastic model

is more suitabie. because the actual output could be quite different from the sin-

gle vaiue a detern.rinisiic model would produce. For example, reasonably good

deterministic models cf daily evaporation at a given location can be deveioped

using energy supply and vapor transport data, but such data cannot be used to

make reliable models of daily precipitation at that location because precipitation

is iargely random. Consequently, most daily precipitation modeis are stochastic.

At the midrile ievel of the tree in Fig. 1.4.1, the treatment of spatial varia-

rion is decideci. Hydrologic phenomena vary in all three space dimensions, but

explicitly accounting ior all of this variation may make the model too cumber-

some for practical application. In a deterministic lumped model, the system is

spatially averaged, or regarded as a single point in space without dimensions. For

example, many modeis of the rainfall-runoff process shown in Fig. 1.2.3 treat the

precipitation input as uniform over the watershed and ignore the internal spatial

variation of waiershed flow. In contrast, a deterministic distributed model con-

siders the hydrologic processes taking place at various points in space and defines

the model variables as functions of the space dimensions. Stochastic models are

classified as space-inclependent or space-colrelated according to whethel or not

random variables at different points in space influence each other.

At the third level of the tree, time variabiiity is considered. Deterministic

models are ciassified as steady-flow (the flow rate not changing with time) or

unsteady-fow models. Stochastic models always have outputs that are variable

in time, Thel' may be classified as time-independent or time-correlated; a time-

at ! .

E

E

!

q

E

oi

> a i

eo

!

o

a

e , l{

c

' c

tr

3

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s

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!

cO

oE

a!

l r O

L2 APPLTED HYDROT-oGy

independent moclei represents a sequence of hydrologic events that do not influ-

ence each other. while a time-correlated model represents a sequence in which

the next event is partially influenced by the current one and possibly by others in

the sequence.All hydrologic models are approximations of reaiity, so the output of the

actual system can nevel be forecast with certainty; likewise, hydrologic phenom-

ena vary in all three space dimensions, and in time, but the simultaneous consid-

eration of ali fil'e sources of'variation (randomness, three space dimensions, and

time) has been accomplisltd for only a few idealized cases' A practical model

usual ly considers only one or two sources of var iat ion '

df the eight possible hydrologic model types shown along the bottom line

of Fig. 1.4.1, ftur are considered in detail in this book. In Fig. 1.4.2, a section of

a river chdnnel is used to illustrate these four cases and the differences between

them. On the right of the figure is a space-time domain in which space, or distance

along the channel. is shown on the horizontal axis and time on the vertical axis

for each of the four cases.The simplest case, (a), is a deterministic lumped steady-flow model' The

inflow and outflow are equal and constant in time, as shown by the equally sized

dots on the lines at x : 0 and x : t. Many of the equations in the first six chapters

of this book are of this rype (see Ex. 1.1.1, forexample). The next case, (b), is a

deterministic lumped unsteady-flow model. The inflow 1(t) and outflow QQ) are

now allowed to vary in time, as shown by the varying sized dots at -x :0 and x: L.

A lumped model does not illuminate the variation in space between the ends of

the channel seciion so no dots are shown there. The lumped modei lepresentation

is used in Chaps. 7 and 8 to describe the conversion of storm rainfall into runoff

and the passage of the resulting flow through reservoirs and river channels. The

third case, (c). is a deterministic distributed unsteady-flow model; here, variation

along the space axis is also shown and the flow rate calculated for a mesh of points

in space and time. Chapters 9 and 10 use this method to obtain a more accurate

model of channel flow than is possible with a lumped model. Finally. in case

(d), randomness is introduced. The system output is shown not as a single-valued

dot, but as a disrribution assigning a probability of occurrence to each possibie

value of the variable. This is a stochastic space-independent time-independent

model where the probability distribution is the same at every point in the space-

time plane and values at one point do not influence values elsewhere. This type

of modei is used in Chaps. 1l and 12 to describe extreme hydrologic events such

as annual maximum rainfalls and floods. In the last three chapters, 13 to 15' the

models developed using these methods are employed for hydrologic design.

1.5 THE DEVELOPMENT OF HYDROLOGY

The science of hydrology began with the concept of the hydroiogic cycle. From

ancient times" many have speculated about the circulation of water, including

the poet Homer (about 1000 e.c.), and philosophers Thales, Plato, and Aristotle

in Greece; Lucretius. Seneca. and Pliny in Rome; and many medieval schol-

I (t)

l ni , \1 t \V \I

U '

-==-=Y-1 -

- nR.\-S\fRrR_0

" \ \ \ \ \ \ \

L

(a) Deterministic lumped steady-flow model, / = 0

QQ)

s

x\\\R.\\\\(\\L O - t A l t + t

Deterministic lumped unsteady-flow model, dSldt = I(t),Q(t)

rNTRoDUCrroN 13

Space-time domain

T ime I

t? ?t ll tI " Io r L

Dis tance. r

Time t

i r. i t^ / l l' ? t

I l .O L

Distance -r

( b )

0 - | A r l * t

Time t

i--1'-�r-?l l

nfff,-ffifrt I I t .0 * i A . r l * L

Distance.r

Time t

(c) Deterministic distributed unsteady-flow model

o

v=€*

r\\\\sR\\\\\\L (Probability distribution)

Disrance -r

(d) Stochastic space-independent time-independent model.

FIGURE I.4.2The four types of hydrologic models used in this book are illustrated here by flow in a channel. For

the three deterministic models (a) to (c), the size of the dots indicates the magnitude of the flow,

the change of inflow and outflow with time being shown on the vertical lines at x = 0 and x : L 'respectivelv. For the stochastjc svstem (d). the flow is represented bv a probabilitv distribution that

l 4 APPLTEDHYDRoT-o( ; \ '

philosopher Anaxasoras of Clazomenae (500-428 e c.) formed a primitive version

of the hydrologic cy'cle. He believed that the sun lifted water from the sea intothe atmosphere. fiom which it feli as rain, and that rainwater was then collectedin underground reservoirs, which fed the river flows. An improvement of thistheory was madc by another Greek philosopher, Theophrastus (c. 312-28"7 s.c.).who correctly described the hydrologic cycle in the atmosphere; he gave a soundexplanation of the formation of precipitation by condensation and freezing. Afterstudying the works of Theophrastus, the Roman architect and engineer MarcusVitruvius, whc iir"ed aboilt the time of Christ, conceived the theory that isnow generally accepted: he extended Theophrastus' explanation, claiming thatgroundwaier was largell, derived from rain and snow through infiltration from theground surface. This may be considered a forerunner of the modern version ofthe hydroiogic ci 'cie.

Independent thinking occurred in ancient Asian civil izations (UNESCO,1914). The Chinese recorded observations of rain, sleet, snow, and wind on An-yang oracle bones as early as 1200 e.c. They probably used rain gages around1000 e c., and established systematic rain gaging about 200 s.c. In India, the firstquantitative measurements of rainfall date back to the latter part of the fourthceniury s.c. The concept of a dynamic hydrologic cycle may have arisen in Chinaby 900 e c ,l in lndia by 400 s c.,2 and in Persia by the tenth century,3 but theseideas had little impact cn Westelx thought.

During the Renaissance. a graduai change occurred from pureiy philosoph-ical concepts of hydrology toward observational science. Leonardo da Vinci(1452-1519) made the first systematic studies of velocity distribution in streams,using a weighted rod held afloat by an inflated animal bladder. The rod wouldbe released at a point in the stream, and Leonardo would walk aiong the bankmarking its progress u,ith an odometer (Fig. i.5.1) and judging the differencebetween the surface and bottom veiocities by the angle of the rod. By releasingthe rod at different ooints in the stream's cross section. Leonardo traced the

I In the volume "Minor Foiksongs" of the "Book of Odes" (anonymous, 900-500 B.C.) is wntten:"Rain and snow are interchangeable and becoming sieet through first (fast) condensation." Also, Fan

Li (400 8.C., Chi Ni tzu or "The Book of Master Chi Ni") said: "...the wind (containing moisture)is ch'i (moving force or energy) in the sky, and the rain is ch'i of the ground. Wind blows accordingto the time of the year and rain falls due to the wind (by condensation). We can say that the ch'i inthe sky moves downwards (by precipitation) while the ch'i of the ground moves upwards (through

evaporation). "

2 Upanisads, dating from as early as 400 B.C. (Micropaedia, Vol. X, The New EncyclopaediaBritannica, p. 283. i974), translated from Sanskit to English by Swami Prabhavananda and Frederick

Manchester, Mentor Btoks. No. MQ92l, p. 69. In this work is written: "The rivers in the east floweastwards, the rivers in the rvest flow westward, and all enter into the sea. From sea to sea theypass, the clouds lifting them [o the sky as vapor and sending them down as rain."3 Karaji, M., "Exiiaction t'f Hidden Water", ca. 1016 A.D., translated from Arabic to krcian byH. Khadiv-Djam. lranian Culture Foundarion, Tehran, Iran. In this work is written: "Springs comefrom waters hidden inside the earth while waters on the ground surface from rains and snows ...

( b )

FIGT,RE T.5.1Leonardo da Vinci measured the velocity distribution across a stream section by repeated experimentsoi the type shown in (a). He would release a weighted rod (&) held afloat by an inflated bladder

and follow its progress downstream, measuring distance with the odometer and time by rhythmicchanting. (Source: Frazier, 19'74, Figs. 6 and 7. Used with permission.)

velocity distribution across the channel. According to Frazier (I914), the 8000existing pages of Leonardo's notes contain more entries concerning hydraulicsthan about any other subject. Concerning the velocity distribution in streams, hewrote, "Of water of uniform weight, depth, breadth and declivity [slope], thatportion is swifter which is nearest to the surface; and this occurs because thewater that is uppermost is contiguous to the air, which offers but little resistancethrough its being lighter than water; the water that is below is contiguous to theearth, which offers great resistance through being immovable and heavier thanwater" (Maccurdy, 1939). Prior to Leonardo, it was thought that water flowedmore rapidly at the bottom of a stream, because if two holes were pierced in awall holding back a body of water, the flow from the lower hole was more rapidthan the flow from the upper one.

The French Huguenot scientist Bernard Palissy (1510-1589) showed thatrivers and springs originate from rainfall, thus refuting an age-old theory thatstreams were supplied directly by the sea. The French naturalist Pierre krrault

rNrRoDUCloN l5

Bladder

,it

it't

4i

(

16 APPLTED HYDRoLOcY

Herecogn i zed tha t ra rn fa l l i sasou rce fo r runo f fandco r rec t l yconc luded tha tthe remainder of the p..tipliutlon was lost by transpiration' evaporation' and

diversion.Hydraul icneasurementsandexper imentsf lour isheddur ingtheeighteenth

..nru.j. New hydraulic frinclptes weie .discovered such as the Bernouili equa-

tion and Chezy's tontluiu, und b""t' instruments were developed' including

the tipping bucket raln gage and the current meter' Hydrology advanced more

rapidiy during rhe nrnete"enih century. Dalton established a principle for evapora-

tion (1802), the tireory ofcapillary'flow.was described by the Hagen-Poiseuille

;;;"ii"" 11S:ll, and iire rational method for determining peak flood flows was

proposed by N{rilvanev (1850)' Darry developed his law -"|-ry:::::"dia

flow

( I 856), Rippl prcsenteO fris diagram ftr determining storage requirements ( I 883) '

)na i lunning proposecl his open-channel f low formuia (1891)'

However" quantltative hydrology was sti i i immature at the beginning of

the twentieth centurv. Empirical appioaches were employed to solve practical

hydrologic al problems. -Ctuauutty

hLydrologists replaced empiricism with rational

analysis of observed o,,tu- cr".n uni Rrpl (191i) developed a physicaily based

model for infiltratron, fluttn tf Si+l introduced frequency analysis of.floo1ryil:

and water Storage requirements, Richards (193 1) derived the governing equatlon

for unsaturated llow, Sherman devised the unit hydrograph methgf to transform

effective rainfall to Oirect runoff (1g32), Horron developed infiitration rheory

(1933) and u O"rcripr,on of drainage basin form (1945)' Gumbel proposed the

extreme value law to.ftl'Otofogic siudies (1941)' and Hurst (195 1) demonstrated

that hydrologic observaiion' 'iuy exhibit sequences of low or high values that

persist over many years'Like many sclences, hydrology was recognized only recently as a separate

discipline. .quout l9oi,1i," 'r, inlt"istates Civil Service Commission recognized

hydrologist as a job classification' The "hydrology. series" of positions in the

iorn*iirion iist of occupations was described as follows:

Thisser ies inc ludesprofessionalsc ient i f icposi t ionsthathaveasthei robject ivethestudy of rhe rnteneiationship and reaction between water and its environment in

thehydroiogrc. ' " i " . t t , . . "posi t ionshavethefunct ionsof invest igat ion 'analys is ,andinterpre iat i "nor t t ' "ph"no,o"nuofoccurrence'c i rcu lat ion,d is t r ibut ion,andq u a l i t y o f w a t e r i n t h e i a r t h ' s a t m o s p h e r e ' o n t h e E a r t h ' s s u r f a c e ' a n d i n t h esoil and rock strata. Such work requires the application of basic principles drawn

from and .uppt"*"nt"O by fields such as meteoiology' geology' soil science' plant

physiology' hydraulics, and higher mathematrcs'

Theadventof thecomputerrevolut ionizedhydrologvandm.adlhydrologicanal-ys ispossib leonalargerscale.Complex. theor iesdescr ib inghydrologicprocessesare now applied ,rrin! lo-puter simulations' and vast quantities of observed data

are reduced to summiry staiistics for better understanding of hydrologic phenom-

ena and for establishi"g nyat"ftgic design levels' More recently' developments in

eiectronics and data trinsmissioi have made possible instantaneous data retrieval

frnm remore recorders and the development of "real-time" programs for forecast-

(

INrRoDUcrloN 17

now provide many hydrologists with new computational convenience and power'

The evolution oi hyOrotogic knowledge and methods brings about continual

improvement in the scope and accuracy of soiutions to hydrologic problems.

Hydrologic problerns directly affect the life and activities of large numbers

of people. ,q,n element of risk is always present - a more extreme event than

any iristoricatly known can occur at any time. A corresponding responsibility rests

uptn the hydrologist to provide the best anaiysis that knowledge and datd' will, { ,

permit. 6t

REFERENCES

Daiton, J., Experimental essays on the constitution of mixed Sases; on the force of steam or vapor

from waters and other iiquids, both in a Torricellian vacuum and in air; on evaporation; and

on the expansion of gasei by heat, Mem Proc' Manch' Lit' Phil soc ' vol' 5' pp 535-602'

1 802.Darcy. H., Les fonraines publiques de la ville de Dijon, V Dalmont, Paris' 1856'

Frazier. A. H., Water.urr"nt..t"rr, Smithsonicut Studies in Htstory and Technolog)- no' 28'

Smithsonian Institution Press, Washington, D.C , 1974'

Green, W. H., and G. A. Ampt, Studies on soil physics, J' Agric

r 9 l l .Gumbel, E. J.. The return

l94 l .period of flood flows, Ann. Math. Stat.,

5a1 . . vo l . 4 . pa r t l . PP . l - 24 .

vo l . 12 , no . 2 , PP . i 63 -190 .

cyiindrischen Rohren, Poggendois

municipal water suPPlY, Trans. Am.

water balance and

in Hydrology, 25.

Hagen, C. H. L., Uber die Bewegung des Wassers in engen

Annalen der Physik und Chemie' vol. 16, 1839'

Hazen, A., Storage to be provided in impounding reservoirs for

Soc. Civ. Eng., vol .17, pp' 1539-1640' l9 l4 '

Horton. R. E., The role of infiltration in the hydrologic cycle, Trans. Am. GeoPhYs. Union, voI

5 0 1 - 5 0 5 , 1 9 3 2 .UNESCO, Contributions to the development of the concept

Conf.804/Col. l, Faris, August 1974.of the hydrological cYcle, Sc.74l

14. pp. 446-460. 1933.

Horton. R. E., Erosional development of streams and their drainage basins; Hydrophysical approach

to quantitative morphology, Bull. Geol. Soc. Am', vol' 56, pp 275-3'70' 1945'

Hurst. H. E., Long-term storage'capacity of reservoirs, Trans Am soc' Civ' Eng'' vol l 16' paper

no. 2447, pp. 170^799, t951.

MacCurdy, E., Tie Notebooks of Leonardo da Vinci, vol' 1, Reynal and Hitchcock' New York'

t 939 .Manning, R., On the flow of water in open channels and pipes, Trans' Inst' Civ' Eng' Ireland' vol'

20, pp. 161-207, 1891; supplement vol . 24, pp' 179-207' 1895'

Mulvaney, i. J., On the ose of self-r*gistering rain and floo6 gauges in making observations of

the relations of rainlall and of flood discharges in a given catchment, Proc. Inst. Cit' ' Eng

Ire land, vol .4, PP. 18-31, 1850.

Richards, L. A., Capillary conduction of liquids through porous mediums' P'r.].lt1s' A Journal of

General and Applied Pftyslcs, American Physical Society, Minneapolis' Minn'' voll' pp'

318-333, July-Dec. 193 l .

Rippl, W., Capacity of storage reservoirs for water supply, Minutes of Proceedings' Institution of

Civ i l Engineers, vol . 71, pp.210-278, 1883.

Sherman. L. K.. Streamflow from rainfall by the unit-graph method, Eng News Rec'' vol 108' pp'

U. S.S.R. Nrt ional Commit teewater resources of lheUNESCO, Par is, 1978

for the International Hydrological Decade, World

earth. Enelish translation, Studies and Reports

Vrnr r rcke . F . . Random F ie ld r : Analr'.sis and Stnthesis' MIT Press, Cambridge' Mass'' 1983

1

CH ER

HYDRGLOGICPROCESSES

Hydrologic processes transform the space and time distribution of water through-out the hydrologic cycle. The motion of water in a hydrologic system is influencedby the physical properties of the system, such as the size and shape of its flowpaths, and by the interaction of the water with other working media, includingair and heat energy. Phase changes of water between liquid, solid, and vapor areimportant in some cases. Many physical laws govern the operation of hydrologicsystems.

A consistent mechanism needed for developing hydrologic models is pro-vided by the Reynolds tronsport theorem, also called the general contol vol-ume equation. The Reynolds transport theorem is used to develop the continuity,momentum, and energy equations for various hydrologic processes.

2.I REYNOI-DS TRANSPORT THEOREM

The Reynolds transport theorem takes physical laws that are normally appliedto a discrete mass of a substance and applies them instead to a fluid flowingcontinuously through a control volume. For this purpose, two types of fluidproperties can be distinguished'. extensive properties, whose values depend onthe amount of mass present, and intensiye properties, which are independent ofmass. For any extensive property B, a corresponding intensive property p can bedefined as the quantity of B per unit mass of fluid, that is B = 6916^. B and Bcan be scalar or vector quantities depending on the property being considered.

The Reynolds transport theorem relates the time rate of change of anextensive property in the fluid, dBldt, to the external causes producing this

APT

2

rEssEs 2l

change. Consider fluid momentum; in this case, B : mY and p : d(mY)ldm:V, the fluid velocity, where bold face type indicates a vector quantiry. ByNewton's second law, the time rate of change of momentum is equal to the netapplied force on the fluid: dBldt : d(mY)ldt: >F. The exrensive properriesdiscussed in this book are the mass, momentum, and energy of liquid water, andthe mass of water vapor.

When Newton's second law or other physical laws are applied to u $tiObody, the focus is on the motion of the body and the analysis follows the lpdywherever it moves. This is the Lagrangian view of motion. Although this cofrceptcan be applied to fluids, it is more common to consider that fluids form acontinuum wherein the motion of individual particles is not traced. The focus isthen on a control volume, a fixed frame in space through which the fluid passes,called the Eulerian view of motion. The theorem separates the action of externalinfluences on the fluid, expressed by dBldt, into two components: the rime rateof change of the extensive property stored within the control volume, and thenet outflow of the extensive property across the control surface. The Reynoldstransport theorem is commonly used in fluid mechanics (White, 1979; Shames,1982; Fox and MacDonald, 1985; and Roberson and Crowe, 1985). Although ithas not been widely used in hydrology up to this time it provides a consistentmeans for applying physical laws to hydrologic sysrems.

To derive the governing equation of the theorem, consider the controlvolume shown in Fig.2.1 .1, whose boundary is defined by the dashed controlsurface. within the control volume there is a shaded element of volume dv. rf thedensity of the fluid is p, the mass of fluid in the element is dm = pdv, the amountof extensive property B contained in the fluid element is dB : Fdm: BpdV, andthe total amount of extensive property within any volume is the integral of theseeiementai amounts over that volume:

B _ (2 . r . r )

where JJJ indicates integration over a volume.Fluid flows from left to right through the control volume in Fig. 2.1. 1, but

no fluid passes through the upper or lower boundary. After a small interval oftime At, the fluid mass inside the control volume at time / has moved to the rightand occupies the space delineated by dotted lines. Three regions of space canthen be identified: region I, to the left, which the fluid mass occupies at time tbut not at t + A|. region II, in the center, filled by the fluid mass at both pointsin time; and region III, on the right, outside the control volume, which the fluidmass occupies at t + Ar but not at t. For the cross-hatched fluid mass initiallywithin the control volume, the time rate of change of the extensive property canbe defined by

d B 1

a :

] : : rn [ (au + Bi l r ) r+r , - (Br + Bi l ) ' ] (2 . t .2)

where the subscripts t and t * At are used to denote the values of the suhscrinted

il[u,*

,trrft\,,,

22 cpprrEo HroRri rocr

(d t F lu ld i r i , r : r . f \ l . , r t r l I l ( lhc conr to l vo lumc) lL t t imc I occup ies regron5l l a n d i i l . L r r i n r c ' - . 1 / .

llllr\

\ ' d A = ! c o s U i . l i . o s 0 < 0 )

{b)Expan, l .d \ Le*of rn l lo! ! regjon.

Outnow

v . d A = Y c o s 0 d A ( c o s g > 0 )

{ . I Expanded v iew ofoutflow region.

I

I

c o s e > 0 f o r o u r f l o w

cos0 =0 ar impermeable boundar ies0 c o s 0 < 0 f o r i n f l o w

(/)c.s0 rs.0 lor innow and oulf low

FIGURE 2.I.1, , . r , , - . i r r r , - . , ' i . n . f r r r p R p l , n o l d ( r r r f q F i n r h F . r e m

23

quantities at these two time points. Rearranging (2.1.2) to separate the extensivepropeny remaining within the control volume (B from that passing across thccontrol surface (81) and (8y11) yields

# = ̂ l*{;t*'r),*,r, - (arr),1 * }lt'u'1,*.,

- ru,l,l} (2. r .3)

As At approaches 0, region II becomes coincident with the connol volume\ andrhe finlst temlm in (2.1.3) becomes the time derivative (dldt) of the amounlof Bstored within the control volume:

;.+f[<r"1,-o, - (arr),] = rn,l I I u*, (2 . r .4 )

In this equation, the total derivative d/dr is used to account for the casewhen the control volume is deformable (i.e., changes in size and shape as timepasses). Ii the control volume is fxed in space and time, the total derivative canbe replaced by the partial derivative d/rl because the focus is on the time rate ofchange of the extensive property stored in the control volume without regard forits intemal spatial distribution.

The second term in Eq. (2.1.3), involving B1 and B1n, represents the flow ofthe extensive property across the control surface. Figure 2. lr l(c) shows a close-up view of region n at the outlet from the control volume. An element of areain the outlet control surface is labeled dA, and the element of volume dy is thevolume of the tube containing all the fluid passing through dA in time Ar. Thelength of the tube is Al = VAr, the length of the flow path in time At. The volumeof the tube is dY : Al cos qdA where 0 is the angle between the velocity vectorV and the direction normal to the area element dA. The amount of extensivepropeny 8 in the tube is Bpdv : BpLlcosqdA. The total amount of f luid inregion n is found by integrating these elemental amounts over the entire outletcontrol suriace. Thus the term in B in (2.1.3) can be written as

(2 . r .5 )

where the double integral .fJ indicates that the integral is over a surface.As At approaches 0, the limit of the ratio AllAt is the magnitude of the fluid

velocity y. Let the normol area vecrpr dA be defined as a Yector of magnitude dAwith direction normal to the arca dA poirrrrti.g outward from the control surface;then the term Vcos 0dA can be expressed as the vector dot product V tlA. So Eq.(2.1.5) can be rewritten to give the flow rate of the extensive property leavingthe controi surface as

[ [ r . o t t * " e a a

t r ) ( fl i m l ; ( 8 r r r ) , , 1 | | l P p V d A

J . V - 0 1 A r I J JIII

A I;q{},",.'} = ̂ ';1',

(2. | .6)

(

24 reeLtro sloe,l,ry

A similar analysis may be made for f luid entering the control volumein reg ion I l scc F ig . 2 .1 .1 (b) ] . In th is case, cos0 is negat ive and dV:. \ l cos(180" A)dA : A l cos 0dA, so tha t

.\r -0 A/

, : i f u 'uoot r ,

Subs t i tu t ing (2 .1 . - l ) . (2 .1 .6 ) , and (2 .1 .7 ) in to (2 .1 .3 ) g ives

( 2 . r . 1 )

( 2 . 1 . 8 )

(2. r .9)

Equation (2.1.9) is the goveming equation of the Reynolds rranspontheorem. It is used a number of times in this book, and it is worthwhile toreview the meaning of each term. As stated previously, the equation wil l be usedto provide a mechanisn for tali ing physical laws norrnally applied to a discretemass and applvine them instead to continuously flowing fluid. In words, theReynolds tmnsport theorcm states that the total rate of change of an extensiteproperry of a fluid is equal to the rate oJ change of extensive propen!- storedit.t the controL volume. dldr [!J ppdy, plus the net ctuffiow of exieniive properrythrough the conttol surfdce, IIBpV.dA. When using the theorem, inflows arealways considered negative and outflows positive.

In the fbilou ing scctions, the Reynolds transpon theorcm is applied todevelop conrjnuily. momentumi and energy equations for hyclrologic processes.

2.2 CONTINUITY EQUATIONSThe conservation of mass is rhe most useful physical principle in hydrologicanalysis and is required in almost all applied problems. Continuiryn equations

l lur, o,"o, roo,

, t B . , f f l t l lj , , 1 , ) t ) B o d v + j ) P e v d ^ - J j p e v d A

I ] i I

d B a t t t l t, ' , l l ) B P d v - l l B P v d A

d r a t ) J J ) I

."-{l,r,r,] :

For fluid entering the controi volume, the angle between the velocity vecrorV, pointing into the control volume, and the area vector dA, pointurg out,is in the range 90" < d < 270" for which cos 0 is negarive [see Fig 2.1.1(@1. Consequenrlv. V,dA is always negarive for inflow. For fluid leavine thecontrol volume cos d is positive, so V.dA is always positive for outflow. Arthe impermeable boundaries, V and dA are perpendicular and thercforc V.dA =0. Thus. the integrals in (2.1.8) over inlet I and outlet III can be replaced bya single integral over the entire control surface representing the outflow rninusinflow, or net outfiow, of extensive property B:

HyonoLocrcprocessrs 25

expressing this principle can be developed for a fluid volume, for a flow cross_section, and for a point within a flow. In this chapter, only the integral eLluatioLof continuiry for a flow volume is developed. The equation for continuity ar apoint will be derived in Chap. 4 to describe flow in a porous medium, and thecontinuity equation at a cross section will be derived in Chap. 9 to describe flowat a river section. The integral equarion of cofltinuir) is rhi basis for the pthertwo lorms. I

Integral Equation of ContinnityThe integral equation of continuity applies to a volume of fluid. If mass is theextensive property being considered in the Reynolds tmnspon theorem, then B =m, and B- dBldn = l. By the law of conservation of mass.dBldt:dmldt=Obecause mass cannot be created or destroyed. Substituting these values into theRe)nold\ t-ransport theorem t2. L9r gives

( 2 . 2 . t )

which is the integral equation of continuity for an unsteady, variable_density flow.If the flow has constant density, p can be divided out of both terms of

(2 .2 . l ) , leav inp

(2.2.2)

The inrtegral JJJdy is the volume of fluid stored in rhe control volume. denotedby S, so the t'ilst term in (2.2.2) is the time rate of change of storage dSldr. Thesecond term, the net outflow, can be split into inflow (l) and outflow e(r):

, t t t f f Io : : l l l p d v + l l o v . d A

d t . t J J ) l '

a l t t l ld , l j ) d v * J J v d A : o

t t 1 ( tI J v d A = l

J v d A + J J v d A = e u t _ r t r l

and the integral eiu",,." ", ."t",r"",,, """

'0"

."*nu""

dS- + Q Q ) - / ( r ) = 0at

ds, : I ( t ) - Q t ! )

at( 2 . 2 . 4 )

which is the integml equation ofcontinuity for an unsteady, constant density flow.used exrensively in rhis book. When the flow is steady, ds/dt : O, afi (2.2.2)reduces t0

( 2 . 2 . 3 )

\

26 rppLteo stot t i , rcr

( 2 . 2 . 5 )

which states that the volumetric inflow rate and outflow rate are equall that is'

I(i) = Q$). A steady flow is one in which the velocity at every Point in the flow

is constant in tirne. A simple way of thinking about this is to imagine taklng a"snapshoC' of the ilow now, and again fiYe minutes later; if the the flow is steady,

the two snapshots will be identicalIf the iotai amounts df inflow and outflow are equal, the system is said to

be cloJed so that

( .2 .2.6)

t ll l v . d A = 0

r rI Kt )dt= l QOdt

J T J '

fja' P^,ds I lvtth - | Qrrtdt

J ! j ) J r J ( / I ) A r

When this condition docs not hold, the system is oper. The hydrologic cycle is

a closed system for water- bui the rainfall-runoff process on a watershed is an

open system, because not all the rainfall becomes runoff; some is returned to the

atmosphere through evaporation.ihe continuity equations above are derived lor single phase flow, that is'

a liquid or a gas. but not both together' In multiphase situations, such as when

watir is evaporating, the liquid and gaseous phases of water must be carefully

distiflguished. A continuity equation should be writtel separately for each phase

of the flow: for each phase dBldt is the rate at which mass is being added to, or

taken from. that Dhase.

2.3 DISCRETE TIME CONTINUITY

Because most h]-drologic data are available only at discrete time intervals, rt is

necessary to refotmulate the continuity equation (2 2.4) on a discrete time basis'

Suppose that the time horizon is divided into intervals of length At, indexed by

;. Lquatlon (2.2.,1) can be rewritten as dS : I(t)dt - QQ) dt afi integrated over

the jth time interval to glve

f ( 2 . 3 . L )

5 r - 5 7 - . . r = l i - Q i

where 1j and Q are the volumes of inflow and outflow in the jth time interyal'

Note rh;t in Ei. Q.2.4), I(t) and QG) are flow iates' having dimensions [L3lT]'while S is u uolu*", having dimensions tl-3l In (2.3.2), all the variables have

dimensions [L3]. If the incremental change in stomge is denoted by AS;, then

one writes AS; = | Q1, and

S ; = S ; 1 + A S ; (2.3.3)

uvonoLocrcpnocrsses 27

If the init ial storage at t ime 0 is Se, then Sr : S0 * 11- 0r, Sz = Sr + Iz- Qz,and so on. By substituting for intermediate storage values, one obtains

s ; : ss+2 r , , - a ,

which is the discrcte-time continuity equation.

Data Representation

The functions 0(t) and 1(t) are defined on a continuous time domain; that ls, a

value of the function is defined at every instant of the time domain, and these

values can change from one instant to the next [Fig. 231(a)]. Figure 23'1

shows two methods by which a continuoLls time function can be represented on a

discrete time domain. The first method [Fig. 2.3. I (b)1 uses a s(tmple ddta functionin which the value of a function O(t) in the ith time interval, Q,' is given simply

by the instantaneous value of 0(t) at time iA/:

Q1 = Q() : Q( iL l ( 2 . 3 . 5 )

The dimensions of Q(t) and 0y are the same, either [LrlT] or LL/TI.The second method uses a pulse data function [Fig. 2.3 1(c)]' in which the

value of the discrete time function Q; is given by the area under the conrinuous

time function;

( 2 . 3 . 6 )

Here Q; has dimensions of [Ll or [L] for 8(r) in dimensions of [L/T] or [L/T]'respectively. Altematively, the dimensions of Qj and 0(/) can be kept the same

il Q1 is calculated as the aterage rQte over the interval:

{ 2 . 3 . 7 )

The two principle variables of interest in hydrology, strcamflow and pre-

cipitation, are measured as sample data and pulse data resPectively. When the

values of streamflow and precipitation are recotded by gages at a given instant'

the sbeamflow gage value is the flow rate at that instant, while the precipitation

gage value is the accumulated depth of precipitation which has occurred up to

that instant. The successive differences of the measurements of accumulated pre-

cipitation form a pulse data series (in inches or centimeters). When divided by

the time inteNal At, as in Eq. (2.3.1), the resulting data glve the precipit.ltion

intensit,t (in inches per hour or centimeters per hour). The continuity equation

must be applied carefully when using such discrete time data.

Example 2,3.1 Calculate the storage of water on a watershed as a function oftime given the data in columns 3 and 4 of Table 2.3 I for incremental precipitation

r ) 1 d \

i' l

F^'0, = | et\dr- JU l ) - \ .

o, = [f)-,,^,o,to'

"li!*

2E eppLleotr toRtr t -ocY

Q(tr

l r ' l r l

t L / r l

o , i

L L / T ]

lr ' / r l

L I 3 / Disc'ete time index J

{.) Pulse dala rcPrcsenlalron

FIGURE 2.3.IA conlinuous tinrt lunction O(0, (.1), can be defined on a discrete dme domain eitier by a sampled

dala syslem (bl. in which instantaneous values of the continuous lime function are used. or by a

pu ]seda taSys lem( t ) . ; n {h i ch the in l eg ra lo rave fageva ]ueo f t he func i i onove r the in l e r va l iSused '

over the *atershed and streamflow measured at its outle!. These data are adapted

ftom a flood that occurred on Shoal Creek at Northwest Park in Austin' Texas on

May 24 15. 1981. The wale$hed area is 7.03 mi2 Assume that the init ial storage

is zero.

Solution. 'the

precipitatjon input is recorded as a pulse data sequence in column 3;

the value shown is the incremental depth for the preceding time interval (e g ' the

value shoqn at t = 0-5 h, 0.15 in, is the precipi iat ion depth occuning during the

first 0.5 h and f ie value shown at t = I h,0.26 in' is the incremental precipi lat ion

HvonoLocrcprocgssrs 29

between l:0.5 h and t = t h, and so on) The streamflow output is recorded as

a sample data sequence; the value shown is the instantaneous flow rate (e g ' the

,t,"urf lor rate i ;246 cfs at r = 0 5 h,283 cfs at I = I h' andsoon) To apply

the discrete time continuity equation (2.3.4), the streamflow must be conlened lo

a pulse data sequence. The t ime interval is At = 0.5 h:0 5 x 3600 s: 1800 s

For each 0.5 h interval, the volume of streamflow is calculated by ave'agrnP the

sfearnilow rates at the ends ol the interval and multiplying by Ar' The equiq]enr

depth over the wate$hed of incremental streamflow is then calculated by div$ing

the sffeamflow volume by the watershed area' 'thich is 7 03 miz:7 03 x '180'�

f t2 : 1.96 x IOE ft2.For example, during the first time interval, between 0 and 0 5 h, the stream-

flows [Col.(4)] ire 0(0) : 203 cfs and 0(0 5 ) = 246 cfs' so lhe incremen-tal volume

in tfris intervai ts Km3 + 246)12l \ L; : 224.5 ^ 1800 : 4 04 x 105 fts Tie

eouivalent depth over the watershed is p, = 4 64 v 105/l '96 x 108 = 2 66 x 10 l

f t = 2.06 x 10-3 x 12 in = 0.02 in' as shown in column 5

The incremental precipitation 1l for the same time inte al is 0 15 in so the

increm€ntal change in storage is found from Eq. (2.3 2) withi = l'

AS1 : 11 Q t

= 0 . l 5 0 . 0 2

: 0 . 1 3 i n

as shown in column 6. The cumulative storage on the watershed is found from

(2.3.3) wi lh j : 1 and init ial storage So : 0:

5 , = J 6 + A S r

: 0 + 0 . 1 3

= 0 . 1 3 i n

as shown in column 7. The calculations for succeeding time intervals are similar'

Table 2.3.1 shows that of the 6.31 in total precipitation, 5 45 in, or 86 percent'

appeared as steamflow at the watershed outlet in the eight hours after precipitation

began. The remaining 0.86 in was retained in storage on the watershed ln columns

5 inrt 6 it can be seen rhat afte. precipitation ceased' all sreamflow was clrawn

directly from storage.ihe values of incremental precipitation and streamflow, change in storage'

and cumulative stonge are plotted in Fig 2'3'2 The critical time is I : 2 5 h'

when the maximum stolage occurs Before 2.5 h, precipitation exceeds streamflow

and there is a gain in sto;ge; after 2 5 h, the reverse occurs and there is a loss in

slorage.

2.4 MOMENTUM EQUATIONS

When the Reynolds lransport theorem is applied to tluid momenlum the ertensive

propeny is B=zV, and B = dBldm:Y By Newton's second law' the time rate

of al-g" of momentum is equal to the net force applied in a given direction'

so dBldt = d@Y)/dt : > F. Substituting into the Reynolds transport theorem

(2.1 .9) , rcsu l ts in

4., I

I L ' ) o , t l - l

t ]-r / Tlo. lL / Tlo, = lotrta, .' ) lor r

(d) Conlinuous time domrrn

Discfele time index /

(r) Sample dala rePtesentalion

30 , l ppL reouyo r .oLocy

TABLE 2.3.1The time distribution of storage on a watershed calculated using thediscrete-time continuity equation (Example 2.3.1)

nronolocrc mocesses 31

t 2Time Timeinierval

(h)

Increm€ntal Instantaneous Incrementalprecipitatior streamflow streamflowIt QO Q1(in) , (cfs) (in)

Incremental Cumulativ€saorage storageAs, 5;(in) (in)

Precip i tat ion

* 1 . 5

0

Storag€ onthe watershed

2.5

I r . 5E

e5 0 .5

0

, 1

f

I

2l

56189

10l l

T 2i 3

l 5

Total

0 .00 .51 . 01 . 5) . o

3 .0

4.0

5 .05 .56 .06 .5'7 .07 .58 .0

0 . l 50.161 . 3 32 ) O2 .080.:00.09

t 203246283828

2323569'7953 i

l 1025823443212246r802t2301 t 339435.1103

0 . l 30.23|.2'�12 .03| . 61

- 0.64- 1 .04- 1 .06- 0.69-0 .36

0.220 . 1 70 . l l

- 0.06- 0.04-0 .04

0.000 . l 30.361 .623 .655.294 .653 . 6 12.551 .85| . 49

l � t 01 .000.930 .890 .86

0.020.030.060 . l 70.4zl0.8,{I . 1 31 .060.690.360.220 .170 . l l0.060.040.04

Changein storag€

E o.o

^ 53

a -

I J

E'\, l

0

, t l l l ( fi | | l vpav+ | l vpv .oed I J J J J J

Time (h)

) r : ( 2 . 4 . t )

the integral momentum equation for an unsteady, nonuniform flow. A nonuniform/ow is one in which the velocity does vary in space; in a unifurm fiow tbere isno spatial vadation.

lf a nonuniibrm flow is steady (in time), the time derivative in Eq. (2.4.l)droDs out. leaving

(2.4.2)

For a steady uniform flow the velocity is the same at all points on the contlolsurface, and therefore the integral over the control surface is zero and the forcesapp l ied to the ' r . tem arc rn equ i l ib r ium:

t l) F = | l v p v . d A

Time (h)0 2 4 6 8

FIGURE 2.3.2The time distribution of storage on a watenhed calculaled usin8 the discrete{ime contiNity equalion(Example 2.3.1).

Storage 5/ = St-t+ Sj

I . = o (2.1.3)

(

32 :pprr lo rr on,,r, rr;r

Steady Uniform Flow in an Open Channel

In this section. the mornentum equation is applied to steady uniform flon tn an

open channel. The more complex case of unstead,v nonunifom flow is treated

in Sec. 9.1. Figurc 2.1.1 shows a steady flow in a uniform channel' lhat is- a

channel whosc cross section, slope, and boundarv roughness do not change along

its length. Thc conrinuity, momentum. and energy equations can be applied to

the control volrt lre between sections I and 2

Continuity. For stead) t ' lbw, Eq (2 2.5) holds ard 8r : p2: for uniform flow'

the velocity is the same everywhere in the flow, so V1= V2 Hence' cross sectlonal

a rea Ar = Qt i l t : Q. iV . - A2 . and s ince the channe l i s un j fo rm. i t fo l lows tha t

the depths arc rlso eqiral. 11 = 11

Energy. The crLcrgl'cquation from fluid mechanics (Roberson and Crowe l985)

is *ritten for iectlons I and 2 as

. ' v i : s : i - ) . 1 . 2 1 8 I ( 2..+.,+)

where : is the bed elclation. g is the acceleration due to gravit.v. and l1l is the

head loss bet\\een the t$o sections. Head loss is the energy lost due to ft iction

effects per unit rveight of f luid. With yl = y2 and 11 : y2, (2 ' l '1) reduces to

h l : a . I - z1 (2. . r .5)

Dividing both sides b] L. the length ol the channel' the following is obtained

(2. .+.6)

The bed slope 5(i = tan d where 0 is the angle oi inclination of the channcl

bed. I f 0 i s smal l ( '< l0 ' ) . then tan 0 : s in 0 = ( . t - .2 ) lL In th is case. the

FIGURE 2.{. ISteady unifofrJr ah\ In Lrf open chrnneL

U _ z t - 2 1L L

nYonoLoctc rncr*ses 33

f rL t r tons l , tpe ,St=ht lL , i sequa l to thebeds lopeSs l t i s assumed in th is ana lys is

ifr"t ,tr. onty source of energl loss is friction between the flow and lhe channel

wall. In general, energy can also be lost because of such factors as wind shear on

the surfaie and eddy motion arising from abrupt changes in the channel Seometry'

but these effects will not be discussed until Chap 9 When wall friction is the

onlv source of energy loss, the slope of the energy grade iine is equal to tlle

frici ion slope S/, as shown in Fig. 2.4 1 i

Momentum. There are three forces acting on the fluid control volumei flittion'

9r"., iry. rnd pressure. Of these. the pres'ure force\ 1I lhe lwo end\^of the 'ection

ire equal und cancel each other for uniform flow (because y 1 : -I2) So the friction

una grouity forces must be balanced, because, with the flow steady and uniform'

iq iz + :i applies (IF = 0) The friction force F; is equal to the product of the

,nLl sh"o, siiss 70 and the area over which it acts, PL ' where P is the wetted

pprimeter ol the cros. section: rhat is. F - IOPL $here^lhe negati!e sign'indicates

that the friction force acts opposite to the directton of f low The weight

of fluid in the control volume is yAl, where y is the specific tteiSil of the fluid

(weight per unit volume): the gravity force on the fluid, Fs'-is.the component of

itr. ri.igh, acting in the direction of flow, that is' F. = 7At sin 9 Hence

L U = O : - r o P L + Y A L s i r t A

When d is small, sin g: S0' so the approximation is made that

yALSs."

PL

( 2 . 1 . 1 )

( 2 . 4 . 8 ): TRS9

where R = A/P is the hydraulic radius. For a steacil uniform flow' S0 = S/' so

to : rRSt Q 19)

By a similar analysis, Henderson (1966) showed that (2 4 9) is also valid

for nonuniform flow, although the bed slope S0 and friction slope S1 are no

longer equal. Equation (2.4 9) expresses a linkage between the momentum and

eneigy principles in that the effects of friction are represented from the momentum

vie*loint as lhe wall shear strcss t0 and from the energy viewpoint as a rate of

energy dissipation S/.

2.5 OPEN CHANNEL FLOW

Open channel f low is channel f low with a free surface, tutn ut 11o* ln a rirer

oi in a partially fuil pipe. In this section the Manning equation to determlne the

vclocitli of open channel flow is derived, on the basis ol the Darq: weisbacll

equation for head losses due to wall fr iction'

In fiuid mechanics, the head loss l ' / over a length I of pipe of diameter D

ibr a tlrw with velocity Vis given by the Darcy-Weisbach equatlon

3,1

| 1 t 2

u - ! ; z s (2 . s .1 )

where J is the Darcy-Weisbach friction factor and I is the acceleration dueto gravity (Roberson and Crowe, 1985). Using the definition of friction slope,St = hy't, (2 5.1) can be solved for V:

(2.5 .2)

The hydraulic radius R of a circular pipe is R = A/P = (lTD2lqlnD = D/4, sothe pipe diameter D can be replaced in (2.5.2) by

D = 4 R

to give the Darcy Weisbach equation:

(2. s .3)

(2 .5 .4 )E-/ +RS.

1 t

Tne Chezy C is defineil as C = u@sli; using this symbol, (2.5.4) is rewritten

which is Clrezy 'r equation for open channel flow. Manning's equation is producedfrom Chezy's equation by setting C : Rli 6/r. where n is the lt4annin1 roughnesscoeffrcient.

p2ll ( l /2

t r - -n

(2.s.6)

Y: C \'Fq

Manning's equation (2.5.6) is valid for SI units, with R in mete6 and y in metersper second (S; is dimensionless). Values of Manning's n for various surtaces arelisted in Tabie 2.5.1. For y in feet per second and R in feet, Manning's equationls rewntlen

v= I !9 p" r "

f I .49 = (3 .281)L ' r and 3 .281 f t = I m l . By compar ing Eqs . (2 .5 .4 ) and (2 .5 .6 ) ,

Manning's a can be expressed in terms of the Darcy-Weisbach friction facror f,as follows:

t d g(2.5.8)

with all values in SI units.Manning's equation is valid for fully turbulent fow, in which the Darcy-

weisbach friction factor f is independent of the Reynolds number Re . Henderson/1965) cives the followins criterion for fullv turbulent flow:

J

Material

srotolocircvxocsssEs 35

TABLE 2.5.IManning roughness coemci€nts for various open channel

surfaces

TypicalMaiuring roughnesscoefncient

0 . 0 1 2

0.0200.0230.033

0.0300.0,100.0500.100

0.0350.0400.0500.0700.100

-t,!r.4 Choq, 1959

Concrele

Gravel bottom with sides - concrete- monared slone- riprap

Naiural str€am channelsCleal!. straighl streamClexn, winding streamWinding wilh weeds and pools

With heavy brush and limber

Flood PlainsPJstureField cropsLight bnish and weedsDense brushDense fiees

tL

,6.u&$ u l�9 x 10- L3 with R in ieet

n61R$ > 1.1 x l0-r3 with R in meters

Example 2,5.1 There is uniform flow in a 200-ft wide reclangular channel wlth

bed slope 0.03 percent and Manning's n is 0.015. l f the deplh is 5 f t , calculate the

velocity and flow rate, and verify thal the flow is fully turbulent so that Manning's

equation applies.

Solut ion. The wetted perimeter in the channel is P = 200 +2x5= 210fl The

hydraul ic radius is R = A/P = 2OO x 5l2lo:4.76 ft . The f low velocity is given

by Manning's equation with 11 : 0-015 and 51 : 56 (for uniform flow): 0.03% =

0.0003.

| . 49Y - - - D2 l rc l 2

1.49: _::_::14.76;r,10.ooo3),,,

= 4.87 ft/s

The flow rare is Q : y7 : 4.8'7 x 2OO x 5 : 4870 cfs. The crirerion for fullyluroulenr flou i\ calculated from 12.5.aa.1:

(2.5.9a)

(2.5.9b)

r \

1

36 rppuro nvonorocv

,6.,,Rs, :(o.ots)6(+.?6 x o.ooo3)r/'�= 4 .3 x l0 - r l

which is -greater than t.9 x l0_13 so the criterion is satisfied and Manning's equationis appljcable.

In the e\ent that the flow is not fuily turbulent, the flow velociry maybe computed \r'ith the Darpy-Weisbach equation (2.5.4), calculating the frictionfactor J as a function ofrthe Reynolds number Re and the boundary roughness.Figure (2.5.1) shows a modified form of the Moody diagram for pipe flow; thepipe diameter D is replaced by 4R. The Reyrolds number is given by

4 Wv

where z is the ki ent($ic l/scosiry of water, given in Table 2.5.2 as a function ofiempemturc. The rekuive roughness e is defined by

(2.s. I r )

where k, is th. size ol sand grains resulting in a surface resistance equivalent tothat obsened in the channel.

Figure 2.5.1 for open channel flow was constructed from equations pre-sented by Chow (i959) and Henderson (1966). For Reynolds number less than2000. the florv is Laminar, and

' R e

where C1= 96 lor a smooth-surfaced channel of infinite width and larger if thesurface is rough (Chov, 1959; Emmett, 1978). As the Reynolds number increasespast 2000, the flow enters a region where both laminar and turbulent effectsgovern friction losses and the friction factor is given by a modified form of theColebrook-White equation (Henderson, 1966):

I ^ , f o , 2 . s l_ - _ r ,ocro l l2R

_ * "C)

T - l

_ l) p ? / + lL " ' V ' l

(2 .5 . r3 )

lrYonoLoctcpcoc:sses 37

4 = 3 ' 55puqgn91 s ,111s13 ) I

+ c ! o^ : - 3 3 E 3 3 3

o o o o o oo c j d o o d d d o

€ t

. - k '- 4 R

( 2 . s . r 0 )

r ) S I ) r

. t

e

o

- 6

. :

o €

-

EJFor large Reynolds numbers, that is, in the upper right region of the Moody

diagram, the flow is fully turbulent, and the friction factor is a function of therelative roughness alone. Eq. (2.5.13) reduces to

:

J -rolrBj uort.ud

-G

a

!L

I - , , . ' / : \

3E

TABLE 2.5.2Physical properties of water at standard atmospheric pressure

nvonolocrcnrc,cessEs 39

The Moody diagram given here for open channel flow has some limitations

First. it accounts for resistance due to friction elements randomly distributed

on the channel wall, but it does not account for form drag associated with

nonuniformities in the channel. Emmett (1978) found that the friction factor for

thin sheet flows on soil or grass surfaces could be as much as a factor of 10

greater than the value for friction &ag alone. Also, the Moody diagram is valid

inly for fixed bed channels, not for erodable ones. The shape of the cross sdition

(rectangular, triangular, circular, etc ) has some influence on the friction lectorbut the effect is not large. Because ofthese limitations' the Moody diagram shown

should be applied only to l ined channels with uniform cross section

2.6 POROUS MEDIUM FLOW

A porous medium is an interconnected structure of tiny conduits of various shapes

ani sizes. For steady uniform flow in a circular pipe of diameter D, (2.4'9)

remains valid:

16 : yRSy (2.6. 1)

with the hydraulic radius R = D/4. For laminar flow in a circular conduit' the

wal1 shear stress is given bY

( 2 6 2 )

Combining (2.6.1) and (2.6.2)where p is the dynamic viscosity of the fluid.gives

=r !

. i

. : iI

; i: r iI,i] ;r i

i i

,.1; i i

t :. t )[i: ,! ,ri'?f.i

4 ;:lt.iitti.

ii ' l?;F,.f i ,;*,ri

imrlItl' I&ll: !!e iiq: l

*ii lt :'diii

[ j l r:$i.iflililriit".lr;i

:$ltl,rIl:$ll t :iili i :

Temperature DensitySpecifrcweight

Dyramicviscosity

Kinematicviscosity

Vaporpressure

kg/-' N s/ml m'�ls N/m'�abs.

0'c5"C

10"c

20"c25"C3o'c35'C

5o"c60'cio"c80"c90"c

roo"c

98109810!sro98009'�7m97819'�1'l I97519'�732969396439594953594679398

10001000r 000

998997

99198898-197891296595E

L 7 9 x l o 'l . 5 l x L0 - jl . 3 t x l o l

l . l 4 x l 0 r

LoO x 10-l8 .91 x l 0 l

?.96 x 10 "

7 .20 x l 0 {

6 .53 x l 0 -15 . 4 7 x l 0 '4.66 x lo o

4.04 x lo '

3 .54 x l 0 I

3 . l 5 x 1 0 '

2 .82 x 10 o

l . ?9 x l 0 o

t . 5 l x l o - oL 3 t x l o ol . l 4 x l 0 6

l.oo x to -6

8 .94 x l o - '7 .99 x 10 ?'7 .24 /. lO 1

6 .58 x l 0 - i5 .54 x l o 7

4 . 1 4 x t 0 '4 . l 3 r l o '

3 .64 x lO r

3 .26 x 10 r

2.94 t lO-7

6 l I812

1230t 70023403170425056307180

12.30020.00031.20047,40070 ,100

r0 r , 300

slugs/ftj lb/fc lb-sec{t: psra

ro" F50' F60- F70' F80' F

l2o" Fr4o 'F160' Fl8o" F2oo" F2 12" F

3 .23 x 10 - r2 .73 x l 0 - r2.36 x l0 J

2.05 x l0 )

L 8 0 x t 0 51 .42 x l 0 5

i . 1 7 x l 0 5

0 .981 x 10 r

0 .838 x l 0 5

0.726 x lo-50.637 x l0-r0.593 x 10 5

l � 66 x l 0 - "1 . 4 1 x l 0 - 't . 22 t t o 5

l . 06 x l o 5

0.930 x to 5

0.739 x I0 5

0.609 x t0 5

0 .514 x l 0 -50.442 /. t0- '0 .385 x l 0 -50.341 x 10 )

0 .319 x 10 I

L . 9 41.9.11.9,1L 9 lI � 9 il . 9 2L 9 lr .901 . i i 81 . 8 7LE6

62.1367.1062.3162.3062.2262.0061 . t26 l . 3 86 l . 0060.5860 . l 259 .83

0.t220 . l 7 E0.2560.3630.5060.9,19t . 692 .89

7 . 5 1i 1 . 5 314 .70

9!lD

which is the Hagen-Poiseulle equation for laminar flow in a circular conduit

For flow in a porous medium, part of the cross-sectional area A is occupled

by soil or rock strata, so the mtio QIA does not equal the actual fluid velocity'

but defines a volumetric flux q called the Darcy ftLr ' Darc,t's law for flow in a

porous medium is written from (2.6.3) asSo!r.4 Roberson. I \. andC T Crow€, Engineering FlDid Mechanics,2nd ed., HoDghron Mifflin. Bosron,1980, Table A 5, p a,+2. Used with pemi$ion.

For this case. the friction factor f can be eliminated between Eqs. (2.5.14) and(2.5.8) to solve for the relative roughness e as a function of Manning's n andhydraulic radius R:

e = 3 l 0 d n r o r r + ^ r ' k t ( 2 . 5 . i 5 )

, = (zPi'\ t,\ 3 2 P l

( 2 . 6 . 3 )

a , , , - k c -

where d : I for SI units and 1.49 for English units. To use the Moody diagramgiven R and l ' , € is caiculated using (2.5.15) with the given value of n, thenthe Reynoids number is computed using (2.5.10) and the corresponding value ofJ read from Fig. 2.5.1. An estimate of Vis obtained ftom Eq. (2.5.4), and the

(2.6.4)

, , q

n(2.6.5)

Darcy's law is valid so long as flow is laminar. Flow in a circular conduit

where r( is the hydraulic conductivity of the medium' K = yD2/32u. Vaiues of

the hydraulic conductivity for various porous media are shown in Tahle 2.6 1

along with values of the porosiry al, the ratio of the voiume of voids to the total

volume of the medium. The actual average fluid velocity in the medium is

40 .\pprrso gr oRorocr

TABLE 2,6.IHydraulic conductivity and porosity of

unconsolidated porous media

Material

Hydraulicconductivity,( (cm/s)

Porosity1 (E")

GravelSandSi l tClay

l o - r l 0 2lo -5 1to -L to -3t,: l -e-lo 5

25--1025-5035-504A-'�10

Sruce Freeze and Cherry, 1979-

(2.6.6)

is less than 2000, a condition satisfied by almost all naturally occurring flows inporous medla.

Example 2.6.1 Water is percolating tkough a fine sand aquifer with hydraulicconductility l0 I cnl/s and porosity 0.4 toward a stream 100 m away. lf the slopeof the waler table is I percent, calculate the travel time of water to the sffeam.

Solution. The Darcy flux q is calculated by (2.6.4) with l( = 0.01 crr,.isec :

8 .64 rn ' da tandS i = l 7o : 0 .01 ;hence q = K \ : 8 .64x0 .01 : 0 .086m/day .The water velocity % is given by (2.6.5): V, = Sln = 0.086/0.4 : 0.216 nvday.The tra\el t ime to the stream 100 m away is 100/% : 100/0.216 :463 days :

1 .3 yea fs .

2.7 ENERGY BALANCE

The energy balance of a hydrologic system is an accounting of all inputs andoutputs of energy to and from a system, taking the difference between the ratesof input and output as the rate of change of storage, as was done for the continuityor mass balance equation in Sec. 2.2. In the basic Reynolds fansport theorem,Eq. (2.1.9). the extensive property is now taken as B : E, the amount of energyin the fluid system, nhich is the sum of rrternal energy Eu, kinetic energy \mV2,and potential energ\ mgz (: represents elevation):

^ \ Dv

Hence,

B : E : E , + ! ^ v 2 + ̂ g z (2.1 . r)

= ,, + )v' + e, (2.7.2)llR

where e, is the intemal energy per unit mass. By the rtrst law of thermodynamics,

(

4 l

,., 1: : 1; ; li l lo ' lo,* l

0 f20

Laent heal of fusion0.33 x 106 J^c

t t-I

t

dtF: * rd r,te'cLrqurd wa ler wJLer

0 20 40 60 80 100 120Tenpe.ature (" C)

I IGURE 2.7.1Specific and latent heats for water, Latent heat is absorbed or given up when water changer its siateof being solid, liquid, or gas.

the net rate of energy transfer into the fl\id, dEldt, is equal to the rate at whichheat is transfened into the fluid, dHldt,Iess the rate at which the fluid does workon its surroundings, dIl/dt:

dB dE dH (2. '7 .3)

Substituting for dBldt ar,d B in the Reynolds transpoft theorcm

d H d w d t t l 1 . . , t t l; - -

= ; J J ) , " , * 1 v ' I s z t o d v t ) ) t e " t v '

I s z ) p v d A . 1 2 . 1 . 4 '

This is the **r, O"-* equation for un unr*ud, vadable-density flow.

Internal Energy

Sensible h€at. Sensible heat is that part of the internal energy of a substancethat is proportional to the substance's temperature. Temperature changes produceproportional changes in interoal energy, the coefficient of proportionality beingthe specifc heat C o

dwdtdtdtdt

The subscript p denotes that the specific heat is measured at constant prcssure.

Latent heat. When a substance changes phase (solid, liquid, or gaseous state)it gives up or absorbs Latent heat. The three latent heats of interest are thosetot fusion, or melting, of ice to water; for vaporization of liquid water to watervapor; and for subLimation, or direct conversion, of ice to water vapor. Figure2.7.1 shows how the internal energy of water varies as the result of sensible

(

42 APPLTED iri i)R(rlo(l\

and latent heitt transfer. Latent heat transfers at phase changes are indicated bythe venical lumps in internal energy at melting and vaporization lnternal energy

changes due to sensible heat transfer ate shown by the sloping l ines.

Phase charges can occur at temperatures other than the normal ones of 0'Cfor melting and 100'C for boil ing. Evaporation, for example, can occur at anytemperature bclow the boiling point. At any given temperature, the latent heat of

sublimation (solid to gas) equals the sum of the latent heats of fusion (solid toliquid) and vaporization ,(l iquid to gas).

Latent hcat ffanslers are the dominant cause of intemal energy changes forwater in most hydrologic applications; the amount of latent heat involved is muchlarger than the sensible heat transfer for a change in temperature of a few degrees,$'hich is the usual case in hydrologic processes. The latent heat of vaporization

l" varies slightly with temperature according to

t, : 2.501 x 106 - 23?0T (J,&g) (2.1 .6)

where I is tcmperatlrre in 'C and I, is given in joules (J) per kilogram (Raudkivi,

1979). A jouie is an Sl unit representing the amount of energy required to exerta force of I newton through a distance of I rneter.

2.8 TRANSPORTPROCESSES

Heat energl transport takes place in thrce ways. conduction, convection, andradiation. Conduction results from random molecular motion in substances; heatis transferrecl as molecules in higher temperature zones collide with and transferenergy to molecules in lower temperature zones, as in the gradual warming alongan iron bar \rhen one end is piaced in a fire. Convection is the transport of heatenergy associated with mass motion ol a fluid, such as eddy motion in a fluidstrcam. Con\,ection transpofis heat on a much larger scale than conduction influids. but ifs extent depends on fluid turbulence so it cannot be characterized asprecisely. Radiation is the direct transfer of energy by rneans of electromagneticwaves. and can takc piace in a vacuum.

The conductjon and convection processes that transfsr heat energy alsotransport niass and nomentum (Bid, Stewart, and Lightfoot, 1960; Fahien,1983). Fo| each of the extensive propenies mass, momentum. and energy, therate of flou of extensive property per unit area of surface through which it passesis cailed the lrr-t. For example. in Darcy's law, volumetric f low mte is O acrossarea A. so the voiumetric f lux is 4 = 0/A: the corresponding mass flow rateis n : pQ. so the mass flux is pQlA. By analogy the momentum flor rate isnV : pQV end, the momentum flux is irVl,4 : pQVlA = pVr. The correspondingenergy flov rate is .18l d/ and the energy flux is (dE ldt)/A, measured in watts permeter squa.ed in the SI system: a watt (W) is one joule per second. In general,

a flux is gircn b1

E I . , -flow rate

area( 2 . 8 . 1 )

H\ OROI Or te PPO. , . \F . 4J

Conduction

ln conduction the flux is directly proportional to the gradient of a potential(Fahien, 1983). For example, the lateml transfer of momentum in a laminar flowis described by Newton's law of viscosifj, in which the potential is the flowvelocity:

du

Here 7 is momentum llux, trr is a proportionality coetlicient called the{ynamicviscosity (measured in lb.s/ft2 or N.s/m2), and, duldz is the gradient of the velocityu as a function of distance z from the boundary. The symbol r is usually used torcpresent a shear stless, but it can be shown that the dimensions of shear stress andmomentum flux are the same, and r can be thought of as the lateral momentumflux in a fluid flow occurring through the action of shear stress between elementsof f luid having different velocities, as shown in Fig. 2.8. 1�

Analogous to Newton's law of viscosity for momentum, the laws of con-duction fbr mass and energy arc Fick's law of diffusion, and Fourier's law ofheat conduction, respectively (Carslaw and Jaeger, 1959). Their goveming equa-tions have the same form as (2.8.2), as shown in Table 2.8.1. The measure ofpotential for mass conduction is the mass concentration C of the substance bcingtranspo(ed. In Chap. 4, for example, when the transport of water vapor in air isdescribed, C is the mass of water yapor per unit mass of moist air. The proportionality constant for mass conduction is the diffusion coeficient D. The measureof potential for heat energy transport is the temperature I and the proponionalityconstant is the heat conductivity & of the substance.

The proportionality constant can also be written in a kinematic form. Forexample, the dynamic viscosity /, and the kinematic viscosity z are related by

t r (2 .8.2)

(2.8.3 ' )

FIGURE 2.8.7The relationship between the momentum flux and the velocily gradient in a free su.face flow.Momentum is transfered b€tween rhe wall and the interior of the flow rhrough molecular andturbulent eddy motion, The sbear sFess in the interior of the flow is the same as the momentum fluxthrough a unit area (dashed line) parallel to the boundary.

-

4i l ePPueo r l ; -eoroc:

. E - a q ) . " h . r e $ r i l l e n5 u L 9 ' ! u - i

du, r = p v 7aa

The d imen. to ' r . o t u . r re l l T l '

dur** : pK^

dz.

Convection r

For convection. transiort occurs thrcugh the action of lurbulent eddies' or the

*"ur, ,nou"."n, of elements of fluid with different velocities' rather than through

,f'r.-rnou"rn"n, ol individual molecules as in conduction' Convection requires a

n"*i.g n"i,t. while conduction does not The momentum flux in a turbulent f low

is not louemerl by Newton's law of viscosity but is related.to the .instantaneous

deoanire, r)i the lufbuient veiocity from its t ime-averaged value lt is convenrent'

h;'"; ' ;r, , r 'r i le cquatton< de'cribing con\eclion in the \ame fonn as.those Ior

conduction. For momentum transfer' the flu\ ln a turbulent t low ls wrl[en as

( 2 . 8 . 4 )

(2.8.s)

TABLE 2.8.Iia"*"r-# .ottaoction and corresponding equations for convection of mass'

momentum, and heat energY in a ffuid

Extensive propertY transported

Momentum Heal energy

Conduction:Name of l.r$

Equatron

Fiux

Constant ofproponronalrty

Potent ia l Sradrent

Conv€ction:

Equation

Dif fusi \ r$ [Lrnl

Fick s

dC- (11

D(diffusion coeff.)

dC

e

dC- 4 ?

K"

Newton s

dtt

aa

r

(v iscosi ty)

dua

aa

K-

dT- tl.'

(heat conductivity)

dTfn = PC pKn t,.

Kn

dTa

(sYonor-oclcPncressrs 45

where K, is the momentum dffisivity' or eddy viscoslry, with dimensions [L 2/T]

f, t fout to six orders of magnitude greater than z (Priestley' 1959)' and

iuibul"n, -oa"n,urn flux is the dominant form of momentum transfer in surface

rui"t no* and in air flow over the land surface Equations analogous to (2 8 5)

can be written for mass and energy transport as shown in Table 2 8 1'

It should be noted that the direction of transport of extensive proPerties

described by the equations in Table 2.8 1 is uansverse to the direction of flow'

For example, in Fig. 2.8.1, the flow is horizontal while the transpoJgrrocess

i, u"rti"ai *rough ihe dashed area shown Extensive propeny transP?ft in the

ai.""tion of -otion is called advection and is described by the term I lBpV dL

ir the Reynolds transport theorem, Eq (2 1 9)'

Velocity Profile

Determinationoftheratesofconductionandconvectionofmomentumrequlresknowledge of the velocity prcfile in the boundary layer' For flow of air over land

o, *u,.t,",tt" Iogarithmic tilocity profle is aPplicable (Priestley' 1959) The wind

velocity n is given as a function of the elevation z by

( 2 . 8 . 6 )

where the shear velocity "- = nFd p (70 is the boundary shear stress and p is

the fluid density), ,t is von Karman's constant (= 0 4), and z0 is the roughness

ieight of the surface. Table 2.8.2 gives values of the roughness height for some

suiaces. By differentiating (2.8 6), the velocity gradient is found to be

. ;

:

i ) '

II

lii

u I - l z \u K \ Z o l

du u'

dz kz(2.8.'�])

turbulent momentumThis equation can be used to determine the laminar and

fluxes at various elevations.

TABLE 2.E.2Approximate values of the roughness height of

natural surfaces

Roughness height zo(cm)

Ice, mud flals

Grass (up to 10 cm high)Grass (10 - 50 cm high)Vegetation (1 - 2 m high)Trees (10 - 15 m high)

-tdu/cei Brursaelt. W., Evaporatiotr into the atmosphere D Reidel' Dor'

drechr, Holland, 1982, Table 5.1, P ll'1(adapted).

0.0010.01 - 0.060 .1 - 2 .0

2040 -

'70

{6 ,\PPLIE|] 31 IJROLO'J\'

Eranrpre 2.8.r rhe urnd'peed ha:,:::;.T:';:lllrlH,l;.,'li.L:';i"rl"t,.,labo!" - .orr .ra" l ield "

:T-'; l ;; lo ... .na rhe rurD-ulenr T91.n,u'o t r J n n r d n J r u r b u l e n l m o m e n r u m ; ' i ;

; , . ; . . i ; i t 0 ' m : : . a n d K - - 1 , 5

f lu r J t I me le 's t ion Fora t r ' P= t z t

ml/s' Ioclt '

Solatior. The shear velocity is calculdted from Eq (2 8 6) using the known ve

r , | ' r l - 2 m :, #= i " l ; )

! i r 2 \- o . 4 l o o l r

5 u . r - , i 0 . : : 6 m ' - , . . ^ . ^ " i n , ) R h , . r . ) r e r d m p l e .' " , ; " 1 . , c r l ) p ro l r l e i s l oundb \ ' uD ' l l t r l r ng \ a l ue ' f o r ' i n ' 2 8 6 ' : i o t e ram

f o r : = 2 0 ! m = 0 2 m t h e n

" 1 i 0 . 2 \

0 2 2 b 0 4 u u l

S . r . . r 9 . r r I I m \ a ! : - 0 2 m 5 r m l z u l r c o m p u t c d r a l u e t f o r o l h e l l a l u e ' o f

: - c p l o l r . ' ' n F ig 2 8 2 r i t t " " ' o t i i 1 : *p "uo ' " * ' r t : - - 0 I m Iq e ' \ en b ) t q

( ' S l l r

= ! , = o " . u ^ = 2 . 8 i s - r

A . 0 ' + ' u l

r,- i,,e rdn,indr .oT"llul tl) : l*''-:l"H",xl,I:'],i: :t,:ll* i'; ,tf,i$L lh rL r den ' r t l ! = I 20 kg /m ano tne

dnj : p v ;

aa

: 1 . 2 0 x 1 5 1 Y 1 9 5 r 2 8 3

:5 .1 x 1o -5 N /m2

ai : = 0.1 m. The turbuient momemum flux is given by Eq (2 8 5)i

", ' , ,0=PK^#

: r ' ) o x l 5 x 2 8 3

=5 l N /mz

rr : = 0 I m rhe ratro 'i,'bl r =r{-'i,ili,1ll;ii 13,"'1";J3";|iii?.l"iJ']i.xlr , ,nenrumf lur - :1" . ' j i o l ' : : ) r : _ ' , : , i , - o : )0. (0. , r .2 , - 0 .28 s- and. r I n i . d . r - l r e rm \ t : - : r r '

: " "

O . :S l ' 0 .S t . \ .m - . No re t hd t r he ra t i o o t t he. " ! . K - , d u r . , - ' - 2 9 . . . 1 . , t : ] , ' " ; i o ; , r s . i o . s r _ r 0 : r h e m o m e n r u m

'ur \e' t ' c momenlum l '"-"t l l l - ' ' r"";"""r to t t t 'o ' ion ;n a logan lhm rc-! e loci i ]i1J\ (or . leJhIIe" ' is in!ersel) proPor ' ru' id'

, tu-^! 1. ' *"1.

' i " " . i ) l l^a surface and

o,L, ' ia ' . t i . .oa.ntum f lux lherelore ts large' l near lhe ground suflac

i l inr inishes as ele\at ion increases

du

d.

(

uyonorocrcpnocesses 47

u l

!1 = o.zs .-'

* = z . s , '

, {

FIGURE 2.8.2 t:Logarithmic velocity profile forroughness height ro = I cmand measured velocity 3 rn/s atheight 2 m in a flo$ of air(Example 2.8. 1) . The resul t ingvelocity gradient and shear stressare inversely proportlonal to2 3 4

Velocilylr (nvs)

Radiation

Wheo radiation strikes a surface (see Fig. 2.8.3), it is either reflected or absorbed.The fraction reflected is called the albedo a (0 < a < 1). For example, deep waterbodies absorb most of the radiation they receive, having a : 0.06, while fresh

snow rcflects most of the incoming radiation, with d as high as 0.9 (Brutsaert,

1982). Radiation is also continuously emitted from alt bodies at mtes dependingon their surface temperatures. The net radiation R, is the net input of radiation at

the surface at any instant; that is, the difference between the radiation absorbed'Ri(l - d) (where R; is the incident radiation), and that emitted, R":

( 2 . 8 . 8 )i R , : r R i ( l - o ) - R ,

Net radiation at the earth's surface is the major energy input for evaporation ofwater.

Emission. Radiation emission is govemed by the Stefan-Boltzmann law

( 2 . 8 . 9 )

where e is the emlssiviry of the surface, o is the Stefan-Boltzmann constant (5 67xl0-8 W/m2'K4) and T is the absolute tempemrure of the surface in degreesKelvin (Giancoli, 1984); the Kelvin temPeraturc equals the Celsius temPeratureplus 273. For a perfect mdiator, or black body, the emissivity is e = l: for watersurfaces e : 0.97 , The wavelengt& l of emitted radiation is inve$ely Proportionalto the surface t€mperature, as given by Wien's law.

. 2 .90 x 10-3T

(2.8. 10)

where ? is in degrees Kelvin and i is in meteN (Giancoli, 1984) As a

48 ,qpprrEl rr onocr ct 49

SPACE

ATMOSPHERE

l 6

vapor. dust. Or

Incoming

radiat ion

100

Oulgoing radiation

Shonwave Longwave

6 1 8 2 6

Ner fadiltion.absotbed iFIGURE 2.8.3Radiation balance rt the surface

Backsca(eredby arr

t --\

: [ J J ]by clouds

Reflectedby clouds

consequence ol Wien's law, the radiation emitted by the sun has a much shorterwavelength than that emitted by the cooler earth.

Reflection and scattering. The albedo a in Eq. (2.8.8) measures the proportionof incoming radiation that is reflected back into the atmosphere. The albedo variessomewhat depending oo the wavelength ofthe radiation and its angle ofincidence,but it is customa$ to adopt a single value typical ofthe type of surface.

When radiation stikes tiny panicles in the atmosphere of a size on thesame order of ma-gnitude as the radiation wavelength. the radiation is scatteredrandomly in all directions. Smail groups ol molecules called aeloiols scatter l ightin this wa1. The addition of aerosols and dust panicles to the atmosphere ftomhuman activit)' in modern times has given se to concem abo:ut the greenhouseej&cr, in u,hich some of the radiation emitted by the earth is sca[tered backby the arnosphere; increased scattering results in a general warming of theeafth's surface. However, the precise magnitude of the earth s warming bY rhismechanism is not yet known.

Net radiation at Earth's surface. The intensity of solar radiation affiving at thetop of the atmosphere is decreased by thrce effects before reaching a unit arerof the earth's suriace: scattering in the atmosphere, absorption by clouds, andthe obliqueness oi the earth's surface to the incoming radiation (a function oflatitude, season, and time of day). The intensity of solar radiation received perunit area of the earth's surface is denoted by R,. The atmosphere also acts as aradiator. especiall i ' on cloudy days, emitting longer wave radiation than the sunbecause its temperature is lower; the intensity of this radiation is denoted R7. Theincoming radiation at the earth's surface is thus Ri : R. + R1. The earth emitsradiation R.. (of a wavelength close to that of the atmospheric radiation). and thenet radiatioo received at the earth's surface is

R , : ( R j + R r ) ( l a ) R " ( 2 . 8 . 1 1 )

The interaction of radiation processes between the atmosphere and theearth's sur-face is complex- Figure 2.8.4 presents a summary of rclative valuesfor the larious components of the annual average atmospheric and surface heatbaiance. Ir can bc seen that fbr 100 units of incoming solar radiation at the top ofthe atrnosphere, about half (51 units) reaches the earth's surtace and is absorbedthere; of these 5l units. 2l are emitted as longwave radiation, leaving a ner

ocEAN, LAND 51 2t 7 2l

FIGURE 2.E.4Radiation and heat balance in the aimosphere and at th€ eanh s surfac€. (Sourc€. "Understandin-e

Climatic Change," p. 18, National Academy of Sciences. Washington. D.C.. 1975. Used wilh

radiation of 30 units at the eanh's surface; 23 units of this energy input are usedto evaporate water, and thus retumed to the atmosphere as latent heat fluxi theremaining 7 units go to heat the air overlying the earth's surface. as sensible heatf lux.

REFERENCES

Bird, R. 8., w E. Stewart, and E. N. Lightioot, Transport Phenomena, Wiley, New York, 1960.BruBlsaeft, W., Evaporation into the Atmosphere, D. Reidel. Dordrecht, Holland, 1982.Carslaw. H. R., and J. C. Jaeger. Conduction of Heat in Solids. 2nd ed., Oxford University Press,

Oxford. 1959.Chow, V. T., Open-channel HJ&aulicr. Mccraw-Hill, New York, 1959.Ernmett, W. W-, Overland flow, rn Hilblope Hydrology. ed. by M. J. Kirkby, Wiley, Chichesler.

England, pp. 145-176, 1978.Fahien. R. W.. Fundamentals of Trarsport Phenomeur, Mccraw Hill, New York, 1983.Fox, R. W., and A. T. MacDonald, lntroduction to Fluid Mechanics,3rd ed., Wiley, New York,

1985.Freeze. R. A., and J. A. Cherry, Grounllwater. Prentice-Hail, Englewood Cliffs, N.J.. 1979.French, R. H., Open-channel Htdraulics. Mccraw Hill, New York, 1985.Ciancoli. D. C.. General Physics, hentice-Hall, Englewood Cliffs, N.J., 198,1.Henderson. F. M., Open Channel Flo\r. Macmiuan, New York, 1966.Oke, T. k., Boundary laler Clinates, Merhuen, London, 1978.Prieslley, C. H.8., Turbulent Transfer in the Iawer Atmosphere. Universiiy of Chicago Press.

1959.

e vapor, CO" La!enr

by c louds

longwave radiation

Sensible

\\\\\\\\\

52 apprrco nrororocv

2.5.11 Use the piirabolic velocity djst.ibution formula for laminar flow in a circularconduir. given in Prob.2.5.10. to establ ish that the wall shear stre(\ is rLr =

8/-. !7D. in which 7 is the average pipe velocity.

2.5.12 A rectangular open channel 12 m wide and 1m deep has a slope of0.001 and isl ined with cemented rubble (n =0.025). Determine (d) i ts maximum dischargecapacity, and (r) the maximum discharge obtainable by changing the cross-secrional dimensions without changing the rectangular form of the section, theslope. and lhe volume of excavation. Hint: the best hydraulic rectangular sectionhas a ninimum yelied perimeter or a width-depth ratio of 2.

2.5.13 Soive Prob. 2.f12 i f the channel is 30 ft wide and 4 ft deep.2,6.1 waier is flolving with a friction slope S/ = 0.01. Detemine ((7) the velocity

of tlow in a lhin capillary tube of diameter I mm (, = 1.00 x 10 6 m:/s), 1b)the Darcy flux O/A and tle actual velocity of flow through a fine sand and (c)gra!el.

2.6,2 Cornpute thc rate of f low of warer at 20"C through a lo-m-lon8 conduit f i l ledwith flne sand of effectjve diameter 0.01 mm under a pressure head dilference of0.5 m between the ends of the conduit. The crcss-sectional area of the conduiti s 2 .0 m2 .

2.6,3 Solve Example 2.6.I in the text i f the water is f lowing through: (d) gravel witha hydraulic conductivity of 10 cm/s and a porosity of 30 percent, (r) silt with ahydraulic conductivity of 10-a cm/s and a porosity of 45 percent, and (c) claywift a hydraulic conductivity of 10 7 cm,/s and a porosity of 50 percent. Compareyour answerc with that obtained in the example.

2.4.1 Air is flowing over a short grass sudace, and the velocity measured at 2 mele\ation is I nl/s. Calculate the shear velocity and plot the velocity profile fromthe surface ro height 4 m. Assume 36: I cm. Calculate the turbulert momentumflux at heights 20 cm and 2 m and compare the values. Assume Ja. = 0.07 m2/sand p : 1.20 kg/m3 for air.

2,8,2 Solve Prob. 2.8.1 i f the f luid is water. Assume K. :0.15 mz/s and p :1000

kg/mr. Calculate and compare the laminar and turbulent momentum fluxes at 20cm ele\at ion i f v - l .5l l0-6 m2/s lor waler.

2.8.3 Assuming the sun to be a black body radiator with a surface temperature of 6000K, calculate the intensity and wavelength of its emitted radiation.

2.8.4 Solve Prob. 2.8.3 for the earth and compare the intensity and wavelength of theeanh s radiation with that emitted by the sun. Assume the earth has a surfacetemperature of 300 K.

2.8.5 The incoming radiat ion inrensity on a lake rs 200 w/mr. Calculare rhe net radiat ioninto the lakc i f the albedo is a = 0.06. the suface temperature is l0'C, and theemissivity is 0.97.

2.8.6 Sol\e Prob. 2.8.5 for fresh snow if the albedo is a =0.8, the emissivity is 0.97,anJ rhe .Lf lrce temperature is 0'C.

2.8.1 Solve Prob. 2.8.5 for a grassy f ield with albedo tr = 0.2, emissivity 0.97, andsuafrlce Iempeaature 30oC.

CHAPTERa1

i {

ATMOSPH€NTCWATER

Of the many meteorological processes occurring continuously within the atmo-sphere, the processes of precipitation and evaporation, in which the atmosphereinteracts with surface water, are the most imponant for hydrology. Much of thewater prccipitated on the land surface is derived from moisture evapomted fromthe oceans and transported long distances by atmospheric circulation. The twobasic driving forces of atmospheric circulation result from the rctation of theearth and the transfer of heat energy between the equator and the poles.

3.I ATMOSPIIERIC CIRCULATION

The earth constantly receives heat from the sun tfuough solar radiation and emitsheat through re-radiation, or back radiation into space. These processes are inbalance at an average rate of approximately 210 W/rn2. The heating of the earthis uneven; near the equator, the incoming radiation is almost perpendicular tothe land surface and avemges about 270 W/m2, while oear the poles, it strikesthe earth at a more oblique angle at a rate of about 90 W/mz. Because the rateof radiation is proponional to the absolute temperature at the earth's surface.which does not vary geatly between the equator and the poles, the earth'semitted radiation is more uniform than the incoming radiation. In response to thisimbalance, the atmosphere functions as a vast heat engine, transferring energyfrom the equator toward the poles at an average rate of about 4 \ lOe MW.

If the earth were a nonrotating sphere, atmospheric circulation would appearas in Fig. 3. l. I . Air would rise near the equator and travel in the upper atmospheretoward the poles, then cool, descend into the lower atmosphere, and rcturn towardthe eouator. This is called Hadley circulqtion.

54 rpeLl tu l l l l , t l l t ' ; r

I IGURE 3.I.1One ceil atmospheic circulalion Pattem for

a nonrotating Phnel

The fotation of the earth from west to east changes the circulation Pattem

e. u rlnfoiui, ut*,t the earth's axis moves toward the poles' its radius decreases'

in o.o"i o,ttuinroin angular momentum the velocity of air increases with respect

;; ;;;;; t;;... thJs producing a westerly air flow The converse is true for

;ri;;;;;t;";r"g towaid the equator it forms an easterly air f low The eifect

oroducinq these changes tn wtno olrection and velocit) ' is known as the Coriolis

force.The actuai pattem ot atmospneric circulation has three ce1ls in each hemi-

,ph"...- ", ,tto*n'in Fig. 3.1.2' l i ttre tropical cell ' heated air ascends at the equa

;;;.-;;;;;;,";"td th'-e poles at upper levels' loses heat ald deTTl:.,t:ward ttte

nrot'J ut t.tltua" 30' l'iear the ground' it branches' one branch moving toward

i;;;;;;il; other towarithe pole ln the polar cell' at rises at 60' and

flows to\\ard the poles at upp""tu"^l ' then cools and flows back to 60' near

i '." *n',, su.fa.". The mlii le cell is driven frictionally by the olher t$o: its

. " ' r " . . , :L i ' t . lov ls toward thepo le 'p roduc ingpreva i l ingwester iya i r f low in themid-latrtudcs.- -il;

uneven distribution of ocean and land on the earth s surface' coupled

*irn,n.i. Jiti.."nt thermal propenies creates additional spatial variation in atmo-

;;;#;';;i;;; 'irt.

"niuoi shifting ol the thermal equator du.e to.the earth's

revolution around the sun causes a c-onesponding oscil lation.of the three cell

.ir*Lorlon pattern. With a larger oscillation' exchanges of air between adjacent

."r1. ion UJ -ut. frequent and complete, possibly resulting in many flood years

Also. monsoons may advance deeper into such countries as.India and Austmiia'

With a smaller oscil latlon' tnt"n" high pressure m'1y buiid up around 30' lati-

tude, thus creating extended dry periods Since the atmosphenc clrculatlon ls very

complicated, only the general pattem can be identif ied.'The

atmosihere is divided verticaily into various zones The atmosphenc

circulatlon described above occurs in the troPosphere' which ranges in height

from ab,lttt E km at the poles to 16 km at the equator' The temperature in the

i..,.,n.nt "r. decreases with altitude at :r rate varying with the moisture content of

55

Tropical St.atosph€re

Troposphere

rropopause -------__\EEI

FIGURE 3,I.2Lalitudinal cross section of lhe geneml almospheric circulation

the atmosphere. For dry air the mte of decrease is cailed the dry adiabatic lapse

r4te and is approxirnately 9.8'C,&m (Brutsaefi, 1982). The saturated adiabatic

lapse rate is less, about 6.5'C/km, because some ofthe vapor in the air condenses

as it rises and cools, releasing heat into the surrounding air. These are average

figures for lapse rates that can vary considerably with altitude. T\e trcpopause

separates the hoposphere from the strdtosphere above. Near the tropopause' sharp

changes in temperature and pressure produce strong nallow air cunents known

as jet streams with speeds ranging from 15 to 50 rn/s (30 to 100 miAr). They

flow for thousands of kilometers and have an important influence on air mass

movement.An air masJ in the general circulation is a large body of air that is fairly uni-

form horizontally in propenies such as temperature and moisture content. When

an air mass moves slowly over land or sea areas, its characteristics reflect those of

the underlying surface. The region where zrn air mass acquires its charactenstrcsis its source region; the tropics and the poles are two source regions Wlere a

warm air mass meets a cold air mass, instead of their simply mixing' a definite

surface of discontinuity appears between them, called a front. Cold air, being

heavier, underlies wa-rm air. If the cold air is advancing toward the warm air' the

leading edge of the cold air mass is a cold front and is nearly vertical in slope lf

the warm air is advancing toward the cold air, the leading edge is a v'drm front'which has a very flat slope, the warm air flowing up and over the cold air.

A cycLone is a region of low pressure around which air flows in a coun-

terclockwise direction in the northern hemisphere, clockwise in the southem

hemisohere. Tronical ctcloner form at low latitudes and mav develop into

56 ,rpprirr in oaoL-ocy

CDld a i r

i.a)

7-- \ Cotd air

/ / i ' ' , \ ' \ ' , ,

t l . / /' , . " ' / - . t / r

/.,,"r!'"1l n r ( e ) ( f )

FIGURE J. I .3A plan vie" ol rhe liic c)clc of r Nofthem Hemisphere frontal cyclonei (d) suriace tronr berscrn loruand wam r i f : ( i , ) wr\c bc-qinning to fonni ( . ) cyclonic c i rcutar ion and wave have devetopedt (u/) tas lermo!ing ciJ|i lrurr rs ovenaking refeating warm front and .educing wann secrori (.) walm sector hasbeen el1mrn,r l . .1 rnd r i r c lc lone is d iss ipat ing.

hurricanes v r)-phoons. ExtalropicaL cycLones are forrned when warm and coldair masses. initially flowing in opposite directions adjacent to one another, beginto interact and r^hirl together in a circular motion, creating both a warm frontand a cold front centered on a low pressure zone (Fig.3.1.3). An anticycloneis a region of high prcssure around which air flows clockwise in the nonhemhemisphere. counterclockwise in the Southem hemisphere. When air massesare lifted in atmospheric motion, their water vapor can condense and produceprecipitation.

3.2 WATER VAPOR

Atmospheric water mostly exists as a gas, or vapor, but briefly and locally itbecomes a liquid in rainfall and in water droplets in clouds, or it becomes a soliciin sno*fall. in haii, and in ice crystals in clouds. The amount of water vapor nithe atmosphere i! less than I part in 100,000 ofall the waters of the earth, but itpiays a vital role in the hydrologic cycle.

Vapor transpot in air through a hydrologic system can be described by theReynolds transport theorem [Eq. (2.1.9)] lett ing the extensive properry B be the

Cold ai r

-/^',,

r#&(.)

/,l""_\r G : z ' \\ \ ' " l h / -\\ ro,/ ,. '

\ \-r"-- --l

V--*^;€"

(

mass of water vapor. The intensive propefiy B = dBldm is the mass of watervapor per unit mass of moist air; this is called the spe cif.c humidity q ", and equalsthe ratio of the densities of water vapor (p) and moist air (po):

p,

By the law of conservation of mass, dBldt = la,, the rate at Fbich watervapor is being added to the system. For evaporation from a watef $irface, rn,is positive and represents the mass flow rate of evaporation; confersely, forcondensation, z, is negative and represents the rate at which vapor is beingremoved from the system. The Reynolds tansport equation for this system is thecontinuity equation for water vapor transport:

( 3 . 2 . 1 )

Vapor Pressure

Daltoll's law of partial pressures states that the pressure exened by a gas (itsvapor pressure) is independent of the presence of other gases; the vapor pressuree of the water vapor is given by the ideaL gas Lar, as

( 3 . 2 . 2 )

e = p,R,T ( 3 . 2 . 3 )

p - e = p a R a T ( 3 . 2 . 4 )

r:fn * (u,"-.Jl^,.

q" : O.6229' p (3.2.6)

. : l f f f l1 | | | o "o " av + | | q ,o "v 'a td t J J J " - J J

'

where Z is the absolute temperature in K, and R" is the gas constant for watervapor. If the total pressure exened by the moist air is p, then p - e is the partialDressure due to the drv air. and

where p7 is the density of dry air and Ra' is the gas constant for dry air (287J&g K). The density of moist air po is the sum of the densities of dry air andwater vapor, that is, po = p,t + pu, and the gas constant for water '"apor isR" = Ra10.622, where 0.622 is the ratio of the molecular weight of water vaporto the average molecular weight of dry air. Combining (3.2.3) and (3.2.4) usingthe above definitions gives

By taking the mtio of Eqs. (3.2.3) and (3.2.5), the specific humidity 4, isapproximated by

Also. (3.2.5) can be rewritten in terms of the gas constant for moist air. Ro, as

P : P . R . T ( 3 . 2 . 1 )

58 epprreo nt i :eLtoc,

The relationship between the gas constanB for moist air and dry air is given by

R,=Ra(1 + 0.608q")

:281( t + 0.608q") J&g K

The 8as constanr of moist air increases with specific humidit] ' , but even fora large specific hum;dity (e.g., 4, = 0.03 kg warer/kg of moist air), the differencebetween the gas constants for moist and dry air is only about 2 percent.

For a giren air temporature, there is a maximum moisturc content the aircan hold, and the corcponding vapor prcssure is called the satLtration vaporpressure e,. At this vapor pressure, the rates of evaporation and condensationare equal. O\er a water surface the saturation vapor pressurc is reiated to the airtemperature as shown in Fig. 3.2.1; an approximat€ equation is:

( 3 . 2 . E )

l t 1 2 1 T \e r : o l I exp l i ; : _ l

\ t J t ) - f t l( 3 . 2 . 9 )

where e, is in pascais (Ih : N/m2) and Zis in degrees Celsius (Raudkivi, 1979).Some values of the saturation vapor pressure of water are l isted in Table 3.2.1.

The gradient A : de,ldT of the saturated vapor pressure curve is found bydiffercntiatins (3.2.9)i

" 4098e,- (23'�7 .3 + D1

(3 .2 . r0 )

t lhere ,\ is the gradienr in pascals per degree Celsius.The relati,,e hwnidtty Rp is the ratio of the actuai vapor pressure to its

saturation value at a given air temperature (see Fig. 3.2.1):

( 3 . 2 . 1 t )

The temperature at which air would just become saturated at a given specifichumidity is lts de)\,-point temperature Td.

Example 3.2.1 .\r a climate station, air pressure is measured as 100 kpa, air tem-perature as 20"C. and the wer-buib, or dew-point, temDeraiure as l6oc. Calculate

. t C

; T

10 0 i0 20 10 40Tcmpenrure 1"C)

FIGURE 3.2.1Saturated vapor pressure as a function oflempeftture over waier. Point C has vaporpressure ? and temperalure 7, for which thesahrated vapor pressure is ar. The relativehumidity is R, = €/€". The temp€ralure a!\ Ih ich the " i r r :5atuared for !apor prer \ure,is the dew-poinr temperature 7,./.

59

the coresponding vapor pressure, .eiative humidity, specific humidity, and ajrdensity.

Solution. The saturated vapor pressure at I: 20oC is given by Eq. (3.2.9)

1 1 1 . 2 7 f 1e J : o t r e x p l ^ - - - -

t ] . 2 1 . 2 0=o i I exP l -L . r? - r + 20 l

:2339 Pa

and the actual vapor pressure € is calculated by the same method substituting thedew-point temperature Ta: 16'C:

I t t - t ' t r te - 6 l l e \ p

' , , ' " ,

' I

\23'7 .3 + Ttl

:e ,r *r l{fr1{1: 1 8 1 9 h

The relat ive humidity from (3.2. I l ) is

l 8 l 92339

: 0 . 7 8

= 18Vo

tt

-.1

TABLE 3.2.ISalurated vapor pressure of $alervapor over liquid water

fn"""tu.eSaturated Vapor Pressur€Pa

-:0-10

U5

i01 5202530l5.10

1252866 1 1g'�t2

tz2'7t"/04

3t6142435624'73'�78

S,arcer Btu$aert, 1982, Table 3.:1. p.41. Used with peG

60 eppruo svonoLocr

and the specif ic humidity is given by (3.2.6) with p : lqg kPa = 100 x 103 Pa:

q' = o 6zz!p

=u.urzl l8le , l- \ 100 x 103/

:0.0113 kg water,&g moist air

The aif densiry is ialculared from the ideal gas law (3.2.7). The gas constanr Rois g iven b ! (3 .2 .8 ) w i th 4 . = 0 .0113 kg /kg as R, = 287( l + 0 .6084, ) :28 '7 (1 +0.608 x 0.01 13) = 289 J,&C.K, and I = 20'C = (20 + 2'13) K = 293 K, so rhar

- p' ' R . T

100 x l0l= 2Bg " zg3

= 1 .18 kg /m'

Water Vapor in a Static Atmospheric ColumnTwo laws govern the properties of water vapor in a static column,law

p = p"R"T

and the ,l)d7-osrafic pressure law

dp

dz PaE

The variation of air temperature with altitude is described by

dT

the ideal gas

{3.2. t2)

( 3 . 2 . 1 3 )

( 3 . 2 . 1 4 )

where a is the lapse rate. As showl in Fig. 3.2.2, a l i l€ar temperature variationcombined with the two physical laws yields a nonlinear variation of pressure withaltitude. Density and specific hurnidity also vary nonlinearly with altitude. From(3.2.12), p" = plR.T, and substituting this inro (3.2.13) yields

dp -pc

dz R.T

OI

ap l_s \ ,p \n"rJ "'

Substitutins Ll; : dTla from (3.2.14):

Column

6l

FIGURE 3.2.2Pressure and temperature variation in an amospheric column

d p I s \ a ri

: \"R.1 ,

and integmting both sides between two levels I

'H =(*)'l f 'dan'p , = P t \ i l

and 2 in

(''lthe atmosphere gives

(3 .2 .1s )

Precipitable Wate r

The amount of moisture in an atmospheric columr is called its precipitable v'ater'

Consider an element of height dz in a column of horizontal cross+ectional area

A (Fig.3.2.2). The mass of air in the element is paAdz and the mass of water

contained in the air is q,poAdz. The total mass ofprccipitable water in tbe column

between elevations zt and 22 is

From (3.2.14) the temperature variation between altitudes z I and 12 is

T z : T r - q ( 2 2 - z ) (3.2. t6)

*o = [-',' n,r"oa, ( 3 . 2 . 1 7 )

62 lppuro Hronorocr

The integral (i.2.17) is calculated using intervals of height Az. each withan incremental mass of precipitable water

Lmp = q,P.AL. ( 3 . 2 . l 8 )

where 4" and po are the average values of specific humidity and air density overthe interval. The mass increments are summed over the column to give the totalprecipitable water.

Example 3.2,2. Cflculate the precipitable water in a saturated air column 10 kmhigh above I nr of ground surface. The suface pressu.e is 101.3 kPa, the surfaceair temperature is 30'C, and the lapse rate is 6.5'C,&m.

Solution. The rcsults of the calculation are summarized in Table 3.2.2. The incre-ment in clevation is taken as Az = 2 km = 2000 m. For the f irst increment, atz r = 0 m , I r = 3 0 o C = ( 3 0 + 2 ' t 3 ) K = 3 0 3 K ; a 1 ; r = 2 g g g r n . b y E q . ( 3 . 2 . 1 6 )using a -� 6.5'CAm : 0.0065'C/m.

T 1 : T t - a ( a | - z \ )

:30 , 0.0065(2000 0)

: l'l'C

TABLE 3.2.2Calculation of precipitable water in a saturated air column (Example 3.2.2)

Elevation Temperature

(k'') fc) fK)

4Airpressure

(kh)

Density Vaporpressure

Gdm) Gh)

30 303t'7 2904 21'�79 261

35 238

02

6E

l 0

1 0 1 . 380.4

,19.13'�7 .628 .5

l . 1 60.9'70. '790 .65o.52o.42

1.24l .940 .810 . 1 I0 . 1 00.03

1Specifirhumidity

{kc^c)

10IncrementslmassLm(kc)

E 9Average overlncrementq, p.(ks&c) Ggmr)

l l1o oftotalmass

0 .02610 .01500.00800.00390 .00 r70.0007

0.02050 . 0 1 1 50.01J600.00280 .00 r2

| . 010 .880 .720.590.4'7

13.120.28 . 63 . 3L L

7'�7.0

51

i 142

erraosensnic urrrn 63

= (r7 + 273) K

:290 K

as shown in column 3 of the table. The gas constant R, can be taken as 28? J,&g.K inthis example because its variation with specific humidiry is smq l5sq Eq. 13.2.8)1.The air pressure at 2000 m is then given by f3.2.15) \aith 8/a4, = 9.81/(0.0065 x28'l) = 5'26' u'

1- 1r,*. il t 2 \

P , : P t l V l t

r ? O O , 5 2 6ro l . 3 l= I

\ J U J /

= 80.4 kIh

as shown in column 4.The air density at the ground is calculated ftont t3.2.14.

^ : p'" R.T

_ 101.3 x 10r(287 x 303)

: 1 .16 kg /m3

and a similar calculation yields the air density of 0.97 kg/m3 at 2900 m. The averagedensity over the 2 km increment is therefo.e p, = (1.16 + 0.91)12: 1.07 kg/ml(see colurnns 5 and 9).

The saturated vapor pressure at the ground is determined using (3.2.9):

e=6rr exp (:42\

=orr "*p (#ll*l

:4244 Pa

= 4.24 kPa

The corresponding value at 2000 m whete T: l7'C, is e = 1.94 k])a (column 6).The specific humidity at the ground surface is calculated by Eq. (3.2.6):

q , , :0 .6229P

:0'622 * !4'1 0 1 . l

=0.026 kC&c

Ar 2000 m q" = 0.015 kg/kg. The average value of specific humidity over rhe 2,km increment is therefore q": (0.026 + 0.015y2 : 0.0205 kg,kg (colurnn 8).Substituting into (3.2.18), the mass ofprecipitable water in the first 2-km incremenr is

)

64 rpprrEi, . :r reorocr

- '=f f i ; : r" : r .o7xrx2ooo

= 43 .7 kC

By adding ihe incremental masses, lhe total mass of precipitable water in the columnis found io be mo = 17 kg (column l0). The equivalent depth of l jquid wrter ismp 1p " .1 = 7 l ' ( 1000 x 1 ) = 0 .077 m = 77 mm.

The numbers in column l l of Table 3.2.2 for percent of total mass in eachincrement show that more than half of the precipitable water is located in the fi.st2 km ebove the hnd surface in this example. There is only a very small amountof precipitable water above l0 km elevation. The deptb of precipitable \\ater inthis column is suff icient to produce a smail storm, but a large stonn would fequireinflo\r of moisturc from surounding areas to sustain the precipitation.

3,3 PRECIPITATION

Precipitation includcs rairtfa]i, snowfali, and other processes by which water fallsto the land surface. such as hail and sleet. The formation of precipitation requiresthe lifting of an air mass in the atmosphere so that it cools and some of itsmoisture condenses. The three main mechanisms of air mass l ift ing are /ron-tal lifting. *here *arm air is lifted over cooler air by frontal passage: orographiclft l,,rg, in $hich an air mass rises to pass over a mountain range; and conrectieIrf i izg, where air is drawn upwards by convective action. such as in the centerof a thunderstom cell. Convective cells are init iated by surface heating, whichcauses a rertical instabil ity of moist air, and are sustained by the latent heat ofvapor iza t i . rn g i ren up ar wa ler \apor r i ses and condense. .

The fbmation of precipitation in clouds is i l lustrated in Fig. 3.3. l. Asair rises and coois. water condenses from the vapor to the l iquid state. If thetempemturc is below the freezing point, then ice crystals are formed instead.Condensation requires a seed calied a condensation 'aclers around which thewater molecules can attach or nucleate themselves. Particles of dust floating inair can act as con,:lensation nuclei; panicles containing ions are effective nucleibecause the ions electrostatically attract the polar-bonded water molecules. lonsin the atmosphere include particles oi salt derived from evaporated sea spray,and sulphur and nitrogen compounds resulting tiom combustion. The diameters ofthese particles range from l0

-l to l0 pm and the particles are known as aerosols.

For comparison, the size of an atom is about l0 ag,m. so the smallest aerosolsmay be coniposed of just a few hundred atoms.

The tiny droplets grow by condensation and impact with their neighborsas they a-re caried by tu.buient air motion, until they become large enoughso that the force of gravity overcomes that of friction and they begin to fall, furtherincreasing in size as they hit other droplets in the fall path. However, as the dropfalis, water evapofates from its surface and the drop size diminishes, so the dropmay be reduced to the size of an aerosol again and be carried upwards in the

Rain Drops( 0 . 1 I m m )

FIGURE J. ] .1l,!a!er droplets in clouds ar€ formed by nucleation of vapor on aerosols, lhen go through many

condensation-evapo.ation cycles as ftey circulate in tie cloud. unil they aggregate into Iarge enough

drops to fall through the cloud base.

cloud through turbulent action. An upward cunent of only 0.5 cm/s is sufficient tocarry a l0 pm droplet. Ice crystals of the same weight, because of their shape andlarger size, can be supported by even lower velocities. The cycle of condensation,falling, evaporation, and rising occurs on average about ten times before the dropreaches a critical size of about 0.1 mm, which is large enough to fall through thebottom of the cloud.

Up to about I mm in diameter, the droplets rcmain spherical in shape, butbeyond this size they begin to flatten out on the bottom until they are no longerstable falling through air and break up into small raindrops and droplets Normalraindrops falling though the cloud base are 0.1 to 3 mm in diameter.

Observations indicate that water droplets rnay exist in clouds at subfreezingtemperaturcs down to -35'C. At this temperature, the supercooled droplets willfreeze even without freezing nuclei. The saturation vapor pressure of $,ater vaporis lower over ice than over liquid water, so if ice particles are mixed withwater droplets, the ice particles will grow by evaporation from the dropletsand condensation on the ice crystals. By coll ision and coalescence, ice crystalstypically form clusters and fall as snow flakes. However, single ice crystals maygrow so large that they fall direcdy to the earth as hail or sleet.

o5

l

' - 1

I

II

by,/r./cd.ton - condensingof vapor on tiny solid particlescalled aerosols (0-00i l0 pm)

66 ,qppLteo Hr lnlrooY

Cloud seecling is a process of artiiicially nucleating clouds to induce

precipitation. Silve; iodide is a common nucleating agent and is spread from

ui...uft in which a silver iodide solution is evaporated with a propane flame to

produce particles. While there have been many experiments wherein cloud seed-

ing was ionsidered to have induced precipitation' the great variability of meteo-

rolgical processes int olvecl in producing precipitation make it difficult to achieve

consrstenl results.

Terminal !'elocitY t

Three forces act on a falling raindrop (Fig 3.3 2): a gravity force Fs due to rts

weight, a buo,vancy force F6 due to the displacement of air by the drop' and a

il,"i force F,r duc to friction between the drop and -the sunounding air' If the

d.oi is a ,ph"r" of diameter D, its volume is (trl6)D3 so the weight force is

F8

and the boul ancy force is

(3 .3 . r )

r , : p .e l : lo '\ o /

( 3 . 3 . 2 )

ivhere p*, and p, anci are the densities of water and air, respectively The friction

drag force is given bY

v2F1-- Cap.A

,( 3 . 3 . 3 )

where Ca is a climensionless drag coefrcient, X = (n/4)D2 is the cross-sectional

area of the drop, and y is the fall velocitylf the drop is rcleased from rest, it will accelerate until it reaches its teminal

relocir.v V,, at which the three forces are balanced. In this condition'

F a = F c - F t

Hence, le t t ing V: y , in Eqs (3 .3 .1 -3) ,

FIGURE 3.3.2Forces on a falling raindropl F! : weight; F, = buoyancyr Fd =

drag force of surrounding air.

= r*(!)o'

F,1

(

, ', ll2. t n \ r l

cap,D, l ; l ; :\ ' / -

which, solved for V,, is:

, = lffi(fr -')] ' (3.3.4)

The assumption of a spherical raindrop shape is valicl for drops qp fq J mm

in Ciameter. Beyond this size, the drops become flattened on the bottom a*d moreoval in cross section; then they are characterized by the equivalent diaineter ofa sphericai Iaindrop having the same volume as the actual drop (Pruppacher and

Klett, 1978). Raindrops can range up to 6 mm in diameter, but drops larger than

3 mm are unusual, especially in low-intensity rainfall.For tiny droplets in clouds, up to 0.1 mnl diameter, the drag force is

specified by Stokes ' law for which the drag coe ftic ient is C 7: 241 Re , where Re is

the Reynolds number poWl 1t o with p, being the air viscosity. Fall ing raindropsale beyond the range of Stokes' law; values of C4 developed experimentally b)'obsenation of raindrops are given in Table 3.3.1.

Example 3.3.1. Calculate the terminal velocity of e I -mm dlameter raindrop fall ingin sti l l air at standarcl atmospheric pressure ( lO l.3 kPa) and temperalure l0oc

Solution, The terminal velocity is given by Eq. (3.3.' l) with Cr = 0 671 fromTable 3.3.1. At 20"C, p- = 998 kg/mr. and p,, : 1.20 kg/m3 at pressure lot 3kPa:

=4.02 m/s

Values of V, similarly computed for various diameters are plotted in Fig.i.3.3. It can be seen that the terminal velocity increases with drop size up to

a plateau level of about 5 mm drop size, for which, the terminal velocity is

approximately 9 nts.

TABLE 3.3,IDrag coefficients for spherical raindrops of diameterD, at standard atmospheric pressure (101.3 kPa) and20'C air temperature

67

"r)' o"'l';)o'

I r l / l

' 4 s D l P * ' \L s c r \ p , l )

_lI

Drop diameter D (mm) 0.2

Drug coef f ic ient C? 4.2

0 .4 0 .6 0 .8 1 .0 2 .0 3 .0 ' 1 .0 5 0

1 .66 1 .07 0 .815 0 .671 0 .517 0 .503 0 .559 0 .660

, + x 9 . 8 1 x 0 . 0 0 13 x 0 .671

. l . r / .e: Mso., 1957, Table 8 2, p. 136.

The picccding computations are for sea level conditions. Higher in theatmosphere. the air density po decreases, and Eq. (3.3.4) shows thar there wil l bea corresponding increase in V,; raindrops fall faster in thinner air. At air pressure50 kPa and tempemture 10"C, the plateau velocity of large drops increases from9 n/s to a littie more than 12 m,/s.

Thunderstorm Cell Model

The mechanisms underlying air mass lifting and precipitation are illustrared byconsidering a scherlatic model of a thunderstorm cell, as shown in Fig.3.3.,1.The thunderstorm is visualized as a vefical column made up of three pans, aninfow region near the ground where warm, moist air is drawn into rhe cell,an uplijl region in the middle where moisture condenses as air rises, producingprecipitation. and an outflow region in the upper atmosphere where outflow ofcooler, dryer air occurs. Outside the cell column, the outflow air may descendover a wide area, pick up morc moisture, and reenter the cell at the bottom. Thisentire pattern. called convective ceLI circuLation. is driven by the vast amount of

68 , lppueo rr: : ,eoroc,

2 4 6

FIGURE 3.3.3Tenninal velocity of raindrops as calcularedfrom Eq. (3.3.4) using drag coefficientsin Table (3.3. l ) . Resul ts arc for s iandardatmospheric conditions al sea level.

FIGURE 3.3.4A conveclive thundefsrorm cell visualizedas a cylindrical column of diameterD having inflow, uplift, and outflow

*L)i t -+ - l , t - ; - .1

(

( 3 . 3 . 6 )

p.i A = (q,p.\afiD A. | - (q,pa\a21rD Lzl ( 3 . 3 7 )

( i . 1 . 3 )

0: [p.(l - q,)VAz)2tD - lp.(t - q,)VLz)1rD

l I s, , \tpav{z)2: 1,0, Vl- t1l . I

\ . t -

Q, t li 3 . 3 . 9 )

( 3 . 3 . 1 0 )

69

heat energy released by the condensing moisture in the uplift region. Observationsof cumulonimbus clouds producing thundentorms indicate that the elevation ofthe top of the convective cell ranges from 8 km to 16 km (5 to l0 mi) in theatmosphere (Wiesner, 1970), and at times the tops of these ciouds may evenpenetrate through the tropopause into the stratosphere.

The thunderstorm is analyzed using the continuity equation for water vapor:

if prccipitation of intensity i (ir/h or cm/h) is faliing on an area A beneath thestorm cell, the mass flow rate of water leaving the cell is z, : -p,"r'A, wherep,, is the density of liquid water. Under the assumption of steady flo\!. the timederivative term in (3.3.5) is zero, and the mass flow rate of precipitation is equalto the difference between the mass flow rates of water vapor entering the cell ( 1)and leav ing (2 ) (see F ig .3 .3 .4 ) , so

t f f f ( f I

^ , : , ) ) ) , , , . 0 , - J 1 , , , " , o o ' j , : , s ,

l t t l= ) )a,o,v 'ae + J Jc,r ,v ar2 l

t l0 = | l p , v d A

The cell is a cylinder of diameter D, and air enters through height incrcment.\:1 and leaves though height increment Azz. If air density and specific humidityare assumed constant within each increment (in the manner shown in Example3.2 .2) , then

A continuity equation may be written similarly for the dry air canfing the vapor:

where p,i is the density ofdry air, which may be expressed using Eq. (3.2.1) aspa: p.(l q,). Substituting into (3.3.8):

Substituting (3.3.9) into (3.3.7) and noting that the area on which precipitationis falling is A : (Tl4)D2, it follows that

. 4p. ,V1Lz1lq" , c , fr = - : � l - . - l

p n D \ t - s , , /

Example 3.3.2 A thunderstorm cell 5 km in diameter has a cloud base of 1.5 km.and surface condit ion\ recorded nearbv indicate saturated air condit i .rns with air

70 APPLIEIJ H\ DROI i]CT

temperature 30'C. pressure l0l.3 kPa, and wind speed 1 m/s. Assuming a lapse rareof 7.5'C'km and an average outf low elevation of l0 km, calculare the precipitat ionintensitr fronr this storm. Also determine what proportion of the incoming moistureis precipitaled as air passes through the storm cell and calculate the rate of releaseof iatent heat through moisiure condensation in the column.

Solut ion. The precipitat ion intensity is given by (3.3. l0) where Vr = i rn/s, A; r =

1500 n. p" = 1000 kg/mr, and D = 5000 m. The quanti t ies p"rq, l ,and q,. arcfound b) the methdd outl ined in Ex. 3.2.2 using a:0.0075oc/m for the lapse rare.A tablc may be lbt up for the required values at z = 0, l �5, and l0 km.

Eleralion lemperature

( k m ) r ' ' C ) ( K )

Air Air VaporPressure Density Pressure(kh) (kg/mr) (kpa)

SpecificHumidity(kCAig)

0

1 .,5

t 0

85 .6

21 .1

0.026r

0 .0160

0.0002

l0 303

t9 292-..15 228

t 0 l . 3 L 1 6 4 .24

1 .02 2 .20

0.12 0 .01

Ffom the table. 4,. = 0.0002 kg/kgt the values for pn and q,. are takenas evc r rges bc tween 0 and 1 .5 km: p , r = ( 1 .16 + 1 .02 ) /2 : 1 .09 kg 'm1 , andq , = (0 .0261 + 0 .0160 ) /2 :0 .021 kg ,&g . Subs t i t u t i ng i n to (3 .3 .10 ) :

4p",VtA.t iS, - s,;\' = o o \ r " . 1

: 1 x L 0 9 x l x 1 5 0 0: 1000 " 5000

:2 .?2 x l0 -5 m/s

= 9.8 cm,/h

The mass f low rate of precipitat ion is given by m, = p"iA, where A =(Ti l) t-): = (-. ' .1) x 5000: : 1.96 x 107 m2 and p. = 1000 kg/mr; dp: 1000 x2 ]7 / 10 , . > r L96 x t 07 = 5 .34 x 105 kg / s .

The mass f low rate of incoming moislure is given by

m,, = (.p.q,VL:)1rD

= i . 0 9 x 0 . 0 2 1 x 1 . 0 0 x ! 5 0 0 x 2 x 5 0 0 0

= 5.19 x loj kgis

The pfoport ion of the incoming moisture precipitated is mplm,t = (5.34 I105 r , ' i 5 . 19 x t 051 : 9 .99 t

The rate of reiease of latent heat due to moisture condensation is l,rnr, wherel, is lhc lalenr heat of vaporizal ion of water. 2.5 x 106 J/kg:

i 0 .0210 - 0 .0002\\ 1 0.0002 l

I ,m r :2 .5 r . 106 x 5 .3 ,1 x 105

= 1.335 x lor ' � w

7 l

: 1 . 335 ,000 MW

This heat energy can be compared to large lhermal power plants. which may havea capacity of 3000 MW. It can be seen thar the energy released in thunderstormsls rmmense.

Variabil ity of Precipitation

Precipitation varies in space and time according to the general panb4fof atmo-spheric circulation and according to local factors. The average over'a numberof years of observations of a weather variable is called its normal va1ue. Figure3.3.5 shows the nornal monthly precipitation for a number of locations in theUnited States. Higher prccipitation occurs near the coasts than inland because theoceans supply the bulk of the atmospheric moisture for precipitation. Areas tothe east of the Cascade mountains (e.g., Boise, Idaho) have lower precipitationthan those to the west (e.g., Seanle, washington) because much of the moisturein the predominantly westerly air flow in the mid latitudes is extmcted as the airrises over the mountains.

Pronounced seasonal variation in precipitation occurs where the annualoscillation in the atmospheric circulation changes the amount of moisture inflowover those regions (e.g., San Francisco and Miami). This pattern is i l lustrated inFig. 3.3.6, which shows the normal monthly precipitation for various locations inthe United States. Precipitation is very variable in the mountain states in the westwhere orographic effects influence precipitation. Precipitation increases going eastacross the great plains and is spatially more uniforn in the east than in the west.Precipitation variabil ity for the world is shown in Fig. 3.3.7. The average annualprecipitation on the land surface of the eanh is about 800 mm (32 in), but greatvariabil ity exists, from Arica, Chile, with an annual average of 0.5 mm (0.02in) to Mt. Waialeale, Hawaii, which receives 11.680 mm (460 in) per year onaveruge.

3.4 RAINFALL

Rainstorms vary greatly in space and time. They can be represented b,\ isob-et.tlmaps; an isohyet ts a coltour of constant rainfall. Figure 3.4. I shows an isohyetalmap of total rainfall depth measured for two storms: one a storm of May 3O-June1, 1889, which caused about 2000 deaths in Johnstown, Pennsylvania, followinga dam failure, and the other a stofm of May 24-25, 1981 , in Austin, Texas, whichcaused 13 deaths and $35 million in property damage (Moore, et al., 1982). TheJohnstown storm is plotted on a scale 50 times larger than the Austin storm. Themaximum depth of precipitation in both storms is nearly the same (= 10 in). butthe Austin storn was briefer and more localized than the Johnstown storm. TheAustin storm was caused by a convective cell thunde$torm of the lype analyzedin Example 3 .3 .2 .

Isohyetal maps are prepared by interpolating rainfail data recorded at gagedpoints. A rain gage record consists of a set of minfall depths recorded for

3

4

b

r Ad

a

EE

3

: F

.9- A

! , 1

- a

C ( ? E

6 Z u r

1 t , .

(

E

f

a-

b

E

E

:

, & P

r ; 3 . 8; = i

: E ;

!EffiWI

/ 5

a&.4

t

.=

E

E=E

;a

I c !

9 . q 4

!MEffi

It

f!gDq

,uI

\

l 0 m i +

,,,'1..r'^:i,, //

t '

r s . . l 0 8 - )

. - . : , . _ . - , ' " 4

( a ) S r o r m o f \ { a } 3 0 J u n e l . 1 8 8 9 , s h i c hproduced th€ Johnsto\n nood in Pennsylvanla.Maximum rainfa l l of 9.8 in. reco.ded o!er l8 hourp e r , o d , r { . , l . b o r o . P e . l n ' ) r \ J n i " . l . o n ) e r ' r e ; '

inches dcpth of lota l rarnfal l in lhe s lonn. (S.)r r ( . :

U.S. Arm) Corps ol Engineers, 1941.)

(h) Storm of Mly 1,1 25, l9El . In Aust in. Teras.Maxinum rainfa l l of l l in . recorded ovet 3 bours.hohyets are in inches dcpth of total raintall 'n lhcstolm. (Sol i .?: Massey. Ree!es. and Le!r . 1982.1

E

FIGURE 3.4.IIsohyetal maps for two slorms. The storms have about the same maximum deplh of point rainfall.

but the Johnstown storm covered a much larger area and had a longer duralion than did the Austrn

successive increments in tine, as shown in Table 3.4.1 for the data in 5 minuteiocrements from gage I -Bee in the Austi n storm. A roinf.tll hyetograph is a plot ofrainfall depth or intensity as a function of time, shown in the form of a histogramin Fig. 3.4.2(n) for the l-Bee data. By summing the minfall increments throughtrme. a cumulative rainfall hyetograph, or rainfaLl mass curve. is produced. as

shown in Table 3.4.1 and Fie.3.4.2(b).The maximum rainfall depth, or intensity, (depth/time) recorded in a given

time interval in a storm is found by computing a series of running totais of rainfalldepth for that time interval starting at various Points in the storm, then selectingthe maximum value of this series. For example, for a 3o-minute time interlal.Table 3.4.I shows running totals beginning with 1.17 inches recorded in the first

?'

3oz:

76 ,rpprrro sr oirot.oor

TABLE 3. '1.1Computation of rainfall depth and intensity at a point

Running TotalsTime Rainfalt

( in)Cumulativerainfall 2 h

05

l 0t 52025l035,104550556065l07580859095

r001051 l { Ji 1 51201 2 5] ]0I -151.10l,l5r50

0 .010.1.10 . l 00.0.+0 . 1 90..180 . ) { l0 .500 . 5 r

0 _ l0 .660 . l o0 .190 .160 .5 i0 .760 . 5 r0 . t 10 t-i0 . i 50 . t l0 . l a0 0 90 .0 !0 l l0 . 010 . 0 10 .010 .0 r

0.000.020 .360..+6'0.t00 .69| . l '71 .612 . t 71 .681 .84l . l 53 . 8 11 . t 1,1.561.92

6.'�73" / . 17

1 .121 . 6 77 .898.0.+8 . l 3

8.1.18 .378 .388..108 . . 11

L 1 7L65l . 8 l2 .222 .342.462.642.502 .39

).623.072.923.00: . 862.152.431 .82L401 .05o.920.700 .190.360.28

3 8 r

.1.204.464 .965 .535.565 .505 .254.995 .054 .894 .32 8 . l 34.05 8.203.78 7.983 .45 t . 9 l2 .92 7 .882 . 1 8 1 . 1 11 .68 1 . 24

Max. depth 0. '-6Max, intensir\( i dh ) 9 . r l

3.01

b . l 4

5 .56

5 .56

8.20

4 . 1 0

30 minutes. 1 65 inches from 5 min to 35 min, l.8l inches from l0 min to 40min, and so,ln. The maximum 30 minute recorded depth is 3.07 inches recordedbetween 55 min and 85 min. co[esponding to an average intensity of 3.07 in/0.5h : 6.1,1 inrh over this interval. Table 3.4.I shows similarly computed maxrmumdepths and intcnsities for one and two-hou intervals. It can be seen that as thetime period increases. the avemge intensity sustained by the storm decrcases (5.56in-/h for one hour, .1.10 in/h for two hours), just as the average intensity overan area decrei,ses as the area increases. as shown in Fig.3.4.1. ComDutations

77

0 .8

I: l

t

E 0 .6

i

a

E

-

Marimum depth recorded inspecified time inlerval

\ l3 0 m i n r I

5.56 in.

t n*, I

0 30 60 90 t20 150Time in minutes

\h )

FIGURE 3.4.2Incremental and cumulative rainfall hyetographs at gage l_Bee for storm of May 24 25' 198i inAustin, Texas.

of ma"rimum minfall depth and intensity performed in this way Sive an rndex

of how severe a particular storm is, compared to other storms recorded at the

same location, and they provide useful data for design of flow control structuresAn important fact to be determined from historical rainfall records is the avemgedepth of rainfall over an area such as a watelshed.

78 rpprreo r, crot oct

Areal Raintall

The arithntt,ric-mean method is the simplest method of determining areal averagerainfall. It i irrolves averaging the rainfall depths recorded at a number ot gages

[Fig. 3.4.3!r]11. This method is satisfactory if the gages are uniformly distributedover the area and the individual gage measurements do not vary greatly about themean.

If sonrc gages are considered more representative of ihe area in question

than others. Ihen relati!€ weights may be assigned to the gages in computing theareal average. The T,b€ss€x method asstmes that at any point in the watershedthe rainfall is the same as that at the nearest gage so the depth recorded at r girengage is appLied out to a distance halfway to the next station in any direction.

The relative i,eighL' for each gage are determined from the couesponding areas

of applicalion in t Thie;sen pol,]\,got network, the boundaries of the polygons

being formed bl thc perpendicular bisectors oi the l ines joining adjacent gages

[Fig. 3.4.3(&]1. If there are -/ gages, and the area within the watershed asslgnedto each is -1,. and P; is the rainfall recorded al the jth gage. the areal averageorecipitation for the watershed is

D

where the \\otershed area A : ) j= L Ar. Tn" Thiessen method is generally more

1't0 0

Average rainfxl l = 1'10.0/.1= 35.0 mm or tn

r - l= ) A , P ,

P 1

P 5

FIGURE 3.{.3(.)Computal ion of aferl .r \eragera;nfau by the arifinelic mean method.

79

( i . 4 . l )

accurate than the arithmetic mean method. but it is inflexible. because a newThiessen network must be constructed each time there is a change in the gagenetwork, such as when data is missing from one of the gages. Also, the Thiessenmethod does not directly account for orogmphic influences on rainfall.

The isohyetal method overcomes some of these difficulties by constructingisohyets, using observed depths at rain gages and interpolation between adjacentgages [Fig.3.4.3(c)). Where there is a dense network of raingagesl.isohyetalmaps can be constructed using computer programs for automated, c{ntouring.Once the isohyetal map is constructed, the area A; between each pair disohyets,within the watershed, is measured and multiplied by the average Pj of the rainfalldepths of the two boundary isohyets to compute the areal average precipitation byEq. (3.4.1). The isohyetal method is flexible, and knowledge of the storm patterncan influence the drawing of the isohyets, but a fairly dense network of gages isneeded to correctly construct the isohyetal map from a complex storm.

Other methods of weighting rain gage records have been proposed, suchas the rcciprocal-distance-squared method it which the influence of the rainfaliat a gaged point on the computation of rainfall at an ungaged point is inverselyproportional to the distance between the two points (Wei and McGuinness, 1973).Singh and Chowdhury (1986) studied the various methods for calculating arealaverage precipitation, including the ones described here, and concluded that allthe methods give comparable results, especially when the time period is long;

Observed ra in ia l l wirhLn

or c lose Io rne aleu

20.0

' , . , , , ,

FIGITRE 3.4.3(r)Computation of areai average rainfall by the Thiessen method.

Obs€rved Weighled.ainfall Area ralnfall

(mrn or in) (km2or rnir) (mm or in)

30.0

40.0

50.0

P I

. P jP 2

P 3

2 .2

80.4

40.5

64.0

91.5

0.22

4.O2

1 .35

1 .60

r.95

10 .0

20.o

30.0

40.0

50.0

9 .14 28 '1 .6

Average rainlal l = 284.6/9.14 = 3l. l mm or rn

E0 eppLteo Htoeorocr

(

Sensiblen"ui lo ut'

N"r ,adiurion vapor flow rateH" R, f t , =p.ar

8 1

Isohvets A.ea€ncloseo

(rnm or in) (km2or miz)rainlhll

Rainfall

105 .4

549

r0.6

t. .1"

6t

*Esr imated,

Average ra infa l l = 255.2 / 9.14 = 27.9 mm or in

FIGURE 3.4.3(r)Computation of arerl average rainfall by lhe isohyeral merhod.

that is, the diiterent methods vary morc from one to another when applied todaily rainfall data than when applied to annual data.

3.5 EVAPORATION

The two main factors influencing evaporation from an open water surface arethe supply of energy ro provide rhe larenr heat of vaporization and the ability totmnspot the vapor away from the evaporative surface. Solar mdiation is the mainsource of heat energy. The ability to tanspon vapor away from the evaporarilesurface depends on the wind velocity over the surface and the specific humiditygradient in the air above it.

Evaporation from the land surface comprises evaporation directly from thesoil and vegeration surface, and transpiration through plant leaves, in whichwater is extracted by the plant's roots, transported upwards through its stem,and diffused into the atmosphere through tiny openings in the leaves calledstomata. The processes of evaporation from the land surface and tmnspirationfrom vegetation are collectively termed evapotranspiration. Evapotmnspiration is

influenced by the two factors described previously for open water evaporarion, andalso by a third factor, the supply of moisture at the evaporative sudace. Thepolential e,-apotranspiration ts the evapotranspiration that would occur from awell vegetated surface when moisture supply is not limiting, and this is calculatedin a way similar to that for opelt water evaporation. Actual evapotmnspintiondrops below its potential level as the soil dries out.

Energy Balance Method

To develop the continuity and energy equations applicable for evaporation. con-sider evaporation from an evaporatioa parz as shown in Fig. 3.5 . I . An evapora-tion pan is a circular tank containing water, in which the mte of evaporation ismeasured by the rate of fail of the water surface. A control surface is drawnaround the pan enclosing both the water in the pan and the air above it.

Continuity. Because the contlol volume contains water in both the liquid andvapor phases, the integral continuity equation must be written separately for thetwo phases. For the liquid phase, the extensive prcperty is I : mass of liquidwater ;p - l ,p= h(he dens i ty o f water ) , anddBld t : - r2 , , wh ich is the massflow rate of evaporation . The continuity equation for the liquid phase ts

FIGURE ] .5.7Controi volume defined for continuilr'and energy equation developmenl fof anevaporation pan.

d l l t tn , = - l I l p , d v + l l p , v d A

A T ) J J J I( 3 . 5 . l )

The pan has impermeable sides, so there is no flow of liquid water acrossthe control surface and JJp.V.al = 0. The rate of change of storage wirhin rhe

23.9

56.0

l 5

25

35

45

5 3 +

0.88

| . 59

2 .24

3 .01

| . 22

0.20

t0

20

30

40

50

9 . 1 4 255.2

Heat conducted to ground

i r

{"182 rpprreo sronorocr

system is giren by (d/dt)!l!pdY = pj\ dhldt, where A is the cross-sectionalarea of the pan and ft is the depth of water in it. Substituting into (3.5.1):

' ldh\m , = p " A l , l

m"= p.AE ( 3 . 5 . 2 )

where E = -dhldt is he evaporation rate.Forthe raporphase, B: mass ofwater vapori B:q", the specific humidity,

p - pa, the air density, and dBldt: rn ", so the continuity equation for this phaseis

t ( f ( ( f. u | , | , . , | |^ , = d , J J J e , o , a v - J ) o , o . v a t (3 . s .3 )

For a steady flow of air over the evaporation pan, the time derivative of watervapor stored within the control volume is zero. Thus, after substituting for m"f rom (3 .5 .2 ) . (3 .5 .3 ) becomes

t lp"AE =

) )e,o"v 'at(3 . s.4)

which is the conrinuity equation for an evaporation pan, considering both waterand water vapor. In a more general sense, (3.5.4) can be used to define theevaporation or evapotranspiration rate from any surface when written in the form

where E is the equivalent depth of water evaporated per unit time (in/day ormrn/day).

Energy. The heat energy balance of a hydrologic system, as expressed byE9. Q.1 .l can be applied to the water in the control volume:

'=(r,)j j '"""*

# -# : :,l"l, l1.* )v' * ;)pavr r , r \

+ | l { e , + - : v ' � + s z ) p v . o e- - \ ' /(3.s.6)

where dH/dr is the rate of heat input to the system from exremal so'$ces, dw/dt

is the rute of work done by the system (zero in this case), e, is the specificinternal heat energy of the water, and the two terms on the right hand side are,respectively, the rate of change of heat energy stored in the control volume andthe net outflow of heat energy caried across the control surface with flowingwater. Because V: 0 for the water in the evaporation pan, and the mte of changeof its eleyation, z, is very small, (3.5.6) can be simplif ied to

I 'L.s.zr

f

R , - H , - G : 1 , m , (3 . s .8 )

By substituting for mu from (3.5.2) with A : I rn2, (3.5.8) may be solved for t:

Considering a unit area of water surface, the source of heat energy is netmdiation flux Rr, measured in watts per meter squared; the water supplies asensible heat flux 11, to the ai.r skeam and a ground heat flux G to the groundsurface, so dHldt = Rn - H, - G.If it is assumed that the temperature of thewater within the control volume is constant in time, the only change in the heatstored within the control volume is the change in the internal energy of the waterevaporated, which is equal to lumu, where lu is the latent heat of vaporization.Hence. (3.5.7) can be rewritten as

d H d U td , = A J ) J e u q i d v

t = * r

, - H , - G ) (3 . s .9 )

which is the energy balance equation for evaporation. If the sensible heat flux11, and the grcund heat flux G are both zero, then an evaporation rate E. canbe calculated as the mte at which all the incoming net radiation is absofted byevaporation:

( 3 . 5 . 1 0 )

Example 3.5.1. Calculate by the energy balance method the evaporation rate froman open water surface. if lhe net radiation is 200 W/m2 and the air lemperalure i525oC, assuming no sensible heat or ground heat flux.

Solutian. From (2.1 .6) the latent heat of vaporization at 25'C is I ,= 25OO 2.36x25 = 2441 ktkg. From Table 2.5.2, water densiry p": 997 kg/m', and substitutioninto (3.5.10) gives

244Ix IO3x99 '7:8.22 x l0-8 m/s

:8.22 x l0-8 x lOO0 x 86400 mrn/da)

= 7.l0 nun/day

t,9"

8:l ,rppLteo sr r.norocr

\er.adiat ionR,,j

I - - - l

t ,

+ Air flow '+

u 1

FIGURE 3.5.?Evaporation firirr Jn opc'r !'"at€r surface.

(3 .5 . r2 )

m f

,f

I t ,

. i \ l \E t \ 5 \

: t \ : \" \ . "

\Air Specific

remperalure humidit)T q ,

Aerodynamic Method

Besides the supply of heat energy, the second factor controil ing the evaporationrate from an open water surface is the ability to transport vapor away from thesurface. The transport mte is governed by the humidity gmdient in the air nearthe surface and the wind speed across the surface, and these two processes canbe analyzed bl coupling the equations for mass and momentum transport in air.

ln the control volume shown in Fig. 3.5.2, consider a horizontal plane ofunit area located at height ; above the surface. The vapor flux z, passing upwardby convection through this plane is given by the equation (from Table 2.8.I with

. . . dq ,m t = -PaK"

,az

( 3 . 5 . l 1 )

where K,, is the vapor eddy diffusivity. The momentum flux upward through theolane is l ikcrvise siven bv an ecuation from Table 2.8.1:

duaz

Suppose the wind velocity r.tt and specific humidity 4"r are measured atelevation : I . and a] and 4,. at elevation .z 2, the elevations being sufficiently closethat the transport rates 'r, and T are constant between them. Then the substitutionsdq,ldz. : (q,. 4, )i(zz- 2r) and, duldz = (uz u)/(.zz ir) can be made in(3.5.11) and (1.5.12). respectively, and a ratio of the resulting equations ta.kento glve

K*(q,., - q",)

K^(u1 - u1 )

85

. K.(.q,, - q,)m ' : r

K ^ @ 2 - u t )

The wind velocity in the boundary layer near the earth s surface (up to about50 m) is well described by the logarithmic profile law [Eq. (2.8.5)]

? 5 1 4 )

where r* = shear velocity = 1Gt A, t< is the von Karman constant, usuafly takenas 0.4, and zo is the roughness height of the surface given in Table 2.8.2. Hence,

u ' 1 , t z t t , . r , 1 lu 2 - u t = - I n l - l l n l l l

/ ( I \70/ -o/ I

" . , i z z \-

i I r t li ( \21l

and

* k(u2 - u1)u = -

l t ( z2 l z t )

But u" : -"Eip. by definition, hence

I t r u , - , , 1 ) 2r = f ," L t n ( z 1 z , r . ]

Substituting this result irto (3.5.13) and reananging gives

K*k2 po@", - q",,)(u2 Lt)(3 .5 .1s )

x^fn kztz)12which is the Thornthwaite-Holzman equation for vapor tmnsport. first developedby Thomthwaite and Holzman (1939). In application it is usualiy assumed that theratto K.lK^: I and is constant. Thomthwaite and Holzman set up measurementtowers to sample q, arLd u at different heights and computed the correspondingevaporation rate, and many subsequent investigators have made simiiar experi-ments.

For operational application where such apparatus is not available and mea-surements of 4, and a are made at only one height in a standard climate sta-tion, Eq. (3.5.15) is simplif ied by assuming that the wind velocity t i1 = 0 atthe roughness height:1 = z0 and that the air is saturated with moisture there.From Eq. (3.2.6), q" = 0.622 elp, where e is the vapor pressure and p is theambient air pressure (the same at both heights), so measurements of vapor pres-sure can be substituted for those of specific humidity. At height 22, the vaporpressure is eo, the ambient vapor pressure in air, and the vapor pressure atthe surface is taten to be eo,, the saturated vapor pressure corresponding tothe ambient air temperature. Under these assumptions (3.5.15) is rewritten as

( 3 . 5 . 1 3 )

u r ' l z \u K \Zol

i86 eppLrro rr l rococr

. 0.622k2 p"(e ., - e.)u2'"' : ,1t" trr,ut

Recalling that rr, is defined here for a unit area of surface, an equivalentevaporation rate Eo, expressed in dimensions of [L/T], can be found by seningh - p , ,En r r J .5 . lb and re , : r rang ing :

( 3 . s . l 6 )

( l . 5 . 1 7 )

( 3 . 5 . l 8 )

* here

Eq . ( 3 .5 . l 7 ) i : u con rmon bas i s f o r many evapo ra t i on equa t i ons , w i t h t hc f o rm

of the vapor translcr coeff icient B varying from one place to another. This type

of equation *as f irst proposed by Dalton in 1802.

Example 3.5.2 Calculate lhe evaporation mte from an open water surface by theaerodynanric method lvi th air lemperature 25"C, relat ive humidity 40 percent, airpressurc 101.3 kPa, and wird speed 3 r l /s. a1l measured a! height 2 m above thewater surirce. Assume a roughness height a0:0.03 cm.

So l l l t i o r . The \ epo r t r ans fe r coe f { i c i en tB i sg i venby (3 .5 .18 ) , us ingk=0 .4 . p ,=

Ll9 kg,nr ibr air at 25'C, and p^ = 991 kg/ml. Hence

0.o2l lro-r.r,

pp, , l ln ( ;1 ,0) i '

0 . 6 2 2 x 0 . , 1 : x 1 . 1 9 x l

l 0 l . J l 0 " a o 7 l n 2 , r l l 0 - - r

: , 1 .54 x l 0 r r m /Pa .s

The evapomtion fale is given by (3.5.17). using e,, = 3167 h at 25"C from Table(3 .2 . I I r nd . f r om (3 .2 . l 1 ) , ea : Rheas = 0 . : 1 x 3167 : 1267 Pa :

E,, = Ble^ - e')

= . 1 . 5 4 x t 0 - r ' ( 3 1 6 7 - 1 2 6 7 )

:8 .62 x l 0 " m /s

" , 1000 mm i80400 s\_ 8 6 t r 0 " . 1 - ) . . I, r m \ o a y /:7..15 mntday

Combined -{erodlnamic and Energy Balance Method

Evaporation rlriL) be .omputed by the aerodynamic method when energy supplyis not limiting and by the energy balance method when vapor transport is not

87

limiting. But, normally, both of these factors are limiting, so a combinarion ofthe two methods is needed. Irl the energy balance method. the sensible heat fluxfl, is difficult to quantiry. But since the heat is hansferled by convection throughthe a overlying the water surface, and water vapor is similarly transferred byconvection, it can be assumed that the vapor heat flux /,,nr" and the sensible heatflux 14 are proportional, the proportionality constant being called. the Bowen ratioB (Bowen. t926): l

t L

€ . 5 . 1 9 )

The energy balance equation (3.5.9) with ground heat flux G = 0 can then bewritten as

R^= l "m,( . I + p) (3.5.20)

The Bowen ratio is calculated by coupling the transport equations for vaporand heat, this is similar to the coupling of the vapor and momentum transportequations used in developing the Thomthwaite,Holzman equation. From Table2.8.1, the transport equations for vapor and heat are

. -- dq,m r : - 0 o K * - -

NTu , - - p " C " K L . �

. a z

where Cp is the specific heat at constant pressure and K7, is the heat diffusivity.Using measurements of 4" and I made at two levels z \ a\d z 2 and assuming thetmnsport rate is constant between these levels, division of (3.5.22) by (3.5.21)gives

Hs C pKh(T2 Tt)

m, K"(q,, - q,,)

Dividing (3.5.23) by l, and substitutin g O.622 elp for q" provides the expressionfor the Bowen ratio B from (3.5.19)

^ C,Khp(72 - Tt)P =

oA2t i . k r - "n

(3.s.24)

where 7 is the psychrometric constant

C oKnPY= o6t,K-

Et

r :3 .5.21)

(3.s .22)

(3.s.23)

(3.5.25)

(

88 :ppLrt l i r l onotrr ; r

The ratio d,, ,(,, 01 the heat and vapor diffusivit ies is commonly taken to be I(Priestlel ' and Taylor. 1972).

If thc two lcr,els L and 2 are taken at the evaporative surface and inthe overli ing air stream. respectively, it can be shown that the evdporJtionrate t, confuted iom the rate of nel radiation [as given by Eq. (3.5.10)] andthe evaporation rate computed from aerodynamic methods [Eq. (3.5. i7)] can becombined to vield a \r 'eighted estimate of evaporation f, by

A NL = . - L , t - L .

. . \ + y f + y - (3 . s .26 )

where y is rirc pslrhrometric constant and I is the gradient of the saturated vaporpressure clif\e at.i ir temperature I,, as given by (3.2.10); the weighting faclorsl / ( l+ i u l rd f i1 + 1 ) sum to un i ty . Equat ion (3 .5 26) i s the bas ic eq t ra t iont i r r the conr i rnar ion l rc thod o f comput ing evapora t ion . wh ich was f i rs t deve lopedb,v-. Penman t l9-181 Its derir.ation is lengthy (see Wiesner. 1970). and wil l not bepresenteo nere.

The conrbinatiol] l method of calculating evaporation from meteorologicaldata is the most accumte method when all the required data are available and theassumptlons lre satisfied. The chief assumptions of the energy balance are thatsteady stafe cnergy, f lorv prevails and that changes in heat storage over time inthe water body arc not significant. This assumption l imits the application ot themethod to daily rime intervals or longer. and to situations not involving largeheat storage capaciti ' . such as a large lake Possesses. The chief assumption of theaerod-vnlnric method is assocjated with the form of the vapor transfer coeil icientB in Eq. (1.5.17). ivlany empirjcal forms of B have been proposed, Locally- f ittedto obser\cd rvind and other meteorological data.

The combinltion method is well suited for application to small areas withdetailed climatological data. The required data idclude net radiation, air temper-ature. hurni,-l i t_v, $ind speed, and at pressure. When some of these data areunavailabir. simpler evaporation equations requiring fewer variables must be used(American Societ) of Civil Engineers. 1973: Doorenbos and Pruitt, 1977). Forevaporation orer r erv large areas, energy balance considerations largely golemthe evaporation rate. For such cases Priestley and Taylor (1972) found that thesecond ternr of the combination equation (3.5.26) is approximately 30 percentof the first. so thrt (3.5.26) can be rewritten as the Prlest1e-i-Ir-r lor etapot"lt ioneqLration

_ a _L = d a - t ]

a r 7(3.s.2 ' �7)

where a : 1.3. orhcr investigators have confirmed the validity of this approach.with the r alue of a varying slightly from one location to another.

Pan eraporation data provide the best indication of nearby open waterevaporation $her. such data are available. The observed values ofpan evapurJtionf| are muLtiplied by a pan factor kp (.O := kp = l) to conlert them to equivalentopen \!,ater evaporation values. Usually kp : 0.7, but this factor varies by season

The formulas for the various methods of calculating evaporation are sum-

marized in Table 3.5.1.

Example 3.5,3. Use the combination method to calculate the evaporation raae froman open water surface subject to net radiatjon of 200 W/ml, air temPerature 25oC,relative humidity 40 percent, and wind speed 3 n/s, all recorded at height 2 m, andatmospheric pressure 101.3 kPa.

l

Sdltr l ior. From Example 3.5. I the e \ aporation rr te conespond ing tJ u 4[t ,naiurionof 200 Wm2 is 8. :7.10 mm/da). and frorn E\ample 3.5.2. the Grodynamrcmethod yields E. = "1.45

mm/day for lhe given air temPerature. humidity, andwind speed conditions. The combination method requires values fbr r! and 7 LnEq. (3.5.26). The psychrometric constant y is given by (3.5.25), using Cp: 1005J/kg K for air, ]( / ,( , , : 1.00. and l , = 2441 x 103 J,kg at 25oC (from Example

C oKnP' 0.622t,K"

1 0 0 5 x l . 0 0 x l 0 l l x l O j: ob r r . , 4 r r {

= 67.1 Pa,i"C

\ is the gradient of the satumted vapor pressure curve at 25"C. given by (3.2.10)

with e. : e,, : 3167 Pa for T = 25"C:

4= 4098e '

(23'7 .3 + 7)l

4098 x 3167

= 188.7 Pt"C

The weighrs in the combinal ion equation, then. are y/(A + 7) = 67 I/(188 7 I

61 .1 ) = 0 .262 and A / (A + y ) : 188 .7 / (188 .7 + 67 . i ) : 0 .738 . The evapo re t i onrate is then computed by (3.5.26)

_ A - ) -A t y A + 7

=0 .738 x 7 .10 + 4 .262 ' / 7 .15

=.1.2 .n'r,iday

Example 3,5.4 Use the Priestley-Taylor method to caiculate the evaporation rate

for a water body with neI radiation 200 Wmr and ait temperature 25"C-

Solutian, The Priestley-Taylor method uses Eq. (3.5.27) with f. = 7 10 mrn/day

from Example 3.5.1, A/(A + y) :0.738 at 25"C from Example 3 5 3 anrt ir :

l -3. Hence,

a + 7

89

90 ,qpprieo sr urorocr

TABLE 3.5.1Summary of equations for calculating evaporationx

(1) Energ! balance method

E. = 0.0353R, (nntday)

R,: net radiation (W/m'�)

(2) Aerodynamicmgtirod

E.= B(e^ e") (mrc'JdaY)

whefe

- 0 . l 02u ru = - ( m n v d a y . p r )

l / : r \ lI I n l 5 ll l

a2 is \ \ ind velocity (nl/s) measured at heighl ! l (cm), and

io is frorn Table 2.8.2. Also,

"..:orr.*pfj l{ l 1nt_ . , . 3 - T

f: air temperature ('C)

e ,. : R he o, (,Pa)

in which R/, is lhe reiative humidity (0 = Ri' < l).

(3) Combination method

_ A _ y _t = - t - + - t l m I T V O a V l

A + ) A + " y -

,1098?".\= --:----r: rPr,oc,

L i r / r r / r _

and

l :66 8 (Pal"C)

(4) Priestley-Taylormethod

A - r y

where n = 1.3

' The values shown ue valid for standard atrnospheric pressur€ and air tem-

perature 20oc.

(

3.6 EVAPOTRANSPIRATION

Evapotranspirotion is the combination of evaporation from the

9l

: 1 .3 x 0 .738 x 7 . l 0

:6.8 mm,/day

which is close to the rcsult from the more complicated combination method shown

in the previous example.

, 1 .

Isoil surface ald

transpiration from vegetation. The same factors governing open water evaporationalso govem evapotralspimtion, namely energy supply and vapor transport ln

addition, a third factor enters the picture: the supply of moisture at the evaporative

surface. As the soil dries out, the rate of evapotranspiration drops below the level

it would have maintained in a well watered soil.Calculations of the rate of evapotranspiration are made using the same meth-

ods described previously for open water evaporation, with adJustments to account

for the condition of the vegetation and soil (Van Bavel, 1966; Monteith, 1980)

For given climatic conditions, the basic rate is the reference crop evapotranspira-/lon, this being "the rate of evapotranspiration from an extensive surface of 8 cm

to 15 cm tall green grass cover of uniform height, actively growing, completely

shading the ground and not short of water" (Doorenbos and Pruitt, 1977).Comparisons of computed and measured values of evapotranspiration have

been made at many locations by the Amedcan Society of Civil Engineers (1973)

and by Doorenbos and Pruitt (1977). They concluded that the combination method

of Eq. (3.5.26) is the best approach, especially if the vapor transpon coefficient

B in Eq. (3.5.18) is calibrated for local conditions. For example, Doorenbos and

Pruitt recommend

B - o . o o 2 7 ( 1 + # ) ( 3 . 6 . l )

in which B is in mm./day Pa and 4 is the 24-hour wind run in kilomete$ per day

measured at height 2 m. The 24-hr wind run is the cumulative distance a particie

would move in the airstream in 24 hours under the prevailing wind conditions.

Note that the dimensions of a given here are not metels per second as used in the

equation for B given in Table 3.5. I , but the resulting value of E o is in mjllimetersper day in both cases.

The potential evapotranspimtion of another crop growing under the same

conditions as the reference crop is calculated by multiplying the reference crop

evapotranspiration E,, by a crop coeficient t", the value of which changes with

the stage of growth of the crop. The actual evapotranspiration E/ is found by

multiplying the potential evapotmnspiration by a soil cofficient k' (0 = k. < I ):

E , : k , k , E r ( 3 . 6 . 2 )

The values of the crop coefficient t" vary over a range of about 0.2 < k. <

I 3 as shown in Fis 3 6. t rl)oorenbos and Pruitt. 197?) The initial value of 't '

( '

92 reprtro sloaor oc r

for well-watered soil with l itt le vegetation, is approximately 0.35. As the vegetation develops, ,t. increases to a maximum value, which can be greater than 1 forcrops with large vegetative cover, such as com, which transpire at a greater ratethan grass. As the crop matures or ripens, its moisture requirements diminish. Theprecise shape of the crop coefficient curve varies with the agricultuml practicesof a region, such as the times oi plowing and harvest. Some vegetation, such asorchards or pemranent ground cover, may not exhibit all the growth stages showni n F i g . 3 . 6 . 1 .

t

Example 3.6.1. (From Gouevsky, Maidment, and Sikorski, 1980) The monthl)-values of reference crop evapotranspimtion Er,, calculated using the combination

m7nm7v

I

Stages oi crop growth

Slag. C.op condiLion

I Inr t ra l nr :e les! than l0% grcund cover.

2 Dcvclopm.n( sragc f ron rn i r ,a l sr lgc to ! l rnmcnrofel fc . r i rc I l l l sround cover (70 - 80' / . ) .

3 Mrd s.Json sta8c irom full ground covcr to

4 Lr tc scrson srxgc fu l l mrLur i ty md harvesl .

FIGURE 3.6. IThe relationship bcn!een rhc crop coeflicrent k, and the stase of c.op gfowth.

(

method, for avemge conditions in Silistra, Bulgaria, are shown in thetable below. The crop coefficients for com (see Fig.3.6.l) are k1 =

0 . 3 8 , f 2 = 1 . 0 0 , a n d t : = 0 . 5 5 ; t r = A p r i l l , t 2 : J u n e 1 , 1 3 - J u l y 1 , r a :September l, and t5 = October l. Calculate the actual evapotranspirationfrom this crop assuming a well-watered soil.

Solution.

Month Apr May Jun Jul Aug Sep Oct Apr-C, total

6.f i 5.94 4.05 2.34 34.3 mm

1.00 1.00 0.78 0.55

6 .60 5 .94 3 . i 6 | . 29 24 ' t mm

T i m e l

<.?"( i' \

7 7Vmm7n

3

h

93

E, (rnrn/day) 4.14 5.45

k. 0.38 0.38

E, (mnr/day) |.57 2.07

5 .82

0.69

4.02

Il;

J'l

Monthly average values of t. are specified foilowing the curve in Fig. 3 6 1 using

the given values. In June,,t . r ises from 0.38 at t? = Jl lne I to 1.00 at t3: July

I, so k. is taken as (0.38 + 1.00y2:0.69 The values ofE,are computed using

Eq. (3.6.2) with f t , = 1 for a well-watered soi l ; that is, E, = ,t .Eh The total

evapotranspiration for the growing season from April to October fot com. 24.'7

mm, is 72 percent of the value a grass cover would have yielded under the same

condit ions.34.3 mm.

REFERENCES

American Society of Civil Engineers. Consumptive use of water and imgation water requircments,

ed-by M. E. Jerlsen, Technical Conmittee on lrriSation Water Reqrir€rner$, New York'

1913.Bo\ren. l. S., The ratio of heat losses by conduction and by evaporation from any water surface,

P hl s. Rev., v ol. 27. r'o. 6, pp.'7'19-'1 87. 1926.Bnrtsaert. W., Evaporation into the Atmosphere, D. Reidel, Dordrechl, Holland' 1982.

Doorenbos, J., and W. O. Pruitt, Crop watel requiremenLs, lniSation and Dninage Paper 24

U. N. Food and Agriculture Organization, Rome, Italy, 1977.

Gouevsky, L V., D. R. Maidment, and w. Sikorski, Agricuhural water demands in ihe Silislra

region, RR-80-38, Int. Inst. App. Sys. Anal-, Laxenburg, Austria' 1980.

Mason, B. J., The PhJsics oJ Clouds, Oxford University Press, London, 1957.

Massey, B. C., W. E. Reeves, and vr'. A. Lear, Flood of May 24-25, l9El, in the Ausiin. Texas'

metropolitan area. HtdroloSic Investigations Attas, HA'656, U. S Geological Surley, 1982'

Monteith. J. L., The development and exEnsion of hnman's evaporalion focmula. in /pplica,io'r

of Soil Phlsics, ed.by D. Hiliel, Academic Press, Orlando, Ra. Pp 24'7 253 ' 1980

Moore, W. L-, el al., The Austin. Texas. flood of May 24 25, 1981, Report for Conmitree on

Natural Disasters, National Academy of Sciences, Washington' D.C ' 1982.

Rnftan, H. L., Natural evaporation from open water, bare soil, and grass, Proc. R. Soc Landon

A, \ ,o1. 193. pp. 120-146, 1948.Priestley, C. H. 8., and R. J. Taylor, On the assessment of surface heat flux and evaporJtron u'ing

iarge-scale parameters, Monthry weather Rev., voi. 100, pp 81 92' 1972

Pruppacher, H. R., andJ.D.Klett, Microphtsics of Clouds and Prccipitarion, D Reidet Dordrechl'

Hol land. i978.

98 eperrro gYoeoi oc,v

3.6.5 Use the aerodlnamic method to calculate the evaporanspiration rate (mm/day)

from a well watered. short grass area on a day when the avenge air temperature

is 25oC, relative humidity is 30 pelcent, 24-hour wind run is 100 km' and normal

atmospheric pressure (10L3 kPa) prevails. Assume the Doorenbos-Pruitt wind

function (3.6.1) is valid. By what Percentage would lhe evapotmnspiration rate

change if the relati\e humidity were doubled and the temperaturc' wind speed.

and air pressure femained constant?

C

i . l

SUBSURFACWATER

Subsurt'ace water flows beneath the land surface. In this chapter, only subsurfaceflow processes important to surface water hydrology are described. The broader

field of groundwater flow is covered in a number of other textbooks (Freeze andCheny, 1979; de Marsily, 1986).

4.1 UNSATURATED FLOW

Subsurface flow processes and the zones in which they occur are shown schemati-cally in Fig. 4. l� I . Three imPortant processes are infhration of surface w e r into

the soil to become soil moisture, subsurfoce flow or unsaturated flow through the

soil, and groundwaler J?olt or saturated flow through soil or rock strata. Soil androck stmta which permit water flow ate called porous media. Flow is unsaturatedwhen the porous medium still has some of its voids occupied by air, and saturatedwhen the voids are filled with water. The water tabLe is the surface where the

water in a satumted porous medium is at atmospheric pressurc. Below the water

table, the porous medium is saorated and at greater pressure than atmospheric.Above the water table, capillary forces can saturate the porous medium for a

short distance in the capillary fringe, above which the porous medium is usuallyunsaturated except following rainfall, when infiltration from the land surface can

produce saturated conditions temporarily. Subsurface a\d groundwater outfow

occur when subsurface water emerges to become surface flow in a stream or

spring. Soil moisture is extmcted by evapotranspiration as the soil dries out.

Consider a cross section through an unsaturated soil as shown in Fig. 4.I 2.

A portion of the cross section is occupied by solid particles and the rcmainder by

HAPTER

4

100 eppr-teo gronorocr

FIGURE 4.I.1Subsudace water zones and processes.

voids. The pctrosiry I is defined as

volume of voids

G.oundwate. outflow

( 4 . 1 . 1 )

(4. | .2)

Sol id part ic les

Air-fll1ed voids

total volume

The range for 4 is approximately 0.25 < 4 < 0.75 for soils, the value dependingon the soil texture (see Table 2.6.1).

A part of the voids is occupied by water and the remainder by air, thevolume occupied by water being measured by the soil moisture contefi e deflnedAS

volume of water

total volume

Hence 0 = 0 = 4; the soil moisture content is equal to the porosity when the soil

is saturated.

nTT

FIGURE 4.T.2Cross seclioo thiough an unsaturaied porous medrum.

sugsuRr:ce r,rrsr 101

d q

- - " q t i "

\ .

---t \

9

I

I'IGURX 4.T.3Control volume for development of the continuily equation in an unsaturated porous medium

Continuity Equation

A control volume containing unsaturated soil is shown in Figure 4' I 3 lts sides

have length .it, dy,anddz, in the coordinate dircctions, so its volume isdxdydz'

and the volume of water contained in the control volume is 0 dx dy dz The flow

of water through the soil is measured by the Darcy fu.x q : QlA, the volumetric

flow rate per unit area of soil. The Darcy flux is a vector, having components

in each of the coordinate directions, but in this presentation the horizontal fluxes

ue assumed to be zero, and only the vertical or z component of the Darcy flux is

considered. As the z axis is postive upward, upward flow is considered positive

and do*nward flow negative.In the Reynolds transport theorem. the extensive propedy B is the mass ot

soil water, hence B = dBldm = l, and dBldt = 0 because no phase changes are

occurring in the water. The Reynolds tmnspon theorem thus takes the form of

the integral equation of continu;,ty (2.2 1)l

(4 . r . 3 )

where p. is the density of water' The first telm in (4.13) is the time rate oi

change of the mass of water stored within the control volume, which is given by

( 4 . 1 . 4 )d0

: p , d x d y d z l

where the density is assumed constant and the partial derivative suffices because

the spatial dimensions of the control volume are fixed The second term in (4 1 3)

i . l

t ( ( l l li l l l p " d r + | l p , " v d . r .t l t J J J J J

t ( l ( A1 | | | p- dv: ]1p"0 dr dy dzlt l I J J J A l

Dr iL ) f lu \

102 appuro ur rroLocy

is the net outflow of water across the control surface. As shovr'n in Fig.4.1.3, thevolumetric inilow at the bottom of the control volume 1s q dx dy and the outflowat the top is lq + (dqldz)dz1dr dy, so the net outflow is

l f , ) n \I l P \ ' d A - P . 1 4 r - d t ) d x d Y - P " q d . r d t

) J \ d : / ( * . , . r )

= p - , t x , 1 1 - d z f ,

Substiruting (+. 1.4) and'(4. 1.5) into (4.1.3) and dividing by p. dr dy rt gives

( 4 . 1 . 6 )

This is the continuitl equation for one-dimensional unsteady unsaturated flow ina porous medium. This equation is applicable to flow at shallow depths belowthe land surface. At g.eater depth, such as in deep aquifers, changes in the waterdensity and in the porosity can occur as the result of changes in fluid pressure,and these must also be accounted for in developing the continuity equation.

Momentum Equation

In Eq. (2.6.'+) Darct s k\t' was developed to rclate the Darcy flux q to the rateof head loss per unit length of medium, 51:

. , - v c , (1 . | .1 )

Consider florv in the veftical direction and denote the totai head of the tlowby l,; then 51 : -nhldz where the negative sign indicates that the total headis decreasing in the dircction of flow because of friction. Darcy's law is thenexpressed as

..dh' d z ( .1. r .8)

Darcy's Law appiies to a cross section of the porous medium found byaveraging over an area that is large compared with the cross section of individualpores and grains oi the medium (Phil ip, 1969). At this scale, Darcy's lawdescribes a steady uniform flow of constant velocity, in which the net force on anyfluid element is zero. For unconfined saturated flow the only two forces involvedare gravity and friction, but for unsaturated flow the suction force binding waterto soil panicles through surface tension must also be included.

The porous medium is made up of a matrix of particles, as shown in Fig.4.1.2. When the void spaces are only panially f i l led with water, the water isattracted to the particle surfaces through electrostatic forces between the watermolecules' polar bonds and the panicle surfaces. This surface adhesion drawsthe water up around the particle surfaces, leaving the air in the center of the voids.As more water is added to the porous medium, the air exits upwards and the area

d0 dq

dt dz

suesunrec: w reR 103

of free surfaces diminishes within the medium, until the medium is saturated andthere are IIo free surfaces within the voids and. therefore, no soil suction force.The effect of soil suction can be seen if a column of dry soil is placed vertically\\'ith its bottom in a contailer of water-moisture will be drawn up into the drysoil to a height above the water surface at which the soil suction and gravity

forces are just equal. This height ranges from a few millimeters for a coarse sandto several meten for a clay soil.

The head ft of the water is measured in dimensiqns of height tfut dan alsobe thought of as the energy per unit weight of the fluid. In an unsaturatd porous

medium, the pan of the total energy possessed by the fluid due to the soil suctionforces is referred to as the suction head ,!. From the preceding discussion, it isevident that the suction head will vary with the moisture content of the medium,

as i l lustrated in Fig, 4.1.4, which shows that for this clay soil, the suction headand hydraulic conductivity can range over several orders of magnitude as the

moisture content changes. The total head , is the sum of the suction and gravity

heads

(4. r .9)

No term is included for the velocity head of the flow because the veloclty is sosmall that its head is negligible.

Substituting for h in (4.1.8)

, l ( t l t + - \-

d?.(4. r. r0)

- t on

10 "

: 1 0 "

o l 0 '

EE

ai

l on t F rcuR.E 4 .1 .4- t 0 '

l 0

Variation of soil suction bead tand hydraulic conductivity K wilhmoisture content 0 for Yolo lightclay- (Reprinted with p€rmissionfrom A. J. Raudkivi, fl]d.olr8?,Copyright 1979, Rrgamon BooksLrd.)

Soil Moisture Content

0

l o - ' n

l 04 ,qpp r reo i r vc ro roc r

= -\-#X. -): -(o( * r\

\ t1Z I( 4 . r . )

where D is the soil v,ater diffusivirJ K(dlldl) which has dimensions [LriT].Substituting this result into the continuity equation (4. 1.6) gives

(4.r .12)

which is a one-dimensional form of Riciard's equation, the goveming equation

for unsteadl unsaturated flow in a porous medium, first presented by Richards( 1 9 3 1 ) .

Computation of Soil Moisture Flux

The flow of moisture tfuough the soil can be calculated by Eq. (4.1.8) given

measurements of soil suction head ly' at different depths I in the soil and klowledge

- 5 0

-80

r00

- 180

2m

- 250

-300

a d0 d l^d0 , . tdt dz\, dz I

I

1 9 8 1

( d )

FIGURE 4.1.5{a)Profiles of total soil moi.tu.e head through time at Deep Dean in Sussex, England (So!rc?: Research

Repon l98l 8-1. Inni tute of Hydrology, wal l ingford, England, Fig. 36, p. 13. 1984. Used wi th

Soil surface

depth 0.8 m

Totr i head at dcpth 1.3 m

105

of the relationship between hydraulic conductivity r( and ry'. Figure 4. 1.5(a) showsprofiles thrcugh time of soil moisturc head measured by tensiometers located atdepths 0.8 m and 1.8 m in a soil at Deep Dean, Sussex, England. The total headh is found by adding the measured suction head { to the depth z at which it wasmeasured. These are both negative: z because it is raken as positive upward with0 at the soil surface, and 0 because it is a suction force which resists flow ofmoisture away from the location.

i loExampte 4,1,1, Calculate the soil moisture flux 4 (crn/day) between $pths 0.8m and 1.8 m in the soil at Deep Dean. The dara for total head at th6se depths

BE

E

EE

,a

20 I I. ^ r I I . I' l

, , , . y ' " r , l L [ . , r r ' . 1 , t 1 . . 1 , l . .50

,100

150

-200

250

-300

l98 l

\ .b )

FIGURE 4.1.5(')Varialion rhrough time of total soil water head h at various depths in a loam soil at Deep Dean,Sussex, England. The infiltration of rainfall reduces soil suction which increases again an evapo'transpiration drjes out the soil. Soil suclion head is the difference betwen the total head and the!alue for evaluation shown on eacb line. (Source: Research Report l98l-84, Institute of Hydrology,*?l l ingford, England, Fig- 36, p.33, 1984. Used wi th permission.)

F

106 APPLTED H\D!! i rrcY

are given rt "rcekiy t ime intervals in columns 2 and 3 of Table '1. l l For lhis

soi l the relat ionship between hydraul ic conductivi ty and soi l suction head is K =

250(- d) : Lr . where K is in centimeters per day and Iy' is in centimeters

Sorrtbr. Equation ('1.1.8) is rewrifien for an average flux q 12 between measurement

p o i n t s l a n d 2 a s

,,nt - n2. ? ! r : - ] \ -

i l - i l

ln rhis case. measurcdtnt point I is at 0.8 m and point 2 at l 8 m. so z t : -80

cm. z: : 180 cm. and ir - l l = -80 - ( 180) : I00 cm The suction head

at each depth is r l t = 1t - z. For example, for week I at 0 8 m' h1 = -145, so

{ t : h t : t = l 4 - 5 - ( - 8 0 ) = - 6 5 c m , a n d r y ' " : 2 3 0 - ( - 1 8 0 ) = 5 0 c m , a s

shown in columns'1 end 5 of the table The hydraul ic conductivi iy f varies with ly ' ,

so the value conesponding to the average of the {r values at 0.8 and I 8 m is used

For week 1, the average suction head is 0"" = [(-50) + (-65)]/2 = -57.5 cm; and

the correspondidg hydraul ic conductivi ty is K :250{-,1'*)-2 t t = 250(57.5) -2 r1 =

0.0484 cm,dai. as shown in colr-rmn 6. The head dif ference h1 h|= f 115)(-230) = 85 cm. The soi l moisture f lux beiween 0.8 and l 8 m for week I is

-h t - hzq : - 1 l -

= 0 .0481=ITJTJ

: -0.0412 cm/day

TABLE 4,I.IComputatioo of soil moisture flux between 0.8 m and 1.8 m depth at DeepDean (Example 4.1.1)

( ofumn: I I J 4Total Total Suctionhead ir head ,1 head {1at 0.8 m at 1.8 m at 0.8 m

Week (cm) (cm) (cm)

SuctionQzat 1.8 m(cm)

8Nloistureflux q

(cnl/day)

Unssturated Ileadhydraulic differenceconductivity /K /i 1 - ,t1?(crn/day) (cm)

l23

56189

l0l ll 2l 3

- r65l l 0I -tcll t 51 0 5l l 51 5 016,i190220

-230- t55-280

-230

235240240

- l,l0230

- 2 1 5210

- )40245

-255

165175285

-65-85

5060

2555'70

-85- 0

140150175200

0 .04120.0221

-0.0585-0.0443-0.0675-0.1492-0.0650-0.0354

0.02230 .01 l 0

-0.0045-0.0038-0 .00 r6-0.0003

-50 0.0484-ss 0.0320

60 0.053260 0.044360 0.0587

-50 0 .1 19335 0 .0812

-50 0.0443-60 0.029'7-6s 0.0200-15 0 .0129

85 0.010795 0.0080

- 105 0.0062

85't0

l l 0100I l 5125808015553535205

sussuxr,Act *Artn 107

as shown in column 8. The flux is negative because the moisture is flowingdownward.

The Darcy flux has dimensions [L/T] because it is a flow pet unit area ofporous medium. lf the flux is passing through a horizontal plane of area A = I m 2,

then the volumetric flow rate in week I is

Q : qA

= _0.0412 cm./day x I s12 , ta

= _4.12 x 10-a m3/day d

: -0.412 l i ters/day (-0.1I gal/day)

Table 4.1.I shows the flux 4 calculated for all time periods, and the comPutedva lueso f4 , K ,and l . I - h2a rcp Io t t ed i nF ig .4 .1 .6 . I n a l l cases t he head a t 0 .8 mis greater than that at 1.8 m so moisture is always being driven downward between

these two depths in this example. It can be seen that the flux reaches a maxlmum roweek 6 and diminishes thereafter, because both the head difference and the hydraulicconductivity diminish as ihe soil dries out. The figure shows the imPortance of thevariability of the unsaturated hydraulic conductivity K in affecting the moisture flux

4. As the soil becomes wetter, its hydraulic conductivity incrcases, because thereare more continuous fluid-filled pathways through which the flow can move.

The complete picture of rainfall on the soil at Deep Dean and the soil

moisture head at various depths is presented in Fig. 4.1.5(b). Rainfall during April

and May flows down into the soil, reducing the soil suction head, but later the soil

dries out by evapotranspimtion, causing the soil suction head to increase again.

The head profile at the shallowest depth (0.4 m) shows the greatest variability

and the fact that it falls below the profile at 0.8 m from the beginning of June

onwards shows that during this period, soil moisture flows upwards between these

two depths to supply moisture for evapotranspiration (Wellings, 1984).

r o ' )0 . 1 2

0 .08

0 0 4

0.00

-0.0,1

0.08

- 0 . 1 2

- 0 . 1 6

I 3 5 ' � ] 9 l l 13

FICURE 4.1.6Computation of the soil moisture flux at Deep Dean (ExamPle 4.1.l).

Head difference i r i, (cm x

Hldraul iL condu(t iv i ry

(

l06 .qpprteo nyororocr

4.2 INFIUI'RATION

lnfiltration is the process of water penetrating from the ground surface into the

soil. Many factors influence the infi l tration rate, including the condition of the

soil surface and its vegetative cover, the propefties of the soil, such as its porositr

and hydraulic conductivity, and the cufent rnoisture content of the soil. Soil strata

with different phlsical properties may overlay each other, forming 'horiiorrs; for

example, a silt soil with relatively high hydraulic conductivity may overlay a

clay zone of lo$' conductivity. Also, soils exhibit great spatial variabil ity e\en

within relativell small iueas such as a field. As a result of these great spatial

variations and the time variations in soil propenies that occur as the soil moisture

content chaoges. infi i tration is a very comllex process that can be described onl.v

approximately rvith mathematical equations.The distribuljon of soil moisture within the soil profi le during the downward

movement of $ater is i l lustrated in Fig. 4 2.1. There are four moisture zones:

a saturated aone ttear the surface, a tranmission:one of unsaturated flow and

fairly uniform noisture content, a v,etting zone in which moisture decreases with

depth, and a vefthg.front where the change of moisture content with depth is

so great as to gire the appearance of a sharp discontinuity beiween the wet soil

above and the dr) soil below. Depending on the amount of infiltration and the

physical propenies of the soil, the wetting front may penetrate from a few tnches

to several feet into a soil (Hil lel, 1980).The inJiLtrarion rdtel. expressed in inches per hour or centimeters per hour.

is the rate at $'hich water enters the soil at the surface. If water is ponded on the

surface, the infjltration occurs at theporential inJiLtration rare Ilthe rate of supply

of water at the surface. for example by rainfall, is less than the potential infiltration

rate then the actual infiltration rate will also be less thao the potential rate Nlost

infiltration equations ciescribe the potential ftte. The cumulatire infLltration F is

the accumulated depth of water infiltrated during a given time period and is cqual

to the integral oL'the infi l tration rate over that penod:

( 4 . 2 . l )

0 \ lo rs rure conLe i

* " -T-

r*"rrt""-"*\ lo rs lu re conLen l

\ Trrn 'u ' ,n.on.

FIGURE 4.2.IMoisture zones during infiltralion.

rto = [,,1r,t a,

(sussuRrecx *arEn 109

where r is a dummy variable of time in the integmtion, Conversely, the infiltration

rate is the time derivative of the cumulative infilfation:

' d t (4.2.2)

Horton's Equation , IOne of the earliest infiltration equations was developed by Horton (1933 1939)'who observed that infiltation begins at some mte /0 and exPonentially decreases

until it reaches a constant rate f, (Fig. 4.2.2):

f@ = f" + (fo - f"), k' (4.2.3)

(4 .2 .4 )

where t is a decay constant having dimensions [T-r]. Eagleson (1970) and

Raudkivi (1979) have shown that Hofton's equation can be derived from Richard's

equation (4.1.12) by assuming that K and D are constants independent of the

moisture content of the soil. Under these conditions (4 1 12) reduces to

at dzz

which is the standard form of a diffusion equation and may be solved to yield

the moisture content 0 as a function of time and depth. Horton's equation results

from solving for the rate of moisture diffusion D(d0lrtz) at the soil surface.

Philip's Equation

Philip (1957, 1969) solved Richard's equadon under less restrictive conditions

by assuming that K and D can vary with the mositurc content d. Philip employed

the Boltzminn transformation B(0) : zt-r/2 to convert (4 1.12) into an ordinary

differential equation in B, and solved this equatior to yield an infinite series for

! -8 6 ,\ i ; " '

= z

Time(d) Variation ofrhe parameler t

FIGURE 4.2.2Infiltralion by Honon's equation.

Time

,br lnf i l l rauon mte Jnd cumulat i \e in l l l t rat i , rn

( t , > k r )

,)

:

110 rpeLteo ur t, i.r,rt ocr

cumulative infi l lration F(t). which is approximated by

F( t ) : S t t tT + K t

where S is a parameter called Jo/plirlry-. which is a function oi the soil suctionpotential, and K is the hydraulic conductivity.

Bv differentiation

sussurrace w;rrn 111

FIGURE 4.3.1Variables in the Green-Ampt infiitration model. The venical axis is the distance from lhe soil

surface. the horizontal axis is the moisture content of the soil.

Continuity

Cqnsider a vertical column of soil of unit hodzontal cross-sectional area (Fig'1.3.2) and let a control volume be defined around the wet soil between tile surfaceand depth 1,. If the soil was initially of moisture content 0i throughout jts entire

depth, the moisture content will increase from 0; to 7, (the porosity) as the wetting

front passes. The moisturc content d is the ratio of the volume of water to the

total volume within the control surface, so the incrcase in the water stored within

the control volume as a result of infiltration is l(4 - d1) for a unit cross section.

By definition this quantity is equal to F, the cumulative depth of water infiltratedinto the soil. Hence

, f u ) : 1 5 , - ' ' * *' )

(4.2.s)

(1.2.6)]L

As r --, ' : , /(11 tcnds to K. The two terms in Phil ip s equation represent the effects

of soi i suction head and gravity head, respectively. For a horizontal column ofsoi l , soi l sucri . in is the only force drawing waier into the column, and Phil ip's

cqus t ron reduc ! ' \ t o r 11 ) = J /

Example .1.2.1. A small tube with a cross-sectional area oi, l0 cml is f i t led withsoi l and laid horizontel ly. The open end of the tube is salurated. and afler 15minures. I00 cmi of water have inf i i t rated into the tube. I f the saturated hydraul icconducti \ i tJ of thc soi l is 0.4 cn/h, determine how much int l l t rat ion would havetaken place in 30 minutes i f the soi l column had init ial ly been piaced upright withits upper surface srtumled.

Solution. Tllc cumuiative infiltration depth in the horizontal column is F = 100cml/40 cm: = 2.5 cm. Foa horizontal inf i l t rat ion, cumuial ive inf i l t ratron rs afunction ol soi l sucl ion alone so lhat after I = 15 min = 0.25 h,

F(t) = St 'r

and

2 5 = 5 1 0 2 5 ) ' / ,

S = 5 c m h - r r 2

For ini i l t rat ion down a vert ical column. (4.2.5) appLies with (: 0.4 cm/h.H e n c e . , r 1 i 0 r r 0 . 5 t"'_:i:i::,*,,

4.3 GREEN-AMPT METHOD

In the previou5 scction, infi l tration equations were developed from approximatesolutions of Richa.d'J equation. An altemative approach is to develop a moreapproximate phlsical theory that has an exact analytical solution. Green and Ampt(1911) proposed the simplif ied picture of infi l tration shown in Fig.4.3.1. Thewetting front is a sharp boundary dividing soil of moisture content di beiow iiomsaturated soil \1 ith moisture content 4 above. The wetting front has penetrated toa depth L in time r since infi l tmtion began. Water is ponded to a small depth le" l$a {,Oil rurftca

where A0 = 4 01.

Mom€ntum

Darcy's law may be expressed

,,dh- d z

( 4 . 3 . 2 )

In this case the Darcy flux 4 is constant throughout the depth and is equal to -/'

F(t): L(n - 0t)

: LAO( 4 . 3 . 1 )

l l2 apprtro st: ,rr , ,L,rr

FIGURE 4.].2Inf i l trat ion into a column of soi l of unit cross-\ecnonalarca for the G.een Ampl model.

located respecti\ ely at the ground surface and just on the dry side of the weiting

front, (.1.3.2) crn be approximated by

. , , h t - h t

l a r z 2 )( 4 . 3 . 3 )

The head it at the surtace is equal to the ponded depth h0 The head 12, in thedry soil belo\\ the welting froot, equals -ry'- l. Darcy's law for this systcnr isw tten

- ,,f ao ( tt/ - D1i/ - ^ -' L L J

( .1.3 .4)i ' t t + t l= K l -I L )

i i the ponded dcpth l i0. is negligible compared to i l, and L. This assumption is

usually appropriate for surface water hydrology problems because it is assumed

that ponded $ater becomes surface runoff- Later. it will be shown how to account

for /rs if it is not negligibie.From (.1.3.1) thc wetting front depth is L = FlA0, and assuming fto = 0'

substitution into (,1.3 .,1) gives

" , ) , l t m + f "' ' l

F )( 4 . 3 . 5 )

Since/: dFidr- (4.3.5) can be expressed as a differential .quation in the one

unknown F:

dt' *f als r r lt u " l F )

To solve for F. cross multiply to obtain

Civen K, t, t! and 40, a trial value F is substituted on the right-hand side (agood trial value is F = Kr), and a new value of F calculated on the left-handside, which is substituted as a trial value on the right-hand side, and so on, untilthe calculated values of F converge to a colstant. The final value of cumulativeinfiltration F is substituted into (4.3.7) to determine the corresponding potentialinfiltration rate /.

Equation (4.3.6) can also be solved by Newton's iteration methotl, whichis more complicated than the method of successive substitution but converges infewer iterations. Newton's iteration method is explained in Sec. 5.6.

Green-Ampt hrameters

Application of the Green-Ampt model requires estimates of the hydraulic conduc-tivity K, the porosity 7,, and the wetting ftont soil suction head I. The variation

and integrate

In the case when the ponded depth lig is not negligible, the value of ry' - ,tr 6 issubstituted for ry' in (4.3.6) and (4.3.7).

Equation (4.3.6) is a nonlinear equation in F. It may be solvedby the methodoJ successive substitution by rearranging (4.3.6) to rcad

sussuer,\ce *,\ren 113

r . l "

*j

to obtain

OI

rfO - rplo{rn [r(t + , /40 - l" ( ,rAg)]

f r l| | L , -

l a f = n a tlF + tlr\9 J

then split the left-hand side into two parts

This is the Green-Ampt equation for cumulative infiltration. Once F is found fromEq. (a.3.6), the infi l tration rate/ can be obtained from (4.3.5) or

t(-il+#) -(ffi11"=-"f F \ t ) f l

I l r ' u o \ d F I x a tJo \ F + 4/i\,0l Ja

F(t ) - q^o h ( r . r #)=

K ' (4 .3 .6 )

r , , , = x l L \ o * t \" \ r ( / ) I( 4 _ 3 . 7 \

F(t1= y, + {Arr ln (r - ffi)( 4 . 3 . 8 )

)

Y(

l14 applrro nrororocv

with moisture content 0 of the suction head and hydraulic conductivity was studied

by Brooks and Corey (1964). They concluded, after laboratory tests of manysoils, that ry' can be expressed as a logarithmic function of an etfectire saturations" (see Fig. 4.3.3). If the residual moisture content of the soil after it has beenthoroughly dmined is denoted by 0., the effective saturation is the ratio of theavailable moisture d 0. to the maximum possible available moisture contentq 0,:

' . = 0 - 0 ,n - 0 ,

where 4 tt, is called the effective porosity 0,.The eflective satumtion has the mnge 0 < s" < 1.0, provided 0. = B < 4.

For the init ial condition, when 0 = di, cross-multiplying (4.3.9) gives 0i - 0.:

s"0", and, thc change in the moisture content when the wetting front passes isL0= n 6, - q- (s"0" + d.); thercfore

A6 = (1 _ re )oc (4.3. 10)

The logarithmic relationship shown in Fig. 4.3.3 can be expressed by the Broo&r-Corey equation

( 4 . 3 . l 1 )

in which ry', and i are constants obtained by draining a soil in stages, measuringthe values of s" and ry' at each stage, and fitt ing (4.3.11) to the rcsulting data.

f,/r ln" = [ 7 ]

(4_3.9)

FIGURE 4.3.3The Brooks-Corey rcladonshipbetween soil suction head andeffeciive saturation. (So&rce.'Brooks and Cor€y, 1964. Fig. 2,p. 5. Used with permission.)

- " o . lE

i

:

b 0 .01

10 100

Soil suclion herd. l// (cm water)

sugsuRracr wATEn 115

Bouwer (1966) also studied the variation of hydraulic conductivity with moisturecontent and concluded that the effectiye hydraulic conductivity for an unsaturatedflow is approximately half the conesponding value for saturated flow.

Brakensiek, Engleman, and Rawls (1981) presented a method for determin-ing the Green-Ampt pafilmeters using the Brooks-Corey equation. Rawls, Brak-ensiek, and Miller (1983) used this method to analyze approximately 5000 soilhorizons across the United States and determined average values of $e Green-Ampt pammeters n, 0", rlt, and K for different soil classes, as shqwr{ in Table4.3. l. As the soil becomes finer rnoving ftom sand to clay the wettinglfroflt soilsuction head increases while the hydraulic conductivity decreases. Talle 4.3.1also shows typical ranges for 11, 0", and ry'. The ranges are not large for 4 and0", but ty' can vary over a wide range for a given soil. As was shown il Example4.1.1, rK varies along with /, so the values given in Table 4.3.1 for both ry'and ( should be considered typical values that may show a considerable degreeof variability in application (American Society of Agricultural Engineers, 1983;Devaurs and Gifford. 1986).

TABLE 4.3.1Green.Ampt infiltration paramet€rs for various soil classes

Soil class Forosity Effectiveporosrty

Wettingfront Ilydraulicsoilsuction conductivityheadI t K(cm) (cm/h)

Sand

Loamy sand

Sandy loam

Loam

Silt loam

Sandy clayloam

Clay loam

Silty clay

Sandy clay

Sr l ty c lay

Clay

o.43'�7(0 . 3 74-{.500)

0.43'�7(0.363-{.s06)

0.453(0.351-{.555)

0.463(0.375-{.551)

0 .501(0.42GJl. s 82)

0.398(0 .3324 .464)

0.464(0 .409-0.5 l9)

o.4'7 |

0 .417(0. 3 s4-{.480)

0.40I(o .3294 .47 3)

0.4r2(0.283 O.541)

0.434

0.486(0. 394-{. 5 78)

0.330(0.235 !.425)

0.309(0.279-0.501)

0.432

0 . 3 2 1

4 .95 I 1 .78(0.97-25.36)

6 . 13 2 .990.35-21 .94)

11 .01 I . 09(2.6'�745.4'�7)

8 .89 0 .34(1 .33 59 .38 )

16.68 0.65(2.92 95 .39\

2 1 . 8 5 0 . l 5(4.42 108.0)

20 .88 0 .10(4.79,9 1 . 10)

21.30 0.10

0.430(0.418--0.524) (0.347,4.517) (5.67 131.50)

23.90(4.08 140.2)

29.22(6.13 139.4)

(6.39-1s6.s)

(0.370-{.4m) (0.207,{.435)0.4'79 0.123

(0.42s-{.s33) (0.334-{.512)0.4'75 0.385

(0.42'74.523) (0.269-{.501)

0.06

0.05

0.03

The numbers in pdetrtheses below each pammeer ee one standard deviation aroun.i lhe pdMete-

value eiven. .toar."r Rawls. Brakensiek. and Mill€r, 1983.

(

l l6 rpprtgo ar ororocv

TwoJayer Green-AmPt Model

Consider a soil with two layers. as shown in Fig '1 3.4. The upper layer has

thickness Ar and Green-Ampt pammeters K1. ry'1' and A0l, and the lower layer

has thickness 112 and parameten K2, t!2, and A02. Water is ponded on the sudace

and the wetting front has penerated through the upper layer and a distance 12

into the lo$er layer (L2 < H2). It is required that KL > K2 for the upper layer

to remain saturated while water infiltrates into the lower layer' By a method

similar to that described, previously for one layer of soif it can be shown that the

infiltration rate is givan bY

sucsuxrecr weren 117

Ad= ( t _ . r . )d .

=(1 - 0 .3X0.186)

= 0.340

and

4/ L0 :16 . "1 x .0 .340

= 5 .68 cm I

The cumulative infiltration at r: I h is calcLriated employing rhjn"tnoa ofsuccessive substitution in Eq. (4.3.8). Take a trial value of F(r) jr(r = 0.65cm, then calculate

Ft ! \=K t + , r , :O l n { r . f #- 0 6 5 ' l - 5 . o s l n r r + 9 6 1 |' J b n

- L l t c m

Substi tut ing F = 1.27 into the r ight-hand side of (,1.3.8) gives F: L79 cm, and aftera number of iterations F converges to a constant value of 3.17 cm. The infihalionrate after one hour is found from Eq. (4.3.7):

r={H ', 5 6 R

=0.651-_ j - 1 l\ J . l /

: 1 .81 cm/h

4.4 PONDING TIME

In the preceding sections several methods for computing the mte of infiltrationinto the soil were presented. All of these methods used the assumption that wateris ponded to a small depth on the soil surface so all the water the soil can infiltmteis available at the surface. However, during a rainfall, water will pond on thesurface only if the rainfall intensity is greater than the infiltration capacity of theso ,. The ponding tin € tp is the elapsed time between the time rainfall begins andthe time water begins to pond on the soil surface.

If rainfall begins on dry soil, the vertical moisture profile in the soil mayappear as in Fig. 4.4. L Prior to the ponding time (1 < tp), the ninfall intensity isless than the potential infilbation rate and the soil surface is unsaturated. Fondingbegins when the rainfall intensity exceeds the potential infiltration (ate. At this

tlme (t = tp), the soil surface is saturated. As rainfall continues (r > /p), thesaturated zone extends deeper into the soil and overland flow occurs from theponded water. How can the infiltration equations developed previously be used

to describe this situation?

f = K t K z t a t + H t L t l' H t K t + L 2 K ,

'

and that the cumulative infi l tration is given by

Prramer.h Kr. %. d02( K r < K L )

11.3. t2)

( 4 . 3 . r 3 )F : H r L + L z L 9 z

By combining Eqs. (4.3.12) and (4.3 13) into a differential equation for L2 andintegmting. one ar.ives at

r d , . l L 2 ILr i , i ' ,_ [ ld :HrKz - lo2Kr{r ) | F l r r l ln I : -1 ; l - , r ] .1 14)-

K2 K Kt I tuz I n I

from which the cumulative infiltration and infiltration rate can be determined.

This approach can be employed when a more permeabie upper soil layer overlies

a less permeable iower layer. The normal Green-Ampt equations are used while

the wett ing front is in the upper layer; ( '1.3 12) to (4.3.14) are used once the

wett ing front (nler\ lhe lo$er layer.

Example 4,3.1. Compute the infiltration rate/ and cumulative infiltration I after

one hour of infiltration jnto a silt loam soil thal initially had an effective saturation of

30 percent. Assume water is ponded to a small but negligible depth on the surface-

Solut ion. Frcnm Table,1.3. l , for a si l t loam soi l 0" :0.486' / = 16 7 cm' and

K = 0.65 cnth. The init ial effect ive saiurat ion is.r" = 0.3' so in (4.3.1{J)

t * t tr * * *i

L pper l rycr paramelers K L | / i . Ag Hr

r lLowe. hycr t , ,

wetting fionr ----l--f-

H. FIGURE 4.3.4Paramelers in a two-layerCreen-Ampt model.

118 epprteo ur . .rr,r-ocr '

0

sussunr,ace wArEn 119

Soi lmois lure

FICURE 4.1.ISoil noisture profiles before. during,and after ponding occurs.

Mein and Lrrson (1973) presented a method for detemining the pondingtime with infi l tretion into the soil described by the Green Ampt equation forrainfall of intensily 1 starting instantaneously and continuing indefinitely. Thereare three principles inrolved: (i) pdor to the time ponding occurs, all the rainfallis infi l trated; (2) the potential infi l tration rate/ is a function of the cumulativeinfi ltration F: lnd (3) ponding occurs when the potential inli l tration rate is lessthan or equai to tl ie rainiall intensity.

In the Green-Ampt equatioo. the infi l tration rate/ and cumulative infi l trationF are related by

f : K i {Ad + l i' \ r I( 1 . 1 . 1 )

rvhere K is thc ht-draulic conductivity of the soil, O is the wetting front capil laq/pressure head. and Ali is the difterence betxeen the init ial and final moistureconrents of ihc soil. As shown in Fig. ,1.,1.2, the cumulative infi l tration at theponding time r" is given by F, : it, and the infi l tration rate by / = i; substi lutingi n t o E q . ( 4 . 4 . l ) .

' "f!t9 "so tv l nS . (1 .1 .2 )

. K , L \ 0' i ( i K)

gives the ponding time under constant rainfall intensity using the Green-Anlptrn f i l t ra t ron eqr , , . ion .

Example J.J,l. Compullr the ponding l ime and the depth of water infi l tfaled etponding i i)f a silr loam soil of 30 percent init iaL effective saturalion, subject torainfall int.rsit ies of (e) I cm/h and (b) 5 cm/h.

FIGURE 4.4.2Iniiltralion rale and cumulative infiltration for ponding under constanr inrensity rainfall.

So l zabn . F rom Examp le 4 .3 .1 , f o r a s i l t l oam so i l 4 rL0 :5 .68 cmand f : 0 .65cm/h. The ponding r ime is given by (4.4.2):

( d ) F o r i = 1 c n / h ,

K l ' 4 0t - : _" i Q - n

0.65 x 5 .68" r .0 ( r .0 - 0 .65)

= 10 .5 h

: 1 . 0 x 1 0 . 5: 10 .5 cm

0.65 x 5 .68'P

5r5 - 0 .65)= 0 .17 h (10 n in )

,, l

{x

..

E

and

( D ) F o r i = 5 c m / h ,

120 .rrprten slr,nr,t oor

and

= 5 . 0 x 0 . 1 7

= 0 .85 cm

In each case lhe inil l tration rate/ equals the rainfall intensity i ai ponding

To obtain the actuab infiltration rate after ponding, a curve of potential infil-

tration is const.ucted beginning at a time t6 such that the cumulative infi l tration

and the infiltration rate at Ip are equal to those observed under rainfall begtnning

at t ime 0 (see dashed line in Fig 4.4 2) Substituting t = lp t0 and F = ,Fp into

E q . ( 4 . 3 . 6 ) g i r c s

sucsuRr.{cr oertR 121

F is obtained by the method of successive substitution in the manner used in Example:f.3.1. The solut ion converges !o F : 3.02 cm. The conesponding rale is given by(4.3. '7):

r=dtlP * tl, 5 6 R:0 .651j j2- + l l

= L87 cm/h

L

For I ) 11,

and subtracting (,1..1.-l) from ('1 4.4),

| , r , r O ' f , . 0 1 0 - F , , 1F | / l u t n l - - 1 . : l - l n l - | K r r r o l

l l d / ' t t t )a

or

F - F r - O Ig i n l f f i l : K t t - r p t ( ' + ' 1 5 )L i l , \ d - r r p l

Equation (4..1.5) can be used to calculate the depth of inf i l t rat ion after

ponding, and lhen ('1.1 7) can be used to obtain the infiltration mte/

Example.l..l.2. Calculare lhe cumulative infiltration and the infihation rare after

one hour of rainfall of intensity 5 crn/h on a silt loam soil with ao initial effeclive

saturation of 30 Perceni.

Sol4/ ior. From Example '1 3 1, fAd = 5.68 cm and i( = 0 65 cnth for this soi l '

and from Example 4.4 1, tp : 0 17 h and Fp = 0 85 cm under rainlal l intensity 5

cm/h. Fo. t = 1.0 h. the inf i l t ral ion depth is given by (4 4 5):

I t s o + r \ - .! F , - a s g l n l d $ ' F ; J - ^ " - ' , '

1 o 8 s , o ' ' n , - t 9 l - L I - o o 5 r l o - o l - r' \ 5 .68 + 0.85/

F - rF p - t t J d l n L l t

f f i l K u , t o )

/ ' - u l 8 l n l l L l - K t t - I o ld'l a0

(4 . ,1 .3 )

(.1.4.4)

Th€se results may be compared wilh the cumulative infiltration of 3 17 cm obtainedin Example 4.3.1 for infiltmtion under continuolls ponding. Less water is infilratedafter one hour under the 5 crn4l rainfall because it took 10 minutes for ponding tooccut, and the infiltration rale during this period was less than its potentinl value

Table 4.4.1 summarizes the equations needed ior computing vanous quan-

tities for constant rainfall intensity; a set of equations is given for each of threeapproaches, based respectively on the Green-Ampt, Horton, and Philip infiltrationequations. Equations (1) and (2) are the methods for computing infiltration underponded conditions. Equation (3) gives the ponding time under conslant rainfall

intensity, and Equation (4) gives the equivalent time origin to, from which the

same infiltration rate and cumulative infiltration as those obseryed at ponding

time would be produced under continuously ponded conditions. After ponding

has occurred. the infiltration functions can be found for Horton's and Philip's

equations by substituting t - t0 into Eqs. (1) and (2). For the Creen-Ampt equa-

tion. the method il lustmted inEx. 4.4.2 can be used. In Eq. (4 4.2) under the

Green Ampt method, ponding time tp is positive and finite only if i > K: pond-

ing will never occur if the rainfall intensity is iess than or equal to the hydraulic

conductivity of the soil. Table 4.4.1 indicates that the same condition holds for

Phil ip's equation, while Horton's equation requires i > fc ro achieve ponding

Il, in the Horton equation, i >/0, ponding will occur immediately and tp = 0.

The condition / < r( holds for most rainfalls on very permeable soils and

for light rainfall on less permeable soils. In such cases, streamflow results from

subsurface flow, especially from areas near the sffeam channelDetermination of pon{ing times under rainfall of variable intensity can be

done by an approach similar to that for colstant intensity. Cumulative infi]tration

is calculated from rainfall as a function of time. A Potential infiltration rate can

be calculated from the cumulative infiltration using the Green-Ampt or other

infiltratio! formulas. Whenever rainfall intensity is grcater than the potential

infi l tration rate, ponding is occuring (Bouwer, 1978;Morel-Seytoux. l98l) For

sites where estimates of a constant infiltration rate arc available, the estimates can

be used as a guide to decide whether surface or subsurface flow is the primary

rnechanism producing flood flows (Rarce and McKerchar, 1979) This subject

is developed further in Chap. 5.

E

- =

E J

3 ;

; :

I

l

suusuxrecl werrx 123

REFERENCES

American Society of Agricuhural Engineers. Advances in infilration, Proc- Nat. Cont'. an Adyancesin InJi l t rut ion, Chicago, M., ASAE Publ . I l -83, St . Joseph, Mich. , 1983.

Bouwer. H.. Rapid field measurement of air enrry value and hydraulic conducdviiv of soil assrgnificant parameters in flow system analysis, Water Resaur. Rer., vol. 2, 129,138, 1965.

Bouwer, H., Surface-subsurface waler relations, Chap.8 in Groundwater Hldlolog),, MbGra$-Hill.New York, 1978.

Blakensiek, D. L., R. L. Engleman, and W. J. Rawls. Variation wirhin rexrure classes &soil waterpafamelers, I ranr. Am. Soc. A8t ic . Ene., \o l . 24, no. 2. pp. 335-339. 1981.

Brooks, R. H.. and A. T. Corey, Hydraulic properties of porous media. H)-drologr P.lpers, no.3.Colorado Stale Univ. , Fo( Col l ins, Colo. . t964_

de Marsily, G.. Quantitative Hydraqeolog-), Academic Press. Orlando. Fta., 1986.Devau|S, M., and C. F. Gi f ford, Appl icabi l i ty of $e Creen lnd Ampr inf i l t r i r ion equar ion to

rangelands, l rydr€r Rrrour. BuLl . , vol .22. no. l , pp. 19 2?, 1986.Eagleson, P. 5., Dynanic HydtoLoq!-, Mccraw-Hill. New York, 1970.Ffeeze, R. A. and J. A. Cherry. Groundwater, Prentice Hall, Engle*ood Clift\, N.J.. 1979.Green, W. H.. and G. A. Ampt, Studies on soil physics. part I, lhe flow of air and rater through

soi ls , J . A8r ic. Sci . . vol .4, no. I , pp. l -24, t911.Hillel , D . . Applications of Soil Pb sics , Academic Pr€ss , Orlando , FIa. . I 980.Honon. R. E., The role of infiltration in the hydroiogic cycle. 7].dnr. An. Geophrs. U ion. ral.

14, pp. 146-460. 1933.Honon, R. E., Analysis of runoff plat expe menis with varying infillration capacity, Irdnr. An.

Geophts. Union, Vol. 20, pp. 693-'711, 1939.\'lein. R. G., and C. L. Larson. Modeling infiltration durins a sready ftirl. wdt?r Resow. Res.,

vol . 9, no. 2, pp. 38f39r1. 1973.Morel-Seytoux. H. J., Application of infiltration rheory for the deremiDation of the excess rainfall

hyetograph, Wd,er Resour. Bul l . , vol . 17, no. 6, pp. 1012 1022. 1981.Rarce, A- J., and A. I. McKerchar, Upsream generation of srorm runoff. Physicdl H,-drologr.

New Zealdnd Experience, ed. by D. L. Munay and P. Ackroyd. New Zealand HydrologrcalSociety, Wel l ington, New Zealand, pp- 165 192, 19?9.

Phi l ip. J . R. , The lheory of inf ikrat ion: l � The inf i l r rat ion equat ion and i rs solut ion, Soi l Sci . , vol .8 3 , n o . 5 , p p . 3 4 5 3 5 7 . 1 9 5 ' 7 .

Philip, J. R., Theory of infiltralion, h Advanres in Hydroscience, ed. by V. T. Chotr. vol. 5, pp.215-296, 1969.

Raudkivi, A. 1., Hydrology, Pergamon Press, Oxford, 1979.Rawls, W. J., D. L. Brakensiek, and N. Miller. Green-Ampt infiltration parameters from soits data,

J. HJdraul . Dir . , An. Soc. Ciy. Eng., vol . 109, no. I , pp.62,70, 1983.R i c h a r d s , L . A . . C a p i l l a r y c o n d u c t i o n o f l i q u i d s t h r o u g h p o r o u s m e d i u m s , P h ) r i c r . v o l . L , p p . 3 l 8

3 3 3 , 1 9 3 1 .Skaggs, R. W., Infiltration, Chap. 4 in Htdroloeic Modellin| of SmaU Watersheds, ed. by C. T.

Haan, H. P. Johnson, and D- L. Bratensiek, Am. Soc. Agr ic. Eng. Mon. no. 5, St . Joseph,Mich. , 1982.

'Itrstriep, M. L., and J. B. Stali, The Illinois urban drainage area simuiatot, ILLUDAS, Bull.58,

Ill. State Water Survey, Urbana, Ill., 1974.$rellings. S. R.. Recharge of the Upper Chalk Aquifer ar a sile in Hampshire, England: L water

balance and unsaturated o\|, J. Htd l., vol. 69, pp. 259-273. 1981.

PROBLEMS

4.1.1 Figure 4.1.5(r) shows the profi les through t ime of soi l moisture head ,t , withvenical l ines ,rr weeklv intcrvrlc Celc[ lare th,. soi l moisr]rre f l , ,r , hpn.,AAn n c

i

l l +

3:

< +

- i l

i = 1 7 a- 4 = a F ! :

'l<I

€ -

;

i " . .f i' . , -1 . .

c -la

.. .,1.

126 ,rpplrro rrr l r l rocr

4.{.,1 Calculate the curnulalive infiltralion and the infiltration rate after one hour ofrainfal l at 3 cnth on a clay loam with a 25 percent ini t iai effect ive salurcl ion.

4.4.5 Compute the ponding t ime and depth of water inf i l t rated at ponding for a si1t1clay soi l !L. irh a 20 percent ini t ial effect ive satuntion subject to a rainfal l intensityof (a) 1 cdh (b) 3 cm/h.

4.1.6 Calculate the cumulative inf i l t rat ion and the inf i l t ral ion rate on a si l ty clay soi lafter one hour of rainfall at 1 cm/h if the initial effective saturation is 20 percenlAssume ponding deprh lro is negligible in the calculalions.

4.J.1 Solre Prub. 4.+.0-assumrng that any ponded water remains stal ionary over the\oi l ,o r^ rr n r "" i b" ac,o"unted lor In lhe ct lLuldt lon\.

4.4.8 Rainfal l of inlensity 2 cm/h fal ls on a clay loam soi l . and pondin8 occurs afterf ive minutes. Calculate the ponding t;me on a nearby sandy loam soi l i f bothso i l r r n r i - l ) h - J t hc . cme e f f ec t i ! e . r l u ra l i on r . .

4.4.9 A soi l has sorpti \ i ty 5 = 5 cm h-r 'r and conductivi ty I( = 0.,f cnl/h. Calculate theponding rinle and cumulative infiltration at ponding under a rainfall of 6 cnth.

,1.4.10 A soi i has Hofion s equation parameters /o = l0 cm/h,J = 4 cm,tr and l : :2h L. Calculate thc ponding l ime and cumulal ive inf i l rat ion at ponding under arainfall oi 6 cm/h.

4.4,11 Show the! ihe ponding t ime under rainfal l oi constanl intensity i for a soi ldescribed bl Phi l ip 's equation with parameters S and ,( is given by

s:(i - &2), r = } ( , i l

{. ,1.12 Show lhrt the ponding t ime under rainfal l ol intensi ly i tbr a soi i described b}Horton s equalion wiih parameters /0. /., and ll is given b1

l i ' ' I' - J - i 7 . ' n i ' o

' l lt r l l

Indicate lhe ran-se of values of rainfal l intensity lor which this equation is \al idand explain wha! happens i f i is outside this range.

CHAPTERF\

4------T-f-

sSURFACE

WATER

Surface water is water stored or flowing on the eanh's surface. The surface watersystem continually interacts with the atmospheric and subsurface water systemsdescribed in previous chapters. This chapter describes the physical laws governingsurface water flow and shows how hydrologic data a{e analyzed to provide inputinformation for models of surface flow.

5.1 SOURCES OF STR.EAMFLOW

The watershed, or catchment, is the area of land draining into a stream at a givenlocation. To describe how the various surface water processes vary through timeduring a storm, suppose that precipitation of a constant mte begins and continuesindefinitely on a watershed. Precipitation contributes to various storage and flowprocesses, as illustrated in Fig. 5.1 .1 . The vertical axis of this diagram reprcsents,relative to the rate of precipitation, the rate at which water is flowing or beingadded to storage in each of the processes shown at any instant of time.

Initially, a large proponion of the precipitation contributes to surJace stor-dge; as water infiltrates into the soil, there is also soil moisture storage. Thercare two types of storage: retention and detention: retention is storage held fora long period of time and depleted by evaporation, and detention is short-termstorage depleted by flow away ftom the storage location.

As the detention storages begin filling, flow away from them occu$: llrJalurated fiow through the unsaturated soil near the land surface, groundwater fowthrough saturated aquifers deeperdown, and overlandfiow across the land surface.

128 rppLteo s, , ,norocr

Ch.rnie l precip i rJ l ion

Channel tlow

Time since precipitalion beg.tn

ffi oerentton f net""tio"

FIGURE 5.I.ISchemaiic illusrf!iion of the disposal of precipitation during a srorm on a watershed.

Channel f ioru is the main form of surface water t low, and all the other surface flowprocesses contribute to it. Determining flow rates in stream channels is a centraltask of sudace water hydrology. The precipitation which becomes streamflowmay reach the stream by overland flow, subsurface flow, or both.

Hortonian Overland Flow

Honon (1933) described overland flow as follows: "Neglecting interceprion byvegetation, surface runoff is that part of the rainfall which is not absorbed by thesoii by infiltration. If the soil has an infiltration capacity /, expressed in inchesdepth absorbed per hour, then when the rain intensity i is less than/ the rain isall absorbed and there is no surface runoff. It may be said as a first approximationthat if i is greater than /, surface runoff will occur at the rate (i ,." Hortontermed this diiierence (l. -rf "rainfall excess." Honon considered surface runoffto take the form of a sheet flow whose depth might be measured in fractions of aninch. As flow accumuiates going down a slope, its depth increases unti l dischargeinto a stream channel occurs (Fig. 5.1.2). Along with overland flow there isdepression storuge in surface hollows and surface detention storage proportionalto the depth of the overland tlow irself. The soil stores infi i trated water and

FIGURE 5.1.2Overland flow on a slope produced by the excess of

outlet rainfail ovef infiltrarion. (After Honon, 1945, Fig.1 3 , p . 3 1 1 . )

then slowly releases it as subsurface flow to enter the stream as baseflow duringrainless periods.

Hortonian overland flow is applicable for impervious surfaces in urbanareas, and for natural surfaces with thin soil layers and low infiltration capaciryas in semiarid and arid lands.

Subsurface Flow

Hortonian overland flow occurs rarely on vegetated surfaces in humid regions(Freeze,1972, 1974; Dunne, Moore, and Taylor, 1975). Under these conditions,the infiltration capacity of the soil exceeds observed rainfall intensities for allexcept the most extreme rainfalls. Subsurface flow then becomes a primarymechanism for transporting stormwater to streams. The process of subsurface flowis illustated in Fig. 5.1.3, using the results of numerical simulations ca.rried outby Freeze (1974). hrt (a) shows an idealized cross secrion of a hillside draininginto a stream. Prior to rainfall, the stream surface is in equilibrium with the watertable and no saturated subsurface flow occurs. Pa(s (b)-(d) show how a seepagepattem develops from rainfall on surface DE, which serves to raise the watertable (e) until inflow ceases (t : 2'7'7 min), after which the warer rable declines01. All of the rainfall is infiltrated along surface DE until r : 84 min, when thesoii first becomes saturated at Di as time continues, decreasing infiltration occursalong DE as progressively more of the surface becomes saturated (g). The totaloutflow (h) partly comprises saturated groundwater flow contributed directly tothe steam and partly unsaturated subsurfaca flow seeping from the hillside abovethe water table.

Subsurface flow velocities are normally so low-that subsurface flow aionecannot conFibute a significant amount of storm precipitation directly to stream-flow except under special circumstances where the hydraulic conductivity of thesoil is very high (karce, Stewarr, and Sklash, 1986). However, Moseley (1979)has suggested that flow through root holes in a forested soil can be much morerapid than flow through the adjacent soil mass.

surrece weren 129

R a i n i a l l , ;

II

I = 0.005cm/sec

(

130 lprlno uroror rcr sr.rr.rce rr,rrrn l3l

Saturation Overland Flow

Soturalion overland fow is produced when subsurface flow saturates the soil nearthe bottom of a slope and overland flow then occurs as rain falls ortto saturatedsoil. Satumtion overland flow differs from Hortonian overland flow in that inHo onian overland flow the soil is saturated from above by infiltration. while insaturation overland llow it is saturated from below by subsurface flow. S3turationoverland flow occurs most often at the bottom of hill slopes and 1ne4, st..umbanks. t

The velocity of subsurface flow is so low thar not all of a watershed cancont bute subsurface flow or saturation overland flow to a stream during a stomt.Forcst hydrologists (Hewlen, 1982) have coined the terrrs variabLe seurce erees.or partial areas, to denote the area of the watershed actually contrjbuting flo$,to the stream at any time (Betson. 1964; Ragan. 1968; Han. l9?7; Rarceand McKerchar. 1979; Hewlett. 1982). As shown in Fig. 5.1.,1, the variabiesource area expands during rainfall and contracts thereafter. The source area forstreamtlow may constitute only l0 percent of the watershed during a storm in ahumid, well vegetated region.

r-\,

Va. inb le source area

i

/\/ . ! -

rr)

/

[ ]GURE 5.1. .1' lhe

smrl l rmo!\s in the hydrographs sho$ how srreamflow increases rs rhe vrr jable soLrrce extend\Inlo swamps, shallow soils and ephemefal channels. The process rcverses as srreamflos declines.(Reprinted fronr Prll.irles of Forest HJdrolo$ by J. D. Hewtett. Coplright 198: rhe Unive.siry of( r r ^ , n , r D f e s q R " . . l n r " l i h ,

r r = 3 1

\I \ , . = : n

l \ r r o =

i )

0 I00 200 ](xr

^ - . 0 .005_ b + L 0

r = 1 1 1 - \ l O . O O t {__- f - F

r = l l 8 : . i 0 0 0 5 -

2i? i-l>- LoF , D

Inr lo* boundary

Outflow

r j = 2 1 8 l j = 2 7 1

100 100

!o.uo: 0 . 1 5

5 o . r oF o.osO 0

FIGURE 5.1.3Saturaled unsaturated \ubsLrface flow in a small ideatized rwo dimensionai fiow system (d) Boundaryand Inr t ra l condi t ions. i r l ( , / ) Transienr hydmul ic head conrours (broken l ine) and s(eam l ines (sol id) .(d) \\ater labie .ise. fl \\:atef table decline. (g) Inflow as a function of timc and position. (rJOudo\t hydroSraph. (Srrr..r Freeze, 1914. p. 644. Copyright by the American ceophysicalU r i o n . )

{ = f f i f f i(.)

I = 2 i l m i n(.LA

(

132 apprttonvonor,rr ' r

5.2 STREAN'IFLOW HYDROGRAPH

A streanflow or discharge hydrograph is a graph or table showing the flow rate

as a function of t inre at igiven location on the stream ln effect, the hydrograph

is "an integral expression of the physiographic ald climatic -characteristics that

gorern the relations between rainfall and runoff of a particular drainage basin"

iCnorr'. 1959). T\.\'o t)-pes ol hydrographs are particularly important: the annual

hldrograph and the storm hydrograph

I

Annual HydrograPh

The annual hyclrograph. a plot of streamflow vs time over a year' shows the

long term balancc of precifitation. evaporation. and streamflow in.a watershed

Exairnples typical ol t i,r"c-tnain ty-pcs of annual hydrographs are shown in Fig'

5 . 2 . 1 .The first h)drograph. from Mill Creek near Bellevil le' Texas' has a pe''en-

nial or continuous flow regime t)'pical of a humid climate The spikes' caused

bv rain storms. are called direct runoJf or quickfow. while the slowly varving

floii in rainless periods is called baseflow' The total volume ol flow under the

annual hldrograph is the lrdsll l -rield. For a river with perennial f low most of the

basin yieid uiuoil5 .orn.. from baseflow, indicating that a large proportion of thc

rainlail is infiltratecl into ihe basin and reaches the stream as subsurface flow'

The seconrl hldrograph. from the Frio River near Uvalde, Texas' is rn

erample of an eplenera! river in an a d climate There are long periods when

the river is dry. llost storm rainfall becomes direct runoff and little infiltration

occurs. Basin yiclcl from this watershed is the result of direct runoff irom iarge

storms.The third h1<lrograph. fronr the East River near Almont, Colorado' rs

produced by a sn.,i"-fed ,iu"r. The bulk of the basin yield occurs in the spring and

earl"l summer ftom sno$ melt. The large volume of water stored i[ the snowpack'

and its steady release. create an annual hydrograph which varies more smoothl!

over the year lhln lor the perennial or ephemeral streams illustrated

Storm HydrograPh

Studl of annual hydrographs shows that peak streamflows are produced infre-

,1u"nily, ond ar" th" r"wiiof storm rainfall alone or storm rainfall and snowmelt

cornUined. Figure 5.2.2 shows lbur components of a streamflow hydrograph dur-

ing a stom. P;ior to thc lime of intense rainiall, baseflow is gndually diminishing

(sJgment AB1. Direct runoff begins at B. peaks at C and ends at D Segment DE

follows as normal baseflorv recession begins again'

Baseflow Separation

A variety of techniques have been suggested for separating baseflow and direct. ---.r r-\n,- ̂1 11,. nl,r. isr ' ' , rhe nrtrntal deoletior, curre described by Honon

Jan Feb Mar Apr May Jun Jul Aug Sep Ocr Nov Dec

(a) Mi l l Creek nc.rr Bchi l lc . Texas (wateNh€d rrcu = 17b mr: l

12

a

,l

100,000

10,000

r.000

100

l 0

I

100.000

10.000

1.000

t00

t 0

I

0 . 1

10.000

r.00t)

t00

t 0

I

0 . 1

!

FIGURE 5.2.1The annual strcamflow hldrographs for i98l from three difierent gaging station! illustralrng themrin rvne\ o fh !d ro log ic req i rner ' ra r nerenn ia l r i ver . {b ) enhemera l r i ver . ( . l in .q f r , l r i ver rDr i la

Jan Fcb Mar Apr May Jun Jul Aug Sep Ocr Nov Dec

1r) Fr io River near Uvalde. Texas (wrLershed area = 661 mi: )

Jdn Feb Var qp t \4dr JLn lu l Aug S(p O ' I No\ De(

(. ) Erst River at Almonl, Colorado (lvarershed area = 289 mir )

sunr.cce wAren 13513.1 rper reo ur oro, , tr

Hydrograph €omponcnB

AB basetlow recessron

BC - r i ! ing l in lb

CD fa l l inS l imb

DE - baseflo$ recessionl. L

{

c'

. !

:

-

FIGURE 5.2.2Components of he streamilo\thydrograph dunng a slorm

rvhere B6 is thc ilow at time 16 and k is an exponential decay constant havlng

rhe dimensions ol t ime tsingh and Stall. l97l) Equation (5.2 1) is l inearized by

plotting the logarithn oi 0(I) against t ime on a l inear scale ln Northland. New

2ealand, a typical value for,t is 6 x l0-3 days, which conesponds to a "half-

l i fe' of t 16 dals (Martin. 1973). The half-l i fe is the time for baseflow to r3cede

lo the point where Q(r)tQ6 : 0 5 The concept underlying Eq (5 2' l) is that of

a litear resenc.,it. whose outflow rate is proportional to the curent storage (see

S e c . 8 . 5 ) :

(1933). The normdl dePletion curve. or master basefow recessiotl curve' \s a

characteristic graph of f low tecessions compiied by superimposing many of the

recession cunes obserled on a given stream Recession curves olten take the

fonn of exponential decaY:

QQ) : Qoe- \ t - t " ) tK ( 5 . 2 . l )

s(t) = kQ(t)

By noting thc periods of time when the streamflow hydrograph is coincident

with the norrnal baseflow recession curve, the points where direct runoff begins

and ceases can be identif ied (B aod D on Fig 5.2.2) Between these points direct

runolf and baset-low can be separated by various methods.Some alternative methods of baseflow separation are; (a) the straight iine

method. (r) the l ired base length method. and (c) the variable slope method

These methods are i l iustrated in Figure 5.2.3The strakht line rtethod, involves drawing a horizontal line ftom the point

at which surfacc runoff begins to the intersection with the recession limb This is

applicable to ephemeral strcams. An improvement over this approach is to use an

inclined line to connect the beginning point of the surface runoff with lhe point on

the recession limb of ihe irydrograph where normal baseflow resumes. For small

forested watersheds in humid rcgions. Hewlett and Hibbert ( 1967) suggested that

baseflow durinq a stgm can be assumed to be increasing at a rate of 0.0055

l /s .ha .h (0 .05 c ts r mi rh ) .

(4) Straighr line meihod.(b) Fixed base method.(.) variable slope method.

FIGURE 5.2.3Baseflo$ separation techniques.

In the jLted base method, the suface runoff is assumed to end a fixed timeN after the hydrograph peak. The baseflow before the surface runoff began isprojected ahead to the time of the peak. A straight line is used to connect thisprojection at the peak to the point on the recession limb at time N after the peat.

ln the variable slope method, the baseflow curve before the surface runoffbegan is extmpolated forward to the time of peak discharge. and the baseflowcurve after surface runoff ceases is extrapolated backward to the time of the pointof inflection on the recession limb. A straight line is used to couect the endpointsof the extrapolated curves.

5.3 EXCESS RAINFALL AND DIRECTRUNOFF

Excess rainfalL, or effective rainfall, is that rainfall which is neither retained onthe land surface nor infilhated into the soil. After flowing across the watershedsurface, excess rainfall becomes direct runoff at the watershed outlet under theassumption of Hononian overland flow. The graph of excess rainfall vs. time, orexcess rainfall hyetograph (ERH), is a key component of the study of rainfall-runoff relationships. The difference between the obseryed total rainfall hyetogmphand the excess rainfall hyetograph is termed abstactions, or losses. Losses iueprimarily water absorbed by infiltration with some allowance for interception andsurface storage.

The excess rainfall hytograph may be determined from the rainfall hyetograph in one of two ways, depending on whether streamflow data are available for

Time

136 eppusn Hyonr)locv

the stom or not. In this section. it is assumed that streamflow data are available.Sections 5.4 and 5.5 show how to calculate abstractions when streamflow dataare not available.

Suppose thar a rainfall hyetograph ard strcamflow hyetogruph are avail-able, baseflow has been separated from streamflow to produce the direct runoffhydrogmph, and rhe excess rainfall hyetograph is to be determined. The param-eters of infi l tration equations,can be determined by optimization techniques suchas nonlinear prolramming,(Unver and Mays, 1984), but these techniques arecomplicated. There is a simpler altemative, called a {-index. The d.index isthat constant rate of abstractions (in/h or cm/h) that will yield an excess rainfallhyetograph (ERH) with a total depth equal to the depth of direct runoff r7 overthe wate$hed. The value of @ is determined by picking a time interval lengthAt. judging the nunrber of intervals M of rainfall that actually contribute to directrunoff, subtracting dA, fi'om the observed rainfall in each interval. and adjustingthe values of @ and M as necessary so that the depths of direct runoff and excessrainfall are equal:

sonpecr *iqrer 137

TASLE 5.3.IRainfall and streamflow data dapted from the storm of May 2rl-25, 1981, onShoal Creek at Northwest hrk, Austin, Texas

ObservedTime Rainfall Streamflow

(in) (cfs)

Excess rainfallTim€ hyetograph @RH)lj hl (in)

Direct runoll

Columni 1

, r : 1 1 n . - 4 ' 1 4

Fhere R,, is the obserled rainfall ( in) in time interval m.

( 5 . 3 . 1 )

Zl May 8:30 P.M.9:009:30

l0:0010i30l l i 00I l :30

25 May 12:00 A.M.l2:30l:00t : i 02:002.303:003 :30,l:004:10

I23456'7

89l0n

Torat 4.80

0 . 1 50.261.332.202.080.200.09

203246283828

569'79531

I10258234432122461802t2301 t3394354303

1 .06L93t . 8 l

428192352919 l ] t

10625183439211846t1{28303 1 3

43550

Example 5.3.1, Deiermine the direct runoff hydrograph, lhe dLindex, and rheexcess rainfall hyetograph from the observed aainfall and streamflow data givenin Table 5.1. l . The watersbed area is 7.03 mir.

Solut ion, The basin-average raintal l data given in column 2 of Table 5.3.1 wereobtained by taking Thiessen-weighred averages of the rainfall data from two rainfallgages in lhe \\ atershed. (ldeally. dara from several more gages would be used. ) Thepulse data represenration is used for rainfali with a time interval of At = l/2 h. soeach value shown in column 2 is the incremental precipitation that occuned duringthe half-hour up to the time shown- The streamflow dala shown were recorded assample da!a: rhe value shown in column 3 is the streamflow recorded at that instantof t ime. The observed rainfal l and streamflow dara are plotted in Fig.5.3.1, fromwhich it is rpperent that rainfall prior to 9:30 p.M. produced a small flow in lhestream (approximatel) 400 cfs) and that the direct runoff occuned following intenserainfal l betreen 9:30 and l1:30 P.M.

The computation of the effective rainfall hyetograph and the dircct runoffhydrograph uscs the following procedure:

Srep 1. Estimate the baseflow. A constanr baseflow rate of,100 cfs is selected.Srep 2. Calculate the direct runoff hydrograph (DRH). The DRH, in column

6 of Table 5.3.1, is found by rhe straight l ine method, by subtracting the 400 cfsbaseflow from the observed streamflow (column 3). Eleven half-hour time intervalsin column ,t are labeled from the first period of non-zero direct runoff. beginningat 9:30 P.M

.lrep J. Compute the \,olLrme y,l and deplh r, of direct runoff.

Excess rainfall = observed minfall - abstradions (0.27 in per half-hour)

Direct runoff = observed streamflow - bas€flow (400 cfs)

, o : \ 9 , o ,

= 43,550 cfs x l/2 h

frj ?6nn . l

s l h t

:7 .839 x 107 f r3

v,ld = ------:=--*

watefsneo area

_ 7.839 x t0? ftr'l .03 mi2 x 52802 lt2ltlti2

: 0.400 ft

:4 .80 in

138 .rpplrro ryororocr

7 8 . , r r 0 n 1 2 r 2 3 4P r,t . A. M.

! rA \ 24 25 . r 98 l

FIGURE 5.3.1Rainfatl and streamilor ibr rhe storm of May 24-25, 1981, on Shoal C.eek ar Norihwest pafk,

Srep 4 Estimate the mle of rainfall absrractions by infiltraiion and surfacestorage in the watershed. Any rainfall prior to rhe beginning of direct runoff istaken as ir i i . r l abstrdction ( i .e., that rainfal l pr ior to 9:30 p.M. in Table 5.3.1).The abstraclion rate d, and M, the number of nonzero pulses of excess rainfall, areibund by triqi and enor.

l . I fM = l . lhe la.gest rainfal l pulse, R. = 2.20 in, is selected, substi tured intoEq. (5.3.1) using 17 : ,1.80 in and At = 0.5 h, and solved for a tr ial value of$:

rt: ),6. dtn

4 8 0 = ( 2 . 2 0 d x 0 . 5 )

d= 5.20 ir/h

which is not phlsicaliy possible.2. If M:2. the one-hour period having the highest rainfall is selected (between

10:00 P M and I l:00 P M.) and substituted into (5.3.l) !o solve for a new rrial

surr,cce lrArEn 139

value of d:g

1= ) , (R, " - $L)

. * = o l o + 2 . 0 8 d x 2 x o . 5 )

6 = _ 0.52 inlh r

again impossible. t

3. l f M : 3, the I j hour period having pulses l �33. 2.20, and 2.08 in Atselected,and the data is substi tuted into (5.3.1):

. { .r d : L 6 . - 6 L t )n = l

4 .80 = (1 .33 + 2 .20 + 2 .08 d x 3 x 0 .5 )

4 = 0 5 4 i n / h

This value of d is satisfactory because it gives {At : 0.27 in, which is greater

than all of the rainfall pulses in column 2 outside of the tkee assumed tocontribute to direct runoff.

Srep J. Calculate the excess .ainfall hyetograph. The ordinates (column 5)are found by subtracting OLt = 0.2'7 in from the ordinates of the observed rainfailhyetograph (column 2), neglecting all inteflals in which the observed rainfall depthis less than dAt. The duration of excess rainfall is 1.5 h in this example (9:30 to

I I r00 P.M.). The depth of excess tainfall is checked to ensure that it equals rr' (total

of column 5 : 4.80 in). The excess portion of Ihe observed rainfall hyetograph rs

cross-hatched in Fig. 5.3.1.

Runoff Coefficients

Abstractions may also be accounted for by means of ran off coefrcients The mostcommon definition of a runoff coefficient is that it is the ratio of the peak rateof direct runoff to the average intensity of rainfall in a storm. Because of highlyvariabld minfall intensity, this value is difficult to determine from observed data.A runoff coefficient can also be defined to be the rutio of runoff to rainfall over agiven time period. These coefficients are most commonly apPlied to storm rainfalland runoff, but can also be used for monthly or annual rainfall and streamflow

data. If > {=, n^ i" the total rainfall and r7 the corresponding depth of runoff,then a runoff coefficient can be defined as

L : M

) n ,

€

.9

=

E

E

(5.3.2)

140(

APPLIED H\ ' ] ]ROLOC\

Example 5,3.2. l letermine the runoff coeff icient for the storm in Example 5.3.1.

Jolalion. Considering only the rainfall rhat occurred after the beginning of diiectrunoff (9: i0 P M.):

\ p r l l + r ) n r r r o , ̂ 1 ^ _ 0 . 0 9

= 5 .90 in

) n -n = l

,1.80

5.90: 0 . 8 l

5.4 ABSTRACTIONS USING INFILTRATIONEQUATIONS

Abstractions include ittterception of precipitation on vegetation above the ground.depression stardge on the ground surface as water accumulates in hollows overthe surface, and infrLtration of water into the soil. Interception and depressionstomge abstractions are estimated based on the naturc ofthe vegetation and groundsurface or are assumed to be negligible in a large storm.

1n the previous section, the rate of abstractions from rainfall was determtneob)'' using a known streamflow hydrograph. In most hydrologic problems, thestreamflow hrdrograph is not available and the abstractions must be determinedby calculating infiltmtion and accounting separately for other forms of abstrac_tion, such as interception, and detention or deprcssion storage. In this section, it isassumed that all abstractions arise from infiltration, and a method for determiningthe ponding time and infiltration under a variable intensity rainfall is developedbased on the Green-Ampt infiltration equation. Equivalent relationships for usewith the Honon and Philip equations are presenred in Tabie 5.4. l� The problemconsidered is: given a rainfall hyetograph defined using the pulse data repre_sentation, and the parameters of an infiltntion equation, determine the pondingtime, the infiltrarion alter ponding occurs, and the excess rainfall hyetograph.

The basic principles used for determining ponding time under constanr rain-fall intensity in Sec. 4.4 are also employed herc: in the absence of ponding,cumulative infiltration is calculated from cumulative rainfall; the Dotential infiltration rate at a given time is calculated from the cumuladve infiitration at tnattime; and ponding has occurred when the potential infiltration rate is less than orequal to the ninfall intensitv.

+ 3

i =

Eg

(

:

u l r - - l :t 5 % l1 6 r l e :

a ;-f _i;_

ni

i lr l

< {:." i l

I

1

< l a: 11+ t -i l

. . l l \ - _ l !

+ . 8: g !

> 3 = s l ; E

, . t , 3 s l c i . , *.E i :

-=- . i 3 l :

i I T I E v t . - i /: 1 ) ; . ! j= - s E r.: {r- f \: .= Lra. q ] -'! .E .ii F +E : E= :\ , , E ( Jn . r

-1",

s)t ::

'"i

+

II

)

F

142 rpprrco sYororocY

Consider a time intervai from , to t + At The rainfall intensity during

this interval is denoted ir and is constant thrcughout the interval The Potential

in f i l tn t ionra teandcumula t ive in f i l tmt ionat thebeg inn ingof the in te rva la re / 'and Fr, respectively, and the corresponding values at th€ "nq

9{ t]t" inteNal are

/, r,t,, and i, *.,. ti i, assumed that F, is known from given initial conditions or

previous comPutation.A flow chart for determining ponding time is presented in Fig 5 4 1 There

are three cases to be considered: (1) ponding occurs throughout the interval; (2)

there is no ponding throughout the intewal; and (3) ponding begins part-way

thrcugh the interval. The infiltration rate is always either decreasing or constant'ith ime, so once ponding is established under a given rainfall intensity' it will

continue. Hence, ponding cannot cease in the middle of an interval' but only at

its end point, when the value of the rainfall intensity changes

Following the flow chart' the first step is to calculate the qrrrent potential

infiltration rate /, from the known vaiue of cumulative infiltration Fr' For the

Green-AmPt method, one uses

sunrece uren 143

FIGURE 5,4.1Flow charl for determining infittration and ponding rime under vafiable rainfalt intensirr

t a _ F

i l(s.4.s)

and the cumulative infiltration F,11, is found by substituting Fr = F, and A/ =At - Al' in (5.4.2). The excess rainfall values are calculated by subhacting

/, = ̂{ry-' ')\ ' t I

( 5 . 4 . 1 )

The result /, is compared to the rainfall intensity i ' If /' is less than or

equal to i,, case (1) arises and there is ponding throughout the interval. In this

"us", fot ih. Green-Ampt equation' the cumulative infiltration at the end of the

interval, F,11,, is calculated from

F,+,s, - \- {Ao rn In'it.rti'l = "t

This equation is derived in a manner similar to that shown in Sec 4 4 for Eq

14.4 .5) .' goth cases (2) and (3) have/, > i, and no ponding at the beginning of the

'nterval. Assume that this remains so throughout the interval; then, the infiltration

rate is l, and a tentative value for cumulative infiltration at the end of the time

interval is

F;+!, : .F. + irA/ (5.4.3)

Next, a corresponding infi l tration rate/i11 is calculated frorn )' l1v If/ ' ,1;is

sreater than i,, case (2) occurs and there is no ponding throughout the interval'

ihus F,1,1, : Fi+,rr and the Problem is solved for this interval

Ifl;-j, is less than or equal to l,' ponding occurs during the interval (case

(3)). The cumulative infiltration Fo ar ponding time is found by setting /t = l,

ia r, = r " in (5.4. 1) and solving for Fp to give' for the Grcer-Ampt equation '

, K4' L0i , K

The ponding time is then I + At', where

(s.4.2)

At trme r, cLimulaiivejnnltration. F, is

(2) No pondirgthroughout

i n t e r v a l :

rarnfal l .

( l ) Ponding occursthroughoul interval :

F/,a, calculaiedby infiltration

(3) Ponding occurs dur ing

Calculate 4, from 1,. findAi = (.Fp Ft)li!

and calculate 4*a, from

by infiltration equation.

(5 .4 .4 )

144 rppr_teo syonorocy

:ffii*5:l?Tilion ftom cumulative rainrall, then taking successive dirrerences

F1lTrj;,: i;l.T,i"ill,ii::x"Jrr,l, qven in coilrmns I and 2 or rrbre 5 4 ?d.t",-'n. rh.-"... '. ;#;;;i #&;1i

"' initial efre'ti\e sarurarion 40 percen'.

Solution. From Table 4.3.1. for aa n d A : 0 . z g r . n , o . E q . 1 a . : . t o ] " n d y l o a m s o i l , , ( : 1 . 0 9 c m / h ,

r y ' = I l . 0 l c m i

Ad: ( l r " )4

= (1 - 0 .aXo:112)

Q A0= 0 .247 x | .O l

lT.:11:-,1*.*l in rabre 5.4.2 , "alq.ln = 0 167 h. corumn 3 orrhe raoresnows Ihe cumulative rainfal l deDth5

*, "*rri

* jf ,lt *q:i:_"--,' ilii{#r qil##'u : fi #;;.';li:l::;"1$jt**T:rj?,':11;t : {:r:} :: ;if*', ;T','i ";;ffi"t t:; j: il;:.,;,:; i,.T..,:,;:,fi[ 3;J:fflil ii ?: :i,iii,;

/;_.,=d_tlg * rl

: t . o s l 2 7 2 * , l\0 .18 , l

::j1,:T::,:-: , ", ,* .".- ln,',tu""T11.,, ,."o.. than 1,; rhererore, uo#:lii't,:[Hf TXt",T;"'T:;"i.?1 -"11*tiue inrilt'arron equai, li,ii,L,'*minutes of rainfal l , U", ", ini . ;" .

rJ .ound that ponrlrng does nor occur up ro 60

rr1# * |I

: r .os l f i . r ):2..1i cmth

uh i ch i . I e . . r han , . . - J g4 c rn , h l o r t h (begins ar 60 *,n rr* e,g s.i.iirl'

''e rnterval from 60 to ?0 minutes' so ponding

O

'1

ri

F-

^ t

d l .

' ; t

. = l

' : l

< t

9 l

- I

' ; lr l , E: l - . E

9 l

- t

I

? l

€ - : : : . r €o r - , c ; c j J

s e i f F :

c . " 1 . : j r l

u i . a I

. r . g l11I

5 1 ! a l aE i d e i : . : : 3 :d t i- t ^ i

I . < ll E E l a ' : a : j ;l . l 9 l I € - , s - ,

III

i i € l s e cl ; E l - ? ^ r ] 5 ^ j ;l r : l -

- - - ^ j il

J Il ll g 1

l 5 F 1 = = a ? i s l1 i ! l - r o d d " j -1 l1 lli t

u t i= L - 1 - ' r ^ - ; :. _ 1 . . - . - :

II

F . : 1 - - _F ! i - : t 3 $ e . 6

i q € €

i .16

During the ponded period, (5.,{.2) is used to calculate infiltration. The value ofFr-r al 70 mlns is given by

f - . ̂ tr r F, aLa'"

f l4r, ' . ,A;,1 _ "_r,

t - . . . 1r r t -- z.'72 h r!)t-*

: :: l - r.oo o. rorI t . t t . t z . t z I

or

/ - = r u s - , , , , " , l t _ - - L l ]- ' - \ 1 . 4 9 I

{ . r infa l l andnf i l t rat ion rate

20

l 8l 6

l 2

l 086

20

i u m u l a t i r e r a i n f a l lInd inf i l t rat ion

,alnnialz z ab(lracrion

N: conrinuing!--.! ahsrracrion

_ + Porenrial

r: e - Rainfall

tr Rainfrll. ln i i l t rat ion

Tl l 3 7 c m

. - lI

L 2

- l o

1

aE- ) )

00 20 .10 60 E0 100 120 140 t60 t80

Time (nin)

{ r l

. IGURE 5.4.2r f i r f r " r ; . n . n n a r . , . ( ! . , . ; - t r l l I n r l - r v r f l ? h l e r r i n f r I i n r F n \ i l v r F y t m n l p s 1 1 i

5.96

stnr,qcr wxrsn 147

which is solved by the method of successive approximation to give.lcr +r'r = 2 21

cm as shown in column 6 of Table 5.4 2 The cumulative excess rainfall (column

7) is found by subfacting cumulative infiltJation (column 6) from cumulative

rainfall (column 3) And tbi excess rainfall values in column 8 are found by taking

differences of successive cumulative rainfall values Ponding ceases at 140 min

when the rainfall intensity falls below the potential infiltratioo rate Aftql 140 min'

cumulative infiltration is computed fiom iainfall by t5 4 3) For ex?mde' at 150

m i n F , - 4 , = F , - i , A , t - 4 5 3 + 0 2 8 : 4 8 1 i n a s s h o w n i n c o l l r n n t l .eJ.lown in Fig. 5,4.2' the lotal rainfall of 11 37 cm is disPos'd of as an

init ialabstractionofl .?7cm(cumulativeinf i l trat ionatpondingtime)'acontinuingabshaction of 3.64 cm (5 41 cm total infilhation 1 77 cm initiai absffaction)'

and an excess rainfall of 5.96 cm.

5.5 SCS METHOD FOR ABSTRACTIONS

The Soil Conservation Service (1972) developed a method for computrng absrac-

tionsfromstormrainfall.Forthestormasawhole,thedepthofexcessprecipita-tion or direct runoff P, is always less than or equal to the depth of precipitation P:

likewise, after runoffiegins, the additional dePth of water retained in^the water-

shed, F., is less than oiequal to some potential maximum retention S (see Fig

s.l. il. irr*" is some amount of rainfalll. (initial abstraction before Ponding) for

which no runoff will occur, so the potential runoff is P - I '' Tt\e hypothesis of

ttre Sis metnoa is that the ratios ofthe two actuai to the two potential quantities

are eoual. that is,F

s P _ 1 "( 5 . 5 . t )

( 5 . s . 2 )From the continuity PrinciPle

P : P " + l o + F '

Combining (5.5.1) and (5 5.2) to solve for P" gtves

FIGURE 5.5.7Variables in the SCS melhod of rainfail

abstractions: 1" = initial absFaction'

P. : rainfall excess, F, : continuing

l J8 rpp r r ro s ro rc :

t D - t \ 2

$hich is the basic equatiol for computing the depth of excess rainfall or directrunoff from a stor.nl by the SCS method.

By study of rcsults from many smali experimental watenheds. an empiricalrelation was developed.

1 , = 0 2 5

( P - 0 2 s FP + 0.8S

(s . s .3 )

(5 .5 . .1 )

( 5 . 5 . 5 )

On this basis I

Pl()tt ing the data l0r' P and P" from many watersheds, the SCS found curvesof the type shown in Fig. 5.5.2. To standardize these curyes. a dimensionlesscurve number CN is defined such that 0 = CN = i00. For impervious andrvater surfaces CN : 100: for natural surfaces CN < 100. As an illustration. therainfall event of Exampie 5.3.2 has P" = 4.80 in. and P = 5.80 in. From Fig.5.5.2. it can be seen that CN = 9l for this event.

0 1 2 3 ' 1 5 6 ' � 1 8CumuLrr i !e rarnla l l P rn inches

TIGURE 5.5.2Solur ion of the SCS fur l , r f f equar ions. (Sorr .er Soi l Conservr l ron Service,t 0 . l t )

- i P r ) 1 5 )'

P r rEs

cu c Number cN = llgll 0 + 5

1972 , F ig . 10 . I , p .

suance wrr=n 149

The curve number and S are related by

s=f,ff-rowhere S is in inches. The curve numbers shown in Fig. 5.5.2 apply for normalantecedent moisture conditions (AMC II). For dry conditions (AN{C I) or werconditions (AMC IID, equivalent curve numbers can be computed Lr!

cN(l) = --i.29x1]I) t

,.. r'tl0 - 0.058CN(l l )

and

CN(IID :23CN(rr)

10 + 0.13CN(I I )15 5 t { \

The mnge of antecedent moisture conditions fbr each class is shown in Table" " 'Curve

numbers have been tabulated by the Soil Conservation Seryice on thebasis of soil type and land use. Four soil groups are defined:Croup A: Deep sand, deep loess, aggregated siltsGroup B: Shallow loess, sandy loamGroup C; Clay loams, shallow sandy loam, soils low in organic content, and

soils usually high in clayGroup D: Soils that swell significantly when wet, heavy plastic clays. and

cetain saline soilsThe values of CN for various land uses on these soil types are given in Table5.5.2. For a watershed made up of several soil types and land uses, a compositeCN can be calculated.

Example 5.5.1 (After Soil Conservation Service, 1975). Compute the runoff from5 inches of rainfall on a 1000-acre wate.shed. The hyd.ologic soil group is 50percent Group B and 50 percent Group C interspersed throughoui the watershed.Antecedent moisture condition II is assumed. The land use is:

40 percent residential area that is 30 percent impervious12 percent residential area that is 65 percent impervious

TABLE 5.5.1Classification of antecedent moistur€ classes (AMC)for the SCS method of rainfall abstractions

Total s-day ant€cedent minfall (in)

AMC group Dormant season Grow;ng s€ason

IIIm

Less than 0.50 . 5 t o l . lOver L l

Less than 1.41 .4 t o 2 .1Over 2.1

l0 rppL teo Hlonorocr

ir"rtjut;1'."" numbers for selected agricultural ̂ suburban' and urban land

-'", t"ni*""aunt moisture condition ll ' I ' -- 0'25)

,-and Use DescriptionHldrol0gic Soi l GrouP

lFor a mole detailed d.i.iip(ion oi rgoculrural rand use curve nunbers reler io Soii Conservation SeNtre Lq?l'

Chap 9

2cood coler is prorectcd ;r!dr grJrrog and liller and btush covcr sorr'

lcune numbeb de conrtulcd assuning tne runoff from fte house and dileway is directed toqards rhe streei

;;;;;';;;;i,;",;,,", '11'"'1g6 ro h*ns 'r"'e addnionar inlrtratron courd occur'

,lThe.emiinine perviou' rr$s rlieir are consroereC to be in good pasture condition fo' hese cu'!e numbers

5ln some *armer cli'r!i!' ni rh' ' )!nrr) I curle nunber of 95 mat be used

. lu l t ivaled land l : $ i thot i t 'onser\ i l t ioq t reatment

wilh c.' n sen- a!ion lreatmenl

Prsturc or range land: Po0r cond'oon

g!\rd contlrtron

luerdo\ ! : good condi l rurr

Wood or forest land: th in n lnd poor cover ' no mulch

gi \ )J co\er l

Open Spaces. la$ns. prrk\ ' goi i 'ourses cemeter ies etc '

good condition: grits\ colel on 75q' of more of rhe area

fair condilion: glit\s co\eI nn j07' to 75q' of the area

a.nrmercia l aod businesi areas i857' impervrous)

12

62 7 l

88

?8

9 l

68

39

'79 86'74

89

80

l0'71 78

25 55

11

70

8l

11

39

49 69

14 EO84

89 92 94

lnd, ,srr i r l drs l r ic ts ( l l ' '_ : Lnrper\ roLrs I 8 l 88 9 1 93

Resident ia i l :

L/E acre or less

l12 ac.e

A!€rage q. impervious4

65

_'i8

t0

l5l0 r\1

90

83

8 1

80

19

92.

ul

86

85

8,1

Paved parhing lots. ioois. dr i lewa-\s e lc l

paved with cu.bs a d \torm icwers5

gravel

989898 98

98

16

12

98

85

82

9E

89

87

98

9 l

89

sr,rrlcr warrn 151

18 percent paved roads with curbs and storm sewers16 percent open land with 50 percent fair grass cover and 50 percent goodgrass cover14 percent parking lots, plazas, schools, and so on (a11 impervious)

Sorrlrbr. Compute the weighted curve number using Table 5.5.2.

Land Use Product % CN Product

Residential (30% impervious)Residendai (657o impervious)RoadsOpen land: Cood cover

Fair coverhrking lots, etc

20 8 l6 9 09 9 84 744 '�79"t

98

50

20 726 8 59 9 84 6 t4 6 9'7 98

50

14405 1 08822M216686

4038

1620540882296

686

4340

Thus,

4ntR - 4140Weighled CN - -:-: - = 8J.8

. - l f f i - , . ,' - c N

- ' "

- .l!90 - 1683 .8

= 1.93 in

^ (P 0 25 r zr . : ''-"i-.J,'i"n"'

5 + 0 . 8 x 1 . 9 3:3 .25 in

Example 5.5.2. Recompute the runoff irom this watershed if the wet conditions ofantecedent moisture condition III are applicable.

Solution. Find a cwve number for AMC trI equivalent to CN = 83.8 under AMCIl using Eq. (5.5.8):

23CN(n)CN(III) :

10 + 0.13CN0I)

23 x 83 .8t 0 + 0 . 1 3 x 8 3 . 8

= 92.3

Itydrologic soil group

152 rpprtto uvon,r ,- 'cr

Tben .

s = f f - r o100092.3

= 0 .83 in

. P.=-#

5 + 0 . 8 x 0 . 8 3

: 4 . i 3 i n

The change in runolT caused by the change in antecedent moisture condit ion is

{ . l J I l r ' o l . i n r l - pe rcen t i nc re3 'e

Urbanization Effects

During the past i5 to 20 years, hydroiogists have paid considerable arrention to

the efiects of urbanization Early works in urban hydrology were concemed with

rhe effects of urbrnization on the flood potential of small urban watersheds The

effecrs of urbanization on the flood hydrograph include increased total runoff

voiumes and pcak flo* rates. as depicted in Fig 5 5'3 In general' the najor

changes in flor. rates in urban watersheds are due to the following:

1. The voiume oi water available for runoff incteases because of the increased

impervious corer provided b1' parking lots. streets, and roofs' which reduce

the amount ol infihration

. | l r f i l raoon bc fo te u .banrza t 'on

. - l i r tL t ru l ion l f l c r u rbanrza t lon

FIGURE 5.5.3Thc effect of urbanization

I

sunr.^.cr weren 153

2. Changes in hydraulic efficiency associated with artif icial channels, curbing,gutters, and storm drainage collection systems incrcase the velocity of flowand the magnitude of flood peaks.

The SCS method for rainfall-runoff analysis can be applied to determine theincrease in the amount of runoff caused by urbanization.

xxample 5,5.3 calculate the runoff from 5 inches of rainiall,or][a 1000-acrewarershed. The soil is 50 percent Group B and 50 percent Grou-f C. Assumeantecedent moisture condition Il. The land use is open land with fair grass coverbefore urbanization; after urbanization it is as specified in Example 5 5.1- Howmuch additional runoff is caused by urbanization?

Solution. The curve numbers for open land wilh fair gtass cover are CN = 69 forGroup B and 79 for Group C, so the average curve number for the walershed is CN= (69 + 19)12 = 74. From (5.5.6), S = (lOD0l'74) - 10 = 3.51 in. The excessminfall or dircct runoff P" is calculated from (5.5.5) with P = 5.0 in:

^ (P - 0.2S)rt " : p r o : s

_ ( 5 . 0 0 . 2 x 3 . 5 1 ) r5 . 0 + 0 . 8 x 3 . 5 1

: 2.37 in (before urbanization)

After urbanization, Example 5.5.I shows P" = 3 25 in. so the impact of urhaniTetionis tocause3.25 2 .37 :0 .88 ino f add i t iona l runof f f rom th is s tom. a27 percentincrease,

Time Distribution of SCS Abstractions

To this point, only the depth of excess rainfall or direct runoff during a stormhas been computed. By extension of the previous method, the time distributionol abstractions F. within a storm can be found. Solving for F o from Eqs (5 5. i)

and (5 .5 .2 ) ,

S , P I \E - " ' - 3

D > tt - cr - t a - J

Differentiating, and noting that 1o and S are constants.

dF"

( s .5 .9 )

s2dPtdtd t l P - 1 . + S ) z

( 5 . 5 . 1 0 )

As P --') -, (.dF.ldt) -> Q as pquired, but the prcsence of dPldt (rainfall intensity)in the numerator means that as the rainfall intensity increases, the rate of retention

of water within the watershed tends to increase. This property of the SCS method

may not have a strong physical basis (Morcl-Seytoux and Verdin, l98l).In application, cumulative abstmctions and minfall excess may be deter-

mjned either from (5.5.9) or ftom (5.5.5).

: ,7 Add i t ion l l cxcess ra in la l l

/ ' z caused br ' u tban iza l ron

Al le r u rbanrza l ron

(

15:l repr-rsn nYoeoio,,t

Example 5'5 {. Stoml rainfalL occurred on a watershed as shown in column 2 ol

Table 3.5.3. Ths lalue of CN is EO and Antecedent Moisture Condidon II appl ies

Calculale the cumulatrre abslract ions and the excess rainfal l h!etogmph

So/.rtr 'or. Fof CN = 80, S : (1000/80) - l0 = 2 50 in; 1, = 0 2J : 0 5 in The

initial abstractiL'n absorbs all of the tainfall up to P = 0 5 in This includes the 0 2

in of rainfall occurring during the first hour and 0 3 in of the rain falling during

the second hour. For P > 0.5 in, the continuing abstraction Fo is computed from

(s .5 .9 ) :

. - s ( P - t " )P - 1 " - 5

_ 2.50(P 0.5)P - 0 . 5 + 2 . 5 0

_ 2 .50(P - 0 .5 )P + 2 . 0

For example. , i l ier !*o hours. the cumulati le rainfail is P : 0 90 in' so

, . - 2 . 5 0 ( 0 9 - 0 5 )o 9 + 2 0

= 0 .34 in

as shown in c\-.lumn'1 ofthe table The excess rainfall is that remaining afier initiai

and continuing abstraclions From (5.5 2)

-omputation of abstractions and excess rainfall hyetograph by the

SCS method (Example 5.5.4)

2 3 4 5Cumulati \ t

Lumulauvecumulatire rb.t.actions rint

exltls.. ^

rainfall P - ralntall r'(ini I. F, (in)

Excessrainfallhyetograph(in)

5.6 FLOW DEPTH AND VELOCITY

SUnn,lCt r,rrex 155

D D - I F

= 0 9 0 - 0 . 5 0 - 0 3 4

: 0.06 in

as shown in column 5. The excess rainfall hyetograph is determined bytaking the difference of successive values of P" (column 6).

I. L

{

The flow of water over a watershed surface is a complicated process varytng ln

all three space dimensions and time lt begins when water becomes ponded on

the surface at sufficient depth to overcome surface retention forces and begins

to flow. Two basic flow types may be distinguished: overiand flow and channel

flow. Overland flow has a thin layer of water flowing over a wide surface'

Channel flow has a much narrower stream of water flowing in a confined path

Chapter 2 gave the physical laws applicable to these two types of flow On a

natural watershed. overland flow is the first mechanism of surface flow but it

may persist for only a short distance (say up to 100 ft) before nonuniformities

in the watershed surfaca concentrate the flow into tortuous channels Gradually'

the outflows from these small channels combine to produce recognizable stream

channel flows which accumulate going downstream to form streamilow at lhe

watershed outlet.Surface water flow is governed by the pinciples of continuity and

momentum. The application of these principles to three-dimensional unsteady

flow on a watershed surface is possible only in very simpiified situations, so one-

or two-dimensional flow is usually assumed.

Overland Flow

Overland flow is a very thin sheet flow which occurs at the upper end of slopes

before the flow concenhates into recognizable channels. Figure 5.6 1 shows flow

down a uniform plane on which rain is falling at intensity i and infilfation

occurring at rate f. Sufficient time has passed since rainfall began that all flo\as

are steady. The plane is of unit width and length 40. and is inclined at a.ngle d

to the horizontal with slope S0 : tan 8.

Continuity. The continuity equation (2.2.5) for steady, constant density flo$ is

The inflow to the control volume from rainfall is lle cos d, and the outflow is

/a cos g from infittration plus y) from overland flow. The depth y is measuredperpendicular to the bed and the velocity Vparallel to the bed Thus the continuityequation is written

Column:

Time(h)

l l v . a , c . = oJ )

0

0.06

0 . 1 l

0 .58

1 . 8 3

0 .56

0 .50

0 .50

0.90

'1.2'�7

1 . 3 L

5 .29

5 .16

0

I

2

3

5

6

7

0

0 .20

0

0.20

0.50

0 .50

0 .50

0 .50

- 0

- 0

0.34 0.06

0 .59 0 .18

1 .05 0 .76

L56 2.59

l . 8 l 3 . 15

1 .65 3 .210 .06

156 ,rpprteo i ;r i ;rorocr

t t

. / , .= r /_!= ( i , f ) locos I

FIGURE 5.6.ISteady flow on a unilbrm plane under rainfall.

Momentum. For unitbrm laminar flow on an inclined plane, it can be shown(Roberson and Crowe. 1985), that the average velocity yis given by

{ lJ J

V dA -&0 cos 0 + W - iLacos d = 0

The discharge p", "ni, *iarl, 40, is given by

qa - \1 = (t -rlo cos d ( 5 . 6 . 1 )

(s.6.3)

where g is acceleration due to gravity and / is the kinematic viscosity of the fluid.For uniform flow, Se : SJ : hy'L, and (5.6.3) can be rearranged to yield

, 2 4 v L V 1' Vy 4y2g

(5 .6 .4 )

which is in the lorm of the Darcy-Weisbach equation (2.5.1) for f low resistance

, . L V 2.'t ' 4R 2g

with the friction factor J = 96/Re in which the Reynolds number Re = 4VRl2,and the hydraulic radius R : y. For a unit width sheet flow, R : area/(wettedperimeter) =} x 1/1 = -v, as required. The flow remains laminar provided Re <2000.

For laniinar sheet flow under rainfall, the friction factor increases with therainfall intensitl. If i t is assumed that J has the form CrlRe, where C1 is a

(

suruacr w,rren 157

resistance coefficient, experimentation canied out at the Univelsity of Illinois(Chow and Yen, 1976) gave

C r = 9 6 * 1 0 8 l o a

where i is the rainlall inrensity in inches per hour.Solving for y from (5.6.5) and using t}'e lact that hy'L

flow, one finds

t v 2

t = 88so

then 4o = Vy from (5.6.2) is used to substitute for V, yielding/ . r \ i t :I tq6 \' - \ sesn /

which specifies the depth of sheet flow on a uniform plane.

(s.6.6)

: Sq for uniformt l

,t!(5 .6 .7 )

( 5 . 6 . 8 )

Example 5.6,1. A rainfall of intensity I in/h falls on a uniform, smooth, impeNiousplane 100 feet long at 5 percent slope. Calculat€ the discharge per unit width. thedepth, and the velocity at the lower end of the plane. Take / = 1.2 x l0

-5 ft:/s.

Soful ion, The discharge per unit width is given by (5.6.2) with i = | in/h :2.32x

10-5 fvs, and "f = 0. The angle d : ran-r(so) : ran r(0.05) = 2.86", socos 0 = 0.999.

40 : (i ,Io cos ,

: ( 2 .32x tO 5 - 0 ) x 100 x 0 .999

= 2 .31 x l 0 3 f r r / s

The Reynolds number is

4V\R? . ""!

v

4qa= ;

_ 4 x 2 . 3 1 x 1 0 r

1 . 2 x 1 0 - 5

and the flow is laminar- The resistance coefficient C. is given by (5.6 6):

c . :96 + 108t01

: 9 6 + 1 0 8 ( l ) u 4

The friction factor is f : CllRe :2041770 = 0.265, and the depth is calcuiatedfrom Eq. (5.6.8),

,)

l5E apprrEo sr oroeoct

l 0 2 b 5 \ \ 2 . J I . t 0 r t : l ' '

| 8 ' 12.2 0.0s l= 0.0048 fr (0.06 in)

The lelocit l V is given by

r , - Q a!

:ili,]"'""'Field studies of overland flow (Emmett, 1978) indicate that the flow is

laminar but that the flow rcsistance is about ten times larger than for laboratorystudies on uniform planes. The increase in flow rcsistance results primarily fromthe unevenness in the topography and surface vegetation. Equatiorl (5.6.8) canbe rewritten in the more general form

y : d q t (5.6.9)

For laminar florv nr : 213 and ct: (J/8gSo)L/3. Emmett's studies indicate thatthe Darcy-Weisbach friction factor / is ir the range 20-200 for overland flow atfield sites.

When thc flow becomes turbulent, the friction factor becornes independentof the Reynolds number and dependent only on the roughness of the surface. Inthis case, Manning's equation (2.5.7) is applicable to describe the flow:

( s . 6 . l 0 )

with R : y, 51 = Se for uniform flow, and qs = 1/y. This can be solved for J toyield

suR,.ac! \,'a,hR 159

y = d 6

: l.80(2.31 x lo-3)2/3

= 0.031 fr (0.4 in)

Velocity V- qol1 = 2.31 x l0-10.031 : 0.075 ftls. It can be seen th?t this flow ismuch deeper and slower flowing than flow on the smooth plane of.E,{imple 5.6. 1.

dExample 5.6.3. Calculate the discharge per unit width, depth, and ftlocity at theend of a 200-ft st ip of asphalt, of slope 0.02, subject to rainfall of 10 in/h, withManning's n = 0.015 and kinematic viscosity v = 1.2 x l0 5 ftzls.

Solutbr. The discharge per unit width is given by Eq. (5.6.2) with i : l0 inft =2.32 x l0 o ft ls,/= g, and d = tan t{0.021 : 1.15', for which cos 0: 1.00:

qo: G - nLo cos 0

= 2.32 x 10-a x 200 x 1.00

: 0.0464 cfs/ft

The Reynolds numberis Re:4qdv = 4x 0.046/(1.2 x 10 1 : 15333, so the flowis turbulent. The depth of f low is given by Eq. (5.6.9) with a: (n/l.49Sl1r/s:[0.015/(1.49 x 0.o2tt2)13t5 : 0.205 and n :0.6:

Also,

y : dq t

: 0.205 x (0.0464)0 6

= 0.032 ft (0.4 in)

" q oJ0.04640.032

: 1.43 fVs.

Channel Flow

The passage of overland flow into a channel can be viewed as a lateral flow in thesame way that the previous examples have considered rainfall as a lateral flowonto the watershed surface.

Consider a channel of length L. that is fed by overland flow ftom a Planeas shown in Fig. 5.6.2. The overland flow has discharge 40 per unit width, sothe discharge in the channel is q = np,. To find the depth and velocity atva.rious points along the channel, an iterative solution of Manning's equatioo isnecessary- Manning's equation is

/ tqo \'= \ t t " l

/ \3/5l n q a \- \r.4es;,,/

( 5 . 6 . I 1 )

which is in the general form of (5.6.9) with a : (n/1.49S1/\3ts ar 'd m = 315.

For SI units. cr = ,0 6/so 3

Example 5.6.2. CalcuLate the depth and velocity of a discharge of 2 31 xl0-3

cfs/ft (\\'idrh) on turf having Darcy-Weisbach J : 75 and a slope of 5 percent

Take r,: 1.2 x 10 5 ft l /s.

So /z f i on . The Reyno lds number i sRe =4qo l v=4 / . 2 .31x10 t l . 2x l 0 5=770

(laminar f low), and a : Lts8so) ur : (75l(8 x 32.2 x 0 05)) r/r = 1.80. From (5 6.9)

with n = 213 g = | 4 9 5 , , r o * r ,n

(s.6.12)

(

l60 . lppr-reour' :ror-oc.,

f 0 ) : Q i - Qis acceptably small. The gradient of/ with respect to ) is

df do,dt, dy,

FIGURE 5.6.2Ove.land f lolr fronr I Dtiroe nro a channel.

Solution of Nlanning,s Equation by Newton,sMethod

There is no geoeral analytical solution to Manning's equation for determining theflov depth given the flow rare because the area A and iydraulic radius n may Uecomplicated funcrions of rhe depth. Newton,s method can be appiied itemtiveiv togrve-a numerical solution. Suppose that at iteration j the depth y, is selecred andthe flow rate Q] is computed from (5.6.12), using the area-and

'hydraulic radius

corresponding ro,),. This O; is compared with the actual f low 0' the objecr is roseiect ) so that the error

FIGURE 5.6.3Newton's melhod extmpolates lhe lan-gent of lhe eraor function at the currentdepth )! io obtain the depth )) - L tbr ihe

This expression for the gradient is useful for Newton's method, where,given a choice ofy;, 1+r is chosen to satisfy

(5 .6 . r8 )

For a rectangular channel A = B*) and R - B*yl(B. * 2y) where B" is thechannel width; after some manipulation, (5.6.18) becomes

/ sa .+oy , \\3), (8" + ,t l

Values for the channel shape function l(213R) (dRldy) + (l/A) (dAld-v)l for othercross sections are given in Table 5.6.1.

lalt 0 -.fL) )r\dr),

: tu, - ),

(s.6. i6)

This y; a 1 is the value of y, in a Plot of/ vs. y, where the tangent to the cune at

-r - -r,y intersects the horizontal axis, as i l lustrated in Fig. 5.6.3.Solving (5.6. 16) for y;a 1,

"fCvi )r!+ ' " u7ta14

( 5 . 6 . l 7 )

whiqh is the fundamental equation of the Newton's method. lteIations are con-tinued until therc is no significant change in y; this will happen when the eEor

/$) is very close to zero.Substituting into (5.6.17) from Eqs. (5.6.i3) and (5.6.15) gives the New-

ton's-method equation for solving Manning's equatton:

I oto,/ t + t - t j

t z - d R - 1 d a \\3R dY A dY t1

sunr,cce wAEn l6l

Jlvj)t

r l

{

( 5 . 6 . l 3 )

(5 .6 . 14)

because Q is a constant. Hence. assuming Manning's n is constant,

I f - r I "9sJ 'a,n j ' lt 7 \ 4 1

I 4 q . . . { 2 4 R r r d R , n 2 j 4 ! ,=

, . o I l - a r - ; ' " 4 r ) ,

=1.19 5uzn p zt t l 2 dR + r dAl (5 6 15)

n " , r i R d . \ A d t '

^ t 2 d R l d A \: a'\,3R dr n

i dr ),where the subscripr j outside the parentheses indicates that the conrents aree v a l u a t e d f o r ! - r )

a

F

: i D , ' �1 , ,t)::i-

,,,/

: r : & 1 . c

a l = -" 1 1 " i

l 66 l ;+ 1 ,

; t 6 :e l f E

l

- s

l \ lt - lk l\lt{i l l *

; l ?l * l

/ t ;

*.1 oqtl r

a 3

i l e-,13l

li, ll^.r t + r +> , L - a -

Y l >- > \ f t > . -

d + qil + qi

a t a

\"

' ;

ti" r l -

&

,al6'it-.f t?l a

{ t \

o - 3

(

sunrlct utnr"n 163

Example 5.6.,1, Caiculate the flow depth in a two-fooFwide rectangulal channel

having n = 0.015, So = 0.025, and Q = 9.26 cfs.

Solution.

n = L !2s^ ' z r8" v r5 l- , n

' " r B ^ - 2 y , t : ,

- l 49 ,n n r . . - l f l ' '

- 0 0 1 5 ' "

" ' - ' 2 - 2 1 t ) '

- 31 4 l ) i 5 /3

1l + _v, )23Also,

2 . l R l d A _ 5 8 " + 6 l j _ l 0 + 6 ) t

3R,lt *

i dy= 3rlB- * 2y,) -

zy,1llzy;

_ 1 6 6 ' 7 + ) " j)i(l + -yi)

From Eq. (5 .6 . l8 )

( t 9 .26 te l ) y t ( t + r )y t+ t = y j -1 .661 + y ,

(s .6.19)

(5 .6 .20)

From an arbitmri ly chosen staning guess of -rr : i .00 ft , lhe solut ion lo three

signif icant f igures is achieved after three i teral ions by successivelv solving (5 6.19)

and (5.6.20) for Q and -vr*1. The .esul l is ) = 0.58 ft .

! L

8!

It€rationj I

,ri (fr)

O, (cfs)

1 .00 0 .60 r 0 .577

t9.19 9.82 9.26

0.511

9 .26

Eaample 5.6.5. Compute the velocity and depth of flow at 200-foot incremenls

along a loo0-fooFlong rcctanguiar channel having widih 2 ft, roughness n = 0 015,

and slope S = 0.025, supplied by a lateral flow of 0 00926 cfs/tt

Sofurton. The method of Example 5.6 4 is appiied repetitively to compule ) for

Q = D.00926L. The velocity is V = QIB *J : Ql2y.

Distance along chanflel, L (fO 0 200 400 600 E00 f000

Flow rate (cfs)

Depth ) (f0

Velocity V (fi/s)

0 1 .85 3 .70 5 .56 7 .41 9 .26

0 0 .20 0 .31 0 .4 i 0 .49 0 .58

0 4.63 5.91 6.86 7.s6 8.02

164 :pprteo sYonor 'rcr

I

0 .E -

sunr,rct onrea 165

TABLE 5.7.1App-ii-"t" ut".uge velocities in fVs of runoffllow for calculating

time of concentration

Slope in percent8-l I

Description of wat€r course0-3

Pdstures

Cultivated

0 -1 .5

0 2 .5

0-3.0

0 tt .5

l . 5 2 . 5

2 .5 3 .5

3 . 0 - , 1 . 5

8 .5 8 .5 13 .5 - l ?

1.25,

l 1

( s .7 . L )

It at'

Jo v (1)

lf the velocit) ' can be assuned constant al v'

i : r , 2 , . . . . 1 . t h e n

3 r,,t = ) _

7r ri

Velocities for use in Eq. (5 7 3) may be conputed using the methods described' ' ' T r h l - ( a l

Ourlet channel determine le loci ty by Manning s fbnnul l t

Nalural channel notwell defined 0-2 24 1

*This .ondltion usuallv occu.s in the upPe! ert.emilies of a wareBh'd p'or to lhe overland flolvs accumular'ng

**These values va.y sith rhe channel size and other conditions' $hee posiible more accurure detemtnarrons

should be rnade fo! paniculff conditrons by the Mannjng channel fonnula for !elocit)

(Soa..!: Drainage Manual- T€xas Highqav Depanmenl, Table VU p ll 18 1970')

Because of the travel time to the watershed outlet' only part ot the watershed

may be contributing to surface water flow at any time t after precipitation beginr'

The growth of the iontributing area may be visualized as in Fig 5 T l If rainfall

oi constant intensity begins and continues indefinitely. then the area bounded by

the dashed line labeled 1r will contribute to sfeamflow at the watershed outlet

after t ime tr; l ikewise, the area bounded by the l ine labeled t2 wil l coniribute to

FIGURE 5.7.1Isochrones at rr and /: define fte area contributing to florv at the oullet for rainfall of duralions '

"rJ r,. f;^. ui.o**,ration r" is the time of flow from the farthest Point in the wateBhed (A) to

the out let (B).

FIGURE 5.6.4

Dimensionless hydrograph of ovcr land t lo$

The srcady flow q, is allained al time oi

equilibrium r"- (Afler lzza.d. l9't6 )

The exanrpLes in thls sectlon nave assumed stearly f low on the watershed'

I" .""iii.';;;; ; con.tunt inten'irv rainfall' the :t"{I il-ol at equiiibrium is

appr*.ft.O asyrllptoticotty in the manner illustrated by Fig 5 6 4 Thus' the flow

ti"'"fine boih in spa." and time on the wate$hed surface and in the stream

channel.

5.7 TRAVEL TIIIIE

The favel time ol iio* from one point on a watershed to another can be deduced

iarn ,tt" no* distance and velocity lf two points on.a strexm,are a distance l,

aDa l tandthe ! 'e ]oc i tya longthePathconf lec t ing themiSV(1) .where l lSd is lanceaiong the path. then the travei t ime ' is given by

dl = v(l) dt

f' I' at1 d t : | - ;

J 0 J 0 V ( / l

(.5 .1 2)

in an increment of length Ali'

( s .7 .3 )

166 , l pp t - t t ou runo rocY

T+:l3,ti;'" in a channel (Example 5.7.1)1000E00600

Distance along (hannel. I rftr 0 20o

200

0

I 86.2

4.63 5 .91 6.86 1 . 56Calculated velocitY Y (ftsl

Average velocity ? (tts)

travel rime At = '\l/V Gl 31 . 7

6.42 1 . 21

2'�7.7 25.'1

(: Ar = 208.5 t

strearnflow aftcr timc 12. The boundaries of these contributing areas arc llnes or

il;iffi;;;; io ttt" outt"t and are called isochrones The time at which all

of the watershed begins to contnoute is the time of concentration f"; this is the

ii-" "f no* iiom tie farthest point on the watershed to the oudet'

Example 5.7.1. Calcuiare rhe ̂ time of concentmtion of a watershed in wh;ch &e

r ^ . " " . 1 1 . o ' o a t h c o l e l s l 0 0 f e e t o f p a s t u r e a i a 5 p e r c e n t s i o p e , t h e n e n t e r s alfttili'�i;d::J;e"L' "i'-*r ha;ins width 2 rt'- roughness '' : 0 015' and

slope 2 5 percent, ano recervrnB a lateral flow of 0 00926 cfs/ft'

Solution. From Table 5 / I, pasture al 5 Percent slope has.a velocity. of flow in

ilt" ,"t*. Z S-: s fVs; use a vilocity of 3 d ft/s The ravel time over the 100 feet

lio^tii"lt,st = alii' =-rool:'o i33 s For the rectangular channel' the velocitv

a I200 t . oo t i n tea l swasca l cu la ted inExamp le5 .6 .5 'The t rave l t imeo le feachin te r va ] i s f ound t romtheave raSere loc i t y i n tha t i n te f \ a l ' Fo rexamp le . f o r t heir.i zoo it, ,lt = All. : 2ool2 i) = 86 i s This yields a total travel rime for the

"i".t"i-"i tot s s' as shown in Table 5 7 2 The time of concentration r' is the

sum of the tmvel tlmes over Pasture and in the channel' or 33 + 209 = 242s: 4

m in .

5,8 STREAM NETWORKS

In f lu idmechan ics . thes tudyof thes imi la r i t yo f f lu id f low. insys temsofd i f fe r .ent sizes is an important tool ln rclating the iesults of small-scale model studies

;J';;";i;' ;;;loiip" uppri'ntion' in hvdrologv' the . seomor ph-oto sv of tJte

*u,.rih"d, or quantitative study of the surface landform' is used to amve at mea-

,"r", J g""-J,". similarity among wate$heds' especially among their stream

networks."fi"e quantitative study of stream networks was originated by Horton (1945)'

H" d;i";J a s)'stem ior ordering stream networks and derived laws relating

ifr. "".Ut'. and length of stteams -of

different order' Horton's stream ordenng

ryr,"-, ut slightly riodified by Stmhler (1964)' is as follows:

| - 6 I , - r . ' l F ( i q - . r P r ^ ' d e ' I t h e ' e c h a n n e l \

suxr'ccr wAren 167

Wherc two channels of order I join, a channel of order 2 results downstream;in general, where two channels of order i join, a channel of order i + I rcsults.

Where a channel of lower order joins a channel of higher order, the channeldownstream retains the higher of the two orders.

The order of the drainage basin is designated as the order of the streamdraining its outlet, the highest stream order in the basin, f. i

An example of this classification system for a snrrll watershed in Te"$ is stro*ni n F i g . 5 . 8 . I .

2 \/ K"v

- - - BoundafyOrder I stte.mOrder I slrerm

- Order I slream

2

/ lIt

I

0 0 .5

(

168 .rpprreo ;r lrorocr

Horton (19'15) lound empirically that the bifurcation r4tio R8' or ratio of

the number,\,. of channels of orrier i to the number Ni+ I of channels of order

i + I is relati\ely constant from one order to another' This is Horton's Law of

Streatn Ntunbcrs.

i = 1 , 2 , . . . , 1 L ( 5 . 8 . 1 )

As an example . in F ig 5 .E l ' Nr = 28 , Nz = 5 ' and N: = l ; so N1/N2 : 5 '6

and N2/N3 .-l :.0. ' I he.th"oretical minimum value of the bifurcation ratio is 2'

and values typicall l l ie in the range 3 5 (Strahler. 1964)

ny m"",ur;ng the length of each stream' the average length oi streams ot

each orde.. l,.canbefound Honon proposed a law of Stream Lengths in which

thc averagc lengths of sireams of successive orders are related by a length rctio

R r :

( s .8 .2 )

By a similar reasoning, Schumm ( 1956) proposed ̂ Lttvr of Strecm AreqJ to relate

the a\eraqe areas A, drained by streams of successive order

( 5 . 8 . 3 )

! = n '

l o E r

t

J : rOrdcr /

3 1Order r

FIGURE 5.6.2-eo.o.ptrologi."l purr*erers fof the Mamon basins ('So!rcdr Valdes Fiallo and RodriguezlNrbe'

p. L 123. i919 Copydghl by the American Geophysical Union )

sux+cr wrrtn 169

These ratios are computed by plotting the values for N1, l;, and Ar on a loga-rithmic scale against stream order on a linear scale, as shown for two Venezu-elan watersheds in Fig. 5.8,2. The ratios R6, R1, and RA are computed from theslopes of the lines on these graphs. The Mamon 5 watershed is a subbasin of theMamon watershed (Fig. 5.8.3). The consistency of R6, R1, and RA between thetwo watenheds demonstrates their geometric similarity. Studies have been madeto relate the characteristics of flood hydrographs to stream networt parameters(Rodriguez-Iturbe and Valdes, 1979; Gupta, Waymire, and Wang, 1980; Gupta,Rodriguez-lturbe and Wood. I98or. A'

Other parameters useful for hydrologic analysis are the drainage density andthe length of overland flow (Smart, 1972). The drainage density D is the ratio ofthe totai lensth of stream channels in a watershed to its area

) ) /1, 1, 'lJ

A1(s . 8.4)

FIGURE 5.E.3Drainage basin of the Mamon watershed in VeDe ala. \Source: Valdes. Fiallo, and Rodnguez-Iturbe, p. 1123, 1979. Copyright by lhe American Geophysical Union.)

(

170 ,rePrtrr .:r lnor,'lr r

where L,, is the lcngth of Ihe jth stream of order l. If the streams are fed byHortonian orcrland llow from all of their contributing area, then the averagelength of oielland flow, lu, is given appfoximately by

I2D

( 5 . 8 . 5 )

Shrevc ( 1966) showed that Horton's stream laws result from the most l ikelycombinations of channels into a network if random selection is made among allposs ib le combina t ionsr

REFERENCES

B e l s o n . R . P . \ \ h l t j r r r l e r s h e d . u n o f f ? J . C € o p l - ) s . R c r . . v o l . 6 9 . n o . 8 . p p . 1 5 , 1 1 I 5 5 2 . 1 9 6 4 .Chow. V. T. . 0tr .hdl i r ( l HvL\ tuLi . r , Mccraw-Hi l ] . New York, I959.Chow. V. T. . . i l l . i ts . C Yen. Urban storn$aEr runof i : delerminat ion of volumes and l lowrates.

reporr EP\ 600/:-16- I 16. Nlunicipal Environmenlrl Research Laboralory. Office of Researchand Dev.k,pmenr. U. S. Environmentai Protect ion Agency. Cincinrat i , Ohio. Ma) 1976.

Dunne, T. . T. R. Nloore. and C. H. Taylor , Recogni t ion and predict ion oi runol i produ. inr zone!in humid rcgions- H:-r l roL. SL\ . BuU.. eal .20. no. l . pp. 305-121, 1975.

Emnett . w W.. O\er l . rnd f low, tn Hi l ls lope H\drolos. ed. by M. J. Kirkby. wi ley. Ne\ i York,p p . l : 1 5 l r 6 . L 9 r S .

Freeze. R. A.. Role of sLrbsudace flow in genenring surface runoff:. Upslream source areas. WirlerR e r . r l , ? . R r r . l o l 3 . n o . 5 , p p . l l 7 : 1 2 8 1 , 1 9 7 : .

Ffeeze. R. A. . Srreami l ( , lv generaho., Re\ ' . Geaphls. Space Pl1,-s. . vol . 12. no. 21, pp.621 611.1 9 7 4 .

Gupla. V. K. . t . $a)m i€. C. T Wang- A represenrat ion ol an rnsrarraneous uni l hydrograph l iomgeomoqrholog). i l i ter Retout . Rsr. , vol . 16. no. 5. pp. 855-862, 1980.

Gupu.V.K. I Rod. igrcr I turb€. and E. F. Wood(eds.) , S.nl . ,Pror l?nr t r ,Y_\drolog), D. Reidel ,Dordrccht l - lo l l l fJ 1986.

Han. R. D. . \ \ ' r rcr l lu \ in soi l and subsoi l on a sreep forested s lope. J. Hx/ro l . , lo l . i i . .17 58.l9 l r - .

Hewlet t , J . D . nnd A. R Hibbert .Factorsaf fecr inglheresponseofsmal lwarenhedstopreLi l r rJ l ionin hurnid i rcas. rn t r | . s ! r ,p. on Farest H)"drL ' lagr, ed. by W. E. Sopper and H. W. Lul l .Perg.rn l ! i r Press. Oxford. pp. 275 290, 1967.

He$1et! .J .D. . Pt inLipLs of Farcst H)"Ll rc loeJ Ur, | ofGeorgia Press, Athens. Ga.. 1981.Honon. R. E.. The role of iniiltration in the hydrologic cycle. ?l?nr. A||'. Ceoph)"s. UnL)tl, \o\.

14. pp. +16 460. 1933.Honon. R. E. . Erosionnl developmenl of st reams .rnd thei f drainage basinsr hydrophysical approach

ro quaniLrxrrre.norphologl . Bul l . Ceol . So. . i ln . . vol . 56. pp. 215 170, 19,15.lzzard- C. F . Hrrlraulics of runoff fron deleloped surfrces. Proc€ed,r8r. 26th Annual iueeting of

the Highlv i r r Rescarch Board. !o l . 26. pp. 129 146, December 1946.Mar l in. G. N . Charrcter izal ion of s imple exponent ia l basef low rccession. J. H)d/ . ,1. (New Zealand),

r o l . l : . n o . 1 . p j r . 5 7 - 6 2 . 1 9 7 3 .Nfo.el Seytoui . H J. . and J. P. Verdin. Ex|ension of the Soi l ConseNat ion Service ra infa l l

runolf nr.Lhodolo.!) for ungaged w.rtersheds. report no. FHW.dRD'8 L/060. Federal HighwayAdnl inL! t r r I ion. \ \ ' r \h inglon- D.C.. avai lable f rom Nat ional Technical Informat ion Service,S p r i n g i . l d . V ! l 1 6 l 6 1 9 8 1 .

Vlosele) , . M P . Si .c. tmi los generat ion in a fore\ ted watershed, New Zealand. Water Reso r . Res.! ( )1. 15. no. 1. rp. 195-806, 1979

Rarce. A. J.. ind A I McKerch,r. Upstream generatron of stofm runoff. in Ph'"sical $)droLoet,l\er Z?aLo"d E\p.rience, ed. by D. L. Munay and P. Acboyd. New Zeatand HydrologicalSocier) . \ \e l l ingl rn. Neq. Z€aland, pp. 165 192. 1979.

sunrece wemn 171

ftarce, A. J., M. K. Stewan, and M. G. Sklash, Storm runoff generarion in humid headwatefcatchments, 1, Where does the water come from? Water Resaur. Rpr., vol. 22, no. 8. pp.1263 12'�72, 1986.

Ragan, R. M., An experimeotal investigation of panial area contibutions, Int- Ass. Sci- Hr-drol.P ubl. 76, pp. 241-249, 1968.

Roberson, l. A., and C. T. Crowe. Engineering fuid nechanics,3rd ed., Houghton-Mifflin, Bosron,1 9 8 5 .

Rodriguez-Iturbe , I. . and J . B . Valdes, The g€omorphologic structure of hydrologic U:esponse. lyorerResour. Res. vol . 15, no. 6, pp. 1409 1420, 1979. r . l

Schumm, S. A., Evolution ofdrainage sysrcms and slopes in badlands at Pbnh Arnbrry, New Jersey.Bul!. Geol. Soc. An., vol. 67, pp. 59'7446, 1956 I

Shreve, R. L. , Stat is t ical law of st ream numbers, J. of CeaL., ro l . '71, pp. 11-3 '7,1966.S ingh. K. P. , and J- B . Stall , Deri vation of base flow reces sion curves and pafameters. lyare. R€rrr/r.

Rer. , vol . 7, nt . 2, pp. 292 303, 1971.Sman, J. S., Channel networks, in Adldnc?r in l1)droscience, ed. by V. T. Chow, Academic Press,

Or lando, Fla. . vol . 8, pp. 305-346, 1972.Soil Conservation Service. Urban hydrology for small waiersheds. rech. rel. no. Jj U. S. Depr. of

Agr icul ture, Washington, D.C., 1975.Soil Conservation Service, Nalional Engineering Handbook, seclion 4, Hydrology. U. S. Depr. oi

Agrlculture. available from U. S. Govemment Printing Offic€. Washington. D.C., 1972.Strahler, A. N., Quantitative geomoehology of drainage basins and channel net$orks, secrion 4-

II, in HanAbook of Applied Hydrolagf, ed. by V. T. Chow, pp. 4-39, 4-76. Mccraw-Hill,New York. 1964.

Terstriep. M. L-. and J. B. Stall, The Illinois urban drainage area simularor. ILLUDAS, BuL!. 58.Illinois Stale Water Survey, Urbana, Ili., 1974.

Unvef, O., and L. W. Mays. Optimal determination of loss rate functions and unit hydrographs,I later Resour. Rer. , vol . 20, no. 2, pp. 203-214, 1984.

Valdes, J. B., Y. Fiallo, and L Rodriguez Iturbe, A rainfall-runoff anaLysis of the geomorpbologicruH, Water Resout. Rcr. , vol . 15, no. 6, pp. l42l 1434, 1919.

PROBLEMS

5.2.1 ll QG) : Qoe-\t-ta]]k describes baseflow recession in a stream, prove that thestorage S(t) supplying baseflow is given by S(t): kQQ).

5.2.2 Baseflow on a river is 100 cfs on July 1 and 80 cfs on July t0. Previous studyof baseflow recession on this ver has shown that i! follows the linear reserroirmodel. If there is no rain during July, estimate the flow rate on Juiy 31 and thevolume of water in subsurface storage on July I and July 31.

5.2.3 The streamflow hydrograph at the outlet of a 300-acre drainage area is as shown:

Time (h) 0 IDischarge (cfs) 102 100

5 6630 460

2 398 2m

4512

7 8 9 l 0 t t 1 2330 210 150 105 75 60

Determine the base flow using the straight line method, the fixed base method,and the variable slope method. Assume N: 5 hours for the fixed base method.

5.3.f For the following rainfail-runoff data, determine the findex and rhe cumulativeinfiltation cufle based upon the dindex. Also, determine the climulatile e\cessrainfall as a lunction of time. Plot these curves. The watershed a.ea is 0.2 mir.

Tim€ (h.) IRainfall rate (intr) 1.05Direct runoff (cfs) 0

2 3 41 .28 0 .80 0 .7530 60 45

5 60.70 0.6030 15

,7

00

r74

5.5,2 Solve Prob. 5.5.1 fbr the l0-year design storm.

5.5,3 Scrl \e Prcb. 5.5.1 for the 10o-year desjgn slorm.

5,5.4 (d) Compute the runoff from a 7-in rainfall on a 1500-acre watershed that hash]droLogic soil groups that are 40 percent group A, 40 percent group B, and20 percent group C intenpersed throughout the watershed. The land use is 90percent residential area that is 30 percent impervious, and l0 percent paved roaclsuith curbs. Assume AMC II conditions.(r) what was the runoff for the same wate$hed and same rainfall before devel-opment occured? The land use prior to development was pasture and ftnge landin poor condit ion.

5.5,5 A 200-acre watershed is 40 percent agricultural and 60 percent urban land. Theagricultural area js 40 percent cuitivated land with conservation treatment, 35percent meadow in good condition, and 25 percent forcst iand with good cover.The urban arca is residential: 60 percent is j-acre lots, 25 percent is l-acrelors, and 15 percent is sffeets and aoads with curbs and stom sewers. The entirewatcrshed is in hydrologic soil group B. Compute the runoff from the wate.shedfor 5 in of rainfall. Assume AMC II conditions.

5.5.6 Solve Prob. 5.5.5 i f the moisture condit ion is (a) AMC I. and (b) AMC III .5.5,7 For the rainfall-runoff data given jn Prcb. 5.3. I , use the SCS method for abstrac-

tions to delermine ihe representative SCS cufle number for this watedhed, assum-jng AMC IL

5.5,8 Considering the rainfal l-runoff data in Prob. 5.3.1 and using the curve numberdetermined in Prob. 5.5.7. determine the cumulative inf i l i rat ion and the cumula-t i \e aainfai i excess as functions of t ime. Plot these curves.

5.6.1 Conrpute the uniform flow depth in a trapezoidal channel having n : 0.025,Jr = 0.000S, and C = 30 cfs. The base width is 4 ft, and the sjde slopes arel : ; = 1 3

5.6,2 Compute the uniform flow depth in a trjanguiar channel having n : 0.025, S0 =

0.0004. 0 = 10 cfs, and side slopes 1:z : l :4.5.6.3 A rainlall of 3 in/h falls on a uniform, smooth, impervior.rs plane that is 50 feet

long and has a slope of I percent. Calcuiate discharge per unit width, depth, andve loc i t y a t i he bo t t om end o f t he p lane . Take l / = l . 2v l 0 s f t sandn :0 .015 .

5.6.4 Sol\e Prob. 5.6.3 i f the rainfal l has jntensity 10 in/h.5,6.5 Solve Prob. 5.6.3 if the rain falls on grass with an jnflhation rate of 0.5 ii-/h

and a Darcy-Weisbach roughness/: 100.5,7 .1, S olve Example 5.7. 1 in the text if the flow length over pasture is 50 ft, and the

channel is 500 feet long.5.8.1 Determine the length ratio Rr for the luiller Creek watershed in Fig. 5.8.1.5.8.2 Determine the drainage density and average overland flow length for the Miller

Creek watershed in Fig. 5.8.1.

CHAPTER

HYDROLOGICMEASUREMENT

Hydrologic rneasurements are made to obtain data on hydrologic processes. Thesedata are used to better understand these processes and as a dircct input into

hydrologic simuladon models for design, analysis, and decision making A rapid

expansion of hydrologic data collection worldwide was fostered by the Intema-tional Hydrologic Decade (1965-1974), and it has become a routrne pmctlce

to store hydrologic data on computer files and to make the data available in a

maciline-readable form, such as on magnetic tapes or disks. These two develop-

ments, the expansion and computerization of hydrologic data, have made avail-

able to hydrologists a vast array of information, which Permits studies of grcater

detail and precision than was formerly possible. Recent advances in electronicsallow data to be measured and analyzed as the events occur, for purposes such

as flood forecasting and flood waming. The purpose of this chapter is to revlew

the sequence of steps involved in hydrologic measurement, from the observation

of the process to the receipt of the data by the user.Hydrologic processes vary in space and time, and ate tandom, or Probabil-

istic, in character. Precipitation is the clriving force of the land phase of the hydro-

logic cycle, and the random nature of precipitation means that prediction of the

resulting hydrologic processes (e.g., surface flow, evaporation, and streamflow)at some futurc time is always subject to a degree of uncertainty that is large in

comparison to prcdiction of the future behavior of soils or building structures, for

example. These uncertainties create a requirement for hydrologic measurement to

provide observed data at or near the location of interest so that conclusions can

be drawn directlv from on-site observations.

(

175

l7{r elrr :eo syotorocrt7'

6.I HYDROLOGIC MEASUREMENTSEQUENCE

Aithough h),droiogic processes vary continuousll. in t ime and space. the!. areusually measured as point sampLes, measurements made through iime at a tixedlocatron ln space. For example. rainfall varres continuousll jn , irc" ouer u *ater_shed. but a rain gage measures the rainfall at a specific point in tie.atenlea. f lerciultrng data form a titne.rerles, which may be sub.jected to statistical analysjs.._ ,t 1"..,], )ears. some progress has b."nn]ud" in,rr"ururing .t i.stributed

::l:I:::_"^".9: 1_]1"" or area in space at a specific point in time. For exampre,e\irmares oI \\ Inter snow cover are made by flying an aircraft over the snow iieldand measuring the radiation reflecred from the siow. The ,"rutiing- Ouru torrn orTacr series. Distributed sarples are most often measured at ,onra ?ir,un"a ,.on,the phenomenon being observed; this js termed remote sensing. Wt "ih", rn.,lutu4.,: neasured as a time series or as a space series, a similar s'equence of steps isfo i lo*'ed.

The .equence of step' commonlv folloued for hydrologi. measurement t\:hu)n ln f ig :6 .1 .J , beg inn ing w i rh rhe ph ls rca l Oev ice uhr i l , "n r " , o . . "n .oto rhe physical phenomenon and ending with the delivery of data io a user. rnesesteps afe now described.

1. Sensing. A sensor is an insrrument that translates rhe le\el or intensit), ofthe phcnornenon jnro an observabte signal. For "^"rpt., ; ;;;;;;r:thermo.et".

:,: li:: ;.:]T'::il: rhrough the erpansion or contraction ofthe volume of mercury

.:.i i, ""i

l^," |toe. a sto.rage rain gege collects the incoming rainfall in a can ortuDe. Sensors 'rray be direct or indirect.

A direct sensor measures the phenomenon itself, as with the storage ralnga,ee: an indirect sensor rneasures a variable related to the pfr"no,rra-n, u, *itf,l l :,:::r"tt

the.momerer. Many hldrologic lariables are measured rndirecrty,rncrudrng streamt'low, temperature, rumidity, and radiation. sensors for the malorhyclrologic i,ariables are discussed in the subsequent ,".rion, oiiflir.i"p,"r.2. Recording. A recorder is a device or procedu." to, f..r".uing^,t. ,,gnnfproduced by the sensor. Manual recording simply involvei un obr"?u", rur.,ngreadrngs off the sensor and taburaring thim foi iuture ..r"r"n."] iaou or tl.

:^.,i,1b1. *iif"i] data are produced by observen who read rhe level in a sroraee

:i'.l.li::":n d"y at a fixed time (e.g-, 9 A.M.). Automatic *""rdi"; r;;;;;;:;oe\ rce whrch accepts the signal from the sensor and storcs it on a paier chart orpuncned tape, or an electronic memory including magnetic disks oi iup"r. eup",

1;1o1oy recuirl a mechanic-al sysrem ofpulleys oir"".i, ," """ri^," tiJlotion or

;:"'.:::',: l": ::ji:i ?1 i p:" 9" a charr or a punching mechanism for a paperrdpc. rrB. o.r.z snows hvdrolopic paper chart and tape recordets in commonuse. Historically, chans were th! first recorders widely used in hydrology; theyare still used when there is a needbur charrs have a grcar ai,uouun,ug.'ln11,uul li'fi::JJ'Jii;ffiJ:::?.T::.j

::"]li,:fl l.d form is a.rediols procedure, invotying manualiy o*tini,i,. rin.on the chart and.recording the points where it changes direction. By contrast,paper tape recorders can be directly read by a com-puter. Si"i..niia"t puo..

Transform rhe i.tenshy olthe phenomenon into anobsenable signal

Make an eleclrorrc orpaper record of the signal

Move lhe record to acenira l processrng sr te

Convert the rccord inloa coripulerized drta sequence

Check tbe dat , . rnd el iminr leerroA and redunda r infornrrtion

Archive rhe dara 0n acompurer rape or or \ r (

Recover the dr ta in

J

FIGLRE 6.T.1The hydrologic measurement

tapes are currently the most widespread form of automatic hydrologic recorders.but electronic storages are beginning to be adopted: their use can be expected tcspread because of their convenience and because they need no mechanical systemto translate the signal from sensor to rccorder.

3. Transmission. Transmission is the transfer of a record from a remoterecording site to a central location. Transmission may be done rourinely, such asby manually changing the chart or tape on a recorder at regular intervals (fromone week to several months in duration) and carrying the records to the centrallocation. A rapidly developing area of hydrology ls real-time transmission oldata through microwave networks, satellites, or telephone lines. The recordersite is "polled" by the central location when data are neededi the recorder has thedata already electronically stored and sends thern back to the central locationimmediately. Microwave tmnsmitters operate with relatively short-wavelengthelectromagnetic waves (10-l to l0-l m) traveling directly over the land surfacewith the aid of reneater stations: satell i te data tmnsmission rrses radio \r 'tvPc rl r^

Hldrologic phenomenon(e.9. precjp i lat ron)

l7E rppL eD Hi oqorocr

179

Ii L

FIcLRE 6.1.2(a)

i:lj'f:lii:iii?fill"i,:.i:,TT,jtJ:itjjjff:"li:i::il;,:T"iiii,;!",fiJ11ii:i::,"" ;il5 ;.-i- :!:i"::il:1[n:*;"ii::^"5'r.::.lll*xifiili n::l ;1,:lll,,*xi{{ff i:fr1n._::#,n"j.::,,#h}lrti[jr.x,:.#*: x ;rk".;ru,1:ror,r ' , i e r.", ,* ln" ",o' , , i #::""-: : i :

nne,ro'r i 'ne "nd one Ior mex.ure'nenr re\et.

fi :i: : j", I ;, i::l,i:ll;it ;jl;i.*ltLriil';.,;** ".i *.liijiiil ;1,,,.

;tJJ#'-i'1":t'il'::J;:';";T1^::":1'l''"'0":" position is rixed rerative toproducrng flood forecasrs and 1o, o..l1tL'1t1"

*'mission of data are valuable forsrtes $hich are difficutt ,o ,"".h ;r"lj;f ,t;;J,tlnuous

access to remote recordingL Translarion. Translation is'l,""", r."'", ;;;; ;;;,;#il:J,:h:?#:'"?::;j,:# iiiilrl lil,1 ilTil.

l:? ^,j11:,:.^ are avaitabte which read l6_tracL r,ya-.o.lo"eri p"p"fi"j" ,""o.d,ano pr,,dLrce Jl electronic signal in aano ",,., r_r,'iu","r. ; ;,;.";;.J[l"il;::iil5

bv compurers ca"seire reade^5. Editiitg. Editjng is the procedure of checking the records translated intorhe compurer ro correct any obvious erors "hi.h h;r,;-;".;;; o"r,,rg un, o,

FIGL,'RE 6.r.2(r)Paper chart recorder attached to a float for recording water level variation. Rises or falls in the floal

level move lhe pen horizontally, parallel to the front of lhe recorder case The paper is driven roward

the back of lhe case continuously at a slow rate, hereby allowing the pen to trace oul a record of

water ]evel against time on the chart. (Source:T. J Buchanan U S Geological Srjrvey )

the previous steps. Common errors include mistakes in the automatic timing

of recorded measurements and information lost in transmission and translation,

which is filled in by directly analyzing the record made at the recorder site.

6. Storage, Edited data are stored in a computerized data archive such as

WATSTORE, operated by the U.S. Geological Survey, or TNR1S (Texas Natuml

Resource Information System). Such archives contain many millions of hydrologic

data systematically compiled into files indexed by location and sequenced by the

time of measurement.'1.

Retrieval. Data are retrieved for users either in a macbine-readable form'

.uch as magnetic lape or disketle. or as a paper Pnntoul

6.2 MEASUREMENT OF ATMOSPHERIC WATER

Atmospheric Moisture

The measurement of moisture high in the atmosphere is made by mears of a

radiosonde. which is a balloon filled with helium that is attached ta a measuring

TABLE 6.2.7Summary of pan coefficients (aft€rLinsley, Kohler, and Paulhus, 1982)

183182 ,: rrreo rr oporocv

transmitted lo a distant receiver. Rain gages commonly have a windbreak device

constructed around them in order to miDimize the amount of distortion in the

measuremenl of rainfall caused by the wind flow pattem around the gage

Radar can be used to observe the location and movement of areas of

precjpitarion. Cefiain types of radar equipment can provide estimates of rainfall

;ates over areas within the range of the radar (World Meteorological Organiza-

tjon, 1981). Radar is sometimes used to get a visuai image of the pattem of

rainfall-producing thunderstorms and is pa iculariy useful for tracking the move-

menl of tornadoes. The introduction of color digital radar has made it possible to

measure rainfal in distant thunderstorms with more precision thao was formerl)'

possjble. The phenomenon upon which weather radar depends js the reflection of

-i.ror. uu., emitted by the radar transmitter by the droplets of water in the storm'

The rlegree of reflection is related to the density of the droplets and therefore to

the rsinlall jntensity.

Snowfall

Snowfall is recorded as part of precipitation in rain gages ln regions where there

is a continuous snow cover, the measurement of the depth and density of this

sno\\, cover is important in predicting the runoff which will result when the snow

corer melts. This is accomplished by means of surveyed szow coarse'r, which arc

secrions of the snow cover whose depth is determined by means of gages that have

been installed prior to the snowfall The density of snow in the snowpack may

be determined by boring a hole through the pack or into the pack and measuring

the amount of liquid water obtained from the sample Automated devices ior

measuring the weight of the snow above a certain point in the ground have been

de\eloped-these include snow pilLows, which measure the pressure ot snow on

a plastic pil low fi l led with a nonfreezing fluid.

Interception

The amount of precipitation captured by vegetation and trces is determined by

comparing the precipitation in gages beneath the vegetation with that recorded

nea.ty unde, the open sky. The precipitation detained by interception is dissipaled

as stem flow down the trunks of the trees and evaporation from the leaf surface

Stem flow may be measured by catch devices around tree trunks.

Evaporation

The most common method of measuring evaporation is by means of an evapora-

tion pan. There are various types of evapomtion pans; however, the most widely

used are the U.S. Class A pan, the U.S S.R. GGI-3000 pan, and the 20-m2 tank

(\\t)rld Meteorological Organization, 1981) The Class A pan measures 2-5 4 cm

(10 in) deep and 120.6'7 cm (4 ft) in diameter and is constructed of Monel metal

or unpainted galvanized iron. The pan is placed on tjmber supports so that arr

Location:Class APan Co€Incient

Felt Lake, California 0.'7 7Ft. Collins, Colorado 0.70Lake Colorado City, Texas 0.12Lake Elsinora, Califomia 0.'77Lake Hefner, Oklahoma 0.69Lake Oke€chobee, Florida 0.81Red Bluff Res., Texas 0-68

circulates beneath it. The U.S.S.R. GGI-3000 oan is a 61.8-cm diamerer tankwith a conical base fabricated of galvanized sheet iron. The surface area is 0.3mr; the tank is 60 cm deep at the wall and 68.5 cm deep at the center. The tankis sunk in the ground with the rim projecting approximately 7.5 cm above groundlevel.

In addition to the pan, several other instruments are used at evaporationstations: (l) an anemometer located I to 2 meters above the pan, for deternriningwind movement; (2) a nonrecording precipitation gage; (3) a thermometer tomeasurc water temperature in the pani and (4) a thermometer for air temperature,or a psychrometer where temperature and humidity of the air are desired.

By measuring the water level in the pan each day, the amount ofevaporarionwhich has occuned can be deduced after accounting for the precipitation dudngthat day. The depth of the water in the pan is measured to the nearest hundredthof an inch by means of a hook gage or by adding the amount of water necessaryto raise its level to a fixed point. The evaporation recorded in a pan is greaterthan that which would be recorded from the same area of water surface in a verylarge lake. Adjustment facton or pan coefficients have been determined to convertthe data recorded in evaporation pans so that they correspond to the evaporationfrom large oper water surfaces. Table 6.2.I lists pan evaporation coefficients forvarious locations.

Evapotranspiration

Evapomtion from the land surface plus transpiration though the plant leaves, orevapotranspiration, may be measured by means of Lysimeters. A lysimeter is atank of soil in which vegetation is planted that rcsembles the surrounding groundcover. The amount of evapotranspiration from the lysimeter is measured by meansof a wafer balance of all moisture inputs and outputs. The precipitadon on thelysimeter, the drainage through its bottom, and the changes in the soil moisturewithin the lysimeter are all measured. The amount of evapotranspiration is theamount necessary to comDlete the water balance.

).E'

(gvonorcorr: ve,qsurErtraur 185

181 t | ' r .r t)c\

Gas cl l inder

- /i , l

O n l i r e

FIGLRE 6.3. l lc lfVr t" , l " r .L nt . r 'u . .n,"r t us ing a bubble gage 'ecor ief Thc $ater level is measured as the bacl

t " r t "* "" ,n. b lbbl ing nrer ; oI g ' {s b} using nrercurJ mnnoneler (So'r" r Rantz eta l \ 'o l

l . F i ! i . t r 5 1 . l 9 8 l l

6.3 I1EASUREMENT OF SURFACE WATER

\later Surtace Elevation

Watersur faccc leva t ionn lgasurements inc ludebothpeak leve ls ( f loodcres te lcvatioirs) and the stage as a function ol t ime These measurements can be made

lnunu"ti1' o, automaically. Crest stage gcges are used to obtain a record of f lood

crests at'sites where recording gages are not installed' A crest stlge gage consists

of " *"nd"n stttff gage or siale' situated inside a pipe that has small holes for

th. "ntry of ruat"..-A .mull amount of cork is placad in the pipe' l1oafs as thc

warrr rises. end adheres to the staff or scale at the highest water level'

N{anual obscrvations of water level are made using staff gages' which are

gracluated boards set in tj ' le water surface. or by means of sounding devices that

ff i l th. l"r"l at which they reach the water surface' such as a weight on I * ire

stupcnriecl from a bridge over the surface of a river-

Autontatic lecords of water levels are tnade at about l0'000 locations ln the

Unired Statcst the bubbLe gage is the sensor most widely used [Fig 6 3'l(4)]

A bubble gage senscs the water level by bubbling a continuous stream ot gas

lusurll,v cri|bc,n dioxide) into the water. The pressure required to-contrnuouslY

push th" gas stteam out beneath the water surface is a measure of the depth of

in" ,r,","r-or.. the nozzle of the bubble stream This pressure is measured by

a nra onctcr in the recorder house [Fig 6 3 l(D)] Continuous recotds of water

levcls rre nreintainecl fof the calculation of stream flow rates The level of water

in a slrcam at any time is referreci to as lhe Sage hright'

FTGURE 6.3.1( t )The mercury manomeler used to measure the gas Pressure in a water lclel recorde'_ A\ the water

level and gas pressure change, itn electric 'nolor drjves a pair of sensof wire\ up or dorn to follow

tie motioi oilhe mercury surface. (S(,af.zr G. N Mesnier, US Geological Survey lISGS Water

Supp. Pap. 1669-2' . F ig. 7. 1963.)

Flow Velocity

The velocity of flow in a stream can be measured with a curtent rnel€r' Curent

meters are propeller devices placed in the flow, the speed with which the propeller

rotates being proportional to the flow velocity (Fig. 6.3 2). The current meter can

be hand-held in the flow in a small stream, suspended from a bridge or cable-way

across a larger stream, or lowered ftom the bow of a boat (Fig. 6 3'3)' The flow

velocity varies with depth in a stream as shown in Fig 6 3 4. Figure 6 1 5 shows

lsovels (l ines of equal velocity) for sections of the Kaskaskia River in Il i inois'

The velocity rises from O at the bed to a maximum near the surface, with an

average value occurring at about 0.6 of the depth lt is a standard practice of the

U.S. beological Survey to measure velocity at 0.2 and 0.8 of the depth when

the depth is more than 2 ft and to average the two velocities to determine the

average velocity for the vertical section For shallow rivers and near the banks

on deeper rivers where the depths are less than 2 ft' velocit,v measurements are

made at 0.6 depth. On some occasions' it is desired to know the tra\el t ime of

flow from one location to another some distance away. perhaps several days flow

svoror-ocrcveesuaertrr 187186 .rtrr i t tr rvirrorocr

- €- _o^

l !

20

40

60

80!

: I

t'r00

0.20FIGURE 6.3,4Typical vertical variation of the flow velocity in a

srream. ( . to! rc?r RaDlz, et a l . - vol . I , F ig. 88, p

1 3 3 , 1 9 8 2 . )

0 . 4 0 0 . 6 0 0 . 8 0 1 . 0 0 1 . 2 0

Rario of poini velocityro mean !e loci ty

FIGLRE 6.3.2Current rll.ttfs fol lneasuring water velocrty The snraller one nounled on the base in the foreground

"li".nla ," " venicat rod-and used when wading across a shallow stl€am. The lafger one rn tbe

;;.G;; *;p.;"d on a wife and used io' grg'ng a *'p'l 't':i ll:,i-i,l'l1se

or hoat Both

. " . i , * n r f o n r l " p . l n . i P l e l h a t t h e s p e e d o i r o t i o n o f $ e c u p " 1 5 p r o p n r o n a l I o r h e f l o w r e l o € r t y

i r r " "p. , , " , ' r " "* i t Aei t r ical $ i res 1olhe two scfews on the vert ical shaf t hold ing the cups Each

il"* ii . ."ps -*pr". 3 rdation. a contacl is ciosed inside the shaft and lhe oper3br hean a click

in he:rdphones 1c \\hrch the wrres are allached B) counting the number of ciicks,in^a fixed time

'" ' " ; . , i ' , . , , ' 4 l l \econdsr. the le loci t t rs determiner l (Sol t r '€ ' T J Buchanan U S Ceological

l " r l " ' t , ' " o"nl 'J .J" ' i rg + rn ' - rechr iques of waler-resources invest igat ions of the Unired

states- Gco;ogiat Survev. Book 3 Chapter A8 1969 )

time. For these purposes a float is used which is carried along with the water at

approximately its average \elocity.Velocity measurements can also be made based upon electromagnetlc

sensing. The Velocity Modified FIow Measurement (VMFM) meter is a velocity-

sensing insrunent based upon such principles (Marsh-McBimey' 1979). The

ponable meter shown in Fig. 6.3 6 has a solid state electrcnjcs system housed

in a small box, an electromagnetic sensor, and connecting cable. The sensor is

placed on the same rod used for propeller-type current meteN and the rod is hand

ietd for making velocity measurements. When the sensor is immersed in flowing

water, a magnetic field within the sensor is altered by the water flow, creating

a voltage variation which is measured by electrodes imbedded in the sensor'

Ttre amplitude of the voltage variation is proportional to the water velocity The

voltage variation is transmitted through the cable to the electronic processor sys-

tetn, *hi.h automatically averages point velocity measurements made at different

locations in a stream cross section. The sensor also monitors water depth using

a bubble gage, and the processor integmtes velocity and depth measurements to

produce discharge data. This meter can also be used to measure flows in sewer

pipes and in other types of open channels.

Stream Flow Rate

Stream flow is not directly recorded, even though this variable is perhaps the

most important in hydrologic studies. Instead, water level is recorded and stream

flow is deduced by means of a rating curve (Riggs, 1985) The rating cun'e

is developed using a set of measurements of discharge and gage height in the

stream, these measurements being made over a period of months or years so as

to obtain an accumte rclationshiP between the stream flow rate' or discharge, and

the gage height at the gaging site.

DISCHARGE COMPUTATION. The discharge of a stream is calculated from

measurements of velocitv and deDth. A marked line is strctched across the stream'

FIGLRI' 6.3,3Cun . , - re penJ(d fo rn b 'q n f a b '1 (

Z€alrnn

II

(Soxr.er Ministry of wo.ks and Development' Ne\|

0 T -

2 )

- ? a -

- l 0 -

1 2 l

0 -

2 - ,

18E :r i ,L-rEo sr o*olocr

i i = 2.82 l t /scc

80 100 120 1.1t1 lo0 l8{ l 100

8 ' r

l 0 -

t2 -.

1 4 '

l 880 100 120

Disltnce, feel

l ,+0 r 60

F IG I RC b . J ,51';:,i;i;;;t; for sections of the Kaskaskia River' lllinois These pfofiles are based upon polnr

[hi.hwereconvertedtononormensionalvelocit iesbydividingthepointveloci l iesby

;'.1i:J;""ilt;; * t..t." rr'' nondimensional velocilies were used ro draw the isolels

ill'"*'"ii"""i'""ii.',t'). in" a;."r'ntg" for the isovels shown was1000 cfs (to'rcer Bhowflnk'

1979. Used \t i th Permrssron )

At regular intervals along the line, the depth of the water is measured with a

sraau-uted rod or by lowering a weighted line from the surface to the strcam bed'

i"Jiir" ".ro.itv i, In.uturid using a cu ent meter' The discharge at a cross

section of area A is found bY

1 8 l

o

o : l l . u ao- t )

i, , !,,

( 6 . 3 . 1 )

MARSH McBIRNEY,Inc.

FIGURE 6.3.6VMFM (Velocity Modified Flow Measurement) meter'Used \r, i th permissjon.)

syonorootcveesureusr.rr 189

Model 201D

(Counesy of Marsh-McBimey. lnc ' 1987.

in which the integral is approximated by summing the incremental discharges

ca lcu la ted f rom each measurement i , i = l ' 2 , . . . ' n , o f ve loc i ty V ; and depth

d; (Fig. 6.3.1), The measurements represent average values over width Aw; of

the stleam, so the discharge is computed as

g : \ v , a , L w 1t = l

Example 6.3,1 At known distances from an initial point oo the stream. bank' the

measured depth and velocity of a stream ale shown in Table 6 3'l Calculate the

corresponding discharge at this locatior.

( 6 . 3 . 2 )

*i o"' .-

D€prh /,

FIGURE 6,3.7Computation of discharge from stream gaging data.

I/r = mean of velochies. at0.2 and 0.8 dePth

190 .qprrrro uyonolocy

TABLE 6.3.IComputation of discharge from stream gaging.

Measure. Distsnce Widthment lromnumber initial

poini Atri (fi) (ft)

Mean Area Dischergevelocity

V dLu, Vl\w(rvs) (ftl (cfs)

Depth

d(ft)

l23

56,7

89

l0 t 21 3

l 5t 6t 1l 8r9202 l

0.00o . 3 70.87r .09

o . 7 l0 .87

2.032 .222 .513.06

2.962.622.04r . 562.04t . 51t . l 80.00

0 6.0 0.0rL t6 .0- ' 3 .132' 20.0 4 452 n .0 4 .612 20.O 5.792 20.O 4.5

|2 20 .0 4 .4l ] 2 20 .0 5 .4152 n .5 6 . i) 6 1 1 5 . 0 5 . 8r82 l 5 .0 5 . ' � 1197 15 .0 5 . I2 r? 15 .0 6 .0221 15 .0 6 .5712 r5 .0 1 .2)57 15.0 ' �7.2

212 15 .0 a .2281 15.0 5.5302 15.0 3.6311 . 5 3 .2i25 4.0 0.0

4.'�7 0.049.6. 18.488 .0 16 .692.0 100.3

I14 .0 152 .890.0 63.9E8.0 ' �76.6

r08 .0 r53 .4106 .8 2 t6 .?87 .0 193 . I85 .5 214 .616.5 234.1m.0 280.897.s 288.6

108.0 283.0108.0 220.3I 2 3 . 0 1 9 1 . 982 .5 I 68 .354.0 84.836.8 43.13 .2 0 .0

Total 325.0 1693 .0 t 06 t . 4

Data trtre pro\ided by l,he U S ceological SuRey from a gaging fiade ontne Coldado River ar Austin, October 5, 1983.

Solution. Each measurement represents the conditions up to halfway between thismeasurement and the adjacent measurements on either side. For example. the firstthree neasurements were made 0, 12, and 32 feet from the initial point, and so1x,, = [(32 - l2)/2] + [(12 - 0)t2l = 16.0 fr. The cofiesponding area increment isd2Awl = J. I x 16.0 = 49.6 ft , , and the result ing discharge increment is V2d2 Al ' 2:0.3' l x 49.6: 18.4 ft3ls. The other incremental areas and discharges are similarlyconputed as shown in Table 6.3.1 and summed to yield discharge O = 3061 ft3h.and total cross sectional area A = 1693 ft2. The average velocity at this cross section;t Y - QIA = 3061/1693 = 1.81 fVs.

There are indirect methods of measuring stream flow not requiring the useof current meters or water level records. These jnclude the dye gaging methotl inwhich a known quantity of dye is injected into the flow at arr upstream site andmeasured some distance downsteam when it has become completely mixed in thewater. By comparing the concentrations at the downstream site with the mass ol

svonorocrc veasuntver"r 191

d l 0

a .

t {

f

5

Gageheight (ft)

Discharge ( 1000 cfs)

Dischafge Oage Discharge(cfs) height (ft) (cts)

L 52 .O2 .53 .03 .54.O

5 .06.0'7 _08 .09.0

20l 3 l307530808

I , 1 3 01,498t . 9 t22.8563,961

6,561

10.0 8,000I 1.0 9.588r2 .0 r1 ,300t 3 . 0 1 3 , 1 0 014.0 15.00015 .0 17 ,0101 6 . 0 r 9 . l l 0t'7 .o 21.340f8.0 23,92019.0 26.23020.0 28,610

FIGURE 6.3.8Rating curve and table for

the Colorado River at Austrn,

Texas, as applicable from

octobe. 1974 to July 1982.(Sox/ccr U. S. GeologicalSurvey, Aust in, Texas.)

the dye injected at the upstream site, the flow rate can be deduced This method ts

paniiulariy suitable for stony mountain streams, where the dye is mixed quickly

and measurements by other methods are difficult.

RATING CURVE. The rating curye is constructed by ploning successive mea-

suremeDts of the discharge and gage height on a graph such as that shown in Fig'

6.3.8. The rating curve is then used to convert records of water level into flow

rates. The rating curve must be checked periodically to ensure that the relation-

ship between tlie discharge and gage height has remained constant; scouring of

the stream bed or deposition of sediment in the steam can cause the ratrng curve

to change so that thi same recorded gage height produces a different discharge

The relationship between water level and the flow mte at a given site can

be maintained consiitently by constructing a special flow control device in the

stream. such as a sham crested weir or a flume.

(

192 .i'ruso ttYoaorocY

6.4 MEASUREMENT OF SUBSURFACE WATER

Soil N{oisture

The amount of moisture in the soil can be found by taking a sample ol soii

;;; ;;;;;;yi"g it. Bv compadns the weisht of the sample before and after

,fr" ar",nn-u"A -"uruiing th" rolume of the sample' the moisture content of

;; ;"il .;" be determined Some recording devices which rccord soil motsture

iir". t ' t in lhe field have been developed particula-rl) for inigarion 5ludie\' The'e

i^.l"d; trorra bloct.r and neurron.probes' Neurron probe' rell on the reflection

oi n"roin, eniined from a probe device inserted in a hole in the ground by the

-"*"." ,n the sunounding soil' the degree of reflection of the neutrons being

proportional to the moisture content (Shaw, 1983)'

Infiltration

Measulementsofinli l tratlonaremadeusingaringinf|trometer'.whichisametal,in* uooro^itut"ty two feet in diameter that is ddven into the soil; water is placed

i;til';. ring ani the level of the water is rccorded at regular time intervals as

iir"..d"r. ThIs permits the construction of the curnulative infiitration curve' and

irorn ,ni, ,n" indltration rate as a function of time may be calculated Somehmes

a sccond ring is added outside the fi$t, filled with water ancl marntatneo at a

.o,rsrunt t"u"i ,o tt ut the infihation from the inner ring goes vertically down into

theso i l . lnSomecases 'measurcmcntso f in f i l f fa t ioncanbemadebyus ing t lacersi"o"o"*o at the surface of the soil and exkacted from probes placed below the

surface-

Ground Water

The level of water in the saturated flow or ground water zone is determined by

means of observation wells An observation well has a float device so that the

ue.ti."l rnou.-.nt of water in the well is transmitted by means of a pulley system

," ,fr" ia."tO* house at the surface' Devices that drop a probe down tbe well on

a wire to sense the water level can be used to obtain instantaneous measurcments'

ill" t"i""i,y ol ground water flow can also be determined by tracers' including

.ornlnon ,uir. A-quantity of the tracer is introduced at an upstream well' and the

time for the pulse of tracer to reach a well somewhat downsteam of the first is

t..urO.a. fnir is the actual relocib' and not the apparent or Darcy velocity' Such

In.urut.."no also assist in determining the amount of dispersion of contaminants

introduced into ground water.

6.5 HYDROLOGIC MEASUREMENTSYSTEMS

Urban Hydrology Monitoring Systems

Urban storrnwater investigations require well-designed data colleation systems and

instrumentation, both for water quantity and water quality Besides conventronai

( "Yonoroctc vresuxuaerr 193

stream gaging and precipitation measurement' elaborate instruments employing

microprocissor technology are used to collect and record information at remote

locations such as in underground storm dralns.An instrumentation package called an urban hydrology monitoring sJstem'

as used by the U. S. Geological Survey for urban stormwater investi.gations' is

shown in Fig. 6.5.1 (Jennings, 1982). This system is designed- to. collect storm

rainfall and irnoff quantity and qualiry data. I[ was specifically designed for flow

gaging in underground storm sewers using a flow constri4io[ as the discharge

ioitri. ff,. sysrem is composed of f ire components: the sYli lem control unit '

rain gages, atmospheric sarnpling, stage sensing, and warer qf,aiity sampling"Tte

system iontrol unit is a miiroprocessor that rccords data at a central

site, controis an automatic water-sampling device, records rain gage readings via

telephone lines, and continuously monitors the stage in the storm sewers' The

IRa'n gages

IArmospheric sampling

FIGURE 6.5.1Typical installation of a U. S Geological Survey urban hydrology monitoring syslem (So!rce-'

Jennings, 1982. Used with permission )

Optional sensorslRainfall, Slage, Temperature,

Conduci iv i ty . pH, etc.

Hvonolocrc N€esuReMerqr 195194 rprrtso ur onor-ocv

control unit operates in a standby mode between storms, so that data are collected

only ii'there is rainfall. The rain gage has an 8-in diameter orifice and a tipping

bucket mechanism coupled to a mercury switch. The buckets are calibrated to tip

after each 0.01 in of rainfall.The atmospheric sampler is used to collect samples of atmospheric con-

stituents affecting water quality, as for acid rain studies. Two rectangular collec-

tors afe used, one for sampling rainfall and the other for dry deposition of dust

and other constituents between rainfalls. Water quality samples are also takenfrom srorm seweri using automatic pump sarnplers. The samples are stored in a

freezcr !\hich mlintains the water temperature at approrrmately 5'C.

Real-time Data Collection Systems for River-Lake Systems

Real-tir)e data collection and transmission can be used for f lood forecasting onlarge river lalie systems covering thousands of square miles, as shown in Fig.6.5.2 lor the lower Colorado River in aentral Texas. The data collection sys-tem used there is called a Hydrometeorological Data Acquisition System (Hy-

dromet. EG&G Washington Analytical Seryices Center, Inc., 1981) and is usedto prolide information for a flood forccasting model (Section 15.5). Thjs infor-matior is of two types: (a) the water surface elevations at va ous locationsthroughout thc river-lake system, and (b) rainfall from a rain gage network forthe ungaged drainage areas around the lakes. The Hydromet system consistsof (a) remote terminai unit (RTU) hydrometeorological data acquisition stationsinstalied at U.S. Geological Survey river gage sites, (b) microwave terminal unit(MTUI microwave-to-UHF radio interface units located at microwave repeatersites, which convert radio signals to microwave signals, and (c) a central controlstation located at the operations control canter in Austin, Texas, which receiresits infornation from the microwave repeating stations. The system is designedto automatjcally acquire river level and meteorological data from each RTU;telemeter this data on request to the central station via the UHF/microwave radiosystern; determine the flow rate at each site by using rating tables stored in thecentral s]-stem memory: format and output the data for each site: and maintain ahistorical file of data for each site which may be accessed by the local opemtor. acomputer. or a remote dial-up telephone line terminal. The system also functionsas a self-reponing flood alarm network.

Flood Early Warning System for Urban Areas

Because ol the potential for severe flash flooding and consequent loss of lifein manl urban areas throughout the world, fiood early warning systems ha.vebeen constructed and implemented. Flood early waming systems (Fig. 6.5.3) arercal-time event rcporting systems that consist of remote gaging sites with radiorepeater sites to transmit information to a base station. The overall system is usedto collect. tnnsport, and analyze data, and make a flood forecast in order tomaxinize the warning time to occupants in the flood plain. Such systems hale

COLORADO R]VER

,4\

d

SANSABA'RIVER

B U U ,Creek

Red BlufiSan Sabr

coLoRADO /RIVER //

R€pealer

Remote term;nal unit

UHF radio transm'ssron

Microwave t ransmission

Columbus

COLORADO RIVER

v/o,t

100

Brrdy

IA

{

mrles

FIGURE 6.5.2Real-lime data transmission net*ork on the lower Coiorado River' Texas waier level and raiDfall

dala are aulomalically transnritted lo lhe coDtrcl cenle. in Auslin every 3 hours to gtlide releases

from the dams. During floods data are updated everJ 15 minutcs-

I96 rpcr-reorronor.ooy

FICURE 6.5.3Eramp,e of a flood early warning systen for urban areas.

beeo jnstal led in Austin and Houston, Texas, and elsewhere (Sierra./Misco, lnc..19 86) .

The remote srarions (Fig. 6.5.4) each have a tipping bucket rain gage, whichgenerates a digital input to a transmitter whenever I mm of rainfall drains thrcughthe funnel assembly. A transmission to the base station is made for each tip ofthe bucket. The rain gage is completely self-contained, consisting of a cylindricalstand pipe housing for the rain gage, antenna mount! battery, and electronics.

Some remote stations have both rainfall and streamflow gages. The remotesiahons can include a stilling well or a pressure transducer water level sensorsimjlar to the one il lustmted in Fig. 6.5.5. The pressure transducer measurescharees oi the v',ater level above the pressure sensor's orifice. The electronicdiffcrential pressure transducer automatically compensates for temperaturc andbarometric pressure changes with a one percent accuracy over the measured range.

lillI

aH"--

f;-.r -t \ \

^, FrW Il1 ,/

ADirect ional antenna

197

FIGURE 6.5.4Remote slation combining precipitation and

sream gages. (Courtesy of Sierra/Misco, lnc.,

I986. Used wi th permisEion.)

Rain gage lop

Main housrDB

Ground level

tr {

t

Automatic repeater stations, located between the remote stations and the base

station, receive data from the remote stations, check the data for validity, and

transmit the data to the base stationIncoming radio signals are transformed from radio analog form to digital

format and are forwarded to the base station computer through a communlcatlons

FIGURE 6.5.5Self-reponing rain and water level gage on the Navidad River'

Texas. (Courtesy of Sierra,Misco, Inc , 1986 Used wrthpermission.)

198 c*riEr Hvororooy

port. After dara quaiity checks are made, the data are formatted and filed on eitherhard or f loppy disk media. Once the data fi l ing is cornpiete, the information canbe displaved or saved for analysis.

The base station has data managcment software whjch can handle up to?00 senso$ with enough on-line storage to store tbrce years of rainfall data. Itcan cqver 12 separate river systems with up to 25 iorecast points possible ineach, each forecast point can receive inflow from up to l0 different sources.Dillerenr future rainfall scenarios can be input for each individual forecast point,ar)d optional faatures can be added to controi pumps. gates, wall maps, rcmorealarms. and yoice synthesized wamings (Sierra/Misco, Inc., 1986).

6.6 NIIIASUREMENT OF PHYSIOGRAPHIC CHARACTERISTICSln lrr,drologic studies in which gaged data are sometimes not available, for cxiun-ple in rainfall-runoff analysis, runoff characteristics are estimated from phys-iog.aphic characterisrics. Watershed physiogr.aphic infqrmation can be obtainedfron maps describing land use, soils maps, geologic maps, topographic maps,and aerial photography. A typical inventory of physiographic characteristics is thefollowing l ist of 22 factors compiled for the USGS-EPA Narjonal Urban StudiesProgram (Jennings, 1982):

1, Total drainage area in square miles (excluding noncontributing areasl.2. Imper\ious area as a percentage of drainage area.3. Effective impervious area as a percentage of drainage area. Only impen ious

suriaces connected directly to a sewer pipe or otber stormwater conveyanceare included.

:1. Average basin slope, in feet per mile, determined from an average of terrainsiopes at 50 or more equispaced points using the best ar ajjable iopographlcmap.

5, Main conveyance slope, in feet per mile, measured at points 10 and g5percent of the distance from the gaging station to the drainage divide alongthe main conveyance channel.

6. Permeability of the A horizon of the soil profile, in inches per hour7. Soil nroisture capacity avenge over the A, B, and C soil horizons, in uches

of water per inch of soil.8. Sojl water pH in the A horizon.9. Hydrologic soil group (A, B, C, or D) according to the U.S. Soil Conserva_

lion Service methodology.10. Population density in persons per square mile.11. Strcet density, in lane miles per square mile (approximarely l2_ft lanes).12, Land use of the basins as a percentage of drainage a{ea includlng: (a)

rural and pasture, (b) agriculrural, (c) low-density residential (] to 2 acresper drvell ing), (d) medium-density residential (3 to 8 dwell ingi per acre),

nyonoroclcMeesunEr,iENr 199

(e) high-density residential (9 or more dwellings per acre), (l) commercial'

igj it;orttiut,'1t 1 under construction (bare surface), (i) vacant land' (j)

wetland, and (k) Parkland

13. Detentiol storage, in acre-feet of storage'

14, Fercent of watershed upstream from detention storage

15, Ercent of area drained by a storm sewer system'

16. krcent of steets with ditch and gutter drainage , 'r.

17. Fercent of streets with ditch and swale drainage. g.*

lE, Mean annual precipitation' in inches (long term)'

19. Ten-year frequency, one-hour dumtion, minfall intensity, in inchcs per hour

(long term.).

20. Mean annual loads of water qualitv constituents in runoff, in pounds per

acre.

21. Mean annual loads of colstituents in precipitation' in pounds per acre'

22, Mean annual loads of constituents in dry deposition' in pounds per acre'

These data are employed in modeling the water quantity and quality charactenstlcs

of urban waterch;ds so that the conclusions drawn from field studies can be

extended to other locations.

REFERENCES

Bhown l l l i k ,N . ,Hyd rau l i c so f f l ow in lheKaskask iaR ive r ' l l l i no i sRep l , r r o f l n res t i ga t i on9 lItlinois State Water Suvey, Urbana, lll.. 1979

EG&G Washingron Analyrical Services Center, Inc I'c'Ner Colorado Rivet Authotuv Sofuare

lJset's Manual, Albuquerque, N Mex., December 1981

Jennings, M. 8., Data collection and instrumentation, in LJrbatt Stormlrater Hydrolog' ed by

b. f. X;ut.r, water Resoulces Monograph 7, American Geophys;cal Union' pp 189-217'

washington, D.C., i982t-inslev, R. Kl M. A. Kohler, and J. L H Paulhus. Htdrclogr'for Engheers' Mccra*-Hill New

York, 1982.Marsh-McBimey, lnc., The UMFM flowmeter, Product brochu'e, Gaithersburg' Md ' 1979'

Rantz,S.E..ei al. , Measuremenr and computation of streamflow' vol ' 1' Measurement of stage

and discharge, wat€r SupPty PaPer 2175, U S Geological Survey' 1982'

Riggs, H. C., Srreamlo w Chaiacterisrics. Elsevier, Amsterdam, Holland' 1985'

Sf,""i.,, g. fra., Hydrohgy in Practice, Van Nostrand Reinhold (UK), wbkingham' England' 1983

Sierra.rMisco, Inc., Ftood earty waming syslem for city of Aus(in, Texas; Berkeley' Calif 19E6'

U. S. Geological Survey , Nationat Handbook oJ Recommendcd Meho^ Iar wat,er data Acquisition'

Offici of Water bam Coordination, U S Geologic4l Survey, Reslon' va ' I477 '

world Meteorological Organizatiofl, Guide to Hydrological Practices'-\d lt Dato Acquisi'ion an4

ProcessinS-, Report no. 168, Geneva. Switzerland,4th ed ' l98l'

PROBLEMS

6.3.1 A discharge measurement made on the Colorado River at Austin' Texas' on June

11, 1981,-yieided the fol lowing results Calculate the discharge in f t3/s'

200 rpprrr L., r r ororc,cv

Distance from bsnk (fa)

Depth (ft)

Velocity (ft/s)

0 3 0 6 J

0 18 .5 2 I . 5

0 0 .55 t . 70

80 lc)o 120 140 160

22.5 T.A 22.5 22.5 22.0

3 .00 3 .06 2 .91 3 .20 3 .36

Distance

Depih

Velocity

260 280

23.0 22.8

1 .94 1 . 67

320 340

19 .2 18 .0

L54 0 .81

180 200 224 21022.0 23.0 22.0 22.5

3.44 2.'�70 2.61 2. 15

300

21.5

| . 44

Distance

Depth

Velocity

360 380 410 450

11.'7 12.0 1t.4 9.0

L t 0 ) . 52 L02 0 .60

4'�70

5 .0

0.40

6 1 5

0

0

520 570

2 . 6 1 . 3

0.33 0.29

6.3.2 Plol a graph of velocity vs. distance from the bank for the data given in prob.6.3.1. Plot a graph of velocity vs. depth of f low.

6.-1.3 Thc observed gage height during a discharge measuremenr of the Colorado Ri\erat Austin is 11.25 ft . I f the measured discharge was 9730 fr ' rs. calculate thepercent dif ference between the discharge given by the rat ing curve (Fig. 6.3.8)and Ihat obtained ;n this discharge lneasurement.

6.3.4 The bed slope of the Colorado River at Austin is 0.03 percent. Delermine, forthe data given in Exarnple 6.3.1, what value of Manning s n would yield theobserved discharge for the data shown.

6.3,5 A discharge measurement on the Colorado River at Austin, Texas, on June 16.1981, yielded the fol lowing results. Calculare the discharge in f t ls.

Distance from bank (ft)

Depth (ft)

Velocily ({Vs)

35 55 75 95 l l 5

18 .0 19 .0 21 .0 20 .5 18 .5

0.60 2.00 3.22 3.64 3.11

0

0

0

135 155

t 8 . 2 1 9 . 5

1.42 3.49

Dislance

Depth

175

20.0

5.O2

21.5

4.92

235

21.5

4.41

I95

2t.5

255

22.03 .94

215 2952r .5 20 .52.93 2.80

Distance

Depth

Velocity

325

l7 .0

2 .80

355

1 3 . 5

1 . 5 2

385

10.6

1 .72

425

9 .0

0.95

6 . 1

0.50

525 575

2.0 0

0 .39 0

6.3.6 If the bed slope is 0.0003, determine the value of Manning's l? that would yieldthe same discharge as the value you found in Problem 6.3.5,

6.3.7 The observed gage height for the discharge measurement jn prob. 6.3.5 was l9. ?0ft above darum. The rating curve at this site is shown in Fjg. 6.3.8. Calculatethe percent difierence between the discharge found from the rating curve for thisgage height and the value found in Prob. 6.3.5.

CHAPTER

7i ' 1

d UNITHYDROGRAPH

ln the previous chapters of this book, the physical laws governing the operationof hydrologic systems have been described and working equations developed todetermine the flow in atmospheric, subsurface, and surface water systems. TheReynolds transport theorem applied to a control volume provided the mathematicalmeans for consistently expressing the various applicable physical laws. It may beremembered that the control volume principle does not call for a description ofthe intemal dynamics of flow within the control volume; all that is required isknowledge of the inputs and outputs to the contol volume and the physical lawsregulating their interaction.

In Chap. l, a tlee classification was presented (Fig. 1.4.1), distinguishingthe vadous types of models of hydrologic systems according to the way eachdeals with the nndomness and the space and time variability of the hydrologicprocesses involved. Up to this point in the book, most of the working equationsdeveloped have been for the simplest type of model shown in this diagram,namely a deterministic (no randomness) lumped (one point in space) steady-flowmodel (flow does not change with time). This chapter takes up the subject ofdeterministic lumped unsteady flow models; subsequent chapters (8-12) cover arange of models in the classification tree from left to right. Where possible, useis made of knowledge of the governing physical laws of the system. In additionto this, methods drawn from other fields of study such as linear systems aaalysis,optimization, and applied statistics are employed to analyze the input and outputvariables of hydrologic systems.

In the development of these models, the concept of controi volume remainsas it was inkoduced in Chap. l: "A volume or structure in space, surrounded bya boundary, which accepts water and other inputs, operates on them internallyand produces them as outputs." ln this chapter, the interaction between rainfall

203202

and runoff on a watershed is anaiyzed by viewing the watershed as a lumpedlinear system.

7.T GENERAL HYDROLOGIC SYSTEM MODEL

The amount of water stored in a hydrologic system, S may be related to the rutesof inflow 1 and outflow I by the integral equation of continuit.v Q.2.1):

4 ! : r od t -

( 7 . r . r )

lmagjne that the water is stored in a hydrologic system, such as a resenoir(Fig. 7. l. I.), in which the amount of stomge rises and falls with time in responseIo I and Q and their rates of change with respect to time.. dlldt, d2lldt2, . . . ,dQidt.d2Qldt2,. . . . Thus, the amount of storage at any time can be expressedby a storagc function as.

, = , ( , 4 ! a ' r n d Q d Q )" ' \ " a , d t2 " ' ' d r ' d r ) l( 7 . t . 2 )

( 7 . 1 . 3 )

in which al, c2, . . ., an, b t, bz, . ., r ' ' are constants and derivatives of higher

order than those shown are neglected. The constant coefficients also make the

system time-invariaat so that the way the system processes inPut!into output does

not change with time.Differentiating (7.1.3), substituting the result for dSld4n (7.1 1), and

rearranging yields

- d Q , - d " t Q * - ^ { g _ . , 4 Q * n =a n

d F + a n - t

d f 1 ' - " ' � d | - " t d t - v -

dt d2t d^- | I d,l, - O r i O ,

a l . b n t ( i l . r - b . -

which may be rew tten in the more compact form

N(D)Q : M(D)I

where D = dldt and N(D) and M(D) are the differential opemtors

1 7 . t . 4 )

(7 .1 . s )The function / is determined by the naturc of the hydrologic system being

examined. For example, the linear reservoir inhoduced in Chap. 5 as a rnodel forbaseflow in streams relates storage and outflow by S = t0, where I is a constant.

The continuity equation (7.1.1) and the storage function equation (7.1.2)must be solved simultaneously so that the output 0 can be calculated given theinput /, where I and Q are both functions of t ime. This can be done in two ways:by dilferenriating the storage function and substituting the result for dsld/ in(7. I . 1), then solving the resulting differential equation in I and Q by jntegrarionlor bl applying the finite difference method directly ro Eqs. (7.1.1) and (7.1.2)to solve them recursively at discrete points in time. In this chapter, the first, orintegral, approach is taken, and in Chap. 8, the second, or differential, approachis adopted.

Linear System in Continuous Time

For the storage function to describe a Linear system, it must be exprcssed as alinear equation with constant coefficients. Equation (7.1.2) can be written

dQ *oJ - a 1 Q + a 2 d t + 4 3 * r "

+ b 1 r + b 2 + + b t + + . . .dt ar

1- u n--::::la f

and

M(D) : -b^# - b^-,# - . . . - u, f i + rSolving (7. 1.5) for Q yields

o<tt: ffixo ( ' t .r .6)

ff= rro Ot,t

QQ)FIGURE 7.I. IContinuity of water stored in a hydrologic sysrerr.

The function M(D)/N(D) is called the transfer function of the system; it describes

the rcsponse of the output to a given input ,equenceEquation (7.1.4) was presented by Chow and Kulandaiswamy (1971) as a

general irydrologic system model. It describes a lumped system because it contains

derivativis with respect to time alone and oot spatial dimensions. Chow and

Kulandaiswamy showed that many of the previously proposed models of lumped

hydrologic systems were special cases of this general model. For example, for a

linear reiervoir, the storage function (7 1 3) has al : k and all other coefficients

zero, so (7.1.4) becomes

r * + a : rdt

( 7 . l � 7 )

204 - L rco r r ur.,r,c '

?.2 RESPUNSI, F.LINC,IIONS OF. LINEAR SYSTEMS

The solution of (7. L 6) for the transfer function of hydrologic systems follows twobasic principles for l inear system operations which are derived ftom methods forsolving l inear differential equations with constant coefficients (Kreyszig. 1968):

1. If a -.olution i0) is multiplied by a constant c. the resulring funcrion .f lO) isalso ai solution {principle of propr.trtionality).

2. If two solutions /r(0) and .fz(Q\ of the equation are added. the resultingfunction/r(B) + lr(C) is also a solution of rhe equarion Qtrincipte oJ additi. inot-sttperpo.rit lon).

The palticular solution adopted depends on the input function N(D)1, and on thespecifiecl i ir l t ial conditions or values of the output vaiables at I = 0.

Impulse Response Function

The response of a l inear system is uniquely characterized by 1ts impulse responsefitnction.If a system receives an input ofunil amount applied instaltaneous]y (aunit impulse) at t ime r, the response of the system at a later t ime r is described bythe unit impulse response function u(t .r): t - 7 is the time lag since the impulsewas applied [Fig. 7.2.](a)1. The response of a guitar string when it is pluckedis one example of a response to an impulse; another is the response of the shockabsorber in a car after the wheel passes over a pothole. If the storage reserloirin Fig. 7.1.i is init ially empty, and then rhe reservoir is instantaneously fi l ledwith a unjt amount of water, the resulting outflow function O(/) is the intpulsere \por - . fun i t ron .

Foilo*' ing the two principles of i inear system operation cited above, i l twoimpulses are applied, one of 3 units at t ime r1 and the other of 2 units ar unre72. the response of the svstem will be 3u(l r) + 2u(t - r!, as shown in Fig.7.2.l(b). Analogously. continuous input can be treared as a sum of infinitesimalimpulses. The amounr of input entering the system between times r and 7 + dris 1(]-)dt. For example, if /(z) is the precipitation intensity in inches pcr hour anddr is an infinitesimal t ime jntervai measured in hours, then 1(7)dr is rhe depth ininches of precipitation input to the system during this interval. The djrect runoff/ - r rime units later resulting ftom rhis input is 1(i)r(t r)dr. The response to thecompletc input rime function /(r) can lhen be found b1, integrating rh..".ponr.to its constituent impulses:

( 1 . 2 . t )

This exprcssion. called the convolution integral, is the fundamental equation forsolution of a l inear system on a continuous time scale. Fjgure 7.2.2 i l lustratesthe response summation process for the convolutjon intggral.

For most hvdrologic applications, solutions arc needed at discrete inteNalsof t ime. because the input js specified as a discrete time function, such as an

Or,l: lom,q, r)dr

205

l V ) , Q t 1 \

fu),Qlt)

7 t r z

Tim

( h )

FIGURE 7.2.7Respoflses of a linear system to impulse inputs. (a) Unil impulse response functio.. (DJ The rerponse

to 1wo impulses is found by summing the individual response functions.

excess rainfall hyetograph. To handle such input, two further functions are need-ed, the unit step response function and the unit pulse response function, as sholl'nin F lg . 7 .2 .3 .

Step Response Function

A unit step inplt is an inpur that goes from a rate of 0 to I at time 0 and continuesindefinitely at that mte thereafter [Fig.

'7.2.3(b)]. The output of the system, itsunir step response function g(t) is found from (7.2.1) with (z) : 1 lot " - U, ut

QQ) : BO : [outt - ,) a, (1 .2 .2)

If the substitution I = t - r is made in ('1.2.2) rhen dr : - dL, the l imit t : Ibecomes I = t - t - 0, and the limit r = 0 becomes I = t - 0 = /. Hence,

I o8( t ) =

) r u t l \d l

or

I

f

Timel

g(0 : ['ou(t)

aL ( 1 . 2 . 3 )

206 cppr-rtr) rr Drorool

-,r ^, _---*lr/ . �

Tirnc rndcx , - ,7 + l

) . l /

t , r l t .

1, ) Discrcrc t imc lurc l r {ms(d l Cor l ,nuous t ime func t jons

FICURE 7.].2The relrt ionship betwcen continuous and discrete convolul ion.

In \\,ords. the value of the unit step response function g(l) at t ime I equa]s theintegral of the impulse response function up to that t ime. as shown in Fig. 7.2.3(a.)and l r ) .

Pulse Response Function

A unit pulse input is an input of unit amount occurring in duration A/. The rateis ,1(r) : 1,'Xl, 0 < r < Ar, and zero elsewhere. The unit pulse response fwtctionproduced b1' this input can be found by the two linear system principles citedearlier. First, by the principle of proportionality, the rcsponse to a unit step inpurof rate l/At beginning at t ime 0 is (l/Ar)gO. If a similar unir step inpur oeganat time A1 instead of at 0, its response function would be lagged by time intervalAr. and would have a value at rime I equal to (l/Arg(. - Ar). Then, using theprinciple of superposition, the response to a unit pulse input duration Ar is loundby subtracting the response to a step input of rate l/At beginning at t ime Ar fronr

-\

1 , r - , r + ) " t /

. - - ' Q , , = L r , , r t , , , . ,

Q \ t ) = I ! \ 1 ) t t . t r ) t t 1

207

j

(d) lmpulsc response. ll(/)

/-\ |/ \._/ourpur

r(/)

Z ,\.- . r'."

h(t)

l r r r = j l v r r r , r r a | l

FIGURE ?.2,3Response functions of a Iinear system. The response funclions in (a). (r). and (c) are on a continuous

time domain and that in (d) on a discrete time domarn.

the response to a step input of the same tate beginning at time 0, so thal the unitpulse response function h(t) 1s

r,<rt: *tetrl s(r - Arl t 7 . 2 . 4 )

: ,.!,[,^r"- 1,-^' ^u*]t F

L.t .J t ^tu\l) dl ( '7 .2.s)

208 .+prteou'rorolocv

As ;hown in Fig. 1.2.3. g(t)- g(t-Al) represents the area under tbe impuise

response function between , - At and 1, and h(r) represents the slope of the unit

step rcsponse function g(t) between these two t ime points.

Example 7,2.1. Delermine the impulse. step and pulse response funclions of a

I incar reservoir with storage constant t(S : tO).

Solution- The continuity equation (7 l l) is

+ = ro_eoand differenrial;ng the stomge function S : kQ ytelds dsldt: kdQitlt. so

^ d ? = t , , ' - Q ' , ,

dQ Ler,, - lt',,d t k - k

This is a first-order linear differential equation. and can be solved by mulliplyingborh sides of the equation by the inte7ratirtg factor ettNl

o , , d e _ r , r , , p , t 1 . l e , , t r . t 1d t k k '

so that the two tems on the left-hand side of the equation can be combjned as

!qg" ,5: le ,hrqt ld t - k

Inlegrating from the init ial condit ions Q : Q. at t = 0

P r ) t ( 'I doe " t - | ' : e - \ h : t d r) o o

- J k

where r is a dummy variable of lime in the integration. Solving.

r ,Q l r ) " " Q o - | l e " l t d r

ano rearangrnS,

f te@: e"e_t tk +

)01" , , " ) , r to)a,

Cornparing this equation with the convolut ion integral (7.2.1), i t can be seen thatthe two equations are the same provided Q": 0 and

r , 1 t - r t - l e " ' (

So if I is defined as the lag time r - r, the impulse response function of a linearreservol l Is

utD = le-R

(

The requirement thal 8. : 0 implies that the system

convolution integral is aPPlied

The unjl step response is given by (7.2 3):

I 'gC1:

J "u0 dt

: I le tkatJ t K

=t "-"-t '= t - e - t / r

The unit pulse response is given by (7.2 4):

. . l .1111 - - l j r 11 B r r A r ) l

1 . Fo r0= t < A t , 8 ( t - AD : 0 ' so

I L ( l - e - ' t k )

n ( r ) : t 8 ( t ) = A , .

2 . Fo r t > A t ,

stans from res! when the

t _ ,h i r , - - r l , t ? ! ' ' t e ' ' u ^ ' l

=1,,o,r: A r \ ' t )

The impulse and step response functions of a linear reservoir with t : 3 h are

fioned';n Fig. '7.2.i, aling with the pulse response function for At = 2 h

o 2 4 6 8 I 0

Tine (h)

FIGURE 7.2.4Response function of a linear reservoir with t = 3 h Puise resPonse function js for a pulse input of

two hours duration. (from Example ?.2. L)

209

Slep response functron

g ( / ) = l - € " ^

Pulse response funclion l(t) fot Lt = 2h

lmpulse response funclron

210 rppueo nvonolocv

Linear System in Discrete Time

The impulse, step, and Pulse rcsponse functions have all been deltned on acontinuous time domain. Now let the time domain be broken into discrete inte als

of dumtion Ar. As shown in Sec. 2.3, there are two ways to represent a continuousdme function on a discrete time domaih, as a puLse data system or as a sampLedota system. The pulse data system is used for Precipitation and the value of itsdiscrete input function for the mth time interval is

P ^ = l I ( r ) d t m = 1 , 2 , 3 , . . .J ln - t ) 6 t

('7 .2.6)

P. is the depth of precipitation falling during the time interval (in inches orcentimete$). The sample data system is used for streamflow and direct runoff,

so that the value of the system output in the 'lth time interyal (l = t? A/) is

g , = Q @ A t ) n : r , 2 , 3 , . . . (1 .2.1)

Oi is the instantaneous value of the flow rate at the end of the ,1th time inter-Ial (in cfs or m3/s.1. Thus the input and output variables to a watershed systemare recorded with different dimensions and using different discrete datarcpresentations. The effect of an input pulse of duration A/ beginning at time

0, - l)At on the output at time t = ,Al is measured by the value of the unit pulseresponse function hlt - (n- I)At1 : 7t7"6, - (m - l\Atl = h[(n-n + l)Lt],given, following Eq. (7.2.5), as

I f {n-u + l )at

h l ( n - m l r A r l = - l u t l t d lAI J (h-hJL l

( '7 .2 .8)

On a discrete time domain, the input function is a series of M pulses ofconsrant ra te : fo rpu lse m, I ( r )= pS611o, (m- l )A t< r<mAt . ( r ) = 61otr > MAr. Consider the case where the output is being calculated after all the inputhas ceased, that is, at t = nAt > MAt [see Fig.'7.2.2(b)). The contribution to theoutput of each of the M input pulses can be found by breaking the convolutionintegral (7.2.1) at / = nAt into M parts:

r l I

A ' J O

P, ro', I

Al Jar

^ ra{t ^ .MLI

, t t l u tn l r - i dr + . . . + r+

| u . r ,Ar - " larL! J\a- | )61 Ar J\M- l ,Ar

where the terms P./At, m = 1,2, . . . , M, can be brought outside the integralsbccause they are constants.

Ineachof these integrals, the substitution I = nAt - "ismade, sodr: dl,rhe l im i t r : (n - l )A tbecomes I : nA, t - (m- l )A t = (n - m + l )A / , and

lnLto- - l l t ' t tu tn \ t - r \d r

Jo

u(nA,t 't) d't + u(nA,t - r)dr + . . . (7 .2 .9)

(

the limit r= rzAr bessmss / : (4-m)Ar. The mth integral in (7.2.9) is now written

D p"A' p fn n)Lr

AI J w- t , l , r Al J \a-h - l )6r

D in-m+ tlA,t

= + | u\t) dt

zt l

!= P^h[(n - rr + l)44 1 {a

by substitution from (7.2.7). After making these substitutions'for each terrn in(7 .2 .9 ) ,

(7 .2.rO)

8n = Pftl(nA't)l + P2hl(n - l)Atl + . . .- P ^ h l ( n - n a t ) A I l - . . .

+PMh[(n- M + t )At ]

( 7 . 2 . l l )

which is a convolution equation with input P. in pulses and output On as a sampledata function of time.

Discrete Pulse Response Function

As shown in Fig. 1.2.3(Q, the continuous pulse response function h(t) may berepresented on a discrete time domain as a sample data function U where

('7.2.r2)

It follows that U" = h!nA,tl, U^-1 = h[(n l)At],. . . , and Un ya1 =

hl(n - M + 1)A4. Substituting into (7.2.11), the discrcte-time version of theconvolution integral is

Q " : P t U , * P 2 U n - 1 + . . . + P n U ^ - ^ + r * . . . * P u U , - u + t

- S o r r- Z - ' n " a - n + l

Equation (7.2.13) is valid provided n > M; if n < M, then, in (7.2.9), one wouldonly need to account for the fi$t |r pulses of input, since these are the only pulsesthat can influence the output up to time n Ar. In this case, (7.2.13) is rewritten

n - \ - rY n 2 - ' ) . U n - ^ + t

m = 1

Combining ('7 .2.13) and (7 .2.14) gives the final result

U n - n + t : h l ( n - m + \ ) A ' t l

\ 1 . 2 . 1 3 )

(7 .2 . 14)

2 l2 , rpp r teosvo to locY

lnpul

4 ,)X

Unir pulsc rcsponsc apptied lo P,

Unr pulsc responsc applicd to P1

'ig

J

FTGURE 7.2.5Application of the discrete convolution equation to tbe oulpul from a linear system

(

o , = > P , u n ̂ + l (1.2.1s)

which is the discrete convolution equation for a linear system The notation

,'= U u" the upper timit of the summation shows that the terms are summed

fot m : 1,2,. . ., n for n = M, btJt for n > M, the summation is l imited to

m = 1 , 2 , . , M .As an example, supPose there arc M : 3 pulses of inpd[ P 1, P2, and P3

For the first time interyal (n = t), there is only one term in thefconvolution' that

f o f m : l l

Q 1 = P 1 U 1 - 1 a 1 : P 1 U 1

For n = 2, there are two terms, corresponding to m: 1,2:

Qz - p t l Jz 111 I P2L/2 ,2a1 - P1U2- l P2U1

For n : 3, there are three terms:

Qz = P: ,Ut - t+ t I P2 t l3 -2a1 4 P3L l3 311: P tUz l P2U2 * P3U1

And for n -- 4,5, . .. there continue to be just three terms:

Q , : P t U ^ ' t P 2 U , - 1 I P l U n - 2

The results of the calculation are shown diagramatically in Fig 7 2 5 The

sum of the subscripts in each term on the righahand side of the summation is

always one greater than the subscript of 0.In the example shown in the diagram, there are 3 input pulses and 6 non-

zero terms in the pulse response function U, so there are 3 + 6 - 1 : 8 non-

zero terms in the output function O. The values of the outPut for the final three

periods are:

Q a : P t U s * P 2 U 5 * P 3 U a

Q7 = P2U6 + P3U5

Qs= PtUo

Q, and P^ are expressed in different dimensions, and U has dimensions

that a.re the ratio of the dimensions of 8,and P.tomale(7 2.15) dimeosionally

consistent. For example, if P. is measured in inches and 0n in cfs' then the

dimensions of U are cfs/in, which may be interpreted as cfs of output per inch of

input.

7.3 TIIE UNIT HYDROGRAPH

The unit hydrograph is the unit pulse rcsponse function of a linear hydrologicsystem. First proposed by Sherman (1932), the unit hydrograph (originaily namedunit'graph) oi a watershed is defined as a direct runoff hydrograph (DRH) resulfing from I in (usually taken as I cm in SI units) of excess rainfall generated

213

unifomrly over the drainage area at a constant rate for an effective dumtion.Sheman originally used the word "unit" to denote a unit of time, but since thattime it has often been interpreted as a unit depth of excess rainfall. Shermanclassified runoff into surface runoff and grcundwater runoff and defined theunit hydrograph for use only with surface runoff, Methods of calculating excessrainfall and dircct runoff from observed rainfall and streamflow data arc prcsentedin Chap. 5.

The unit hydrograph is a simple linear model that can be used to deriverhe hydrograph resuiting from any amount of excess rainfall. The following basicassumpticns are inherent in this model:

1. Tbe excess rainfall has a constant intensity within the effective duratron.

2. Tbe excess rainfall is unifomly distdbuted throughout the whole drainagearea.

The base time of the DRH (the duration of direct runoff resulting from anexcess rainfall of giyen duration is constant,

The ordinates of all DRH'S of a common base time are directly proportionalto the total amount of direct runoff rcpresented by each hydrograph.For a given watershed, the hydrograph resuiting from a gJven excess rainfallreflects the unchanging characteristics of the watershed.

Under natural conditions, the above assumptions cannot be perfectlysatisfied. However, when the hydrologic data to be used are carefully selected sothat they come close to meeting the above assumptions, the results obtained by theunit h-""drograph model are generally acceptable for practical purposes (Heerde-gen, 1974). Although the model was origlnally devised for large watersheds, ithas been found applicable to small watersheds from less than 0.5 hectares to 25km2 (about I acrc to 10 mi21. Some cases do not support the use of the modelbecause one or more of the assumptions are not well satisfied. For such reasons,the model is considered inapplicable to runoff originating from snow or ice.

Conceming assumption (l), the storms selected for analysis should be ofshort duration, since these will most likely produce an intense and nearly constantexcess rainfall rare, yielding a well-defined single-peaked hydrogmph of shorttlme base.

Conceming assumption (2), the unit hydrograph may become inapplicablewhen the drainage area is too large to be covered by a nearly unifonn distributionof rainfall. In such cases, the area has to be divided and each subarea analyzedtb. stoms coveing the whole subarea.

Conceming assumption (3), the base time of the direct runoff hydrographIDRH) is generally uncertain but depends on the method of baseflow separarion(see Sec. 5.2). The base time is usually short if the direct runoff is consideredto inciude the surface runoff only; it is long if the direct runoff also includessubsurface runoff.

Conceming assumption (4), the principles of superposition and proportion-ality are assumed so that the ordinates Q, of the DRH may be compured by Eq.

(

TABLE 7.].1

iorp".iton of linear system ard unit hydrograph concepts

215214 .rpprno uvonolocv

UniL hYdrograPh

I Excess ra infa l lP ' ,

t ' - - r - - " ,| - - -i/----lvarer\hqd it \ \ ' ii- ' \==y+<:_ - - )

Q,,= L P , ,u , , , * r

,zf ',,- Drs.."tc rul'c nmnt

4 .

3 ( / )l

5- Slstem stdns i rom resl .

6. Sysrern is l incar

L Ttunsfcr iunction has constant cocflicients

t t . Slsrcm obcys cont inuiry.

: [ = / 1 , ) O ( / )J ! -

Lrne!r sysrcm

D,rec! runoffg,

N.- , , *.- *** ,","n'/1 , / t ) r t t h ldroeraph ol

/ J f durat ion aI| .t

'r-J .

5 .

t ( / )

I

5. Direcl runofi hydrograph shns irom zero All previolls

minfall is abso.bed by walershed (ininal abstraclron or

6. Direct rrnotT bydrogmph is calculated using principles

of proponionality and supeQosrlron

7. Walcrshed lesponse is iime invarianl, Rol changing from

one storm 10 another'

8. Tolal depths of excess rainfall ,rnd direcr runoff are equa

t , - ) = t P

J ,- r int o, cm'+r e^ces'/ .ainlal l+ '

216 apnum uvonolocv

*c:

,'..

:.

.

277

(7.2.15). Actual hydrologic data are not truly l inear; when applving (7.2.15) tothen. the resulting hydrograph is only an approximation, which is satisfactory inmanl practical cases.

Concerning assumption (5), the unit hydrograph is considered unique fora given $atershed and invariable with respect to time. This ls the principLeof time invariance, which. together with the principles of superposition andproponionaiity, is fundamental to the unit hydrograph model. Unit hydrographsare applicable only when channel conditions remain unchanged and watershedsdo not have appreciable storage. This condition is violated when the drainagearea containl many reservoirs, or when the flood overflows into the flood plain,thereby producing considerab]e storage.

The principles of linear system analysis form the basis of the unit hydrographmerhod. Table 7.3.1 shows a comparison of l inear system concepts with theconcsponding unit hydrograph concepts. In hydrology, the step response functionis crrrrrnonly called the S-hydrogruph, and the impulse response t'unction is calledthe instantaneous unit hydrogrqph which is the hypothetical response to a unitdepth of ercess rainfall deposited instantaneously on the watershed surface.

7...1 UNIT HYDROGRAPH DERMTION

The discrete convolution equation (7.2.15) allows the computation of direct runoff0,, given excess rainfall P, and the unit hydrograph L/,-,11

o : > P t lz , , n e n f i r t

The reverse process, called deconvolution, is needed to derive a unit hydrographgiven data on P, and @,. Suppose that there are M pulses of excess rainfall andN pulses of direct runoff in the storm considered: then N equations can be \\fittenfor Qn,n = 1,2,. . .,N, in terms of N - M + I unknown vaiues of the unith)'drogmph, as shown in Table'7.4.1.

lf Q" and P^ are given and Un-. * 1 is required, the set of equations in Table'7.1.I

is overdetermined, becatse there are more equations (,AD than unkrowns( N M + t ) .

Example 7.4,1. Find the half-hour unit hydrograph using the excess rainfall hyeto-graph and direct runoff hydrograph given in Table 7-4.2. (these werc derived inExarnple 5.3.1. )

Solution, The ERH and DRH in Table 7.4.2 have M = 3 and N : 11 pulsesrcspectively. Hence, the number of pulses in the unit hydrograph is N - M + I =11 - 3 + I = 9. Substituting the ordinates oi the ERH and DRH into rbe equationtin Table 7.4.1 yields a set of 11 simultaneous equatjons. These eqirations may besolved by Gd,,sj elint irarion to give the unit hydrograph ordinates. Gauss eliminationinvolves isolating the unknown varjables one by one and successively solving forthem. ln this case, the equations can be solved from top to bottom, working withjusl the equations involving the firsl pulse Pr, s[arting with

TARLE7.4.1

The set of equations for discrete time conYolution A, = \ r ^u,-^., 'n = l

n = 1 , 2 , . . . , N

'o ; =Pzu, +PtuzA; - p,lJ1 *P2U2 +P\Uj

O" = PMLJl + PM-tU2 +ju * t = o +PMuz +

+ P tUM+ P2UM + P .UM + l

: ' l

O , . ' = 0 - 0 - . - 0b , - o - o - . + o

+ 0 + . . .+ 0 + . . .

i P y U a y I P M - 1 U N M + l+ 0 *PuUu-u ' r

,, =L = 111 = +o+ "wtn- ' P t l � 0 6

, . Qz - PzUt 1923 1.93 x 404 :1079 c fs / in1 . 0 6

529'7 - l.8l x 404 - 1.93 x 10'�79. , Q3 - P3Ut - P2U2

and siml)arl1 for lhe remaining ordinales

1.06

9131 -.. 1.81 x 10'�79 - 1.93 r.2313

1.06

10625 - r.8l x 2343 1.93 x 2506l .06

7 8 3 4 - 1 . 8 1 x 2 5 0 6 - 1 . 9 3 x 1 4 6 0

= 2343 cfs/in

(7 .4 . r ) : 2506 cfs/in

: 1460 cfs/in

= 453 cfs/inL06

TABLE 7.4.2Excess rainfall hyetograph and directrunoff hydrograph for Exampl€ 7.4.1

Time(i h)

Excess rainfall(in)

Direct runoff(cfo

1 .061 .93t . 8 t

I2l

5618

t 0t 1

424t9235?..979.131

10625't834

3921t84614028303 1 3

(2!8 .rppueo uvonorocY

Il,liiJJ;it** derived in Exampre 7.4'1

L, (cfs i in lt 2 3 4404 t0'79 2343 2506

51460

6 7 8 9453 38t 2'14 1'�73

J92 l - t . s l 1460 1 .93 453

1.06

1 8 4 6 - 1 . 8 1 x 4 5 3 - 1 . 9 3 x 3 8 11.06

1 4 0 2 - l . 8 l x 3 8 1 - 1 . 9 3 x 2 7 4

= 2'74 cfslin

= 173 cfs/ in1.06

'fhe derivcd uni! hydrograph is given in Table ? 4 l Solutions may be similafly

obtained by focusing on other rainfall pulses The deplh of direct runolT in the

unir bydrograph can be checked and found to equal 1 00 inch as required ln cases

where the derived unit hydrograph does nol meet this requirement' the ordinates are

adjusted by proportion so that the depth of direct runoff is I inch (or I cm)

In general the unit hydrcgraphs obtained by solutions of the set of equations

in Tablei.4.t for different rainfall pulses are not identical To obtain a unique

solurion a method of successit'e apProximatioz (Collins, 1939) canbe used' which

involves four steps: (l) assume a unit hydrograph, and apply it to all excess-

rainfall blocks of the hyetograph except the largesti (2) subtract the resultlng

hydrograph from the actual DRH, and reduce the residual to unit hydrogmph

t.r-r, (:j compute a weighted average of the assumed unit hydrogmph and the

rcsiduai unit hydrograph, and use it as the revised apProximation for the next

trial; (.1) repeai the privious three steps until the residuai unit hydrograph does

not differ by morc than a permissible amount from the assumed hydrogmph'

The resulting unit hydrograph may show efiatic variations and even have

nesative ralues. Ii this occurs, a smooth curve may be fitted to the ordinates to

p.Jdu.. nn approximation of the unit hydrograph Enatic variation.in the unit

ilydrograph.rniy be due to nonlinearity in the effective rainfall-dir-ect runoff

,etutio-ntt]ip in ihe watershed, and even if this rclationship is truly linear' the

observed data may not adequately rcflect this. Also. actual stomls are not always

uniform in time and space, as required by theory, even when the excess minfall

hyetograph is broken into pulses of short duration'

7.5 UNIT HYDROGRAPH APPLICATION

Once the unit hydrograph has been determined, it may be applied to frnd the

direct runoff und' ,tr*-flo* hydrographs. A rainfall hyetogmph is selected, the

abstractions are estimated, and the excess rainfall hyetograph is calculated as

described in Sec. 5.4. The time interval used in defining the excess rainfall

hyetograph ordinates must be the same as that for which the unit hydrograph was

specified. The discrete convolution equahon

219

a l \ D t lu n 1 , r h t v n - n + |

may then be used to yield the direct runoff hydrograph. By adding an estimated

baseflow to the direct runoff hydrograph, the streamflow hydrograph is obtained

Example 7.5.1, Calculate the sfteamflow hydrograph for a storm of 6 jn excess

rainfall, with 2 in in the first half-hour, 3 in in the second hqlf-hour and I in in

the third half-hour. Use the half-hour unjt hydrograph comprtei{ in Example 7.4 1

and assume the baseflow is constant at 500 cfs throughout the l'lpod. Check that the

total depth of di.ect runoff is equal to tie total excess precipitafion (watenhed area: 7 .03 m i2 ) .

Solution. The calculation of the direct runoff hydrograph by convolution is shown

in Table 7.5.1. The unit hydrograph ordinates frotn Table 7.4.3 are laid out along

the top of the table and the excess precipitat ion depths down lhe left side The t ime

interval is in Ar = 0.5 h intervals. For the f irst t ime interval, ,? = 1 in El (7.5 1) '

and

Q t : P L u t

:2.00 x 401

:808 cfs

For the second time interval

Qz : PzUr i P tUz

: 3 . 0 0 x 4 0 4 + 2 . 0 0 x 1 0 7 9

: 1212 + 2158

TABLE 7.5.ICalculation of the direct runoff hydrograph and streamflow hydrograph for Exampk7.5.1

Unit hydrograph ordinates (cfvin)Dircct Streamfloiv'

Precipitation 1 2 3 4 5 6 7 I 9 r'rofl(in) 404 1079 2343 2506 1460 453 381 274 173 (cfs)

( 7 . 5 . 1 )

Time(i-t'l

(cfs)

| l = t 1 . 0 0 .2 3.rn3 1.00456189

l01 t

808t2 l2 2 t58404 3237

t079

808 13083370 38?08-127 8827

13,110 13.620t2 ,181 i 3 ,281

T792 82923581 4081?111 ?644

316 1519 20495 19

'�193 1293t13 ) ' �73 673

Total 54,438

46867029 50122 3 4 3 " t 5 1 8

250629204380I460

906r359

1 1 4 33 8 t

548822

*Baseflow = 500 cfs.

22{J rplrtEo uronolocv

= 3370 cfs

as shown in the table. For the third time inten'al'

Q ;= PtU t + P1U2 ' r P1U3

:1 .00x40.1 + 3 .00 x 1079 + 2 .00 x 2343

:404 + 3237 -i 4686

= 8327 cfs

The cileulatrons for r = 4,5, ' follow in the same manner as shown in Table

l . : ' l and qraph i .a l l ) .n f i8 - .5 .1 lhe lo l r r d r rec l runo l f vo lume rs

v,= | q,rr

= 5 4 , 4 3 8 x 0 5 c f i h

=5'1'438 x o 5 ftr 'h x 3qo9 s

l h

:9 .80 x l0 r i l

and the crrrrespondrng depth of drrect runoff is found by dividing by the watershed

, r .e . { - r .0 i n i - ' .01 528n f r : - I qb 103 l t :

v,,, ' = a

9.96 x l0r ̂l .9b 108

'

=0 500 ft

= 6 .00 in

3 in ra infa l lexcess

2 in rflinfallexcess

0 1 2 3 4 5Timc (h )

FIGT]RE 7.5.IS t l ean l f l o ' , ' h yd rog raph f ronas to rmw i tbexcess ra in fa l l pu l seso fdu ra l i oo0 '5haDdamoun l2 in .I ;n. rnrl I in. re+ectiuelf . Total streamflo!! = baseflow + direct runoff (Exanlple 7 5 lJ

l 4 l

t 2 l

?

a: E '

a 6 ':

2 )

221

which is equal to the total deptl of excess precipitation as required

The streamflow hydrogFph is found by adding $e 500 cfs baseflow to the

direct runoff hydrograph, as shown on the righFhand side of Table 7 5 1 and

graphical ly in Fig 7 5.1 '

7,6 LTNIT HYDROGRAPH BY MATRIX CALCULATION

Deconvolution may be used to derive the unit hydrograph frPnl{a comPlex mul-

tioeaked hydrograph, but the possibility of enors or nonlinearilY in the data is

gieater than for a single-pealed hydrograph. lrast squares fftting or an opti-

irization method can be used to minimize the error in the fitted direct runoff

hydrogmph, The application of these techniques is facilitated by expressing Eq'

(7.4.1) in matdx form:

P1

0 0 00 0 0

0 0 . . . 0 0 . . . 0P 1 0 . . . 0 o . . . 0P 2 P 1 . . . 0 0 . . . 0

000

.00

Q rQzQt

Qu

?'.'Qx-t

( 7 . 6 . 1 )Py Py-1 P 1,a-20 Py P1a1.

p , n n' . P 2 P t ' . o ir" l:

LP)TL\ : IQl (7 .6 .2)

Given [P] and [O], there is usually no solution for [14 that will satisfy all

1f- equations (?.6.1). Suppose that a solution [{./] is given that yields an estimate

lQl of the DRH as

. . 0 o . . . P u P u - s

. . 0 0 . . . 0 P u

tPlttt = tQl

Q, : p ,u, i Pn- lJ2 * . . . * Pn u+f lu

(7 .6 .3a)

n : 1 , . . . , N ( 7 . 6 . 3 b )

with all _equations now satisfied. A solution is sought which minimizes the error

[0] - [Q] between the observed and estimated DRH's.

Solution by Linear RegressionThe s^olution by linear regression produces the least-squares enor between [O]and [Q] (Snyder, 1955). To solve Eq. (7.6.2) for [L/], the rectangular matrix [P]

222,qelr-trosvpnot-ocv

is reduced to a square mattrx [Z] by multiplying both sides by the transpose of

lPl, denoted by [P]r, which is formed by interchanging the rows and columns of

LPI. Then both sides arc multiplied by the inverse [Z] -r

of matrix [Z], to yield

LUt : LZ I - t tP fLQ l (1 .6 .4 )

where lZl = LPlrlPl- However, the solution is not easy to determine by this

method, because the many repeated and blank entries in [P] crcate difficulties in

the inversion of [Z] (Bree, 1978). Newton and Vinyard (1967) and Singh (1976)

give aitemative methods ofobtaining the least-squares solution, but these methods

do not enque that all the unit hydrograph ordinates will be nonnegative

Solution by Linear Programming

Litrcu progromming is an altemative method of solving for [L]) in Eq ('7 6.2)

that minimizes the absolute value of the eror between [0] and [Q] and alsoensures that all entries of [{,1 are nonnegative (Eagleson, Mejia, and March,1966; Deininger, 1969; Singh, 1976; Mays and Coles, 1980).

The general linear programming model is stated in the form of a linearobjective function to be optimized (maximized or minimized) subject to linearconstraint equations. Linear programming provides a method of comparing allpossible solutions that satisfy the constxaints and obtaining the one that optimizesthe objective function (Hillier and Lieberman, 1974: Bradley, Hax, and Magnanti,t91'�7).

Example 7.6,1. Develop a linear program to solve Eq. (7.6.2) for the unit hydro-graph g iven the ERH P. ,n = t ,2 , . . . ,M, and the DRH Q, ,n = 1 ,2 , . . . ,N

Sotution. The objective is to minimize ) f-, le,l where e,: Q,- Q, Linearprogramming requires that all the vadables be nonnegative; io accomplish this task,<, is split into two cofiponents, a positive deviation 0n and a negative deviation pnIn the case where €, > 0, that is, when the observed direct runoff O, is greaterthan the calculated value Q", 0" = e, and B^ = 0; where e, < o, F^ = -€n and0. = 0 (see Fig.7.6.1). lfe, = 0 then d, = F, = 0 also, Hence, the soiution mustoDey

Q. = Q,- p, + e, n = 1 , 2 , . . . , N

and the objective is

rninimize :(d, + B,)

The constraints (7.6.5) can be written

tQ,l + to,t - tp;=to^l

or, expanding as in Eq. (7.6.3r),

P"Ut + P , - tUz* . . .+ P , -M + tUM + 0 , - p^= Q,

(7 .6 .5 )

("7 .6.6)

(1 .6.1)

n = 1 , . . . , N ( 7 . 6 . 8 )

To ensure that the unil hydrograph represents one unit of direct runoff an additjonai

(

,"= a,,-4,

-C;FIGURF 7.6.7Deviation €" bed,veen observed andestimated di4ct lhnoff hydrographsis the sum of a{lbsitiv€ deviation d,and a negative (Eviation B. forsolution by iinear programming.

constraint equation is added:

where ( is a constant which converts the units of the ERH into the units of the DRH.Equations (7.6.6) to (7.6.9) constitute a linear program with decision variables (orunknowns) ll., 0" and B, which may be solved using standard linear programmingcomputer programs to produce the unit hydrograph. Linear programming requiresall the decision variables to be non-negative, thereby ensuring the unit hydrographordinates will be non-negative.

The linear programming method developed in Example 7.6.1 is not lim-ited in application to a single storm. Several ERHS and their resulting DRHScan be linked together as if they comprised one event and used to find a com'posite unit hydrograph best representing the response of the watershed to this setof storms. Multistorm analysis may also be carried out using the least-squarcsmethod (Diskin and Boneh, 1975; Mawdsley and Tagg, l98l).

In determination of the unit hydrograph from complex hydrographs, tbeabstractions are a significant source of error-although often assumed constant,the loss rute is actually a time-varying function whose value is affected by themoisture content of the watershed prior to the storm and by the storm patternitself. Different unit hydrographs result from different assumptions about thepattern of losses. Newton and Vinyard (1967) account for errors in the lossrate by iteratively adjusting the ordinates of the ERH as well as those of the unithydrograph so as to minimize the enor in the DRH. Mays and Taur (1982) usednonlinear programming to simultaneously determine the loss rate for each stormperiod and the composite unit hydrograph ordinates for a multistorm event. Unverand Mays (1984) extended this nonlinear programming method to determine theoptimal parameters for the loss-rate functions, and the composite unit hydrograph.

7.7 SYNTHETIC UNIT HYDROGRAPHThe unit hydrograph developed from rainfall and sheamflow data on a $atershedapplies only for that watershed and for the point on the sream where the

'r) a

11.6.9)

observed DRHo^

Time

streanilow data were measured. Synthetic unit hydrograph procedures are used todevelop unit hydrographs for other iocations on the strcam it the same watershedor for nearby watersheds of a similar character. There are three types of syntheticunit hydrogmphs: (l) those relating hydrograph characteristics (peak flow rare,base time, etc.) to watershed chamcte stics (Snyder, 1938; Gray, 1961), (2) thosebased on a dimensionless unit hydrograph (Soil Conservation Service. l9j2), and(3) those based on models of watershed storage (Clark, 1943). Types (l) and (2)are described here and type (3) in Chap. 8.

Snyder's Synthetic Unit Hydrograph

In a study of watersheds located mainly in the Appalachian highlands of the UnitedSrares. and varying in size from about l0 ro 10,000 mi? (30 to 30,000 km2),Snyder 11938) found synthetic relations for some chamcte stics of a standardunit b,drograph fFig. 1.7.la). Additional such relations were found later (U.S.Army Corps of Engineers, 1959). These relations, in modified form are givenbelow. From the relations, five characteistics of a requircd unit hydrograph lFig.7.7.lDl for a given excess rainfall duration may be calculated: the peak dischargeper unit of uatershed area, qpRj the basin lag /rn (time difference between thecenhoid of the excess minfall hyerograph and the unit hydrograph peak), thebase time /b, and the widths W (in rime units) of the unir hydrograph at 50 and75 percent of the peak discharge. Using these chamcteristics the required unithydlograph ma), be drawn. The variables are i l lustrated in Fig. 7.7.1.

Snl der defined a standard unit hydrograph as one whose rainfall durarion r.is relatcd to the basin lag /p by

t P : 5 ' 5 t r

For a standard unit hydrograph he found that:

Time

la.)

FIGURE 7.7.1Snyder 's synthet ic uni l hydrograph.hydrograph (t n + 5.5/i)_

(

\1) )

(d) Standard uni t hydrograph l t ,= 5.5t) .

P

ao

I

f:

st )

{ ' t .7 .1)

l - t r

Timc

(

l. The basin lag is

I p = C C , I L L ' \ A J (7 .7 .Z )

where tp is in hours, L is the length of the main stream in kiiometers (or miles)

from th; outlet to the upsheam divide, L,. is the distance in kilometers (miles)

frcm the outlet to a point on the stream nearest the centroid of the watenhed

area, Ct = 0.?5 (1.0 for the English system), and C, is a coefficient derived

from gaged \.latersheds in the :ame region. , ' ,

2. The peak discharge per unit drainage area in m 3/s km 2 (cfs/mi 2) ofJhe standad

unil hydrograPh is

(7 .1 .3)

where C2 = 2.75 (640 for the English system) and Co is a coefficient derived

frorn gaged watersheds in the same region.To compute Ct and Cp for a gaged watershed, the values of /- and

L. are measurcd from the basin map, From a derived unit hydrograph of

the watershed are obtained values of its effective duration la in hours, its

basin Iag lra in hours, and its peak discharge per unit drainage arca, qpR. jn

m3/s km2 cm (cfs/mi2'in for the English system). If ton = 5.5tn, then 1^ =

tr,tpR = tp, and qpR = qn, and C, and Co are computed by E4s (7.7.2) and(' l .1 .3). lt t,R is quite differcnt from 5.5ta, the standard basin lag is

('7 .7 .4)

and Eqs. (7.7.1) and ('7.'7.4) are solved simultaneously for t. and t/". The

values of C, and C, are then computed ftom (7 .1 .2) and ('7 7 .3) with qpR= qp

and tDR = tD..When an ungaged watershed appears to be similar to a gaged watershed,

the coefficients C, ard C, for the gaged watershed can be used in the above

equations to derive the required synthetic unit hydrograph for the ungagedwate$hed.

The relationship between qp and the Peak discharge per unit drainage area qon

of the required unit hydrograph is

sdp (7 .7 .5)

The base time t6 in hours of the unit hydrograph can be determined using the

fact that the area under the unit hydrograPh is equivalent to a dircct runoff of

1 cm (1 inch in the English system). Assuming a triangula-r shape for the unit

hydrograph, the base time may be estimated by

, 1 \

c1c "

t r t R

' 4

4.

. - C:'o

qoo

where C, : 5.56 (1290 for the English system).(r) Required unir

('7 .7 .6)

226

The ividth in hours of a unit hydrograph at a discharge equal to a cenainpercent of the peak discharge 42n is given by

w = c.qi ot (1 .7 .1)

where C, = 1.22 (440 for English system) for the 75-percent width and 2.14(770. English system) for the 5o-percent width. Usually one-third of this widthis distributed before the unit hydrograph peak time and two-thirds after thepeak.

Example 7.7,1. From lhe basin map of a given watershed, lhe fol lowing quanti t iesare measuredr l, = 150 km, a. :75 km, and drainage area = 3500 km'�. Ffom theunil hydrograph derived for the watershed. the following are determined: tR : 12h, i,r = 34 h, and peat( discharge = 157.5 m3/s.crn. Determine the coefficients Crard C,, lor he synthetic unit hydrograph of the watershed.

Solution. Fturn the given data, 5.5te = 66 5, which is quite different from t t (34h). Equation (7.7.4) yields

| 1 ,= J.l + - -:_

Solving (7.7. l) and (7.7.8) simultaneously gives t, = 5.9 h and r, = 32.5 h.To calcuiate C,, luse ( '1.7.2):

, : C €,(LL,)a 3

32 .5 =0 .75C , ( l 50 x 75 )o l

C ' = 2 6 5

The peak discharge per unit area is 4/r = 157.5/3500 :0.045 m3/s krn2.cm. Thecoefficient Cp is calculated by E4. Q .'7.3) w\th qp = qpR, and tp : tpR.

- -9J91\lDK trF

o . o c s = 2 1 5 C ,34.0

CP=0 56

Example 7,7,2. Compute the six-hour synthetjc unit hydrograph of a watershedhaving a drainage area of 2500 km? with | : 100 km and l ," : 50 km. Thiswat.rshed is a sub-drainage area of the watershed in Example 7.7.1.

Solut io , The values C? = 2.64 and C, = 0.56 deternrined in Example 7.7.1can also be used for this watershed. Thus, U. (7.1 .2) gives ,p = 0.75 x 2.64 x(100 )< 50)0:i : 25.5 h, and (7.7.1) gives t, = 25.515.5 = 4.64 h. For a six-hourunir hydrograph, rR : 6 h, and E9. Q.'1.4) gives lpn = tp - (t, - tR)14 : 25.5 -( 4 .61 6 ) i 4 =25 .8h . Equa t i on (7 .7 .3 ) g i ves qp=2 .75x 0 .56 /25 .5 :0 .0604

(1 . ' / .8)

227

m3/s.kmlcm and (7.7.5) gives 4r1:0.0604 x 25.5/25.8 :0.0597 m3/s km ? cm; thepeak discharge is 0.059'7 x2500: 149.2 n:,ls cm. The widths of lhe unit hydrographare given by Eq. ('/ .'/ .'/). At'15 percent of peak discharge, W : |.22qii 08 : L22x0.059?-108 = 25.6 h. A Eimilar computation gives a W: 44.9 h at 50 percent ofpeak. The base time, given by Eq. (7.7.61. is I b = 5. 561 q op = 5.56/0.0597 : 93h.Thg hydro$aph is dmwn, as in Fig. 'l .'J .2, and checked b ensure that it rcpresentsa depth of direct runoff of I cm.

A further innovation in the use of Snyder's method has been theqrelonaliza-

tion of unit hydrcgraph parameters. Espey, Altman and Graves (1977) pveloped

a set of generalized equations for the construction of lo-minute unit hydrographsusing a study of 4l watersheds ranging in size from 0.014 to 15 mi2, and inimpervious percantage from 2 to 100 percent. Of the 41 watersheds, 16 are locatedin Texas, 9 in North Carolina, 6 in Kentucky, 4 in Indiana, 2 each in Coloradoand Mississippi. and I each in Tennessee and Rnnsylvania. The equations are:

Tp = 3. 1 Lo.23 S -0.251-0. l8@1.57

Qp = 31 .62 x lO3 Ao e6T- \ 'o1

Tn = 125.89 x l03Ao--o e5

wn= t6 .22 / to3Aoe3o ae2

(1 .7 .e)

(7.7. 10)

( '7 .7.11)

(7 .'7 .r2)

- i r ^ F 6h

120

9 , *-; 80

E 6 0t9 r ^

20

0 2a 40 60 80 100Time (h)

FIGTJRE 7.7.2Synthetic unit hydrograph calculated by Snyder's m€thod in Example 7.7.2

. rzn = 25.8 h-

Usr 0.60

228 \ lPLlED N \ DPoLocI

rr{Jl 0.03 0.05 0.07 0.09 0.1 | 0.13 0.15 0 l?

Main channel MannjnS , vr lue

E 8lr

3 o l

.9a , ^.E FIGURE 7.7,3

walershed conveyance factor Qas a funciion of channelroughness and walershedimperviousness. (Adapted withpermission fiom Espey,Ai tman. and Gfaves. 1977. )

wts = 3.24 / lo3 Ao Te Q-a

18 (1 .7 .13)

where

a : the total distance (in feet) along the main channel frcm the point being

considered to the upstream watetshed boundary

S - the main channel slope (in feet per foot), defined by H/0.8L, where H is

the clifference in elevation between A and B A is the Point on the channel

bottom at a distance of 0.21 downstream ftom the upstrearu watershed

boundaryi B is a point on the channel bottom at the downstrcam pornt

being considered

1 - the impervious area within the watenhed (in percent). assumed equal to

5 percent for an undeveloped watershed

4) : the dimensionless wate$hed conveyance fa,ltor, which is a function ofpercent impervious and roughness (Fig. 7.7.3)

A : the watershed drainage area (in square miles)

4, : the time of rise to the peak of the unjt hydrograph from the beginningof runoff (in minutes)

C, = the peak flow of the unit hydrograph (in cfs/in)

Ta = the time base of the unit hydrograph (in minutes)

W:0 : the width of the hydrograph at 50 percent of 0p (in minutes)

Wri : the width of at ?5 percent of 0/, (in minutes)

SCS Dimensionless HydrograPh

The SCS dimensionless hydrograph is a synthetic unit hydrograph in which thedischarge is expressed by the ratio of discharge 4 to peak discharge 4/, and the

229

t;me bv the ratio of time , to the time of rise of the unit hydrogmph' Ip Given the

o.ut ait.t *g. una lag time for the duration of excess rainfall' the unit hydrograph

i""'1" "rii_i'r"a rr.m"the synthetic dimensioniess hydrograph for the_given basin.

i",.," l.l.4tal shows such a dimensionless hydrograph, prepared from the unit

ii,tn ""t-rtt lr " "tiety of waiersheds The values of qp and ?p may be estimated

"li le"u i;prin.a moiel of a triangular unil hydrograph as shown.in Figure

;.;. i(;.-ff ir; the time is in hours and the discharge in m3/s cm tor cfsrin r tsoil

Conservation Service, 1972). I""""-fro-

u ,"ui"* of a large number of unit hydrographs' the Sbil tonse ation

S"rui"e ,ogg.rts the time oi recession may be-approximared as -t

'd7 ?P As the

]r"u onO"t-t-n" unit hydrograph should be equal to a direct runoff of 1 cm (or I

in), it can be shown that

where C = 2.OB G83.4 in the English system) and '4 is the drainage area rn

souare kilometers (square miles)."'- '- Funh"r, a srudy of unir hydrographs of many large and small rural water-

sheds indicates that the basin lag tp : 0 6f', where 7. is the time of concentratron

oi1tr" *ut".rit"O. As shown in Fi,g. '7 -7 '4(b), time of rise I? can be expressed in

rcrms of lag time l, and the duration of effective rainfall r'

CA

+ tt)

('7 .'7 .r4)

( 7 . 7 . 1 5 )

r lTt,

{a )\ b )

FIGURE 7,7.4soil consen,arion scrvice synthetic unlt hydrograPhs (a) Dimensionl€ss hydrograph and (') trian-

gular unii hydrcgraph. (So,rrce: Soil Conservation Service' 19?2 )

230 rppueo uyonorocy

Example 7.7.3. Construct a lo-minute SCS unit hydrograph for a basin of area3.0 km2 and t ime of concentrst ion 1.25 h.

So fu r i oz . Thedu ra t i on t r= 10m in =0 .166h , l ag r ime r , = 0 .6 f . = 0 .6x 1 .25 =0.75 h, and r ise t ime fp : t , l2 + tp = 0.166t2 + 0. i5 = 0.g3J h. From Eq.(-/ .'7 .14), qp = 2.08 x 3.0i 0.833 : 7.49 m3/s.cm. The dimensionless hydrograph inFiE.'/.'7.4 may be coriverted to the rcquired dimensions by multiplying the valueson rhe hoizontal axis by f/ and those on the venical axis by q2. Ahematively, thetriangular unir hydrograph can be drawn with tb = 2.6'jT, = 2.22 h. The depth ofdirect runoff is.checked to equal I cm.

7.8 UNIT HYDROGRAPHS FOR DIFFERENTRAINFALL DURATIONS

When a unjt hydrograph of a given excess-rainfall duration is available. the unithydrographs of other durations can be derived. If other durations ale integralmultiples of the given duration, the new unit hydrograph can be easily computedby applicatjon of rhe principles of superposition and proportionality. However.a generai method of de vation applicable to unir hydrographs of any requiredduration may be used on the basis of the principle of superposition. This is rheS-hyd.rograph method.

The theoretical S-hydrograph is that resulting from a continuous excessrainfall ar a consrant rate of I crnlh (or t in/h) for an indefinite period. This is theunit step response lunction of a watershed sysrem. The curve asiumes a deformedS shape and its ordinates ultimately approach the rate of excess rainfall at a timeof equilibrium. This step response function g(r) can be derived from the unit pulsercsponse function ft(l) of the unit hydrograph, as follows.

From Eq. (1 .2.4), the response at t ime tto a unit pulse of duration Albeginnirg at t ime 0 is

h@ = ^:ko- s('- arl (7.8. 1)

Sinilarly, the response at time I to a unit pulse beginning at time Ar is equalto h(t lr. that is, ft(r) lagged by Ar time unirs:

Ih(t - At):

,ts(, A, _ sG - 2\t)l

and the response at time t to a third unit pulse beginning at time 2Al is

h(t - 2LD = jrcfr - 2Ar) - 8(r - 3Ar)l

Continuing this process indefinitely, summing the resulting equations, andreananging, yields the unit step response function, or S_hy&ograph, as shown inF ig . 7 .8 .1 (a) :

(7 .8 .2)

(7.8.3)

8() - At[h(r) + h(t - 6t) + h(t - z\t) + . . .] (7 .8 .4)

Cont inuou\ rJ inta l l dt a sequence of pulses

sQ) = ^t [h\ t )+ht Lt l+h \ t -2L!)+

( ,

( c )

FIGURE 7.8.1Using the S hydrograph to find a unj! hydrograph of duration A1 from a unit hydrograph of durationAr

where the summation is multiplied by A/ so that g(t) will correspond to an inputrate of l, rather than l/At as used for each of the unit pulses.

Theoretically, the S-hydrograph so derived should be a smooth curve'because the input excess rainfall is assumed to be at a consrant, continuous rate.However, the summation process will result in an undulatory form if there areerors in the ruinfall abstmctions or baseflow separation, or if the actual durationof excess rainfall is not the derived duration for the unit hydrograph. A dura-tion which produces minimum undulation can be found by trial. Unduladon ofthe cuwe may be also caused by nonuniform temporal and areal distribution of

' t t l

!

1 1 .g{

Uni! hydrogmph of durarion ar

, ( r r = + ls ( r - r { r - A/ ) l

rajnfall: funhermore. when the natural data are not l inear, the resulting unstablesvsterr .rscil lations may produce negative ordinates. In such cases. an optimizationtechnique ma1 be used to obtain a smoother unit hydrograph.

,\fier the S-hydrograph is constructed, the unit hydrograph of a givenduration can be derived as follows: Advance, or offset, the position of the S-h)'drograph b] a period equal to the desired duration Ar' and call this S-hydrographan ofiset S-hydrograph, e'(t) [Fie. 7.8.1(&)], defined by

232 ri'j'l-lEDLl\')RolocY

s'( I ) : s(r Ar ' ) ( 7 . 8 . 5 )

The diflerence b€tween the ordinares of the original S-hydrograph and the offsetS-hydrograph. divided by Al', gives the desired unit hydrograph lFig. 7.8.1(c)l:

h '@: #ko -s ( / -a r ' ) l ( 7 . 8 . 6 )

Example 7.8.1. Use rhe 0.s-hour unit hydrograph in thble 7.4.3 (from Example7 4. I ) !o produce the S-hydrograph and the I .5-h unit hydrograph for this wate$hed.

Solut ion. The 0.5-h uDit hydrograph is shown in column 2 of Table 7.E.1. The S-hl-drograph is found using (7.8.4) with Ar:0.5 h. For I = 0.5 h, g(r) : Arl ,(r) =0 .5x404=202c fs : f o r I = I h , s ( r ) = A r [ f t ( r ) + l ? ( r 0 .5 ) ]=0 .5x (10 ' / 9+404 )= '142c i s i f o r r= 1 .5 h . g ( r ) = A rU ! ( t + h ( r -0 .5 ) + h ( t - 1 .0 ) l = 0 .5 x (2343 + 1079 +.10.11 = 191j cfs: and so on, as shown in column 3 of Table 7.8.1 . The S-hydrographis offset by AI ' : 1.5 h (colufrn 4) to give 8(r - At ') , and the dif ference dividedb) Ar' to give the 1.5-h unit hydrograph f i ' ( f) (colurnn 5). For exarrple, for r : 2.0r . . ' - l r b 6 . 2 0 2 y . 1 5 - l a 7 6 . t s r n .

' t , | IJLE 7.8.1Calculation of a 1,5-h unit hydrograph by the S-hydrographmcthod (Example 7.8.1)

s(r)(cfs)

I' I

ime

I

20.5"h unithydrographt1\t )(cfs/in)

3S-hydrograph

4 5Lagged 1.5-h unitS-hydrogrsph hydrographC(t Ar') h'( l(cfs) (cfs/in)

0 5

t 0: . 5l 03 -_{.1.0, i .5-r,05. - i

,10.1

1 0 7 9ll,l3r5061,i60

381

t13

00

202

! 9 1 33 1 6 638964t2343 t3

4531453'�7453'�74537

0202142

I 9 1 3316638964 )2343134450453"t

135195

12"/5t9'762103

369276t49)a0

233

Instantan€ous Unit HYdrograPh

If the excess rainfall is of unit amount and its duration is infinitesimally smali.

the resulting hydrograph is an impulse response function (Sec. 7.2) called the

instantaneous unit hydrograph (IUH). For an IUH, the excess minfall is applied

to the drainage area in zero time. Of course, this is only a theoretical concept

and cannot be realized in actual watersheds, but it is useful because the IUH

characterizes the watershed's response to rainfall without reference tg the rain-

fall duration. Therefore, the IUH can be related to watershed geqmalphology(Rodriguezlturbe and Valdes, 1979; Gupta, Waymirc, and Wang, 19p).

The convolution integral (7.2.1) is

(7 .8 .1)

If the quantities l(.r) afi 8(t) have the same djmensions, the odinate of theIUH must have dimensions [t-l]. The properties of the IUH are as follows, with

0 < a(l)< some positive peak value for I ) 0

Q@ : fou(t - r) I(r) dr

'':

u(t) = 0

,r(l)---> 0

Jo u(L) dl = | and

f o r l < 0

as I ---r "c (7.8.8)

f u(t1t t t t = t1

The quantity 11 is the lag time of the IUH. It can be shown that lL gives thetime interval between the centroid of an excess rainfall hyetograph and that ofthe corresponding direct runoff hydrograph. Note the difference between tr andthe variable t, used for synthetic unit hydrograph lag tirne-ro measures the timefrom the centroid of the excess rainfall to the peak, not the centroid, of the directrunoff hydrograph. The ideal shape of an IUH as described above resembles thatof a single-peaked direct-runoff hydrograph, however, an IUH can have negativeand undulating ordinates.

There are several methods to determine an IUH from a given ERH andDRH. For an approximation, the IUH ordinate at time / is simply set equal to theslope at time / of an S-hydrograph constructed for an excess rainfall intensity ofunit depth per unit time. This procedure is based on the fact that the S-hydrographis an integral curve of the IUH; that is, its ordinate at time t is equal to the integralol rhe area under the IUH from 0 to ,. The ILrH so obtained is in general onlyan approximation because the slope of an S-hydrograph is difficult to measureaccuralely.

The IUH can be determined by yarious methods of mathematical inversion,using, for example, orthogonal functions such as Fourier series (O'Donnell, 1960)or Laguerre functions (Dooge, 1973); integral transforms such as the Laplacetransform (Chow, 1964), rhe Fourier transform (Blank, Delleur, and Giorgini,

HAPTER

8C

LUMPEDFLOW t

ROUTING

FLow routittg is a procedure to determine the time and magnitude of flow (i. e.. theflow h\ dfograph) at a point on a watercourse from known or assumed hydrographsat one or more points upstream. If the flow is a flood, the procedurc is specificallyknown as fr-rod routitrg. ltr a broad sense, flow routing may be considered as alanalysis to trace the flow through a hydrologic system. given the input. Thedifference bet\'"een lumped ard distributed system routing is that in a lumpedsystem model, the flow is calculated as a function of time alone at a particularlocation. \lhile in a distributed system routing the flow is calculated as a functionof space and time througbout the system. Routing by Jumped system methods issometirnes called htdrologic rorlinS, and routing by distributed systems methodsis somctimes rel'erred to as hydrauLic routing. Flow routing by distributed systemmethods is describecl in Chaps. 9 and 10. This chapter deals with iumped systemroutlng.

8.I LUMPED SYSTEM ROUTING

For a hldro)ogic system, input 1(I). output O(/), and storage S(r) are related bythe corlinuitv equation i2.2.4):

$ : rro o(r)at

(8. 1 . 1.)

If the jnllorv h)'drograph. ./(/), is known, Eq. (8.1.1) cannot be solved directlyto obtain the outflow hydrograph, O(t), because both Q and S are unknown. .{second relationship, or storage function, is needed to relate S, /, and Q; coupling

( 'ruvpeo rrow nolrr.rc 243

the storage functjon with the continuity equation provides a solvable combination

of two equations and two unknowns. In general, the storage function may be

wriften as an arbitrary function of I, Q, and their time derivatives as shown by

Eq. 0 t .2):

^ d Q d l Q I' " . 4 . 8 . )( 8 . 1 . 2 )

In Chapter T, these two equations were solved by differentiating a linearizb formof Eq. (8.1.2), substituting the result for dsldt into Eq. (8. 1.1), then intplratingthe resulting differential equation to obtain 00) as a function of (r). In thischapter, a finite difference solution method is applied to the two equations. Thedme hodzon is divided into finite intervals, and tbe continuity equation (8.1.1)

is solved recursively from one time point to the next using the stomge function(8.1.2) to account for the value of storage at each time point.

The specific form of the storage function to be employed in this proceduredepends on the nature of the system being analyzed. In this chapter, thee partic-ular systems are analyzed. First, reservoir routing by the leveL pooL method, inwhjch storage is a nonlinear function of 0 only:

s = f(Q) ( 8 . 1 . 3 )

and the iunction l(O) is determined by relating reservoir storage and outflowto reservoir water level. Second, storage is Jinearly related to 1 and Q in theMuskingum method for flovt routing in channels. Finally, several Linear resemoirmodels are analyzed in which (8. 1.2) becomes a linear function of 0 and its timederivatives.

The relationship between the outflow and the storage of a hydrologic systemhas an important influence on flow routing. This relationship may be eitherin ariabLe or variabLe, as shown in Fig, 8.1.1. An invariable storage functionhas the form of Eq. (8.1.3) and applies to a reseryoir with a hodzontal watersurface. Such reservoirs have a pool that is wide and deep compared with itslength in the direction of flow. The velocity of flow in the reseryoir is very low.The invariable storage relationship requires that there be a fixed discharge fromthe reseryoir for a given watcr surface elevation, which means that the reservoiroutlet works must be either uncontrolled. or controlled by gates held at a fixedposition. If the control gate position is changed, the discharge and water surfaceelevation change at the dam, and the effect propagates upstrcam in the reseryoir tocreate a sloping water surface temporarily, until a new equilibrium water surfaceelevation is established throughout the rcseryoir.

When a reservoir has a hodzontal water surface. its storase is a function ofits water surface elevation, or depth in the pool. Likewise. rhe

-outflow discharge

is a function of the water surface elevation, or head on the outlet works. By com-bining these two functions, the rcservoir storage and discharge can be related toproduce an invariable, single-valued storage function, S =l(O), as shown in Fig.8 l.1(c). For such reservojrs, the peak outflow occuff when the outflow hydro-graph lntersects the inflow hydrograph, because the maximum storage occurs

,:,1,r,,#,

244

(d) lnvrr iable relal 'onsh'p

FIGURE E.1.1Relationships between discharge and

( r ) Var iablc rc lat ionship

when dSldt : I - Q = 0, and the storage and outflow are related by S =/(0)This js indicated in Fig. 8. l � l(a) where the points denoting the maximum storage,R. and maximum outflow. P, coincide.

A variable storage-outflow relationship applies to long, nanow reserroirs.and to open channels or streams, where the water surface profile may be signif-icantly curved due to backwater effects. The amount of storage due to backwaterdepends on the time rute of change of flow though the system. As shown in Fig.8.1.1(r), the resulting relationship between the djscharge and the system storageis no longer a single-valued function but exhibits a curve usually in the form ofa single or t\4'isted loop, depending on the storage chamcteristics of the system.Because of thc retarding effect due to backwater, the peak outflow usually occurslater than the time when the inflow and outflow hydrographs intersect, as indi-cated in Fig. 8.1.1(b), where the poirts R and P do not coincide. If the backwatereffect is not lery significant, the loop shown in Fig. 8.1.1(b) may be replacedby an average curve shown by the dashed line. Thus, level pool routing methodscan also be applied in an approximate way to routing with a variable discharge-stomge relationship.

The preceding discussion indicates that the effect of storage is to redistdbutethe hy-drograph by shifting the centroid of the inflow hydrograph to the position ofthat of the outflow hydrograph in a time of redistribution, In very long channels,the entire flood wave also travels a considerable distance and the centroid ofits hydrograph may then be shifted by a time period longer than the time ofredistribution. This additional time may be considered as time of tanslation. Asshow:.r in Fig. 8. 1.2, the total ,ime of fiood movementbetween the centroids of theoutflow and inflow hydrographs is equal to the sum of the time of rcdist butionand the time of tmnslation. The process of redistribution modifies the shape ofthe h)drograph, while translation changes its position.

LLIMPED FLO\\ TOT,IIC Z5

, I ' IGURE E.1.2' Conceptual interpretation of the time of flood

8,2 LEVEL POOL ROUTING

Level pool routing is a procedure for calculating the outflow hydrograph from areservoir with a horizontal water surface, given its inflow hydrogmph and storage-outflow characteristics. A number of procedures have been proposed for thispurpose (e.g.. Chow, 1951, 1959), and with the advance of computerization,graphical procedures are being replaced by tabular or functional methods so thatthe computational procedure can be automated.

The time horizon is broken into intervals of duration Ar, indexed by j, thatis, t = 0, At, zLr, . . . . j At,(j + l)At,. . ., and the continuity equation (8.1.1)is integrated over each time interval, as shown in Fig. 8.2.l. For the j-th timeinterval:

(

(8 .2 . r )

The inflow values at the beginning and end of the j-th time interval are 1; and1j+L, respectively, and the corresponding values of the outflow arc Qj arld Qj +r.Here, both inflow and outflow are flow rates measured as sample data, ratherthan inflow being pulse data and outflow being sample data as was the case fortire unit hydrogmph. If the variation of inflow and outflow over the interval is

li ,1.

d

P l t t ) ^ t f , r t -

J , ds -

) ,o , I (ud ' -

l ,o , Qt t td t

)

?16 .rp;,r r i : i r sroroLocv

FIGURE 8.2,IChange oI sloruge during a routing period At

approxinately l inear, the change in storage over the interval, S;+r- S;, can be

found b1 reu'r i t ing (8.2.1) as

I ; t I ; . n1 Q i + Q i . t57*r - S; : : - - - - -1: L t -

f t t ( 8 . 2 . 2 )

The values of /i and /i + L are known because they are prespecified. The vaiues of

0; and S- are known at the Jlh time interval from calculation during the previous

time interval. Hence, Eq. (8.2.2) contains two unknowns, 0; + L and S; a L whichare isolared by multiplying (8.2.2) through by 2/Al, and reananging the result toproduce:

) \ , 2 s , \t T - Q , t ) = t r . * / ' , , ' { ; - o , J\ J '

(8.2.3)

In order to calculate the outflow, Oi+ r, from Eq. (8.2.3), a storage-outfow

function Telating 2SlAt + Q and 0 is needed. The method for developing thisfunction using elevation-storage and elevation-outflow relationships is shown inFig. 8.2.2. The relationship between water surface elevation and reserYoir storagecan be derived by planimetering topogmphic maps or from field surveys. Theelevation-discharge relation is derived from hydraulic equations relating head anddischarge. such as those shown in Table 8.2.1, for larious types of spil lways

I Sr- S, l

t-ut'lPeo tuw noulNc 247

u,*ou*flrction

FIGURE 6.2.2Development of the srorage outflow functjon for ievel pool routing on the basis of storage-elevationand elevalion-outflow cu es.

and outlet works. The value of At is taken as the time interval of the inflowhydrograph. For a given value of water surface elevation, the values of storage Sand discharge Q are determined [pans (a) and (b) of Fig. 8.2.2], then the valueof 2SlA/ + O is calculated and plotted on the horizonral axis of a graph with th€value of the outflow 0 on the vertical axis lpart (c) of Fig. 8.2.2].

In routing the flow thrcugh time interval j, all terms on the right side ofEq. (8.2.3) are known, and so the value of 25;+r/At * Qial can be computed.The corresponding value of Q;a1 can be determined from the stomge-outflowfunction 2SlA/ + 0 versus Q, either graphically or by linear interpolation oftabular values. To set up the data required for the next time interyal, the valueof 2Sr+r/Ar - Q+1 is calculated by

/ 2 S , - , \ / 2 S , . , \

\-* - o,- I

- l-^,

' Q, ' t)- zQ,- , (8 .2 .4 )

The computation is then repeated for subsequent routing periods.

Example 8,2.1. A reservojr for detaining flood flows is one acre in horizontalarea, has vefiical sides, and has a 5-ft diameter reinforced concrete pipe as the

248 a:,plieo rrYoror,oor

TABLE 8.2.ISpillu'ay discharge equations

Spiulrat type Equation Notation

:!- ^\

Uncontrolled over

Gate conlrolled ogee

l.tE A \E A \

Morning gloD spi l l

=--:>r ' l'-

1 iCulve (subnrergedin let contro l )

Q: ii2scLi,Hl''� - 41'�)

Q = C"(2nR,)H311

Q = CdWDJW

I :discharge, cfs

C = variable coefficient ofd;scharge

a = effective length of cresl/9 =total head on the cresr

including veloci ty ofapproacb head.

I/l =total head to bolom ofrne openlng

H1=total head to top of theopening

C = coefficienr which differs with gate and creslarangeinent

C, = coefficient related to Hand R,

R. = radjus of the overflo!! crest

H : total head

W = entrance width

D = height of opening

C,, = discharge coefficient

Sauft: Desigh af Snall Dadr. Burenu of Reclanation. U. S. Department of rhe Interior, 1973.

outlet structure. The headwater-discharge relatjon for the outlet pipe is given iocolumns I and 2 of Table 8.2.2. Use the level pool routing method to calculate lhereservoir outflo\^, ftom the inflow hydrograph given in columns 2 and 3 of Table3.2.3. Assume that the reservoir is ini t ial ly empty.

Solution. The inflow hydrograph is specified at l0 mjn time intervals, so Ar = 10min = 600 s. For all elevations. the horizontal area of the reservoir.,rater surfaceis I acfe = 43,560 ft2, and the stomge is calculated as 43,560 x (depth of water).Fo. example, for a depth of 0.5 ft , S = 0.5 / 41,560 = 21,780 ft j , as shownin column 3 of Table 8.2.2. The corresponding value of 2S/At + O can then bedelermined. For a depth 0.5 ft , the discharge is given jn column 2 of Table 8.2.2as 3 cfs. so lhe stomse-outflow function value is

2 5 ^ 2 < 2 r , 7 8 0A r + q = 6 0 0

+ J = / b c r s

(

TABLE 8.2.2ii""lloprn"nt of the storage-ou^tflow function for aJelentibn reservoir (Example E'2' l )'

LUMPED FLo\\ nor,rrrrc 249il

':

;

j

i, '

i

i

rrt)

iii

i i

r l

r 2 3 4Elevation Discha.ge Storage (25/Nl* + QH Q S(ft) (cfs) (ft ) (cfs)

0.00 .51 . 01 .52 .O2.53.0

4 .0

5 .0

6 .06 .51.$'7.5

8.08 .59.09 .5

I0 .0

038

l 13043607897

1 t7137156

190205218

253264

0 02t,'780 '�76

43,560 rs365,340 2358'�7,120 320

108,900 406f30,680 496ts2,460 586174,240 678196,020 '7'70

211.8(n 863239,580 955261,3Q 1044283.140 r 134304,920 t22l326,'�7fi) 1307348,480 1393370,260 14'�76392,040 i 5604t3,820 t643435,600 t727

*Time interval Al = I0 min.

as shown in column 4 of Table 8.2.2. The storage-outflow function is plofted inF ig . 8 .2 .3 .

The flow rouring calculations are carried out using Eq. (8.2.3). For the firsttime inte.val, S r : 0t = 0 because the reservoir is initially empry: hence (2S /At -

0r) = 0 also. The inflow values are /r : 0 and 1z = 60 cfs, so (/r + 1, = 0 + 60 : 60cfs. The value of the storage-outflow function at the end of the time interval iscalculated from (8.2.3) with j = 1:

l \ . , o ) = , t + 1 , r + { 3 1 - o , )\ a r

- ' \ t r

=60 + 0:60 cfs

The value of p;11 is found by ljnear interpolation given 2S./+ r/Aa + 0j, r. If thereis a pair of variables (r,)), witi known pairs of values (r r, ) r) and (12, )r, then theinterpoiated value ofy corresponding to a given value of .r in the range x t< , = x2is

\ Y 2 - Y t ) ,) ' : y l f - l x r t , /

l r2 - r l ,

250

TABLE 8.2.3Routing of flow through a det€ntion reservoir by the level pool method(Example 8.2,1). The computational sequence is indicated by the arrows inthe table.

Columnr

Time Time

index j (min)

3

lnflow

(cfs.)

4 5? (

L + t , * = o .A1

- '

(cfs) (cfs)

6 7

9a * o,- oumowar -'

(cfs) (cts)

r o a oI t o L 6 o

l 2 0 t m4 30 1805 40 24t)6 50 300'7 60 3608 70 3249 80 280

t0 90 240l l r00 20012 l l 0 16013 t20 120t4 l l 0 8015 t10 4016 150 0l t 1601 8 i 7 019 18020 r902t 20022 2tA

+-__2 o.o -__ o.o= tn - ss.: *__)_9!j -.-.-------.---.-----.-lj_>180 201. t 235.2 17.I300 378 .9 501 .1 6 l . I4m 552.6 798.9 123.2540 728.2 t092.6 182.2660 927.5 1388.2 230.3680 1089.0 160?.5 259.3600 1149.0 1689.0 27A.O520 | 134.3 i669.0 26'�7.4440 1061.4 t514.3 254.9360 954.I t424.4 235.2280 820.2 1234.t 206.9200 683.3 1020.2 t68.5120 555 I 803.3 124.140 435.4 595.1 79.80 338.2 435.4 48.6

272.8227 .3194.9169.'�7

272.8221 .3194.9t69.'�7

22 .8t6.212.69 .8

338.2 32.1

^ 200'!

; 160

! 1 2 0

c 8 0

0 0 .2 0 . , 1 0 .6 0 .8 1 .0 1 .2 1 .4 1 .6 L2S /A / + O ( c f sx l 0 r )

FIGURE 8.2.3Storage-outflolr tunct;on ibr a detention reservoir (Example 8.2.1).

luupeo n-ow nocrrsc 251

In rhis case, r : 2SlAt + Q a'rld J : Q. Two pairs of values around 25/A. + g : 60

are selected from Table 8 2.21they are (-rr, )r) = (0.0) and (x2-y) = ('76'3). The

value of ) for .r = 60 is, by linear interpolation'

) :o + &+(60-o)= 2.4 cfs

So, Q, = 2.4 cfs, and the value,of 2S1Al - 02 needed for the nexi itt.ation rs

found using Eq (8 2.4) withi : 2 4,

, 2 S r ^ \ , "l ; 0,)= i Q,\ 2Q,

: & 2 x 2 4

: 5 5 . 2 c f s

The sequence of computations just described is indicated by the arows in the firsttwo rows of Table 8.2.3.

Proceeding to the next time interval, (12 + I) = 60 + 120 : I80 cfs' andrhe routing is performed with j - 2 in (8.2,3).

t s . , ) s . \l : ' o . l - t l , + L t - l - : r - o , l\ A r - ' l ' A r

: 180 + 55 .2

= 235.2 cfs

By linear interpolation in Table 8,2.2, the value ol Qr= 11.1 cfs and by Eq(8.2.4),2S3lLt - Qt = 2Ot.t cfs, as shown in the third row of Table 8.2.3. Thecalculations for subsequent time intervals are performed in the same way, with thercsults tabulated in Table 8.2.3 and plotted in Fig. 8.2.4. The peak inflow is 360cfs and occurs at 60 mjn; the detention reseNoir reduces tbe peak outflow to 270 cfsand delays it until 80 min. As discussed jn Sec. 8.1, the outflow is maximized atthe point where the inflow and outflow are equal, because storage is also maximized

20 40 60 80 r00 r20 r40 160 180 200

Time (mrn)o lnflow . Oulflow

I.IGURE t.2.4Routing of flow through a detenlionreservoir (Example 8.2. l).

iii

l ;trt.

1ihl: l

252 ,qppueu svoror-oor

at that time, and there is a single-valued function relating storage and outflow fofa reservoir with a level pool.

The maximum depth in the storage reservoir is calcuiated by linear interpo-lation from Table 8.2.2 as 9.'17 ft at the peak discharge of 270 cfs. If rhis deprh istoo great, or if the discharge of 270 cfs in the 5-ft outlet pipe is too large. eitherthe outlet slfucture or the sulface area of the basin must be enlarged. An equivalentsize of elliptical or arch pipe would also tend to iower the headwater elevalion.

8,3 RUNGE-KI'TTA METHODt

An altemative method for level pool routing can be developed by solving thecontinuitl equation using a numerical method such as the Runge-Kutta method.The Runge-Kutta method is more conplicated than the method described in theprevious scction. but it does not require the computation of the special storage-outflow Iunction, and it is more closely related to the hydraulics of flow throughthe reservoir. Various orders of Runge-Kutta schemes can be adopted (Carnahan.et al., 1969). A third order scheme is described here; it involves breaking eachtime interval into thrge incrgments and calculating successive values of watersurface elevation and reservoir discharge for each increment.

The continuity eqsation is expressed as

where S is the volume of water in storage in the reseryoir; 1(r) is the inflow intothe resenoir as a function of time; and 0(ll) is the outflow from the rcservoir.which is determined by the head or elevation (1{) ir the reservoir. The change involume, dS, due to a change in elevation, dH, can be expressed as

dS = A(EtdH

I = ,t't- arndt

IQt \ - OtH,rA H ' = - A , r. A(H)

( 8 . 3 . 1 )

( 8 . 3 . 2 )

where A (,FI) is the water surface area at elevation 11. The continuity equation isthen rewritten as

I ( t ) ' Q Q '

A(n ( 8 . 3 . 3 )

The solution is extended forward by small increments of the independent variable,time, using known values of the dependent variable 11. For a third order scheme,there are three such incrcments in each time interval At. and three successiveapproxinrations are made for the change in head elevation, dH.

Fig. 8.3.1 illustmtes how the thrce apprcximate values AI1t,AH2, and AH]are defined for the j{h interval. The slope, dHldt, approximated by AHlAr, isfirst evaluated at(H1,t1), then at (Aj + ^HJ3,tj + A/3), and finally at (Hj +2AH2l3,tr + 2,\t l3).h equations,

dH

dr

Ht

(

t t r , + *

LUM?ED FLow Rourh-c 253

FIGURE 8.3.1Steps to define elevation incremenls inthe third-order Runge-Kutta method.

i,i

1'

t ,! i

. ;j;;i

i:!

; i

lu.:.j

tii ;i )

IIi,1,1

i t,t

2 L H .' ' l 3

(u ,

Alf r \-) l

o ( u . *

LEr3 )o,

4,,* 4)-a(n' *T)

(8.3.4b)

, , * ) , t * '

(8.3.4a)The value of l1;+ r is Biven by

2LHz3

A' (8.3.4c)

ln i t ia l condi l ionsReservoir w:rler surface eleva-

r i o n . r = 0 , j = 1 . 1 Y , = I

V a r y t i m e r + r = r + d r

ttt)). rat + +), t q + + N )

Derernrine O(/') from elevarion-dischdrge rclalionship

. . i t r t , t Q \ H ) l

I A(H) l

A HDeLermine o(Hr +

I )

t " , ^ H , Ii r l , r i r O ( H t + ; ) |r d : = - - - - - -

l a rA (H/ + -r) l

Dcterrr, ine QWt + +^H7)

/(,/ + +^/) Qrni + ]A@! + + ^H2)

A H , 1L H = 4

+ i L H l

FIGURE 8.3.2E l - , ! . L , - , . . . r - r . r r t ^ n h , , ! i n r ^ , r i n n , i . ; n d r h i r 6 i r , ^ r , . t " r p , , n - "

L l . = H . + ^ H" l + |

aHt 3Altl4 4

luupeo now nounrc 255

(8.3.s)

(8.3.6)

A flowchart of the third order Runge-Kutta method is shown in Fig. 8.3.2.

Example 8,3.1, Use the third order Runge-Kutta method to perform $etSeservoirrouting through the one-acre detention rese oir with vertical walls, as Eeglribed inExample 8.2.1. The elevation-discharge relationship is given in columnfl and 2of Table 8.2.2 and the inflow hydrograph in columns I and 2 of Table 8.3.1.

Solnrion. The function A(d) relating the water surface area to the .eservoir eleva-tion is simply A(fl ) : 43,560 ft2 for all values ofa/ because the reseryoir has a basearea of one acre and vertical sides. A routing interval of A/ : l0 min is used. Theprocedure begins with the determination of (0), 1(0 + l0/3), and (0 + (2/3) x ( t 0)),which ar€ found by linear interyolation between the values of 0 and 60 cfs foundin column 2 of Table 8.3.1i they are 0, 20, ̂ trd 40 cfs, respectively. Next, All ris computed using Eq, (8.3.4a) with A, = l0 min :600 s, A = 43,560 ft ' � , and1(0) : 6 "lt since the reservoir is initially empty, Hj : 0 and Q(H) : o:

n r , _ I ( t , \ - QtHt t n ,A t H . t

f O - O l= - -

x {50043.560

= 0

TABLE 8.3.1Routing an inflow hydrograph through a detention reservoir bythe Runge-Kutta method (Example E,3.1),

ColunDi 1 2Time lnflo\{(min) (cfs)

6 7Depth Outflow(ft) (cfs)AHr AH: AHr

0 0l0 6020 12030 18040 24050 30060 36070 32080 28090 244

100 200 0 160120 t2n130 80l4t) 40150 0

0 0 .28 0 .540. '79 1.04 t.24t . 41 L5 l 1 .59L61 | . 62 1 .61L59 1 .58 1 .60| . 62 r . 66 1 .12| . ' t 9 1 .42 l � 130.84 0.57 0.360.15 -0.05 -0.21

-0.37 -0.52 -0.63-0.?5 ,0.86 -0.94-1 .03 -1 .10 -1 .14-1 .19 -1 .2 t - 1 .21

t . 2 \ - 1 .? .O -1 .18- 1 . 1 5 - t . t 2 - 1 . 1 I

0 00.40 2.41 .53 ) ' � 7 .93.08 62.84.69 t24.56.28 182.6'7.98 230.49.27 259.09.15 269.59.63 266.89.06 2s4.38 .17 234 .17 .05 206.45.85 167.84,66 123.53.54 80.0

256 eprirm rvonolocv

For ihe nexr {ime incrcment. using (8.3.4r) with {0 + 10/3) : 20 fr '/s.

l r + 4l- oln, + Yt\,rs^ = _l---i l- ll--- ,r,

ela, + f)(20 - 0)= _ x 6 0 043,560

, : 0.28 fr

For rhe lasr ilcremenr, Itj + (213)AH2 = 0 + (2/3) (0.281 = 6.16 ft. By linearinleryolation from Table 8.2.2, Q(0.18) = l. l0 cfs. By substitution in (8.3.4c)

l, _ 2At\ nlrr ! 2aHr\' l ' / ' l "' , - ' J ' . .

t 'u ' !!! ' t \" \ ' 1 3 /

(40 1.10) x 60043,560

: 0.54 ft

The values of Ai/1. Al12 and A-F13 are found in columns 3, 4, and 5 of Table 8.3. I .Then, for the whoie ten-minute time interval, Al1is computed using Eq. (8.3.6):

AHr 3AH34 * 4

0 3= - + - ( 0 . 5 4 ) : 0 . 4 0 f t4 4

So. H at 10 min is given by H2 = H1+ Altr = 0 + 0.40 = 0.40 ft (column 6),and the coresponding discharge from the pipe is interpolated from Table 8.2.2 as

Q = 2.4 cfs (column 7).The routing calculations for subsequent periods follow the same procedure,

and the solution, extended far enough to cover the peak outflow, is presented inTable 8.3.1. The result is very similar to that obtained in Example 8.2.1 by the

t - Q

FIGURE 6.4,1Prism and wedge storages in a

r-oureo rlow rournrc 257

level pool louting method. As before, the peak inflow of 360 cfs al 60 min is

reduced to 270 cfs occuffing at 80 minutes.

8.4 HYDROLOGIC RIVER ROUTING

Thrc Muskingum method is a commonly used hydrologic routing method for han-

dling a variable discharge-stomge relationship. This method models the storage

volume of flooding in a river channel by a combination of wedge and p.Jism stor-

ages (Fig. 8.4.1). During the advance of a flood wave, inflow excepds'putflow,pioducing a wedge of storage. During lhe recession. outflow excee$ inflou.

iesulting in a negative wedge. In addition. there is a prism of storage which

is formed by a volume of constant cross section along the length of prismatic

channel.Assuming that the cross-sectional area of the flood flow is directly propor-

tional to the discharge at the section, the volume of pdsm storage is equal to

KQ where K is a propofiionality coefficient, and the volume of wedge storage is

equ^l to KX(l - Q), where X is a weighting factor having the rcnge 0 < x < 0.5.

The total stomge is therefore the sum of two components,

S = K Q + K X ( t - Q ) ( 8 . 4 . l )

which can be rearranged to give the storage function for the Muskingum method

. s = K l X l + ( r - n Q l

and rcpresents a linear model for routing flow in streams.

( 8 . 4 . 2 )

The value of X depends on the shape of the modeled wedge stomge Thevalue of X ranges from 0 for reservoir-type storage to 0.5 for a full wedge. WhenX = 0, there is no wedge and hence no backwater; this is the case for a level-poolreservoir. In this case, Eq. (8.4.2) results ill a linear-reservoir model, 5 : KQ.In natural streams, X is between 0 and 0.3 with a mean value near 0.2. Greataccuracy in determining X may not be necessary because the results of the methodarc rclatively insensitive to the value of this parameter. The parameter x'is the

time of travel of the flood wave through the channel reach. A procedure called theMuskingum-Cunge method is described in Chapter 9 for determining the valuesof K and X on the basis of channel characteristics and flow rate in the channel.For hydrologic routing, the values of K and X are assumed to be specified andconstant throughout the range of flow.

The values of storage at time j and j + I can be written, rcspectively, as

and

S i = K I X I I + ( r - k Q i l

51+t : KIXI i+r + (1 -40;+r l

(8 .4.3)

(8.4.4)

Using Eqs. (8.4.3) and (8.4.4), the change in storage over time interval A/ (Fig.8 . 2 . 1 ) i s

S ; + r - S ; : K { f x t ; + r + Q - b Q i * i - [ x l j + ( t - X ) Q j l l ( 8 4 s )

(258 eppr tr, nYoaoL.ocv

The changc in storage can also be expressed. using Eq. (8 2 2)' as

( 1 . l , , t ) , Q ' Q ' \i . ' 5 .

2 l , - - - - - 1 '

Combining t8.4.i) and (8.4.6) and simplifying gives

Q.1+t = Ct l l+ t + C2 l i + C3Qj

which is thc routing equation for the Muskingum metbod where

( 8 . 4 . 6 )

(8 .21 .7)

(8 . .1 .8 )

(8 .4 .9 )

(8. ,1. r0)

Lt zKX2K(1 l) + Ar

A,t + 2KX2K(.1 & + Ar

2 K ( t - n - A , t2 K ( l - n + L t

Nole lha l a C: ' C . LIf observed inflow and outflow hydrographs are available for a rjver reach,

the values of r( and X can be determined Assuming various values ofX and using

known values of the inflow and outflow, successive values of the numemtol and

denominator of thc following expression for l( ' derived from (8 4.5) and (8 4 6)'

can be computed.

0 . 5 A r [ ( 1 r r r + 1 1 ) - ( Q 1 " t + Q ) \ ( 8 . 4 . 1 1 )X ( t 1 " 1 - I 1 ) + ( l - E ( Q i * t - Q i )

The computed vaiues of the numerator and denominator are plotted for each

time inrerval, with the numerator on the venical axis and the denominator on

the horizontal axis. This usually produces a graph in the foIm of a loop The

value of X $at produces a loop closest to a single line is taken to be the conect

value for the reach. and K, according to Eq (8.4 11), is equal to the slope ol

the line. Since K is the time required for the incrcmental flood wave to traverse

the rcach. iis value may also be estimated as the observed time of travel of peak

flow through the reach.If observed inflow and outflow hydrographs are not available for delermin

ing K and X, their values may be estimated sing the Muskingum-Cunge method

descr ibed in Sec .9 .7 .

Example 8.,1.1. The inflow hydrograph to a river reach is given jn columns I and2 of Table E.4.1. Determine the outflo$ hydrograph from this reach if ( = 2.3 h'X - 1 , .15 . rnd l t I h . Ihe in i l . r l ou t f iow is 85 h 's .

Sot/t ior. Delermine the coefficients Cr. C2. and Cr using Eqs. (8.4.8) - (8.4 10):

- | 2 (2 . -1 ) (0 l5 l 0 3 l. _ _ = _ 0 . 0 6 _ r l- '

2 ( 2 . 3 X r 0 . l 5 ) + I 4 . 9 1

^ t + 2(2.3X0.15)", = --*IT

2 ( 2 . 3 X 1 0 . 1 5 ) - l 2 . 9 1= _ : 0.592'74 . 9 1

nupeo ruow nourtrc 259

4 . 9 1

4 . 9 1

Check to see that the sum of the coefficients Cr, C:, and Cr is equal to l.

C t + C 2 + C : : 0 . 0 6 3 1 + 0 . 3 4 4 2 + 0 . 5 9 2 7 : 1 0 0 0 0

For the first time interval, the outflow is determined using values for /fand 12 from

Table 8.4.1, the init ial outf low Q1 : 85 cfs, and Eq (8 4 ?) withi = l*

Qz : CJz ' r C2 I1 + C3Q1

= 0.0631(137) + 0.3442(93) + 0.5927(85)

= 8 .6 + 32 .0 + 50 .4

: 9l cfs

as shown in columns (3) to (6) ofTable 8.4 1. Computations for the following time

intervals use the same procedure withi :2,3, to produce the results shown

in Table 8.4.1. The inflow and oudlow hydrographs are plotted in Fig. 8 4.2 lt

can be seen that the outflow lags the inflow by approximately 2.3 h, which was the

value ol ( used in the computations and rePresents the travel time in the reach.

TABLE E.4,IFlow routing through a river reach by the Muskingum method

(Example 8,4.1),

Column: (l)RoutinBPeriodi(h)

(.2)InflowI(cfs)

(3)

c r L * t

(s)

CtQi(C, = 0.0631) (Cz = 0.3a4D (.Ct= 4 592'7)

(o)Outflow0(cfs) i

, . i

, !

t l

, l

! j

: ,

:

: i

859 1

I 1 4r59233

4m509578623642635603546419

3412142t5r70

(4)

czl j

1 9 32 1373 2084 3205 4426 546'7 630I 6789 691

10 675l t 63412 57 t13 47714 39015 329t6 24717 1841 8 1 3 4t9 10820 90

8 .61 3 . l20.227 . 934.539 .842 .843.642.640.036.030 .124.620.8l 5 . 6l l . 68 . 56 .85.'�7

32.O4't.2'71 .6

1 1 0 . 1t52 .1187 .92 t6 .8233.4231 .8232.3218.2196.5\u .2134.2I t3.285.063.346 .1

50 .454.0

94.5r 37 .8192.3248.93 0 1 . 4342.8369.4380.4316 .135'�7 .3323.6283.'�7244.5202.2t62 .4127.6

7 m I

600

a 5 m ln; 400 -

: roo -.17a zoo -l

260 ^rrL rec rrr ororocv

0 2 4 1 6 8 l 0 1 2 1 4 1 6 l 8 2 0

Ti lne (h)

o Inflow . Oulflov

100

FIGURE 8.5.ILinear rc!enoi$ in series

FIGURE 8.4.2Routing of flow througb a riverreach by the Muskingum method(Example 8 4. l ) .

8.5 LINEAR RESERVOIR MODEL

A linear resentLir is one whose storage is linearly related to its output by a

storage constont k, which has the dimension of time because S is a volume while

B is a flow rate.

(8 . s .1 )

The linear reser\oir model can be derived from the general hydrologic system

model [Er1. (7.1.6)] by letting M(D) = I and letting N(D) have a root of - t/k by

making N(D) = 1 + kD. It can be shown further that if, in Eq (7. 1 6)' M(D) = l

and N(D) has n real roots -l l /.r, l /k2,. . .,- l lk^, the system described is a

cascade of n l ineal reservoirs in series, having storage constants kr'&?' . .,fr,,'I''NHydrographs

e l=t

/\ i-uupeo n-ow eournc 261

respectively. The concept of a linear reservoir was first introduced by Zoch ( 1934,

1gi6, 193'7) in an anaiysrs of the rainfall and runoff relationship. A single linear

rese oir is a simplified case of the Muskingum model with X : 0. The impulse,

pulse, and step response functions of a linear reservoit are plotted in Fi9,7.2,4.

Linear Reservoirs in Series

A watershed may be represented by a series of ,? identical linear reservoirs (Fig.

S,S.t) "u"ir having the same storage constant k {Nash. 1957). ny toL,i jg u uni-

volume inflow through the z linear reservoirs, a mathematical modl for the

instantaneous unit hydrograPh (IUH) of the series can be derived. The impulse

response function of a linear reservoir was derived in Ex. ('1.2.1) as ,(t - ") :

(l/t)exp[-(t- r)/k], This wil l be the outflow from the first reservoir, and

constitutes the inflow to the second reseryoir with t substituted for r - r, that is

for rhe second reseryoir 1(r) = ( 1/k) exP ? r/k). The convolution integral (7.2. I )pives de outflow from the second reservoir as

q2Q)= | I(r)u(r r)dr. , Jo

t: - ? - | / k

k2'

This outflow is then used as the inflow to the third rcservoir. Continuing thisprocedure will yield the outflow q, from the t-th rcseruoir as

u(t) : q.(t): ub(;)'-'" -^ (8. s.3)

F t t \- | l i l " - n , ' . e - t t - k d r { 8 . 5 . 2 r

J 0 \ i ( / K

where f(z) = (n - 1)! When n is not an integer, I(n) can be interpolated from tablesof the gamrna function (Abramowitz and Stegun, 1965). This equation expressesthe instantaneous unit hydrcgraph of the proposed model; mathematically, it isa gamma probability distribution function. The integral of the right side of theequation over t from zero to infinity is equal to l.

It can be shown that the fiISt and second moments of the IUH about theoigin t = 0 are r€spectiYely

and

The first moment, Ml , reprcsents the lag time of the centxoid of the area under theIUH. Applying the IUH in the convolution integral to relate the excess rainfallhyetograph (ERH) to the direct runoff hydrograph (DRH), the principle of linearity

M t : n k

M z = n ( n + l ) k z

(8.5.4)

( 8 . 5 . 5 )

requires each infinitesimal element of the ERH to yield its conesponding DRHwith the same lag time. In other words, the time difference between the centroidsof areas under the ERH and the DRH should be equal to Mt.

By the method of moments, the values of k and n can be computed from agiven ERH and DRH, thus providing a sirnple but approximate calculation of theIUH as expressed by Eq. (8.5.3). If M/r is the first moment of the ERH about thetime origin divided by the total effective rarLnfall, and M p, is tbe first moment ofthe DRH about the time origin divided by the total dircct runoff, then

262 APPLTEDHYDR.-,LoGY

Mo, M1, = nk

MO, Mh = n(n + 1)k2 + 2nkM1 (8 . s .7 )

Since the r alues of M t,, M g,, M 4 and Mp. can be computed from given hydrologicdata, the \rlucs ofr and i can be found using Eqs. (8.5.6) and (E.5.7), rhusdetermining the IUH. It should be noted that the computed values of n and t mayvary son'ic$hat even for small errors in the computed moments; for accuracy. asmall t ime interval and many significant f igures must be used in rhe compurarion.

(8.5.6)

If M/. is the second moment of the ERH about the time origin divided by thetotal excirss rainfal1, and M9, is tte second moment of the DRH about the timeorigin dirided by the total direct runoff, it can be shown that

a

I S 1 5 2 t 2 7 3 1 3 9 4 5 5 t 5 7Time

FIGURE 8.5.2Excess rainfall hyetograph (ERH) and direct runoff hydrograph (DRH) for calcularion of r and X ina l inear reservoir model (Example 8.5.1).

luvpro now nourrxc 263

Example 8.5.1. Given the ERH and the DRH shown in Figure 8.5.2, determrne n

and t for the IUH

Solution. Determjne the moments of the excess minfall hyetograph and the direct

runoff hydrograph. Each block in the ERH and DRH has du.ation 6 h :6 x 3600 s :

21,600 s. The rainfall has been converted to units of m3/s by multiplying by tbewatershed area to be dimensionally consistent with the runoff. The sum of the

ordinates in tie ERH and in the DRH is 700 m3/s, so the area under eachigraph= 700 x 6 : 4200 (m3/s) h.

. . \- | incremenral .", rnor"nt -n Itolal area l

6- - [ r 0 0 , 3 + 3 0 0 " 9 - 2 0 0 1 5 , r 0 0 . 2 t ]

4IUU

= 11 . 5 ' 7 h

The second moment of area is calculated using the parallel axis theorem.t . -

Mb =J> lincremental area x (moment arm)2]- t -

+ ) lsecond moment about cenfoid of each increment] | f ,orut ur.u) /

r I- , ; l 61 100 r J / . 300 ' 91 200 r 15 : - t00 2 t2 l

l - l

126rl loo ioo - 2oo + lool lj

= 166.3 h2

By a sin'ilar calculation for the direct runoff hydrograph

Solve for nt using (8.5.6):

Ma) : 28'25 h

M o . : 8 8 2 8 h 2

n k = M 9 , - M 1 ,

: 28.25 - 11.5'7

: 16 .68

Solve for r and I using (8.5.7):

Ma,- M,,: 4(4 + DP + 2nkMll

: n2l? + nkx k + 2nkM\Hence

882.8 - 166.3 = (16 .68) ' � + 16 .68k + 2 x 16 .68 x 11 .57

and solving yields

264 appr-rrr;gvororocv

t = 3 . 1 4 h

Thus

, _ 16 .68

1 6 . 6 8

These values ofi and t can be substituted into Eq. (8.5.3) to determine the IUHof rhis walershJd. By using the melhods descrbed in Sec 7.2, the corresponorngunit hydrograph can be determined for a specified rainfall duration.

Composite Models

In hydrologic modeling, l inear rcservoirs may also be l inked in parallel. Linearreservoirs may be used to model subsurface water in a saturated phase (Krai-jenhoff van der Leur, 1958), as well as surface water problems. Diskin et al.(1978) presented a parallel cascade model for urban watersheds. The input to themodel is the total rainfail hyetograph, which feeds two parallel cascades of linearreservoirs. for ihe impenious and pervious areas of the watenhed, respectively.Separate ercess rainfall hyetographs are determined for the impervious and pervious areas. and used as input to the two cascades of linear reservoirs.

Linear reserr,oirs in series and parallel may be combined to rrrodel a hydro,logic systenr. The use of Iinear reservoirs in series represents the storage effgctof a hydrologic sl stem, resulting in a time shift of rk between the centrojd of theinflow and that of the outflow as given by Eq. (8.5.6). A l inear channel is anidealized channel in which the time required to translate a discharge through thechannel is constant (Chow, 1964). To model the combined effect of storage andtranslation. the linear reseryoir may be used jointly with a linear channel. Othermore elaborate composite models have been proposed. Dooge (1959) suggesteda series of alternating linear channels and linear reservoirs. For this model, thedrainage arca of a watershed is divided into a number of subueas, by isochrones,which are lines of constant travel time to the watershed outlet. Each subarea isrepresented by a linear channel in series with a linear reservoir. The outflowfrom the lincar channel is represented by the ponion of a time-alea diagram cor-rcsponding to the subarea. This outflow, together with outflow from the precedingsubareas. serves as the inflow to the linear reservoir.

Randomized linear reservoir models have also been developed, in which thestonge constant k is related to the Horton stream order (Sec. 5.8) of the subareabeing drained. By considering the network of streams draining a watershed asbeing a random combination of linear reservoirs, with the mechanism of thecombination being govemed by Honon's stream ordering laws, it is possibleto develop a geomorphic instantaneous unit hydrograph, the shape of which isrelated to the stream pattem of the watershed (Boyd, et a1., 1979', Rodriguez-

(

Irurbe, and Valdes, I979;Wood, 1986)

Luvpeo tow nourl\c 265

Gupta, et al., 1980; Gupta. RodriguezJ turbe, and

REFERENCES

Abramowitz, M., and L A. S\eg\\ , Hanlbook oJ Mathenaticat Futlctionr. Dover. New york, 1965.Boyd, M. J., D. H. Pilgrim. and L Cordery, A srorage routing model based on catchment geomor

phology, J. Hydrol., \ol. 42, pp. 42,709,230, t9'79. :l

Carnahan, B., H. A. Luther, ana J. O. Witkes, Aplilietl wunerical Merhods, Wile!, $ew yort,

la6s. '

tChow, V. T., A practical procedrre of flood routing, Cir. EnR. and public ltorks Rey., yol. 46.

no. 542. pp. 586-58R. Augu,ir lc5'Chow. V. T., Open-chaaael Hfiraulicr, Mccfaw-Hijl, New Yo.k, p_ 529, 1959.Chow, V. T., Runoff, in Handbook of Applied Hldrolo?\,, sec. t4, pp. 14 30, Mccmw-Hill, New

York, 196.4Diskio, M. H., S. lnce, aod K. Oben-Nyarko. Parallel cascades model for u$an $,arersheds. J.

Hy!1. Div. An. Soc. Civ. Eng., vol. 104, no. HY2, pp. 261 276, February 1978.Dooge, J. C. I., A general theory of the unit hydrograph, J. Geophys. Rer., vol. 64. no. l. pp.

24r-256, 1959.Gupta. V. K-, E. Waymire, and C. T. Wang, A representation of an instanEneous unit hydrograph

fronr g€omorphology, Water Resour. R€r-., vol. t6, no.5, pp. E55,862, 1980.Gupta, V. K.. l. Rodriguez lturbe, aod E. F. Wood, Scate probtens in Hydrotogl, D. Reidel.

Dordrecht. Holland, 1986.Kiai.ienhoff van der Leur. D. A., A study of non-steady ground{arer ilow with special reference ro

a res€rvoir-coel'ficient. De lngnieur, vol. 70, no. i9, pp. BE7-B94, I958.Nash, J. E., The form of the instanraneous unir hydrograph. IASH pubticarion no. 45, vo1. 3 4, pp.

114-t21. 1957.Rodriguez Itufbe, I., and J- B. Valdes, The geomorphologicaj strLrcture of hydrologrc response.

Water Renur. Res., vol. I5. no. 6, pp. 1409-t420. t979.Zoch, R. T., On the relat ion berween rainfal l and st,ram f low. l ,4Dnthl\ ,Weather Re\' . . \ols.62,

6a . 05 : pp J l 5 - . , .U2 . 105 . t 2 t . t . r 5 t 47 : t a14 t a3o t o r ' . r e .p ; c ' , ve t )

PROBLEMS

8.2.1 Storage vs. outflow characteristics for a proposed reservoir are given below.Calculate the storage-outflow function 2SlA, + p vs. Q for each of the tabuiatedvalues if Ar = 2 h. Plot a graph of the storage-outflow funcrion.

Storage (106 mr) 75 81 87.5 100 | 4 . 2Oulf low imr/sJ 5' 221 5J9 l j lo 2210

8.2,2 Use the level pool rouring rnethod to route the hydrograph given below through thercservoir whose storage-outflow characteristics are given in prcb. 8.2.1 . What isthe maxhnum reservoir discharge and storage? Assume that the rcseryolr init;all)contains 75 x 106 ln3 of stoaage.

Time (h) O 2

Inflow (mrhec) 60 lcro

t 0 1 2 1 4

t , 310 1 , 930 r , 460

4 6

232 30D

t 6 l 8

930 650

8

520

l

HAPTER

9C

DISTRIBUTEDFLOW

1

ROUTING

The flow of water through the soil and stream channels of a watershed is adistibutcd process because the flow rate, velocity, and depth vary in spacethroughout the watershed. Estimates of the flow rate or water level at imponantlocations in the channel system can be obtained 'rsi\E a distributed fo14 routingmodel. This type of model is based on panial differential equations (the Sain!Venant equations for one-dimensional flow) that allow the flow rate and waterlevel to be computed as functions of space and time, rather than of time alone asin the lumped models described in Chaps.7 and 8.

The computation of flood water level is needed because this level delineatesthe flood plain and determines the required height of structures such as bridgesand levees; the computation of flood flow mte is also important; first, because theflow rate determines the water leyel, and second, because the design of any floodstorage stmcture such as a detention pond or reservoir requires an estimate ofits inflo\\' hydrograph. The alternative to using a distributed flow rcuting modelis to use a lumped model to calculate the flow rate at the desircd location, thencompute the coresponding water level by assuming steady nonuniform flow alongthe channel at the site. The advantage of a distributed flow routing model over thisalternative is that the distributed model computes the flow mte and water levelsimultaneously instead of separately, so that the model more closely approximatesthe actual unsteady nonuniform nature of flow propagation in a channel

Distributed flow routing models can be used to describe the transformationof storm rainfall into runoff over a watershed to produce a flow hydrograph for thewatershed outlet, and then to take this hydrograph as input at the upstream end ofa river or oioe svstem and route it to the downstream end. Distributed models can

o srctgurro crow norrnc 27J

also be used for routing low flows. such as irrigation water deliveries through acanai or river system. The true flow process in either of these applications variesin all thrce space dimensions; for example, the velocity in a river varies alongthe river, across it, and also from the water surface to the river bed. However,for many pmctical purposes, the spatial variation in velocity across the channeland with respect to the depth can be ignored, so that the flow process can beapprcximated as varying in only one space dimension-along the flow channel,or in the direction of flow. The Soint-Venant equations, first developed.b/.Barrede Saint-Venant in 1871, describe one-dimensional unsteady open chanhellflow,which is applicable in thjs case, t

9.1 SAINT.VENANT EQUATIONS

The fo)lowing assumptions are necessary for derivation of the Saint-Venanr equa-tions:

1, The flow is one-dimensional; deprh and velocity vary only in the iongitudinaldirection ofthe channel. This implies that the velocity is constant and the watersurface is horizontal across any section perpendicular to the longitudinai axis.

2. Flow is assumed to vary gradually along the channel so that hydrostaticpressure prevails and vetical accelerations can be neglected (Chow, 1959).

3. The longitudinal axis of the channel is approximated as a straight line.4. The bottom slope of the channel is small and the channel bed is fixed: that is.

the effects of scour and deposition are negligible.5. Resistance coefficients for steady uniform turbulent flow are applicable so that

relationships such as Manning's equation can be used ro desiribe reristanceenects.

6. The fluid is incompressible and of constant density throughout the flow.

Continuity EquationThe continuity equation for an unsteady variable-density flow through a controlvolume can be written as in Eq. (2.2.1):

( 9 . 1 . 1 )n f r r f f

0 = ; l l l p d v + | l p v . d AA I J J J J J '

. i

i1r l

Iiii

Consider an elemental conkol volume of length dx in a channel. Fig. 9. l. 1shows thrce views of the control volume: (a) an elevation view from the side, (b)a plan view from above, and (c) a channel cross section. The inflow to the conftolvolume.is the sum of the flow 0 ente ng the control volume at the upstream eDdot the channel and the larcral infow q entering the control volume as ; distdbutedrlo\^ along the side of the channel. The dimensions of 4 are those of flow perunit length of channel, so the rate of lateral inflow is qdx and the mass inflowrate is

i ii l j

- - rl '78

274 'rrrt-i"o sroaorocY

n * ] 9 a ' ,

FIGURE 9.I.7,c"'"i.;"tuf reuch of channel for derivalion of the Saint-venant equations'

f ff l p v . d A = - p t Q + q d x \ (9. i .2)

This is negative because inflows arc considered to be negative in the Reynolds

*n,oor,,i"ur"rn. The mass outflow from the control volume ls

TII

lI

l,l'nuo =,(o. ff*)

-.- - -- -- _t Enersy gDde rine

i

_ - - - _ - - - - l- - - - - - - . - - - - - _ ! - n z_ I

( 9 . 1 . 3 )

( o,rr^r"rtao arow nourrxc 275

where dQldx is the mte of change of channel flow with distance. {he volume of

the channel eiement is A dt, where A is the average cross-sectional area, so the

rate of change of mass stored within the control volume is

a ff l^.,._ apaartt u J J J P " ' - a t

( 9 . 1 . 4 )

where the panial deriyative is used because the control volume is defifed to be

fixed in size (though the water level may vary within it). 1h. lel ohflow of

mass from the control volume is found by substituting Eqs. (9. 1.2) to (*{ .4) into( 9 . 1 . l ) :

r y - d e + s d x ) . 0 ( a . # * ) : oAssuming the fluid density p is constant, (9.1.5) is simplified by di{iding throughby pdx adad rearranging to produce the conservation forn of the contmulty equa-ron,

dQ dA

which is applicable at a channel cross section. This equation is valid for aprismatic or a nonpritmatic channel. a prismatic channel is one in wnich lhe cross-sectional shape does not vary along the channel and the bed slope rs constant

For some methods of solving the Saint-Venant equations, the nonconserva-tion form of the continuity equation is used, in which the avemge flow velocityy is a dependent variable, instead of Q. This form of the continuity equation canbe derived for a unit width of flow within the channel, neglecting lateml inflow-as fo l lows. Fora un i t w id tho f f lowA:yx | -yand Q=UA= W Subst i tu t inginto (9. 1.6).

( 9 . 1 . 7 )

(e . 1 .5 )

(e. 1 ,6)

( 9 . 1 . 8 )-.dy

4W) . dy- f - = ( ,

dx At

av d\

Nlomentum EquationNewton's second law is wrinen in the form of Reynold's transport theorem as inEq. ( .2 .4 .1 ) :

>' = fi[|["'v n [[vov dt' ( 9 . 1 . 9 )

This states that the sum of the forces applied is equal to the 1a1" of change of

276 AP|L EL, HYDkoLui :Y

momentun] storcd within the control volume plus the net outflow of momentu6

across thc control surface. This equation, in the folm IF = 0, was applied tosteady unilbrm flow in an open channel in Chap. 2. Here, unsteady nonuniform

flow is considered.

FORCES, There are five forces acting on the control volume:

l F = F i + F l + F c + F . + F p ( 9 . I l 0 )

\rhere F\ rs the grctir\ force along the channel due to the weight of the water

in the control robme, F1 is the friction Jorce along the bottom and sides ofthe control volume, F" is the contractionlexpansion force produced by abrupt

change:i in the channel cross section, F,is the wind shear force on the water

surface. and F, \s the unbalanced pressure force [see Fig 9.l.1 (D)]. Each ofthese fivc forces is evaluated in the following paragraphs.

Gravitj. The volume of fluid in the control volume is -,4 d.r and its weight is pgA

dr. For a small angle of channel inclination d, S, = tin 6 and the gravity force

is given b1'

FE : pgAdx.sin 0 : pgAS,,dx

where thc channel bottom slope Su equals -dzlrr.

F , = -pgAS"dx

where S. is the eddy loss slope

K, dt.QtA)2

( 9 . I l l l

Friction. Frictional forces created by the shear stress along the bottom and sidesof the control volume are given by - r0Pdr, where zs is the bed shear stress andP is the wetted perimeter. From Eq. (2.4.9), rq = TRS/ = p8(A/P)S/, hence thefriction force is written as

F1 : - PSASI dx (9. r . l2)

wherc tlie friction slope 57 is derived from resistance equations such as Manning'sequauon.

Contractior/expansion. Abrupt contmction or expansion of the channel causes

energy loss through eddy motion. Such losses are sirnilar to minor iosses in apipe system. Tbe magnitude of eddy losses is related to the change in velocityhead l'2i2g : (QlA)r/2g through the length of channel causing the losses Thedrag forces creating these eddy losses are given by

( 9 . 1 . 1 3 )

( 9 . r . t 4 )29 dx

in which K" is the nondimensional expansion or contraction coefficient, negativefor channel expansion l'ttherc d(QlA)2ldx is negativel and positive for channelcontraction.

DrsrxlBLTED FLo\,! nourrrc 277

Wind Shear. The wind shear force is caused by frictional resistance of windagainst the free surface of the water and is given by

(9. r . 15)

where r, is the wind shear strcss. The shear stress of a boundary'on a fluid maybe wdften in general as

- PC rlv,)v,. ( 9 .1 .16 )t , l

where 4 is the velocity of the fluid relative to the boundary, the notatio{llV,]V.is used so that r, will act opposite to the direction of V,, and Cr is a shear sresscoefficient. As shown in Fig. 9.1.1(, ). the av€rage warer velociiy is 0/A, and thewind velocity is V. in a direction at angle ar to the water velocity, so the velocityof the water rclative to the air is

The wind force is, from above,

F.

v ,= l - v , "o , ,

Nflv,lv,Bdx2

= -WFPdt

(9 . r . 17 )

i 9 . r . 18 )where the wlrd shear factor WJ equals CllV,lV/2. Note that from this equationthe direction of the wind force will be opposite to the direction of the water flow.

Pressure. Referring to Fig.9.1.1(r), the unbalanced pressure force is the resul-tant of the hydrostatic force orl the left side of the contol volume, Fp1, thehydrostatic force on the right side of the control volume, Fr., and the pressureforce exerted by the banks on the conhol volume, Frr:

Fp: Fpt - Fo, + Fo, (9, 1 . 19)

As shown in Fig.9.1.1(c), an element offluid of thickness dw at elevationw from the bottom of the channel is immersed at depth y - w, so the hydrostaticprcssurc on the element is pg$ - w) and the hydrostatic force is pg$, - w)b dw,where, is the width ofthe element across the channel. Hence, the total hydrostaticforce on the left end of the control volume is

t\Fp=

)opsQ - w)bdw

The hydrcstatic force on the dght end of the control volume is

/ , r r . \Fr : lFo , * Td , l p . t . z t )

\ d . r lwhere dFo/dx is determined using the Leibnitz rule for differentiation of anlnlegral (Abramowitz and Stegun, 1972):

(9. 1 .20)

278 .rpprrr a tLioro ocv

o F - f a P , 4 o *a ; ) n P q - u a "

- ) a p C o

t x

a ' | . d b .= P g A - d x + ) n q S O

- w l * a u

(9. t .22)

because A : Jo t at'. The force due to the banks is related to the rate of changein width of fhe channel, dbldx, thro|ugh the element d{ as

^ l f ' d b , ) .. FeD =

I)o Peb A d'd''N)dx

Subst i tur i rg Eq. fs l .zr l into (9.1.19) givest " f \

f o = f n t - l r o ! + ^ * a x ) + r r ' b' \ d . x l

dFoL ' '

dx

Now substituting Eqs. (9.1.22) and (9.1.23) into (9.1.24) and simplifying gives

f , : PSA?dr (9. 1 .25)

r tI l vpv oe = - d7vo + Bv.q dx)

J J{9.1.2"7)

where pBl'Q is the momentum entering from the upsfeam end of the channel,and pBv,qdx is the momentum entering the rnain charinel with the iateral inflow,which has a velocity v, in the r direction. The term B is known as the momentumcoeffcient or Boussinesq coefrcient, it accounts for the nonuniform distributionof velociry at a channel cross section in computing the momelltum. The value of

B is given by

t 9 . r . 2 l )

(9.r .24)

Thc sum of the five forces in Eq. (9.1.10) can be expressed, after substi-t u t i n s ( 9 . 1 . I l ) . ( 9 . 1 . 1 2 ) , ( 9 . 1 . 1 3 ) , ( 9 . 1 . l 8 ) , a n d ( 9 . 1 . 2 5 ) , a s

A\l2F : pgAS"dx - pgASydx - pgAS,dr - WTBpdt - pBA;dt (9.1.26)

MOMENTLT\{. The two momentum terms on the right-hand side of Eq. (9.1.9)represent the rate of change of storage of momentum in the contlol volume. andthe net outflow ol momentum across the control surface. resDectivelv.

Net momentum outflow, The mass inflow mte to the control volume [Eq.(9.1.2)l is d.Q + qdt), rcprcsenting both stream inflow and lateral inflow.The corresponding momentum is computed by multiplying the two mass inflowrates by their respective velocities and a momentum collection factor B:

( ,rsmmurlo rl-ow r.outtr.'o 279

t f fa= | , ; ) ) , ' u (e r 28 )

where v is the velocity through a small element of area dA in the channel cross

secdon. The value of p ranges from 1.01 for straight prismatic channels to 1.33

for river valleys with floodplains (Chow, 1959; Henderson, 1966).The momentum leaving the contlol volume is

The ner outflow of momentum across the control surface is the sum of (9.I .27)

JJuouun : olovo. ry4

aad (9 .l .29)l

Jfnru uo --ptBve + Bv,qdrt * ,lpro - Y*)

=-olo"'a-Y)*

@.les)

( 9 . 1 . 3 0 )

ltomentum storage. The time mte of change of momentum stored in the contolvolume is found by using the fact that the volume of the elemental channel is .4d{, so its momentum is pA dxV, or pQdx, and then

*JII",*=,#* ( 9 . l . 3 1 )

After substituting the force terms from (9.1.26), and the momentum terms from(9.1.30) and (9.1.31) into the momentum equation (9.1.9), it reads

pgAS6dx - pgAsr d,\ - pgAs?dt WFpd\ pgA q-.d"

' d x

f ̂ d(.pvql de= - 0 1 0 v , d , l * . p i *

(9.r .32)

Dividlng through by pdr, replacing V with Q/A, and rearranging produces theconsenation form of the momentum equation:

9 * a r 1 ? r a t - R A l i l - s " + \dt dx \dr

S/ + S"/- Pqv' I WtB = O 19 1 33)

The depth y in Eq. (9.1.33) can be replaced by the water surface elevarion h,us ing [see F ig . 9 .1 .1 (a) ] :

h : y + z (9.1 34)

where a is the elevation of the channel bottom above a datum such as mear sea

2E0 eppr iep u.ronorocr

level. The derivative of Eq.along the channel is

(

(9.1.34) with respect to the longirudinal distance.r

(9.1.35)

B]ut AaiA\ : -S,, so

( 9 . l � 3 6 1

The momcntum equation can now be expressed in terms of fr by usiog (9.1.36)in (9 . 1 .33) :

+ . w#A " *(X"s/ + s")- pqv, + wrB = o ( 9 . t . 3 7 )

ah AJ dzdx ax dx

d h :

Ax * s .

The Saint Venant equations, (9.1.6) for continuiry and (9.1.37) for momen-tum, are the governing equations for one-dimensional. unsteady flow in an openchannel. The use of the terms ̂ S/ and S" in (9.1.37), which represenr the rateof energl' loss as the flow passes through the channel, iilustrates the close rela-tionship betrveen energy and momentum considerations in describing the flow.Strelkofl ( I 969) :howed that the momentum equation for the Saint-Venant equa-tions can aiso be derived from energy principles, rather than by using Newton'ssecond law as presented here.

The noncoirservation form of the momentum equation can be derived in asimilar manner to the nonconseryation form of the continuity equation. Neglectingeddy losses, wind shear effect, and lateral inflow, the nonconservation form ofthe momentum equation for a uIlit width in the flow is

9.2 CLASSIFICATION OF DISTRIBUTED ROUTINGMODELS

The Saint-Venant equations have various simplified forms, each defining a one-dimensional distributed routing model. Variations of Eqs. (9.1.6) and (9.1.37) inconservation and nonconservation iorms, neglecting lateral inflow, wind shear,and eddl iosses. are used to define various one-dimensional distdbuted routingmodels as shown in Table 9.2. l �

The momentum equation consists of terms for the physical processes thatgovem thc flow momentum. These tgfms arct the local accelerctio, term, whichdescribes the change in momentum due to the change in velocity over time, theconrecti'e acceLeration term, which describes the change in momentum due tochange in velocity along the channel, the pressure force term, proportional to thechange in the water depth along the channel, the graviry force term, proportionalto the bed slope Sr, and the friction force term, proponional to the friction slope

u: - f! * sf{ s, - s,l o {e r.r8d t ox \ dx

' l

I

TABLE 9.2'ISummary of the Sainl-Venant equalions*

orsrqrsurro FLow lourNc 2E1

Continuiry equatiotl

Conservation form

Nonconservation form

Monentum equation

Conservalion form

L d a - I _ L l L r l ,A at A dx\A I

Local Convectiv€

acceleration acceleration

term rcrm

dQ dA

, d J d V d y -

d \ ' d x

& -

Pressureforce

' I

i {d

s('t, - S7)

Gravity Friclion

tetm terTn

Nonconservation form iunit width element)

lrY ' vdY . d y+ 8-tJ y 6 \ e . o f )

r\lnematlc wave

8(s" s , : 0

Ditlusion waveDynamic wave

* Neslecting lateral inflow, sind sh€ar, and eddy tosses, and assuming p = l.

57, The local and convective acceleration terms represent the effect of inertialforce. on the flow.

When the water level or flow rate is changed at a particular point ina channel carrying a subcritical flow, the effects of these ihanges propagareback upstream. These backwater effects can be incorporated inlo distributedrouting methods through the local acceleration, convectiue acceleration. andpressurc terms.. Lumped routing methods may rlot perform well in simu]atingthe flow conditions when backwater effects are signihcant and the river slope ismild, because these methods have no hydraulic meihanisms to descdbe upsteampropagaion of chalges in flow momentum.

As shown in Table 9.2.1, alternative disnibuted flow routins models areproduced by using rhe full continuity equation while eliminatins som! rerms of themomenlum equation. The simplest distributed model is the k inimatic wave model,which neglects the local acceleration, convective acceleration, and pressure termsin the momentum equation; that is, it assumes S, = 57 and th^e fricrion andgravity forces balance each other. The difiusion wave midel neglects the localand convective acceleration terms but incorporates the press ure teril. Tlte aynamtcwave model considers all the acceleration and pressuie terms in the momentumequation.

. The momentum equation can also be written in forms that take into accounawnether the flow is steady or unsteady, and uniform or nonuniform. as shown

in Eqs. (9.2.l). In the continuity eqi]ation, dA/dt = 0 for a steady flow, and thelateral inflow q is zero for a uniform flow.

Conservation form:

2E2 eppuro svonorocr

/ = 3 A /

1 = 7 L l

1 = O

I do I dto2lA) dv_ _ | \ : S /gA dt gA dx

Nonconservation form:

_Lavg d t f ln s . : s ,

)Observer sees this for

vdv

(9 .2 .1a)

(9.2.rb)g d x

I Steady, uniform flow

Steady, nonuniform flow

Unsteady, nonuniform flow

9.3 WAVE MOTION

Kinematic waves govem flow when inertial and pressure forces are not important.Dynamic waves govern flow when these forces are important, such as in themovement of a large flood wave in a wide river. In a kinematic wave, the gmvityand friction forces are balanced, so the flow does not accelerate appreciably.Fig. 9.3.I illustrates the difference between kinematic and dynamic wave motionwithin a differential element from the viewDoint of a stationarv observer on the

Observer sees ftis foroynamrc wa!e

FIGURE 9.3. IKinematic and dynamic waves in a short reach of channel as seen by a slalionary observer.

aQ * ouoa-,(dQ\ _ o

d x \ d t I(9 .3.7. )

2E3

river baDk. For a kinematic wave, the energy grade line is parallel to the channel

bortom and the flow is steady and uniform (S, = Sf) within the differential length,

while for a dynamic wave the energy grade line and water surface elevation are

not parallel to the bed, even within a differential element.

Kinematic Wave CeleritY

A wave is a vadation in a flow, such as a change in flow rate or wateF trfaceelevation, and lhe wave celerity is the velocity with which this variation gavels

along the channel. The celerity depends on the type of wave being confidered

and may be quite different frcm the water velocity. For a kinematic wave the

acceleration and Pressure terms in the momentum equation arc negligible, so the

wave motion is described principally by the equation of continuity. The name

kinematic is thus applicable, as kinematics rcferc to the study of motion exclusive

of the influence of mass and force; in dynamics these quantities are included.The kinematic wave model is defined by the following equations.Continuity:

ao dAd r d t '

( 9 . 3 . 1 )

( 9 . 3 . 2 )

r q l l )

(e.3.4)

Momentum:

s 9 = s rThe momentum equation can also be expressed in the form

A : a e p

For example, Manning's equation written wift S, : 51 and l? : AiP is

() _ | .4gs:12 a5,3- np2t3

--

which can be solved for .4 as

/ - . , J 5I nP" | - , ,

\ t . 4evS ; / -

dA = . .66o r (dQ\d t ' -

\ d t l

and substituting for AAldt in (9.3.1) to give

so a = [nP2/31Q.49 f")10 6 anO F = 0.6 in this case.Equation (9.3.1) contains two dependent variables, A and p, but A can be

eliminated by differentiating (9.3.3):

( 9 . 3 . 5 )

(9.3.6)

I: l

:

2E4

Kinenratic waves result from changes in 0. An increment in flou', dQ, canbe written as

d Q , d Q .av

dxax '

dt al

Dividing through by dr and realranging produces:

dQ dt dQ dQ

, 0x dt dt d\

Equations (9.3.7) dhd (9.3.9) are identical if

d Q - "dx

and

Differentiating Eq. (9.3.3) and rearranging gives

d Q _ IdA qBQu l

and by comparing (9,3.I I) and (9.3. 12), it can be seen that

# - #OI

d x l

A: "Bap '

19.3.9)

(9.3. 10)

(9.3.8)

1 9 . 3 . )

(9.3. r 3)

( 9 , 3 . 1 5 )

(9 .3 . r2 )

- d Q _ d xaA dt

(9.3. 14)

where ci is the kinematjc wave celerity. This implies that an observer movingat a velocity dxidt =ck with the flow would see the flow mte increasing ata rate of dQldr :q. If q =9, the observer would see a constant discharge.Eqs. (9.3.10) and (9.3.14) are the charocteristic equations for a kinematic wave,two ordinary differential equations that arc mathematically equivalent to thego\ern inF "on t inu i t ) and momentum equat ions .

The kinemaiic wave celerity can also be expressed in terms of the depth )as

l d o" r n , b

where dA = B d,y .Both kinematic and dynamic wave motion are present in natural flood

waves, Iir many cases the channel slope dominates in the momentum equalion(9.2.1): therefore. most of a flood wave moves as a kinematic wave. Lighthil l

orrrusurso FLow noLrNc 2E5

and Whitham (1955) proved that the velocity of the rnain pan of a natural flood

wave approximates that of a kinematic wave. If the other momentum terms Idyl dt,

V(dVldx), and (llg)dy/dx) are not negligible, then a dynamic wave fronr exists

wirjch can propagate both upstream and downstream from the main body of the

flood wave, as shown in FiE. 9.3.2. Miller (1984) summarizes several qiteria

for dercrmining when the kinematic wave approximation is applicable, but there

is no single, universal criterion for making this decision.As previously shown, if a wave is kinematic (S/ : S,) the kinemafic lrave

celerity varies wrth dQ/dA. For Manning's equation, wave celerity incpasesas p increases. As a result, the kinematic wave theoretically should advance

downstream with its rising limb getting steeper. Ho$,ever. the wave does notget longer, or attenuate, so it does not subside, and the flood peak stays atthe same maximum depth. As the wave becomes steeper the other momentumequation tenns becorne more imponant and introduce dispersion and attenuadon.The celerity of a flood wave departs from the kinematic wave celerity becausethe discharge is not a function of depth alone, and, at the wave crest, 0 and )do not rcmain constant.

Lighthill and Whithan (1955) illustmted that the profile of a wave front canbe determined by combining the Chezy equation (2.5.5)

O = CAJRE

with the momentum equation (9.2.1r) to produce

Lo = c A t R t S ' 4 - Y d v - t d v \

1 \ " d x q d x S a t l

in which C is the Chezy coefficient and R is the hydraulic radius.

( 9 . 3 . l 6 )

( 9 . 3 . 1 7 )

Dynamic Wave Celerity

The dynamic wave celerity can be found by developing the chamcteristic equa-tions for the Saint-Venant equations. Beginning with the nonconservation form of

Main body of flood wavrkinemaric in nature

FIGURE 9.3.2Molion of a flood wave.

and

286 epprreo svonoloor(

the Sainr Verart equations (Table 9.2.1), it may be shown that the con-espondingcharacterjstic equations are (Henderson, 1966):

da- : V = cdAL

:(v ) 2c) = e6" s,at

in which ca is thedynamic wave celerity, given for a rectangular channel by

ca = "6 (9 3.20)

where I ir the depth of f low. For a channel of arbitrary cross section, cd =

.,?7/r. tni, celerity c7 measures the velocity of a dynamic wave with respecrto sti l l water. As shown in Fig.9.3.2, in moving water there are two dynamicwaves. one proceeding upstream with velocity V- cd and the other proceedingdownstream with velocity y + cd. For the upstream wave to move up the channelrequires V{ cd, or, equivalently, that the flow be subcritical, t n"" y = .,/gy isthe critical veLocitlt of a rectangular, open-channel flow.

Example 9.3.1. A rectangular channel is 200 feet wide, has bed slope 1 percenland Manning roughness 0.035. Calculate the water velocity y. the kinematic anddynamic wave celerities cr and co,, and the velocity of propagation of dynamicwales yl- c,i at a point in the channel where the flow rate is 5000 cfs.

So/urror. Manning's equation wiih R - ], Sa: S/, and channel width B is written

g = !42 5-'' aP"

t 4q . . -=

-s;. \By\\t ' '

which is sohed for y as

/ t ", = l ' Q . I-

\1 4esii '�B

/i 0.035 x 5000 l\ 1 4 9 ) 0 0 1 r 2 2 0 0

:2 .89 f t

Hence. the water velocity is

200 x 2.89:8.65 ft/s

( 9 . 3 . 1 8 )

( 9 . 3 . 1 9 )

' , oBr

(

The kinematic wave celeri t ] cr is given by (9.3.15):

orsrmgureo FLow nolmxc 287

t d o'^

B dv

, o l t .aos) ,a \= ai\ , ,'.1\ /

/r.+rsj' ' \,r, .,. | \_ l _ x : ) r r tI n A l l -\ ' I '_ 1 .49 x 0 .011 i2 x 5 x (2 .89)43

0.035 x 3= 14.4 fVs

The dynamic wave celerity is

"o: "6=,ln1a16: 9.65 ftls

The velocity of propagation of the upstream dynamic wave is

V c 7 : 3 . 6 5 - 9 . 6 5 = 1 0 f t / s

and that of the downstream dynamrc wave rs

V + cd = 8.65 + 9.65 : 18.3 ft /s

In interpreting these rcsults with Fig. 9.3.2, it can be seen that a flood wave travelingat the kinematic wave celerity (14.4 f,, will move down the channel faster thanthe water velocity (8,65 fvt, while the dynamic \raves move upstream (-1.0 fvs)and downsheam (18.3 ft/s) at the same time.

ln the event that the approxjmation S, : Sl is not valid, the various velocjtiesand celerities can be determined using the full momentum equation to describe Syas i n Eq . ( 9 .3 .17 ) .

9.4 ANALYTICAL SOLUTION OF THEKINEMATIC WAVE

The solution of the kinematic wave equations specifies the disfibution of the flowas a function of distance .r along the channel and time /. The solution may beobtained numerically by using finite difference approximations to Eq. (9.3.7), oranalyicaliy by solving simultaneously the characteristic equations (9.3.10) and(9.3.14). ln this section the aoalytical method is presented for the special casewhen lateml inflow is negligible; numerical solution is {iscussed in Sec. 9.f.

' The solution for Q6,t) requires knowledge of the initial condition Q@,O),or ihe value of the flow along the channel at the beginning of the calculations, andtbe boundary condition Q(0,t), the inflow hydrograph at the upsteam end of thechannel. The obiective is to determine the outflow hvdrosraoh at the downsrream

288 apr ie:, rroxolocv

end of thc chrnncl, 0(t, r), as a function of the inflo\r' hydrograph. any latera)flou' occurring along the sides of the channei, and the dynamics of f low in thecharnel i is expressed by the kinematic wavs equations.

lf rhe lateral f low is neglected, (9.3.10) reduces to dQ/dx :O, or Q =7

constanr. Thus, if the flow rate is known at a point in time and space, this flo\tvalue can be propagated along the channel at the kinematic wave celerity. asgjven b)

( 9 . 4 . l )" , = d Q = d "dA dt

The :olution can be visualized en an x t plane, as shown in Fig. 9.4.1(b),where dis,i ince is plotted oD the horizontal axis, and time on the venical axis.Each poirt in the ,y-t plane has a value of Q associated with it, which is theflow ratc al that location along the channel, at that poiDt in time. These valuesof Q may bc thought of as being plotted on an axis coming out of the pageperpendicuirr to the -r-r plane. In particuJar, the inflow hydrograph 0(0, 0 isshown in Fig. 9.4. I(c) folded down to the left, and the outflow hydrograph Q(L, t1is shown in Fig 9.4.1(c) folded down to the ght of the.x-l plane. These two

1 , )Outflow hydrogriph

' I i f tc tn in)

L A 1 2 1 4 5 6Flow rate

(cfs, rhousands)

6 5 4 1 1 1 0 0

(c f r . lho ! ' rndr )

FIGURE 9.1.IKinematic *a\e routing of a flow hydrograph through a channel reach of length L using propagatiot

of the flo\\' iLlonS characteristic lines in lhe.r-t planc. lf flow raie were plotied on a lhird axrs.perpendicular to the j l plane (b), then the inflow hydrogEph (d) is the variation of flow throughdme at r : 0 folded down !o the left of the;r-/ planei the outflow hydrog.aph (c) is the varialion

of flow rdtc through tilne at -r = a and is folded down to the righl of the i-l plane in the figure.

The dashed iires jndjcate lhe propoeation oi sDecific flow rales along characteistic lines in the -r-rp lane.

orsrnBureo rlow rourNc 289

hvdrographs are connected by the character[stic /lzes shown in part (6) of the

figure. The equations for these lines are found by solving (9.4. t):

1".. | . ,.J , u - 1 ,

c t a t

or

. (2.4.2)

so the time at which a discharge 0 enteing a challne] of length L at time/o wiu

appear at the outlet Is

L( 9 . 4 . 3 )

The slope of the chamcteristic l ine is c1 = dQldA for the panjcular value

of f low rate being considered. The lines shown in Fig. 9.4.1(b) are straightbecause 4 = 0, and Q is constant alo,rg them. If 4 + 0. Q and c* vary along thecharacte stic l ines, which then become curved.

Rainfall-runoff Process

The kinematic wavc method has been applied to describe flow over planes, as amodel of the minfall-runoff process. In this application the laterai flow is equalro the differeflce between the rates of rainfall and infiitration. and the channeiflou is taken to be flow per ulit width of plane. The characteristic equations canbe solved analytically to simulate the outflow hydrograph in response to rainfallof a specified duration. By accumulating the flow from many such planes laidout over a \{atershed, an approximate model can be developed for the aonversionof rainfall into streamflow at the watershed outlet.

The kinenatic wave model of the rainfall-runoff process offen the advan-tage over the unit hydrogmph method that it is a solution of the physical equa-tions goveroing the surface flow, but the solution is only for one-dimensionalflow. whereas the actual watershed surface flow is two-dimensional as the waterfollows the land surface contour. As a consequence, the kinematic wave param-eters, such as Manning's roughness coefficient, must be adjusted to produce arealistic outflow hydrograph. Eagleson (1970), Overton and Meadows (1976),and Stephenson and Meadows (1986) present detailed information on kinematicwave models for the rainfall-runoff process.

Example 9,4.1, A 2oO-foot-wide rectangular chamel is 15,000 feet long. has abed slope of I percent, and a Manning's roughness factor of 0.035. The jnflowhydrograph to the clrannel is given in columns 1 and 2 of Table 9.4.1. Calculatethe outflow hydrograph by analytical solution of the kinematic wave equalions-

Solution. The kinerralic wave celelity for a given value ofthe flow €te is calculatedrn the same manner as shown in Example 9.3.1, where it was shown that for thiscbannel, c, : 14.4 ftls for Q = 5000 cfs. The coresponding values for the other flow

290 rpl lrro Hyororocy

TABLE 9.4.IRouting of a flow hydrograph by analytical solution of thekinematic nave (Example 9.4,1).

Colunrni 1 2Inflow Inflo\.yTime Rate

(min) (cfs)

3Kinematic

4 5Travel Outffow*time time

(min) (min)c€lerity(f1l0

0i 2)1t64860128.196

r08t2t)

2000' 2000r 3ooo

4000s00060005000400030002An2000

r0.0t0.0I t . 1r 3.214.115 .514 .41 3 . 2| 1 . 1I0 .0l 0 .0

25 . 125 .121 .3l9 .01'�7 .4r 6 . l1 1 . 4l 9 .02 ) . 325. )2 5 . 1

25. L37 .145.355 .065 .416 .189.4

r03.011'�7 .3133 . I145 .1

*Outf low t inre = Inl low rime + Travel t iDrc.

mtes on rhe inflo\\, hydrograph are shown in column 3 of rable 9.4.1. The rraveltrme through a reach of length I is l / . i so lor | : 15,000flandc.r: I4.4 f t /s.the travel t ime is 15,000/14.4 = 1042 s = 17.4 min, as shown in colunrn 4 of thetable. The l ime when this discharge on the r ising l imb ofthe hydrograph wil l ar iveat tbe outlel of the channel is, by Eq. (9.4,3) t = [ " + L/ c k = 48 + l?.4 = 65.4 min.as shown in column 5. The inflow and outflow hydrographs for this example arepior{ed in Fig. 9.4.1. l t can be seen thar the kinematic wave is a wave of tmnslat ioDwithour attenuation; rhe maximum discharge of6000 cfs is undiminished by passagethrough the channel.

9.5 FINITE-DIFFERENCE APPROXIMATIONSThe Saint-Venant equations for distributed routing are not amenable to analvti_cal solution except in a few special simple cases. They are panial differentialequations that. in general, must be solved using numedcal methods. Methodsfor solving partial differential equations may be classified as dircct numericaLmethods a\d, characteristic methods.In direct methods, finite-difference equa_tions are formulated from the original panial differential equations for continuityand momentum. Solutions for the flow rate and water surface elevation are thenobtained for incrcmental times and distances along the stream or river. ln charac_teristic mcthods, the partial differential equations are first transformed to a char-acteristic iorm, and the characteristic equations arc solved analytically, as shownpreviously for the kinematic wave, or by using a finite-difference reprcsentation.

In numeical methods for solving panial differential equations, the calcula_tions a-re performed on a grid placed over the .r-/ plane. The i_r grid is a networkof points defined by taking distance increments of length & and time incrementsof duration Al. As shown in Fig. 9.5.1, the distance points are denored by index

orsrrrgLrruo rlow nourrilc 291

( j + t ) ^ l

O , , - l ) A . r t ^ x ( t + l ) A r L

Dis lance.r

i and the time points by indexj. A time line is a line parallel to the r axis throughall the distance points at a given value of time.

Numerical schemes transform the goveming partial differential equarionsinto a set of algebraic finite-differenca equations, which may be linear ornonlinear. The finite-difference equations reprcsent the spatial and temporalderivatives in terms of the unknown variables on both the curent time line,j + I, and the preceding time line, j, where all the values are known from pre-vious computation (see Fig.9.5.l). The solution of the Saint-Venant equarionsadvances from one time line to the next.

Finite Differences

Finite-difference approximations can be derived for a function a(l) as shown inFig.9.5.2. A Taylor series expansion of u(x) at x + An produces

u(x + Lx) = u ( r i + Axu ' (x ) + !A .x2u, , tx ) + lAr r r i , , , t - r t + . . . (9 .5 .1 )

wherc u'(x): du/ dx, u" (i: ful df ,. . . , and so on. The Tay)or series expansiona t r - A { i s

, ( r - dn) : u (x ) - Ax u ' (x ) + :A- r ' �4 " (x l -

) *3 u , , ,1x1 + . . . ( .9 .5 .2 )z 6

. -A central-differel.?ce approximation uses the difference defined by subtract_ing (9 .5 .2 ) f rom (9 .5 .1 )

u(x + Lr) - uG Ar) : 2 A_rr,(x) + 0(Ar3) (9.5.3)

where 0(A!3) represents a residual containing the third and higher order terms.Solving lor a'(n) assuming 0(Ar3; = g t"ruy,r tn

u(x -l Lr) - u(x - Axlu \ x ) = -

2 L \

3F

!

h

g

o

: }t {

d

FIGURE 9.5.7The grid on $e ,r r planeused for numerical solution ofthe saint-venanl equationsby finite d ifferen ces.

(9 .5 . ,1 )

292 ^ppr,;r rr op..ocr

FIGURE 9.5.2Finite difference approximations forthe funclion ll(.r).

which has an enor of approximation of order A-r'. This approximation error. dueto dropping the higher order terms, is also refe[ed to as a truncation error,

A forv,ard difference aPproximation is defined by subtracting a(x) from(9 .5 . r ) :

u(; + At) - a(r) = Atr',", * O,*') ( 9 . 5 . 5 )

Assuming second and higher order terms are negligible, solving for ri'(.;r) gives

u' (x) -a ( r + A r ) - r ( . r )

) + Aj Distance r

u(x) - u(x Ar) = A{''(x) + 0(Ar'�)

so that sol\' ing for r/'(r) gives

u'(x) :A * r

( 9 . 5 . 6 )

which has an error of approximation of order Ar.The backward-dffirence approximation uses the difference defined by sub-

tracting (9.5.2) from u(x),

(9 . s .7 )

(e .5 .8 )AT

A finite-difference method may employ either an explicit scheme ot animplicit s(heme for solution. The main difference between the two is that in theexplicit mehod, the unknown values are solved sequentially along a time linefrom one distance point to the next, while in the implicit method the unknownvalues on a given time line are all determined simuLtaneously. The explicit methodis simpler but can be unstable, which means that small values of A-r and Atare required for convergence of the numerical procedure. The explicit method is

prsrnreureo rLow rourNc 293

convellient because results are given at the grid points, and it can treat slightly

varying channel geometry from section to section, but jt is less efficient than the

irnplicit method and hence not suitable for routing flood flows over a long timeperiod

The implicit method is mathematically more complicated, but with the useof computers this is not a serious problem once the method is programmed. Themethod is stable for large computation steps with little loss of accuracy anq henceworks much faster than the explicit method. The implicit method can a$o landlechannel geomety varying significantly from one channel cross section to tryt next.

Explicit Scheme

The finite-difference representation is shown by the mesh of points on the time-distance plane shown in Fig.9.5.1. Assuming that at t ime r (t ime line j), rhehydmulic quantities u are known, the problem is to determine the unknownquantity at point (i, j + l) at t ime t +At, that is, , l1*I.

The simplest scheme determines the partial derivatives at point (i,j + l) interms ofthe quantit ies at adjacent points (i - I, j),(t. j), and (l + l, j) using

AI

ul*t - ul- t

( 9 . 5 . 9 )

(9 .5 . j 0 )

A forward-difference scheme is used for the time dedvative and a central-difference scheme is used for the spatial derivative.

Note that the spatial derivative is wriften using known terms on time linej. Implicit schemes on the other hand use finite-difference approximations forboth the temporal and spatial derivatives in terms of the unknown time line/ + l .

The discretization of the x-r plane into a grid for rhe inregmtion of thefinite-difference equations introduces numerical enors into the coaputation. Afinite-differcnce scheme is stable if such elTorc are not amplified during succes-sive computation from one time line to the next. The numerical stability of thecomputation depends on the relative grid size. A necassary but insufficient con-dition for stability of an explicit scheme is the Courant condition (Courar\t andFriedrichs, 1948). For the kinematic wave equations, the Couant condition is

a r < - (9.s. 1 l )

where cr is the kinematjc wave celerity. For the dynamic wave equations, c^ isrePlaced by y + cd in (9.5.1 l). The Courant condition requires that the rime step

dr

and

a"l _Ax

29,1 .Apt: ti:1, ittor{c,rocY

be less than the time for a wave to travel the distance A{i If A/ is so large that

the Courant condjtion is not satisfied, then there is' in effect, an accumuiation or

piling up of water. The Courant condition does not apply to the implicit scheme.'

For computational purposes in an explicit scheme' Ar is specified and kept

fixed throughout the computations, while At is determined at each time step To

do this. a lri just meeting the CouGnt condition is computed at each grid point

i on time llne j, and the smallest Ar1 is used. Because the explicit method is

unstable Lrnlcss Aa is small, i t is sometimes advisable to determine the minimum

Ar, at a time line j then reduce it by some percentage. The Courant condition

does not guarantee ttability, and thereforc is only a guideline

Implicit Scheme

lnrplicit schcmes use finite-difference approximations for both the tempofal and

spatial derivative in terms ofthe dependent variable on the unknown time line. As

a simple elample the space and time derivatives can be wdtten for the unknownpo in t ( l - i . I + l ) as

d u j , i l , u i , I l - u ! * 1dx An

and

duj,i\ ', _ ut,It, u!n,

dt A/

This schene is used in Sec.9.6 for the kinematic wave model. In Chap l0 a

more complex in]plicit scheme, refened to as a weighted 4-point implicit scheme,

is used ior the full dynamic wave model.

9,6 NUN{ERICAL SOLUTION OF THEKINENIATIC 1VAVE

As shown in Eq. (9.3.7), the continuity and momentum equations for the kjne-

matic \r'a\ e can be combined to produce an equation with 0 as the only dependent

variable:

d Q * o g 6 o - t d Q = oax dt

( 9 . 5 . l 2 )

( 9 . 5 . 1 3 )

( 9 . 6 . 1 )

The objective of the numerical solution is to solve (9.6.1) for 0(.t, l) at each point

on the -r-I grid. given the channel parameters a and B, the lateral iDflow 4(f)'and the initial and boundary conditions ln particular, the purpose of the solution

is to detemine ihe outflow hydrograph Q(L,t). The numerical solution of the

kinematic $ave equation is more flexible than the analytical solution described

in Sec. 9.4t it can more easily handle variation in the channel properties and in

the initial and boundary conditions, and it serves as an intrcduction to numerical

solution of the dlnamic wave equations, presented in ChaP l0.

295

To solve Eq. (9.6.1) numerically, the time and space derivatives of Q are

approximated on the ,r-1 grid as shown in Fig. 9.6. l. The unknown value is @,1] i.The values of 0 on the jth time line have been prcviously determined, and so has

Q,r . Two schemes for setting up the finite difference equations are described in

this section; a linear scheme, in which Q,l i j is computed as a linear function ofthe known values of Q, and a nonLinear scheme, in which the finite-differcnceform of (9.6.1) is a nonlinear equatron.

I l,

Linear Scheme €'

The backward-difference method is used to set up the finite difference equations.The finite-difference form of the space derivati ve of Q I I ] | s found by substitutingthe values of 0 on the (/ + l)th time line into (9.5.12):

/ i +t nl +)ou rzi r t ui

dx Ar

The finite-difference form of the time derivative is found likewise by substituting

( / + 1 ) A /

{9.6.2)

! , o ,

aQ o ' , : o i '; ;

= ^r

o t _ , n to g _

_ , + : , +

a i A r

^ a i + a i - t- 7

O Known value ofQ

E Unknown value ofQ

Distance r

( r + 1 ) A r

FIGURE 9.6.1Finite difference box for solurion of rhe linear kinematic wave equation showing the fiDite differenceequalions.

(9.6.3)

If the \alue of 0ll l were used for Q in the telm \BQB-1 tn Eq. (9.6.i). rheresulting equation would be nonlinear in p]l j . fo create a l inear equation, thevalue of Q used in dpQp-' is found by averaging the values across the diagonalin the bo \ shown in F ig . 9 .6 . l :

t O ' , + O t ' 'o = - ' - ' - '' 2 \ 9 . 6 . 4 )

296 epiLrn s- 'nroroc r

the valucs of 0 on the ( i + l) th distance l ine into (9.5.13):

d o _ o : + ) - q , , ,dt Lt

The ralue of latcral inflow q is found by averaging the values on the (j + l)thd i i tan . e , . , lhe 'e a re assumed Io be g i ren rn the prob lemr .

4,it, + 4, ",s : 2 (9.6.5)

By substituting Eqs. (9.6.2) to (9.6.5) into (9.6.l), the finite-difference form ofthe l inear kinematic wave is obtained:

q i Q - t ^ t 8 - t + O ' ' \ u ' l g - , i A t!

_ "p \ ,_ ) \ r , r=f i ) L ' n u u '

This equation. solved for the unknown 0,1;j, is

g , + ) :f; + "Fl

do 1 ldA\

Q _ P \ , A )

(9.6.1\

A flow chart for routing a kinematic wave using this scheme is given ioFig. 9.6.2. Q was chosen as the dependent variable because this results in smailerrelative eirors than if cross-sectional arca A were chosen as the dependent variable(Henderson, 1966). This is shown by takjng the logarithm of (9.3.3):

l n A = l n a + B h Q

and differentiating:

(9.6.8)

(9.6.9)

to define the relationsbip between the relatjve errors dNA and dQlQ. Using eitherManning s equation or the Darcy-Weisbach equation, B is generally less than 1,and it follows that the discharge estimation error would be magnified by the Iatio

Compute jn i l ia l condi t ions dei ined b)basef low, l ime r= 0 on t ime l ine/= L

Apply Couranl condition (9.5.1 1) to eachgrid poim on time line, and cboose the

smal lesl r dr : min / i .

Advance to next time slep:t = t + L t , j = j + l .

Use inflor hydrograph lo determin€dischargc p, /nr ut upslream boundar, , l = I

lncrement to nexl inte. ior point ( i + l ) on r ime l ine i + l .r = x + Ax. Solve for the discharge

0 l : i u s i n s E q . ( e . 6 . ? ) .

Downurge

6 Slop

oisrnrsureo Fl-ow noLrrNc 297

1 1 "6-l

FIGURE 9.6.2Flowchart for linear kinematic wave computation.

1/B if the cross-sectional arca were the dependent variable instead of the flowrate.

Example 9.6,1, Using the same data for the rcctangular channel in Exar,nple 9.4.1(width = 200 ft, length = 15,000 ft, slope = I percent, and Manning's r = 0.035),develop a linear kinematic wave model and route the inflow hydrograph gi\en incolumns I and 2 of Table 9.4.1 through the channel using Ar : 3000 ft and Al :

3 min. There is no lateral inflow. The initial condition is a uniform flow of 2000cfs along the channel.

298 orr, r i o rnorrrc,.r '

Sol i t t ion. The vaJue of B is 0.6, and cr is found using n = 0.035, P = B : 200 ft .and 5, = 0.1.) l fol lowing Eq. (9.3.5) as

II nP" \ io.oj5 ' (2oor?' l . '' - \ r / q t ' , / - l

l r s , o o l , ' ' I - " '

\ " /

F o r l r = 3 m i n = 1 8 0 s a n d A { = 3 0 0 0 f t , E q . ( 9 . 6 . 7 ) w i t h q : 0 g i v e s

[,0**n'.'- , r . 4q , ,0 6 ,0 to j n '

, ' " I- l

[# .o.*,..o t1 n''' ! n''

)'"" ")

Ihis pfoblcm is solved fol lowing the algori thm given in Fig. 9.6.2. Cornpu-tat ions p|occcd from upstream to downstream as shown in Table 9.6.1. in whichthc di i lence axis is laid out horizontal jy, i = l ,2, . . . , 6, while the t ime axisrurs lcrt ical l ! down t lre page. j = 1,2, . . . . The init ial condit ion is Q: = 2000cfs. covering the first row of discharge values. The upstream boundary conditioo isthe intlow h\.drograph pj in the first column of discharge values. The lnflows att : A. 12,21, . . - min are obtained from Table 9.4. l . and the remainder arc f i l ledin b1 l inear interpolat ion between the tabulated values.

The first time the inflow departs from 2000 cfs is after t : 12 min, sothe calculat ions for 0 on the l5-min l ime l ine are used as an i l lustrat ion. Thecompulational sequence is indicated in Table 9.6.1 by the sequence of boxes. WlthJ : 5 and i = l , the f irst unknown value is Qlj : 0!, whlcfr is rhe discharge atdistance 3000 ft on the l5-min t ime l ine. I t is found as a functio| of Qji . : Qtr=2000 cfs. thc discharge at 3000 ft on the l2-min t ime l ine, and of 01+' : Ob =2250 cfs, ihe value of the inflow hydrograph at r = 0 on the l5rnin rime hne.Substil!ting these values into the finite-difference equation:

f -*,"^., rrro, r r :.+o r' 0.6,' zooo,1 200a;-2250, 0 " I

t ' " "

f*s . o ort u,(2ooo + 225olo6 r)] : 2095 cfs

as shown in the table. Moving along the 15-min t ime l ine (/ = 6), the secondunknown is O!, the value at djstance 6000 ft, calculated as a function of 01+t =

Oi = 2000 cfs, now at 6000 ft on the l2-min t ime l ine, and of Qjt+t = Qt =20q5 ;{s, the ralue just computed for 3000 ft at 15 min. The same procedure asabore produues Q\ = 2036 cfs as shown in Table 9.6.1. Al l the unknown valuesare determined in the same manner. The outflow hydrograph is the column of flowrates for i = 6 at 15.000 ft.

The values of At : 3 min and Ar = 3000 ft were chosen so tiat the Courantcondil ion (9.5.1I) would be satisf ied everywhere in the r-r plane. As shown lnTable 9.4.1, the maximum wave celeri ty is 15.5 fr ls, for a discharge of 6000 cfs,Herc,!t /Ar = 3000/180: 16.7 fvs, which is greater than the maximum wavecelefiry, thus sarisfying the Courant condition throughout.

0 I 2cf|/) 2OOO 2OOO 20OO 2O0O tr'ZOOO

b,7

89

10l t

25c02150 2449 2246 212'73000 26t2 2414 22383250 2910 2613 238s 22283500 3158 2A36 2s66 236n

20102030206721292218

prsrnisurEo Filw toulNc 299

TA.BLE 9.6'1Numerical solution of the linear kinematic wave (Example 9.6.1.). Valuessiven in the table are flow rates in cfs. The italicized yalues show thepropagation of the peak discharge. The boxes show the computationalsequence for obtaining th€ flows along the 15-min time line,

Time Time(min) index

2l4

5

3

9

\ 2

Distance along channel (ft)6000 9000 120q) r5000

I ' r 6

15

l 821

21

2252 2l 18 2053 202320622129

485 l

60

6669't2

78

1 71 8l9202122

262'�7

50005250550057506061)575055005250500047504500

449s49525209546s5'720573456235441523850124711

437446384902

542155'�73s597

53905Zt35011

4t)3'�7130745'�7848485 t 1 8533254575469544353355184

369439654239451641935043523753565397536E5281

3358362038924t714452

49615145526353125298

t44147150

4950) t

2col2001200r

2028201920t3

200820052no4

200020002000

2061 21332049 21012036 20't6

Fig. 9.6.3(d) showsplots of the columns of Table 9.6.1, the flow hydrographsat the various points along the channel. It can be seen that the teak dischargediminishes as the wave passes down the channel, also marked by the italicizedvalues in the table. Fig. 9.6.3(r) shows plots of rcws from Table 9.6.1 , representingthe disribution of flow along the channel for various points io time, which showsthe rise and fall of the flow as the wave Dasses down the channel. Fis.9.6.4is a comparison of the analytical solution calculated in Example 9.4.i wlth rwo

300 ,c r ' p r t r oHvono r -oc l

a 3

E 2

3

^ 5

i z

tE 27T .

Disiance downsrream:3,000 ft6,000 fr9.000 fi12.000 f!15,000 fr

0 20 .10 60 80 I00 120 140

Time (m'n)

(dr Sol!r ior lhrcugh Lime at dif ferenl poinls rn space.

0 1 . . 1 6 8 1 0 1 2 1 4

Dislance along channel (fi, thousands)

o O m i n o 3 0 o 6 0 a 9 0 x i 2 0 v 1 5 0

1r l Solu i ion rhrough space al d i f ferenl porn$ rn nme.

FIGURE 9.6.3Numerical solution of thelinear kinematic waveequation in space and time(Exampl€ 9.6. l ) .

nurne.ical solut ions, the one calculated here with A.r = 3000 ft and At:3 min,and another solution simila.ly computed with A-r :1000 ft and At = I min. It canbe seen that the numedcal scheme introduces dispersion of the flood wave into thesolution, with the degree of dispersion increasing with the size of the Ar and A/increments.

Nonlinear Kinematic Wave Scheme

The finite-difference form of Eq. (9.6.1) can also be expressed as

g , - l - Z t - t . A i , : ' , - A j , , , q ' , l i * d , , ,A , A , ,

(e.6.10)

As for the linea.r scheme, 0 is taken as the independent variable; using Eq.(9 .3 .3 ) .

( a

(h

0

orsrnrnwep ri-ow rormrc 301

60 80 r00Time (min)

TIGURE 9.6.4Routing of the kinematic waveby analytical and numericalmetbods. The analytical solulionshows Do wave aBenuallon,while the numerical sllulionsdisperse the wsve, tha d*reeof dispersioo rncreisrigtith thesize of the time and disrLrc€steps. (d) Inflow. (r) Numericalsolution, using Ar= I min andAr= 1000 ft. (c) Numericaisolution, Ar=3 min, Ai= 3000 ft.(d) Analytical solulion.

140120

and

e,7',: "@,11)P

Aji *, = ol8, *t)F

Eqs. (9.6.11) and (9.6.12) arc substituted into (9.6.10) to obtain, after rearrang-ing'

{u+i *,|ati f = Hn,- + o\d,n)B . ̂ , :ny-) {e.6.r3)\ . t

This equation has been ananged so that the unknown discharge 0,1]i is on tneleft-hand side, and all the known quantities are on the right-hand side. It isnonlinear in Qill; ro a numerical solution scheme such as Newton's methodwiil be fequired (see Sec. 5.6 for an introduction to Newton's method).

The known righrhand side at each finite-difference grid point is

, = ! .g ' \ r o te l t1P + A, f4r l l4 t - ' )r t

( 9 . 6 . 1 4 )

from which a residual error f(Qt,li) is defined using Eq. (9.6.13) as

f ,O, . i = yE,: i . a@t -t tP . cA-r

The first derivative of/(0jl i) is

f , q , I i = ! * o 1 t d , t l 1 ,

The obiective is ro find Ol- I lhat forces f(Ol-lt ro eoual 0.

(9.6. r r)

(9.6.r2)

(9 .6 .1s )

(9.6. l6)

( 9 . 6 . 1 8 )

where e is an error criterion. A flowchart for the noniinear kinematic wave schemeis p resented in F ig .9 .6 .5 .

The inirial estimate for Q,/ ] j is important for the convergence of the itemtivescheme. One approach is to use the solution from the iinear scheme, Eq (9.6.7),

as the first approximatjon to the nonlinear scheme. Li, Simons, and Stevens(1975) performed a stabil ity analysis indicating that the scherne using Eq. (9.6.13)

is unconditionally stable. They also showed that a wide range of values of Ar/Arcould be used without introducing large errors in the shape of the djschargehydrograph.

9.7 MUSKINGUM.CUNGE METHOD

Several variations of the kinematic wave rcuting method have been proposed.Cunge (1969) proposed a method based on the Muskingum method, a methodtraditionally applied io iinear hydrologic storage touting. Refering to the tlme-space computational grid shown in Fig.9.6.1, the Muskingum rcuting equation(8.1.'/) can be written for the discharge at r = (i + l)A,r and t = 0 + 1) At:

302 ,rppuel urororocv

Llsing Newten's method with iterations k = l, 2, .

t l nt +l\( ) 1 " - 0 l

- l - r ' Y t l *v _ . ( - r \ z t . t r

f , E ; , _ l ^

The conre-gence criterion for the iteralive process is

l lQ"Ii)r*, =,

Q! ;l = c t8'*' + c2di + c3gi +t (9. ' �7 . t )

in which C; . C2, and C3 are as defined in Eqs. (8.4.8) through (8.4.10). In thoseequations. ,( is a storage constant having dimensions of time, and X is a factorexpressing the relative influence of inflow on storage levels. Cunge showed thatwhen K and A, are taken as constant, Eq. (9.7.1) is an apprcximate solution ofthe kinematic wave equations [Eqs. (9.3.1) and (9.3.2)]. He further demonstmtedthat (9.7.1) can be considered an approximate solution of a modified diffusionequation (Table 9.2.1) if

-. at

ck

Aj(

dQtdA

(9.6. l7)

(e.1 .2)

and

( 9 . ? . 3 )

where cr is ttle celerity corresponding to Q and B, and B is the width of thewater surface. The right-hand side of (9.?.2) rcpresents the time of propagation

t l o \Y = - | - |'

2 \ - Bc lS,Ar i

Compul! .,ritial condilions defined bybasef low at t ime / = 0 on r ime l inej = L

Advance to next lime step:| = t + A t , i - j + l

Use inflow hydrograph to determine dischargeOr'-' at upstream boundary.

Incrcment ro n€,(t inlerior poinl- r =.r + A-r oni ime l inej+1.

Solve for in i t ia l est imate of QL-, =Q, ' i ,

using the lin€ar scheme eslimale lrom (9.6.7);f ind / (0() forr =, using {9.6.15).

Det€rmine / (0 i ) usi rs (9.6.16).

S o l v e t o r 0 r * r u l i n g { 9 . 6 . 1 7 r .

Derermine/(0r + L) using (9.6.1s) .

Check convergence;

r l t p ' " t t < e t

). Id

FIGURE 9.6.5Flowchart for nonlinear kinematic wave computation.

304 rr: , : i i rr rrvceor.oor

of a givcn discharge along a reach of length Ar. Cunge (1969) showed that fo1numericel rtabiLit) it is required that 0 = X < l/2.

Muskingum-Cunge routjng is carried out by soll ing the algebraic aquation(9.7.1). The coefficients in Eq. (9.7.1) are computed by using Eqs. (9.7.2) a\d(9.7.3) aiong with Eqs. (8.4.8) through (8.4. i0) for each time and space point ofcomputation. sjnce K and X both change with respect to time and space.

The Muskingum-Cunge method offers two advanrages over tie standardkinematic $ave methods. First, the solution is obtained through a l inear algebraicequation (9.7.l) instead of a finjte difference or characteristic approxjrnation ofa panial clfferential equaiion: this allows the entire hydrograph to be obtainedat required cross sEctions instead of requiring solution over the entire iength ofthe channcl for each time step, as in the kinematic wave method. Second, thesolution using (9.7.1) wil l tend to show less wave attenuation, permitting a moreflexible choice ol t ime and space iDcrements for the computations as comparedto the kincmatic wave method.

The comprehensive Brit ish Flood Studies repon (Natural EnvironmentResearch Council, 1975) concluded that the Muskingum-Cunge method is prefer-able to methods using a diffusion wave model (see Table 9.2.1) because ofits simplicity; its accu.acy is similar. Disadvantages of the Muskingum-Cungemethod are that it cannot handle downstream disturbances that propagate upstreamalrd that it does not accurately predict the discharge hydrograph ar a downstreamboundary. rvhen there are large variations in the kinematic wave speed such asthose which result from the inundation of large flood plains.

REFERENCES

Abramo\,litz. \}{ and L A. Sreevn. Handbook oJ Mathematical Funclkrts, Dovef. Ne$. york_ 1972.Chow, V. T.. Opcn.hanneL H)-rlruuLics, Mccra\r-Hiil, New York. 1959.Courant, R.. and K. O. Friedrichs, S&]'ersonic FLow an.l Shock Wdyer, Inlerscjence publishers. New

York. 19,18.Cunge. J. .\ , On the subject of a flood propagalion nethod (Muskjngum method). J. Hrdldrlicr

Rer..r'.r, Intcmarional Associarion of Hydraulics Research. vol. 7, no. 2, pp. 205-230,1 9 6 9 .

Eagleson, P. S . Dr dnic H'"drolo , Mccraw-Hi l i , Ne\r York, 1970.Henderson. F. M., Open Cha neL Flo|', Macmitlan. New York, 1966.l - i . R. M.. D. B. Simons, and M. A. Slevens, Nonl inear k inemat ic wave aDDroximat ion f in watet

r o u l r - h d r . , & t o " , R c " . . r o l . t t . n o 2 . p p 2 4 5 - 2 5 2 t o / 5Lighthi l l . \ '1 . J . and G. B. Whirham, On k inenal ic waves. I : f lood mo!,ement in tong r ivers, prdc.

R . S o c . L a t l d o A , v o l . 2 2 9 . n o . t 1 7 8 , p p . 2 8 l 3 1 6 , M a y , 1 9 5 5 .Mi l lef , J . E. . Basic concepts of k inemat ic-wave models, U. S. Geot. Surv. Prot ' . pap. 13A2,1984.Natural En!ircnmenr Research Council, FLaod Studies Report, Vot.t||, Ftood Routile Studies,

lnsl r tute of Hidrolog! , Wal l ingford. England, 1975.Ovenon, D. E , and \,1. E. Meadows, Stormwater Modcllrg, Academic pfess, New york, 1976.Sain!-Venani. Bare de, Theory of unsteady walel flow, wirh application ro rjver floods and to

propagarron of tides in rive. cbannels, Fr€nc, Acadenr of kience, vol. 73. pp. 148-154,2 l ? 1 , 1 0 . I 8 1 1 .

Stephenson. D., and M. E. Meadows. Kinematic Hldnnogy and Modeling, Developmenrs in WaterScien.e 26. Elsevier . Amsterdam, 1986.

Srrelkoff. T . Onc drmcnsional equations of open channel ftav!. J. Hfdr. Di,., An. Soc. Cn,. Eng.,vol . ! i . ro. HYl . Do 86i-876. 1969.

DTTIRIBLiTED FfO\r nOurt^lC 305

BIBLIOGRAPHY

Explieit solution of the Saint-Venant equations

Anein. M., and C. S. Fang, Streamflow routing (with application ro North Carolina rivert. reD.no. 17, Warer Resources Res. Inst. of the Univ. of Norrh Carolina, Raleigh, N. C.. I969.

Garrison. J. M., J.-P. P. Granju, and J. T. Price, Unsready flow simularion in livers ano reservorrs.l . fudr. Div. , An. Soc. Civ. EnB., vol . 95, no. HY5. pp. t559-1576, 1969.

Isaacson, E., J. J. Stoker, and A. Troesch, Numerical solution offlood prediction and riverrequtaronproblems, reps. IMM 205, IMM-235, Inst. for Math. and Mech.. New yo* Uri!!. NewYork. 1954, 1956. I .1

lsaacson, E., J. J. Stokef, and A. Troesch, Numerical soiution of flow problems in rirers, d Hvd.Div. , An. Soc. Civ. Eng., vo| .84. no. HY5, pp. t -18. 1958.

Johnson, B., Unsteady flow computations on the Ohio-Cumbertand-Tennessee-Missjssippi riversystem, tech. rep. H-74-8, Hyd. Lab., U. S_ Army Eng. Warerways Exper. Sta., Vicks-burg. Miss. , 1974.

Ligge . J . A. , and D. A. Woolhiser, Di f ference solur ions of the shal low waler equat ions, J. Erp.Mech. Di i , . , Am. Soc. Ci t , . E| l8. , vol . 93, no. EM2, pp. 39 7] , 1967.

Manin, C. S., and F. c. De Fazio, Open channel surge simulation by digiral computer, J. d_vd.Dir . , Am. Soc. Civ. Eng., ro l . 95, no. HY6, pp. 2049-2070, t969.

Ragan. R. M., Synthesis of hydrographs and warer surface profites lor unsteady open channel flowwith lateral iDflows, Ph. D. dissertation. Cornelt Universjty. tthaca, New york. I965.

Stre lkof f , T. , Nume. ical solut ion of Sainlvenant equat ions. . I . l i_)d. Div. ,An.Soc.Ci En! . , \d.0 6 . n n . H Y I . p p . 2 2 3 - 2 5 2 . 1 9 i 0 .

Sroker. J. J., Numerical solurion of flood prediction and river regulation problems. rep. IMM-2trO.lnsl . for Math. and Mech.. New Yor l Univ. , New York, I953.

Kinematic wave models of overland ffow and river flow

Borah. D. K., S. N. Prasad, and C. V. Alonso, Kinematic wave routing jnco.porating shock fifiins.Woter Resour. R?s. , voi . i6 , no. 3, pp. 529,541, 1980.

Brakensiek, D. L., A simulated warershed flow system for hydrograph predictjon: a Krnematrcapplication, Proceediner, Inremational Hydrology Symposium. Fort Collins, Colo., !ol. i.pp. l . I J .7. Seprember lao7.

Conslantinides. C. A., Two-dimensional kinematic overland florv nlodeling, prr.eedingr, Sec,ond International Conference on Urban Sbrm Drainage, University of Illinois al Urbana-Champaign. ed. by B. C. Yen. vot- I , pp.49-58, June 1981.

Dawdy, D. R., J. C. Schaake, Jr., and W. W. A ey. User's guide for disrributed rouring rainfall_runoff model, Water Resources lnvestigation Z8-90, U. S. ceojogical Survey. Seprembert978.

Devries, J. J., and R. C. MacArthur, Introduction and application of kinemaric wave routrnstechniques using HEC-I, training document no.10, Hydrotogic Engineering Cenrer, U. SlArmy Corps of Engineers, Davis, Calif., May 1979.

Gburek, W. J., and D. E. Ovenon, Subcritical kinemalic flow in a srable slream, J. Hjtl. Dt\,., An.Soc. Ci t . EnB., vol .99, no. Hy9, pp. 1433-t44' t , 1973.

Harle}, B. M., F. E. Erkins, and P. S. Eagleson. A modular distributed model of carcnmentdynamics, reporr 133, Ralph M. Farsons Lab., Mass. lnst. Tech.. December 1970.

Hende|son, F. M., and R. A. Wooding, Overland flow and groundwater from a steady rainfall of,-. finiie duration, J. Geophys. Res., vot_ 69, no. 8, pp. l53t 1540, April, 1964_Kibief, D. F.. and D. A. Woolhiser, The kinematic cascadi as a hydrologic model. hydrolog) paper

no. 39, Colorado Stare Univers i ty , Fon Col l ins, Colo. , March l9?0.Leclerc, C., and J. C. Schaake, Methodology for assessing the potential impact of urbao development

on urban nrnoff and the efficiency of runoff conrrol altemarives, rcpolt no. 167, Ralph M.hrsons Lab. . Mass. Inst . Tech. , 1973_

Monis. E. M-, and D. A. Woolhiser, Unsteady one-dimensional flow over a plane: parrrat equrtrb-r ium and recession hydrographs, Watet Resour. ncr . , vol_ t6, no.2, pp.355 360, 1980.

CHAPTER

10DYNAMICWAVE I

ROUTING

The propagation of flow in space and time through a river or a network ofrivers is a complex problem. The desire to bujld and live along ivers crcatesthe necessity for accu.ate calculation of water levels and flow rates and providesthe impetus to develop complex flow routing nlodels, such as dynamic wavemodels. r\nother impetus for developing dynamic wave models is the need formore accurate hydroiogic simulation, in panicular, simulation of f low in urbanwatersheds and storm drainage systems. The dynamic wave model can also beused for routing low flows through rivers or inigation channels to provide bettercontrol of water distribution. The propagation of flow aiong a river channel or anurban drainage system is an unsteady nonuniform flow, unsteady because it variesin time, nonuniform because flow prope ies such as water surface elevation,velocity, and discharge are not constant along the channei.

One dimensional dist buted routing methods have been classified in Chap. 9as kinematic wave routing, diffusion wave routing, and dynamic wave routinS.Kinematic waves govem the flow when the inertial and pressure forces arenot important, that is. when the gravitatiolal force of the flow is balanced bythe frictional resistance force. Chapter 9 demonstrated that the kinematic waveapproximarion is useful for applications where the channel slopes are steep andbackwater effects are negligible: when pressurc iorces become important butinertial forces remain unimportant, a dilfusion wave model is applicable. Boththe kinematic ware model and the diffusion wave model are helpfui in describingdownstream wave propagation when the channel slope is greater than about0.5 ft/mi (0.01 percent) and there are no waves proPagating upstream due to

ovN,clr,4lc wevp aolrflrc 311

disturbances such as tides, t butary inflows, or resen'oir operations. When both

inertial and pressure forces are important' sucb as in mild-sloped vers, and

backwater effects from downsteam disturbances are not negligible, then both the

inertial force and pressure force terms in the momentum equation are needed'

Under these circumstances the dynamic wave routing method is required, which

involves numerical solution of the full Saint-Venant equations Dynamic routing

was fi$t used by Stoker (1953) and by Isaacson, Stoker, and Ttoesch (1954, 1956)

in their pioneering investigation of flood routing for the Ohio River. This thapterdescribes the theoretical development of dynamic wave routing modelsl using

implicit fnite-difference opproximationt to solve the Saint-Venant equati$s

10,1 DYNAMIC STAGE-DISCHARGERELATIONSIIIPS

The momentum equatjon is written in the conseNation form [from (9 1.33)] as

a Bo2lA\dx

Uniform flow occurs when the bed slope So is equal to the friction slope S/ and

all other terms are negligible, so that the relationship between discharge. or flow

mte, and stage height, or water surface elevation, is a single-valued function

derived from Manning's equation, as shown by the uniform flow rating curve

in Fig. 10.1 . L When other terms in the momentum equation are not negligible.

the stage-discharge rclationship forms a loop as shown by the outer curve in

Fig. 10.1.1, because the depth or stage is not just a function of discharge, but

also a function of a variable energy slope. For a given stage, the discha.rge rs

usually bigher on the rising limb of a flood hydrograph than on the recesslonlimb. As the discharge rises and falls, the rating curve may even exhibit multipleloops as shown in Fig. 10.1.2 for the Red River (Fread, 1973c). The ratjng curve

for uniform flow is typical of lumped or hydrologic routing methods in which

dQ

At. t ^ ( q * - t " + s i + s " ) - P q v ' + w r B = o ( r 0 . r . l )

ra l ing curvc

{Kinemaric wave

Loop rlring curve(Dynamic and

FIGURE 10.1.1Loop rating curves. The uniformflow raling cur.r'e does not reflectbackwater effects, whereas lheloopd curve does.Di \ch . rge

d i f fu \ ion $ave mode l \ )

312,qppi-rEo rvotrot-o<;Y

8 0 -

:10

0

7 0 -

6 0 -

5 0 -

,10 60

Dischargc ( I 000 cfs)

o Obsewed - Conrpuled

FIGURE I0.1,2Looped strg.-discharge relation for the Red River. Alexandria Louisiara (Ma] 5 June 17,1961. Sor'r rr: Frcad. 1973c).

S : /(Q). while the loop rating curve is typical of distributed or hydraulic routing

methods.Flo\a' prcpagation in natural rivers is complicated by several factors: junc-

rions and tributaries. variations in cross section, variations in resistance as a

fulcrion both of flow depth and of location along the dver, inundated areas, and

meandering of the dver. The interaction between the main channel and the flood

plain or inundated valley is one of the most imponant factors affecting flood

propagation. During the dsing Part of a flood wave, water flows into the flood

plain or lalley from the main channel, and during the falling flood, water flows

from the inundated valley back into the main channel. The effect of the valley

storage is to decrease the discharge during the falling flood. Also, some losses

occur in the valley due to infiltration and evaporationThe flood plain has an effect on the wave celerity because the flood wave

progresses more slowly in the inundated valley than in the main channel of a

river. This difference in wave celerities disperses the flood wave and causes flow

from the main channel to the flood plain during the rising flood by creating a

transverse water surface slope away from the channel. Dudng the falling flood'

the transverse slope is inward from the inundated valley into the main channel,

and water ttlen moves from the flood plain back into the main channel [see Fig.

10 .1 .3 ( . r ) and ( r )1 .Because the longitudinal axes of the main channel and the flood plain valley

are rarei-"- parallel, the situation desc bed above is even morc complicated in a

oyrueutc wevr rorrrrc 313

(a) Tfansverse slope during rising flood.

r-- v r{a . = atu%'wm! 'Tq | f ' ' ! !M

A A"((

{}) Transverse slope during falling flood.

II

(.) Main channel parallel to valley.

FIGURE T0.1.3Aspects of flow in nalural rivers,

meandering river. For a large flood, the axis of the flow becomes parallel tothe valley axis [Fig. 10.1.3(c) and (4]. The valley water slope and valley watervelocity (if depths are sufficieat) can be greater than in the main channel, whichhas a longer flow path than the valley. This situation makes it difficult for flow togo from the main channel to the flood plain valley during the rising flood and vicevelsa during the falling flood. Flood wave propagation is more complex when theflow is varying rapidly. The description is also more complicated for a branchingriver system with tdbutades and the possibility of flood peaks from differenttdbutaries coinciding. Also, with tributades, the effects on flood propagation ofbackwater at the junctions must b€ considered.

When backwater effgcts exist, the loop rating curve may coasist of a seriesof loops, each corresponding to a differenr fearure conrolling wdter level in thechannel (see Fig. 10.1.4). Backwater effects of reservoirs, channel junctions,mrrowing of the natural river channel, and bridges can demonstrate this charac-teristic.

(d) Meandering main cbannel.

316 ,q.rrurir rr onorocv

cfs 3470

M

Je A

'0 .54-t . + 0.54r '

31863500 cfs'

r r + lDislance-r

FIGURE 10.2.2Values of the flow rate at four points in the -r-r plane (Example 10.2.I).

Example 10,2,1. The values of flow rate C at four points in the space-time grida.e as shown in Fig. 10.2.2. Using Ar = I h, A.t = 1000 ft , and 0: 0.55, calculatethe values of dQldt and AQ/drby the four-point implicit method.

Solution. As shown in Fig. 10.2.2, the values of flow rate at the four poinls are

0/ = 3500 cfs, 8l +t : 3386 cfs, Q/ +' = 3583 cfs, and p/ff =3476 "L. 11'"

t ime deri\ ,at ive is calculated using (10,2.1) with u = O and Ar = I h = 3600 s:

dQ

ao!' ' +oi: i 01 Ql*,

2A t

3 5 8 3 + 3 4 7 0 - 3 5 0 0 - 3 3 8 6

2 x 3600= 0.023 cfs/s

The spalial derivative is calculated using (10.2.2):

de ^o i i i 0 ! ' ' . , ^Q! , o !. ^ = o L ,

' ( ' - o r A r

^ -- 134?0 - 3583y ^ -- (1186 3500)- ' u . r ) r t 0 0 0

| i r - u . r ) ) - - 1 6 -

= - 0.1 13 cfs/ft

10.3 FINITE DIFFERENCE EQUATIONS

The conservation form of the Saint-Venant equations is used because this formprovides the versatility required to simulate a wide range of flows from graduallong-duration flood waves in rivers to abrupt waves similar to those caused by adam failure. The equations are developed from (9.1.6) and (9.1.37) as follows.

Continuity:

Momentum:

dodx

oyr.rar,uc weve nounrc 317

( 1 0 . 3 . 1 )

(tp.3.2)

f

q

h

s,

s"B

W1

BI

a Q ^ Q i : t t - Q ! ' t

. . ^ O l - , O l, ; = V n -

- r r - U J - - ^ ; l -

d h ^ h i : 1 1 h l - t . . ^ . h ! . , - h l

; = s - - - - - - - - - - - \ r v ) - - . -dX Nt A-ri

and the time dedvatives are estimated using (10.2. t):

t A + A d ! t t r ( A - A o t l l , ' - r a - A o t l - \ A r A a t l . l

( 1 0 . 3 . 3 )

( 10.3.4)

dtA + Ao) _dt

# - *#t * se(! * sr * s") Bqv, +wyB = o

where

r = longitudinal distance along the channel or river

/ = time

A - cross-sectional area of flow

= cross-sectional area of off-channel dead storage (contdbutes to continuity,but not momentum)

= lateral inflow per unit length along the channel

: water surface elevation

= velocity of lateral flow in the direction of channel flow

: friction slope

= eddy loss slope

= width of the channel at the water surface

: wind shear force

= momentum correction factor

= acceleration due to gravity.

The weighted four-point finite difference approximations given byfus. (10.2.1) - (10.2.3) are used for dynamic routing with the Saint-Venantequations. The spatial derivatiyes dQlAx and Ahl dx arc estimated between adjacenttime lines according to (10.2.2):

2 Atj( 1 0 . 3 . 5 )

( t0 .3.6)

adjacent time

( 10.3.7)

( t 0 . 3 . 8 )

- oAl'' + tr - olA',where 4i and ti indicate the lateml flow and cross-sectional area averaged overthe rcach .\.ri.

The finite-difference form of the continuity equatjon is produced by substi-ru t ing Eqs . (10 .3 .3 ) , (10 .3 .5 ) , and (10 .3 .7 ) in to (10 .3 .1 ) :

31E eppr rro ltr'oqol-ocr

d o , o ! . I + o ! j ] - o l - o ! . l

at 2 Atj

The nondcrivative terms, such as 4 and A, are estimated betweenl ines us ing (10 .2 .3 ) :

^q ! " -q l ' , ' ^q , ' q i t

, 1 w 2

, , , " , 2

, = q q l ' ' + ( t - 0 ) 4 1'

t t - t 1 4 ) - , ) A i . A l , ,A _ A # . i l - d ) ^ -l z

eei 4 - o. , , l , , ' u , lo i - ' - o ' - o . \

\ r r I l t u , l

tA Ao t t (A - Aa ) i + t - tA ' Ao t l tA - Ao t ; - l

( 1 0 . 3 . 9 )

2 Atj

Similarly, the finite-difference form of the momentum equation is w tten as:

0 , ' Q ! - ' , - O i - Q ! , r . ^ l r F Q ' � t t t , " t , - t P Q 2 l A t l - l

, A,, - "L *

s; inii-l.J

+rs,r' rs",")-,ra*ti

. ^ l , F Q ' , a t , ' * t - ( B Q z t A t , t = , l n i , . - n l: ( l - t r l ^ . ' 8 . . t , 1 - l ;

L - ,i '-\J ,- -,rl- \Ps"'), * \wrB ),

)= 0

+ r r - r j + r l+\wP )t

]

t- \i ,, r;\* ls4, * [sr,/

( 1 0 . 3 . 1 o )

Thc four-point finite-difference form of the continuity equation can be fur-ther modified by multiplying Eq. (10.3.9) by Ari to obtain

. , { ( * " ) : : ,

t------1t '- ---r i I-\Fe,,). tt*, + \wfB) Lx t J

= 0

where the avemge values (marked with -) over a reach are defined as

^ B i - F , . th = 2

A,= \:t-^ B , t B i - lo, - --

2

q, = L!-Qt:t2

Also,;

( r 0 . 1 . l 7 )B i

for use in Manning's equation. Manning's equation may be solved for S/ andwdtten in the form shown below, where the term l0 0 has magnitude Qr and sign

Positive or negative depending on whether the flow is downstream or upstream'respectively:

ovrevrc wavE rourrrc 319

e . O ! j l - Q l - ' - q l * ' A r r ) + ( 1 q ( Q l * t - Q i - 4 ! t ' x )

A Y .

+ # l tA +A , t , i + t + (A +A) ! I t t -@ +A) ! - (A + A . ) ! +1 j1= o' ( 1 0 . 3 . 1 l )

Similarly, the momentum equation can be modified by multiplying by Ari to

, l

l l o t ' ' o ; : l o i - o - , , i

l# l : +sAi ln i : , t / , ' - ' . ( i , : - 0 . " , ls . ' - a- , , ]

- . t - t - - J I I-\$q,J, Lx, - \wJBl, Ar,l

-,,,-rl(4q, ),.,- (#)i .'^llri. ' - r,i + (sr)ja", . (tJl-,]

( 1 0 . 3 . l 2 )

( r 0 . 3 . r 3 )

( 10.3. l4)

( 1 0 . 3 . 1 5 )

( 1 0 . 3 . r 6 )

320 ,rprlLr! sr onorc,cr

The minor head losses arising from contraction and expansion ol the channelare propo ional to the difference between the squares of the downstrcam andupstream !elocities, with a contmction/expansion loss coefficient K":

/-\ n1lA,Q'\ J J , =

, - r .2.208A; R,

' ( r 0 . 3 . l 8 )

I (",:#[?):., (f),] ( 10.3. l9)

The veloclty of the wind rclative to the water sudace, V,, is defined by

, 1 n I

14) : l : l ' t 4 , , , . o ' - ( 1 0 . 3 . 2 0 )

(-t) : tc;,1(v1,1[v;, ( 1 0 . 3 . 2 1 )

where ra is the angle between the wind and the water directions. The wind shearfaator is ihen siven bv

where C, is a friction drag coefficient tC"= Cy'2 given in (9. 1 . I8)1.Thc terms having superscript j in Eqs. (10.3.11) and (10.3.12) are known

either from initial conditions, or from a solution of the Saint-Venant equarionsfor a prerious time line. The terms I, Ari, Fi, K", C*, and V, are known andmust be specified jndependently of the solution. The unknown terms are Bf

*r,

Q! | , t , h ! - ' .h l i , ' , A i . ' , A ! i , t , B ! * I , andBf f , r . However , a l l rhe te rmscan be expressed as functions of the unknowns, O!

- ' , O! ii , hl + | , and hl lrt,so there are actuaily four unknowns. The unknowns are raised to powers otherthan unity, so (10.3.11) and (10.3.12) are nonlinear equations.

Th€ continuity and momentum equations are considered at each of theN - I rectangular grids shown in Fig, 10.2.1, between the upsteam boundaryati = I and the downstream boundary at i : N. This yields 2N - 2 equations,There are two unknowns at each of the N grid points (0 and i), so there are2N unknowns in all. The two additional equations required to complete thesolution are supplied by the upstream and downstream boundary conditions. Theupstream boundary condition is usually specified as a krown inflow hydrograph,while the downstream boundary condition can be specified as a known stagehydrograph. a known discharge hydrograph, or a known relationship betweenstage and discharge, such as a rating curve.

10.4 FINITE DIFFERENCE SOLUTION

The following discussion for the solution of a system of finite dilfercnce equationsfollows that of Fread (1976b). The system of nonlinear equations can be expressed

ovl,qurc w,wE notlrllo 321

in functional form in terms of the unknowns h and Q at time levelj + l, as

follows:

U B ( h t , A ) = o

C : ( h ' Q t ' h z ' Q ) : o

M t ( h , Q t ' h z , Q ) - - 0

:

C, (h i , Q i , h i * t , Q i * t ) = O

M i ( h i , Q i , h i * t , Q i * t ) = 0

'

C y - t ( h N - t , Q n - t , h N , Q t i = 0

M y - t ( h N . - t , Q N - t ' h n , Q = 0

DB (kv, Qn) : o

upstream boundary candition

continuity for gnd I

momentum for grid 1

continuity for grid i

momentum for grid i

r {

f lro.q. rt

continuity for grid N - I

momentum for grid N 1

downstream boundary condition

residual for upstream boundarycondition

residual for continuity at grid I

residual for momentum at grid I

residual for continuity at grid I( ro 4 .2 )

residual for momentum at grid i

This system of 2N nonlinear equations in 2N unknowns is solved for eachtime step by the Newton-Raphson method. The computational procedure for eachtimej + I starts by assigning tial values to the 2N unknowns at that time. Thesetrial values of 0 and ft can be the values known at time j from the initial condition(ifj = l) or from calculations during the previous time step. Using the rial valuesin the system (10.4.1) results in 2N residuals. For the ,tth iteration these residualscan be expressed as

U B ( h \ , Q D : R U B K

c\(h\, o\. ht, oi) - RC\Mr(h\. a\, hi, ab = RM\

:

c, e!, q, hf * t, ef * ) = RCkt

u, \n!. e!, hl * 1, e!, ) = Rrut

:

c \ | ( , f - 1 , O N - , , h h , Q h \ : R C h , r e s i d u a l f o r c o n t i n u i t i a t g d N - I

Mn-r(hl"-r' Ok-t, h1iu' Qb = nl6-t residual for momentum at grid N - I

DB (h1i, Q*) - RDBk residual for downstream boundarycondition

1 ) )

The solution is approached by finding values of the unknowns 0 and ,t so tharthe residuals are forced to zero or very close to zero.

The Newton-Raphson method is an iterative technique for solving a systemofnonlincar algebraic equations. It uses the same idea as was presented in Chap. 5for the derermination of flow depth in Manning's equation, except that here thesolution is for a vector of variables mther than for a single variable. Consider thesystem of equations (10.4.2) denoted in vector form as

"f(r) - o ( 1 0 . 4 . 3 )

where x: (Q1. hy, Qr, hz, . . .,QN,hi is the vector of unknown quantit ies andfor ireration k,xk - (Q\,hkt,Qt,h\, ,Ok,hb The nonlinear system can beiinearized t0

Jlxk + t1 - 1xk1 + -/(.rr)(.rt +1 - .rr) (10 .4 .4 )

where J(/) is the Jacobian, which is a coefficient matrix made up of the firstpartial deri\atives of /(x) evaluated at xx. The right-hand side of Eq. (10.4.4)is the l inear vector function of;k. Basically, an iterative procedure is useo todetermine xt+l that forces the residual error/(-t+r1 in Eq. 110.4.41 to zero.This can be acconrplished by sening/(;"*') :0 rearranging (10.4.4) to read

" i ( - / t t . / * l l1= fqxk l ( 1 0 . 4 . 5 )

This systerr is solved for (xt+r x9: Arr, and the improved estimate ofthe solution. rr 'L, is determined knowing A.rr. The process is repeated unti l(.t' + | - l) is smaller than some specified tolerance.

Thc system of i inear equations represented by (10.a.5) involves "r(xt, theJacobian of the set of equations (10.4.1) with respect to h and Q, and /(r),the vector of the regatives of the residuals in (10.4.2). The resulting system ofequatlons ls

dVB an + a.{/_B 46, = puskd n t d A t -

dc) ,,

aMt , ,

:)a ;ta )f-- , Al l | :^ AU, | - -*dn, - | a

dfr, d\!, - dni + |

aM,, dM, ,^ dM: , ,-, dt L , ;- du, + #dh +dff, dV, - d4r +l

frap, * fian,* #'on' =

#on, * ffian,*

- ^ L i

RM\

# i o n , + t = - � R C !

#ion'* ' --RM!

ffiao,:

I

ffior*-, + ffiaen-t+ ffitanu * nfuton, = -nci,-,d M N - t , ,

ffioo,-, + ffianx* %Fon,= -nMh-t

ov*-avrc v"evr nor,nrc 32J

Upslream

CL

Cl

CZ

fr,"

iL3 h t

I tutL a / , ,

A U B

acL acr acrA

-al2 ao,

ac2 ac2 ac2 ac1ah, eo, ak ao-.aM2 ai,t2 aM2 aM26k ;a. 6E io,

-RC

-RCr

dCt 6Ct

ak iO.aM1 aMl;/'. tA

?Cr dcr

aMt ai.t1ahr ro"

-RM:

ac1 ac| acA ac4 iah" ;o" ah, aO, IaMA ar44 aMA ar44ah4 aQ4 ah\ aQs )

aDB aDB In & ; l- - l

ffioo**ffi*6 =-*ourIn Fig. 10.4.1 these equations are presented in matrix form for a rlveli.dividedinto four reaches (five cross sections). The partial derivative terms arb {escribedin de ta i l in App. 10 .A.

Gaussiaa elimination or maaix inversion can be used to solve this setof equations (Conte, 1965). The Jacobian coefficient matrix is a sDarse matrixwith a band width of at most four elements along the main diagonal. Fread(1971) developed a very efficient solurion technique to solve such a systemof equations taking advantage of this banded (quad-diagonal) structure. Solving(10.4.6) proyides values of d,h; and dQi. The values for the unknowns at rhe

t*"rl

ar|t aMt aMlao' ;k ;0,

Cross sccrion I 2 1 4 5#

R e a c h t 2 3 4

I'IGURE IO.4.T.Syskm of linear equations for an iterarjon of rhe Newton-Raphsonreaches (five cross sections).

( 10.4.6)

method for a riler with four

324

srar r t r i rh !a rues o f r r= ta l .n l , . . . ,a1 ; . tkom in i r ia l condir ions, previous l ime step,

or irom an exLrapolation procedurt

Sol!. ior thc paflirl dcrivative terms lorhe Jacobian coefficienl marrir

using rhe values for xt

Conrporc rhe residuals RUB(, RC1,RMi. . . . .Rci 1. RM,l , 1. and

RDB^ f rom (10.4.2) .

So! \ c syncm of€qual ions for, / l r rnd. /p, usint Ga!ssian el ' 'n inal ion.

Dercrmine.alues of l jnrand O, i+ i using Eqs. (10.4.7) and

rr i ) .1.8): ' r - =roi ' l r i - l . . . .01,, . i r , l . ' r .

Rcra) to sla'1 ncxt nme slep.

FIGURE I0.4.2Procedure ibr solvinS a syst€m of difference equalioos al one time srep using the Newton,Raphson

/

(t + l)th itemtion are then given by

h!* t = n ! +an '

Q ! * ' : Q o t + a Q ,

oyr^lrrc weve aorrrrc 325

( 10.4.'�7 )

( 10.4.8. )

The flow chart in Fig. 10.4.2 outlines the procedure for solving the system

of difference equations for one time step using the Newton-Raphson method.

10.5 DWOPER MODEL

In the early 1970s, the U.S. National Weather Service (NWS) HydrologicResearch Laboratory began to develop a dynamic wave routing model based uponthe implicit finite-difference solution of the SaincVenant equations described inthe previous section. This model, krown as DWOPER (Dynamic Wave Opera-tional Model) has been implemented on various rivers with backwater effects andmild bottom slopes. It has been applied to the Mississippi, Ohio, Columbia, Mis-souri, Arkansas, Red, Atchafalaya, Cumberland, Tennessee, Willamette, Platre.Kamar, Verdigris, Ouachita, and Yazoo rivers in the United States (Fread, 1978),and has also been used in many other countries.

One of the DWOPER applications descdbed by Frcad (1978) deals withthe Mississippi-Ohio-Cumberland-Tennessee system, a branching river systemconsisting of 393 miles of the Mississippi, Ohio, Cumberland, and Tennesseerivers as shown in Fig. 10.5.1. Eleven gaging stations located at Fords Ferr)',Golconda, Paducah, Metropolis, Crand Chain, Cairo, New Madrid, Red Rock,Grand Tower, Cape Girardeau, and Price Landing were used to evaluate thesimulation by comparing the obseRed and calculated water levels and flow mtesat those locations. Figure 10.5,2 shows the observed vs. simulated stages at CapeGirardeau, Missouri, and at Cairo, I l l inois, for a flood in 1970.

In applying DWOPER to this system, the main stem river is consideredto be the Ohio-Lower Mississippi segment, with the Cumberland, Tennessee,and upper Mississippi rivers considered firsGorder t butades (Fread, 1973b). Thechannel bottom slope is mild, varying from about 0.25 to about 0.50 ftlmi (0.005-0.01 percent). Each branch of the river system is influencad by backwater fromdownstream branches. Total discharge through the system varies from low flowsof approximately 120,000 cfs to flood flows of 1,700,000 cfs. A total of45 crosssections located at unequal interyals ranging from 0.5 to 2l miles were used todescribe the system. Three months of simulation time, compari[g 20 observedand computed hydrographs using 24-hour time steps, required 15 seconds of CPUtime on an IBM 360/195 computer.

Another application by Fread (1974b) on the lower Mississippi illustratesthe utility of DWOPER simulating floods resulting from hurricanei. Figure 10.5.3

*A computer program for DWOPER can be obtained from the Hydrologic Research Laboratory.Offrce of Hydrology, NOAA. National Weather Service, Silver Spring, Maryland, 2091-.

tr t

F

J26 appL . s ro r , ' c r

r ! !1 . l { l l 6 5 l Shawneetown

=

=a

! t

Lpper \J i isrssippi

Crand Chr in

Cairo (mi le 955.8)

Ne{ Madr id

carurhersvi l le (mi ie 8,16.4)

Barkley Dam(mi le 10.6)

Kenlucky Ddm(mi le 12.41

al

z

' .

=

f;64.-;;;;l' on PUIdr ro ld l node

i- ( ompu ld l i ^nh l r rode I

+ Labrat inflow

FIGURE 10.5.1Schematic of the Mississippi-Ohio-Cumberland_Tennessee river system. The numbers shown arcfive. niles lronr the outh. (Sorrce: Fread, 1978. Used with pemission.)

shows the stage and discharge hydrographs on the lower Mississippi River at

Carrol l ton. Louisiana, duing hurricane Camil le in 1969. The f igure shows a

brief period of ncgative discharge resulting from the hurricane-generated flood

wave forcing waier to flow up the Mississippi River.

10.6 FLOOD ROUTING IN MEANDERING RIVERS

The dynamic wave model developed in the prcvious section can be expandedto consjder flood routing through meandering rivers in wide flood plains (Fig.

10.6.1). The unsteady flow in a river which meanden through a flood plain is

complicatcd b), f ive effects: (l) differences in hydraulic resistances of the mainriver channcl and the flood plain; (2) variation in the cross-sectional geometries

of the channel and the plain; (3) shod-circuiting effects, in which the flow leavesthe meandcring main channel and takes a more direct route on the flood plain: (4)

'14(l

l:18

3 : l 3 oe9 i28B 326c 321

6 3703 1 8

ovnerrrc uvs nourxo 327

l0 20 l0 40 50 60

Time (days)

- Observed . . . Simulaled

(a) Cape Girardeau, Missoun.

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0

Time (days)

- Observed , . . Simulaled

(b) Cairo. l l l inois.

FIGURE I0.5.2Observed vs. simulaled stages a! Cap€ Girardeau. Missouri, and at Cairo. lllinois, for 1970 flood

see Fig. 10.5.1 for location of these stations. Cairo is on the Ohio River and Cape Girardeau on

the Mississippi River. (Sorrce: F.ead. 1978. Used with permission.)

portions of flood plain acting as dead storage areas in which the flow velocity is

negligible; and (5) the effect on energy losses of the intemction of flows between

the main channel and the flood plain, depending upon the direction of the lateral

exchange of flow. Because of these differences, the attenuation and tnvel time

of flow in the channel can differ significantly ftom that in the flood plain

Fread (1976a, 1980) developed a model for meandering rivers, distinguish-ing the left flood plain, the right flood plain, and the channel, denoted by

the subscripts l, r, and c respectively. The continuity and momentum equa-

tions, neglecting wind shear and lateral flow momentum, are expressed as

; l

f"

124

3 1 8

9 1 1 6

E 3 t o9 308

{ :oog 304

t loo298

328

t0 20 30 40 50 60 70

Tirne (hn)

ob.crved - conrpured-- -' (a ) Stage hydrograph.

1969 Hunicane Cami ' leCarrolllon

20{)

l0t)= u

100

240

-300

( l 10 20 30 40 50 60 70Tjme (hrs)

- compuLed+ di\.hargc is rssociated wilh a velocily dircclcd downslream.

dischr+cd is associa led wi th a !e loci ty d i .ecled upsrrerm.

(, ) Discharge hydrograph.

FTGURE 10.5.3Stage and d;scha.ge hldrographs for lhe 1969 hunicane Camille at Carrolhon on the iower Missis-sippi River. Canolhon is at mile 102.4 from lhe mourh of the Mississippi. The root-mean-squareeffor for the simulation using DWOPER was 0.34 ft. (Fread, 197E. Used wirh p€rmission.)

a(KcQ) d(KtQ) .)(K,Q)a t a - o r , - a " ,

aQ

d ( 4 . + A t + A , + 4 . , )* - - - -

* - q 0 { 1 0 . 6 . 1 1

and

dtt lo2 /A.t aKl02tA,\+ ' - ' + ' - '

dx, dxr

dQlQ2tA,)- d " , -

-'^,\.* +s/. +s") * ro,l#, * sr,) * *,(ff,+s,,) = o( 10.6.2)

orreurc weve noullrc 329

Flood pla'n

FIGLTRE IO.6.IMeandering river in a flood plain. Unsteady flow in natu.al meandering rivers in wide flood plains

is complicated by large differences in resistance and cross-sectional geometries of the river and lheflood plain. As shown, further complications are due to the shod circuiting of flow along the moredirect roure of the flood plain. As a result, lhe wave attenuation and travel time differ belween theflood plain and the channel.

The total cross-sectional area of f low is the sum of A", At, A,, and A,. Theconstants Kc, K1, and K, divide the total flow Q into channel flow, left floodplain flow, and right flood plain flow, respectively, and are defined as r(. :=QctQ, KL - QLlQ, and K, = QlQ.

The flow is assumed one-dimensional, so the water surface is horizontalacross the three sections and the head loss /y incurred in traveling between two

ver cross sections is the same no matter which flow path is adopted. Hence,ht = St Lx in each section of the flow (left, channel, and dght), ar'd S1 : hy' Lr.By taking the mtio of the flows computed by Manning's equation in this manner,i1 is canceled out; the rutios of the flows in the left and right overbank areas tothat in the channel are

(10.6.3)

( 10.6.4)

The friction slope is also defined for the left flood plain, (Sft), right flood plain,(,!,), and channel (St using Manning's equationi for example,

li l

d'

and

Qr ," 1,, I n,'\''' I Lr,\t'to"

: tA"\&/ \A, , /

O, ,, t, I a,'1213 | t ,1"'a

: ; , r \R . / \ a , . /

330 errrrro uvonorocY

nL KtQlKtQ

2.2rA7R!t3

10.7 DAM-BREAK FLOOD ROUTING

( 1 0 . 6 . 5 )

The weighted four-point implicit scheme can be used to soive this model

for the unknowns h arrd Q The dynamic wave model described above by

Eos. ( l0.6. l) and ( 10.6.2) is incorporated into the National Weather Ser!ice com-

pui.r'prog.u* DAMBRK; DAMBRK is a program for analyzing the floods that

could rcsult from dam breaks.

Forecasting downstream flash floods due to dam failures is an application of flood

routing that bas received considemble attention The most widely used dam-breach

modeiis the National weather Service DAMBRK* model by Frcad (1977, 1980,

1981). -l 'his

model consists of three functional parts: (l) temporal and geometric

description of thc dam breach; (2) computation of the breach outflow hydrogmph;

and (3) routing the brcach outflow hydrograph downstrcam.

Breach Jormation, or the growth of the opening in the dam as it fails, is

shown in Fig. 10.?.1. The shape of a breach (tdangular, rectangular, or trape-

zoidal) is specified by the slope z and the terminal width 8" of the bottom of the

breach. The DAMBRK model assumes the breach bottom width starts at a point

and enlarges at a linear rate until the terminal width is attained at the end of the

failure tinre interval T. The breach begins when the reseryoir water surface ele

vation lt cxceeds a specified value /r". allowing for ovenopping failure or ptping

failure.

*A computer program for DAMBRK can be obtained from the Hydrologic Research Laboratory'

Oitlce oi uyaiology. hwOAA, Nationai wea$er Ser!ice, Silver Spring, Maryland, 20910'

FIGURE IO.7. IBreach fornration Breach formation in a dam faiiure is described by the failure lime f' the size'

and rhe shape. The shape is specified by z, \rhich defines lhe side sloPe of the breach TyPically'

O < a < 2. The bottom width b of th€ breach is a function of time' wilh the terminal widd being

8, , . A l r iangular breach h3s B = 0and ! >0 A rectangular breach has 8 ' >0 andz:0 Fora

l r a p e z o i d a l b r e a c h , 8 . > 0 a n d . > 0

T: T

+ r( 0< b< 8 , . )

ovsrurc wave noulrc 331

Reseryoir outflow consists of both the breach outflow O, (broad-crestedue i r f low) and sp i l lway ou t f low Q" :

Q = h + Q , ( 1 0 . 7 . 1 )

The breach oudlow can be computed using a combination of the formulas for abroad-crested rcctangular weir, gradually enlarging as the breach widens, and atrapezoidal weir for the breach end slopes (Fread, 1980):

: (to. z.zr

where t, is the time after the breach starts forming, C,, is the correction factor forvelocity of approach, K, is the submergence correction for tail water effects onweir outflow, and ,6 is the elevation of the breach bottom. The spillway outflowcan be computed using (Fread, 1980):

gs : c ,L , (h -&, ) ' t * , f i c r , t r {n - h )o5 +cdL, {h -h , j )o5 +e , Oo j l .3 )

where Cr is the uncontrolied spillway discharge coefficient, l, is the uncontrolledspillway length, ilr is the uncontrolled spillway crest elevation, C" is the gatedspillway discharge coefficient, As js rhe area of gare opening, h" is the center-line elevation of the gated spillway, C,/ is rhe discharge coefficieni for flow overthe dam crest, Z7 is the length of the crest, ft2 is the dam crest elevation, and 0ris a constant outflow or leakage.

The DAMBRK model uses hydrologic storage routing or the dynamic wavemodel to compute the reseryoir outflow. The reservoir outflow hydrograph is thenrcuted downsteam using the full dynamic wave model described in Sec. 10.3;altematively the dynamic wave model described in Sec. 10.6 for flood routingin meandering rivers with flood ptains can be used. The DAMBRK model cansimulate several reseryoirs located sequentially along a valley with a combinationof reservoirs breaching. Highway and railroad bridges with embankments can betreated as internal boundary conditions.

Internal boundary conditions are used to describe the flow at locations alonga waterway where the Saint-Venant equations are not applicable. In other words,there are locations such as spillways, breaches, waterfalls, bridge openings,highway embankments, and so on, where the flow is rapidly rather than graduallyvaried. Two equations are required to define an intemal boundary condition,because 1wo unknowns (Q and h) are added at the intemal boundary. For example,to model a highway stream uossing (Fig. 10.7.2) with flow through a bridgeopening Q6., flow over the embankment Q.n1, and flow tfuough a breach, Qy,the two intemal boundary conditions are;

Q ! * ' = Q o , + Q " + Q t

Q l l , ' = Q l * '

( 1 0 . 7 . 4 )

( 1 0 . 7 . 5 )

I h h . t ) 5

Qb .3 . lB^ tbc .K , : : - - - - :D ' + 2 .452C.K . th - hb t2 )

and

J32 api,- rn nr on"roer

r - l t i + l i + 2

(b) Cross scct ion local ions.

C . M C . M C . M C . M C . M

i + 1

i + 2 i + 3

8 b . = c , \ t A i - r r ' h : ' - h i - i ' -

Q.^= K" L"^c" \1 r j + t - 1 r , ^ '3 '2

J + |

(.) Finire difference grid.

FIGURE - I0.7.2lnremal bounJaq cordilion for highway embanlments. C refers to the continuity equation and M|o lhe momcnlum equatron.

The breach flow Q6 is defined by Eq. (10.7.2); Q6, is defined by a rating curveor by an orif ice equation. such as

( 10.7.6)

*here Ce is a bridge coefficient; the embankment overflow Q". is defined by abroad-crested weir tbrmuia

( 1 0 . 7 . 7 )

lnremal boundJry condiuon equdrron\

where C.n,. L",. and K". arc the discharge coefficient, length of embankment,

ovNevrc wave rourwc 333

and submergence corection factor, and ,em is the eleyation of the top of theembankment.

Referring to Fig. 10.7.2, the finite-difference grid illustrates thar for a giventime iine, the continuity and momentum equations are witten for each grid wherethe Saint-Venant equations apply, and the internal boundary conditions are wrinenfor the grid from r' to i + l, where the highway crossing is located.

Teton Dam Failure i I,

The DAMBRK model has been applied (Fread, 1980) to rcconstruct de down-s[eam flood wave caused by the 1976 failure of rhe Teton Dam in lda]ro. TheTeton Dam was a 300-foot high earthen dam with a 3000-foot long crcst. As aresult of the breach of this dam, I I people were killed, 25,000 people were madehomeless, and $400 million in damage occurred in the downsheam Teton-SnakeRiver valley. The inundated area in the 60-mile reach downstream of the damis shown with 12 cross sections marked in Fig. 10.7.3. The downstream valleyconsisted of a narrow canyon approximately 1000 ft wide for the first 5 miles, andthereafter a wide valley that was inundated to a width of about 9 mi. Manning's nvalues ranged from 0.028 to 0.047 as obtained ftom field esrimates. Inrerpolatedcross sections developed by DAMBRK were used so that cross sections wercspaced 0.5 miles apart near the dam and 1,5 miles apart at the downstream endof the reach. A total of 77 cross sections were used.

The computed reservoir outflow hydrograph is shown in Fig. 10.7.3 wirha peak value of 1,652,300 cfs. The peak occurred approximately 1.25 hoursafter the breach opening formed. It is interesting to note that the peak reservorroutflow was about 20 times greater than the largest recorded flood at the site. Thecornputed peak discharges along the downstream valley are shown in Fig. 10.7.4,which illuskates the mpid attenuation of the peak discharge by storage of water inthe flood plain as the flood wave progressed downstream through the valley. Thecomputed flood peak travel times are shown in Fig. 10.7.5, and the computedpeak elevations are shown in Fig. 10.7.6. The maximum depth of flooding wasapproximately 60 ft at the dam- The 55-hour simulation of the Teton flood usedan initial time step of 0.06 h. Execution time would be less than l0 seconds onmost mainframe computers.

FLDWAV Model

The FLDWAV model* (Fread, 1985) is a synthesis of DWOPER and DAMBRK.and adds significant modeling capabilities not available in either of the othermodels. FLDWAV is a generalized dynamic wave model for one-dimensional

+The computer program FLDWAV can b€ obtained frcm the Hydrologic Research Laboratory. Officeof Hydrology, Nationa.l Weather Service, NOAA, Silver Spring, MD, 20910.

3J4 aprrrt , ' ar ororocr

McrJ i Bu l rcs

\ Rexburs2 2 . 5

\ SNAKERIVER

* Miteage is valley miles downstream

kom Teton Dan

.t. ./ / ) 1 )

i/-'21

\ Shcl l ) geging srr t ion3

Hour

FIGURE 10.7.]iiooaea ar.u downsrream of Teton Dam and computed outflow bydrograph a! lhe dam (.sorrce:

Fread, 1917).

t . 8

1 . 2

t . 0

0 .E

r0 20 30 40 50 60

Disrance downsfeam of darn (mi)

e Observed _ Computed

t . 5.;

9 , ^

;

'd

Teron Dam

TIGURE T0.7.4Profile of peak discharge fromTelon Dam failure (totrce: Frcad, 19'71).

0 .60 .4

0 .2

d!i

a

40

I l 0E 5

l0 20 30 40 s0 60 70Distance downsream of dam(mi)

o Observed - Computed

oyrevrc weve rournc 335

FIGURE I0.7.5Travel time of flood peak from Teton Dam failure$ource: Frcad, 19'71).

I

a!

unsteady flows in a single or branched waterway. It is based on an implicit (four-point, nonlinear) finite-difference solution of the Sainr-Venant equarions. Thefollowing special features and capacities are included in FLDWAV: variable A/and Ai computational intervals; irregular cross-sectional geometry; off channelstoragei roughness coefficients that va.ry with discharge or water surface elevation,and with distance along the waterway; capability to generate linearly interpolated

5 t00

l0 20 30 40 50 60

DistaDcc downsrream of dam(mi)o Observed - Computed

FICURE 10.7.6Profile of peak flood elevatior from TetonDam failure (.tonrrer Fread, 1977).

336 Apprrto uYorot-ocr

cross seclioiis ancl roughness coefficients between input crcss sectionsi automatic

computation of initial steady flow and water elevations at all cross sections along

the waterway; extemal boundaries of discharge or water surface elevation time

series (hydrographs), a single-valued or looped depth-discharge relation (tabular

or computcd); time-dependent lateral inflows (or outflows); intemai boundaries

enable ihe treatment of time-dependent dam failures, spillway flows, gate con-

trois, or bridge flows, or bridge-embanl(ment overtopping flo\'; sho(-circuiting

of flood plain flow in a valley with a meandering river; ievee failure and/or over-

topping: i special compuational technique to provide numerical stability when

,r"n,in! Rn*t that thange from supercritical to subcritical, or convenely' with

time a;d distance aiong the waterway; and an automatic calibration technique for

determining the variable roughness coefficient by using observed hydrographs

along the waterwaY.FLD\VAV is coded in FORTRAN IV, and the computer program is of

modular design, with each subroutine requiring less than & kilobytes of storage'

The overall piogram storage requirement is apProximately 256 kilobytes Program

array sizes are variable, with the size of each array set intemally via the input

pararDeters used to describe each particular unsteady flow aPplication lnput data

io the program is free- or fixed-format. Program output consists of tabular and/or

graphical displays, according to the user's choice.

APPENDIX 1O.A

The following equations descdbe the partial derivative terms in the system ol

eouarions (10.4.6) (Fread, 1985) For the continuity equation (C), the terms

dependent on f i+1 and 0r * r in Eq (10 3 '11) cont r ibu te to the der iva t ives

For AClAn, the product rule dCldh = ACIdA x dAlAh: BACIdA is used The

derivatives are;

A C A - t , , ^ ^ r r + rdh,

: 26r,'o

- oo '

dC

dQi

d h i n t= * ,n +r , l j l , '

I A \

dC

( r 0 . A . 1 )

(10.A.2)

( 1 0 . A . 3 )

( 10.A.4)o Q i * t

where B, is the top width of the off-channel dead storage cross-sectional area'

For the momentum equation (,14)' the terms dependent on hi + I and Qi + \

in Eq. ( 10.3.12) contdbute to the derivatives, which are

. , ( * ) , . ' * , ]:#,-'l 'u#li '"' [ ' t1*t, *' I K-ozB \-

\g A"A ' / '( r0 .A .13 )

+4)fn1 ;; - ni. '

oyravrc wevr norrrc 337

+ {Sl ),/ * | 4., * (s"ri -'A,,] . )l*,#)".']

#,:&.4'(f)i" .'"t"V#):,. '*' .(#)., ,,*'l

.(#),-''*, .(*):.;+ is")1.'r",] . :(rr,#),:,'

: & .'14#)i :,' -',ul.'[(#),:'-, . (#)',, *,]]

( 1 0 . A . 5 )

.4; l r i : l -11'r +(sr)1 *ra-r,

( 1 0 . A . 7 )

d Q i * r

( 1 0 . A . 8 )

fhe derivatives of .S/ are found by differentiating Eq. (10.3.18), and are

#=41ry):,' +rei.'[r

- t \d S , ) i ^ ^ l l d n i 5 8 ; . , r I d B , t \- = -t)t)t =

-:- _: 1- - -:- i

d n i ) t \ n ; q n r - t 6 P , ; r F . d h t . . 1\ " " ' � _ r I

* r lA, l

, (sr) , .=. l t or , 1 \* ,

_ , " , , \ a a O , _ _ O * , )

The derivatives of S,, found by differentiating (10.3.19), arc

*10.A.6)

at,-lI

*,J

a s r ) , = r , " , ( , - o r , - 5 8 , - | d B t \- i i , = r , " , , , r_-^ - * , j , * )

- l, ( s r , = l t d n ,du,

\n ,da ,

\ * ) ,

( i0 .A.9)

( 1 0 . A . 1 0 )

( 1 0 . A . 1 1 )

( 1 0 . A . l 2 )

3JE ,qperrriD uyoRorocv (

/ - \l as " \ l -K"Q 'B\\ d t ' l

= \ r r * , 1 , . ,\ / r + |

/ - \/ a i " I l - x "o \l a o l = \ " 4 . ' a ' l\

- / i

l x " o \\ e A r A r / . , ,

+ l

if the upsiream boundary condition is a discharge hydrograph.

dDB

dhN

dDB

aQ, = "

il the dou nstream boundary condition is a stage hydrograph, but

ADB

dDB

oQn '

if the downstream boundary condition is a discharge hydrograph, and

I ds" i\ a o l

( 10 . A . 11 )

( r0 .A . l 5 )

( 10 .A . 16 )

The partial derivatlves for the UB and DB functions arc evaluated as follows:

AUBdht

dUBdQt

( r0.A. l7)

( 1 0 . A . l 8 )

( r0 . A . r9 )

( r 0. A.20)

( 1 0 . A . 2 1 )

( 10.A.22)

(10.A.23)

(1O.4.24)

4DB _ _Qt+ t Qtdhp h* *t - h*

dDB

dQtt

where ,( is the iteration number, if the downsteam boundary condition is a stage-discharge rating curve.

REFERENCES

Conte, S D., Elenentary Nunerical Anaryrir, Mcc.aw-Hill, New Yotk. 1965.Fread, D. L., Discussion of "lmplicil flood rouling in natural channels, by M. Amein and C. S.

Fary. J.HJd.Div.. An. Soc. Civ. En8., vol. 97, no. HY7, pp. 1156-lI59, l97l.Fread, D. L.. Effect of t ime step sjze in implici l dynamic routing, Water Resaw. Bull . , . !o1.9, no.2.

pp. i38 351, 1973a.

urreurc wevl xorn^*c 339

Fread, D. L., Technique for implicit dynamic routing in rivers with major tributarjes, Woter Resour.Rer., vol. 9, no.4, pp.918-926, 1913b.

Fread, D. L., A dynamic model of stage-discharge relations affected by changing discharge, Fig.1l, p. 24, NOAA Technical Memorandum NWS HYDRO-16, Office of Hydrology. NarionalWeather Service, Washington, D.C., 1973c.

Fread, D. L., Numerical properties of implicit fou!-point finite difference equations of unsteadyflo\\, NOAA technical memorandum NWS HyDRO.lg, National Weather Service, NOAA,U. S. Depanment of Commerce, Silver Spring, Md., 19'74^.

Fread, D. L., Implicit dynamic routing of floods and surges in the lower Mississippi, presenred arAm. Geophys. Union national meeting, Apr;l 1974. Washington, D. C., 197+.'!

Fread,D.L., Discussion of "Comparison of four numerical methods for flood,ro1iting." byR. K. Price, J. Hld. , iv., Am. Soc. Ci\ ' . Eng., vol. lOl. no. HY3. pp. 565-5679'1975.

Fread,D.L., Flood routjng in meandering rivels with flood plains, Riveru '76, Symposium onlnland Waterways for Navigation, Flood Control, and Waler Diversions, Augusr 10-12.19?6, Colorado State University, Fon Coli ins, Colo., vol. I , pp. 16 35, 1976a.

Fread, D. L., Theoretical development of implicit dynamic routing model, Dynamic Routing Semina.at L-ower Mississippi River Fo.ecast Center, Slidell, Louisiana, Narional Wearher Service.NOAA, Silver Spring, Md., December 1976b.

Fread, D- L., Th€ development aod testing of a dam-break flood forecaaring model. Praceedingr,Dam-break Flood Modeling Workshop, U- S. Water Resources Councii, Washingron, D.C ,pp. 164-191 , 197'7.

Fread, D. L., NWS operational dynamic wave model, Prcceedings,25th Annual Hydmulics DivisionSpecialty Conference, Am. Soc. Civ. Eng., pp.455+64. August 1978.

Fread, D. L., Capabilities of NWS model to for€cas! flash floods caused by dam failures, P/€pritUoLume, Second Conference on Flash Floods, March l8 20. Am. Meteorol. Soc.- Boston,pp . l 7 l 178 , 1980 .

Fread, D. L., Some limitations of dam-breach flood routing models, Preprint, Am. Soc. Civ. Eng.Fali Conven(ion. Sr. Louis. Mo.. Ocrober 1981.

Fread, D. L., Channel routing, Chap. 14 in Hydrological Forcc.tsting, ed. by M. C. Anderson andT. P. Burt, Wiley, New York, pp.437-503, 1985.

lsaacson, 8.. J. J. Stokei, and A. Troesch, Numerical solution offlood prediction and river regrlar;onproblems, reps. IMM-205, IMM-235, Inst. for Math. and Mech.. New York Univ., NewYork. 1954. 1956.

Sloker, J. J., Nrimerical solution of flood prediction and riyer regularion problems, rep. IMM-2oO,Inst- for Math. and Mech., New York Univ.. New York, 1953.

BIBLIOGRAPHY

Implicit solution of the Saint-Venant equstions

Abbott, M. B., and F. Ionescu, On the numerical computation of nearly horizontal flows. J. I/j|drd!/.Rer., vol. 5, no.2, W. 91-l \7, 196'1.

Amein, M., An implicit method for numerical fiood routing, Waret Resour. Res., vol.4, no.4,pp . 'l t9-126 , 1968 .

Arnein, M., and C.S.Fang. Implici t f lood routing iD natural charnnels, J.Hld.Di\ ' . , An.Soc.Ci\, . En8., \ to1.96, no. HY12, pp. 2481-2500, 1970.

Ame''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''n, M., Computation of fow through Masonboro Inlet, N.C., J. Watenaays, HarDors, onaCoastal Eng. Dir. , Am. Soc. Civ. Eng., vol. l0l , no. WW1, pp. 93-108, t9?5.

Amein, M., and H.-L.Chu, lmplici t numerical modeling of unsready f lows, J.Hyd.Dir. ,An. Soc. Ciy. Eng., vol. l0l , no. HY6, pp.717-'731, 19'75.

Baltzer, R.A., and C.Lai, Computer simulat ion of unsteady f lows in walerways, J.HyA.Dir..An. Soc. Civ. Ens., vol. 94, no. HY4, pp. 1083-l I17, 1968.

Chaudhry, Y. M., and D. N. Contractor, Application of the implicit method ro surges in channels,Water Resour. R€s., vol.9, no.6, pp. 1605 1612, 19'73.

(

CHAPTER

1 1HYDROLQGICSTATISTICS

Hydrologic processes evolve in space and time in a manner that is partly prc-

dictable. or detenninistic, and partly random. Such a process is called a stochrzsricproce,l. In some cases, the random variability of the Process is so large comparedto its deterministic variability that the hyd.ologist is justified in treating the prc-

cess as purely random. As such, the value of one observation ol the process is nor

coneLated $ith the values of adjacent observations, and the statistical propenies

of all observations are the sam9.When tbere is no corelation between adjacent observations, the output

of a hydrologic system is trcated as stochastic, space-independent, and time-

independent in the classification scheme shown in Fig. I 4.1 This type of reat-

ment is appropriate for observations of extreme hydrologic events, such as floods

or droughts, and for hydrologic data averaged over long time inte ais, such as

annual precipitation. This chapter describes hydrologic data from pure mndomprocesses using statistical parameters and functions. Statistical methods are based

on mathematical principles that describe the random variation of a set of observa-tions of a process, and they focus attention on the obse ations themselves rather

than on the physical processes which produced them. Statistics is a science of

description, not causalitY.

11.I PROBABILISTIC TREATMENT OFHYDROLOGIC DATA

A random variable X is a variable described by a probabiLity distibution. T-be

distribution specifies the chance that an observation )i ofthe variable will fall in

351

a specified range of X. For example, if X is annual precipitation at a specifiedlocation, then the probability distribution of X specifies the chance that the

observed annual precipitation in a given year will iie in a defined range, such as

less than 30 in. or 30 in-40 in, and so on.A set of observations ri, -r2, , xn of the random variable is called

a sample. It is assumed that samples are drawn from a hyPathetical infinitepopulation possessing constant statistical properties, while the properties of a

sample may vary from one sample to another. The set of all posiblc, samplesthat could be drawn from the population is called rhe sample space, an$at eventis a subset of the sample space (Fig. l l . l . l). For example. the sarfiple spacefor annual precipitation is theoretically the range from zero to positive infinity(though the practical lower and upper limits are closer than this), and an event Amight be the occurrence of annual precipitation less than some amount, such as

30 in.T\e prob(.bilit!- of an event, P(A), is the chance that it will occur when

an obseNation of the random variable is made. kobabilities of events can beestimated. If a sample of r observations has z4 values in the rarge of event A,then the relative frequencl of A is naln. As the sample size is incrcased, therelative frequency becomes a progressively better estimate of the probability ofthe event. thar is.

P(A) = lim nA ( l | 1 . r )

Such probabilities are called objectbe or posterior probabilities because theydepend completely on observations of the random variable. RoPle are accustomedto estimating the chance that a future event will occur based on their judgment

and experience. Such estimates are called subjective or prior probabilities.The probabilities of events obey certain principles:

1, Total probability. If the sample space C) is completely di vided into z nonover-lapping areas or events A1, 42, . . . , A^, fher'

P \ A t ) , P \ A + . . . + P ( A 4 t = P t O ) = I ( r 1 . 1 . 2 )

2. Complementarity. It follows that if t is the complement of A, that is' A=,0 - A, then

P ( A ) = 1 - P 1 o , ( r l . r . J )

Sample spaceO

FIGURE 11.1.1Events A and B are subsets of the sample space fl-

352 ,rrrrrro uvoaorocr

3. ContlitionaL proba&llil"". Suppose there are two events A and B as shown inFig. l l . l . l . Event A might be the event that this year's precipitation is lessthan .10 in, while I might be the evenl that next year's precipitation willbe less than 40 jn. Their over)ap is A n B, the event that A and B bothoccur. two successive yea$ with annual precipitation less than 40 in/year.lf P(B A) is the conditional probabiLity that B will occur given that A hasalready occurred, then the joiD, probability that A and B will both occur,P (A n B). is the product of P(B A) and the probability that A will occur, thatis, P(A o B) = e(AlA)P(A), or

I P 6 n B \P(B A) =

p6.) ( 1 r . 1 . 4 )

If the occurrence of B does not depend on the occurrence of A, the eventsare said to be independent, and P(B A) - P(B). For independent events, from( r l . r .4 ) ,

P G N B \ - P ( A ) P ( B ) ( l 1 . 1 . 5 )

If, for tbe example cited earlier, the precipitation events are independent fromyear to _rear, then the probability that precipitation is less than 40 in in t\ osuccessive years is simply the square of the probabil ity that annual precipitationin an1 one 1e" r rn i l l be les ' than 40 in .

The notion of independent events or obseryations is critical to the couectstatisticai interpretation of hydrologic data sequences, because if the data areindependent they can be analyzed without regard to their order of occurrence. Ifsuccessivc observations are cofielated (not independent), the statistical methodsrequired are more complicated because the joint probability P(A n 8) of succes-sive events is not equal to P(A)P(B).

Elample l l. l .1. The values of annual precipitation in College Statjon, Texas,from 1911 to 1979 are shown in Table i i.1.1 and plotted as a time series in Fig.I I .I 2(d). what is the probability that the annual precipilation R in any year willbc Less than 35 in? Greater than 45 in? Between 35 and 45 in?

TABLE I1 .1 .1Annual Precipitation in College Station, Texas, 1911-1979 (in)

Year 1910 1920 1930 1940 1950 1960 1970

39.9l i . 012.342.1. + l . l28 . lr 6 . 83 1 . l

48.1M . l42.848.434.2

46.438 .9

50 .6

44.834.045.63'�7 .343.'�741 .84 i . l

35 .23 5 . l

49.344.241.130.853 .634 .550 .343 .82t.641 .1

21 .O3'�1 .046.826.925.423.056 .543.44 1 . 3

46.044.337 .829.63 5 . 119.136.632.56t.141 .4

33 .93t. '73 l . 559 .650.538 .6

28.112 .05 1 . 8

uYonOrOCrCSr,rrlSrrCS 353

Solution. There are n = 79 l l + I =69data. Le!A be the eventR< 35.0 in. Bthe event R > 45.0 in. The numbers of values in Table I L1.I fal l ing in these rangesarc nA :23 and np: 19, so P(A) - 23169: 0.333 and P(B) = 19169 - O.rrt .From Eq, (1 | .1 .3), the probabil i ty that the annual precipitat ion is between 35 and45 in can no$ be calculated

P(35 0 < R < ,' o'"_:l _ ;::;:'�;:;,

"'^ ""]

I: 0392 I t

nExample 11.1.2. Assuming that annual precipitarion in College Statidn is an inde-pendent process, calculate the probability that there Iill be two successive years ofprecipitation less than 35.0 in. Compare this estimated probability with the relativefrequency of this event in the data set from l9l I to 1979 (Table I l . l . l ) .

Solution. Let C be the event that R < 35.0 in for two successive years. FromExample 1l. l . l , P(n < 35.0 in) : 9.33r, and assuming independent annualprecipitatjon,

P(C)= lP(R < 35.0 in)l '

: (0.333)' �

: 0 1 1 1

From the data set. there are 9 pairs of successive years of precipitation less than35.0 in out of 68 possible such pairs, so from a direct count i t would be estimated

1t)

65

60

20

1 5

t 0

5

I

\ , 1 , ir\

' t J \ l V\ hv 1

Y y I r \ lI

r9t I 20 30 40 50 60Year

(r) Annual precipitat ion.

'70

60

55-50

€ t '

! : s3 : ov0. 25

50

a 4 5; a o9

+ . "i z s

20

l 5

l 0

0 2 4 6 8 t 0 1 2 t 4 1 6Number of data

(r) Frequency hislogram.

FIGURE 11. T.2Annual precipilation in College Stalion, Texas, 1911 1979. The frequency hjstogram is form€d byadding up the number of observed precipitation values falling in each interval.

354 eprrrrr,rrorot-ocr

rhatpic) = nsln=9168=O.L32, approximately rhe value found above bv assu|ninernoepenoence.

Probabil it ies estimated from sample data, as in Examples I L l.I and 11.1.2,

are approximate, because they depend on the specific values of the observations ina sample of iimited size. An altemative approach is to fit a probability distributionfunction to the data and then to determine the probabilities of events tiom thisdistribution function.

11.2 FREQUENCY AND PROBABILITY FUNCTIONS

If the observations in a sampie are identically disributed (each sample vaiue

drawn from the same probability distribution), they can be ananged to form a

frequency hittoplram. First, the feasible range of the random vadable is divided

into discrcte intervals, then the number of observations fall ing into each interval rscounted. and finally tbe rcsult is plotted as a bar graph, as shown in Fig. I1.1.2(b)for annual precipitation in College Station. The width Ar of the interval used insetting up the frequency histogmm is chosen to be as small as possibie while stillhaving suiTicient observations falling into each interval for the histogram to havea reasonabl). ' smooth variation over the range of the data.

If the number of observations ni in interyal l, covering the rangc

[r, - A.r. -r;], is divided by the total number of observations i?, the resu]t is calledthe relat,t, lrequettct function f,k\:

JJxi) = -n

( 1 1 . 2 . i )

which, as in Eq. (l l . l . i), is an estimate of P(.r1- A-t = X = x), the probabil ity

that the rardom variable X will lie in the interval fxi - Ar, -rl. The subscript sindicates that the function is calculated from sample data.

The sum of the values of the relative frequencies up to a given point is thec umul,.tt iv e f| eq uency function F'(x):

l = l

This is an estimate of P(X = x), the cumulaxive probabiLi4^ of x1.The relative frequency and cumulative frequency functions are defined for

a sample, conesponding functions for the population are approached as limits asr ---':c :rnd Ar --) 0. In the limit, the relative frequency function divided by theinteryai length trr becomes the probabilitl density function f(x):

f ( - \

f tr t : l im J-3'

"-.d A-x(. t r .2.3)

The cunrulati\re frequency function becomes the prcbabitity distribution functionF(x).

(1r.2.2)

355

( l r . 2 . 5 )

For a given value of .r, F(-{) is the cumulative probabiliry P(X < r). 4)d it can beexpressed as the integral of the probability density function over thb r{hge X < x:

p (x1)= P(x1 -Ax_ � X< x i )

F(r) = ii 'o P'tt'

t:;whose derivative is the probability density funcrion

: F ( r i ) - F ( . r ; - A r )

= F(x;) - F(xi-1)

dF(x)

dr

;r) = F(-r) - f ;r.,ta,

f (.x) dr

( 1 1 . 2 . 4 )

( r 1 . 2 .1 )

( r l . 2 .6 )

wherc a is a dummy variable of integration.From the point of view of fitting sample dara to a theoretical disrriburion,

the four functions - relative frequency /,(r) and cumulative frequency Fdx) forthe sample, and probability distribution F(x) and probability density /(.r) for thepopulation-may be arranged in a cycle as shown in Fig. I 1.2. 1� Beginning in theupper left panel, (a), the relative frequency fundion is computed ftom the sampledata divided into inte als, and accumulated to form the cumulative frequencyfunction shown at the lower left, (b). The probability disrdburion function, ar rhelower right, (c), is the theoretical limit of the cumulative frequency function asthe sample size becomes infinitely large and the data interval infinitely small.The probability density function, at the upper right, (d), is the value of the slopeof the distdbution function for a specified value of r. The cycle may be closedby computing a theoretical value of the relative frequency function, called theincremental probabilitv function:

r'= | f(x) dx

f', f', -4.

= | rata'- |

The match between p(.ri) and the observed relative frequency function /d.ri) foreach xi can be used as a measure of the degree of fit of the distribution to thedata.

The relative frequency, cumulative frequency, and probability distrjbutionfunctions are all dimensionless functions varying over the range [0,1]. However,since dF(r) is dimensionless and dx has the dimensions of X, the probabilitydensity function/(i): dF(x)/dx has dimensions [X]-r and varies over the range[0,.o]. The relationship dF(x) = f{x) dx can be des$ibed by saying that /(r)

la) Relotir? rt'?quenq) function

\h) ( rnuta rc f teque\.y lunc oa

356 epnr-rrogYutror-ocv svororccrcsrmrsrrcs 357

- _ x - l to

( 1 1 . 2 . 9 )

The coresDondins standsrd normal distibution has orobabilitv densitv function

where u is a dummy variable of integration, has no analytical form. Its values aretabulated in Table l l.2.l, and these values may be approximated by the followingpolynomial (Abramowitz and Stegun, 1965):

IB = ; t r + 0. l9685alzl + 0.n5r9alzl2 + 0.0003441e 3 + 0.0t9527)zl4l-4

' ( 1 1 . 2 . 1 2 a )

where Jzl is the absolute value of z and the standard normal distribution has

Sample Populat ion

l ( z ) ! _ e , . _ : . < z < c (- 1 -

,.1 zIf

( 1 1 . 2 . 1 0 )

which depends only on the value of e and is plotted in Fig. I L2.4. fue standardnormal probability distribution function P

rut = [ -L ,-,',, our_@ \lz;(1r.2.r1)

.;

E

F ( z ) : B f o r z { 0

= l - B f o r z > 0

(rr .2. t2b)

( t t . 2 . I zc )

The erlor in F(z) as evaluated by this formula is less than 0.00025.

Example 11.2.1. What is the probabiljty that the standard nomal random variablez wil l be less than -2? Less than I? What is P(-2 < z < 1) ' !

FIGURE 1I.2.1Frequency funclions from sample data and probability functions from ihe populalion.

rcpresenrs the "density" or "concentration" of probability in the inte al [-r, t +

d\1.One of the best-known probability density functions is that forming the

familiar bell-shaped curye for the nomal distribution:

I im 4(. r ' ) = F(. r , )

t I t" ir)'�]l(xj= -exp - ----:-----

,Ti, | 20' )

0 .5

0.4

0 .3

0 .2

0 . 1

0

l -

( r r .2 .8)

where trr and o are parameters. This function can be simplified by defining thestandard normaL variable z as

FIGURE II.2,2The probability density function for the standard normal distribution (p: 0, o = l).

3.0

358 .rpt,L.reosvonorocv

TABLE II.2.1Cumulative probability of the standard normal distribution

0.0 0.5000 0.50,10 0.5080 0.5120 0.5160 0.5199 a.5239 0.5219 0.5319 0.53590.1 0.5398 0.5,138 0.5478 0.5517 0.555? 0.5596 0.5636 0.5675 0.5714 0.57530.2 0.s?93 0.s832 0.s871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.61410.3 0.6179 0.62r? 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.65170.4 0 6554 0.659r 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879

0.5 0 6915 0.6950 0.6985 0.?019 0.7054 0.7088 0.1123 0.1151 4.1190 0. ' �72240.6 0.1257 0.729h 0. ' �7324 0. ' �7157 0.1389 0.1422 0.1454 0.1486 0.1511 0.15190.7 0.7580 0.7611 0.1642 0. '7673 0. ' �7704 4.1'134 0.1' �764 0.7794 0.7823 0.?8520.E 0.1831 0.7910 0. ' �7939 0. ' �7961 0.7995 0.8023 0.8051 0.8078 0.8106 0.81330 .9 0 .8 r59 0 .8186 0 .8212 0 .8238 0 .8264 0 .8289 0 .8315 0 .8340 0 .8365 0 .8389

1 .0 0 .3113 0 .6 ,138 0 .8461 0 .8485 0 .8508 0 .8531 0 .8554 0 .8577 0 .8599 0 .8621l. l 0 Eb43 0.3665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.88301.2 0.8819 0.8869 0.88E8 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.90151.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.911' l1.4 0.9192 A.92A1 0.9222 A.9236 0.9251 0.9265 0.92'�79 0.9292 0.9306 0.9319

1.5 0.9132 0.93215 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.94411.6 0 94i2 0.9463 0.94'�74 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.95451.1 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.96331.8 0 9611 0.9649 0.9656 0.9664 0.967r 0.9678 0.9686 0.9693 0.9699 0.97061.9 0 9?11 0.9119 0.9'726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767

2,0 09tr2 0.9178 0.9783 0.9788 0.9193 0.9798 0.9803 0.9808 0.9812 0.98172.1 0.9S2t 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.98572.2 0.9861 0.9864 0.9868 0.98?l 0.9875 0.9878 0.9881 0.9884 0.9887 0.98m2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.99162.4 099r8 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936

2.5 0.9938 0.9940 0.9941 0.9943 0.9945 A.9946 0.9948 0.9949 0.9951 0.99522.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.99642-7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.99'�71 0.99'�72 0.9973 0.99142.E 0.9974 0.997s O.9976 0.99-1'�7 0.9977 0.9918 0.99'�79 0.99'�79 0.9980 0.99812.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986

3.0 0.9987 0.9987 0.998? 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.99903.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.99933.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.999s3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.99973.4 0 9997 0.9997 0.9991 0.999"t 0.9997 0.9991 0.9991 0.9991 0.999? 0.9998

Souce: Granr- E L,lndR.S. I'avenwonh. Stdrir,;calQuoLiry and Contrcl, Table A, p.&3, Mccmw-Hi l l . Ne* lbrk, l9?2 Used * i th pemissjon.

IB = ; t l + 0 . 1 9 6 8 s 4 x 2 + o . n s t g 4 x e ) 2

+ 0.000344 x (2)3 + 0.019527 x (2)11 I

= 0.023

Ffom ( l L2 . l2 r ) . F \ -2 ) - B - 0 .023.P{Z = l) = F(l), and from (11.2 12a)

IB =

rtl + 0. 196854 x l + 0.1 15194 x (1) ' �

+ 0.000344 x (l)3 + 0.01952? x (1)41 I

= 0 . 159

F r o m ( 1 1 . 2 . 1 2 c ) , F ( l ) : | - B : I - 0 . 1 5 9 = 0 . 8 4 1 .Finally,

P( -2 < Z < 1)=r (1 ) - F ( -2 )

:0 .841 0 .023

= 0 . 8 1 8 .

11.3 STATISTICAL PARAMETERS

The objective of statistics is to exhact the essential information ffom a set ofdata, reducing a large set of numbers to a small set of numbe$. Statistics atenumbers calculated from a sample which summarize its important characteristics.Statistical parameterJ are chamcteristics of a population, such as ,u and o in Eq.( 1 1 . 2 . 8 ) .

A statistical parameter is the expected value E of some function of a randomvariable. A simple parameter is the ruean /r, the expected value of the rarldomvariable itself. For a random variable X, the mean is EQf, calculated as theproduct of r ard the corresponding probability density /("r), integrated over thefeasible range of the random variable:

Solution. P(.2= 2) = F(.-2), and from Eq. (11.2.12a) with lrl = -21 :2,

E(t) : t t=f-rrur* (1 r . 3 . 1 )

.09.08.03. 01.00

359

, lI t

t1

ffi- E

*. s.FI;I

itti

:

:t

i

i

Iil

iiI

E()O is the first moment about the origin of the random variable, a measure ofthe midpoint or "central tendency" of the distribution.

The sample estimate of the mean is the average i of the sample data:

l \ -

Table 11.3.1 summarizes formulas for some population palameters and theirsample statistics.

( I l � 3 . 2 )

To employ the tabl€

4 ( z ) = I F , ( ? )where 4(13 ) is $e

(360 ,rpeueo uronoroov

TABLE 11.3.IFopulation param€ters and sample statistics

Population parameter Sample statistic

Arthnielic mean

s : E ( X ) :

Me dian

-\ such $ai F(r) : 0.5

Gcometr ic mcan

ant i log [E( iog x) ]

2. Varidbilia^

o1 = El( t - t t l ' l

S landard de\rat ion

o = lEl\r p)rl j v2

Coeilicient of variation

o

11

3. S\nnrct\

Coefficient of skewness

t[(n ,4)]lY = d (n l)(n - 2)r l

Tne variability of data is measured by tlte variance o2, which is the secondmomenl about tne mean:

| , f v td t

50!h percentile value of dala

I T _ T Il t t x , I

, I \ -n t - .

"=[*r'" -"']J

;

rEl r t - 1 t t ' l = q ' =

) , ,1x - y t ' [ (x \d - r ( 1 I . 3 . 3 )

.a

(

The sample estimate of the variance is given by

svonoroctcsmrlsrrcs 361

( 1 t . 3 . 4 )r -I-

s- = _______: ) txi_x)-

in which the divisor is z - I rather than rx to ensure that the sample statistic isunbiased, that is, not having a tendency, on average, to be higher or lower thanthe true value. The variance has dimensions ln2. The ltandard deigtion o is ameasure of variability having the same dimensions as X. The qurint$r o is thesquare root of the variance, and is estimated by s. The significance of 6e standarddeviation is illustrated in Fig. ll.3.l(a); the larger the standad deviation, thelarger is the spread of the data. T\e coeficient of variation CV = crl p, estimatedby r/i, is a dimensionless measure of variability.

T1\e symmetry of a distribution about the mean is measured by the s/<ewnesswhich is the third moment about the mean:

El(, - tD3l= J*" - p)3-f6) dx ( l l . 3 . s )

The skewness is normally made dimensionless by dividing (11.3.5) by o'to givethe coeficient of skewness y'.

v : \ z l ( , - r " ) ' l(f'

A sample estimate for 7 is given b1.

( 1 1 . 3 . 6 )

( l r . 3 . 7 )( n - l ) ( n - 2 ) s 3

(a) Slandard deviation o.

FIGURE I1.3.1The effect on tbe probabiliry density function of changesof skewness.

(r) Coefficient of skewness cJ.

in the standard deviation and coefficient

362 rplrrm nvonorocl

OI

. : (

Column: IIear

, l , \ , ,, , 1 F * , l - 3 , 1 t . l l F , , l r z l I , , l

\ i ' \ , - ' l \ ' r / , - t I

nln lxn - 2)-rr

(t - t)1

( l 1 . 3 . 8 )

As shown in Fig. 11.3.1(b), for positive skewness (7 > 0), the data areskewed to the right, with only a small number of very large values: for negativeskewncss (y < 0), the data are skewed 10 the left. If the data have a pronouncedskewness. the small number of extreme values exert a significant effect on thearithmetic mean calculated by Eq. (11.3.2), and alternative measures of centraltendenc) xle approprjate, such as the median or geometric mean as listed in Table . 3 . 1 .

Example 11.3.1. Calculate the sample mcan, sample standard deviation. and sam-plc coefficienl of skewness of the data for annual precipitalion in College Station,Teras. fron 19'7A ro 19'79. The data are given in Table 11.1.1.

Solution. The values of annual precipitalion from 1970 to 1979 are copied incolumn 2 of Table I ].3.2. Using Eq. (l L3.2) the mean is

r +

401.7= I o

= 40 .17 ln

Thc squares of the deviations from the mean are shown in column 3 of the table,

TABLE 11.3.2Calculation of sample statistics for College Stationannual precipitation, 1970-1979 (in) (Example 11.3.1).

Pr€cipitation r (r-;)2

197019 t I)912191:�l19141975r97619111978t979

Total

33 .93 t . 13 r . 559 .650 .538 .643.428.732.05 1 . 8

40t.1

39 .3' 7 r . 7

15 .2317 .5106 .7

2 .510 .4

r 3 t . 666 .7

t35 .3

1 0 1 6 . 9

246.5-401.6-.65t.1' t335.3

I102 .3-3 .933.7

- 1s09.0-545.31573 .0

6,180.3

to ta l ing 1016.9 jn2 . From (11 .3 .4 )

syonorocrcsrerrstrcs 363

? | \ - .

1016.99

: 113.0 in2 l ldThe srandard de\ ial ion is

. r : ( l l 3 . o ) r / :

= 10 .63 inThe cubes of the deviation from the mean are shown in column 4 of Table 11.3.2,totaling 6480.3. From (11.3.7)

( n - l ) ( n - 2 ) s 3

l0 x 6480.3

9 x 8 x ( 1 0 . 6 3 ) 3= O.149

11.4 FITTING A PROBABILITYDISTRIBUTION

A probability disribution is a function representjng rhe probability of occurrenceof a random variable. By fitting a distdbution to a set of hydrologic dara, a greardeal of the probabilistic information in the sample can be compactly summarizedin the function and its associated parameters. Fitting distributions car be accom-plished by the method of moments or the merhod of maximum liketihood.

Method of Moments

The method of moments was first developed by Karl harson in 1902. He consid-ered that good estimates of the parameters of a probability distribution are thosefor which moments of the probability density function about the origin are equalto the conesponding moments of the sample data. As shown in Fig. 11.4.1, ifthe data values are each assigned a hypothetical ,,mass', equal to their relativefrequency of occurrence (l/n) and it is imagined that this system of masses isrotated about the origin .r = 0, then the first moment of each obsenation 'rr; aboutthe origin is the product of its moment arm r; &nd its mass 1/n. and the sum ofthese moments over all the data is

n',-.\- ri

:-. n

364 lrrlno Hvorolocr

..d--.&t5t.i,l.

i

nvorolocrcsrensrrcs 365

probability density function is /(r) : Ie e for .x > 0. Determine rhe relationshipbetween the parameter i and the first moment about the origin, p.

Solut ion. Ustng Eq. ( 11.4.1),

r . t : E(r) : f ,oo*

t : flfr,,t

_ l ' -- J n

I

l"

F

'Mo inent a rm

(b) Sample data.

FIGURE I I . { .1The nretlod of nloments selecG raiues for the paramelers of lhe probability densily function so lhat

its nroments are equal to those of lhe sarnple da!a.

the samFle mean. This is equivalent to the centroid of a body. The correspondingcentroid ol the probability density function is

u: f , -x f {x)ax ( 1 r . 4 . 1 )

Likervise, the second and third moments of the probability distribution canbe set equal to their sample values to determine the values of parameters of theprobabilig distribution. karson originally considered only moments about theorigin. but later it became customary to use the variance as the second cerll'o1moment. d = E[G- p)?], and the coefficient of skewness as the standardized thirdcentral Eroment. t = El(x -,u)l]/d, to determine second and tl ird parametersof the distribuiion if required.

Example 11.4.1. The exponential distribution can be used to describe variouskincls of hldrologic data, such as the interaffival tines of rainfa]l events. Its

which may be iniegrated by parts to yield

ln this case I = 1//-., and the sample estimate for l is l/;.As a matter of intercst, it can be seen that the exponentiai probability density

function/(-r) : ,tre-a and the impulse rcsponse function for a linear reseNoir (seeEx.'1.2,1) u(l) = ( l lk\e-ttk are identical i fr = l and i = 1/t. ln this sense,the exponential distribution can be thought of as describing the probability of the"holding time" of water in a liDear reservoir.

Method of Maximum Likelihood

The method of maximum likelihood was developed by R. A. Fisirer (1922). Hereasoned that the best value of a parameter of a probability distribution should bethat value which maximizes the likelihood orjoint probability of occurrence of theobserved sample. Suppose that the sample space is divided into intervals of lengthdr and that a sample of independent and identically distributed observations x r,x2, , i, is taken. The value of the probability density for X : ;r; is /(-ti),and the probability that the rcndom variable will occur in the interval including.r1 is fl,r1) dr. Since the observations are independent, their joint probability ofoccurrence is given from Eq. ( 1 1. 1.5) as the product/(x 1) dr f (x i) dx . . . f lx ) dx: tn l= r "f(;rr)l dx", and since the interval size dr is fixed, maximizing the jointprobability of the observed sample is equivalent to maximizing the likelihoodfunction

(d) Probabiliiy density function.

i t . t . . l+

t = L *r, (firsr momenL aboui fie orisin)

t i l i l t t l lt i l i l t t t ll l l l l l l l l

Because many probability density functions are exponential, it is sometimes moreconvenient to work with the los-likelihood function

(rr.4.2)

yn l : l r n y i . r1 ) l

" 1, , ' r , ' i r r . r l omer rbour I ' re a ! r/ 2

( l l . 4 . 3 )

366 ^ri , l i i 'o srorrorocr

Example 11.,1.2. The fol lowing data are the obsened t imes belween rainfal l eventsat ir given location. Assuming that the interarr ival t ime of rainfal l events fol lows anexponenrial disiribution, determine the parameter i for this process by the methodof maximum l ikel ihood. The t imes between rajnfal ls (days) atet 2.40, 4.25. 0. '7'1,1 3 . 3 1 , 3 . 5 5 . a n d L 3 7 .

Solut ion, For a given value x;, the exponential probabil i ty density is

f(r ' ) = i"-* '

so. from Eq. (11,.4.3), the logJikel ihood iunction is

l n r = t i n l f ( . r , r t

= ) tn 1,1"-^,1

s - .- ' ( l n l , d r , )

= n t n , f - I ) ; ,

nraxinrum value of ln I occurs when { ln L)/d,I , = 0; that js, when

This is the same sample estimator for l as was produced by the method of moments.ln ihis case. i = (2.40 + 4.25 + 0. '7'7 + 13.22 + 3.55 + 1.3' l) /6 : 25.5616 =

4.28 da,vs. so i = 114.28 : 0.234 day-r. Note that d2(ln L)ldl2: .2, whichis negative as required for a maximum.

The \alue of the logJikelihood function can be calculated for any value ofi . For example, for t : 0.234 day-r, the value of the logl ikel ihood function is

l n L = n l n l - 1 i , .

:6 ln (0 .234; 0 .234 x 2s .56

rYonolocrcsrersrrcs 367

- 1 4 . 6

1 4 . 8

- 1 5 . 0

15.2

- 15.4

, 1 5 . 6

- 1 5 . 8

1 6 . 0

t 6 . 2

- t 6 . 4

o.r0 0.20 0.30 0.40l" (days ,)

FIGURE II.4.2The loglikelihood fitnction for an exponential distribution (Example 1l.4.2).

Figure 11.4.2 shows the variat ion of the log-l ikel ihood function with i , wit i themaximum value at t :0.234 day-r as was detemined analyt ical ly.

The method of maximum likelihood is the most theoretically correct methodof fitting probability distributions to data in the sense that it produces the most

fficient parameter estimates-those which estimate the population parameterswith the least average ellor. But, for some probability distributions, there is noanalytical solution for all the parameters in terms of sample statistics, and thelog-likelihood function must then be numerically maximized, which may be quitedifficult. In general, the method of moments is easier to apply than the methodof maximum likelihood and is more suitable for practical hydrologic analysis.

Testing the Goodness of Fit

The goodness of fit of a probability dist bution can be tested by comparingthe theoretical and sample values of the relative frequency or the cumulativefrequency function. In the case of the relative frequency functlon, the X2 lestis used. The sample value of the relative frcquency of interval I is, from Eq.(l1.2.1), i(xi) = n1/n; the theoretical value from (11.2.7) is p(x) = F(x) -F(-r; 1). The f test statistic yl is given by

y: $ nff"(ri) - p(xi)12

?t P(xi)(1r .4.4)

J ' l! {

a

d\ ln L ) n ia i A L

l l -

I

t

where z is the number of intervals. It may be noted that nfdx) = n1, theobserved number of occurrences in interval i, and np(x) is the correspondingexpected number of occu[ences in interval i; so the calculation of Eq. (11.4.4)is a matter of squaring the difference between the observed and expected numbers

r n r = n r n r . - l ! - . ,= 6 l n l _ 2 5 . 6 1 ,

: -14 .10

368 eprueo urororocv

of occunences. djviding by the expected number of occurrences in the interval.and summing thc result-over ali intervals-

To describe the f test, the / probability distribution must be defined. A

f distribution with v de gree s of freedom is the distribution for the sum of squaresof / independent standard normal random variables ai; this sum is the randomvariable

r \ - , ( 1 1 . 4 . 5 )

The f distr ibutiof function is tabulated in many stat ist ics texts (e.g., Haan.

1977). In the / test, v: m- p- I , where f l is the number of intervals as before,

and p is the number of pammeters used in fitting the proposed distribution. A

confdence level is chosen for the test; it is often expressed as I - .v, where

a is tcrnrcd the signifcance leveL. A typical value for the confidence lcvel is

95 percent. The null hypothesis for the test is that the proposed probability

distribution fits the data adequately. This hypothesis is rejected (i.e.. the fit isdeemed inaciequate) i f the value of X: in ( l l �4.4) is larger than a l imit ing value,

f, . ,-". determined from the;2 distdbution with r degrees of freedom as thevalue haring cumuiative probabil i ty I - d.

Exampl€ 11.4.3. Using the meftod of moments, fit the normai distribution to theannual precipitat ion at College Station, Texas, from l91l to 1979 (Table 1l. l . l ) .Plot the rclative frequency and incremental probability functions, and the cumulativefrequcncy zrnd cumulative probability functions. Use the 12 test to detennine whetherthe normal distribution adequately fits the data.

Solrtion. The range for precipitation R is divided into ten intervais. The first rntenalis R < 20 in, the last is lR > 60 in, and the intermediate intervals each cover arange of5 in. By scanning Table 11.1.1 the frequency histogram is cornpi ied, asshown in column 2 of Table 11.4.1. The relat ive frequercy function/,(,r ,) (column3) i s ca l cu la l ed by Eq . ( 11 .2 .1 ) w i t h n :69 . Fo r examp le , f o r i : 4 ( 30 -35 i n ) ,n, - 1,1, and

1 4

69

= 0 .203

The cumulative frequency function (column 4) is found by su ming up the relatrvefrcqucncies as in Eq. ( I 1 .2.2). For i = 4

F,(,ra): >/,(-rj)

= F,(.{1) + /,(-r4)

= 0 .130 + 0 .203

svonolocrcsr.qrrsrrcs 369

TABLE 11.4.1[itting a normal distribution to annual DreciDitation at ColleseStation, Texas. l9lI-1979 (Example 1L4.3).'

Column:

Interval Range(in) nt fdtl F"{x) F(xt Ati)

I23

56,7

89

l0

Total

< m l20-25 225-30 630-35 1435-40 I I4t)-45 1645,50 l050-5s 555-60 3>60 l

69

0.014 0 .0 t40.029 0.0430.08? 0.1300.203 0.3330.159 0 .4930.232 0.7250.145 0.8700.072 0.9420.043 0.9860.014 1.000

1.000

- 2 .157 0 .0 i5- L61 t 0 .053- l .065 0.144-0.520 0.301

0.026 0.5r00 .571 0 .7161 .117 0 .868t.662 0.9522.208 0.9862.753 L000

0.0r 5 0,0040.038 0.rr47r0.090 0.00*10 .158 0 .8910.209 0.8050.206 0.2220 .15 t 0 .0190 .084 0 .1140.034 0. t630.014 0.004

|.un 2.37'�7

Mean 39.77Slandarddeviarion 9.17

:0 .333

It may be noted that this is P(X s 35.0 in) as used in Example I l . l . I .To fit the normal distribution function, the sample statistics t : 39."j'l in and

r = 9.17 in are calculated for the data from lgll to 1979 in the manner shown inExample l l .3. l , and used as estimares for 4 and c. The standard no.mal va atez corresponditg to the upper limit of each of the data intervals is calculared by(l1.2.9) and shown in column 5 of the table. For example, for i = 4,

- = '--s

_ 35 .0 - 39 ."179 . 1 7

: -0.520

The corresponding value of the cumulative normal probability function is given by(11.2.12) or Table 11.2.1 as 0.301, as l isted in column 6 of Table 11.4.1. Theincremental probability function is computed by (11.2.7). For i = 4,P-'-l*lli?'-'

: 0 . 1 5 8

and similarly computed values for the otber intervals are shown in column 7.The relative frequency funcrions/"(.r, andp(r, from Table 11.4.I are plotted

in Fig. 11.4.3(a), and the cumulative frequency and probabil i ty distnbution func-tions I'r(ri) and F(.r) in Fig. 1 1.4.3(D). From the simiiadry of the rwo funcrions

0 2.1

0.20

a

a

.a=E-

370 qr-r, i .rouronoroor

0 .90 .8

0 7iJ.6

0 . t -0 . 0

<20 15 30 35 40 45 50 55 60 >65Annual precipitation lin)

E sL rmp lc , / . ( r , ) n Fn rcd . r ( r , ]

( /) Relal ive ircqucncy iunction.

Normaldist r ibut ionw i l h g = 3 9 7 7 , o = 9 l 7

0.00

0 .5 -

u..1 l

0 . t -0 . 2 l

20 :5 30 .15 40 45 50 55 60 >6s

Annual precipital ion ( in)

. Slnlplc. 4(J') - Fiued.F(.r i)(, ) CLrmularive frequency funcrion

FIGURE I1.4.3Frequenc\ functions for a normal distribution fitted to annuai precipitation in College Station, Texas(Examp lc l l 4 .3 ) .

shown in each plot, it is apparent that the normal distrjbution fits these annualprccipitat ion data very well .

To check the goodness of f i t , the 12 test stat ist ic is calculated by (11.4.4).F o r i = 4 ,

nv,@i - p\il'� '�69 x(0.20290 0.ts'7'/'1)2P\r4) 0. ts777

= 0 . 8 9 1

as shown in column 8 of Table I 1.4.1. The total of the values in column 8 is xl =2 .377 .Theva lueo fx2 , , - " f o racumu la t i vep robab j t i t yo f l - a=0 .g5anddeg reeso f

nyonorccrcsr.qrrsrrcs 371

f r cedom /= m-p - I : l 0 -2 - 1 :7 i s 1 l . o " r : l 4 . l (Ab ramow i t z and S regun ,1965r. Sjnce rhis \alue rs prearer (han qj, ine nutf hyporhesis lrhe dislr ibul ionfits the data) cannol be rcjected at the 95 percent confidence level: the fit of thenomal distribution to the College Station annual precipitation data is accepted. Ifthe distribution had fitred poorly, rhe values of /r(.ri) and p(.ri) would have beenquite different from one another, resulting in a value of X: larger rhan 14.1, inwhich case the null hypothesis would have been reiected.

rT.5 PROBABILITY DISTRIBUTIONS FORIIYDROLOGIC VARIABLES

' |l l ,

i*

In Sec. 11.4, the normal distribution was used ro describe annual precipitation atCollege Station, Texas. Although this disbibution fits this set of dita particuladywell, observations of other hydrologic variables follow different distributions. Inthis section, a selection of probability distributions cornmonly used for hydrologicvadables is presented, and examples of the types of variables to which thesedistxibutions have been applied are given. Table 11.5.1 summarizes, for eachdistributior, the probability density function and the range of the variable, andgives equations for estimating the distribution's parameters from sample moments.

Normal Distribution

The normal distribution arises from the cental limit theorem, which states tbat ifa sequence of random variables Xi are independently and identically distributedwith mean g, and variance o2, then the dishibution of the sum of a such randomvariables, l^: ) i= 1 X;, tends towards the normal distribution with mean rp andvartance na'as I becomes large, The important point is that this is true no marterwhat the probability disbibution function ofX is. So, for example, rhe probabilitydisftibution of the sample mean i = l ln2 i=1x1can be apDroximated as normalwith mean g, and variance (1ln)2no2 : dln no run.r *hai rhe distrjbution of ,ris. Hydrologic variables, such as annual precipitation, calculated as the sum ofthe effects of many independent events tend to follow the normal distuibution. Themain limitations of the normal distribution for describing hydrologic variables arethat it varies over a continuous range [-co, c.], while most hydrologic variablesare nonnegative, and that it is symmetric about the mean, while hydrologic datatend to be skewed.

Lognormal Distribution

If the rundom variable I : log X is normally distributed, then X is said to belognormally distributed. Chow (1954) reasoned rhat this distribrltion is aDDlicableto hydrologic variables formed as the products of other variables sinci-if X :X1X2X3 . . . & , then y = log X: ! i= , log X l = 2 i= ry1 , wh ich tends tothe normal distribution for large z provided that the X; are independent andidentically distributed. The lognormal disrribution has been found to describethe distribution of hydraulic conductivity in a porous medium (Freeze, 1975),

1I

6

I

ot' LCa- :_ Lcq - Fa a ^ > , a r l- 1

, : - d l Y ' u . r l -; l ! . * i ! - u i E \ - ' ' I

l l | [ ] t a I t l-< \ ! ,< co- vg n :

8

8l

E e

: I

| r l

I

.

r i d

t t r - li t i ts t ! t ^

T t ; i l c aG l I : c l r . ' {

r l I e51 5l l l- < t - < l :

ce e .\ \ -

F * ! ? =- u EO E E d t r6 : { l p F

9 ;A EE A

E F

g€E

g0

i rE ' -

3 ^ L

t \I \ )

- r x , r lV i r l1 , il l l

- t lt - .

l : r il t '

I

8 A

c 9

q r ( . fI

^ ^ l

al :rl ' t-t b l ^ . r d

i l ' 5 t i' l | . 3

* f ' x | l Ed ^ l ^ ;

I t s l l s e - i l g ' t o- r a l - l > [ f * , . 1 L I

6 l P . : " l Ir r l r r gx x\ \ \ \

F s

i E E

o!

; i: . -

n^'

374 ,qppirlo svororocv

the disrrlbution of raindrop sizes in a storm, and other hydrologic 1'a.iabies. Thelognormal distribution has the advantages over the nomal dist bution that it isbounded (X > 0) and that the log transformation tends to reduce the positiveskewness commonly found in hydrologic data, because taking logarithms reduceslarge nunrbers proportionately morc than it does small numbers. Some limitationsof the lognormal distribution are that it has only two paramete$ and that it requiresthe logarithms of the data to be symmetric about their mean.

Exponential Distribution

Some sequences ofthydrologic events, such as the occunence of preciprrarron,may be considered Poisson processes, in which events occur instantaneously andindependcntiy on a time horizon, or along a line. The time between such events.or intererrivaL ,r'me, is described by the exponential distribution whose parameteri is the mcan rate of occurrence of the events. The exponential distribution isused to describe the interardval times of random shocks to hydrologic systems,such as slugs of poliuted runoff entering streams as rainfall washes the pollutantsoff the land surface. The advantage of the exponential distribution is that it iseasy to cstimate ,\ from observed data and the exponential distribution lends itselfwell to theoretical studies, such as a probability model for the linear reservolr(,{ = l i l . rhere t is the stomge constant iD the l inear reservoir). Its disadvantageis that it requires the occunence of each event to be completely independent ofits neighbors, which may not be a valid assumption for the process under study-for example. the arrival of a front may generate many showers of rain-andthis has led investigators to study various forms of compound Poisson processes,in which ,\ is considered a random va able instead of a constant (Kavvas andDelleur, 1981;\\raymire and Gupta, l98l).

Gamma Distribution

The time taken for a number p of events to occur in a Poisson process is describedby the gamna distribution, which is the dist bution of a sum of B independent andidentical exponentially distributed random variables. The gamma distribution hasa smoothly varying form like the typical probability density function illustatedin Fig. 1 1.2. I and is useful for describing skewed hydrologic variables withoutthe need for log transformation. It has been applied to describe the distribution ofdepth of precipitation in storms, for example. The gamma distdbution inyolves thegamna func t io r l (B) , wh ich is g iven by f (B) = (p - 1 ) l = (B-1) (B-2) . . .3 .2 .1for positive intcger p, and in general by

nyoeor-oorcsrrrrsrrcs 375

Rarson Type III Distribution

The karson Type III distribution, also called the three-porameter gamma distri-bution, introduces a third pammeter. the lower bound €, so that by the methodof moments, three sample moments (the mean, the standard deviation, and thecoefficient of skewness) can be ransformed into the three parameters A, B, and€ of the probabil ity distribution. This is a very flexible distribution, assuming anumber of different shapes as n, B, and e vary {Bobee and Robitait le, 1977)-

The karson system of distributions rncludes seven lypes; lthey are allsolutions for /(r) in an equation of the form d

dvG)l _ f6)6-d)dx Ca + Ci + C zx2

where d is the mode of the distribution (the value of r for which/(,r) is a maximum)and C6, C1, and C2 are coefficients to be determined. When C2 = 0, the solutionol ( I 1.5.2) is a Fearson Type III distribution, having a probabil ity densjty functionof the form shown in Table 11.5.1. For C1 = (, = 0, a normal distributionis the solution of (11.5.2). Thus, the normal distribution is a special case ofthe Fearson Type III distribution, describilg a nonskewed variable. The FtanonType III distribution was first applied in hydroiogy by Foster (1924) to describethe probability distribution of annual maximum flood peals. When the data arevery positively skewed, a log transfonnation is used to reduce the skewness.

Log-Pearson Type III Distribution

If log X fbllows a Pearson Type III distribution, then X is said to follow alog-Pearson Type III distribution. This distribution is the standard distributionfor frequency analysis of annual maximum floods in the United States (Benson,i968), and its use is described in detail in Chap. 12. As a special case, u,hen logX is symmetric about its mean, the log-Pearson Type III distribution reduces tothe lognormal distribution.

The location of the bound e in the log Pearson Type III distribution dependson the skewness of the data. If the data are positively skewed, then log X - € and

TABLE IT.5.2Shape and mode location of the log-Fearson Type III distribution"r , fun"tion of itt o.

Shape parameter g I< ln l0 - ln l0< [<0 I> 0

0 < p < I

p > l

( 1 r . 5 .2 )

rG)= f u|-te-.,du ( . 5 . 1 )No modeJ shaped

Unimodal

Minimum mode No modeU-shaped Reveise J-shaped

No mode UnimodalReverse J-shaped(Abramowltz and Stegun, 1965). The two-parameter gamma distribution (parum-

eters B and ,\) has a lower bound at zero, which is a disadvantage for applicationto hydrologic variables that have a lower bound larger than zero. San e Bobee, 1975.

376 qpurl rr ololoL, v

e is a lo$er bound. wtrile if the data arc negatjvely skewed, log X < e and e is anupper bound. The log transformation reduces the skewness of the transformed dataand may produce transformed data which are negatively skewed from original datawhich are positively skewed. In that case, the application of the log-FtaISon TypeIII distribution would impose an artificial upper bound on the data. Dependingon the vaiues of the paramete$, the log-Fearson Type III distribution can assumemany different shapes, as shown in Table 11.5.2 (Bobee, l9?5).

As describcd previously, the log-Pea$on Type III distribution was devel-oped as a method of fitting a curve to data. Its use is justified by the fact rhat ithas been found to yield good results in many applications, particulariv for floodpeak data The fit of the distribution to data can be checked using the 1l rest, orby using probabil ity plotting as described in Chap. 12.

Extreme Value Distribution

Extremc ialues are selected maximum or minimum values of sets of data. Forexample, the annuai maximum discharge at a given location is the largest rccordeddischarge value during a year, and the annual maximum discharge values for eachyear of historical record make up a set of extrcme values that can be analyzedstatisticalll . Distributions of the extreme values selected from sets of samplesof any probabil ity distribution have been shown by Fisher and Tippett (1928) roconverge to one of three forms of extreme value distibutions, called Types I,II, and III. respectively, when the number of selected extreme values is large.The properties of the three limiting forms were further developed by Gumbel(1941) for rhe Extreme Value Type I (EVI) distribution, Frechet (1927) for rheExtreme Value Type II (EVII), and Weibull (1939) for the Extreme Value TypeIII (EVIN).

The three l imiting fornrs were shown by Jenkinson (1955) to be special casesof a single distribution called the General Exteme yalue (GEV) distriburion. TheDrobabil itv distribution function for the GEV is

REFERENCES

Abramowitz, M., and l. A. Sreg}n, Handbook of Mathematical Functions, Dorer, New York,p.932, t965 .

Benson, M. A., Uniform flood-frequency estimating methods fo. fedelal zgencies, Woter Resour.R?J., vol. 4, no.5, pp. 891-908, 1968.

Bobee, 8., The log-harson Type III distribution and its application in hydrology, Water Resour.Rer., vol. 11, no. 5, pp. 681 689, 1975.

Bobee, B. 8., and R. Robitaille, The use of the karson Type 3 and log ksrson Trye 3 disiributionsrc\ i . i ted. Wat?t R?sour.\Rer.. \ol. IJ. no. 2. pp. 427443. )q11. I t

Chow. V. T.. The log-probabil iay law rnd i tr engrneenng applcarions. Prot. A.z' l .soc. Ci. Ene.,vo l . 80 . pp . 1 .25 . 1954 . '

Fisher, R. A.. OD the mathemat;cal foundations of theoretical statistics, Tans. R. Soc. Iandon A,vol. 222, pp. 309-36E, 1922.

Fisher, R. A., and L. H. C. Tippet!, Limiting forms ol the frequency distribudon of the largest orsmallest member of a sample, Proc. Canbridee Phil. Soc., vol. 24. pan Il, pp. 180-191,1928.

Fosler, H. A., Theoretical frequency curaes and their application to engineering problems, frans.Am. Soc. Civ. Eng., vol. 87, pp. 142-113, 1924.

Frechet. M., Sur la loi de probabilit€ de I'ecai maximum ("On th€ probability law of maximumvalues"), Annaler de Ia societe Polonaise de Mathenatique, vol.6, pp. 93 116. h'rakow,Foland, 1927.

Freeze, R. A., A stochastic-conceptual analysis of one-dimensional groundwater flow in nonuniformhomogenous media, Water Resour. Res., vol. 11, na.5, pp.'125-'141, 19-15.

Gumbel, E. J., The relum period of flood flows, The Annak of Mathenatical Statistics, rol. ).2,no. 2, pp. 163-190, June 1941.

Haan, C. T., Statistical Methods in HydroloS), Iowa State Univ. Press, Ames, Iowa, 1977.Jenkinson, A. F., The frequency disrributioo of the annual maximum (or minimum) values of

meteorclogical elemenls, Quart. Jour. Roy. Met. Soc., vol. 81, pp. 158-171, 1955.Kavvas M. L., and J. W. Delleur, A stochastic cluster model of daiiy rainfall sequences, lydter

Resour. Res., vol. 17, no.4, pp. l l5l-1160, i981.Fbarson, K., On the systematic fitting of curves to observalions and measurements, Eiom€trikz, vol.

l , no. 3, pp. 265-303, 1902.Waymire, E., and V. K. Cupta, The mathematical slructure of rainfall rcpresentations L A review

of the stochastic rainfall models, Water Resour. R€r., vol. 17, no. 5, pp. 1261-1294, 1981.weibull, W., A statistical theory of the strength of materials, ln|eniors Vetenskaps Akademien (me

Royal Swedish Instiiute for Engineering Research), proceedings no. 51, pp. 5-45, 1939.

PROBLEMS

11.1.1 The airnual precipitation data for College Station, Texas, from l91l to 1979 aregiven in Table ll.l.l. Estimate from the data the probability that the annualprecipitation will be greater than 50 in in any year. Calculate the probability thatannual precipitation will be greater thar 50 in in two successive yea$ (a) byassuming annual precipitation is an iDdependent process; (r) directly from thedata. Do the data suggest there is any tendency for years of precipitation > 50in to follow one another in College Station? l.

11.1,2 Solve Prob. 1l.1.1for precipitation less than 30 in. Is there a tendency for yearsof precipitation less than 30 in to follow each other more than independence ofevents ftom year to year would suggesi?

ll.3.l Calculate the mean, standard deviation, and coefficient of skewness for CollegeStation annual precipitation from 1960 to 1969. The data are given in Table1 1 . 1 . 1 .

F(.r) = "^o [-1' -o])"*1 ( l l . s . 3 )

where k. r.r. and a are parameten to be determined.The three l imiting cases are (1) for I : 0, the Extreme Value Type I

dist butjon, for which the probabil ity density function is given in Table I 1.5. l,(2) for t < 0, the Extreme Value Type II distdbution, for which (11.5.3) appliesfor (u + alk) = -r < 6, and (3) for k > 0, the Extreme Value Type IIIdist bution. for which (11.5.3) applies for -.c < r = (u + alk).In all threecases, a is assumed to be positive.

For the EVI distribution -r is unbounded (Table I I .5.1), while for EVII, .ris bounded from below (by u + o'lk), and for the EVIII disrribution, x is similarlybounded from above. The EVI and EVII distributions are aiso known as theGumbel awJ Frechet distributions, rcspectively. lf a variable ;r is described bythe EVIII distribution. then -x is said to have a W€lbrll disriburion.

180

160180

160

rnrqcrercrexer-vsIs 3El

Relativ€ frequencY

(b) Relalive frequencY function

CHAPTER

T2FREOUENCYANALYSIS

^ 140

e 100

& 8 06

i 6t)a + o

;. r40

; 120

I ' o o* 8 0

.v 6040

Hydrologic s)stems are sometimes impacted by extreme events. such as severestorms, floods. ard droughts. The magnitude of an extreme event is inverselyrelated to its frequency of occurrence, very severe events occuning less frequentlythan more moderate events. The objective of frequency analysis of hydroJogicdata is to relate the magnjtude of extreme events to their frequency of occurrencethrough the use of probability distributions. The hydrologic data analyzed a.eassumed to be independent and identically distributed, and the hydrologic systemproducing them (e.g., a storm rainfall system) is considered to be stochastrc.space-independent, and time-independent in the classification scheme shown inFig. 1.4.1. The hydrologic data employed should be carefulLy selected so thatthe assumptions of independence and identical distribution are satisfied. In prac-tice, this is often achieved by selecting the annual maximum of the variable beinganalyzed (e.g., the annual maximum discharge, which is the largest instantaneouspeak flow occuning at any time during the year) with the expectation that suc-cassive observations of this variable from year to year will be independent.

The results of flood flow frequency analysis can be used for many engi-neering purposes: for the design of dams, bridges, culve(s, and flood controlstructures; to determine the economic value of flood control projects; and todelineate flood piains and determine the effect of encroachments on the floodplain.

12.1 RETURN PERIOD

Suppose that an extreme event is defined to have occurred if a random variableX is greater than or equal to some level .x7. The recurrence i?1ten cl r is the time

FIGIJRE 12.1.1Annual maximum discharge of the Guadalupe River near Victoria, Texas.

between occurences ofX >.r1. For example, Fig. 12.1.1 shows the record of

annual maximum discharges of the Guadalupe River near Victoria, Texas, from

1935 to 1978, plotted from the data given in Table 12 1.1. If;rr = 50,000 cfs'

it can be seen t-hat the maximum discharge exceeded this level nine times during

the period of record, with recurrence intervals ranging from 1 year to 16 years'

as shown in "fable 12.1.2.Tlte return period ? of the event X > Jr is the exPected value of r, E(z),

its avemge value measured over a very large number of occurrences' For the

Guadalupe River data, there are 8 recurence intervals covering a total period

of 41 years between the first and last exceedences of 50,000 cfs, so the retum

period of a 50,000 cfs annual rnaximum discharge on the Guadalupe River is

TA3LE 12.1.1Annual maximum discharges of the GuadalupeRiyer ne&r Victoria' Texas, 1935-1978' in cfs

2D

01930 1940

Year

(a ) Time serjes ofannual maximum discharges

0 55.900 13,300 23,7N 9'1901 58,000 12,300 55,800 9'1402 56,000 28,400 10,800 58'5003

'7, ' �710 11,600 4,100 33'1004 12,3ffi 8,560 5,720 25,2N5 38,500 22,OOO 4,950 15,000 30'2006 1?9,OOO 17,900 r,730 9,190 14'1001 n,mo 46,000 25,300 ?0,000 54'5008 2s,4oo 6,9"10 58,300 44,3N l2"7AJ9 4,940 20,600 10,100 15,200

0 0.08 0.16 0.24 0.32

382 eprrrso urocorocv rnequercv,llrervsts 3E3

What is the probability that a T-year return period event will occur at leastonce in N years? To calculate this, first consider the situation where no l-yearevent occurs in N yeam. This would require a sequence of N successive "failures,"

so that

PrX < x1 each year fo r N years r = t l p rv

The complement of this situation is the case required, so by (l l. l .3)

P(X > x7 at least once in N years, = I - (l l)N t (12.1.3)

Since p : l/7,

P(X > x7 at least once in N years) =

TABLE T2.T.2Years rvith annual maximum discharge equaling or exceeding 50,000 cfs onthe Guadalupe River near Victoria, Texas, and corresponding r€currenceintervals

Exceedenceyear 1936 1940 l94l 1942 1958 l96t 1961

i n t e r v a l 4 l 1 1 6 3 6 5(years)

1972 19'7'7 Averuge

uppro"inru,"ly ; ='+l/8 = 5.I yea$. Thus the retum period of an event of a givenmagnitude may be defined as the average recurrence interr4l between eventsequalLing or exceeding a specified magnitude.

The probability p = P(X > rr) of occurrence of the event X = x7 1nany observation may be related to the return period in the following way. Foreach observation, therc are two possible outcomes: either "success" X > ,r7(probability p) or "failure" X { x7 (probability I - p). Since the observations areindependent, thc probability of a recurrence interval ol duration r is the productof lhe probabil it ies of z- i failures followed by one success, that is. (l -p)'-tp,

and the expected value of z is given by

( t 2 . 1 . 4 )

Example 12,1.1. Estimate the probabil i iy that the annual maximum discharge Qon the Guadalupe River will exceed 50,000 cfs al least once dudng the next threeyears.

Solution. From the discussion above, P(C = 50,000 cfs in any year) : 0. 195. sofrom Eq. (12. L3)

P(O > 50,000 cfs at least once during the next 3 years) : I - (1 - 0. l95l I

:0 .48

The problem in Example l2.l. l could have been phrased, "What is theprobability that the discharge on the Guadalupe River will exceed 50,000 cfsat least once during the next three years?" The calculation given used only theannual maximum data, but, altematively, all exceedences of 50,000 cfs containedin the Guadalupe River record could have been considered. This set of data iscalled the portial duration series.It will contain more than the nine exceedencesshown in Table 12.1.2 if there were two or more exceedences of 50,000 cfswithin some single year of record.

Hydrologic Data Series

A complete duration series consists of all the data available as shown in Fig.12.1.2(a). A partial duration series is a series of data which are selected sothat their magnitude is greater than a prcdefined base value. If the base value isselected so that the number of values in the series is equal to the number of yearsof the record, the series is called an annual exceedence series: an example isshown in Fig. l2.L2(b). An extreme value series includes the largest or smallestvalues occurring in each of the equallylong time intervals of the record. Thetime interval length is usually taken as one year, and a series so selected is calledan qnnuaL series. Using largest annual values, it is an annual maximum seriesas shown in Fig. I2.1.2(c). Selecting the smallest annual values produces anannual minimum series.

' - ( ' - ; ) "

( 1 2 . 1 . t a )=p + 2(t - p)p + 3(t - p)zp + 4(t - p)3p + . . .

= p I t + 2 ( t - p ) + 3 ( t , p ) 2 + 4 ( t - p ) 3 + . . . l

The expression within the brackets has the form of the power series expansion(1 + r)' ' : I + N + ln(n 1)12)x2 + [n(n - l)(n-2)/61x3 + . . ., with x : (l -p)and n 2 . so r12 . L Ia ) may be rewr i t ten

L \ r ) - r ,

l t - \ t - p ) r

p( r2 . t . t b )

Hence E(r) - T : llp, that is, the probability of occurrence of an event in anyobservation is the inve$e of its retum oeriod:

P(x> x7) = |

( r2 . | . 2 )

For example, the probability that the maximum discharge in the Guadalupe Riverwil l equal or exceed 50,000 cfs in any year is approximately p = l l i : 115.1 =0 . 195.

384 ,qpprrro xvonorocr

The annual maximum values and the annual exceedence values of thehypothetical data in Fig. 12.1.3(a) orc arranged graphically in Fig. 12. 1.3(D) inorder of magnitude. In this particular example, only 16 of the 20 annual maximaappear in the annual exceedence series; the second largest value in several yearsoutmnks some annual maxima in magnitude. However, in the annual maximumseries, these second largest values are excluded, resulting in the neglect of theireffect in the analysis.

The retum period Ir of event magnitudes developed from an annual excee-dence series is rclated to the conesponding retum period f for magnitudes dedvedfrom an annual maximum series by (Chow, 1964)

rneeuercver,ralvsts 385

Rank of values(d) Original dala.

l0 20Rank of values

(r) Annual exceedence and maximum values.

FIGURE T2.1,3Hydrologic data arranged in the order of magnitude. (So,rrce. Chow, 1964 Used with permission.)

ildependent-the occurrence of a large flood could well be related to saturated

soil conditions produced during another large flood occucing a short time ea.rlier.

As a result, it is usually better to use the annual maximum series for analysis

In any case, as the return period of the event being considered becomes large'

the rcsults from the two approaches become very similar because the chance that

two such events will occur within any year is very small.

12.2 EXTREME VALUE DISTRIBUTIONS ..The study of extreme hydrologic events involves the selection of a sequence of

the largest ot smallest observations ftom sets of data. For example, the study

of peak flows uses just the largest flow recorded each year at a gaging stalion

out of the many thousands of values recorded. In fact, water level is usually

recorded every 15 minutes, so there are 4 x 24 :96 values recorded each day,

100

3 4 0

r00

80

60

: 6 0

s 4 0

l l .t

(r) Annual exceedences.

k ) Annual maxima. Time

^ : [ ' ( * ) ] -

" E 0

: 6 0

> 4 0

20

0

FIGURE I2.1,2Hydrologic data arranged by time ofoccurrence. (Sorrce: Chow, 1964.Used with permission.)

(12.r.s)

Although the annuai exceedence series is useful for some purposes, it islimited by the fact that it may be diflicult to verify that all rhe observations are

Base for annLral exceedence values

3E6 qppuro lrvorr:rocr

and 365 x 96 = 35,040 vaiues recorded each year; so the annual maximum flowevent used for flood flow frequency analysis is the largest of more than 35,000obsenations du ng that year. And this exercise is carried out for each year ofhistorical data.

Since these observations are located in the extreme tail of the probabilitydistribution of all observations from which they are drawn (the parent population),it is not surpdsing that their probability dishibution is different from that of theparent population. As described in Sec. 11.5, there are three asymptotic formsof the distributions gf extreme values, named Type I, Type II, and Type III,rcspectivei!. t

The Extreme Value Type I (EVI) probability distribution function is

nnrquercv ll'ralvsrs 387

I FIGURE 12.2,1For each of lhe three typ€sof extreme value distributions

III the variale.x is plo$ed againsta ieduced variaE ) calcuiatedfor rhe ExIeme value Type Idjstribqiorll, The Type Idislributiot{is unbounded in r,while lhe Type II distributionhas a lower bound and theType III distribution has anupp€r bound. ("Soar.r. NaturalEnvironment Research Council,1975 , F ig . 1 .10 , p . 41 . Usedwith pennission.)

( t 2 . 2 . 7 )

( . r2 .2.8)

r . _ , , II r r r - e x r l - " , , p { - : l

] - c c s . r s m

The paramete$ arc estimated, as given in Table l l.5.l, by

( i 2 . 2 . 1 )

The pararneter ri is the mode of the distdbution (point of maximum probabilitydensitv). A reduced ,-ariate y can be defined as

- 7 - 1 0 t 2 3 4 5 6Reduced variale. )

so

T - |F(x) = ::------:

and, substituting into (12.2.6),

[ , r \ ly r = - r n l l n l - l l

t \ , - r / r

For the EVI distdbution, rr is related to yr by Eq. (12.2.4), or

t;v o s7l

u=7 0.5172a

j : ' u

a

Substituring the reduced va ate into (12.2.l) yields

(r2.2.2)

(12.2.3)

Solv ine fu r r ;

F(x) = exP [ -exP(*Y) ]

,=-'['(#)]

l: = P t x > x " lT

= l - P ( x < x r )

= 1 F (xr)

( t 2 . 2 . 4 )

| l ) 5 \

( 12.2.6)

Let (12.2.6) be used to define y for the Type II and Type III distributions.The values of r and I can be plotted as shown in Fig. 12.2.1. For the EVIdistribution the plot is a straight line while, for large values ofy, the correspondingcurye for the EVII distribution slopes more steeply than for EVI, and the curvefor the EVIII distribution slopes less steeply, being bounded from above. Figure12.2.1 also shows values of the return period Z as an altemate axis to y. Asshown b1 Eq. (12.1 .2),

Extreme value distributions have been widely used in hydrology. Theyform the basis for the standardized method of flood frequency analysis in GreatBritain (Natural Environment Research Council, 1975). Storm rainfalls are mostcommonly modeled by the Extreme Value Type I distribution (Chow, 19531Tomlinson, 1980), and drought flows by the Weibull distribution, that is, theEViII distribution applied to -x (Gumbel, 1954, 1963).

Example 12.2.1. Annual maximum values of lo-minute-duration rainfall atChicago, Illinois, from l9l3 to 194'7 are presented in Table 12.2.1. Develop amodel for storm rainfall frequency analysis using the Extreme Value Type I disui-bution and calculatg the 5-, l0-, and 50-year retum pe od maximum values of 10-minute rainfall at Chicago.

Solution. The sample moments calculated from the data in Table 12.2.1 are i :0.649 in and s = 0.171 in. Substituting into Eqs. (12.2.2) and (12.2.3) yields

o:! ! ln

The probabil ity model is

. [ / x - 0 . 5 6 9 \ lF(r ) - e \p l . "p : l

To determine the values of-rr for va ous values of retum period f, it is convenientto use the reduced variate J]r. For f = 5 years, Eq. (12.2.7) gives

. f . r r r lv ? - _ t n l t n l _ l lL L r - r / l

[ , < ' l= - t n l t n l : j - l l

L ' ) t t )

= 1 .500

and Eq. ( l l .2 8 ) y ie lds

xT: u + dyT

: 0 . 5 6 9 + 0 . 1 3 8 x 1 . 5 0 0

= t J . / 6 r n

So thc lo-minute, 5 yea. stom rainfail magnitude at Chicago is 0.78 jn. By thesame merhod. rhe 10- and 5o-yeaj values can be shown to be 0.88 in and 1� l I in,

TABLE T2.2.7Annual maximum l0"minute rainfallin inches at Chicago, Illinois, 1913-1947

rnrquei,rcraralvsrs 3E9

respectively. It may be noted from the data in Table 12.2.I that the so-year retumpedod rainfall was equaled once in rhe 35 years of dala (in 1936), and rhat the l0-yea! rctum perjod minfall was equaled or exceeded four times during this period,so the frequency of occurence of observed extreme rainfalls is approximately aspredicted by rhe model.

12.3 FREQTMNCY ANALYSIS USING FREQUENCYFACTORS ' ;

Ca.lculating the magnitudes of extreme events by the merhod outlined in Example12.2.1 requires that the probability distribution function be invertible, that is,given a value for T or lF(x) = f/Q - l)1, the conesponding value of 17 canbe determined. Some probability distriburion functions are not readily invertible,including the Normal and karson Type III distributions, and an altemativemethod of calculating the magnitudes of extreme eveDts is required for thesedistributions.

The magnitude -rr of a hydrologic event may be represented as the rnean p,plus the departure Ar1 of the variate from the mean (see Fig. 12.3.1):

x T = l L + L t r ( 1 2 . 3 . 1 )

The departure may be taken as equal to the product of the standard deviation o andaflequency factor Kr; that is, Axz : KTo. The departure A-t1and the frequencyfactor (r are functions of the return period and the type of probability distriburionto be used in the analysis. Equation (12.3.1) may rherefore be expressed as

38E eeprtto ;r '' otorocv(

_ G x 0 . 1 7 717

: 0 . 1 3 8

= 0 . f l 9 - 0 . 5 7 7 2 x 0 . 1 3 8

:0 .569

0 .340.700 .570.920.660 .650 .630.60

x T : t t + K r q

x 7 : 7 * K 7 s

(12.3.2)

(12.3 .3)

l 9 l 0

which may be approximated by

In the event that the vadable analyzed is y = logx, then the same method isapplied to the statistics for the logarithms of rhe data, using

Y r : i * K r s . ( l 2 . 3 . 4 )and the requAed value of ,r7 is found by taking the antilog of y7.

The frequency factor equation (12.3.2) was proposed by Chow (1951), andit is applicable to many probability distributions used in hydrologic frequencyanalysis. For a given distribution, a K-I relationship can be determined betweenthe frequency factor and the corresponding return period. This relationship canbe expressed in mathematical terms or by a table.

Frequency analysis begins with the calculation of the statistical parametersrequired for a proposed probability distribution by the method of momenrs fromthe given data. For a given rctum period, the frequency factor can be determinedfrom the K-f relationship for the proposed distribution, and the magnitude ,r1computed by Eq. (12.3.3), or (12.3.4).

The theoretical K-T rclationships for several probability djstributions com-monly used in hydrologic frequency analysis are now described.

0..190 .660 1 60 . 5 80.. i 1

0 .11

0 .5 30 .160 .570 .300 6 60 .680 .680 . 6 10 .880.,19

0 .330 .960 .940 .800 .62o . 1 1r . 0.640 .520.64

Standard dev l i r rn = 0 . l l r in

390 .qppuo rronolocr

Then, subsrituring w :j.tto (12.3.'7)

Kr: z

rnequ*cr rxnrrsrs 391

:2.79'1 | -2.51 557 + 0.80285 x 2.797 I + 0.01033 x (2.79'1 | ).

r.xtreme valre i lype I dlstnbu-

f

r . = - $ l o I r ' ' ln 1

51 t2 * ' � " l ' � ^ | . - , , ] t

(12 3 8 )

To express f in terms of r(7, the above equation can be written as

| + 1.432'79 x 2.'79't | + 0.18921 x (2.'79'�71)' + 0.00131 x(2.?971)3

FIGURE 12,3.1The magnilude of an extremeevent :rr expressed as a deviationKrr frcm the mean p. where l(ris $e frequency factor.

EXTREME VALUE DISTRIBUTIONS. For thetion. Chow ( I 953 r derived the expression

NOR\4AI- DISTITIBUTION.( 1 2 . 3 . 2 ) a s

o

f . r l r l ' '

' = l ' " \ t / l ( o < P = s 5 )

then calculating; using the approximation

2.515517 + 0 .802853w + 0 .01032811,2

The frequency factor can be expressed from Eq.

( r2 .3 .s )

( t 2 . 3 _ 6 )

(12.3.'�l)

. | [ , r K ' \ l lr - e x p l - e x p l - t y + | ' l l l

r L \ v o / r l

( r2 .3 .9 )

This is the same as the standard nonnal variable z defined in Eq. (11.2.9).The \alue of z coresponding to an exceedence probability of p (p = U7)

can be calculated by finding the value of an intermediate variable w:

where y = 0.5772. When xr : tt, E4. (12.3.5) gives Kr : 0 and Eq. (12.3.8)gives f : 2.33 years. This is the retum period of the mean of the Extreme ValueType I distribution. For the Exffeme Value Type II disriburion, the logarithm ofthe variate follows the EVI distribution. For this case, ( 12.3 .4) is used to calcularey7, using the value of K1 from (12.3.8).

Example 12.3,2. Determjne the 5-year retum pedod rainfall for Chicago usingthe frequency factor method qnd the annual maximum lainfall data gjven in Tablet 2 . 2 . 1 .

Solution. "fhe mean and standard deviarion of annual maximum rainialls ar Chicagoare i : 0.649 in and r : 0, 177 in, respectively. For T = 5, Eq. (12.3.8) gjves

" u 7 t T - - 'nr = . - l 0 .5 i72 + l n l rn l= -L l l fL t - | J )

- 4{o.rrr , , r f , ' r -s r l ,? r L l 5 l . l j

: O. " t t9

B y ( 1 2 . 3 . 3 ) ,

xr:i + Krs- ' � 0 . 6 4 9 T 0 . 1 1 9 > o . t ' 7 1

=0.78 in

r . de ter rn ined In t \ample 12 .2 . L

LOG-PEARSON TYPE UI DISTRIBUTION, For rhis distriburion, the first stepis to take the logarithms of the hydrologic data. v = log -!. Usuallv losarithms to

t + 1.432'788v,/ + 0.189269w2 + 0.001308w3

When p > 0.5, I p is substituted for p in (12.3.6) and the value of z computedby (12.3.7) is given a negative sign. The error in this formula is less than 0.00045in z (Abramowitz and Stegun, 1965). The frequency factor K1 for the normaldistribution is equal to z, as mentioned above.

For the lognormal distribution, the same procedure applies except that it isapplied to the logarithms of the variables, and their mean and standard deviationare used in F,9. {12.3.1).

Example 12.3.1. Calculate the frequency lactor for the normal distributjon for anevent wllh a return period of 50 yea$.

Solution, For I = 50 years, p = l/50 = 0.02. From Eq. (12.3.6)

[ , r , l r ' ?= l t n l - l lL \p - l l

| ' ,1]"': l l n l - l l

L ' r o . o 2 r l l= 2.'7971

P(X ? .rr )

J92 ,rpp,,r: ur ororocv

base l0 arc used. The mean t, standard deviation .ty, and coefficient of skewnessC, are calculated ior the logarithms of the data. The frequency factor dependson the retum peiod I and the coefficient of skewness C,. When C' : 0, thefrequenc5, factor is equal to the standard normal variable ;. When C, I 0, Kr isapproximated by Kite (1977) as

^ I l .K t : ( a / - l l l . | ; ( z r - b z ) k ' - t z ' - l ) k ' i z k ' t i t ' r 1 2 . 3 . 1 0 . 1' t J

where li -.- C,i6. a

TABLE I:.].1Kr values for Pearson Type III distribution (positive skew)

Return period in years

Skewcoefnci€ntCr or C"

r0 25 50Exceedence probabilily

reequr',Lcv nuerr srs 393

The value of z for a given retum period can be calculated by the procedureused in Example 12.3.1. Table 12.3.1 gives values of the frequency factor forthe Fearson Type III (and log-Pearson Type UI) disfiibution for various valuesof the retum period and coefficient of skewness.

Example 12.3.3. Caiculate the 5- and so-year relum pedod annual maximumdischarges of the Guadalupe River near Victoria, Texas, using the lognormal andlog-Pearson Type III distributions. The data frcm 1935 to 19?8Fre given in Table1 2 . t . t . I r

4TABLE 12.3.1 kon!.\lKr yalues for Pearson Type III distribution (negative skew)

Retum period in years

l0 25 s0Eice€dence probebility

r00200100 Skew

cmffcientCs ot C- 0 .100.50 0.20 0.04 0.02 0.01 0.005

0.20 0.10 0.04 0.02 0.01 0.00s

3 . 02 . 92 . 82.'�72 .62 .52 .42 .32 .22 . 12 .O1 . 9l � 8| . ' 7L 6l � 5t . 4L 31 . 2l . l1 . 00 .90 . 8a. '70 . 60 .50 .40 .30 .20 . I0 .0

{.396 0.420{.190 0.440o.3t.1 0.460

4.316 0.419-0.368 0.499{).360 0.518o .351 0 .537o.341 0.5550 .330 0 .574{ .3 t s 0 .592{.30t 0.609

4.291 0.62' lo.2E2 0.643{.268 0.660o.251 0.6150.240 0.690{.225 0.?05{ .210 0 .719{ .195 0 .132Lr.180 0.745rl.16.1 0.758{.1,18 0. ' /69{ . i 32 0 .780{ . i l 6 0 .7900.099 0.800o.08r 0.808{ .066 0 .816{1.050 0.824

-o.033 0.830{ .017 0 .8360 0.842

r.180 2.2'781 .195 2 .211LL210 2.2151.224 2.2' �72r.238 2.26'�71.250 2.2621.262 2.2561.27 4 2.248r.284 2.2401.294 2.2301.302 2.2t91 .310 2 .2011 .318 2 .193t . 324 2 .179t.329 2.163L333 2 .146L337 2.128l . 339 2 .108t.340 2.087l.341 2.066L340 2.043L339 2 .018L336 1.9931 .333 | . 967L328 1 .939| . 323 1 .910| . 3 t ' 7 1 .880L309 r.8491 .301 t . 818t.292 1.7851.282 1.751

3 . i 52 4 .0513 .134 4 .0133 . I 14 3 .9733.093 3.9323 .071 3 .8893.048 3.8453.023 3.8002.997 3. '7532.910 3.7052.942 3.6562.912 3.6052 .881 3 .5532.848 3.4992 .815 3 .4442.' t80 3.1882.743 3.3302.706 3.21r2.666 3.2112.626 3.1492.585 3.0872.542 3.0222.498 2.9512.453 2.8912.407 2.8242.359 2.'�7552.3t1 2.6862.26t 2.6t52.21 2.5442.159 2.4722.to7 2.1U)2.054 2.326

0.017 0.8460.033 0.8500.050 0.8530.066 0.8550.083 0.8560.099 0.8s70 . i 16 0 .8570.132 0.8560 .148 0 .854o.t64 0.8520 .180 0 .8480.195 0.8440 .2 r0 0 .8380.225 0.8320.240 0.825o.254 0.8170.268 0.8080.282 0.7990.294 0.7880.307 0.71'�70 .3 t9 0 .1650.330 0. '7520.341 0. '7390.351 0. ' t2s0 .360 0 .7110.368 0.6960.376 0.68r0.384 0.6660.390 0.6s10.396 0.636

1 .2 ' � 70 | . 716l � 258 L6801.245 i .6431.231 1.606| . 216 1 .5671.200 1.528l . 183 r . 4881 . t 66 | . 448| . 141 | . q7l � t 28 1 .3661.107 1.3241.086 1.2821.064 |.2401.041 L l981 .018 1 . t 5 ' � 70 .994 L l160.970 1.0750.945 1.0350.920 0.9960.89s 0.9590.869 0.9230.844 0.8880 .819 0 .8550.795 0.823o.'771 0.793o.' t47 0.7640.'724 0.?380.702 0. ' |20.68r 0.6830.666 0.666

2.000 2.252 2.4821 .945 2 .118 2 .3881.890 2.to4 2.2941.834 2.029 2.201|.77' �7 L955 2.10Et. '720 l .880 2.0161.663 1.806 t.9261.606 1..'733 L8311.549 1.660 1.7491 .492 1 .588 1 .6641 .435 1 .518 1 .5811 .319 t . 449 1 .5011.321 1.3E3 \.424t . 2 ' 70 1 .318 1 . 351|.21'7 | .256 t.282r .166 t . 19 ' 1 1 .216t . 1 1 6 1 . 1 4 0 1 . 1 5 5r.069 1.0E7 1.0971.023 1.037 1.0440.980 0.990 0.9950.939 0.946 0.9490.900 0.905 0.9070.864 0.867 0.8690.830 0.832 0.8330.798 0.799 0.8000.?68 0.769 0. ' �7690.?40 o. ' t$ 0. '7410 .714 0 .1 t4 Q . ' � 7140.689 0.690 0.6900.666 0.667 0.667

4.9704.9094.8474.7834 . 7 t 84.6524.5844 .5154.4444.3724.2984.2234. t4'74.0693.9903 .9 t03.8283.'�7453.6613.5753.4893.4013.3123.2233.t323.0412.9492 .8562.'�7632.6102.516

-o. I4 .2-.0.34 .4

-o.8-o.9-1 .0- l . l

- l . 31 . 4

- l . 6

-1 .8- l . 9) .o-2 .1-2.2

-2.6

2 .8-2.9-3,O

sr!/ce: U. S. waler Resources Council (1981).

J94 lprL rr: , sr oaorocv

Solution. Tl\e logarirhms of the discharge values are taken and their statistrcscalcuiated: I = :1.2743.r, : 0.402?,C, = -0.0696.

Ialnormal distribution. The frequency factor can be obtained irom Eq.( 12.1.7). or from Table 12.3.I for coeff icient of skewness 0. For I = 50 years, -K7was computed in Example 12.3.1 as l(50 = 2.054; the same value can be obtainedfrom Table 12.3.1. By (12.3.4)

h:t + l(.r,

lso= 4.2'743 + 2.054 x 0.4027: 5 . 1 0 1

Then

r5e = (10)5 ror

= 126,300 cfs

Sinri lar l l , (s = 0.842 from Table 12.3.1 , \ ' \ = 4.2'143 + 0.842 x 0.4021 = 1.6134,and - r5 = (10 ) r6 r : r a : 41 ,060 c f s .

Lag Pearson T:-pe IIl distribution. For C, : -0.0696, the value of .K50 isobtained by rnterpolat ion from Table 12.3.1 or by Eq. (12.3.10). By interpolat ionw i t h T = 5 0 ) r s l

, ) O O ) O 5 4 ,( . , , - 2 .054 |

- - "^ . - : "

\ 0 .0bqo 0 j 2 .01b( - u . r - u J

So rr," = j + K5as, = 4.2'l 43 + 2.016 x 0.402'l = 5.0863 and rso = (10) 5 086r -

l2l . 990 cfs. By a similar calculation, f5 = 0.845, yr:4.6146, uno rs = 41, 170 cfs.The results for estimated annual maxjmum discharges arc:

Retum hriod

5 years 50 yeaIs

raeeuelrcreNer-rsrs 395

that linearizes the distribution function. The plotted data are then fifted with astraight line for interpolation and extrapolation purposes.

Probability PaperThe cumulative probability of a theoretical dist butior may be represented graph-ically on probability paper designed for the distribution. On such paper the ordi-nate usually represents the value ofx in a cenain scale and the abscissa representsthe probability P(X > x) or P(X < r), the return period I, orihdrreduced variateyr. The ordinate and abscissa scales are so designed that the daq| to be fitted areexpected to appear close to a straight li!e. The purpose of usirfg the probabilitypaper is to linearize the probability relationship so that the plotted data can beeasily used for interpolation, extrapolation, or comparison purposes. In the caseof extrapolation, however, the effect of various errors is often magnified; there-fore, hydrologists should be wamed against such practice if no consideration isgiven to this effect,

Plotting Fositions

Plotting position refers to the probability value assigned to each piece of data to beplotted. Numerous methods have been proposed for the determination of plottingpositions, most of which are empirical. If r is the total number of values to beplotted and ,n is the rank of a value in a list ordered by descending magnitude,the exceedence probability of the ,nth largest value, -r., is, for large n,

P(X :- x^1 - !)n

However, this simple formula (known as Califomia's formula) produces a prob-ability of 100 percent for m: n, which may not be easily ploned on a probabilityscale. As an adjustment, the above formula may be modified to

t u - |P ( X > x ; : : : - :

While this formula does not produce a probability of 100 percent, it yields a zeroprobability (for rl1 = t;, which may not be easily plotted on probability papereither.

The above two formulas represent the limits within which suitable plottingpositions should lie. One compromise of the two formulas is

m n \P\X >- x; = ::-----:::

n( 2 . 4 . 3 )

which was first proposed by Hazen (1930). Aaother compromising formula(known as Chegodayev's) widely used in the U.S.S.R. and Eastern Europeancountries is

m - 0 . 3

( t 2 .4 . t )

Lognormal 41,060(c. : 0)

Log Pearson Type I i I 4 l , l '70(c. : -0.07)

126,300

r2 i , 990(12.4.2)

It can be seen that the effect of including the small negative coefficient of skewnessin the calculations is lo alter slightly the estimated flow with that effect being morepronounced at I = 50 years than at T = 5 years. Another featuae of the resultsis that the 50-year return period estimates are about three times as large as the 5-year return period estimates; for this example, the increase in tie estimated flooddischarges is less than proportional to the increase jn.eturn period.

12.4 PROBABILITYPLOTTING

As a check that a probability distribution fits a set of hydrologic data, the datamay be plotted on specially designed probability paper, or using a plotting scale P(X :- x^) =

n + 0 . 4(r2.4.4)

39E eppr-rro rrororccv

f.t,l1 0 o.)

rneeuercr,lx,+vsrs 399

accurately from small samples. Because of this, the Water Resources Council

recommended using a generalized estimate of the coefficient of skewness' C,"'

based upon the equation

C " = w C ' + l t - W C ^ ( 1 2 . 5 . 1 )

where ty is a weighting factor. Cr is the coefficient of skewness computed using

the sample data, and C^ is a map skewness, which is read from a map such as

Fig. 12.5.1. The weighting factor lV is calculated so as to 4in\nize the Yariance

of C.. as explained next. ,;Tbe estlmates of the sample skew coefficient and the m$ skew coefficient

in Eq. (12.5.1) are assumed to be independent with the same mean and differentvadances, V(CJ and y(C.). The variance of the weighted skew, V(C"), can be

expressed as

v(c*) = w2v(c,) + (1 w2v(c ̂) ( r 2 . 5 . 2 )

The value of W that minimizes the variance C. can be determined by differenti-atinl 02.5.2) with respect to w and solvjng dLv(C.)lldw : 0 for W to obtain

v(c ̂) ( 1 2 . 5 . 3 )v(cr) + y(c.)

a

0 0 0 t 0 0 1

The second derivative

#=2vG)+v(c , , , ) l ( t 2 . s .4 )

is greater than zero, confirming that the weight givel by (12.5.3) minimizes the

variance of the skew, Y(C.).Determination of l{ using Eq. (12.5.3) requires knowledge of y(C; and

V(C,). KC.) is estimated from the map of skew coefficients for the United Statesas 0.3025. Alternatively, V(C ̂ ) can be derived from a regression study relatingthe skew to physiographical and meteorological chamcteristics of the basins (Tung

and Mays, l98 l ) .By substituting Eq. (12.5.3) into Eq. (12.5.1), the weighted skew C". can

be written

^ v(c ̂)c, + v(c,\c^" " : - v r c . l + v t , c " l

The variance of the station skew C, for log-Pearson Type III randomvariables can be obtained from the results of Monte Carlo expedments by Wallis'

Matalas, and Slack (1974). They showed that y(Cr) of the logarithmic statron

skew is a function of record length and population skew For use in calculatingC,, this function can be approximated with sufficient accumcy as

( 1 2 . 5 . 5 )

Srandard normaL var iabLe :

" ' ' - . - r _Rcrurn period T (years) l r0 100 500

Excccdcncc prcbabilry (l / T)

0 . 9 0 . 7 0 . 5 0 . 3 0 . 1 0 . 0 t 0 . 0 0 2

0. | 0.3 0.5 0.? 0.9 0.99 0.998Cumulative probabil i ty

FIGURE 12.4. IAnnual maximum discharge for the G adalupe River near Vicloria, Texas, plotled usjng Blom'sformula on a probabili!) scale for the lognormal distribution.

12.5 WATER RESOURCES COUNCIL METHOD

The U. S. Water Resources Council* recommended that the log-karson TypeIII dist bution be used as a base distribution for flood flow frcquency studies(U. S. Water Resources Council, 1967, 19'76, 19'7'7, and l98l; Benson, 1968).Their decision was an attempt to promote a consistent, uniform approach to floodflow frequency determination for use in all federal planning involving water andrelated land resources. The choice of the log-Fearson Type III distribution is,however, subjective to some extent, in that there are several cdteda that may beemployed to select the best distribution, and no single probability distribution isthe best under all criteria.

Determinatior, of the Coefficient of Skewness

The coefficient of skewness used in fitting the log-Peamon Type III distdbutionis very sensitive to the size of the sampie and, in particular, is difficult to estimate

*The U.S. water Resources Council was abolished in 1981. The Council's work on guidelines fordetermining flood flo!! frequency was taken over by the lnteragency Advisory Cornmittee on WalerData, U.S. Geological Survey, Reston, Virginia.

Y(Cr) : 10'a-B locro("/ ro) (12.5.6)

4 - - o . 3 3 + 0 o 8 l c s l

o r A = - 0 . 5 2 + 0 3 o l c r l

B= 0 .94 -0 .261C,1

reeQutr'cv rret-vsls 401

i f 1c,1 < o.9o (r2 s"]a)

i f lc, l > o.9o 02 s 7b)

i f lc, l < 1.s0 (12 s 7c)

6 Z

3 e

E

E K6 -: ;. 9 t

= q,r

i ( ,

, " i

i : aE -

E '

' 6 3

5 6

u; .q ,i! ; g

; 9 !- ta , j

C5# z

q =- l

? : l

o r B = 0 5 5 i f l c ' l > 1 5 0 Q z 5 ' 7 4

in which lC,l is the absolute value of the station skew (used' a1 an estimate of

population skewtand r is the record length rn 5ear' t

J

Exsmpre l2.s.r. o*"*'": 'T i':;::t:";:'$ :"Jt[d"i"]:;,'ill'i ji'T

il:iJJl::::,"J"[]"ft'?""1"t-.1';;;; ;'rnuic""r' rt Mdnin Luther King Br\d

in ,A.ustln, t *ut as l isled in Table 12 5 l

Solut ioa. The sample data snown in columns I and 2 ofTable 12 5 l colef n = 16

years, from 1967 to 1982'

Slep I Transtorm the sample data' r" to their logarithmic- values' li; lhat is'

let ) , - log ! for i = l n' a5 shown in column 3 of lhe table

TABLE T2.5.7

iii."-r",i"t tr ""istics for logarithms of annual

il;;;;;;h;'ces for walnut creek {Exampl€

12.s.r)Column: | 2

Flo\x rYear (cfs)

3 4

0 - r ) - (J - t)3

t9611968tq691970l9? lt9'�7219'�73t9'l41975t9'�1619'�t'�71978r9191980r9811982

3035,640I,0506,0203,1404.5805 ,140

10,560I2,8405 ,1402,5201,730

12,4003,400

14.3009,540

2.4814

3 .O2 lZ3.'t'1963.51293.66093 .71 l 04.O23-t4 .10863 . 7 1 l 03.40143.23804.09343 .53154.15533.9't95

L33950.01270 .38140 .01980.00430.00050.00520 .14810.22010.00520.05640. 16060.206'�10 . 0 1 1 50.26680 . I l 6 l

- l .55020.0014

4.23s60.0028

-o.00030.00000.00040.05700.10370.0004

-o.01344.uu40.0940

-0.00120.13780.0396

Total 58.2206

t = 3.6388

2.9555 - i 4280

402 ,o ,pp l r t o r vo *o roov

Jr€p :. Compute the sample statistics The mean of log-transfonned vaiues

t sr -1- iR ))

i - 1 ) ' , - - - : i : : - i 6 l q1 6

- - -

Using column 4 of the table, *,. Lunauta deviation is

/ ' - ^ \ ' 's ' = l , ) 0 ' - r ) ' l

\ n - ' ; It

t r ' l l ' :: l : r q 5 5 5 l- \ , t s ' ' - ' - t

= 0.4439

li\ inL column 5 of the table, the skew cocfficienl is

l6 . ( - 1 .4280115 / H r rcA4r r t

= - ' r " "(n - 1)(n 2)s,3

SrsI ,1. Compute the weiShted skew The map skew is 0 3 from Fig. 12 5 1

at Austin. Texas. The variaoce ofthe stat ion skew can be computed by Eq. (12 5 6)

as fol lows. From (12.5.7r) with lC, > 0 90

A : -0.52 + 0.30 - 1.2441 = -0.14'7

F rom (12 .5 .7c ) w i t h C , < 1 .50

B :0 .94 - 0 .261 - 1 .244 = 0 .61 '7

Then us ing ( 12 .5 .6 )

K C , ) : ( 1 0 ) o 1 4 7 0 6 1 7 1 0 9 ( 1 6 / 1 0 ) - 0 5 3 3

The variance of the generalized sker is V(C.) : 0 303. The weight to be applied

to c. is w = v(C ̂ ) l lWC ̂ ) + v(C")l = 0 303/(0 303 + 0533) = 0 362, and the

complementary weight to be applied ro C. is I W = I -0362 = 0 638 Then,

f r on ( 12 .5 .1 )

C" :WC ' + (1 -W)C^

: 0 . 3 6 2 x ( 1 . 2 4 4 ) + 0 . 6 3 8 x ( - 0 . 3 )

: 0 .64

.srep 4. Compute the frequency curve coordinates. The log-karson TyPe

lll l'requency factors 1(1 for skew coefficient values of -0 6 and 0.7 are found

in fable 12.3.1. The values for C" = -0.64 are found by l inear interpolat ion

as in Example 12.3.3, with rcsults presented in column 2 of Table 12.5.2. The

corresponding value of y7 is found from Eq (12.3.4), and i ts anti logarithm is

talien to determine the estimated flood magnitude. For example, for f = 100 years,

t r r : l � 850 and

mreorr.rCv.rrru-vsrs 403

TABLE 12.5.2Results of frequency analysis using the Water ResourcesCouncil method (Examples 12.5.1 a\d 72.5.2)

Column: 1 2 3Return Frequencyperiod factorT Xr loq Qr

4 5FloodEstimolesQr Oi

oears) (cfs) (cf9 1--.'--.''--'.2 0 .1065 0.857

l0 1 . r 9325 t.5t250 L697

r00 1.850

3.686 4,900 5,500d4.019 10.500 10,00f4.169 14,7@ 13,2tn4.310 20,400 17,6004.392 24,1@ 20,m04.4$ 28,900 24,26

The values iD column 4 are tbose computed withour adjustment lor outlieG od thosein column 5 afrer outlier adjuslment.

v 1 - y K 1 r ,

= 3 . 6 3 9 + 1 . 8 5 0 x 0 . 4 4 3 9

= 4.460

and Qr = (10)4 4@ = 28, 900 cfs, as shown in columns 3 and 4 of the table. similarlycomputed flood estimates for the other required return pe.iods are also shown.

As was shown in Example (12.3.3), the increase in flood magnitude is lessthan directly proportional to the increase in retum perjod. For example, increasingthe retum period from l0 years to 100 years approximately doubles the estimatedflood magnitude in the table. As stated previously, flood magnitudes estimated usingthe log-Rarson Type III disfjbution are very sensilive to the value of the skewcoefficient. The flood magnitudes for the longer retum pedods (50 and 100 years)are difficult to estimate reliably from only 16 years of data.

Testing for Outliers

The Water Resources Council method recommends that adjustments be made foro\tliers. Outliers are data points that depart significantly from the trend of theremaining data. The retention or deletion of these outliers can significantly affectthe magnitude of statistical parameters computed from the data, especially forsmall samples. Procedures for teating outliers require judgment involving bothmathematical and hydrologic considerations. According to the Water ResourcesCouncil (1981), if the station skew is greater thar +.0.4, tests for high outliersare considered first; if the station skew is less than -0.4, tests for low outlie$are considered fiIst. Where the station skew is between a0.4, tests for bothhigh and low outlie$ should be applied before eliminating any outliers from thedata set.

/

404 ,qPPueo tl ionolocY

The following frequenc-v equation can be used to detect high outliers:

- r 1 2 5 8 )r ' ; = i | ^ " \ '

YIT', }x. ;,5iiil"",-H: ?:Tl'"""" ;l +:? :'l'i''T'.,:;l:'i il'llJll':i,l,' o'",*' ."ii,.,' ':l'-t;n'U'1,":' ;1, I X$l':;qi{i

ojX,,il5

f';; "'i,Ji, l?i1ll"J'iJ:'-'""llli"l.:3: i:i'i:J'fff fi;il''nto*^'on{:*,lli"l;.1t,1,3:.'fi:::':"",:[;i."1."''"i"""*'* "" y""*"1:, ti:'#iii;ll:l';n**"rqqiF'."|ffi Til:lix,TllfiJ"i:'"'J;#xilo'.-, un "' ' 'nd"J period,?r..t-':,:; ;l:;"' ' i; i";"." rs not available

'J..lil,'iT:[,::iiJ' il :i:;,1 :iti::t:t*:;';u"lilL''':':"'Hil:'""o'do

,,*lu,. equation can u" ";:rj"r,:Tj,l"w outliers: (12.5.e)

;l;r,i "r; *:"lll ;!ti:'i::ii;j{*l i ;;rllilti ,l:*'rll.'""'':1i:}'i.t.riuto ut the Water Resourc

E \am pre r 2's 2''1"9- :::':i':,tl:-':"X':"i"i":"'#':Xill':' l::':'l'i :;l;

l:jilT;:',:l ji:l'fi"'lii Ji'"*'i" n""o trequenc) cur\ e

*&!Eilq#$1at'iir9

*

Htfltf;tf;!gtrffitrfi'lI$

ItI

FREQUENCYAT_ALYsls 405

Solution,

S/ep L Determine the threshold value for high outliers. From Table 12.5.3,Kn = 2.2'79 for n = 16 data. From Eq. ( 12.5.8) using -t and s y from Example 12.5.I

)r=t + K,.r" = 3.639 + 2.2'79(0.4439) = 4.651

Then

g" :110 )o 65 ' : 44 ,735 c f s

The largest recorded value (14,300 cfs in Table 12.5.I ) {oeilnot exceed the thresh-old value, so there are no high outliers in this sample.

fStep 2. Determine the thJeshold value for low outliels. The same iKn value is

usecl:

lL:, - K,sy: 3.639 - 2.2'79(0.4439) = 2.62'7

Qt: (10)1621 = 424 cfs

The 1967 peak flow of 303 cfs is less than Q1 and so is considered a low outlier.

Step J. The low outlier is deleted ftom the sample and the freqLiency analysisis rcpeated using the same procedure as in Example 12.5.1. The statjsljcs for thelogajithms of the new data set, now rcduced to 15 values, are t = 3 716, sr:0.3302, and C, : -0.545. It can be seen that the omission of the 303 cfs valuehas significantiy altered ihe calculated skewness value (from the 1.24 found inExample 12.5.1). The map skewness remains at -0.3 for Austin, Texas, and therevised weighted skewness is C, = -0.41. Values of Kr are interpolated from Table12.3.1 at the requircd retun periods. and the conesponding flood flow estimatescomputed as Oi, listed in column 5 of Table 12.5.2. By comparing these valueswith those given in column 4 for the full data set, it can be seen that the effect ofremovilg the low outlier in this example is to decrease the flood estimates for thelonge! rctum periods,

Computer Program HECWRC

The computer program HECWRC (U. S. Army Corps of Engineers, 1982) per-forms flood flow frequency analysis of annual maximum flood series according tothe U. S. Water Resources Council Bulletin l7B (1981). This program is avail-able from the U. S. Army Corps of Engineers Hydrologic Engineering Center inDavis, Califomia, in both a mainframe computer version and a microcomputervelslon.

12.6 RELIABILITY OF ANALYSIS

The reliability of the results of frequency analysis depends on how well theassumed probabilistic model applies to a given set of hydrologic data.

Confidence Limits

Statistical estimates are often presented with a range, or confidence intenal,within which the true value can reasonablv be exDected to lie. The size of the

$iilrt,11;'Jir"'^r*, =,__,: ;","ni;Sample t"il'

K, size , K"sample

samP* r sire n K"

L' \'IE N.i* , K" -''" ' : "_;:

," i.ool "o I llt2.q62 .18 i : : l i i u .soo'o t9:: ii ill. i; l:ll ii iill

t ni r ifri 1fr * *il[ 1*t ''.*t ffift i*i i .l.: i! ;li. ii "o'

Ssmple

406 rppueo nvonorooY

confidence intenal depends on the confdence level p The upper and lower

boundarl' values of the confidence interval are called confdence limits (Fig'

1 2 . 6 . 1 ) .Conesponding to the confidence level p is a signifcance lelel a, given by

(12.6.1)

For example , r t B :90 percent ' then a- (1 09) /2 = 005 or5 percent '

For estimating the event magnitude for retum period T, the upper limit U7 "

and lo\\'er limit lr." may be specified by adjustment of the frequency factor

eauatioir:(r2.6.2)

( r2 .6 .3 )

where (| ^ and Kl - are the upper atrd lower confidence limit factors, which can

be determined foi'normally distributed data using the noncentral l disftibution

(Kendall and Stuart, 1967). The same factors are used to construct approxlmate

confidence limits for the Rarson Type III disribution Approximate values for

these factors are given by the following formulas (Natrella, 1963; U S Water

Resoutces Council, 1981):

K r i

K 7 -

U r . o

Probabi l i ry P = l -2o

in which

and

- 2

rnreuarcvarelvsis 407

(12.6.6)

I _ B- 2

U7,"= j+ srKf."

L 7 , , = Y + s r K L 7 . "

7 2

o = r i - i(r2.6.7)-. I

l . t

d

and

The quantity zo is the standard normal variable with exceedence probabiliry c

Example 12.6,1. Determine the go-percent confidence limits for the 100-year dis-charge for Walnut Creek, using the data presented in Example 12.5 1. The IoB-arithmic mean, standard deviation, and skew coefficient are 3.639' 0.4439, aDd-0.64, respectively, for 16 years of data.

Solution. For B = 0.9, a : 0.05 and the required standard nomal variable z " hasexceedence probability 0.05, or cumulative probability 0.95 From Table ll.2 l'the required value is z o : I .645. The frequency factor r(7 for I = 100 years wascalculated in Example 12.5 1asKrm= I 850. Hence, by Eqs (12 6 4) to (12'6 ' l)

z2^ , t (A5t2n - l - " - l - : i : - : : : = 0 9 0 9 8

2 \ n - l t 2 ( 1 6 l /

^ ' a . r 1 .645 t ' �t - x i i : { 1 . 8 5 0 ) 2 - i 6 -

= 3 . 2 5 3

Kr* � 1 .850 + [ (1 .850)? - 0 .9098 x 3 .253] "20.9098

l �850 - l ( | .850) ' � -0 .q098 x l .253 l r ' ' �0.9098

(Y*,0.0,

= |.286

The confidence limjts are computed using Eqs. (12 6.2) and (12 6 3):

Uroo.oos =I + rr1(, '*.0 o,

=3.639 + 0.4439 x 2.781

: 4.8,74

Lrm,o.os =! - rrfff*,6 0,

=3 .639+0.4439x1.286

: 4.210

The conesponding discharges for the upper and lower limits are (l}r4 814 : '74,820

cfs, and (10)a2'0 = 16,200 cfs, respectively, as compared to an estimated eventmagnitude of 28.900 cfs from Table 12.5.2. T\e confidence interval is quite wide

(12.6.4)

( r2.6.5) = 2.' l8l

K 7 -xt*,0.0,

FIGURE 12.6.1o€finitiol of conf&rlf€ | inits

I(, - ab

K2, - ab

(40E erpr-iEo uYonoloct

in lhis case because the sarnple size is small As the sample size increases' lhe width

oi the conl idence interval around the estimated f lood magnitude wil l diminish

Standard Error

The statuiard error of estimate s" is a measute of ths standard deviation of event

.rugni,"A"t .o.put"d f.o- samples about the true event magnitude Fofmulas

for-thc standarcl'error of estimate for the normal and Extreme Value Type I

distributions are (Kite, I977):

Normal 1

12 + 11 t ' 2- , - i rt t l

Extrenrc Value TYPe I

f r . . - ^ ^ - . " , 1 t ' 'a : l ; ( t

+ I I 3e6K7 '+ l . l ooo r< i ) l s

where .r is the standard deviation of the original sample of size l1 Standard errors

lnuu U" ut.O to consfuct confidence limits in a similar manner to that illustrated

in i"u*pt" 12.6.1, except that in this case the confidence l imits for signif icance

level c are defined as xT ! seza

Example 12.6.2' Determine the standard effor of estimate and the 90 percent con-

f idenc; l imits of the 5-year-return-period, 10-minute-duration rainfal l at Chicago'

l l l inois. From Example 12.3 2, the estimated s-year depth is lr = 0 78 in: also'

s = 0 . I 7 7 i n , K r : O ' 7 1 9 ' a n d a : 3 5

solr. l ior,.ThestandarderroriscomputedfortheExtlemeValueTypeldistr ibutionus ing Eq . ( 12 .6 9 )

I r . ",'1"'r " : l l ( r + l . t 3 9 6 K r + l � l o o o K ; ) l s'

1 r ' )

The 90 percent confidence l imits, with z" : | 645 for a = 0 05' arcrr l : J"zd =

0.?8 i b.046 x 1 .645 = 0.?0 and 0 86 in Thus the 5 year' lO-minute rainfal l

est imate in Chicago is 0.78 in with 90 percent confidence l imits [0 70' 0 86] 'n'

Expected ProbabilitY

Eryecled prohabitity is defined as the average of the true exceedence probabilities

of all nagnitude estimates that might be made ftoT -:u::"tjil-". samples of a

speciliecl s]zc for a specified flood frequency (Beard, 1960;U S WaterResources

( 1 2 . 6 . 8 )

( l 2 . 6 . e )

. l ] , , ' , . , . , n 0 " o . r ' n L t 0 0 0 ' r 0 . 7 t 9 r ' � , j ' t

o ' "l 15 J

:0 .046 in

rnequrrcr'arnrvsrs 409

Council, t98l). The flood magnitude estimate computed for a given sample isapproximately the median of all possible estimates; that is, there is an approxi-mately equal chance that the true magnitude will be either above or b€low theestimated magnitude. But the probability distribution of the estimate is positivelyskewed, so the average of the rnagnitudes computed from many samples is largerthan the median. The skewness arises because flood magnitude has a lower boundat zero but no upper bound.

The consequence of the discrepancy between the nFdl?n and the mean floodestimate is that, if a very la.ge number of estimates of fnood magnitude are madeover a region, on average more 100-year floods wil l occufthan expected (Beard,1978). The expected probability of occurence of flood events in any year can beestimated for events of nominal rctum period T by the following formulas, whichare derived for the normal distdbution, and apply approximately to the FbarsonType III distribution (Beard, 1960; Hardison and Jennings, 1972).

The expected probability for the normal distribution is expressed for asample size of n as

( r2.6. l0)

where z is the standard normal variable for the desired probability of exceedenceand /n-l is the student's t-statistic with n - I degrees of freedom. Calculationcan be performed using the appropriate tables for /n r and z. These computationscan also be caried out using the following equations (U. S. Water ResourcesCouncil, l98l; U. S. Army Corps of Engineers, 1972).

f o/ears) Exceedence probability Expected probability E(PJ

l t ' \ t / 2E\P = Pl t " - r > z( - - - l

L \ ' ? + l /

1000

100

20

l0

3 . 3 3

0.00I

0 . 0 1

0.05

0 . 1 0

0.30

/ rrn \0 . 0 0 1 1 1 . 0 + a I

\ z l . 55 /

0 . 0 1 1 1 . 0 * r o

]\ ,? l 16 l

/ 6 \0 . 0 5 i 1 . 0 + - I

\ n l 0 4 /

/ ? \0. l0l 1.0 + --a I

\ a r . 04 /

(12.6. I la)

(12.6. t tb)

(12.6.1Ic)

( r2.6. I td)

(12.6.1 le)o. :o / r .0 . 0 a6 l\ n 0 o 2 5 /

CHAPTER

I 3HYDROLOGICDESIGN

Hy.�drologic design is the proccss of assessing the impact of hydrologic events on awater resource system and choosing values for the key variables of the system sorhar ir wil l perform adequately. Hydrologic design may be used to develop plansfor a new structure, such as a flood control levee, or to develop managementprograms for better control of an existing system, for example, by producinga flood plain map for limiting construction near a river. There are many factorsbesides hydrology that bear on the design of water resource systems; these includepubiic weifare and safety, economics, aesthetics, legal issues, and engineeringfactors such as geotechnical and structural design. While the central concern ofthe hydrologist is on the flow of water through a system, he or she must also beaware of these other factors and of how the hydrologic operation of the systemmight affect them. In this sense hydrologic design is a much broader subject thanhydrologic anaiysis as covered in previous chapters.

13,1 HYDROLOGIC DESIGN SCALE

The purposes of water resources planning and management may be groupedroughl,v- into two categories. One is water control, such as drainage, flood con-trol, pollution abatement, i lsect control, sediment control, and salinity control.The other is v'eter use and management, such as domestic and industdal watersuppl),, inigation, hydropower generation, recreation, fish and wildlife improve-ment. low-flow augmentation for water quality management, and watershedmanagement. In either case, the task of the hydrologjst is the same, namely,

:.:*..lnt*

a design inflow, to route the flow through the system, and to check

417

whether the output vaiues are satisfactory. The difference between the two casesis that design for water control is usually concemed with extreme events of shonduration, such as the instantaneous peak discharge during a flood, or the minimumflow over a period of a few days during a dry period, wirile design for water useis concemed wjth the complete flow hydrograph over a period of years.

The hydrologic design scale is the range in magnitude of the design variable(such as the design discharge) within which a value must be selected to determinethe jnflow to the system (see Fig. 13. I . I ). The most impo{tait factors in selectingthe design value are cost and safety. It is too costly ro he$gn small structuressuch as culverts for very large peak discharges; however,lffi a major hydraulicstructure, such as the spillway on a large dam, is designed for too small a flood,the rcsult might be a catastrophe, such as a dam's failure. The optimal magnitudefor design is one that balances the conflicting considemtions of cost and safety.

Estimated Limiting Value

The practical upper limit of the hydrologic design scale is not infinite, since theglobal hydrologic cycle is a closed system; that is, the total quantity of wateron earth is essentially constant. Some hydrologists recognize no upper limit, butsuch a view is physically unrealistic. The lower limit of the design scale is zero inmost cases, since the value of the design variable cannot be negative. Althoughthe true upper linrit is usually unknown, for practica) purposes an estimated upperlimit may be determined. This estimated Limiting vaLue (ELV) is defined as tfreIargest magnitude possible for a hydroLogic event at a given location, based onthe best available hydroLogic information. The range of uncenainty for the ELV

r00 Esrimated limilin8 value (ELv)

t000

500

200

100

50

25

l 0

5

2

'lcr 70

Eg 6 0

50

g

JT :

e

FIGURE 13.1.1Hydroiogic design scale. Approximatemnees of the deqgn le\e l for d i f ferenr typesof structures are shown. Design rray b€based on a percentage of the ELV or ona desig! retum period. The values for thelwo scales shown in the diagmm are illus'lrarive only and do noi correspond directly

TABLE 13.1.IGeneralized design criteria for water-control structures

419418 ' . r r r rmr r ro ror -oo t

depends on the reliabil i ty of information, technical knowledgc, and accuracl ofanall,sis. As information, knowledge, and analysis in]prove, the estimate betterapprorimatcs the true upper l imit, and its range of uncertainty decreases. Therehave bcen cases in which observed hydroiogic events exceeded their previouslyestimatcd l imiting values.

The concept of an estimated l inrit ing value js jmplicit in the commonlywed probable naximum precipitatto, (PMP) and the corespond\ng probablematiirLun food (PMF). The probable maximum precipitation is defined by theWorld Meteorological Organization (1983) as a "quantity of precipitation that isclose to the physical upper limit for a given duration over a pa-rticular basin."Based on worldwide records, the PMP can have a retum period of as long as500.000,000 years, conesponding approximately to a frequency factor of 15.Howcver. the retum period varies geographically. Some would arbitrari ly assigna rcturn period. say 10.000 years, to the PMP or PMF, but this suggestion hasno pirl sical basis.

Probabil ity-Based Limits

Because of its unknown probabil ity, the estimated )imiting value is useddeterniinistically. Lower down on the design scale, a probabil it l '- or frequenclbased approach is comrnon]y adopted. The magnitudes of hydrologic events atthis level are smaller, usually within or near the range of frequent obserr ations.As a result. their probabil it ies of occurrence can be estimated adequateiy whenhydrologic records of sufficient length are available for frequency analysis. Theprobabil istic approach is less subjective and more theoretically manageable thanthe dctcrministic approach. Probabil istic methods also lead to logical ways ofdeternrining optimum design levels, such as by hydroeconomic and risk analyses.which wil l be discussed in Sec. 13.2.

For a densely populated area, where the failure of water-control workswould result ln loss of l i fe and extensive properly damage, a design using theELV ,night be justified. In a less populous area where failure would result onlyin minor damage, a design for a much smaller degree of protection is reasonable.Betu'ecn these extremes on the hydrologic design scale, varying conditions existand r arying design values are required. When the probabilistic behavior of ahydrologic e\ ent can be deterrnined, it is usuaily best to use the event magnitudefor a specified retum period as a design value.

Based on past experience and judgment, some generalized design criteriafor $'ater-control structures have been developed, as summarized in Table l3. 1. luAccording to the potentiai corsequence of failure, structures are classified asmajor. intennediate and minot; the corresponding approximate ranges on thedesign scale are shown in Fig. 13.1.1. The criteria for dams in Table 13.1.1pertain to the design of spillway capacities, and are taken from the NationalAcaderny of Sciences (1983). The Academy defines a small dam as having 50-1000 acre ft of storage or being 25+0 ft high, an intermediate dam as having1000-50,000 acre.ft of storage or being 40 l00fthigh, and a large dam as havingmore rhan 50,000 acre ft of storage or being more than t00 ft high. In general,

Type of structure Return period (years)

Highway culvertsIrw trafficIntermediate trafficHigh traffic

Highway bridgesSecondary systemPrimary system

Farm dra'nageCulvertsDitches

Urban drainageStorm sewers in small citiesStom sewers in large cities

AirfieldsLow trafficInlermediate trafficHigh traffic

LeveesOn farmsAround ciries

Dams with no likelihood ofloss of life (low hazard)Small dansIntermediate damsLarge dams

Dams with probable loss of Iife(significan! hazard)Small damslntermediate damsLarge dams

Dams with high likelihood of considerableloss of life (high hazard)Small damsIntermediaie damsLarge dams

5 -10lG-2550,100

l0-5050-100

5-505 5 0

25-50

5- 1010 2550 100

2 5 050 200

50 100I00 +

100 +

*-t*O"

5Ost50- 1007e1CAq.

50-1009ol0OVolOOTc

j . l

there would be considerable loss of life and extensive damage if a major structurefailed. In the case of an intermediate structure, a small loss of life would bepossible and the damage would be within the financial capability of the owner.For minor structures, there generally would be no loss of life, and the damagewould be of the same magnitude as the cost of replacing or repairing the structure.

Design for Water Use

The above discussion applies to the hydrologic design for the control of excessivewaters, such as floods. Design for water use is handled similarly, excapt thatinsufficient rather than excessive water is the concem. Because af the long time

427420 .rppriEouronot-ocY

span of droughts. there are fewer of them in historical hydrologic records than

,t*" "* "-tErn. noods lt is therefore more difficult to determine drought design'""^ "r-

,-rt-rg-tt.-it"q""ncy- analysis, especially if th€ design event lasts several

v e a r s . a s i s s o m e t r m e s t h e c a s e l n * u t " r s u p p l y d e s i g n . A c o r n m o n b a s i s f o rii. i"rG"-"i.""i"ipui *ut"t supply sysrcms is tlte criical drouS.ht^Qf record'

iir"t ", irt. *-u recorded drought -

The design is considered satisfactory if it

ffi r;o;i; *"i"t ut tr,l. requireJ rate throughout an equivalent- critical pedod

irr. iitili"i* of the criticil-period approach is that the risk level associated

iln;;*u;. d"sign on this single hiitorical event is unkrown To overcome

irrir-ri-i",1"", nreth;ds of syntheti; streamflow generation have been developed

urlng ao,t-tpu,"r, and random number generation to prepare synthetic- streamflow

,".o'ra, ,ftit are statistically equivalenito the historical record Together with the

ti.tori.uf ,."o.0, the synthetic records provide a probabilistic basis for design

us;int, oruucl.t, events iHirsch, 19?9; Salas' et al ' 1980)"-

nvlr"i'"ti. design for water use is closely reguiated by the legal framework

"f -",;;';;;,-.tp.!i"lly in uaa regions The law specifies which users will

r,uu" ,nro itlo"ution, reduced in the lvent of a shonage. In an effort to protect

ifre fisn and wildlife of a stream, methods have been developed in rccent years

i"-q""",tify theLr need for inst-ream flow (Milhous and Grenney' 1980) Uniike

flood control and water supply, ror which sufficient hydrologic information ls

prJa"d'[y n"* rate and water level' instream flow neecis are influenced also

irn nrrbirtirv. temDeraturg' and other water quality variables in a complex manner

;;t;;;; f;t'';6..i.r,o unoth"' watei resources systems are subject to the

demands of competmg users' me need to maintain instream flow' and competing

i"-""Jt ra""oL flJod control Hydrologic design must speciiy the appropnate

design ievel for each of these factors

13.2 SELECTION OF TIIE DESIGN LEVEL

A hyd.rologic design level on the design scale is the mae1it1de.1f,1h; hydrologic

"""i, ," uJ .*tia""red for the design if a structure or project As it is not always

economical to desrgn strucrures ind pmjects for the estimated limiting value'

ii," i-ii L ofren m-odified for specifii design purposes. The final design value

","1 1,. ftt,n., *"dif ied according to engineering judgment and lhe experience

oi ir,r" i"ttgn"t or planner' Three approaches are commonly used to determine a

ivit"i"g"?.trg" 'ialue: an empiricai approach' risk analysis' and hydroeconomic

analysis.

Empirical APProach

Dudng the early years of hydraulic engineering practice' around the early 1900s'

r cnill\,,av desiened to pass a riood SO to tOO percent larger than the largest

;.;5;;;i";;;i"d of perhaps 25 vears was considered adequate This design

"ii "-lon i, no nror" than a rule of thumb involving an arbirary factor of safety. As

in .iurrrpt. of tn" inadequacies of this criterion' the Republican River in Nebraska

il- r s3!-'.^p.ti.n""d a dood over l0 times as large as any that had occuned on

that river during 40 prior years of record. This design practice was fbund to beentirely inadequate, and hydrologists and hydmulic engineels searched for bettermethods,

As an empirical approach the most exfeme event among past observationsis often selected as the design value. The probability that the most extreme eventof the past N years will be equaled or exceeded once during the next l? years canbe estimated as

l r .f

( 1 3 . 2 . l )

Thus, for example, the probability that the largest flood observed in N years willbe equaled or exceeded in N future years is 0.50.

If a drought lasting m years is the critical event of record over an N-yearperiod, what is the probability P(N,m,n) that a worse drought will occur withinthe next lt years? The number of sequences of length m in N years of record isN-m+ l , and inzyearso f recordn-z * l . Thus the chance tha t the wors tevent over the past and future spans combined will be contained in the n futureyears is given approximately by

P(N, m, n):( n - m * l )

( N - m + l ) + ( n - m + l )( r3 .2 .2 )

n - m + |( n > m )

N - f n - 2 m + 2

which reduces to (13.2.1) when n - l.

Example 13.2.1. If the critical drought of recotd, as determined from 40 years ofhydrologic data, lasted 5 years, what is the chance that a more severe drought will

occur during the next 20 years?

Solution. Uslng Eq. (13.2.2),

2 0 5 + lP(40,5,20) :

4 0 + 2 0 - 2 x 5 + 2= 0 .308

Risk Analysis

Water-control design involves consideration of risks. A water-control structuremight fail if the magnitude for the design retum period ? is exceeded within theexpected life of the structure. T\is natural, or inherent. hydroiogic risk of failurecan be calculated using Eq. (12.1.4):

P : 1 - 1 1 - P I Y > - r - t 1 4 ( 1 3 . 2 . 3 )

where P(X > ri = UT, and tl is the expected life of the structure; R representsthe probability that an event x > rr will occur at least once in z years. This

422 qrlr-teo svorolocr

2 5 l0 20 50 100 200Design life n ()earc)

FIGURE 13.2.1Risk of al least one exceedence of the design event during the design life-

relationship is plotted in Fig. 13.2.1.If, for exampie, a hydrologist wants to be

app.oximatell 90 percent certain that the design capacity of a culvert will not be

exceeded during the structurc's exPected life of 10 years, he or she designs for

the ioo-year peak discharge of runoff. If a 4O-percent risk of failure is accePtable'

the design return period can be reduced to 20 years or the expected life extended

to 50 years.

Example 13,2.2. A culvert has an expected life of l0 years. If the acceptablerisk of at least one event exceeding the culvert capacity during the design life is l0percent, what design return period shouid be used? What is rhe chance that a culvertCesigned for an event of this retum period will not have its capacity exceeded for50 years ?

Solution. By Eq. (13 .2 .3)/ I fR = r - l r - ? l

o . r o : r - ( r - | ) '

and sol\ ing yields f: 95 years.

If f: 95 years, the risk of failure over,r : 50 years is, r r50

n = r - l r - r l\ - 9 5 /

= 0 . 4 1

So the probability that ihe capacity will not be exceeded during this so-year periodis I - 0.41 : 0.59, or 59 percent. . t

i 1

It can be seen in tsig. 13.2.1 thal, for a grven risk of4uilrr.. the rcqurreddesign return period I increases linearly with the design life r, as 7 and z becomelarge. Under these conditions, what is tbe risk of failure if the design return periodis equal to the design l ife, that is, T = n'! By expanding Eq. (13-2.3) as a powerseries, it can be shown that for large values of n, I - (l - l l7)' : | - e-n11,so, for I: n, the risk is I - e

-r = 0.632. For example, there is approximatelya 63-percent chance that a 100-year event will be exceeded at least once duringthe next 100 years.

Although natural hydrologic uncefiainty can be accounted for as above,other kinds of uncertainty are difficult to calculate. These are often treated usinga safety fqctor, SF, or a salery, margin, SM. L€tting the hydrologic design valuebe ,L and the actual capacity adopted for the project be C, the factor of safety is

" o : 9' ^ L

and the safety margin is

S M = C - l

423

r00

a

&E r 0

(r3.2.4)

( 13 .2 . s )

The actual capacity is larger than the hydrologic design value because it hasto allow for other kinds of uncertainty: technological (hydraulic, structural,construction, operation, etc.), socioeconocic, polit ical, and environmental.

For a specified hydrologic risk R and design life n of a structurc,Eq. (13.2.3) can be used to compute the rclevant retum period ?'. The hydro-logic event magnitude I corresponding to this exceedence probability is foundby a frequency analysis of hydrologic data. The design value C is then given byt multiplied by an assigned factor of safety, or by L plus an added margin ofsafety. For example, it is customary to design levees with a safety margin of oneto three feet, that is, one to three feet of freeboard above the calculated maximumwater surface elevation.

Ilydroeconomic Analysis

The optimum design rctum period can be determined by hydroeconomic analysisif the probabilistic nature of a hydrologic event and the damage that will resultif it occurs arc both known over the feasible range of hydrologic events. As thedesign retum period increases, the capital cost of a structure increases, but theexpected damages decrease because of the better protection afforded. By summing

, R = r - ( r - i l

- - R = 0 . 6 3 f o r n = 7

ano rarge 'l.

)

421

the capital cost and the expected damage cost on an annual basis. a design retumperiod having minimum total cost can be found.

Figure 13.2.2(a) shows the damage that would result if an event, such asa flood. ha\' ing the specified retum period were to occur. If the design eventmagnitude is r1, the structure wil l prevent all damages for evenrs with .r < 17.but none for -r > ,rr, so the expected annual damage cost is found by takingthe product of the probabil ity f(x)dx thal an event of magnitude -r wil l occurin any, given year, and the damage D(,r) that would result from that event, and

t Rcrurn period {yca.s)

i 2 5 r 0 2 5 5 0 t 0 0 2 0 a

(

5 ( L

I 30lle

o

t0r:l

60

1l)

?? 11)

2tr

0

0.04 0.02 0.01

/ Damagc risk cosll/ = shaded area in (a)

l : 5 1 0 2 5 5 0 r 0 0

Design return period (years)

o Risk cosr c Capital cosi ^ Toral cost

( r) Hydroeconomic analysis.

FIGURE T3.2.2D€tefrnrDarlon of the oprimum design retum period by hydroecononic analysis (Exanpte 13.2.3).

0 . 2 0 . I

Annual exceedence probability

. , D-r , r fe ' fo ' evef l ' . f \drod. rerJrn pe' .od. .

Op' imum design rerurn\1

Per iod (25 Years) i

(rsrq.r 425

integrating for x) x7 (the design level). That is, the expecred annual cost Dr is

o,: f ,o61.11'1a, (13.2.6)

which is the shaded area in Fig. 13.2.2(.a).The integml (13.2.6) is evaluated by breaking the range of -I ) x1 into

interyals and computing the expected annual damage c_osl for events in eachi n t e r v a l . F o r x i l = r < r i , ) {

r , foo, :

),, ,DG)fe) dx ( 1 3 . 2 . 7 )

which is approximated by

nr, =lD(", -{:?lii' I i'" , , '= L 2 l j , , Iu tdx

( 1 3 . 2 . 8 )D k , , t I D t r . ) .= -

, - l P k . . x , \ P ( x s x i t t

Bu t P( r < 11) - P( r < r i - r ) : [1 - P(x > - r ) ] - [ l p (x = x i - � ) ] =P(x > xi-1) - P(x > xi), so (13.2.8) can be written

1gti = D(.xi 1)-+ D(xi) Ipi" =,r, ,; p(x> xi) ( 1 3 . 2 . 9 )

and the annual expected damage cost for a structure designed for retum period Tis siven bv

D. = i l^-+*r]lr1.r =.r, ,; - p(,, =.ri)l

By adding Dr to the annualized capital cost of the structure, the total cosr canbe found; the optimum design retum period is the one having the minimum totalcost.

Example 13.2.3. For events of various retum perjods at a gjven locarion, thedamage costs and the annualized capital costs of structures designed to control theevents, are shown in columns 4 and 7, respectively, of Table 13.2.1. Dete.minethe expected annual damages if no shucture is provided, and calculate the optimaldesign return period.

Solution. For each retum period shown in column 2 of Tabie 13.2.1. the annualexceedence probability is P(.r > r1) = l/f. The corresponding damage cost A, isfound using Eq. (13.2.9). For example, for the interval i : I between T = l yeara n d f = 2 y e a r s ,

f ̂ Drr , r ' lLD t - 1 "-:Y)L: :!::.: I p,r: ,,, pq,2 ,r,L 2 t '

( 1 3 . 2 . l 0 )

426 ,v,r,r,rr:o nr onor-ocr (

TABLIT 1-1.2.1Calculalion of the optimum design return period by hydroeconomicanalysis (Example 13.2.3)

Column: IIncre-ment

2ReturnperiodTLyearsJ

l 4Annual Dsmege

probability($)

lncremental Damageexpected riskdsmage cost($/yearl ($/year)

7 8Capital Totalcost cost

($/year) ($/year)

, Il 2

l r 0

5 2 06 2 5I 5 08 1009 200

t.0000.5000.2000 .1000.0670.0500.0400.0200 .0 r00.005

020,000 5.00{)60,000 12,000

t40,000 10,000l?7.000 5,2832t 3,000 3,250250,000 2,315300,000 5,500,100,000 3,500500,000 2.250

49.098 044,098 3,00032.098 r4,00022.098 23,00016,815 25,000r3,565 27,000I 1,250 29,0005,750 210,0002,250 60.000

0 80.000

49,0984'�7 ,09846.09845,0984 1 , 8 1 540,56544,25045.75062,25080,000

Annual expected damage = 549,098

_ r 0 2 0 . 0 0 0 r , ^ ^ .2

_ / { t u _ u . ) r

= $5,000/year

as shown in column 5 of fte table. Summing these incremental costs yields an annuale\pected damage cost of $49.098/year if no structure is builr. This represents thear erage rnnual co.l of f lood damaCe o\ er many ) ears. arsuming constan( econom,.conditions. This amount is the damage risk cost corresponding to no structure, andis shown in the first l ine of column 6 of the table.

The damage lisk costs diminish as the design return period of the controistiucture incrcases. For example, if f = 2 years were selected. the damage risk cosl\\ould be.19.098 Aa, : 49, 6n, 5, 000 : $44,098/year. The values oldamagerisk cost and capital cost (column 7) are added to form the total cost (column 8);thc rhree costs are plotted in Fig. 13.2-2(b). It can be seen from the table and thefigure that the oplimum design return period, the one having minimal roml cost, is25 years, for whjch the total cost is $40,250/year. Of this amount, $29.000/yearrl l percent) is capilal cost and $l1,250/yed (28 percent) is damage risk cost.

Hldroeconomic analysis has been applied to the design of flood controlreserroirs, Ievees, channels, and highway stream crossings (Corry, Jones, andThonpson, 1980). For a flood damage study, the duration and extent of floodingmust bc determined for events of various retum periods and economic surveysmust be taken to quantify damages for each level of flooding. The social costs offloodjng arc dilTicult to quantify. The U. S. Army Corps of Engineers HydrologicEngineering Center jn Davis, California, has available the following computerprograms for hydroeconomic analysis (U. S. Army Corps of Engineers, 1986):

HyuRor-ocrc oesrcN 427

DAMCAL (Darnage Reach Stage-Damage Calculation), EAD (Expected AnnualFlood Damage Computation), SID (Structure Inventory for Damage Analysis),AGDAM (Agricultural Flood Damage Analysis), and SIPP (Interactive Nonstruc-tural Analysis Package).

13,3 FIRST ORDER ANALYSN OF UNCERTAINTY

Many of the uncertainties associated with hydrologic systemq ar\ not quantifiable.For example, the conveyance capacity of a culven wilh an uholism.rcted entrancecan be calculated within a small margin of error. but durifi a flood, debrismay become lodged around the entrance to the culvert, reducing its conveyancecapacity by an amount that cannot be predetermined- Hydrologic uncedainty maybe broken down into three categories; natural, or inherent, uncertairu1.,, whicharises from the random variability of hydroiogic phenomena; model uncertainty,which results from the approximations made when representing phenomena byequations; and parameter uncertainty, which stems from the unknown natureof the coefficients in the equations, such as the bed roughness in Manning'sequation. lnherent uncertainty in the magnitude of the design event is described byEq. (13.2.3); in this section, model and parameter uncertainty will be considered.

The first order analysis of uncertairD, is a procedure for quantifying theexpected variability of a dependent variable calculated as a function of one ormore independent variables (Ang aIId Tang, 1975; Kapur and Lamberson, 1977;Ang and Tang, 1984; Yen, 1986). Suppose w is expressed as a function ofx:

Therc are two sources of error in lr: first, the function /, or model, may beincorrect; second, the measurement of -r may be inaccurate. In the followinganalysis it is assumed that there is no model error, or Dics. Kapu and Lamberson(1977) show how to extend the analysis when there is model error. Assuming,then, that /( ) is a correct model, a nominal value of x, denoted .r-, is selected asa design input and the corresponding value of w calculated:

w = f(.x) ( r 3 . 3 . 1 )

t u = f G )

)t | )2f

w = f t x ) - ? r r r ) + ) , " j r x z l + . . .4X Zt aA'

( 1 3 . 3 . 3 )

( l l . l . 2 ,

If the true value of .r differs from t, the effect of this discrepancy on p can beestimated by expanding /(;r) as a Taylor series around -r = ':

where the derivatives df/dx, d2fldt2,. . . , are eyaluated at r : r. If second andhigher order terms are neglected, the resultingfrst order expression for the errorr n w i s

dfw _ 8 , = - : l x _ 7 ) ( 1 3 . 3 . 4 )

The variance of this error is sl = El(w - fr)21where E is the expectation operatorlsee Eq. (11 .3 .3 ) ] t tha t i s ,

128

( l 3 . 3 . 5 )

where sj is the variance ofr.Equatirin ( 1 3.3.5 t gives the variance of a dependent variable w as a lunction

,'f rhe rariance of un independenr variable.r, assuming that the fuDctional rela-tjonship !r = /(r) is correct. The value s\| is the standard error of estimate of w.

If rf. is dependent ol1 several mutualLy independent variables .!1, ;t2, . . ,.r,,. i l can be shown by a procedure similar to the above that

| ^ . \ )+ : t . r :

\.dx" I ^, ( r3 .3 .6 )

,i:'{l#--"]'}

':"=(ff)":

, " " 2 , - " , 2t t d f \ t l d t \ "" ; = l , r r / ' : + \ ; l ' ; . +

Kapur and^Lamberson (1977) show how to extend (13.3.6) to account for theef iec t on s j o f cor re la t ion be tween. r r , x2 , . . . , xn , i fany ex is ts .

First-Order Analysis of Manning's Equation: Depth as theDependent Variable

N4anning's equation is widely applied in hydroiogy to determine depths of flowfor specified flow rates, or to determ;ne discharges for specified depths of flow,iaking into account the resistance to flow in channels arising from bed roughness..\ common application, such as in channel design or f lood plain delineation, isto calculate the depth of flow y in the channel, given the flow rate Q, roughnesscoefficient l, and the shape and slope of the channel as determined by design or bysurveys. Once the depth of flow (or elevation of the water surface) is known, thevalues of the design variables are determined, such as the channel wall elevationor the flood plain extent. The hydrologist faced with this task is conscious of theuncartainties involved, especially in the selection of the design flow and Manningroughness. Although it is not so obvious, there is also uncertainty in the value ofihe friction slope S7, depending on how it is calculated, ranging from the simplestcase of uniform flow (,S, : 57) to more complex cases of steady nonuniformflow or unsteady nonuniform flow [see Eq. (9.2.1)]. The first-order analysis ofuncertainty can be used to estimate the effect on ) of unce ainty in Q, z, and Sp.

Consider, first, the effect on flow depth of variation in the flow rare Q.N{anning's equation is wdtten in English units as

s: !12 s,,,1or,, ( 1 3 . 3 . 7 )

ivhere A is the cross-sectional area and R the hydraulic radius, both dependent onthe flow depth ). If variations in y are dependent only on variations in 0, then,

by (13 .3 .s ) ,

uvonolocrc orsrc^- 429

( r 3 .3 .8 )

( 1 3 . 3 . l 0 )

" 2 _-]l , \ 2

\ d Q l ' v

where dy/dQ is the mte at which the depth changes with changes in e. Now, inChap. 5, it was shown tEq. (5.6.15)l rhat rhe inverue o.f $is Jerivatiie, namelydQldy, is glven for Manning's equation by I j

* = ol iT . \a] u

1 ' . . . n;dt -l3R dy A dy )

Table 5.6.1 gives formulas for the channel shape funcrion (Z/3R)(dRldg +(l/A)(dA/dy) for common channel cross sections. Substituring into (i j .:.A),

" 2 _ s[

^ ,1 2 dR I dA\ l" \ t q 6 i . ' - ' A d y )

B"r.: g/9 . - CVp, the coefficienr of variation of the flow rare (see Table I I .3. I ),so (13.3.10) can be rewritren

( 1 3 . 3 . 1 l )

(13.3.12)

,, CVL'

l z a n t d A \ 2

\ 1 R 6 - i d ' - Jwhich specifies the variance of the flow depth as a function of the coefllcient ofvariation of the flow rate and the value of the chalnel shape function. To takeinto.account also the uncertainty in Manning's roughness r ind the friction slope.S7, it may be simila-rly shown, using Eq. (13.3.6), rhar

, cv | + cv; + ( l /4)CVi .s - - = _ - _ - -'

l 2 d R l d A \ ' �

\1R6 -

i dj)giving the variance of the flow depth y as a function of the coefficients of variationot tlow rate, Manning's I and fricrion slope, and the channel shape function.

Example 13,3,1, A 5o-foot wide rectangular channel has a bed slopg of onepercent. A hydrologist estimates that the design flow rate is 5000 cfs a;d that theroughness is |l : 0.035. If the coefficients of variation of the flow estimate and theroughness estimate a.e 30 percenr and 15 percent, lespecfively, what is the standarderror of esrimate of the flow depth )? If houses are.built neit to this channel withltoor elevaoon one fool above the water surface eievation calculated foa the designevent, estimate the chance that these houses wiil be flooded during the design eventdue to uncenainties involved in calculating the wate. Ievel. AssLime uniforrn flow.

Solution. For a \ridth of 50 feer, A : 501 and R = 50-r/(50 + 2f); rhe flow depthfor the base case is calculated from Manning's equation:

430(

\I)FLIED HYDROLOGY

o: L/9 sI/1ARLtj- n i

I 4 q . 5 O r5000 -

O Orr ,0.0 l ' r t rSOl r ( r j !

rvhich is solved using Newton's i teral ion technique (see Sec. 5.6) to yield

) = ? 3 7 f t

The standard eror of the estimate is Jl . calculated by Eq. (13.3.12) wjth CVO =0.30, C\j , = 0.15, and CVt = 0. From Table 5.6.1. for a rectangular channel,

l 2 d R 1 d A \ t 5 8 + 6 , . r ,\3R dr

' i dy l= 3r(B + 2r

5 x 5 0 + 6 x 7 . 3 7

esrcr 431

It is clear from Example 13.3.1 that reasonable amounts of uncertainty inthe estimatjon of Q and n can produce significant uncertainty in flow depth. Al5-percent error in estimating n : 0.035 is an error of 0.035 x 0.15 = 0.005.This would be indicated from a measurement of 0.035 10.005, which is aboutas accurate as an experienced hydrologist can get from observation of an existingchannel. A 3o-percent error in estimating Q is 5000 x 0.30 = 1500 cfs. Anestimate of Q : 50001 1500 cfs may also reflect the conect order of uncertainty,especially jf the design retum period is large (e.g., 7 = 1Q0 fpars).

The use of the channel shape funcrion t2l3Rt\dRldjt I tUANdA/dy) in(13.3.12) depends on knowledge of dR/dy and, d.A/dy. whichftnay be difficult toobtain for inegularly shaped channels. Also, the assumption that y depends onQ alone may not be valid. In such cases, Eq. (13.3.6) can be used to obtain s,trcating ) as a function of Q and, n, and a computer program simulating flow inthe channel can be used to estimate the required panial derivatives dyldQ and

4,/ rr by rerunning the program for various values of Q and n and reading off thecomputed values of flow depth or water surface elevation. Figure 13.3.I showsthe results of such a procedure for the channel and conditions given in Example13.3.1. The gradients dylAQ and fil0n are approximately linear for this example;this validates the use of only first-order terms in the analysis of uncefiainty (ifthe lines were significantly curved, analysis would require keeping the second-order terms in the Taylor-serics expansion).

Example 13.3.2. For the sarne conditions as in Example 13.3.1 (B : 50 fr, Q =5000 cfs, S., = 0.01, n = 0.035), the variation of f low rate with flow depth at thebase case level has been found from Fig. (l 3.3. l) to be dQl at : I 02E cfs/ft, andthe var ia t ion o f nwi th f lowdepth ,6n /ay=g.gg7r t - ' . I f CVa = 0 .30 and CV, :0.15, calculate the standard error of y.

Solution. Frcri, Eq. (13.3.6),

In this case, rO : 5000 x 0.30 = 1500, r, = 0.035 x 0.l5 : 0.0053; also, o!*ld8:1l 1028, }yl dn = 1/0.0072. Thus,

, t ' r l"i - [r;J x (1500)2 + 10n072-,J

x (0.00s3)'�

or Jr = 1.63 ft as computed in Example 13.3.1.

First-Order Analysis of Manning's Equation: Discharge asthe Dependent Variable

Another application of Manning's equation is the calculation of the discharge orcapacity C of a stream channel or other conveyance structure for a given depth,roughness coefficient n, bottom slope, and cross-sectional geometry. Manning'sequation (i 3.3.7) can be expressed using R = A/P as

3 x 7 . 3 7 ( 5 0 + 2 x 1 . 3 1 )

_ w.206

So' C \ 6 + 6 Y t r l 4 ' C V ;

, ; = _ t ; ; R , t d A , : _\,1n 4- - V^

( 0 .30 ) , + (0 .15 ) ,

(0.206)r

or s, = L63 ft .lf the houses are built with their floors one foot above the calculated water

surface elelation, they will be flooded if the actual deprh is greater than 7.37 r1.00 = 8.37 fl. lf the water sudace elevation ) is normally dislributed, then theprobabili!) that they will be flooded is evaluated by converting ), to rhe standardnormal variable z by subtracting the mean value of _! (7.37 ft) from both sides ofthe jneqLral i ty and dividing by the standard enor (1.63 ft) :

P 0 > 8 . J / r p t ' - J ! > 8 1 7 - - 1 7l � O J l � O J

t \ ) - 7 1 ' � 7 \: P { : > 0 . 6 1 3 |

:P ( i > 0 613 )

: I F , (0 613 )

where F- is the standard normal distr ibution funcrion. Using Table 11.2.1 or themethod employed in Example 11.2.1, the resul i is F:(0.613) = 0.73, so PLv >E.37) = 1 - 0. '73:0.2' l . There is approximately a 27 percent chance that thehouses rvill be flooded during the design event due ro uncertainties in calcutatingthe rvater level for that event.

This e\ample has trcated only parameter uncenainty in the caiculations. Thetrue probability that the houses will be flooded is greater than that calculatedhere, because the critical flood may exceed the design magnitude (due to naturaluncenainty).

t=(fii1"".(!^)":

0= 50{)0 crs. tr = 0.035

432 .crprieouvonot-ocv

o.o4o iI

0.035 oo. ;

5 6 1 8 9 l 0Deplh r- (it)

F-tGURU l3.3. l\rrr iat ion of the f low deplh with f low rate and wiLh Manning's n. Rectangular channel wirh width-s0 f l , bed slope 0.01. Unifonn f low assumed. (Exanrple 13.3.2).

in which P ls the wetted perimeter. Ferforming first-order analysis on (13.3.l3),rhe coefficient of variation of the capacity can be expressed as

C: O = L32StnAs,p-213- n ( 1 3 . 3 . r 3 )

I ^CV!=cv ; n au r , ( 1 3 . 3 . i 4 )

o = r.o(L e7' or'�B + |e5o'3

r;2n)sltz ( 1 3 . 3 . 1 s )

assuming CVa :0 and CVr : 0 .Manning's equation for a chalnel and flood plain (overbank) can also be

expressed as (Chow, 1959)

in which z. and n6 are the roughness coefficjents for the channel and the flood-plain, respectively ar\d 4,, P,, A6, and, P6 are the cross-sectional areas andthe wetted perimete$ ofthe channel and the overbank flow, Equation (13.3.15)assumes that the cross-sectional shape of the channel and the flood plain are bothsymmetrical about the channei center line. This equation can be used to evaluateIevee capacity (the flow rate the levee can ca[y without overtopping). The leveecapacity can be considered a random variable related to the independent randomvariables n", n6, and S7. Applying first-order analysis, the coeificient of variationof the capacity is (Lee and Mays, 1986)

cvi = Jcvl, + $cvl.. (?;'.'t, ( 1 3 . 3 . 1 6 )

, nyDRolocrc DEsIcr-

where CV,a.. CVp, CV^r, and CV2,, have been assumerJ negligible, and

1 P : l +

433

R : P { - < 1 l\ t , I

: P ( C - L < 0 )

( r 3 .3 . 17 )

(13.4. t )

,l " 11at'1''' 1 P, 1'')\ n 6 t \ A , | \ P 6 l

In studies of flood dara on the Ohio River. Lee and Mays (19g6) concludedthat uncertainties in the roughness coefficients and the friction slope accounrfor 95 percent of the uncerrainties in compuring rhe capacily. The), presented amethod for determining the uncartaint)' in the friction shpdi using-rhe observedflood hydrograph of the river. F

13.4 COMPOSITE RISK ANALYSISThe previous sections have introduced the concepts of inherent uncertainty dueto the natural variability of hydrologic phenomena, and model and paramercruncertainty arising from the way the phenomena are analyzed. Composite riskanalysis is. a method of accounting for the risks resulting from the various sourcesof uncertainty to produce an overall risk assessment foi a panicular design. Theconcepts of loading and capacity are central to this analysis.

The Loading, or demand, placed on a system is the measure ofrhe rmpact ofextemal events. The demand for water supply is determined by the peopie whouse the water. The magnitude of a flash flood depends on the characteristics oftbe storm producing it and on the condition of thi watershed at rhe time of thestorm. The capociry, or resistance, is the measure of the ability of the sysrem towithstand the ]oading or meet the demand.

. If loading is denoted by L and capacity by C, then the risk of failure R isgiven by rhe probability that a exceeds C, or

The risk.depends upon the probabil ity disrriburions of Land.C. Suppose that theprobability- density function of L islil/. This function could be, for example, anExtreme Value or log-harson Type III probabiljry density function for exremevalues, as described earlier. Giver f(L), the chanie that tire loading will exceeda fixed and known capacity C* is (see Fig. 13.4.1)

P(L>c*)= f.rrrro, , l3 .4.2)

. TIe true capacity js not ktown exactly, but mhy be considered to haveprobability density function g(C), which could be the normal or lognormal distri-bution arising from the first-order analysis of uncertainty in the syite- capacrry.

f-11 exlmnte. if Manning s equation has been used to deiermine the capacity of a

nydraulrc srructure. the uncertainty in C can be evaluated by first-order analysisas described above. The probability that the capacity lies within a small ranse

0 .5

ll :1

0 . l

0 .2

43il :pprreos'rorouocr

u 0 I 0 2 .0 3 .0 4 .0 5 .0 6 .0 7 .0 8 .0Loading r. and capacity C

{

| , , _ . , 2 n

-\i z7r

The risk R is evaluated using (13.4.2) with C *: 5:

R= I f tL i dL

= | L e - ' L 1 P / 2 d L

J ' 'E-'

- ( t r ^R = l - l : e \ L 1 t ' , 2 d L

J _. ,|-2tr

The integral is evalutated by converting the variable of integration to the standardnormal variable: u = (L - trL]tl oL : Q - 3)l l = L - 3, so dL = du, and L =5 b e c o m e s r = 5 - 3 = 2 : 1 , : - . ! b e c o m e s l / = - @ , a n d t h e n

f 2n- r | -L " - "ou

t - aFz"= I F.(2)

\rhere F: is the standard normal distributron tunction. From Table I 1.2.1, F-(2) =0.9'7'7 , ^nd

R= | - O.g l i:0.023

The chance that demand will exceed supply for a fixed capacity of 5 is approximately2 percent.

(r) The capacity now has a normal distribution with p6 = 5 and oc = 0.15.Hence, its probability density is

lr (C l= . - - ! e - ' ' " ' : ' "1

\J znEc

: t "

(c-5)2t2xlo15))

.l-z;1o.lsl1 . 3 3 1 , - _ . 1 , , .

\,t zjf

and the r isk of fai lure is given by Eq. (13.4.3), with/( l) as before:( - | t - l

R - | ) l I t L , d L B \ C ) d CJ _ r l J c l

r - f r 1 . . . -I I I - ! " ' t - t i ;41 l t : !E"- r s , , t ts46r " l r ( v 2 ' ' I . E

syorolocrc oesrcn 435

FIGURE 13.4.1Composire risk analysis. Areasbaded is the risk Rj ofthe loading exceeding a fixedcapacity of 5 units. Tberisk that the loading willexceed lhe capacity when thecapaciry is random is given byR = I -L I: f(L)dLl stc)dc.The loading and capacityshown are both normallydisrributed (Example1 3 . 4 . 1 ) .

i l

N

dC around a value C is g(C)dC. Assuming that a and C are independent randomvariables, the composite rij& is evaluated by calculating the probability thatloading wil i exceed capacity at each value in the range of feasible capacities,and intesratins to obtain

- r l r Io : ) .1, f tL\dL f tc \dc

The reliabiLity of a system is defined to be the probability that a ststem v,illperform its required function Jor a specified period of time under stated conditions(Harr. 1987). Reliabil ity R is the complement of risk, or the probabil jty that theloading wil i not exceed the capacity:

( 1 3 . 4 . 3 )

( t 3 . 4 . 4 )- r - o

o = f -l[' nr>otfroo, (r3.4 5)

Example 13.4.1. During the coming year, a city's estimated water demand is threeunits. with a standard deviat ion of one unit. Calculate (a) the isk of demandexceeding supply if the city's water supply system has an estimated capacity of 5units: (r) the risk of failure if the estimate of the capacity has a standard error of0.75 units. Assume that loading and capacity are both nomally dist.ibuted.

Solution. (,a) The loading is rormally distributed with p, : 3 and ('r = 1� Itsprobabil i ty function, frorn Eq. (11.2.5), rs

!f&):

Le tL t 1t14

rL'2trttt

436,cppi-rEoHvonorocv

The integral is evaluated by computer using numedcai inlegration to yield R=0.052.Thus, the chance that the city's water demand will exceed its supply during thecoming year, assuming the capacity to be normally distributed with mean 5 andstandard devjatjoD 0.75 is approximately 5 percent; compare this with the result of2 percent when the capacity was considered fixed at 5 units.

It is clear from Example 13.4.1 that calculation of the composite risk offailure can be a complicated exercise requiring the use of a computer to performthe necessary integration. This is especially true when more realistic distributionsfor the loadigg and capacity are chosen, such as the Extrcme Value or log-harsonType III distributions for loading, ard the lognormal distribution for capacity. Yenand co-workers at the University of Illinois (Yen, 1970; Tang and Yen, 1972;Yen. et al., 1976) and Mays and co-workers at the University of Texas at Austin(Tung and Mays, 1980; Lee and Mays, 1986) have made detailed risk analysisstudies for various kinds of open-channel and pipe-flow design probiems.

The composite risk analysis described here is a staric analysis, whicb meansthat it estimates the dsk of failure under the single worst case loading on thesystem during its design life. A more complex dynamic dsk analysis considersthe possibility of a number of extreme loadings during the design life, any oneof which could cause a failure; the total risk of failure includes the chance ofmultiple failures during the design l ife (Tung and Mays. 1980; Lee and Mays1 9 8 3 ) .

I3,5 RISK ANALYSIS OF SAFETY MARGINS ANDSAFETY FACTORS

Safety Margin

The safety- margin was defined in Eq. (13.2.5) as the difference between theproject capacity and the value calculated for the design loading SM=C - t.From ( l3.4.1), the risk of failure R is

R = P ( C - Z < 0 )( 1 3 . 5 . I )

=p(SM < 0)

If C and L are independent random va ables, then the mean value of SM is givenb1

and its variance by

l t S M : l t c - l t L

4 r = 4 n " 1so the standard deviation, or standard error of estimate, of the safety margin is

or * :1o /+ o | ' l ' ' ( 1 3 . s . 4 )

( 1 3 . 5 . 2 )

( 1 3 . s . 3 )

If the safety margin is normally distributed, then (SM-psM)/ds14 is a standard

uvonoloorc orsrcr 437

normal variate z . By subtacting p5y from both sides of rhe inequalit] in ( t 3.5. l )and dividing both sides by o5y, it can be seen that

n:p/sM- l rst . - l isu l

\ osm osv /

- P [ ' - g i ! ' I\ asM/

i

=F f - l i !41- ' \ o s t /

where F- is the standard normal distribution function.

i ( 1 3 . 5 . 5 )

Exsmple 13.5.1, Calculate rhe risk offailure ofrhe warer supply system in Example13.4.1, assuming that the safety margin is normally distributed, and ttrar p. - 5units, oc = 0.75 units, lr. = 3 units, and o. : 1 unit.

Solutian.- Fr-om Eq. (13 .5 .2), p"sM : Itc - ILL : 5 3=2. From (13.5.4), rsM:(dc + 4) t /2 = (12 + 0 .75 \ t t2 : 1 .250. Us ing (13 .5 .5 ) ,

= F " l - _ - l. \ 1 .250/

: r . ( 1 .60)

which is evaluated using Table 11.2.1 ro y;eld n: O.OSS, which is very closeto the value obtained in Example 13.4.1 by numerical integration (an inherentlyapproximate procedure). The risk of failure under the stated conditions is R :0.055,

Note that this method of analysis assumes that the safety margin is normallydistributed but does not specify what the distributions ofloading and capacity mustbe, Ang (1973) indicates that, provided R > 0.001, R is not grearly influenced bythe choice of distributions for l, and C, and the assumption of a normal distributionfor SM is satisfactory. For lower risk than rhis (e.g., R = 0.00001). the shapesof the tails of the distributions for L and C become critical, and in this case, thefull composite risk analysis described in Sec. 13.4 should be used to evaluate therisk of failure.

Safety Factor

The safety factor SF is given by the rati,o ClL and, the risk of failure can beexpressed as P(SF < 1). By taking logarithms of both sides of this inequaliry

,L

n=r.f - p")\

OSM /

43E oplrrto rtvonorocr

( 1 3 . 5 . 6 )

=P ih 9 < o l

If rhe capacit! and loading are independent and lognormally distributed, ihen therisk can be expressed (Huang, 1986)

-'"[,,+(+#)"]I In l(1 + cvZxr + cv?rl] " ' ( 13 . s .7 )

R: P(SF < l )

= P(ln (sF) < ol

Example 13.5.2. Solve Example 13.5.1 assuming capacity and loading are bothlognon]1ally distributed.

. so l r l t i r , r ! . F romExamp le l3 .5 . l . 1 t5 :5ando6=0 .15 , and hence CV c : 0 .75 , / 5 =

0 15 . L i kew ise , pL = 3 and oL : l , so CVr = l / 3 = 0 .333 . Hence . by Eq . ( 13 .5 .7 ) ,

the r isk is

- . (

, . (

-'"{;[;gg]']l l n i r l + ( 0 . I 5 ) : r ( l + r 0 . 3 l 3 r 2 r l ' '

: F , ( - 1 .5463 ) : 0 .061

The r isk of fai lure under the above assumptions, then, is 6.I percent. For the sameproblem (Example l3.5.1) assuming that the safety margin was nomally distr ibuted,the risk was found to be 5.5 percent; the risk level has not changed greatly withuse of the lognomal instead of the nomal distribution.

Risk-Safety Factor-Return Period Relationship

A common design practice is to choose a retum period and determine the corre-sponding loading I as the design capacity of a hydraulic structure. The safetyfactor is inberently built into lhe choice of the return period. Altematively, theloadin-e value can be multiplied by a safety factor SF; then the structure is designedfor capacity C : SF x L. As discussed in this chapter, there are various kindsof uncertainty associated both with t and with the capacity C of the structure asdesigned. By composite risk analysis, a risk of failure can be calculated for theselecied return period and safety factor, The result of such a calculation is shown

uyonor-ocrc oesror 439

" . !i r

0.00002 0.0001 0.001 0.01 0. t 1.0

Risk of fa i lure (yeaar)

I IGURE 13.5.1The risk safety faclor-retu.n period relationship for culvert design on the Glade River near Reston,Virginia. The probability distribution for loading used to develop thjs figure was the Exrreme ValueType I distribution ofannual maximum floods- A lognormal distribution for the culven capacity wasdeveloped using first-order anaiysis of uncenainty. The risk level for given retum pe.iod and safetyfactor was determined using composite risk analysis- (Sorrce: Tung and Ma.v-�s, 1980.)

in Fig. 13.5.1, which shows a risk chart applying to culvert design on the GladeRiver near Reston, Virginia. The risk values in the chart represent annual prob-abilities of failure. For example, if the retum period is 100 years and the safetyfactor 1.0, the risk of failure is 0.015 or 1.5 percent in any given year, while ifthe safety factor is increased to 2, the risk of failure is reduced ro R = 0.006, or0.6 percent in any given year.

Current hydrologic design practice copes with the inherent uncertainty ofhydrologic phenomena by the selection of the design retum period, and withmodel and parameter uncertainty by the assignment of arbikary safety tactors orsafety margins. The risks and uncertainties can be evaluated more systematicallyusing the procedures provided by first-order analysis of uncertainty and compositerisk analysis as presented here. However, it must be bome in mind that just asany function of random variables is itself a rcndom vadable, the estimates of riskand reliability provided by these methods also have uncertainty associated withthem, and their true values can never be determined exactly.

REFERENCES

Ang, A. H.-S., SFuctural risk analysis aDd reliability-based design, J. Structural D.., Am. Soc.Ci ' . Eng. , vol .99, Do. ST9, pp. 1891-1910, 1973.

Ang. A. H.-S., and W. H. Tang, ProbabiLity Concepts in Engineering Plannine and Design, \d.I , Basic Pr incip les, and vol . I I , Decisron, Risk and Rel iabi l i ty , Wi ley. New York, 1975 and19E4. resDect ivelv.

CHAPTER

I 4DESIGN

STORMS

A desi.qu stornt is a precipitation pattern defined for use in the design of a hydrologic srstenr. Llsually the design stornl serves as the systen input, and the result-ing rares of f low through the system are calculated using rainfall-runoff and flowrouting procedures. A design storm can be defined by a value for precipirationdeprh lt a poinr, by a design hyetograph speciiying the tirne distribution ofprecipitation during a storn], or by an isohyetaj map specifying the spatial pattern ofthe pre.rprtation.

Dcsign srorms can be based upon historical precipitation data at a site or canbe constructed using the general characteristics of precipitation in the surroundingregion. Their application ranges fiom the use of point precipitation values in therationai method for determining peak flow rates in storm sewers and highwaycuiverts. 1o the use of storm hyetographs as inputs for rainfall-runoff analysisof urban detention basins or for spil lway design in large reservoir projects. Thischapter colers the development of point precipitation data, intensity-durationfrequercv relationships. design hyetographs. and estimated l imiting storms basedon probable maxirnum precipitation.

14.1 DESIGN PRECIPITATION DEPTH

Point Precipitation

Poinl precipitation is precjpitation occurring at a single point in space as opposedto areal precipitation which is precipitation 6ver a region. For point precipitarionfrequency analy-sis, the annual maximum precipitation for a given duration is

oesrcr.l sronus 445

seiected by applying the method outl ined in Sec. 3.4. to all stoms in a year,

for each year of historical record. This process is repeated for each of a series

of dumtions. For each duration, frequency analysis is performed on the data,

as described in Sec. 12.2, to derive the design Precipitation depths for various

return periods: then the design depths arc converted to intensities by dividing by

the preciPitation dumtion.By analyzing data in this way, Hershfield (1961) developed isohyetal maps

of design rainfall depth for the entire United States; thes€ qeq published inU, S. Weather Bureau technical paper no. 40. commonly calledril{40. The mapspresented in TP 40 are for durations frorn l0 minutes to 24 h&rs and returnperiods fiom I to 100 years. Hershfield also furnished interpolation diagrams formaking precipitation estimates for durations and rctum periods not shown on themaps. Fig. l4.l. l shows the TP 40 map for 100-year 24-hour rainfall. The U. S.Weather Bureau (1964) later published maps for durations of 2 to 10 days.

In many design situations, such as storm sewer design, durations of 30minutes or less must be considered. In a publication commonly known as HYDRO35 (Frederick, Meyen, and Auciello, 1977), the U. S. National Weather Servicepresented isohyetal maps for events having durations from 5 to 60 rninutes,partially superseding TP 40. The maps of precipitation depths for 5-, l5-, and60-minute durations and retum periods of 2 and 100 years for the 37 eastem states

FIGURE 14.1.1The 10o-year 24-hour rainfall (in) in the United Shtes as prcsented in U. S. weather Bureau technicalpapet 40. (Source. Hershfield. 1961.)

-t:l

: 1

i :

Eat

-"3 !t !g

a

!

!

?

e.a

I

X ;: . 9

,-i�

;

cI

I

€ ^ s

.!t

: 1

:''

\

! i

l r

r !

F

I

9 =:

g

6

6

€ e; i u

- ;

.:

!

E

(

- .Et

! .

=

2 .2:

t!

tr

!

--

;:'

' r f i F lt : i : i! q E li r Ft a :9 F :! *

:

)

452 .rplueo syonorocr

are sho*'n in Fig. 14.1.2. Depths for l0- and 3O-minute durations for a gilenreturn pcriod are obtained by interpolation from the 5-, 15-, and 60-minute datafor the same reium pefiod:

orsrcr sror<us 453

Areal Precipitation Depth

Frequency analysis of precipitation over an area has not been as well developedas has analysis of point precipitation. In the absence of information on the trueprobabitity disribution of areal precipitation, point precipitation estimates areusually extended to develop an average precipitation depth over an area. The arealestimate may be either storm-centered or location-fixed. For the location-fixedcase, one accounts for the fact that precipitation stations are sonptimes near thestorn center, sometimes on the outer edges, and sometimes inlbelween the two.An averaging process results in locadon-fixed depth-area curvqt relating arealprecipitation to point measurements. Fig. 14.1.3 provides curves for calculatingareal depths as a percentage of point precipitation values (World MeteorologicalOrganization, 1983).

Depth-area relationships for various durations, such as those shown inFig. 14.1.3, are derived by a depth-area-duration analysis, in which isohyetalmaps are prepared for each dumtion from the tabulation of maximum n-hourrainfalls recorded in a densely gaged area. The area contained within each isohyeton these maps is determined and a graph of average precipitation depth vs. areais Dlotted for each dumtion.

P16 . ; n = 0 .41P5 m in + 0 .59Pr5 m in

P3e -;n = 0.51P15 - i" + 0.49P60 mi"

For rerurn periods other than 2 or 100 years, the following interpolation equationis used. with the appropdate coefficients a and b from Table 14.1.1.

( 1 4 . 1 . 1 a )

( 1 4 . l . 1 b )

. Pr y, = aP2 r, -l bPrco,, ( 14. 1 .2)

\'Iiller, Frederick, and Tracey (1973) present isohyetal maps tbr 6- and 24-hour druations for the I I mountainous states in the westem United States; thesesupei sede the corresponding maps in TP 40.

I.-xample l , l . l . l . Determine the design rainfal l depth for a 25-year 30-minute stormir Oklahoma Ciiy.

So/Ilrior. Oklahoma City is locafed near the center of the state of Oklahoma andthc values of l5- and 60-minute precipitation for 2- and 100-year retum periodsa re read f r om F ig . 14 . l - 2 as Pz . r r = 1 .02 i n , P roo . r5 = 1 .86 i n , P r .6o : 1 .85 i n ,and P16x.66 = 3.80 in, respectively. Using ( l4. l . l r) , the values for 3o-minuteprecrpilation depth aje calculated

P36.6 = 0.51P15 min + 0.49P60 min

F o r l = 2 y e a r s , P : r o : 0 . 5 1 x 1 . 0 2 + 0 . 4 9 x 1 . 8 5 = 1 . 4 3 i n .

Fo r I = 100 yea rs , P1n6 .36 = 0 .51 x 1 .86 + 0 .49 x 3 .80 = 2 .81 i n .

T h e n ( 1 , 1 . i . 2 ) i s u s e d w i t h c o e f f i c i e n t s d : 0 . 2 9 3 a n d & = 0 . 6 6 9 f r o m T a b l e t , 1 . 1 . lto gire the 25-year 30-minute precipitat ion depthl

P2s,rq:aP1.36 * bP1j9.q6

=0 .293 x t . 43 + 0 .669 x 2 .8 t

= 2 .30 i n

TABLE 14.I.1Coefficients for interpolatingdesign precipitation depths usingEq. (14.1.2)

Return period 7 ay€4rc

A r e a ( m i r )

145 lgt 24t 290 318I+,l

i,itjl

1i:i.::*jj

ll

.jl

5l 02550

q 9 0g

.z

€ 8 0e'€

.E 70

& 6 0

0.6140.4960.2930 . 1 4 6

0.2780.4490.6690.835

500

Area (km 2)

FIGI,'RE 14.1.3DePth_area curves for reducing point rainfall to obtain areal average values. (.torrce: world Meteo_rological Organization, 1983; originally published in Technical hper 29, U. S. Weather Bureau,19s8.)

5o! / . . F rcderc t . Myer , and Auc ie l lo , 1971.

,154 eprrrro syonolocy (

1:I.2 INTENSITY.DURATION.FREQUENCY RELATIONSHIPSOne of the first steps in many hydrologic design projects, such as in urbandrainage desjgn, is the determinatjon of the rainfall event or events to be used.The most common approach is to use a design storm or event that involves arelationship between rajnfall intensili (or depth), duration, and the frequency orrerum period appropriate for the faciliry and site location. In many cases, thehydrologist has standard intensity-duration-frequency (IDF) curves avaiiable forthe site and does not have to perform this analysis. However, it is worthwhile tounderstand the procedure used to develop the relationships. Usuallv. the informa_tion is prernted as a graph, with duratjon ploned on the horizontal axis, rrrrensrryon the vertical axis, and a series of curves, one for each design rctum period, asil lustrated for Chicago in Fig. 14.2.1

The inrensity is the time rate of precipitatior, that is, depth per unir trmeaorm/h or in/h). It can be either the instantaneous intensity or the average iltensityor cr the duration of the rainfall. The average intensity is commonly used and canbe expressed as

( 1 4 . 2 . t )

IO

5

2

Ta

Return

I00 !€ars5025

) b | 10 20 40 60 80Duration (min)

FIGURE I.{.2.1Intensit) duration frequency curves of maximum rainfall in Chicago, U. S. A.

orsrc^- sronus 455

where P is the rainfall depth (mm or in) and fi is the duration, usually in hours.

The frequency is usually expressed in terms of return period, Z, which is the

average length of time between precipitation events that equal or exceed the

design magnihlde.

Example 14,2.1. Detennine the design precipitation intensity and depth for a 20-

minure dumtion storm with a 5-year retllm period in Chicago.

Solut ion. From the IDF curves for Chicago (Fig. 14.2. l) , i l rel6esign intensity fora s-year, 2o-minute stom is i = 3.50 in/h. The conespondin&frecipitation depthi s g i ven by Eq . ( 14 .2 .1 ) w i t h Td :20 m in=0 .333h . '

= 3 .50 x 0 .333

= 1 . 1 7 i n

Example 14.2.2. Use the U. S. National Weather Seryice maps (Fig. 14.1.2) andEqs. (14.1.1) and (14.1.2) to plot IDF curves for Oklahoma City, Oklahoma, forreturnpedodsof2,5, 10,25,50, arld 100 years. Consider rainfal l durations rangin8from 5 minutes to I hour.

Solut ian. The six maps presented in Fig. 14.1.2 show precipitat ion for 5-, l5-,and 60-minute durations and 2- and 100-year retum periods. The six values forOklahoma City from these maps are: Pr.5 : 0.48 in, Pr0o.5 = 0.87 in, Pr.15: 1.02in, Proo,rs : 1,86 in, P2,60 : 1.85 in, P16x,66 : 3.80 in. For I = 2 and 100 yr,the precipitation for 10- and 3o-minute durations is obtained by intelpolation fromihe 5-, l5-, and 60-minute values using (14.1.1) as i l lustmled in Example 14.1.1.For each duration, the values for return period I : 5, 10, 25, and 50 yr areobtained usjng the values at T: 2 and 100 yr by interpolat ion using (14.1.2)and Table 14.1.1, as also i l lustrated in Example 14.1.1. The results are shown inTable 14.2.I in tems of precipitation depth, and they are convefied to intensity bydividing by duration. For example, P:r,:o = 2.30 in, so the corresponding intensityis i : PlTa = 2.30 inl0.50 h : 4.60 inlh. The resulting precipitation intensities foreach duration and return period are plotted in Fig. 14.2.2.

TABLE 14.2.7Design precipitation depths (in) at Oklahoma City forvarious durations and return periods (Example 14.2.1)

Duration Ia, (min)Return period I(yr)

25

l 02550

100

4.480.570.63o. '720.800.87

0 .800.941 .051 . 2 1l . 3 31 .45

1.021 . m

1 .541 .701.86

. 1.852.302.623.083.443.80

t . 431. ' �74t . 912.302.562 . 8 1

The vslues in iralics de rcsd from Fig. 14.1.2; rhe remainder are obtained by inteFpo la t ion us ins Eqs . (14 .1 .1 ) and (1 .1 .1 .2 ) .

456 , i r . p r - reoxvo ro rocv

IDF Curves by Frequency Analysis

When local rainfall data are available, IDF curves can be developed using fre-qucncl analysis. A commonly used distribution for rainfall frequency analysisis thc Extreme Value Type I or Gumbel distdbution as discussed in Sec. i2.2.For each duration selected, the annual maximum rainfall depths are exftactedfrom historical rainfall records, then frequency analysis is applied to the annualdata. In somc situations, pafiicularly when only a few years of data arc available(less than 20 to 25 years), an annual exceedence series for each duration may bedetennrned by rahking the depths and choosing the N largest values from a recordof A'rcars. SuCh a series is shown in Table 14.2.2 for a rain gage at Coshocton,Ohio. In the table, the lines connect prccipitation data for variJuJ durations ofthesame storm event. The design prccipitation depths determined ftom the annualexceedence series can then be adjusted to match those derived from an annualmaxinrum series by multiplying the depths by 0.88 for tbe 2-year retum periodvalues, 0.96 for the s-year return period values, and 0.99 for the l0-year retumperiod values (Hershfield, 1961). No adjustment of the estimates is required forloneer return periods.

| . 0l 0 09 . 0t i 0

1 . 0

6 .0

.1 0

Rctum period (years)

L{ l

] r0o50

) z sl 0

E ? . 5

l 5 1 0 1 5 3 0D0ration (min)

FIGURE I4.2.2Intensity-duratioo-frequency cu es for Oklahoma City (Example 14.2.1).

oesrcr sronvs 457

TABLE 14.2.2Annual €xceedence series of rainfall data for Coshocton, Ohio

Rank ReturnperiodwU

Maximum depth (in) and dat€ for duration sho*n

15 min 30 min 60 min 120 mitr

l

2

3

25.00

12.50

8.33

6.25

5.00

4 . t 7

3 . 5 1

3 . l l

2.14

2.50

2.2'�7

2.08

| . 9 2

1 . 7 9

1 . 6 7

1 . 5 6

1 . 4 1

l . l 9

1 . 3 2

1 . 2 5

l . 1 9

L t 4

r.09

t .04

1.00

5

6

1

9

t 0

t 2

t 3

l 4

l 5

t 6

1,1

I E

t 9

20

2 l

22

23

25

1r.423 f---

0.940'7^t5l

l 0 e 2 o| 6/t2r590 .9105]r3t640.8906/271750.8846t231520 .8m814/73

l o . s r o| 1t211690.8056122/510.7836t24/564.7'708 5/75a.1l t )'7t22t58

0.750'7^0n3

0.7506^7[74o.1337/r91670.7327t30t580 . 7 r 0713t52o.'707aly630.7007D4t6a0.7006t4t630.7006n2/(fl4.6924nn40.6888t27t74

1 0 . 6 8 7| 9^2t51

0.6704/t3t55

Solf.p Wenzel. Iq82. Coplri8hr b) the American Geophysicrl Union

458 iPrLrLD HYDR.,L,GY

(

Fl\ampfe 14,2.3. Using the data presented in Table 14.2.2, determine the 2-)ear and 25-year precipitation depth estimares for a ls-minute duration storm rnCoshocton, Ohio. Assume the Extreme Value Type I (Gumbei) distribution jsapplicable.

Solutiotr. The design rainiall depth for a given retum period ? is derelrnined byEq . ( 12 .3 .3 ) :

xr.r.t = ,T,t + Krsr.t

where t7., and Jr./ are the mean and standard deviation of the rainfall depths for aspccjf ied drrat ion 77, and KTis the frequency factor given by Eq. (12.3.8):

t ; lx, = l ! lo.srrz + rn lrn -LI

7 r l \ f l /

Co sidcr a l5-minute durarion for example; the mean and standard deviat ion ofthel. i minule precipitarion data in Table 14,2.2 ate i6 = 0. '799 in and s15 = 0.154in , r espec r i ve l y . Us ing (12 .3 .8 ) , K2= - ( \ r 6 tT \e .5 ' / j 2 + l n { t n tT / (Z - 1 ) l } ) =-i r6i:n)\0.5772 + lnUn[2/l ] ] = 0.164; the corresponding value for I = 25 yr

is i(25 = 2.044. Then, using (12.3.3) for a two-year retum pe.iod,

.r2.1. =;t5 + K2str

=0 .799 0 .164 x 0 .154

=0.7'/4 in

llecause the data in Table 14.2.2 are an annual exceedence series. this vatue lsDLLlt ipi ied by 0.88 to obtajn the design precipirarion depth 0. ' / ' /4 x 0.88 = 0.68 infbr a two-year retum peaiod. For a 25-year retum period

.r25 15 =i15 + r(25s15

=0 . ' 199 +2 .044x0 .154

= l . l l i n

This valLre is not adjusted because its retum pedod is greater than l0 yean.

Equations for IDF Curves

Intensity-duration frequency curves have also been expressed as equations toaloid having to read the design rainfall intensity from a graph. For example,Wenzel (i982) provided coefficients from a number of cities in the United Sratesfor an equation of the form

( t4 .2.2)T c , + f

where r is the design rainfall intensity, Q is the duration, and c, e, and/ arecoefficients rarying with location and retum period. Table 14.2.3 shows valuesof these coefficients for a lO-year retum period in l0 U. S. cit ies.

It is also possible to extend (14.2.2) ro include the return period Z using rheequatron

TABLE 14.2.3Constants for rainfall equationtl4.2.D for l0-year return periodstorm intensities at yarious locations

Location

oesrcs sronvs 459

AtlantaChicagoClevelaDdDenverHoustonLos AngelesMiamiNew YorkSanta FeSt. Louis

9'�7.594.9

96.697.420.3

r24.278 . r62.5

104.'�7

0 .83 6 .880.88 9.040 .86 8 .25o.9'7 13.90o.7'7 4.800.63 2.060 . 8 1 6 . 1 90 .82 6 .570 .89 9 .100.89 9.44

, l

\ l

Con.rdntr corrc-pond to i In in.hes per hour dnd fdin minures. Soar.a: Wenzel. 1982, Copyright by rheAmerican Geopbysical Union.

cT'(14.2.3)

T a + f

. cT'n

T a + f

Example 14.2.4. Determine and compare theintensit ies in Los Angeles and Denver.

Solution. The design rainfall intensity is computed using Id = 20 min, and thevalues of the coeff icients for Los Angeles (c : 20.3, e = 0.63, and/ = 2.06) fromTable 14.2.3, for which Eq. (14.2.2) gives

: ,r; +f

20.3= ,006 f, .06

:2.34 inlh

Similarly, for Denver I = c/(7., + f) = 96.61(21a.e7 + 13.90) = 3.00 in/h. Thedesign intensity in Denver is greater by 3.00 - 2.34 : 0.6 inh. or 28 percent.

r4.3 DESIGN HYETOGRAPHS FROM STORMEVENT ANALYSISBy analysis cfobserved storm events, the time sequence ofprecipitation in typicalstorrns can be determjned HrJff /1967) dcvelnned rjmp .tj.r.il,,,ri^- fclAliArf, W

( r1 .2 .4 )

10-yr, 2o-minu|e design rainfall

heavy storms on areas ranging up to 400 mi2 in Il l inois. Time distribution parrernswere developed lbr four probability groups, from the most severe (first quaniie) tothe least severe (fourth quartile), Fig. 14.3,l(c) shows the probability distributionof first-quartile storms. These curves are smooth, reflecting average rainfalldistribution with time; they do not exhibit the burst characteristics of observedstorms. Fig. 14.3.1(b) shows seiected histograms of f irst-quarti le storms for l0-,50-. and go-percent cumulative probabilities of occurrence, each illustrating theperccntage of total storm rainfall for l0 percent increments of the storm dumtion.The 50 percent histogram reprcsents a cumulative minfall pattem that should beexceeded in about half of the storms. The 90 percent histogram can be interpretedas a storm distribution that is equaled or exceeded in l0 percent or less of thestomrs. The first quartile 50-percent distribution has been used in the ILLUDASstorn drainage simulation model by Terstriep and Stall (1974).

Thc [t. S. Depanmenr of Agriculture, Soil Conservation Service (1986)

460 rrpuEo svonorccv

0 20 40 60 80 100Cumulative perccnt of storm t im€

t a )

0 20 40 60 80 100

Cumulaljve percenr of sto.m time

\b )

t0{J

' 0

; 4 0

7 tt'E

0

FIGURE I4.3. I(d) Time disribu(ion of first-quartile storms. The probability shown is rhe chance that the observedstorn pattern sill l;e lo the left of the curve. (D) Selecred histograms for frrst-quanile storms.(So"/.. Huff, 1967, Copyrighr by the American Geophysical Union.)

oesrcr sroavs 461

developed synthetic storm hyetogmphs for use in the United States for stormsof 6 and 24 hours duration. These hyetographs were derived from informa-tion presented by Hershfield (1961) and Miller, Frederick, and Tracey (1973)

and additional storm data. Table 14.3.1 and Fig. 14.3.2 prcsent the cumulativehyetogmphs. There are four 24-hour duration storms, called Types I, IA, II, andIII, respectively; Figure 14.3.3 shows the geographic location within the UnitedStates where they should be applied. Types I and IA are for the Pacific maritimeclimate with wet winters and dry summers. Type III is for the Qu{ of Mexico andthe Atlantic coastal areas, where topical storms result in larlef4-hour rainfallamounts. Type It is for the remainder of the nation. C

Pilgrim and Cordery (1975) developed a hyetograph analysis method thatis based on mnking the time intervals in a storm by the depth of precipitationoccurring in each, and repeating this exercise for many storms in the region.By summing the ranks for each interval, a typical hyetograph shape can beobtained. This approach is a standard method in Ausralian hydrologic design(The Institution of Engineers Australia, 1987).

TABLE 14.3.7SCS rainfall distributions

24-hour storm 6-hour stofm

h/P24

Hont t tlU Tt?e I TWe IA Type II Type III Hour f

0 0 02.0 0.083 0.0354.0 0.167 0.0766.0 0.250 0.12s'7.O 0.292 0.1568.0 0.333 0.1948.s 0.354 0.2199.0 0.375 0.2549.s 0.396 0.3039.' � t5 0.406 0.362

i0.0 0.4r '7 0.51510.5 0.438 0.583I L0 0.459 0.624l L5 0.4'79 0.654l l .?5 0.489 0.66912.0 0.500 0.682t2.5 0.521 0.70613.0 0.542 0.72113.5 0.563 0. ' �74814.0 0.583 0. ' t6716.0 0.66'1 0.83020.0 0.833 0.92624.0 r.000 1.000

00.0500 . 1 1 60.2060.2680.4250.4800.5200.5500.5u0.51'�70.60 r0.6240.6450.6550.6&0.6830.?010.'7190.'t360.8000.9061.000

0o.o220.0480.0800.0980 .1200 . 1 3 30 . t470 .1630.l ' t20 . l 8 lo.m40.2350.2830.35?0.663o.135o.172o.7990.8200.8800.9521.000

00.0200.M3o.o"t20.0890 . 1 1 50 .1300 .148o.16' l0 . 1 7 80 . 1 8 90.2t60.250o.2980.3390.5000.'7020 .7510.7850 . 8 1 10.8860.9511.000

00.601 .20i . 501 . 8 02 . 1 02.282.402.522.64

3.003 .303.603.904.204.504.805.4n6.00

0 00.10 0.040.20 0. t0o .25 0 .140 .30 0 .190 .35 0 .310.38 0.440.40 0.53o.42 0.600.44 0.630.46 0.660.50 0.700.55 0. '750.60 0. '790.65 0.830 .70 0 .860 .75 0 .890 .80 0 .910.90 0.96r .0 1 .00

Solrce: U. S. Dept. of Agriculturc, Soil Consenation Senice, 1973, 1986.

462 .,n,rrlu utonoLocv (

Triangular Hyetograph Method

A triangle is a simple shape for a design hyetognph because once the design precipitation deptb P and a dumtion Td are known, the base length and heighl of thetriangle are determined. Consider a tdangular hyetogmph as shown in Fig. 14.3.'1.The base length is 7d and the height h, so the total depth of precipitation in thehyetograph is given by P: +Tdh, from which

J P

T ,( 1 4 . 3 . 1 )

A stonn adraficement coe.fJicient r is defined as the ratio of the time before thepeak /, to the lotal duration;

, : n

(14.3.2)

The

the rccessjon time t6 is given by

t6 :71 - to( r 4 .3 .3 )=( l - r )Ta

A ralue for r of 0.5 corresponds to the peak intensity occurring in the middleof the storm. while a value less than 0.5 wil l have the peak earlier and a value

0 I 6 9 1 2 l 5 1 8 2 l

Time (hours)

FIGURE 14.3.2Soil Conserlarion Service :4-hour rainfal l hyetographs. (Sr!/ .p: U. S. Depl. of Agriculture, Sol lConsef!at ion Service, I986.)

H

;

?

e

(,

\ - -

! ; Eflc

_ g t - 8 . 8 "x , > , > , > . >

= . : - F F F

5€ lffi lr

46 r l , r pP l t eouYono r -oCr

0 2 4 6 8 t 0 \ 2 1 4

Timc (m'n)

\1r)

FIGURE 14.3.4Triangulff des jgn hyetographs.

kr) A general triangular designhyetograph. (r) Triangular designhyetograph for a s-year ls-nrinutestom in Urbana. I l l i io is (Exampler 4 . 3 . 1 ) .

e

E ,: ' 6

. r

e l, 93 2

I

0

greater than 0.5 wil l have the peak later than the midpoint. A suitable value ofr isdetermined by computing the mtio of the peak intensity time to the storm duntionfor a series of storms of various durations. The mean of these ratios, weightedaccording to the duration of each event. is used for r. Values of r reponed inthe l iterature are presented in Table 14.3.2, which shows that in many locaiionsstonns tend to be of the advanced type, wilh / Iess than 0.5.

Yen and Chow (1980) anaiyzed 9869 storms at four locations: Urbana,Il l inois; Boston, Massachusetts; Elizabeth City, New Jersey; and San Luis Obis-po. California. Their analysis indicated that the t angular hyetographs for mosthea\ y storms are nearly identical in shape, with only secondary effects from stormduration, measurement inaccuracies, and geographic location.

Example 14.3.1. Determine the triangular hyetograph for the design of an urbanstorm sewer in Urbana, Illinois. The design return period is 5 years, and the design

j 7 min . +9.3 rn in

TABLE T4.3.2Values of the storm adyancement coelficient rfor various locations

Location Ref€renc€

oesrcr,r sror<vs 465

BaltimoreChicagoChicagoCincinnattClevelandcauhati, IndiaOntafloPhiiadelphia

0.3990 .3750.2940.325o .3150 .4160.4800 .4 t4

McPherson (1958)Keifer and Chu (1957)McPherson (1958)Preul and hpadakis (1973)Havens and Emerson ( 1968)Bandyopadhyay ( I972)Marsalek (19?8)McPherson (1958)

. lI t

d

s"!/..e: Wenzel. 1982, Copyrighr by the American ceophys;cal Unjon.

rainstorm duration has been set at 15 minutes. The storm advancement coefficjenri s r : 0 . 3 8 .

Solution, FromFtg. 14. L2, for precjpitatjon at Urbana (jn central lllinors.), p".15 =0.88 in, and p166.15 = 1.70 in.,The d€pth^for a tyear retum period is given byEq . ( \ 4 .1 .2 ) w i t h a : 0 .674 and b : 0 .2 ' 18 f r om Tab le 14 . t . 1 :

Ps.ts = 0.674P2,6 + 0.278Proo.r5

:0 .674 x 0 .88 + 0 .2 i 8 x 1 . . /O

: 1 .07 i n

The peak intensiry f t is calculated using (14.3.1) wirh Z,i = 15 min =0.25 h:

, 2 P 2 t 1 . 0 7" :

h - - 1 5 - d r o I n / h

The t ime r, to rhe peak inrensjry is calculared by (14.3.2):

h = tTd = 0 .38 x 0 .25 = 0 .095 h : 5 .? m in

The recession time 16 is

t b = T d - t a = 0 . 2 5 - 0 . 0 9 5 = 0 . 1 5 5 h : 9 . 3 m i nThe resulring design hyerogmph is plorred in FiE. 14.3 .4lb). Values of preciprrationintensity at regular intervals can be calculated and converted to precipi'tatron depthfor rainfall-runoff analysis for the stodn sewer.

14.4 DESIGN PRECIPITATION HYETOGRAPHSFROM IDF RELATIONSHIPSln the hydrologic design methods developed many yea$ ago, such as the ratronall"h?9, 9."ty rhe peak discharge was used. There was no consideration of thexme disrnbutjon of discharge (the discharge hydrograph) or the time distriburion

9llrecrpitation (the.precipirarion hyetograph). However, more recently developed

desrgn methods, using unsteady flow analysis, require reliable prediction of thedesign hyetograph to obtain design hydrogiaphs.

466

Alternating Block Method

The alrernating bLock method is a simple way of developing a design hyetographfron an intensity-duration-frequency curve. The design hyetogmph produced bythis mcthod specifies the precipitation depth occurring in r successive time inter-vals of duration Ar over a total duration Td = n\t. After selecting the designretum period, the intensity is read from the IDF curve for each of the durationsAr, 2lr, 3A/, . . . , and the corresponding precipitation depth found as the productof intensity and duration. By taking differences between successive precipitationdepth \alues. the,amount of precipitation to be added for each additional unit oftime '\, is found. These increments, or blocks, are reordered into a time sequenqewith the maximum intensity occurring at the center of the required duration Idand the remaining blocks arranged in descending order altemately to the right andleft of rhe centml block to form the design hyetograph.

Example 14.4.1. Determine, in lo-minute increments, the design precipitatronhyetograph for a 2-hour storm in Denver with a l0-year rcturn period.

Solutbn. As was illustrated in Example 14.2.4, the l0-year precipilation inlensit)for a given duration can be calculated for Denver by Eq. (14.2.2) with c : 96.6,e = 0.9'l, andf : 13.90. For a duration of 20 minutes, this computation yieldsi = 3.00 inft. The values for other dumtions at intervals of l0 minutes are shown incolumn 2 of Table 14.4.1. By multiplying intensity and duration, the corespondingprecipitalion depth is calculated. For a 20-minute duration P : lIa = (3.00 in/h) r0.333 = l�001 in: similar calculations for the other durations are shown in column3 of Table 14.4.1. It can be seen that the cumulative depth increases with durationup to the 2-hour or l2O-minute l jmit.

TABLE I4.4.IA design precipitation hy€tograph developed in 10-min increments for a 10-year 2-hour storm at Denyer, using the alternating block method (Examplet4,4.1)

Column: 1Duration

(min)

2Intensity

(in&)

6Precipitation

(i!)

Cumulative Incrementaldepth depth(in) (in)

Time

(min)

l 02030405060'74

8090

1001 r 0r20

4 .1583.0022.3571.943r .6551.443| .279L l491.O140.9560.8830.820

0.6931 .001t . 1 7 8t.296|.3'�79|.443|.4921 .533l.566|.5941 .618L639

0.6930.3080.1780 . 1 1 70.0840.0630.0500.0400.0330.0280.0240.021

0 I0 0.024t0-20 0.0332c_30 0.05030-40 0.0844!-50 0.17850-60 0.69360-70 0.30870-80 0. r l?80,90 0.06390 100 0.04n100-110 0.028I l0-120 0.021

ursrt;"- sronus 467

The lo-minute precipitation depth is 0.693 in compared with l�001 in for 20minutes, so during the most intense 20 minutes of the design storm, 0.693 in willfall in 10 minutes, while 1.001 - 0.693 : 0.308 in wil l fall in the remaining10 minutes. Using this reasoning and taking successive inc.ements of cumulativeprecipifation depth, as shown in column 4 of Table 14.4.1, the precipitation depthadded by each additional increment of stom du.ation is found. In column 6 theprecipitation depths, are ordered so that the maximum block (0.693 in) falls at 50-60 min; the next-largest block (0.308 in) is placed ro the.ightioflFe maximumblock, at 60-70 min (making 1.00 in between 50 and 70 min. Lsjequired), thethirdJargest block (0.178 in) is placed lo rhe lefr of rhe maximudblock (40-50min), and so on. The rcsults are plotted in Fig. 14.4.1, and the descending orderof the blocks to the ieft and right of the maximum block can be seen. A designhyetograph built up in this way represents a lo-year event both for the 2-hour totalduration and for any period within this duration centered on the maximum block.

Instantaneous Intensity Method

Ifan equation defining the intensity-duration-ftequency curve is known, equationscan be developed for the variation of intensity with time in the design hyetograph.The principle is similar to that employed in the altemating block method, namelythat the precipitation depth for a period of duration 7a around the storm's peak isequal to the value given by the IDF curve or equation. The difference from thealternating block method is that precipitation intensity is now considered to varycontinuously throughout the storm.

Consider a storm hyetograph as shown in Fig. 14,4.2. The dashed horizontalline drawn on the hyetograph for a given precipitation intensity i will intersectthe hyetograph before and after the peak. Measured from the time of the peakintensity, the time of inte$ection before the peak is labeled ra and that afterthe peak tr. The total time between intersections is labeled 7d so that

T a = t a * t 6 ( r4 .4 .1 )

o . 7

0 .6

^ 0.5

. E 0 4E: n l

* 0 .2

0 . 1

0

FIGURE 14.4.1A lo-year 2-hour designhyetograph developed forDenver usiDg the altematingblock method (Exarnple14 .4 . r ) .

468 .qrpL.rrouvonorocr

* 7,1 -

) T , t - : : * ( t - t ) T t -

The storm advancement coefficient r is defined as before, as the mtio of the timebefore the peak to the time between intersections.

, : bTa

It follou's from (14.4.1) and (14.4.2) that

- t a t b- r l r (14.4.3\

As shown in Fig. 14.4.2, a pair of curves, ia : f(.t) and ib = f1i,are assumed to fit the hyetograph precipitation intensities, where 1o and i6 areprecipitatjon intensities before and after the peak, respectively. Thus, rhe totalamount of rainfall R within time Zd is given by the area under the curves:

n = lo'' f,,.,a," - lo'

"' 1,,0,0,, ( r4.1.4)

Noring that/0a):/(rb) for any given fa', differentiating Eq. (14.4,4) with respectto ?d gives

( 14.4. s)

( 14 .4 .6 )

( t4 .4 .7 )

( t4 .4.2)

dR_ . _ f ( t " ) - f ( t . \

If the average rainfall intensity for the duration I7 is i.,., then

R = Tai*,

Differentiating (14.4.6) with respect to Ia gives

#= L. .+ r ,#: : f ( ta)=f( tb)

oesrcr srorvs 469

Keifer and Chu (1957) developed a synthetic hyetograph of this type for usein sewer system design for Chicago. They defined the average rainfall inrensityi ̂ ," as in Eq. (14.2.2):

r ; + f (14 .4 .8 )

By differentiating (14.4.8) and substituring the resulr inro (14.4.?) it can be shownthat the intensity i for which the line inrenects the hyetograbh for a duration t2is given by {

c i ( \ - e)T: + f li : ' " , " ' (14.4.9)

The equations for the intensities io and i6 in terms of l, and t6 are found bysubstituting for Td in (14.4.9) using (14.4.3).

Example 14.4.2, Develop a 2-hour design hyetograph for a lo-year rerurn periodstorm in Denver using a storm advancement coefficient of r = 0.5.

Solution. A value of r:0.5 and a stom dumtion of 120 minutes will put the peakintensity at time t = 60 min measured from the beginning of the storm. The relativetimes ,o and r, befo.e and after the peak time arc shown in column 2 of Table14.4.2. A 2-minnte time interval is used near the peak for increased accuracy, anda lo-minute time interyal for the remainder. The rainfall intensities are calculatedby Eq. (14.4.9) using values c : 96.6, e :0.9'7 , and f : 13.9 from Table 14.2.3;r : 0 ,5 ; and Q g iven by (14 .4 .3 ) .

c [ ( t - e ) r j + f )

Before the peai, Td : tolr, so for t = 50 min, for example, r, = 60 - 50 : l0 minand. Td = ta/r = 10/0.5 : 20 min; hence,

. _ . " _ e6 .6t(t __9 :Two

n_! ]:21(200 e1 + 13 .9)1

: | .348 inlh

Similarly calculated values for all time interyals are shown in column 3 of Table14.4.2. The values of intensity affer tle peat are calculated by the same procedureusing IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIId = tb/(l - r). Fot example, at r = 70 min, tt = 1O - 6O: 10 min andTa = lOl(I - 0.5) : 20 min, so ia = 1.348 in& for thjs case also.

The values of precipitation intensity so determined are instantaneous valuesithe precipitation depth in each time interval may be calcuiared usjng the uapezoidalrule. For example, refering to Table 14.4.2, the precipiration deprh during the firstl0 minutes is [(0.118 + 0.156)12] x (10/60):0.023 in as shown in column 4.The sum of ail the precipitatior increments is 1.697 in, which is slighrly higherthan the actual precipitation depth of 1.64 in for a lo-year 2-hour stom in Denver,calculated in Example 14.4.1. The difference a.ises from the discretization of thecontrnuous time scale in this example and the use of a simple rEpezoidal rule to

lr; + f)'

(r: * f)'

470 rpplreo avonor-ocv

TABr-E 14.4.2Calculation of the lO-year 2-hour design hyetograph atDenver bv the instantaneous intensity method (Example14.4.2\

Column: ITimettmln,

2Relative time

(min)

3Intensity

(in/h)

Incrementalprecipitation(ini

d 6 0!0 50z0 4030 3040 2050 l 05 2 85,1 65 6 4 ts i ) l t

6 0 0 t62 2 th6 4 4 r6 6 66 8 870 l080 2090 30

100 4t)l l0 50120 60

0 . 1 1 80 . 1 5 6o.2190.3340.5851 .3481 . 6 9 12.t932.9754.3036.9504.3032.9152 .193L 6911.3480 .5850.3340 .2190 .1560 . 1 1 8

Total depth (in)

0.0230 .03 l0.0460.0770 . 1 6 10 .0510.0650.0860 . I 2 10 . 1 8 80 . 1 8 80 . l 2 l0.0860.0650 .0510 . t 6 10.0110.0460 .0110.023

| . 697

calculate precipitation depth. The difference would be diminished if iwo-minuteincremenls werc used throughout the time horizon, instead of just around the peakinlensir] in Table 14.4.2.

Figure 14.4.3 shows the hyetograph developed in this example along with asimilarly computed hyetograph for an advancement coefficient of 0.25. I1 can beseen rhat the effect of changing lhe adlancemenl coeff icient is simply to changethe localion of the peak jntersity bur not change its magnitude.

1,1.5 ESTIMATED LIMITING STORMS

The estimated limiting values (ELV'S) commonly employed for water controldesign are the probable moximum precipitatlon (PMP), the probable maximumsrornr (PN{S), and the probable maximum f.ood (PMF), altt}ough many otherpossible ELV's could be developed as criteria for various types of hydrologicdesign. The PMP provides only a depth of precipitatjon, the time distribution ofwhich musr be defined to form a PMS. The PMS can be employed as the input

ossrcr sroevs 471

€

= 4

; "

0

Ercr,nrlri.rDesign hyedraphs for a l0-yearstorm at Denver developed bythe instanlaneous intensity methodwith advancement coeffi cientsr = 0.25 and 0.5 (Examplet4.4.2) .

to a rainfall-runoff rnodel of a drainage basin system which can then be used todevelop a PMF for the design of runoff control structures.

Probable Maximum Precipitation

The PMP is the estimated limiting value of precipitation. Consequently, the PMPmay be defined as the analytically estimated greatest depth of precipitation fora given duration that is physically possible and reasonably chamcterisric over aparticular geographical region at a certain time of year. In practice, no allowanceis made in the estimation of the PMP for the effects of long-term climatic change.It must be remembered that the concept of PMP is not completely reliable becauseit cannot be perfectly estimated and its probability of occunence is unknown.However, in operational application, the PMP has been found to be useful, andits use will continue because of public concern for safety of projects such as largedams.

There are a variety of methods for determining the PMP. Because of theuncertainties and limitations of data and knowledge, the PMP must be consideredan estimate and judgment must be used in setting its value, Methods of estimatingPMP are discussed below.

1. Application of storm models. Storm models may be employed to esti-mate PMP where there is insufficient or nonrepresentative storm data, or whercthere is rugged topography that complicates the storm phenomenon and makesprecipitation measurement difficult. The convective cell model for thunderstormspresented in Sec. 3.3 is one example of a storm model, 4nd Wiesner (1970) haspresented a similar model for the precipitation adsing from orographic lifting ofmoist air over hills or mountains. When orographic and convective effects occursimultaneously, the two types of models can be superimposed.

The use of storm models is more successful in determining PMP over largeareas than over small ones. The principle of continuity is more easily applied

4 / I . PPI rFn .1 r DRO| OCY

using mean ielocities of inflow and outflow, and the moisture content of theinflowing air can be defined from the dew-point temperature persisting for aconsiderable time (usually 12 hours) over a large area. These factors are moredifficult to define for extreme precipitation rates that occur locally and over shortperiods. Therefore, although storm models may indicate the magnitude of theprecipitation to be expected, they shoujd be carefully calibrated with data fromobsc.\ ed storms in the region of intercst before use in design.

2. Marinization of octuqL storms. The chief deficiency of the storm modelsis their olersimplified representation of the actual storn. If records of actualstornrs are alq,i lable, they may be maximized to obiain the PMP values. Thisprocess invoives increasing the observed storm precipitatjon by the ratio of theacturl rnoisture inflow to the storm to the maximum moisture inflow theoreticallypossiblc at the site.

if there arc no adequate records of storms in the project basin. it is possibleto l lrrrsposc stonns Irom other areas to the pro.ject basin for thc contputatlonof drc Pl\lP if these storms could have occurred in the basin. The procedure ofstorn transposjtion involves the selection of adequate storms for transposirion,determination of the orientation of the stornls crit ical to tbe basin, and adjusrrnentsfor differences. if any, in dew-point temperature, elevation, prevail ing wind, andorographic effects.

The world's greatest recorded rainfalls. according to the World Meteoro-logical Organization (1983), are approximated by the equation

P = 42)To 115 ( 1 4 . s . r )u'here P is the precipitation depth in rnil l imeters and T7 is the durarion in hours.The equation was obtained by fitting data from observed extreme rainfalls at manylocations for durations ranging from one minute to several months. This equationis an estimate of the precipitation depths that could occur under very extremecircunlstances.

3. GeneraLiz.ed PMP charts. The estimates of PMP may be made either forindividual basins or for large regions encompassing nunerous basins of varioussizes. In the latter case, the estimates are refered lo as generalized estimates andare usually displayed as isohyetal maps that depict the regional I'ariation of PMPfor some specified duration, basin size, and annual or seasonal variation. Thesemaps are commonly known as generaLiTed PMP charts.

The most widely used generalized PMP charts in the United States arethose contained in U. S. National Weather Service hydrometeorological report no.5l, commoDly known as HMR 51 (Schreiner and Riedel, 1978), for rhe UnitedStates east of the l05th meridian. These maps specify the probable maximumprecipitation depth for any time of the year (referred io as an all-season estimate)as a function of storm area ranging from L0 to 20,000 mi2 and storm dumtionranging from 6 to 72 hours. For regions west of the 105th meridian, the NationalAcademy of Sciences (1983) has prepared the diagram shown in Fig. 14.5.1.which specifies the appropriate National Weather Service publication from whichprobable maximum precipitation data nay be obtained.

oesrcr srorus 473

Easlof the l05th Meddian

HtdrometeorologicalRepon 33

and HMR nos.5 r ( 1 9 7 8 ) .5 2 ( 1 9 8 2 ) ,

&53 ( r980)

HydromereorologicalRepon 49 (1977)

HydrorneieorologicalRepon 16 (1961)

Unpubiished UpperRio Crarde

Ba$in Srudy (1967)

HydrometeorologicalRepon 43 ( i966)

Technical paper 38(1960)

f , rI a ]i u lI

NwS referenc€s for:

Alaska 'Techrical PaDer 47. USWB (1963)Hawajian Islands -Hydrometeorological Report 39 (1963)Pueno Rico -Technical PaDef 42, USWB (1961)and Virgin Islands

FIGURE 14.5.1Sources of informarion for probable maximum precipitation computation in the United States-(Sorrce: National Academy of Sciences, 1983. Used with permission.)

The Probable Maximum Storm

The PMS, or probable maximum storm, involves the temporal dist bution ofrainfall. The PMS values are generally given as maximum accumulated depthsfor any specified duration. For example, given depths for 6, 12, 1 8, and 24 hourstypically represent the total depth for each duration and hot the time sequence inwhich the precipitation occurs.

In order to develop the hyetograph of a PMS, one needs to know the spatialand temporal distfibution of the PMP. One procedure to determine the PMShas been outlined in hydrometeorological report no.52 (HMR 52) by Hansen,

4 7 4 . D P L r r D r : D R o r u ' , I

Schreirer, and Miller (1982) for areas of the United States east of the 105thmeridian.

For the purpose of modeling maximum runoff, different critical timesequences of PMP increments should be investigated. In general, the critical timedist bution of PMP increments is determined by judgment from experience andavailable information, such as ftom weather maps of cdtical histodcal storms.A commonly adopted sequence is the most advanced time disribution, that is,beginning with the highest amount and continuing with decreasing increments.

The Probable'Maximum Flood

The PMF is the greatest flood to be expected assuming complete coincideoce ofall factors that would produce the heaviest rainfall and maximum runoff. Thisis derircd from a PMP, and hence its frequency cannot be determined. Fromthe ecoromic viewpoint, it is usually prohibit ive to design a structure for PMF,except for large spillways whose failure could lead to excessive damage and lossof life. Hence, a pmgmatic approach for many design situations is not to definethe design flood as an estimated ljmiting value, but to scale it downwards by acenain percentage depending on the type of structure and the hazard if it fails. Forthis purpose. the flood event actually used in design is often the greatest floodthat nlay reasonably be expected, taking into account all pertinent conditionsof location, meteorology, hydrology, and topography. This design flood maybe determined analytically as the flood caused by a ransposed historical largesrstorm and its magnitude may be a ftactjon of the ELV.

In practice, the design flood is commonly called the standard project food(SPF). The SPF is estimated using ninfall-runoff modeting by applying the unit-hydrograph method to Ihe standerd project storm (SPS), which is the greateststorm that may be reasonably expected. The SPS can be derived from a detailedanalysis of storm patterns and transposition of storms to a position that wouldgive maximum runoff. For a panicular drainage area and season of year in whichsnow melt is not a major concern, the SPS estimate should represent the mostsevere flood'producing depth-area-duration rclationship and isohyetral pattem ofany storm that is considered reasonably characteristic ofthe region. The watershedrunoff characteristics and any water-contol structures in the basin should beconsidered. When melting snow may constitute a substantial volume of runoff tothe SPF hydrograph, appropriate allowances for snow melt should be includedin the estimate. Where floods are predominantly the result of melting snow,the SPF estimate should be based on the most critical combinations of snow,temperaturc, and water losses considered reasonably characteristic. Past estimateshave indicated that SPS magnitudes and SPF discharges arc generally in the rangeof 40 to 60 percent of the ELV for the same basins.

In some cases the SPF estimate may have a major bearing on the design ofa panicular project; in other cases the estimate may serve only as an indication ofthe partial degree of protection proposed for the project. SPF estimates are usually

( oeslcr srotns 475

made only for rnajor and intermediate structures since they require considerable

effon in PreParation

Computer Programs for PMS and PMF Development

The U. S. Army Corps of Engineers Hydrologic Engineering Center (1984)

descitles a computer program called HMR 52 which computes the basin-average

Drecipitation for rhe probable maximum storm in accordance witfr the cri leria in

irydrometeorological repo( No 52 lnput to thi5 program inchide!.the PMP esti-

mates from HMR 5l;the prcgmm computes the spatially averagpd PMP for any

of the subbasins or combinations of subbasins in the drainage basin The program

selects the storm area size and orientation so as to produce the maximum basin-

average precipitation. Storm centering and time distribution must be provided by

the u;er. This progmm can be used in conjunction with the U S Army Corps

of Engineers HEC-I (Sec. 15.2) rainfall-runoff model to compute the probable

maximum flood.

14.6 CALCULATION OF PROBABLE MAXIMUMPRECIPITATION

Estimates of the runoff from the PMP are required in the design of spillways and in

dam safety studies. In order to carry out a rainfall runoff analysis using the PMP

both temporal and spatial distribution of the PMP estimates is required A stepwise

procedure to derivi the distribution of the PMP for the United States east of the

i05th meridian has been developed by Hansen, Schreiner, and Miller (1982) in

hydrometeorological rcport No. 52. This procedure is used for the detemination

oi the probable maximum storm, which is required to determine peak discharge

and to develop the probable maximum flood hydrograph through a rainfall-runoff

analysis. The-procedure is based uPon information derived from major storms of

record and is applicable to nonorognphic rcgions of the eastern United States'

The procedure for determining the probable maximum precipitation over a

watershed has five important etements: (l) depth-area-duration czrves' which

specify the PMP for a sPecified storm area and duration; (2) a standard isohyetal

pirera distributing the precipitation spatially in the form of 3n slliFse; (3) an

orientation adjustient fictor, which reduces the PMP estimates if the standard

isohyetat panem's longitudinal axis is not oriented in the direction of normal

atmospheric moisturc flow in the region , (4) a criticql storm area, which generates

the largest PMP over the watenhed; and (5) an isohyetal adjustment factor, which

specifies the percentage of the PMP depth that applies on each contour of the

standard isohyetal pattem.

Depth-area-duration curves. In probability-based point precipitation eslimates'

there are three variables to be considered: intensity (or depth), duration, and

frequency of occunence. For the PMP, the ftequency of occurrence is replaced

by storm area as the third variable. Figure 14.6.1 shows the PMP for the eastem

476 ,rpprrso Hyonolocy

FTGURE r4.6.r(d)Al l 'season PMP(in) for6 hours. l0 mi ' � . (Sourcet Hansen, Schreiner, and Mi l ler . 1982.)

FIGURE r4.6.1(r)All season PMP (in) for 12 hours, 10 mi'�. (,to{rc€: Hansen, Schreiner. and Miller, 1982.)

oesrcr sronvs 477

FIGURE 14.6.1(c)All-season PMP (in) for 6 hours. 200 mi'�. (Sorrc€: Hansen, Schreiner, and Milier, 1982.)

TIGURE 14.6.1(d)All-season PMP (iDi for 6 hours, l0O0 mi'�. (.Sorrce: Hansen, Schreiner, and Miller, 1982.)

$4ls?I

il*t$tst*l

#::€i€'s,l

si#){l*l*iihl*fi

478 (

FIGURE 14.6,r(?)All'season PMP {in) for 6 hours, 5000 mi'�. (Sorr.r: Hansen, Schreiner. and Mille., 1982.)

Unitcd States for a 10-mi2 area and durations of 6 aod 12 hours. For example,for Chicago. Fig. 14.6.1(a) indicatcs a PMP for 6 hours of approximately 26 in,and Fig. 1.1.6.1(&), gives the PMP for l2 hours as approximarely 30 in. In a 12-hour probable maximum storn over a l0-mi2 area ar Chicago. then, 26 in wouldoccur in the most intense 6 hours and 30 - 26 = 4 in of precipitation would occurin the remaining 6 hours. For a given location, the depth-area-duration graph isa plot of depth vs. stonn a.rea with curves for various storm durations.

Standard isohyetal pattern, A standard elliptical storm pattern adopted in HMR52 is shown in Fig. 14.6.2. It has 14 contours, labeled A through N, eachof u,hich bounds a specified area: contour A contains 10 mi2, contour B 25mi2. and so on. The ratio of the lengths of the major ancl minor axes of theellipses shorvn is 2.5 to l

If a and b are the lengths of the semimajor andsemiminor axes in Fig. 14.6.2, the area of the ell ipse is given by

( r 4 . 6 . I )

Since a = 2.5r, subsiitution into (14.6.1) gives rhe length of the semiminor axrsA S :

i ^ \ l l 2

(r4.6.2)

For example . fo rA = l0 miz , b : ( lo l2 .5n) t /2 : l .13 mi , a ld a = 2 .5b :2 .5 xl� l3 = 2.82 mi. The length r for a ndius at angle d with the major axis is given by

; e ! 3 = i g c F F F E 3,9 eH , l l " i l ' r l i- g < c c ! r a l r i u . ( , ! - - t 4 ) i F z

,9

I

9

E

4E0 ellueo uvonor-ocv

Orientation adjustment factor. The orientation of the isohyeral pattem mosriikely to be conducive to a PMP event for any place in the United States eastof the l05th meridian is indicated in Fig, 14.6.3. Nonh is considered as 0" forthis figure, arlu angles are measurcd clockwise from this direction. This figureis based upon averages of isohyetal orientations for major stolms in the UnitedStates, whjch predominantly have moisture flowing from the south and west

FIGURE 14,6.3Anal)sis of isohyeral orienrations for selected major storms, adopred as recommended onentatronfor PMP. within I 40'. (Srrrc?r Hansen, Schreiner. and Miller. 1982.)

oesrcr srorvs 4E1

(180'to 270'). It is important to note that the moisture flow for PMP at any loca-tion is not restricted to any one orientation; it is reasonable to expect the moistureflow to occur over a range of orientations centered on the values in Fig. 14.6.3.If the longitudinal axis of the storm pattem is oblique to the rccommended ori-entation, the PMP estimate is reduced by a percentage depending on the anglebetween the two directions. The adjustment factor for orientation differences isg iven in F ig . 14 .6 .4 .

i I

Critical storm area. The critical storm area is that fo, *friJl $" precipitationfrom the depth-area-duration curye yields the greatest PMP over the watershed,

iI

l

-1: lli l;i,]

:1il

..

,-l,.1

il

a2b2- 2 _

q2 sin2 0 + b2cos20

For example, for 0 = 45", a - 2.82, and D - Ll3 mi,1 . 132 t (2.827 sin? 45' + 1. 1 32cos 245') = 2.20 mi.

( 1 4 . 6 . 3 )

r2 = 2.822 x

99

98

91

96

95

94 :)

o r E

9 l ?

0 r o i 4 51 4 0

1 50 155orientarion difference (")

1 6 0 1 6 5 t o1 9 0

FIGURE T4.6.4Adjustment facrors for isohyelal orientation differ€nce. This figure is used to determine the adjust-ment factor to apply to isohyet values in Fig. 14.6.2 when the pattem is placed at an o.ientatronmore than 40" away from that recommended in Fig. 14.6.3. (So&rc?: Hansen, schreiner, and Miller,1 9 8 2 . )

P

'

s

d

A Z

5 )

? *

t .

t

I\

I \ \ fI

', I, .

\ I

\ N\l

12'

1,% + ) \

4, '{ \.,1\.'l\

1+, \' \,..i -b

tur ..\ \l\ \

\ah- ll i

a,\

a/, \ \ l ) +-t-\

t\ t q --

\ ) Jtl

(7 !u, ezrs ParP ujJots

oesrcr sronvs 483

taking into account the fact that the watershed area js not shaped elliptically likethe standard isohyetal pattem.

Isohyetal area factor. The value of the PMP represents the average preciprtation

depth over a specified area for a given duration. When a storm is represented bythe standard isohyetal pattem, there will be regions of greater pfecipitation depthnear the center of tle pattern and of lesser depth near the edges. Figure 14.6.5shows, for the most intense 6-hour storm inteNal, the Perqe4 of the specifiedPMP depth to be applied to each contour of the standard isohydal pattern to getthe correct spatiai dist bution. For example, using the PMP &r an area of l0mi2, according to the graph, contour A (area l0 miz) gets 100 percent of thePMP depth, and contour B (area 25 mi2) gets 64 percent. So. since the 6-hourPMP depth for 10 mir at Chicago is 26 in, as found earlier. this storm wouldhave depth 26 x 0.64 : I6.4 in over 25 mi2. which represents a volume of 25mi2 x 16.4 in - 416 mi2 in of precipitation, of which 10 mir x 26 in = 260mi2 in. or 63 percent. occurs in the most intense l0-mi2 area.

20000

r0000

l 5 25

PMP (in)

4000030000

500040003000

2000

^ 1000

; sooI 400< 300

200

100

4030

z0

t 0l 0

FIGURE 14.6.6Depth-area-duration curves for 31"45'N, 98'15'W applicable to the Leon River, Texas. drainage-(Soarcer Hansen, Schrein€r, and M;l ler, 1982.)

\

\ \

,, 1 ) \--\- - ..

\

\ \

484 applro sronorocy

Example 14.6,1. Determine the spatial distribution and the average depth overthe \raiershed of the 6-hour probable maximum precipitation on the Leon Riverdminage basin above Belton Reservoir in Texas. The drainage basin area is approximatel l 3660 mi2 with the drainage center at 31o45'N, 98'15'W. (This example isa simplified version of a design example presented in HMR 52).

Solution.L DeveLop the depth-area-duration cur|e. The 6-hour PMP depth at Leon

River is read for various storm arcas from the standard PMP maps, ;ncluding thoseshown in Fig. 14.6.1. PMP depth is plotted as a function of stom arca and durarionas sbown ih Fig. 14.6.6. In this example, only the 6-hour curve wil l be considered.The plotled points fiom the maps are connected with a smooth line to form rhedepth-area curve for this duratioo. Then the PMP depths for a number of stormareas are read from this curve as shown in column 2 of Table 14.6.1.

2. Adjust Jor watershed orientation. The standard isohyetal pattem js cen-tered oler the l-eon River drainage basin as shown in Fig. 14.6,7. The longitudinal axis of the wate$hed mns southeast (134') to nonhwest (3140), obl ique to theprefer.ed direction at this location, from the southwest (208'). The maD shown inFjg. 11.6.4 appl ies to an o. ientat ion range of 135'to 315'. so rhe prefined orien-rarjon (208") is subrracted from the watershed axis direction (314') falling in therange 135'to 315', i .e. 314'- 208'= 106'. When rhe orientarion di i lerence rsgreater than 90', the value of l80o minus the orienration difference is used in read-ing this f igure. that is, in this case 180o 106'= 74'. The values of the orienra,tion adjustment factor for 74o and the vadous storm areas are read off Fig. 14.6.,1and enlered in column 3 of Table 14.6. l � Mult ipl icat ion of columns 2 and 3 yieldsthe adjusted depths shown in column 4.

3. Find the critical storm area. For a given storm area, the isohyetaledjustment factors for each contour are read from Fig. 14.6.5 by finding tle storm

TABLf 14.6.1Six-hour probable maximum precipitation for Leon River,adjusled for orientation of the watershed r€lative to thedirection of maximum atmospheric moisture flow (Example14.6 .1)

(

Column: INomimlstorm

(mi')

6-h PMPfrom depth-area-dumtion

Orientation Adjust€dadjustm€nt 6-h PMPfactor

(in)(in)

I,000I , 500? , 1 5 01,0004,5006,500

r 0,000L 5,000

1 6 . 1t4.412.9I 1 . 59 .88 .5' 7 . )

5 .9

0.9610.9330.89'�70 .8500 .8500.8500.8500.850

t5.4713.44t ) .519.788 .331 .236.045 .02

,-:

g

E

'?

E

,E

, -

: 3q ? .

^"(.;Y S

"9s'F

7

t-

i

,' l,' ,,/

, r , . ,

) . ' , , z ,

I

6

486

TABLE 14,6.2Computation of isohyetal and areal average depths for the 6-h probablemaximum precipitation at Leon River for a nominal storm area of 1500 mil(Example 14.6,1)

Column: 3Isohyetprecipitationdepth *

(in)

Incrementslvolume ofprecipitation

(in mi?)

Isohyet Isohyetlabel adjustment

factor

4 5Average Incremenialdepth ar€a withinbehreen watershedisohyets A4

(in) (mi'�)

BCDEFGHIJKL\ lN

1 . 6 2| . s 2| . 42I . 32t . 22| . 1 2l � 0 50.960 .880 .800 .560 .410.260 . l 6

21.'�l'�720.4319.08t1 . 1416 .4015 .05t4 .1112 .901 1 . 8 310.75'7 .535 . 5 13.492 . 1 5

2t.1' � |2 t . r 0t9./61 8 . 4 it'7 .0715.7214.58I 3 . 5 112.3611.299 . t 46 .524.502 .82

Total

1 0152550"15

t25r50250211393488582131489

3660

211 .7316 .5493.9924.6

I , 280 .21,965.62 , t 81 .43,3' �76.83 .350 .94,436.8

3,193.73 . 3 1 8 . 3I , 380 .2

31 ,498 .5

Average PMP deplh over walershed (in) = 31,498.5 / 3660 = 8-61 in

-Tlre P\.1P ior 1500 mi'� is 13.4,1 in.

a.ea on the vertjcal axis ard reading hodzontally to the curve for that isohyet. Forexample, for a 1500 mi2 area, the graph gives an adjustment factor of 162 percent,or 1.62, for isohyet A. The adjustment factors for all isobyets for a 1500-mi2 srormarea are shown in column 2 of Table 14.6.2. The adiusted PMP deoth for 1500-mi2 storm area is 13.44 in from Table 14.6.1. and th]s vaiue is muli ipl ied by theisohyetal adjustment factor to give the precipitation depth for each isohyet in coiumn3 of Table 14.6.2; for example, on isohyet A, the depth is 1.62 x 13.44 = 2l.1' / in.The average precipitation depth P"" between adjacent isohyets is shown in column.1. For isohyet B, P^, : (21.71 + 20.43)/2:21.10in, for example.

The incremental areas M added by including each successive isohyet areshown in column 5 of Table 14.6.2. As shown in Fig. 14.6.7, isohyets A throughH are completely contained within the watershed, so the incremental area assignedto them is simply the increase in area from the previous isohyet, with the isohyetaiareas being shown on Fig. 14.6.2. For example, isohyet B encompasses 25 mi2,of which 10 mi2 falls within isohyet A, so the incremental area for isohyet Bis 25 - 10 = 15 mi2. For isohyets I through N, the incremental area in column5 js that part of the isohyetal area tlat is also contained within the watershgd.The inc.emental volume Ay added for each additional isohvet is calculated as

oistcr' sronus 487

Ay: P""M. For example, the volume within isohyet A is 21.77 in x l0 mi::21'7 . '1in.mi ' � , whi le isohyet B includes an addit ional 21.10 in x 15 mj2 : 316.5 in.mi2as shown in column 6 of the table. By summing the incremental volumes for allisohyets containing at least part of their area witiin the watershed, the total volumeis found, and it is divided by the watershed arca to give the average precipitationdepth. In this case, the aveEge depth is 31,489.5 in.mi ' � l3660 mj' � : 8.61 in.

The prccedure described above js repeated for all the storm areas in Table14.6.1 to find the critical storm area: that area which yields lhejgreatest averagePMP over the watershed. In this case the 1500-mi2 area giveslthd.largest averagedepth and is the critical storm area. So the isohyetal precipitatiolv$attem of the 6-hour PMP at Leon River is given by column 3 of Table 14.6.2, and the watershedaverage 6-hour PMP is 8.61 in.

This procedure can be repeated for various durations and the corespondingisohyetal and ave.age depths obtained. A storm hyetograph can be developed fromthese depths by the altemating block method (Sec. 14.4) or other methods. lf apercentage of the probable maximum precipitation is required for a design (e.9.,50 percent, as shown in Table l3.l.l for some darn designs), the values of theisohyet precipitation depths in column 3 of Table 14.6.2 are reduced to the desiredpercentage of the values shown.

REFERENCES

Bandyopadhyay, M., Synthetic storm pattem and runofffor Gauhati, India., J. H))d. Dit,., Am. Soc.Civ. En8., vol. 98, no. HY5, pp. 845-857, l9?2.

Frederick, R. H., V. A. Myers, and E. P. Auciello, Five to 60-minute precipitation frequency forthe eastem and central United States. NOAA technical memo NWS HYDRO 35, NalionalW€ather Service, Silver Sprjng, Maryland, June 1977.

Hansen, E. M., L. C. Scfueir€r, and J. F. Miller, Application of probable maximum precipitarionestimates-United States east of the l05th meridian, NOdA hydrometeorological repon no.52, National Westher Service. Washington, D. C., August 1982.

HaveDs and Emerson. consulting engineers, master plan for pollution abatement. Cleveiand. Ohio,Juiy 1968.

Hershfield, D. M., Rainfall frequency adas of the Uniled States for duralions from 30 mrnues to24 hours and retum periods from I to 100 years, tech. paper 4t), U. S. Dept. of Comm.,Weather Bureau, Washington, D. C., May 1961.

Huff, F. A., Time distribution of rainfall in heavy storms, Water Resour. Rrr., vol. 3, no. 4, pp.1007-1019, 1967.

Institution of Engircers Australia, A6tralian Rdtnfall ann Runof. vo]. I ed. by D. H. Pilg.im, vol.2 ed. by R. P. Canterford, Canberra, Australia, 1987.

Keifer, C. J., and H. H. Chu, Synthetic storm pattem for drainage des;gn,,I- HJd. Dit,., Am. Soc.Cir. Eng.,vol.83, no. HY4, pp. I 25, 1957.

Marsalek, J., Research on lle desigt storm concept, tech. memo. 33, Am. Soc. Civ. Eng., UrbanWater Resour. Res. Prog., New York, 1978.

McPherson, M. B., Discussion of "Synthetic stom pattem for drainage design," J. Hyd. Div., Am.Soc. Civ. Eng.. vol. 84, no. HYl, pp. 49-60, 1958.

Miller, J. F., R. H. Frederick, and R. J, Tracey, Precipitation-frequency atlas of the conlermrnouswestem United States (by sta!cs), NOAA atlas 2, l1 vols., National Weather Service, SilverSpring, Maryland, 1973.

National Academy of Sciences, SafetJ of Eristing Dams: Evaluation and lmprovement. NattonalAcademy Press. Washington, D. C., 1983.

492 ,rlrrrrr alonolocv

li.r000

-s000100i)t{100

:lxl0

^ i000' :-E; 5ooe too

100

I0t)

i 0

l 0

It)

0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0PMP (in)

FIGLRE I4.P.2Depth arei-durarion curves for 34"36'N. 93o27'W, applicable ro the Ouachita River, Afl(ansas,drainage. lSrrrcc: Hansen, Schreiner, and Mil ier, 198?.)

di.ection 95'/275". The watershed contains all of the contours A through H ofthe slorm pattern. Contours I through L add the following incretnental areas (inmir)): 1,2,12: I , 242t K, 221, L, 192.

14.6.7 Sohc Prob. 14.6.6 to determine the average depth of the 6 hour probablemaximum precipitat ion of the Ouachita River, Arkansas. above Rennel Dam(Fig. 14.P. l) considering a storm of 1500 mi2.

14,6.8 Consider a number of storm areas to determine the critical stom area for the 6-hour PMP over the Ouachita River watershed (Prob. 14.6.6).

CHAPTER

1 5I

DESIGNFLOWS

Hydrologic design for water control is concemed with mitigating the adveneeffects of high flows or floods. A flood is any high flow that overtops eithernatural or artificial embankments along a stream. The magnitudes of floods aredescribed by flood discharge, flood elevation, and flood volume. Each of thesefactors is important in the hydrologic design of different types of flow controlstuctures. A major ponion of this chapter deals with development of the designflow or design flood for flow regulation structures (detention basins, flood conholreservoirs, etc.) and flow conveyance structwes (storm sewers, drainage channels,flood levees, dive$ion structures, etc,). The purpose offlow regulation structuresis to smooth out peak discha-rges, thercby decreasing downstream flood elevationpeaks, and the purpose of flow conveyance structures is to safely convey the flowto downstream points where the adverse effects of flows are controlled or areminimal. This chapter discusses methods and simulation models that can be usedin the hydrologic design of flow control structures from urban drainage systemsto flood conhol reservoirs.

Hydrologic design for water use is concerned with the development of waterrcsources to meet human needs and with the conservation of the natural life inwater environments. As population and economic activity incrcase, so do thedemands for use of water. But these must be balanced against the finite supplyprovided by nature and the desire to mairtain healthy plant and animal life inrivers, lakes, and estuaries. Hydrologic information plays a vital role in managingthe balance between supply and demand for water resources and in planning waterresource development projects. In contrast to hydrologic design for water control,which is concerned with mitigating the adverse effects of high flows, hydrologic

494

design for watcr use is directed at udlizing average flows and with m;tigating theeffects of extremely low flows.

15.1 STORM SEWER DESIGN

Population growth and urban development can create potentially severe problemsin urban water management. One of the most important facilities in preservingand inproving the urban water environment is an adequate and properly function-ing storm water drainage system. Construction of houses, commercial buildings,parking lots, par.cd roads, and streets increases the impervious cover in a water-shed, and reduces infiltration. Also, with urbanization, the spatial pattern of flowin the $,atenhed is altered and there is an increase in the hydraulic efficiencyof flow through artificial channels, curbing, gutters, ald storm drainage and col-lection systems. These factors incrcase the volume and velocity of runoff andproduce larger peak flood discharges from urbanized watersheds tban occurred inthe preurbanized condition. Many urban drainage systems constructed under onelevel of urbanization are now operating under a higher level of ubanization andhave inadequate capacity.

One view of the typical urban drainage system is shown in Fig. 15.1.1. Thesystem can be considered as consisting of two major types of elements:, locationelements and. transfer elements. Location elements are the places where the water

FIGURE I5.1.1Typical irban drainage system. (.tour.?r Roesner, 1982, Copyrighl by the American ceophysicalUnion.)

oesrol nows 495

stops and undergoes changes as a result of humanly controlled processes, forexample, water storage, water treatment, water use, and wastewater treatmgnt.Transfer elements connect the location elements; these elements include channels,pipelines, storm sewers, sanitary sewers, and streets. The system is fed byrainfall, influent water from various sources, and imponed water in the pipesor channels. The receiving water body can be a river, a lake, or an ocean. Figure15.1.1 shows a storm sewer system for collection of storm drainage in a pipenetwork and discharge to a receiving water body. This sectign lgonsiders thedesign of a sewer system for storm drainage. 1 ;

System concepts are increasingly being used as an aid io {inderstandingand developing solutions to complex urban problems. These problems involvedistributed systems, and must be aaalyzed to account for both spatial and temporalvariations. Urban watersheds vary in space in that the ground surface slopeand cover, and the soil type, change from place to place in the watershed.They vary in time in that hydrologic characteristics change with the process ofu$anization. The mathematical formulation of models for urban water systemsdistributed in both time and space is a complicated task. Consequently, spatialvadation is sometimes ignored, and the system is treated as being lumped. Somespatial variation can be introduced by dividing the watershed system into severalsubsystems that a.re each considered lumped, and then linking these lumped-system models together to produce a model of the entire system.

Models can be used as tools for planning and management. In particular, anumber of computerized watershed simulation models have been proposed. Thedetermination of the runoff volume and peak discharge rate are important issues inutban stormwater management, and methods for calculating these variables rangefrom the well-known rational formula to advanced computer simulation modelssuch as the Storm Water Management Model (SWMM; see Huber, et al., 1975).

Design Philosophy

A, storm sewer system is a network of pipes used to convey storm runoff in acity. The design of storm sewer systems involves the determination of diameters,slopes, and crown or invert elevations for each pipe in the system. The crownand invert elevations of a pipe are, respectively, the eievatioos of the top and thebottom of the pipe circumference.

The selection of a layout, or network of pipe locations, for a storm sewersystem requtes a considerable amount of subjective judgment. Hydrologists arcusually able to investigate only a few of the possible layouts. Generally, manholesare placed at sheet intersections and at major changes in grade, or ground surfaceslope, and the sewers are sloped in the direction of the grouqd surface, so asto connect with downsteam submains and trunk sewers. Once a layout hasbeen selected, the rational method can be used to select pipe diameters. Thisconventional design approach is based on a set of design standards and criteria,such as those set forth by the American Society of Civil Engineers (1960) andvarious planning agencies.

496 .rprlrso sroaorocr

Storm drainage design can be divided into two aspects: runoff predictionand sl stem design. In recent years, rainfall-runoff modeling for urban watershedshas been a popular activity and a va.riety of such rainfall-runoff models arero*' available, as described by Chow and Yen (1977), Heeps and Mein (1974),Braudstener (1976), McPherson (1975), Colyer and Rthick (1977), Yen (1978).and Kibler (1982). Computer rnodels are described more fully in Sec. 15.2.

The following constraints and assumptions are commonly used in stormsewer design practice:

l. Irree-surfacg flow exists for the design discharges; that is, the sewer system isdesigned fol "gravity flow"; pumping stations-and pressurized sewen are notconsidered.

2. The sewers are of commercially available circular sizes no smaller than 8inches in diameter.

3. Thc design diameter is the smallest commercially avajlable pipe having flowcapacity equal to or greater than the design discharge and satisfying all theappropriate colstraints.

4. Storm sewers must be placed at a depth such that they will not be susceptibleto frost, will be able to drain basements, and will have sufficient cushioningto prevent breakage due to ground surface loading. To these ends, minimumcover depths must be specified.

5. The sewers are joined at junctions such that the crowll elevation of theupstream sewer is no lower than that of the downstream sewer.

6. To prevent or reduce excessive deposition of solid material in the sewers, aminimum permissible flow velocity at design discharge or at barely full-pipegravity f low is specified (e.g., 2.5 fVs).

7, To prevent scour and other undesirable effects of high-velocity flow, a maxi-mum permissible flow velocity is also specified.

E. At any junction or manhole the downsteam sewer cannot be smaller than anyof the up*ream sewers at thar junclion.

9. The sewer system is a dendritic, or branching, network converging in thedownstream direction without closed loops.

Rational Method

The rational method, which can be raced back to the mid-nineteenth century, isstill probably tire most widely used method for design of storm sewers (Pilgrim,1986; Linsley, 1986). Although valid criticisms have been mised about theadequacy of this method, it continues to be used for sewer design because ofits simplicity. Once the layout is selected and the pipe sizes determined by themtional method, the adequacy of the system can be checked by dynamic rouringof flow hydrographs through the system.

The idea behind the rational method is thar if a rainfall of intensity i beginsinstantareously and continues indefinitely, the rate of runoff will increase until

oesrorl nows 497

the time of conceniration t", when all of the watershed is contributing to flow atthe outlet. The product of rainfall intensity i and watershed area A is the inflowrate for the system, r4, and the ratio of this rate to the rate of peak discharge O(which occu$ at time tc) is termed the runofi coeficient C (0 < C < 1). This isexpressed in the rational formula:

( 1 5 . 1 . 1 )

Commonly, Q is in cubic feet per second (cfs), i is in inches pe4 hdpr, and A is inacres, and the conye$ion (l cfs = 1.008 acre.in/ir) is considerbdJo be includedin the runoff coefficient. The duration used for the determinatiof,'of the designprecipitation intensity i in (15.1.1) is the time of concentration of the watershed.

In urban areas, the drainage area usually co[sists of subareas or subcatch-ments of different surface cha.racteristics. As a result. a comDosite analvsis isrequired that must accounr for the various surface characterist ics. The areas ofthe subcatchments are denoted by A; and the runoff coefficients of each subcatch-ment are denoted by C;. The peak runoff is then computed using the followingform of the rational formula:

o = r t c . A ,

where m is the number of subcatchments drained by a sewer.The assumptions associated with the rational method are:

l. The computed peak late of runoff at the outlet point is a function of the averagerainfall rate during the time of concenaation, that is, the peak discharge doesnot result frcm a more intense storm of shoter duration, during which only aportion of the watershed is contributing to runoff at the outlet.

2. The time of concentration employed is the time for the runoff to becomeestablished and flow from the most remote part of the drainage area to theinflow point of the sewer being designed.

3. Rainfall intensity is constant throughout the storm duration.

Runoff Coetfrcient

The runoff coefficient C is the least precise variable of the mtioDal method. Itsuse in the formula implies a fixed ratio of peak runoff rate to rainfall mte for thedrainage basin, which in reality is not the case. Proper selection of the runoffcoefficient requires judgment and experience on the part of the hydrologist. Theproportion of the total rainfall that will reach the storm drains depends on thepercent imperviousness, slope, and ponding character of the surface. Impervioussurfaces, such as asphalt pavements and roofs of buildings, will produce nearly100 percent runoff after the surface has become thoroughly wet, regardless of theslope. Field inspection and aerial photographs are useful in estimating the natueof the surface within the drainase arca.

I, li

( 1 s . 1 . 2 )

498 qrprro uvororocv

The runoff coefficient is also dependent on the character and condition of thesoil. The infiltration mte decreases as rainfall continues, and is also influencedby the antecedent moisture condition of the soil, Other factors influencing therunoff coefficient are minfall intensity, proximity of the water table, degree ofsoil compaction, porosity of the subsoil, vegetation, ground slope, and depressionstorage. A reasonable coefficient must be chosen to represent the integrated effectsof all these factors. Suggested coefficients for various surface types as used inAustin, Texas are given in Table 15.1.1.

TABLE I5.I.I rRunoff co€fficients for use in th€ rational method

Return Rdod (yea$)

ChAracier of surface 2 5 10 25

Develop€dAsphalt ic 0.13Concrete/roof 0.?5

Grass areas (lawns, parks, etc.)

Fl^t, U2% 0.25A\erage.24qa 0.33Steep, over 77. 0.31

Flat, Vzqa 0.21Aveftge,2 7% 0.29Steep, over 7% 0.34

Undeveloped

Cultivated Land

F1ar, O-2q. 0.31A\er^Ee,2-'7qa 0.35Steep, over 79o 0.39

Pasture/RaDge

flu, Vzqa o.25A.verage,2-'7Eo 0.33Steep. over 7% o.37

ForesflVoodlands

F1at, U2% 0.ZzA\era9e,2J% 0.31Sleep, over ?% 0.35

Poar condition ($ass cover less than 50% of the area)

Flat,o 2q. O.32AveftEe,2:qc 0.3'7Steep, over 77a 0.4O

Fair conditian (Erass cover on 50% to ?5% of lhe area)

0.30 0 .340.38 0.420.42 0.46

Coad condition (g^ss cover larger than 7570 of the area)

0.77 0.81 0.86 0.90 0.95 1.000.80 0.83 0.88 0.92 0.9' �7 l �00

0.34 0.3' �7 0.40 0.44 A.4'7 0.580.40 0.43 0.46 0.49 0.53 0.610.43 0.4s o.49 0.52 0.s5 0.62

0.3? 0.41 0.530.45 0.49 0.580.49 0.53 0.60

0.320.420.47

0.43 0.41 0.570.48 0.51 0.600 .51 0 .54 0 .61

0.280.360.40

0.230.320.3'7

0.340.380.42

0.280.360.40

0.25o.140.39

0.25 0.290.35 0 .390.40 0.44

0.36 0.400.41 0.440.44 0.48

0.30 0.340.38 0.42o.42 0.46

0.36 0.490.46 0.560.51 0 .58

0.41 0 .530.49 0.5E0.53 0.60

0.39 0 .480.4't 0.560.52 0.58

0.28 0.310.36 0.400.41 0.45

0.370.450.49

0.350.430.48

Ndrc: The values in the table arE ihe stardards used by tie City of Austin, Texas. Used widr pennission.

I . = t o + t l

The flow time is given by Eq. (5.7.3):

. _ s L i, J _ Z - V

oeslcl nows 499

Rainfall Intensity

The rainfall intensity i is the average rainfall rate in inches per hour for a particulardminage basin or subbasin. The intensity is selected on the basis of the designrainfall duration and retum period as described in Sec. 14.2. The design durationis equal to the time of concentration for the drainage area under consideration,The retum period is established by design standards or chosen by the hydrologistas a design parameter.

Runoff is assumed to reach a peak at the time of conceS.ation t" when theentire watershed is contributing to flow at the outlet. The time f concen[ationis the time for a drop of water to flow from the remotest point iri the watershedto the point of interest. A trial and error procedure can be used to determinethe critical time of concentration where there are several possible flow paths toconsider. The time of concentation to any point in a storm drainage system isthe sum of the inlet time ,o (the time it takes for flow from the remotest point toreach the sewer inle0. and the flow time tr in the uDstream sewers connected tothe outer point:

( 15. 1 .3)

( ls .1 .4)

where li is the length of the lth pipe along the flow path, and V; is the flow. , a l ^ ^ i r . , i - r l . � - ^ : ^ -

The ir et time, or time of concentration for the case of no upsheam sewers,can be obtained by experimental observations, or it can be estimated by usingformulas such as those listed in Table 15.1.2. There may exist several possibleflow routes for different catchments drained by a sewer; the longest time ofconcentration among the times for different routes is assumed to be the criticaltime of concentration of the area drained.

Because the arcas contributing to most storm sewer inlets are relatiyelysmall, it is also customary to determine the inlet time on the basis of experienceunder similar conditions. Inlet time decreases as the slope and imperviousness ofthe surface increases, and it increases as the distance over which the water has totmyel increases and as retention by the contact surfaces increases. All inlet timesdetermined on the basis of experience should be verified by direct overlaad flowcomputation.

Drainage Area

The size and shape of the catchment or subcatchment under consideration mustbe determined. The area may be determiaed by planimetering topographic maps,or by field surveys where topographic data has changed or where the mappedcortour interyal is too geat to distinguish the direction of flow. The drainage area

?

o

iE

O

g E E : : 3 E 8t E t s F e F ;

i lse:f ;f ,E!iEit EE ,f i r q e f : r6 E E : i H i r g- 3 ' g r i a a : H

BE- iE; E E; =

eigiigEifB

- ? - c i l i i g - E : -; n = ; 2 l " z z i : 5 . = ' i E

.EF=E! EIEFg!F EE€:{FE€Er= e g$;;$ Hirlr€t l t I

l t f l I I ! { r - ! ' l q ! r l >

&F

* € E

I r E! . 8 b

9 0 g

l B e

d 9 . E8 . 9 Et ! - -I F -

E e ;s ! . tr d F

8 . 3 +o > g

a

i - p 3' j € > . - - " o "

eEBisEf i ggFg > < ? t =9 U

l lg

q

j

I

e = l b = 3 _ 9 " . 8

E : : s ; j * 3= S r o r - t ' E :

; s E ; E iE *i : E i i € i ;* E : . ! { , Z z .

E g e " E E s ! iF q E: E E"i t; : T i i i ; ;F G g l g e T P

€ E i i ; : E != ; i F t I { i sE i t s g 8 E r r le t 9 E q \ / r e f i i EA P g E € 3 A : E :

9 d ! , t : 5 6

b 6 : ; r = i: : i ; i E 3= - = . 3 . 2 a d

E 6 - " ' 3 ! E =a I : i 9 3: T E ; E I EF : : ? : 9 a9 l i t i ' 6 -9 - A = :-

- - : e

x i : z ! E cE 6 - E e E =

! E; e p i 9 , : €

3 € g E i 4 , 5; E E * 5 Y : iq : c ; c - ; \ o

l i c q d : R+ g E ; E E ; R; > ! : ! 7 1 :; € b = ; ; I : i : . 'i r ? = ' : f f . E e

s E : E f t E: = H € s - < F: 5 E i & 5 ; ! s{ E : E E - E = 3 , i; = H ! t - ; g ! iEEEg iEEPg l t t

l t t l! - o - l ? 1 * . ( J . ! e )

c =

. ! E< E ^

- . u € i nz < =

ao..,

! i € 6* ; - d N :

a- $'.E # Es e f r € € td E 3 3 : e P

- = a E E g l i AJ E _ 9 ) ; _ 6 6

l l l l'3 'l \

9 3 iI ! o- n ' I

r ' + I i- - . = E ;; " 9 - o ^

l

t

iq 5

E : ! ig - & rGU

q o'.: >,

l d E

< =?q )

502 ,rrrurco nvonorocv

contributing to the system being designed and the drainage subarea contributingto each inlet point must be measured. The outline of the drainage divide mustfollow the actual wateNhed boundary, ruther than commercial land boundaries,as may be used in the design of sanitary seweff. The drainage divide lines areinfluenced by pavement slopes, locations of downspouts and paved and unpavedyards, grading of lawns, and mary other featues intoduced by urbanizatron.

Pipe Capacity

In choosing stqrrn sewer pipe diameteN, the mimimum requted diameter is com-puted, and the next larger commercially available size is selected. Commercialpipes are available in diameters of 8, 10, 12, 15, 16, and 18 in, at incrcments of3 in between 18 and 36 in, and at incremeBts of 6 in between 3 ft and 10 ft.

Once the design discharge p entering the sewer pipe has been calculatedby the rational formula, the diameter of pipe D required to carry this dischargeis determined. It is usually assumed that the pipe is flowing full under gravitybut is not pressurized, so the pipe capacity can be calculated by the Manning orDarcy-Weisbach equations for open-channel flow. For Manning's equation, thearea is A - nD2/4, and the hydraulic radius is R = Alp = (jrDz/4)l1tD = Dl4.The friction slope Sy is set equal to the bed slope of the pipe, S0, thus assuminguniform flow, and the discharge is computed for full pipe flow as

^ 1 . 4 9U= -ySIt2ARa3

n '

- | .49 ,t z1 rD2 \1 D 1/3n

- o \ + / \ e /

= 9J!l qlrzD&l,

' u "

This is solved for the required diameter D as/ \ : /8

^ l 2 . r6Qn IU : | - |

\ . & /(1s .1 .6 )

which is valid for Q in cubic feet per second, and D in feet. When using SIunits, with Q in cubic meters per second and D in metels, the coefficient 2.16 isrep laced by 2 .16 >< 1 .49 :3 .21 i r Eq . (15 .1 .6 ) .

Using the Da{cy-Weisbach equation (2.5.4), with A, R, and S/ as forManning's equation,

( l s . 1 . s )

t ̂ \ l l 2

O=rlYns,l\ /

- l

o P z l S g D - \ t ' '= 4 \74"/

( 15. r .7)

(

Solved for D, this gives

orsror n-ows 503

( 1 5 . l . 8 )

where/is the Darcy-Weisbach friction factor and g is acceleration due to gravity.Equation (15.1.8) is valid for any dimensionally consistent set of units

Assessment of the Rational Method ' ',

The mtional method is criticized by some hydrologists U""uor" di its simplifiedapproach to the calculation ofdesign flow mtes. Nevertheless, the rational methodis still widely used for the design of storm sewer systems in the United States andother countries because of its simplicity and the fact that the required dimensionsof the storm sewe$ are determined as the computation proceeds. More realisticflow simulation procedures involving the routing of flow hydrographs require thedimensions of the flow conveyance structures to be predetermirled. The stormsewer system design produced by the rational method can be considered as apreliminary design whose adequacy can be checked by routing flow hydrographsthough the system.

The uncenainties involved in the rational method can be examined by therisk analysis procedures described in Chap. 13 (Yen, 1975; Yen, etal,,1976iYen, i978). In this case, the loading on the system is described by the rationalformula (15.i.2) and the capacity by the pipe conveyance equations (15.1.5) or(15.1.7). Problems 15.1.8 to 15.1.16 address this subject.

Example 15.1.1. A hypothetical dminage basiD comprising seven subcatchmentsis shown in Fig. 15.1,2. Determine the required capacity of the storm sewer EBdraining subarea III for a five-year rctum period storm. This subcatchment hasan area of 4 acres, a runoff coefficient of 0.6, and an inlet time of l0 minutes.

[f F]I \ W I

FIGURE 15.1.2The drainage basin and stom sewer system foaExamples 15.l . l and 15.l .2.

,.', - /o.sr r,fQ2 1ri 5

" \ 8so l

l ll u ll - lB

vrl Is , ic

o

[-]li lt \ l

--1l u ' l lr \l I

I

I

504 eprr-ep Hvonolocv

The desi$ precipitation intensity for this location is given by i = l2}To)1sl(Td +27), where i is the intensity in inches per hour, I is the return period, and Zl isthe duration in minutes. The grcund elevations at points E and B are 498.43 and.195.55 ft above mean sea level, respectively, and the length of pipe EB is 450 ft.Assume MaDning's, is 0.015. Calculate the flow time in tie pipe.

Sd4tior. The time of concentmtion for flow into sewer EB is simply the lo-minuteinlet time for flow from subcatchment III to point E, So, Zd= 10 min and the designrainfall intensity with I : 5 years is

' l21To.t15

1 i(Td - 21)

_ l2o(5)o rt5

(ro + 2'1)= 4.30 inft

The design discharge is given by Eq. (15.1, 1):

Q = C i A: 0 . 6 x 4 . 3 0 x 4

:10 .3 c fs

The slope of the pipe EB is the difference betwe€n the ground elevations at pointsE and B divided by the length of the pipe: 56 = (498.43 - 495.55)/450 : 0.0064.The required pipe diameter is calculated from (15.1.6):

r 1 0 . 3 x

'6-ooA

" _ (

_ (_ \

\3/82.16On I

2 . 1 6 x 0.015i'= 1 .71 f t

The diameter is rounded up to the next commercially available pipe size, l�75 ft or21 in.

The flow velocity through pipe EB is found by taking the nominal diameter(1.75 ft), and assuming the pipe is flowing full with 0 : 10.3 cfs. Hence, y =

QIA = 10.31(n x 1.75214) = 4.28 fys. ' fhe f low rime is L/y: 450/4.28 = 105 s =1.75 min. It should be loted that a slight enor in the computed flow time ,scaused by assuming that the pipe is flowing full. The velocity for paftially-full-pipeflow can be determined using Newton's iteration technique preseBted in Chap. 5, ifnecessary,

Example 15.1.2, Determine the diameter for pipes AB, BC, and CD in the 27-acre drainage basin shown in Fig. 15.1.2. The area, runoff coefficients, and inlettime for each subcatchment are shown in Table 15.1.3, and the length aod slope foreach pipe arc in columns 2 and 3 of Table 15.1.4. Use the same rainfall intensityequation as in Example l5.l.l and assume the pipes have Manning's z = 0.015.

orsrcrv nows 505

Solulion. The same method as was illustrated in Example 15.1.1 is used for eachpipe, except that now the time of concenbatiotr must include both inlet time aDdflow time through upsffeam sewe$. The rcsults obtained in Example 15,1.1 forprpe EB arc shown in the first row of Table 15.1,4.

Pipe AB. This pipe drains subcatchments I and II. From Table 15.1.3, A t: 2acres, Cr = 0,7, and the inlet time is tr : 5 min, while An = 3 acres, Ctr = 0.7,and h = 7 min. Hence, the total area drained by pipe AB is 5 acres and ICA :CtAr 1 CtAr = O.'7 x 2 + O.'7 x 3 = 3.5. The time of concengation used is 7rnin, the larger of the two inlet times, The calculations for th! rfouired diameterare carried out in the same way as in Example 15. l. l; the results qrc shown in $esecond row of Table 15.1.4. The calculaled diameter, 1.94 ft, istounded up to acomrnercial sizn of 2-O ft. (24 in) for pipe AB.

Plpe BC. This pip€ drains subcatchments I tfuough V: subcatchments I and IIthrough pipe AB, subcatchment III through pipe EB, and subcatchments IV and Vdirectly. There are thus four possible flow paths for water to rcach point B; the timeof concentration is the largest of their flow times. The flow time for flow comingfrom pipe AB is 7 minutes inlet time plus 1.76 minutes travel time, or 8.76 minutes;for the flow from pipe EB it is l0 minutes inlet time plus l�75 minutes flow time,or 11.75 minutes; and the inlet times fo! subcatchments IV and V are 10 min and15 min, respe.tively. Thus, the time of corcentration for pipe BC is taken as 15 min.

For subcatchments I and II, ICA :3.5, as shown previously. Fo! subcatcb-ments III to V, the values of the runoff coefficient aDd catchment alea are given inTable 15. 1.3; at point C, using these values, !C.4 = 3.5 + 0.6 x 4 + 0.6 x 4 +0.5 x 5 : 10.8. Proceeding as in Example 15.1.1, the calculated pipe diameterjs 2.87 ft, which is rounded up to 3.0 ft (36 in) for pipe BC (third row of Tablel5 , r .4 ) .

Pipe cD. This pipe drains all seven subcatchments. Using the same methodas for the previous pipes, its time of concentration (to point C) is found to be 15minutes (to point B) plus 1.2 minutes flow time in pipe BC, or 16.2 minutes, and\CA = 15.3, The calculared diameter, 3.22 ft, is rounded up ro a commercial sizeof 3.50 ft (42 in) (fourth row of Table 15.1.4).

The required diameters for pipes AB, BC, and CD are 21,36, u\d 42 iD,,rcspectively.

TABLE 15.1.3Characteristics of the drainage basinfor Exarnple 15,1,2

Catchment Arer Runolfcoefficient

A C(acr€s,

hlettimeti(nin)

ITmry

vtvu

234454 ,54 ,5

0.70.70 .60.60 .50 .50.5

110t015l 5

l 5

9 o

q c !f ,x

^ . E E t E- l - ' ! . 1 9

I E F E

=E is t, 8 f i E e

6 . , EF i 5 ^

o , j s € 9

3!c !.E {

- A * * 9

a ' 4 i. 5 ; €

F a< .= - ' 9

Uvrhl

- e * . F

d l { g

(

.i .i di di

8 8 8 3

e e s e

oesrcr ruows 507

15.2 SIMI.,ILATING DESIGN FLOWS

Since the early 1960s, a host of deterministic hydrologic simulation models havebeen developed. These models incltde event simulation models for modeling asingle rainfall-runoff event and continuous simulation models, which have soilmoisture accounting procedures to simulate runoff from rainfall in hourly ordaily intervals over long time periods. Examples of event simulation modelsinclude: the U. S. Army Corps of Engineers (1981) HEC-I fiood hydrographmodel; the Soil Conservation Service (1965) TR-20 computerFrdgram for Projecthydrology; the U. S. Environmental Protection Agency (197$ SWMM stormwater management model; and the Illinois State Water Survey ILLUDAS modet,by Terstriep and Stall (1974). Examples of continuous simulation models include:the U. S. National Weather Service runoff forecast system (Day, 1985); the U. S.Army Corps of Engineers (1976) STORM model; and the U. S. Army Corps ofEngineers (1972) SSARR sheamflow synthesis and reservoir regulation model.This is by no means a complete list of available models, but it covers most of themodels commonly used in hydrologic practica. The HEC-I model is probably themost widely used hydrologic event simulation model. The acronym HEC standsfor Hydrologic Engineering Center, the U. S. Arrny Corps of Engineen researchfacility in Davis, Califomia, where this model was developed.

HEC-I Model

HEC-1 is designed to simulate the surface runoff resulting from preciPitationby representing the basin as an interconnected system of components. Eachcomponent models an aspect of the rainJall-runoff process within a subbasinor subarea: components include subarea surface runoff, stream channels, andreservoirs. Each component is represented by a set of paramete$ that specifiesthe particular characteristics of the component and the mathematical relationsdescribing its physical processes. The end result of the modeling process is thecomputation of direct runoff hydrographs for various subareas and sbeamflowhydrographs at desired locations in the watershed.

A subarea land surface runoff componenr is used to represent the movementof water over the land surface and into steam channels. The input to this com-ponent is a rainfall hyetograph. Excess minfall is computed by subtracting infil-tation and detention losses, based on an infiltration function that may be chosenfrom several options, including the SCS curve number loss rate as prcsented inSec.5.5. Rainfall and infiltration are assumed to be uniformly disnibuted overthe subbasin. The resulting rainfall excesses are then applied to the unit hydro-graph to derive the subarea oudet runoff hydrograph. Unit hydrogmph oPtionsinclude the Snyder's unit hydrograph and the SCS dimeniionless unit hy&ogaPhpresented in Chap. 7. Altematively, a kinematic wave model can be used to findsubbasin runoff hydrographs.

A stream routing component is used to represent flood wave movementin a channel. The input to this comporent is an upstream hydrograph resultingfrom individual or combined contributions of subarea runoff, streamflow

50E qrpr rErJ Hvonoroc., (

routings, or diversions. This hydrograph is routed to a downstream point, usingthe characteristics of the channel. The techniques available to route the runoffhydrograph include the Muskingum method, level-pool routing, and the kinematicwave method.

A suitable combination of subarea runoff and streamilow routing compo-nents can be used to represent a rainfall-runoff and stream routing problem. Theconrectivity of the stream network components is implied by the order in whichthe input data components are arranged. Simulation must always begin at theuppermost subarpa in a brarch of the strcam network, and proceed downstreamuntil a confluEnce is reached- Before simulating below the confluence, all flowsabove it must be rcuted. The flows are combined at the confluence, and thecombined flow is routed downsteam.

Use of a resemoir component is similar to that of a strcamflow routingcomponent. A reservoir component represents the storage-outflow characteristicsof a reservoir or flood-retarding structure. The resewot component functions byreceiving upsteam inflows ald routing them though a reservoir using storagerouting methods. The reservoir outflow is solely a function of storage (or watersurface elevation) in the reservot and is not dependent on downstream contols.Spillway chamcteristics are entered along with top-of-dam chamcteristics forovertopping. A simplified dam-break option is also available.

Example 15.2.1. (Adapted from Ford, 1986.) A rainfall-runoff model using theHEC- I computer program is to be developed for the Castro Valley Creek catchment,shown in Fig. 15.2. I in order to analyze the effects of urbanization. The catchment

FICURE 15.2.1The Castro Valley watershed (Example 15.2.1).(Souce: Ford, 1986.)

or_srcx nows 509

is divided into four subcatchments; a schematic diagram of the watershed is shownitt Fig. 15,2.2. Subcatchment 4 is uDdergoing urbanization through developmenr ofa new rcsidential arca, and a derention reservoir in subcatchment 4 and downsteamcbannel rnodifications are being investigated, the purpose of which is to reduce theeffects of the additional flow resulting ftom the development. The objective of theproblem is to calculate the ruroff hyfuogaph at the catchment outlet for thee d.if_fereft conditons: (l) the existing condition throughout the catchment, (2) the exisungconditioD in subcarchments I to 3 with subcatchment 4 urbanized, aad (3) the sameaE (2) but with a modified channel and a reservoir in subcatchmenl4. $ubarea runoffcomputations are performed using Snyder's synthetic unit hydrcgratsr with rainfallloss rates detemined using the SCS cufte number method, channel routine is cardedout by the Muskingum method, and rcuting lhrough lhe reservoir by rhe level-poolmethod.

The following Table presents the existing characteristics of the subcatch_ments. The total watershed area is 5.51 mir.

Subcrtlhm€nt Wstershed Iangth to SCS curvelength cenhoid numb€rL k ^ C N(ml) (mi)(mi2)

I234

t , 522 . 1 70.960.86

2.651 .85l � 1 3t.49

1.400.680.60o. '79

'70

8480'70

The pa.rameteN for Snyder's synthetic unit hyd.ogaph fo. the existing condition are9p

= 0.25 and C, = 0.38. The flood wave travel time (Muskingum coefficienr,efor the stleam reach passing through subarea 3 is estimated as 0.3 h, and the traveltime for subarea I is e$timated as 0,6 h. The Muskingum X haE been approximatedas 0.2 for each of the two stream rcachcs.

_ - The desi8n rainfall is a hyporherical 100-year-rerum-peliod srorm defined byt}le following depth-duratioD data,

Components

f--_l subareai sunace

ll Reservoir

->.

oFICURE 15.2.2Schematic diagram of Castro Valley watershed showitrg components of fmc-l analysis.

Channelrouung

Analysis poinrand hydrograph

OUT

510 arpruo rivonorocv

Duration

Rainfall (in) 0.38 0.'74

t h

1 . 3 0

2 h 3 h 6 h 1 2 h 2 4 h

1.70 2.10 3.00 5.00 7.00

A residential development in subcalchment 4 will increase the impervious area so

that the developed SCS curve number will be 85 The unit hydrcgraph patamete$

are expected to change to C, = 0 19 and Cp : 0 5 Modification of the channel

through subcatchment I will change its Muskingum routing parameters to tr : 0 4 h

and i = 0.3. The detention rcsefloir to be constucted at the outlet of subcatchment

4 has the follo*ing characteristics:

Lo* level oudet

Dramelef

Cross-sectional area t9.63 rt2

Orifice coefficient 0 71

Ceoterline eievation 391 fl(above MSL)

Overflow spillway (ogee tYPe)

Lenglh 30 ft

Weir co€fficient 2.86

Crest elevation 401.8 ft(above MSL)

Reservoircapacity Elevation(acre'ft) (ft above MSL)

06

l 2182330

388 .5394.2398.2400.840 t . 8405.8

Solution. The parameters used for Snyder's unit hydrograph in HEC-I are tp and

Ce; rl, is calculated for the existing condition using Eq. (?.7.2) with C I : 1.0' aad

C,. L. and Ia| as given above. For example, for subcatchment l,

tp : C,(Lk)a3 = 0 38(2 65 x 1 40)03 = ! J( 5

The results of this calculation for the four subcatchments arel

c, 0.38 0.38re 0.56 0.41

0.38 0.3E0.34 0.40

t 2 3 4 4urbanized

0 .190.20

The HEC-I input for the Castro Valley Creek catchment is shown ln

Table 15.2.1. The data file has been annotated iD the figure so that it can be under-

stood better. Use of HEC-I'S multiplan option enables the runoff hydrographs for

all three conditions to be calculated in one computer run. Plan I is for existing

conditions, plan 2 has subcatchment 4 urbanized, and plan 3 inhoduces the reser-

voir and channel modifications.Each component oPention begins with a KK card. The input has been set

up so that the runoff from subcatchment 4 is determined first, then rcuting througb

the proposed detention reseryoir js performed, followed by the Muskingum rcut-

ing through subcatchment l. Next, the rainfall-runoff computation is performed for

al

?.et

t

I

< :. i e

I !.2(J

,_ ff

?s 9E €

' ; 6 :

b E E€ E . =

E,]

(,

)

d i t r - " rE X

'6g

O F

^ . ; _F g 3

€ . I E : EF A 1 \ Po v : : u x ; ie - ! ! a i . !E E - � s F t :

ua= lF$: ;E : E ; = = ^ FI i i e q u -

F F . ; n g g Eg e . ; e 6 6 4 t

.:

2 -

9 o

D

z

1 6 € r

U

3 6 - g o e . . ' o R - o o2 < c j d d d

t

. q a3 o

9 E ;3 ? .8F t s +6 + E- e ; *E s gg E g

z

ztr2 2o z .o F aT i Z E Tll. IJI

: E PrEEe :E ' 'E t sf r 7 = 'C/ Y !-- ut u)K F I - ^ A A. ) > ; 7EEE P , A E b Ei J J - ; = Z Z

t t 7 F 2 x z> . r , < ^ ' u e ef , U U o H H

4 ; E F e ? ,E F : - - i l g Si A Z > l l l l l lf 9 H z z z

J J J^ ^ A . - 0 r O i o . Y i E A 3 3 Q 3 3 g 3 3 V i l e l E

\o F co o, o = s I : ! I 5 : g R _ & R

!

If

e

. E ;

E .--

O <

FzlI1

- 8' r * " r

4 ! , 0

; Z E< o9 k pl r -d - &

9 g ' !r t u t az l \ z

r - = . . r o * \ o o r ; = d

a A 2 4 0 0 a A6

- A U

J

pop

oz;)C N

Fz

UI

F ! 4 *

v > E v > o v > < h ( , v E u Ny v : c y v r 0 J f ! Z Y E N

f l f l ; 3 3 S f r 9 8 6 S €E

O

ai

l l

i =

l l t

I

E

zlll

9(_l

p

z

D

U

a : )

] ^

z

Fz'!

EU

U

F

Z

z

I

Uf

B

x >U

F Fl ;

3 6

E !

) &. 4 N 1 9

2 4lfl Y

n ' o + Fk - - e ot ) <r o o

k t ! - - - , - t . r o \ o5 F C o r r h7 = - Z ; d

.t

r 6X E

Z t z Q - i F : P -t 3 ; . e E : i n -

[ _ _ € e i F = z d - g: e l q - i P : ^ g F E - E* - . : o P : : : r o E t 1i : b d ! : : ^ d I ?E 9 : . = : i 3 e E ! E= . . : e _ 1 : - G ^ r " I E

C i E € E ! ' E ; g > E- 3 ; ! E a . E ! i i .t - d - > , 9 d _ 2 9 L a a Z

x ' l € g

r > a > 0 . > x > < o ! / )l \ Z \ Z e , Y o < l v E o - :

f l \ o F @ a , o

5 Z > U X > < o q l

U

t

* :t l ei l

: a

514 epplno nr:onolocr

subcatchment I and the resulting runoff hydrograph added to the runoff hydrographfrom subcatchment 4. Next, mi all-runoff comPutations are performed for sub-catchment 2, and this runoff is routed through subcatchnent 3 and added to theoutlet hydrograph. The final step is to perform the rainfall-runoff comPutations forsubcatchment 3 and to add this result to the outlet hy&ogaph

The resulting runoff hydrcgraphs at the oudet of subcatchment 4 and at theoutlet of the entire catchment for each of the three plans arc shown in Fig. 15.2.3The peak discharge f.om subcatchment 4 under existing conditions is 271 cfs, andunder urbanized conditions, 909 cfs. The detention resenoir reduces the peak dis-charge to 482 Cfs. The peak watet lurface elevation in the resenoir is 402 88 ftabove mean sta level (MSL) at time 12.67 h. The peak discharges at the outlet are1906 cfs fo. existing conditions, 2258 cfs fo! urbanized conditions' and 2105 cfsfor urbanized conditions with the reservoir and channel modifications.

Hydrograph at RES4

l

I

l

1000

800

600

400

200

0

E

a

0 2 , 1 6 8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4

Time (hrs)

Hydrograph at ortlct

8 t0 12 14Time (hrs)

T,IGURE 15.2.3Discharge bydrographs atRES4 and at outlet (Example15.2.1). Plan I is forexisting conditions, Plan 2 hassubcatchment 4 urbaniz€d, alldPIan 3 introduces a rcservoirand channel modrficationsdownsfeam of subcatchmentt6 t8 20 22 24

ucsrcr prows 515

Urban Storm Drainage Models

The first computerized models of urban storm drainage were developed duringthe late 1960s, and since that time a multitude of models have been discussedin the literature. The models applicable to design of storm sewer systems can beclassified as design models, fow prediction models, ar,d planning models.

Design models. These models determine the sizes and other geonpdc dimen-sions of storm sewen (and of other facilities) for a new system or an Lxfnsion orimprovement to an existing system. The design computations are usuafy carriedout for a specified design retum period.

Design models may be classified further into hydraulic design modcls and.least-cost opt rnl design models. Hydraulic design models range from the simplerational method to much more sophisticated flow simulation models based uponsolving the dynamic wave equations. One example of a hydraulic design modelis ILLUDAS (Illinois U$an Drainage Area Simulator), developed by Tenniepand Stall (1974), which is popular both in the United Stares and abroad. Thismodel is an extension of the British TRRL (Tiansportation and Road ResearchLaboratory) model (Watkins, 1962) to include both paved-area and grassed-areahydrcgraphs. A flow chart for the ILLUDAS program is given in Fig. 15.2.4.

l-east-cost optimal design models are intended for determining the lowest-cost storm sewer layout and pipe diamebrs that will convey storm drainageadequately. These models are based on optimization techniques such as linear pro-gramming, dynamic programming, nonlinear programming, heuristic techniques,or a combination of these. The flow simulation for the se*er network is con-sidered a part of the optimization. One of the more comprehensive models ofthis type is a dynamic programrning model called ILSD (Illinois Sewer Design)developed by Yen, et al. (1976).

Flow prediction models. These rnodels simulate the flow of storm water inexisting systems of known geometric sizes or in proposed systems with prede-termined geometdc sizes. Most flow prediction models simulate the flow for asingle minfall event, but some can simulate the response to a sequence of eyents.The simulation might be for historical, real-time, or synthetically-generated stormevents. At least some simple hydraulics is considered in most models. A modelmay or may not include water quality simulation. The purpose of a flow simula-tion may be to check the adequacy and performance of an existing or proposedsystem for flood mitigation and water pollution control, to provide informationfor storm water management, or to form part of a real-time operational controls) stem.

An emerging design philosophy is to use either traditional (rational method)or more advanced optimization methods for designing a storm sewer system, thenchecking the frnal design by detailed hydraulic simulation and cost analysis. Anexample of this approach is a British design and analysis procedure called theWallingford Storm Sewer Package (WASSP; see Price, l98l).

516 sPPr-teo urororocY

R€ad basic data

and design storm

Combine olher hydrognpnstributary io lhis Pomr

FIGLTRE 15'2 4- n r rmAs nrhan siorm drainage model (sourc€i Terstriep and stall' 1974)

:,

( DEslcxnows 517

Phnning mod€ls. These models are used for broader planning studies of urbal

il;;;?", problems, usually for a relatively large space frame and over a

reiutiuety ton g P.'i"d-"1,'lT:: T: m*i:,*"S,1;y"::1ffi.ff:T ;'il,11in a gross manner. consroenDg onIY

withoutconsidennglneoynamlcsoftheirmotiontbloughthesystem.PlanninglrJiiir t" ".pt"ied foi such tu'L' "s 'ntai"t of receiving water qlalitv and

[i"i#r,t iu. iii"t. Thgv !o 1ot requ'ue detaited geometric information on tie

&ainase facilities as do me rrsl ;;il;; of irodels TyPical, examples of

lir"ii* r.a.r, .*: (t I sronu (stoiaft' tttu*"nt' overfloq RlpoffModel)'

#;;ii; the u s A,.v corps of"Engineers ( t9'76)l (2) 'slrMM (storm

fr"i"r-rr.l^it"e"."* vtodel),-oevelopeJiv lrietcalf and Eddv' Inc''the Universitv

;i1#;, ;J wut", n".oor""r'Engi,ieers, In" , (Mgtcall Td Eddv' 1971;

ii. 3. Joui-".*"r protecfion ee*ci, i977); t3) RUNQUALTRunoff Quality),

;;i;;ild"t the hvdraulic poiloo'ot the swMM RUNoFF- model and the

,"i"*'*^"t q*rrty'.oo"r Quei-n in'*tn"t' Giguere' and Davis' 1977);

iiiiiiprliiyo."."mp Simulaiion hosmm-Fortran) developed-b-v Johnson' et

^1 /loRo\ which is a fu,", u"r.ion^oF tft" Stanford Watenhld Model: and (5)

iiiiEei'irtiri "",.i"t."i *oo"ij iv ittev, Ferkins' and Eagleson (1e70)'

15.3 FLOOD PLAIN ANALYSIS

A flood plain is the normally dry land area adjoining rivers' sreams' lakes'

i,""r. rt J""*t that is inundated iuring flood events The most common causes

:iffi;;; th" on"rflo* "r t'i"^it "'a rivers and "ul"y4il high tides

Iro'r",i""-g iio* ,"""t 'tot-t rrt" noJplain can include the full width of narrow

s[eam va]leys, or broao arcas u,ong 'ui"tt in wide' flat valleys As shown tn

Fie. 15.3.1. the channel -d n"Jpl;i"; borh integral parts oJ the natural

:;il;"*" oi'u,o.u' nre noli ;i;;; ;;"' n": i" :i::':,oi.'he channer

.rt".iv ;u the greater rrtt aitti"tii'ilt r"rther the extent of flow over the

flood Plain."""- ifr"

- n", step in any flood plain analysis is to collect. data' including

.p"gr"pii"'-"pr, flood ilow -a""'ii " g"giitg station is n€€rby' rainfall data

if flood flow data arc not u"u"uUf",- -f, t'"t"iyed cross sections and channel

,oueho"t, estimates at a number of points along the stream''""t'"e, A"*r-inu,ion "i tft" n."i'iit.;;;"

"f"t the desired return period is

required. If gaged flow ,"co'd'-il ;"uil;;", " flood flow frequency analysis

.-*U. p.tf"i"Za. If gaged data are oor auailaute' then a minfall-runoff analysts

,n"uri # p.rtor*.0 ,o o"t.*tt tftt nood di'cttatge' The rainfall,hYetograPh ts

determined for the d"rirrO ,"to. piioO, a synthetiJ unit hydrograph is developed

for each subarea of oe otainage'ias-i; ant t}le direcr runoff hydrogaph from

each subarea is calculated. m" ttt'*"-olt*t runoff hydrographs are routed

O"*"rr"". -O ^OOed to determine the total direct runoff hy&ograPh at the mosr

downsheam part of the u,*ug"'ilu'1", "" *ai itiuttratea ti:i111* 15 2'l for

castro valley. The peat oiscrrariJ "i'rrr" -"" downstream hydrograph is used

as the design flood discharge'

518 rpplreo ur onoroor

Once the flood discharge for the desired retum pe od has been determined,

the nc

step is to determine the profile of water surface elevation along the

channeL. This analysis can be canied out assuming steady, gradually-varied,

nonuniibrm flow using a one-dimensional model such as HEC-2 (U. S. Amy

Corps ol Engineers, 1982), or a two-dimensional model based upon either finite

differences or f inite elements (Lee and Bennett, 1981; Lee, et ai , 1982; Mays

and Tasr, 198,1). One-dimensional modeis allow the flow propefiies to vary along

the channel on1y, while two-dimensional models account for changes acrcss the

channei as well. Alternatively, an unsteady flow analysis can be performed to

identifl the maximum water surface elevation at various cross sections during the

propagation of the flood wave through a stream or river rcach, using DAMBRK,

DWOPER. or FLDWAV, as described in Chap. 10. Unsteady flow models are

necessary for flood plain delineation in large lakes because the storage in the lake

alters tbe shapc and pcak discharge of the flood hydrograph as it passes through.

FIGURE 15.3.1Typical seciions and profiles in an unobstructed reach of stream valley. (Sorrc€: Waananen, et al ,1977. Lsed b! permission.)

( otsrc;r r.rows 519

After the water surface elevations have been determined, the a.rea covered

by the flood plain is delineated. The lateral extent of the flood plain is determinedby finding ground points on both sides of the stream that conespond to the

flood profile (water surface) elevations. Ground eleYations in the flood plain

can be determined from topographic maps, street maPs, or stereo aerial photos.

Topographic maps are the most convenient, with the eleYations given by contourlines. The flood plain boundary is determined by following the contour line that

corresponds to the flood profile elevation for a panicular areq. Qf course, theflood plain delineation is only as accurate as rhe topograPhic *ra$S used. After

flood levels have been determined for a particular reach of str&m the actuallocation of the flood plain boundaries should be checked by field surveys.

In order to provide a standard national procedure, the 100-year flood has

been adopted by the U. S. Federal Emergency Management Agency (FEMA)

as the base flood for purposes of flood plain management measures. The 500-year flood is also employed to indicate additional areas of flood risk in the

community. For each skeam studied in detail, the boundaries of the l0O- and

500-year floods are normally delineated using the flood elevations determined at

each cross section, Between cross sections, the boundaries are interpolated using

topogmphic maps at a scale of 1:24,000 with a contour interval of l0 feet or 20

feet. In cases where the 100- and 500-year flood boundaries are close together,

only the 100-year boundary is shown.Encroachment on flood plains, such as by artificial fill material, reduces

the flood-carrying capacity, increases the flood heights of streams, and increases

flood hazards in areas beyond the encroachment. One asPect of flood plain

management involves balancing the economic gain from flood plain developmentagainst the resulting increase in flood hazard. For purPoses of FEMA studies, the

100-year flood area is divided into a foodway and a foodway fringe, as shown

in Fig. 15.3.2. The floodway is the channel of a stream plus any adjacent flood

(

FIGURE 15.3.2Definition of floodway and floodway fringe. The floodway fnnge is the area between the designated

floodway limit and the limit of ihe selecred flood. The floodway limit is defined so that encrcachment

iimited to the floodway fringe will not significantly iDcrease flood eleva[ion. The 100-year flood is

cornmonly used and a one-foot allowable increase is standard in the Unit€d StaEs.

/

water surfrce of selected flood wiihencroachrnenr in fringe areas

Natural water sudace of selected flood

520 qpprtro nr onorocr

plain areas that must be kept free of encroachment in order for the 100-year floodto be carried without substantial increases in flood heights. FEMA's minimumstandards allow an increase in flood height of 1.0 foot, provided that hazardousvelociries are not prcduced. The floodway fringe is the poftion of the flood plainthat could be completely obstructed without increasing the water surface elevationof the 100-year flood by more than 1.0 foot at any point.

FIGURE 15.3,3Flood hazard map for Napa, California. (.torrce: Waananen, et al., 197? Used with permission.)

ossror nows 521

Two types of flood plain inundation maps, flood-prone area and flood hazardmaps, have been used. Flood-prone area maps show areas likely to be tloodedby virtue of their proximity to a river, stream, bay, ocean, or other watercouneas determined from readily available information. Flood hazard maps such asFig. 15.3.3 for Napa, Califomia, show the extent of inundation as determinedfrom a thorough techllical study of flooding at a given location. Flood hazardmaps are commonly used in flood plain information reports and require updatingwhen changes have occurred in the channels, on the flood plainsq aip in upstreamareas. These changes include structural modifications and chandel $r flood plainmodifications in upstream areas. Development of new buildingd on the floodplain, obstructions, or other land use changes can affect the strcam discharges,water surface elevation, and flow velocities, thereby changing the elevationprofile defining the flood plain.

15.4 FLOOD CONTROL RESERVOIR DESIGN

Urbanization increases both the volume and the velocity of runoff, and effonshave been made in urban areas to offset these effects. Storm water detention basinsprovide one means of managing storm water. A storm watgr detention basin canrange from as simple a structure as the backwater effect behind a highway or roadculven, up to a large reservoir with sophisticated control devices.

Detention is the holding of runoff for a shon period of time before reieasingit to the natural water course. The terms "detention" and "retention" are oftenmisusedl retention is the holding of water in a storage facility for a considerablelength of time, for aesthetic, agricultural, consumptive, or other uses. The watermight never be discharged to a natural waterrcourse. but instead be consumed byplants, evaporation, or infiltmtion into the ground. Detention facilities generallydo not significantly reduce the total vohme of surface runoff, but simply reducepeak flow rates by redistributing the flow hydrograph. However, there are excep-tions: for example, the reduced surface runoff volume from land areas that havebeen contour-plowed, and the reduced surface runoff from detention basins ongranular soils.

On-site detention of storrn water is storage ofrunoff on or near the site whereprecipitation occurs. In some applications, the runoff may first be conducted shoftdistances by collector sewers located on or adjacent to the site of the detentionfacility. On-site detention is distinguished from downstream detention by itsproximity to the upper end of a basin and its use of small detention facilities asopposed to the larger dams normally associated with downstream detention.

The concept of detaining runoff and releasing it at a regulated rate is animportant p nciple in storm water management. In areas having appreciabletopogaphic relief, detention storage attenuates peak flow rates and the highkinetic energy of surface runoff. Such flow attenuation can rcduce soil erosion andthe amounts of contaminants of various kinds that are assimilated and transportedby urban runoff from land. pavements, and other surfaces. Several different

522 ,urr,ro sronorocr

methods exist for the detention of storm water, including underground storage,storage in basins and ponds, parking lot storage, and rooftop detention.

Tl]ere are seveml considerations involved in the design of storm waterdetentior facilities. These are: (1) the selection of a design rajnfall event, (2) thevolume ol storage needed, (3) the maximum permitted release rate, (4) pollutioncontrol requirements and opportunities, and (5) design of the outlet works forreleasing the detained water. Flow simulation models such as the HEC-1 modelcan be used to perform reservoir routing to check the adequacy of detention basindes igns .

The hypothqtical ponded area in Fig. 15.4.1 serves as an example ofadetention pond. Figure 15.4.2 provides a comparison of the outflow hydrographsfrom this detention pond with the corresponding inflow hydrographs for variousflow volumes. In all cases, the detention pond reduces the flood peak discharge,but less so when the volume of runoff is large than when it is small.

Modified Rational Method

The modified rational method is an extension of the rational method for rainfallslasting longer than the time of concentration. This method was developed so thatthe concepts of the rational method could be used to develop hydrographs forstorage design. rather than just flood peak discharges for storm sewer design.The modified rarional method can be used for the preliminary design of detentionstorage fbr watersheds of up to 20 or 30 acres.

The shape of the hydrograph produced by the modified mtional methodis a trapczoid, constructed by setting the duration of the rising and recession

oesrcx nows 523

I000 cfsOuiflow peak =

401 cfsInflow

29.04

' !6Cto

e

0 20 40 60 80Time (min)

lnflow peak = 1000 cfsOulflow peak = 538 clslnflow volLrme = I01.26 acre . fi

- Inflow hldrograph--- Outflow hydrograph

FIGURE I5.4,7Schematjc representation of wedge-shaped ponding a.ea with box culvert ourlel. (So,/rre: Craig andRankl, 1978. Used wirh permission.)

0 m 40 60 80 r00 r20 140 160 r80 200 220 240

Time lmin)

FIGTJRE I5.4.2Comparison of inflow and oudlow hydrographs for a detention basin. The inflow peaks are all 1000cfs; however, the inflow volurnes vary. The ponding area is a hyPothetical wedge-shaPed sbragearea (Fig. 15.4.l), and a 4 ft x 4 ft box culven serves as the outlet The pond widlh is 60 ft witha slope of 0.02 ft/ft. The flow with $e largest voiume results in the higlest oudlow €te from thepond. (Sorrce: Craig and Rar*|, 1978. Used with permission.)

^ 600e

E o *

he igh llengihwidth

lnflow peak = 1000 cfsoutnow peak = 471 cfslnf low volume = 54.21 acre. f t

524 eppr-r:o gvtnoloov (

limbs equal to the time of concentration I., and computing the peak dischargeassuming rarious rainfall dumtions. Figure 15.4.3 illustrates modified-rational-method hydrognphs developed for a drainage basin that has a lO-mirute timeof concentration and is subject to rainfall of various durations longer than l0minutes. For example, consider the tallest trapezoid in the figure. Its rainfallduration is Ta = 20 min, and the conesponding rainfall intensity i is used in therational fbrmula (15.1.1) to compute the peak discharge. The hydrograph riseslinearly to this discharge at the time of concentration (10 minutes), is constantuntil the rainfail ceaEes (20 minutes), then recedes linearly to zero discharge at 30minutes. The hydqgraphs for longer rainfall durations have lower peak dischargesbecause their rainiall intensities are lower.

If an allowable discharge out of a proposed detention basin is known, suchas from a requirement that the peak discharge from the detention pond not begreater than the peak discharge from the area under predeveloped conditions, thenthe required deteotion storage for each rainfall duration can be approxinated bydetermining the area of the fiapezoidal hydrograph above the allowable discharge.By calculating the storage for hydrogmphs of rainfalls of various durations,the hydrologist can determine the critical duration for the design storm as theone requiring the greatest detention storage. This critical duration can also bedetermined analytically.

( oPstcr nows 525

Figure 15.4.4 is a representation of inflow and outflow hydrographs for a

a"t"ntio"n Gsin design ln thi' figo'"' a is the ratio of the peak discharge before

;;i;;;; O^ (Jr peak discharge from the detention basin)' and the peak

dischaige after develoPment, 0p:

Qt' Q p ( r s .4 . 1 )

T h e r a r j o o f t h e t i m e s t o p e a k i n t h e t w o h y d r o g r a p h s i s T . . V ' i s . t h e , v o l u m e o filiffi;;;velopment. The volume of tto*et % needed in th! brsin is the

accumulated volume of inflow mrnus outflow diring ttre period whefhe inflow

iut" "*.""0, the outflow mte, shown shaded in the figure''"- -u;il;" g;;metry of the -trapezoidal

hydrograPhs' the. ratio of the volume

of storage to the volume ot runor, rz"/%' cante ditermined (Donahue' McCuen'

and Bondelid, 1981):

(15 .4.2)

where T2 is the dumdon of the precipitation and Io is the time to Peat of the

inflow hy&ograPb.

# = '- "l' . ?(' - -")]

a ' ?

FIGURE I5.4.3Typical storm water runoffhydrographs for the mod;6edrational method with variousminfall durations.

FIGURE 15.4.4l n f l o w a n d o u t f l o w h y d r o g r a p h s f o r d e t e n t i o n d e s i g n ' T h e o u t n o w h y d r o g r a p h i s ' . b a s e d o D l h einflo\^ h\doqraph for predeveloped condrtroni o' on'ortt"' tott tttt|.ittive oudloq cnreria r5orr'e:

o"""rt"J, uioi*. and Bondelid l98l Used wrth Perm;srion r

f , = 10 minules

Time (min)

526 , r i ' r l r pouvo to roc r

wbere i is rainfall intensity and a and , ale coefficients. The voiume of runoffafter development is equal to the volume under the inflow hydrograph:

, a

T , / + b( 1 5 . 4 . 3 )

V, : QpTa ( r s .4 .4 )

The volume of storage is determined by substituting (15.4.4) into (15.4.2), andrearranlrnq to get

, . ._ s l l l, , -eorolr -a,Lr | f r l r - 3, ,

r a e r - e a r a - e J p - * * - + ; :

(

Solution. The critical duration is found from Eq. (15.4.'l):

- i ( 1 3 . s x 0 . 8 2 5 1 2 5 . 0 r r 0 6 . 6 1 1 r r

- | - ^ t l 8 r 2 r 2 o r I' 'o zto. rx;Gsxsb. or

:2' / .23 min

Example 15.4.2. Determine the maximum detention storagedescribed in Example 15.4.I i f f : 40/ZO : 2.

Solution. 'lhe peak discha.ge for the duration of 27.23 min is

t " \Op= cAl;:- l' \ t d + o l

=(0.825x25)(r;T1ae)

:48.44 cfsBy Eq. (15.4.6), then,

v . : (2 i .23) \48 .44) - ( r8 ) (2 i .23) - ( r8 ) (20) + r rs t t :o i { l )+ ( l8 l ] i20) I

2 ) 18.44= 895.77 cfs min x 60 s/min

=53.746. l t3

As a compadson, from (15.4,4), V, = QpTa : 48.44 /. 27.23 = 1319 cfs ' min :'79,140 ft ' , so V,IV,= 53,'746/19, 140:0.68. Hence the detention pond wil l store68% of jts inflow hydrograph in this example.

15.5 FLOOD FORECASTING

Flood forecasting is an expanding area of application of hydrologic techniques.The goal is to obtain real-time precipitation and stream flow data thrcugh amic{owave, radio, or satellite communications network, insert the data into rainfall-runoff and stream flow routing programs, and forecast flood flow rates andwater levels for periods of from a few hours to a few days ahead, depending onlhe size of the watershed. Flood forecasts arc used to provide wamings for peopleto evacuate areas threatened by floods, and to help water management personneloperate flood control structures, such as gated spillways on rcservoirs. The datacollection systems used in flood forecasting are described in Chap. 6.

The components involved in a flood forecasting model for a large reservotrsystem can be illustrated by considering a model developed at the University ofTexas at Austin for the Highland Lakes reservoir system on the lower ColoradoRiver basin in central Texas (Fig, 15.5.1; see Unver, Mays, and Lansey, 1987).This system is characterized by integrated operation of several reservoirs for

DESI6\ FrOWS 5-Z /

f& t*e watershed

Consider a rainfall intensity-dumtion relationship of the form

( 15 .4 . s )

( 1s.4.6)

where a has been replaced by QolQp.The duration that results in the maximum detention is determined by sub-

stituting Op : CiA : CAal(Td + ,), then differentiating (15.4.6) with respect toId and settjng the derivative equal to zero:

dv , ^ _dQ, Q i r r l a t t ,Q , t l' - \ i = I ) - 1 t ,

d r a - ' "

d T " 2 l d f d )

-TaC Aa c Aa ^ Q'rTr

l l a ' b ) - 1 1 1 b 2 ( A a

bC Aa(Td + b)2

rt2 r

2C,^"

where it is assumed that Q6 To, and 7 are constants. Solving for 27,

The time to peak To is set equal to the time of concentration.

Example 15.4,1. Determine the cri t ical duration fd ( i .e., the one thal requiresrhe maximum detention storaSe) for a 25-acre watershed with a developed runoffcoefficient C = 0.825. The allowable discharge is the predevelopment dischargeof l8 cfs. The time of concentmtion for the developed conditions is 20 min, andfor undeveioped conditions is 40 min. The applicabie rainfall-intensity-durationrelat ionship is

96.6

- | bcAal d =

| "I a)z TI n v A ' P

\ * , 2CAA

t /2

- b ( 1 s . 4 . 7 )

Ta + 13.9

T528 ,qp l i r eo rvo lo r -oc r

FIGLRE 15.5 .1Lo\ref Coloiado Rilef basin. (Sor'..: Unver. \4ays, and Lansey. 1987.)

multiple objectives. The portion of the river basin controlled by the LowerColorado River Authority (LCRA) extends from the headwaters ofLake Buchanandownstream to the mouth of the Colorado River at the Gulf of Mexico. TheHighland Lakes sl,stem consists of seven reservoirs that are serially connected.

Urbirn de\elopment jn the flood plain of the HighJand Lakes has restrictedthe range within which the reservoirs may be operated during floods. As allexample, the original design of Lake Travis in the 1930s calied for a releaseof 90,000 cfs during severe flooding conditions. Subsequent construction in theflood plain dorvnstream of the lake has reduced the safe release (without flooding)to less than 30,000 cfs. Flood control operation of rhe Highland Lakes is alsocomplicated because only two of the lakes. Buchanan and Travis, can store anysignificant f lood volumes. The other lakes are held at constant level during normaloperation.

The ilood fbrecasting model for the Highland Lakes system can be used ina real-tinre framework to make decisions on reservoir operations during flooding.

EI

E ;9 6

e 3

. : R

I ; . -

J ' E g

O *

E >E E

i p .

! -

6 . 9 -

e - 3 ,

i ' l p . 6 Lq ; _ l- 9 - :

a 6 e ?

prcdicled conditions

o r :; : ;

Esl imatedr"noif ro lakes

EsLimated

Buchandn Dam (Lake Buchanan)lnk\ Dam (Lakc Inks)

S L J r c l c D , n r ( L , , f . L M J , h l c f r l l s l

Mrnsl ic ld Dam (Lrkc 1-ruvisrTonr M' l ler Darn (Lrkc Ars l iJr)

Lorghonr Darn {Town Lrkc)

3 0

* E 6

529

530 ,rppriL r, rr oror,,rcr

This model js an integrated computer prcgram with components for flood routing,rainfall-runotT modeling, and graphical display, and is controlled by interactivesoftware. Input to the model includes automated real-time precipitation and saeamflow data from various locations in the watershed, as was shown in Chap. 6.

The overall model structure is shown in Fig. 15.5.2. Real-time data are inputto the model hom the data collection network. The real-time flood control moduleincludes the following submodules: (1) a DWOPER submodule, that is, the U. S.National Weather Service Dynamic Wave Operational Model for unsteady flowrouting; (2) a GATES submodule, which determines gate operation informationfor DWOPER, such es the gate discharge as a function of the head on the gate; (3)a RAINFALL RUNOFF submodule which is an SCS-type rainfall-runoff modelfor the ongaged drainage area surounding the lakes for which stream flow datais not avajlable; (4) a DISPLAY submodule, which contains grapbical displaysoftware; and (5) an OPERATIONS submodule which is the user control softwarethat interactively operates the other submodules and data fi les.

The irput for this flood forecasting model includes both the rcal-time dataand the ph)'sical description of system compotents that remain unchanged duringa flood. The physical data include: (1) DWOPER data describing stream crosssection information, roughness relationships, and so on; (2) characteristics of thereservoir spillway structures for GAIES; and (3) dminage area description andhydrologic parameter estimates for RAINFALL-RUNOFF. The entire river-lakesystem contains 871 cross sections for DWOPER. It is divided into five subsys-tems because running the model for the whole system at once is usually notnecessary, sjnce floods tend to be localized within one or two of the subsystems.The real-time data include: (1) stream flow data at automated stations and head-water and tailwater elevations at each dam; (2) rainfall data at recording gages;(3) information as to which subsystem of lakes and reservoirs will be consideredin the routrni,. anrl r,1r reservoir operations.

15.6 DESIGN FOR WATER USE

The primarv variables to be determined in a water-supply reservoir design arc thelocation and height of the dam, the elevation and capacity of the spillway, andthe capacit] and mode of operation of the discharge works. Two hydrologicalvariables .ue paramount: the storage capacity in the reservoir and the frm yieldor mean annual rate of release of water through the dam that can be guaranteedfrom an analysis of historical data. There can be a great deal of uncertainty in thefinn yieid, especially for a short histo cal period. Natumlly, the storage capacityand firm yield arc interconnected because the larger the storage, the greater is thefirm yield, with the limit that the firm yield can never be greater than the meanannual inflow mte to the reservoir.

A reservoir may be a stngle-parpose structure, such as for water supplyor flood control, or it may be a muLtiple-purpose reservoir, in which zones ofstorage are identified for different purposes (Fig. 15.6.l).

\ Flood control slorage l,-+I

Spi l lstW" o 7

Dead

:-...-

orsrcr nows 531

Intlow 4 Evaporation e/

Spil ls 0/

Discharye fd,

i l

FIGL'RE 15.6.1Zones of storage in a multipur?ose reservoir,

Overview of Design Process

Hydrologic design of a reservoir for water use involves four steps:

1. Prcjection into the fuhrle of the water demand to be met by the reservof;

2. Determination of the location and elevation of the dam, and calculation of itssurface area-storage capacity curve for present and future conditions;

3. Computation of the firm yield of the reservoir for present and futurc conditions;

4. Comparison of the water demand and firm yield of the reservoir to determineits service lift, or peiod of years during which the reservoir will be adequateto meet the demands.

This process is illustrated by the following examples, which involve the designof a new water-supply reservoir for the city of Winters, Texas (Henningson,Durham, and Richardson, 1979). The reservoir considered here was completedand went into service in 1981.

Projection of Demard.

Example 15.6.1. Project the water supply needed for the city of Winters, Texas,from 1980 to 2030, given the historical population and water use data in the tablebelow.

Yesr 1910'Fopulation 1347

1950 19602676 3266

r9m 19301509 2423

r9402335

r9't0 19802E)'7 3061

So&rrroa. A least-squares linear regrcssion equation is fitted to the population data,taking the form

( r 5 .6 . r )where P(r) is the estimated population in year r, and the coefficients are ao :- 48, l'l O and a r = 26.02 year t. For example, for I 980, PG) = - 48, l'l 0 + 26.02 x1980 : 3350. Using (15.6.1), the population is projected into the tuture, yielding

532 ,lpplno uvonolocv

the follovring estimates: 1990 - 3610; 2000 - 3870i 2010 - 4l3O;2020 - 43901and 2030 - 4651. These projections are checked against the population projectionsof local and regional government entities, taking econorric and demographic factorsinto account, and are accepted as adequate.

The populatio[ projections arc then converted into water use projections. In1978, a use rate of 175 gal per capita per day (8pcd) was observed; this is projectedto increase to 200 gpcd in the year 2000 and to 225 gpcd in2030. Hence, for a 1980population of 3350, the water demand is I/: 3350 x 175 = 0.586 MGD (milliongallors per day); to this demand is added an additional amount to meet the needs ofthe rural population arourd the city of Winters. The lesulting total demand is 0.66MGD in 1980,growing to 1.36 MGD in 2030 Gig. 15.6.2). For reservot waterbalance computations, it is more convenient to express the demands in acre ff/year(l acre.ftyear = 8.92 x 10-4 MGD); the demands in 1980 and 2030 are prcjectedto be 740 ard 1520 acre.ft/year, rcspectively. The water use forecasting medloddescribed here is very simple. More comprehensive methods of accomplishing thistask are also available, such as the IWR-Main model (Dziegelewski, Boland, andBaumann, 1981) and statistical time series models (Maidment and hrzen, 1984).

Storage-area curve. Once the location and elevation of the dam are known,its storage-areo curve car' be detennined. Associated with each water surfaceelevation ftt in the reservoir is a surface area Aj and a storage volume S;. Therelationship between sj and Aj colstitutes the storage-area curve.

To determine the storage-area curve, the surface area Aj is determined bymeasurement on a topographic map of the area enclosed within the contour lineat eleyation i; (see columns 2 and 3 of Table 15.6.1). The horizontal slice ofstorage between elevations h; al:Ld h1+t has a.o average arca of (Aj + A1+)/2and a thickness of h1+r - hi, so the storage at the upper level j + I is (see Fig.15 .6 .3 ) :

(

TABLE 15.6.7Storage-srea comPutation for reservoir

oesrcx n-ows 533

Columo: | 2Ixveli Elevation rtl

(fr sbovc MSL)

3 4Surfsce ares Al Storage St(rcr€s) (sc.€'fl)

I2345618

l0l lt 2

t152r756t1fnt't64t't68117217761780t'7E417881'�7r'tot

01

l97'�l

146221262341430573643805

02

234680

142624D436105t52715883'l49822

' I

1 rd

Norer The lasl two incremcnts us€ an elevation differenc€ of ? ft instead of 4 ft b€caus'

elevation 1790 is the cresl of the spillway

(rs .6.2)

where Si is the storage at the lower level j of the slice For example' in Table

15.6.1, s3 = 2 + (17ffi - 1756X1 + 19)12: 42 zcrc'ft Reserv-otr storage can

also be comeuted bv the conic nethod for which (A; 'l A;+)12 in (15'6 2) is

replaced by i l,, + i i*, + ' la1e1t13) (u. S. Army Corps of Engineers' 1981)'

The storage-area curve so computed conesponds to the topographic condi-

tions when the reservoir is first conatructed. After the reservoir has been in use

for some years, its storage-area curve may be modified by- sedim€ntation in the

reservoir, which can reduce both the storage and the area for a given water sr[-

face elevation. Specific studies are needed at each site to determine the rate of

sedimentation and bow the deposited sediment is distributed within the reservoir'

eoe>9 r 0

-

c = ( . + ( h i . t - h i ) ( A i - A i - 1 )

e r i ) 2

I Service.- t i fe

-'-7- -' , >'-

F n n y i e t d / , - - _ - - ' f '

Projecled ure

,.'----V-,,.,,,/ Hisrorical use

1940

s , . ' = l .

1960

FIGTJRE 15.6.2Comparison of plojerted water use and finn yield of water supply for WiDlers, Texas.

flGIJRE 15.6.3Calculation of storage volume of a reservoir'

53.{ e+rro uronolocv {

Firm,yield. The firm yield of a reservoir is the mean annual withdrawal rate thatwould lower the reservoir to its lcrj ricar drousht or re"".a-n. ..r,ilI'l'il:ffJ [:xo.,i"f :?Jff T."J,H:,,,I;that contains sustained low rainfalrecords of rainfall, sream oo*.'l-To -:1t1t

flows and for which hydrological,it.. lh";;;;i; ;;;#;# 1"" evaporation exist at or near the reservoirusingronrhly time inrervats ,, , =tI,i:t:"]i'?:

of the reservoir water balanceThe monthly data on rcservoirinflow I and net evaporation e r (evaporahonminus precipitation on the rcservorr water surface) are obtained from measure_menrs at or near the reservoir site for a long period of ,""oJ, t oj"fuiiy in"fuaing

:f::' ' l l:u droughr. Recordsof warer use ai ir,. ,i,.,o t. iuppii"i.,.i,i. ,r"eu,ionared. erc. ) are anatyzed ro determine rhe rario d, ", *.;;;;;i;.;arcr dse tomean annuat warer use. The variabte.//, catred the dr;;;'i;;;,;',t r:pn"nnu ,r,

l^1"^1"::".:l ,h" annual firm yietd needed.in .on,r, ,. ri,"il r*inj"wrtr a fu|reseryolr, the reservoir water balance is calculated f"r*ard ;; ;;;.;",

S , = S r r + I t - y d t - A , e , - e , t = 1 , 2 , . . . , 7 ( t 5 . 6 , 3 )where S,. and S, are the storages at the heoinni"" ,^A ^^a ^. _ .the. surrace area and e, *" *,,i,'.trti,,1lil"ilt_T;:*r,m;o:th r, ,4, iswithdrawar rate rne iurrace ir;;;;"5;il;ii,"i?# il:"*j;r::i.": :ill:g i \en ,5 ,_ r and S, . The un i ts o f (15 .6 .3 ) a re acre . f t o r mJ.

, " . ,'J,';;iil,::?:.;!';"i "$':lii,j"'j:Tru fi:Tfu 1,",i;;lll,[T1\ rrtr 5, S,,. for aU orher monrhs i.-Normally. spiniay fiow f,i_ bl'ilu,arr;nepe ods of high inflow it may occur rhat S, as compured f.oii eq..tls.o.:) isgrealer than S-,; in this case, O, = .!, _ S* and "'";;;; #S)'= ,.,,,,useo ln the nert computational slep.

Pxamel: 15,6-'?.. _Compute rhe reser\oir warer balance for t,e Winrers EIrh Creeki::::]"1f".i

1940 .hydrorogicar condirjons as siven in Table rS.o.j, i i tn" *n,dra\at rate f is 1240 acre fvyear.

Sorlrbr,. The given duru to...",1?lll:1and ner evaporarion for 1940 are gjven::-"^",'.r:T l l*

5,-respecrively, of Table 1s.6.2. rr," .onir.,ri a""_un'J ra"t*, a,

:1.:y",_""1T#,1ii;,3;;,llll'*f *:f :,:;:lli:;,,lllttiil;r?*'5iffTii?l",i::rfJ,"#TniiT,ffi::'l'jT; r'"'' "'r'"1".*"#,".i*., ,"t ","4:i;.:liil;3'iili''fo'1*" is a ruri reservoir, so= 5--,=8374 acre.rt:5:=8196+191 -68-51 -o =t"u; j l ' ^ -o; !196 acre f r . fo l 'a 2 (Februatv) ,computed by rinear in"*","." o",i_i_""oill"rt ili i,lJl;Jl!.':ffiT*i[i;'i1il';'l;i::1it""'"':"""';,'l!::!1* Y?ul'i: azr'""i" i,, ""'o'iv ri""-,o, in practice, (15.;:; ;ui;:?,b9."o190.*"1

A' and s, are interdependent'adJusrmenls to .4? until a 6nu1 ,,or1t^10 lt",tl1u"ly,

for each month, making snrail

f.t;:ini:*:;il. ""*1f i iliT,il !'"1""",:'-'*T fi .":,:',';:11#t,ii1;1,i1

9 d

E Ea.l oq &

* 59 ; :Q r.r_ D !

T . :l r E

6 E

= F

. = +

S Er ! ^

d , r

, 3= F

. 9 !

. E ;u) It

\o

-l

9 p ?

: - 9

* . " o d E

a q )

. a-€=*E

* z a ,-9

F

, t = E s !

F "-3

E !- A E s

- 5 *-.9

t l E' o F - o , 9 : l €

F 8 € 3 3 S X K S ; h E x; ; a R e g g F g F i i el /

:=

6 - o o o o . 1 o

i r + ,, d

9 E S g X T X H S H : Hc i i d 3 3 d c o d c ; - j j

E E E 8 3 : ! t € 3 B s l c

c ; R H S f i X E S S € : 1 5

F € S 3 : 5 R ; 5 g F F i ?l d

' a . 8 * R - g g o s - l :- l d

€ € G G $ 3 G 6 c ; G 3

- -

& 4

9 :

.E 'g

- c P9 : 9

n j Y€ t r r ii a e

6 t

- i

a- OtS

$ € " t g

t -

E ! iz a ; =

S E s g

; :

E :x ; * -

;E

I N -

|.

g

:

a

€

,E

n

oesrcr rlows 537

Example 15.6.3. Determine the firm yield of the Winters Elm Creek Resenoirgive the bydrologic data in Table 15.6.3.

So/ztioz. The water balance simulation illustrated in the previous example for 1940is carried out sequentially for hydrologic data from the years 1940 to 1969, wbichcontain the critical drought of record for this region, in years 1950 to 1956. Thereare 30 years x 12 months/year : 360 montbs of simulation, The minimum storageS.i, is48 acre.ft for this reservoir, and is reached in ApriJ, l95l (r,:136). Thewater balance simulation for the first six months of l95l is shown in lable 15.6.3.It tums out that the v alte Y : l24O acre.ftlvea.r is the co(ect firm rlield. so mar rneminimum storage is reached just once during rhe 360-month perild6f record. Inpractice, repeated simulations are made using the whole hydrological record withvarious lrial values of y until the maximum withdrawal rate satisfying the requiredcondition on minimum storage is found.

Balance of supply and demand. The service lifeof the reservoir is rhat durationfor which the firm yield (or reservoir supply rate) is greater than the expecteddemand. For the Wintem Elm Creek example, the firm yield is Y : 1240acre.tt/year : 1.106 MGD. As shown by the plot on Fig. 15.6.2, the conditionof just-balanced supply and demand is expected to occur in apprcximately year2014, or 34 yean from the year of reservoi construction.

It may be noted that while the demands are projected forward in time (from1980 to 2030), the firm yield is calculated on the basis of past hydrologic data(1940 to 1969 in the examples). The underlying assumption of this analysis is thatpast hydrologic conditions are typical of patterns that could be repeated in anyfuture sequence of years. So, the firm yield can be thought of as a mean annualsupply rate that could be withdrawn constantly, year after year, even in the faceof future conditions equivalent to the cdtical drought of record. Of course, thefirm yield is not absolutely guaranteed because a future drought may occur whichis more severe than the critical drought of record. Studies of the thickness ofannual tree rings indicate that in some regions droughts have occured in pre\ iouscenturies more severc than those recorded in this centurv. for which rainfall aldflow data are available.

REFERENCES

American Society of Civil Engineers and Water Pollution Control Federation, Design and construc-lion of sanitary and storm sewers, ASCE Manual and Reports on Engineering Practice, na.37. New York. 1960.

Aron, G., and C. E. Egborge, A practical feasibility study of flood peat abatement in urban areas.report, U. S. Army Corps of Engineers, Sacramento Districl, Sacramento, Calif., Marcht973.

Brandslett€r, A., Assessmeni of mathematical models for uban stoam and combined sewer manage-nmelrt, Environmentol Prctection Technology Series, EPA-g.n/2-16-115a, Municipal Environ-mental Research Laboratory, USEPA, August 19?6.

Chow, V. T., and B. C. Yen, Urban storm water runoff-detennina(ion of volumes and flow rates,Envircnmental Protection Technology Series, EPA-6UJ/2-16-116, Muricipal EnvircnmentalResearch Laboratory, USEPA, May 1976; available from NTIS (PB 253 410), Spring-field, Va.

Craig, G. S., and J. G. Ranki, Analysis of runoff from small drainage basins in Wyoming, U. S.