Flow Model For Open-Channel 

Reach Or Network

U.S. GEOLOGICAL SURVEY PROFESSIONAL PAPER 1384
Flow Model For Open-Channel 

Reach Or Network

By RAYMOND W. SCHAFFRANEK

U.S. GEOLOGICAL SURVEY PROFESSIONAL PAPER 1384

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON: 1987
DEPARTMENT OF THE INTERIOR 

DONALD PAUL MODEL, Secretary

U.S. GEOLOGICAL SURVEY 

Dallas L. Peck, Director

Library of Congress Cataloging-in-Publication Data

Schaffranek, Raymond W.

Flow model for open-channel reach or network.

(U.S. Geological Survey professional paper ; 1384)

Bibliography: p.

Supt. of Docs, no.: I 19.16:1384

1. Channels (Hydraulic engineering)-Mathematical models. I. Title. II. Series.

TC175.S34 1985 627.1'0724 85-600018

For sale by the Books and Open-File Reports Section, U.S. Geological Survey, 

Federal Center, Box 25425, Denver, CO 80225
CONTENTS

Page

Abstract _____________________________________________ 1

Introduction ______________________________________________ 1

Terminology ______________________________________________ 1

One-dimensional unsteady-flow equations _____________________________ 2

Model formulation __________________________________________ 2

Finite-difference technique ___________________________________ 3

Coefficient matrix formulation _______________________________ 3

Equation transformation procedure ______________________________ 4

Boundary conditions ______________________________________ 4

Solution method _______________________________________ 5

Model applications __________________________________________ 5

Pheasant Branch near Middleton, Wis. ____________________________ 6

Potomac River near Washington, D.C. ____________________________ 6

Model use _____________________________________________ 10

Summary and conclusions ______________________________________ 10

Acknowledgments __________________________________________ 10

References ____________________________________________ 11

ILLUSTRATIONS

Page

FIGURE 1. Space-time grid system for finite-difference approximation ___________________________________ 3

2. Plots of measured water-surface elevations and computed discharges for Pheasant Branch near Middleton, Wis. _____ 7

3. Map of Potomac River near Washington, D.C. ____________________________________________ 8

4. Schematization of the tidal Potomac River system for the branch-network flow model ____________________ 9

5. Model-generated plot of computed versus measured discharges for the Potomac River at Indian Head, Md., on June 3-4,1981 9

6. Time-of-travel plot of injected particles for the Potomac River from midnight of November 30, 1980, to noon of

December 8, 1980 ___________________________________________________________ 10

TABLE

TABLE 1. Applications of the branch-network flow model ______

in
IV SYMBOLS

INTERNATIONAL SYSTEM OF UNITS (SI) AND 

INCH-POUND SYSTEM EQUIVALENTS

SI unit Inch-pound equivalent

Length

centimeter (cm) = 0.3937 inch (in)

meter (m)= 3.281 feet (ft)

kilometer (km) = 0.6214 mile (mi)

Area

centimeter2 (cm2 )= 0.1550 inch2 (in2)

meter2 (m2 )= 10.76 feet2 (ft2) 

kilometer2 (km2 )= 0.3861 mile2 (mi2)

Volume

centimeter3 (cm3)= 0.06102 inch3 (in3) 

meter3 (m3) = 35.31 feet3 (ft3)

= 8.107x 10'4 acre-foot (acre-ft)

Volume per unit time

meter3 per second (m3/s)=35.31 feet3 per second (ft3/s)

= 1.585x 104 gallons per minute (gal/min)

Moss per unit volume

kilogram per meter3 (kg/m3) = 0.06243 pound per foot3 (lb/ft3) 

gram per centimeter3 (g/cm3) = 6.243 x 10'* pound per foot3 (lb/ft3)

Temperature

degree Celsius (°C) = (degree Fahrenheit-32)/1.8 (°F)

SYMBOLS

Symbol Definition

A Area of conveyance part of cross section

B Total top width of cross section

Bc Top width of conveyance part of cross section

Cd Water-surface drag coefficient

dA A finite elemental area

/ A function

/(/) Functional representation of a dependent variable

g Gravitational acceleration

i Subscript index that denotes a function's spatial location

j Superscript index that denotes a function's temporal location

k Flow-resistance coefficient function

q Lateral flow per unit length of channel

Q Flow discharge

Qm Flow discharge of mth branch at a junction

R Hydraulic radius of cross section

S Vector of state

t Time

At Time increment

Atj Time increment of jth interval

u Flow velocity at a point

u' X-component of lateral flow velocity

u Transformation matrix

M(O Transformation matrix of ith segment

!* Transformation matrix of nth branch

U Mean velocity of flow

U Transformation matrix

U(n Transformation matrix of ith segment

Symbol Definition

U, Transformation matrix of nth branch

Ua Wind velocity

Wk Nodal flow at kth junction

x Distance along channel thalweg

Ax Distance increment

Ax, Distance increment of ith segment

Z Water-surface elevation

Zm Water-surface elevation of mth branch at a junction

a Angle between wind direction and x-axis

/3 Momentum coefficient

7 Flow-equation coefficient

5 Flow-equation coefficient

f Flow-equation coefficient

f Flow-equation coefficient

rj Flow-resistance coefficient similar to Manning's n

B Time weighting factor for spatial derivatives

X Flow-equation coefficient

H Flow-equation coefficient

£ Wind-resistance coefficient

g Water density

Q a Atmospheric density

ff Flow-equation coefficient

X Weighting factor for function values

\l/ Space weighting factor for temporal derivatives

co Flow-equation coefficient

	Superscript notation used to signify local constants
FLOW MODEL FOR OPEN-CHANNEL REACH OR NETWORK

By RAYMOND W. SCHAFFRANEK

ABSTRACT

Formulation of a one-dimensional model for simulating unsteady flow 

in a single open-channel reach or in a network of interconnected channels is presented. The model is both general and flexible in that it can 

be used to simulate a wide range of flow conditions for various channel 

configurations. It is based on a four-point (box), implicit, finite-difference 

approximation of the governing nonlinear flow equations with userdefinable weighting coefficients to permit varying the solution scheme 

from box-centered to fully forward. Unique transformation equations 

are formulated that permit correlation of the unknowns at the extremities of the channels, thereby reducing coefficient matrix size and 

execution time requirements. Discharges and water-surface elevations 

computed at intermediate locations within a channel are determined 

following solution of the transformation equations. The matrix of 

transformation and boundary-condition equations is solved by Gauss 

elimination using maximum pivot strategy. Two diverse applications 

of the model are presented to illustrate its broad utility.

INTRODUCTION

In the past, the utility of numerical-simulation modeling was often limited by imposition of certain simplifying assumptions that were both necessary and justifiable 

at the time-necessary because numerical methods and 

(or) computer capacity were deficient and justifiable 

because parametric evaluation techniques and (or) equipment were lacking or inadequate. Today, for the most 

part, advances in numerical methods, computer technology, and hydrologic instrumentation have enabled 

model engineers to reduce the number of such restrictions, thus producing models that are more nearly formulated on pure hydraulic considerations and have a greater 

potential to provide more comprehensive flow information. Consequently, the scope and complexity of hydrodynamic problems that are now tractable have expanded.

This expansion of the role of numerical-simulation 

modeling has stimulated the need for rapid, economical, 

and efficient techniques to compile and appraise prototype data and model results. Thus, it is insufficient for 

a numerical scheme to be developed merely to the state 

of being a model program. To achieve a state of usefulness as an operationally oriented investigative tool, the 

model program must be supported by a comprehensive

user-oriented data system and must provide a ready 

means of presenting output results in varied graphical 

forms.

In view of this need, the U.S. Geological Survey has 

developed a comprehensive, one-dimensional numericalsimulation model that is fully supported by a user-oriented 

system for modeling. The branch-network flow model, as 

it is called, is capable of simulating unsteady flow in a 

single open-channel reach or throughout a network of 

reaches composed of simply or multiply connected onedimensional flow channels governed lay various timedependent forcing functions and boundary conditions. 

Operational modeling capability is achieved by linking the 

model to a highly efficient storage-and-retrieval module 

that accesses a data base containing time series of boundary values and by including an extensive set of digital 

graphics routines. These features help transform the 

model into a comprehensive tool for practical use in the 

conduct of hydrologic investigations.

Two illustrative applications of the model are presented. 

Application to a 274-m reach of Pheasant Branch near 

Middleton, Wis., demonstrates its capability in computing 

unsteady flow in short, upland-river reaches that can be 

highly responsive to climatological conditions. Application 

to a 25-branch schematization of the 50-km tidal river part 

of the Potomac Estuary near Washington, B.C., illustrates its feasibility in simulating tidal flows in 

estuarine-type network environments that are frequently subject to extreme freshwater inflows and variable 

meteorological influences.

Graphical capabilities of the model are also identified 

and discussed. One particular form of output is presented 

to illustrate how the model can be used to track and 

display the movement of neutrally buoyant conservative 

substances through a riverine system and thereby evaluate its transport and flushing properties.

TERMINOLOGY

To facilitate further discussion of the application of the 

model to either a single riverine channel or a system of
FLOW MODEL FOR OPEN-CHANNEL REACH OR NETWORK

channels, a few definitions are necessary. The terms 

"reach" and "branch" are used somewhat interchangeably 

to mean a length of open channel. The primary subdivision of a reach or branch is referred to as a "subreach" 

or "segment." A "network" is defined as a system of open 

channels either simply connected in treelike fashion or 

multiply connected in a configuration that permits more 

than one flow path to exist between certain locations in 

the system.

ONE-DIMENSIONAL 

UNSTEADY-FLOW EQUATIONS

One-dimensional unsteady flow in open channels can be 

described by two partial-differential equations expressing mass and momentum conservation. These well-known 

equations, frequently referred to as the unsteady flow, 

shallow water, or St. Venant equations (Baltzer and Lai, 

1968; Dronkers, 1969; Strelkoff, 1969; Yen, 1973) can be 

written

(1)

and

3Q dt '

dx

dZ qk

-qu'-£BJJl cos a = 0, (2)

in which the momentum coefficient, /8, the flow-resistance 

function, k, and the wind-resistance coefficient, £, are 

defined as

k =

1.49 (or, in SI units, as fc = r?2),

and

(4)

(5)

In these equations, formulated using water-surface elevation, Z, and flow discharge, Q, as the dependent variables, 

distance along the channel thalweg, x, and elapsed time, 

t, are the independent variables. (Longitudinal distance, 

x, and flow discharge, Q, are positive in the downstream 

direction.) Other quantities in the preceding equations are 

defined as follows:

A, area of conveyance part of cross section;

B, total top width of cross section;

Bc , top width of conveyance part of cross section;

Cd , water-surface drag coefficient;

g, gravitational acceleration;

q, lateral inflow per unit length of channel

	(negative for outflow); 

R, hydraulic radius of cross section; 

u, flow velocity at a point; 

u', ^-component of lateral flow velocity; 

U, mean velocity of flow, =Q/A; 

Ua , wind velocity;

a, wind direction measured from positive x-axis;

r;, flow-resistance coefficient similar to Manning's

n;

Q, water density; and

Q a , atmospheric density.

Although hydraulic radius (R) is used in equation 2 and 

in subsequent expansions throughout this development, 

the commonly used substitution of hydraulic depth is 

employed in the model. This approximation (R»A/B) is 

assumed valid for shallow water bodies, that is, channels 

having a large width-to-depth ratio.

The momentum coefficient, /8, also called the Boussinesq 

coefficient, is present in the equation of motion to account 

for any nonuniform velocity distribution. (See eq. 3.)

Equations 1 and 2 are, in general, descriptive of 

unsteady flow in a channel of arbitrary geometric configuration having both conveyance and overflow (or only 

conveyance) areas and potentially subject to continuous 

lateral flow and (or) the shear-stress effects of wind. In 

their formulation, it is assumed that the water is 

homogeneous in density, that hydrostatic pressure 

prevails everywhere in the channel, that the channel bottom slope is mild and uniform, that the channel bed is 

fixed (i.e., no scouring or deposition occurs), that the reach 

geometry is sufficiently uniform to permit characterization in one dimension, and that frictional resistance is the 

same as for steady flow, thus permitting approximation 

by the Chezy or Manning equation.

MODEL FORMULATION

Numerous varied mathematical methods and corresponding numerical schemes exist that render approximate solutions of the flow equations. However, new 

methods and alternative schemes that provide more accurate approximations and are inherently more flexible 

and efficient are continually being sought. In the branchnetwork model formulation, the flow equations are expressed in finite-difference form using a weighted fourpoint (box) scheme. This technique, also used by Fread 

(1974) and by Gunge and others (1980), permits the model 

to be applied using unequal segment lengths and boxcentered to fully forward discretizations. A unique 

transformation operation is applied to the segment flow 

equations in the branch-network model, however, to lower
MODEL FORMULATION

the order of the coefficient matrices and thereby reduce 

computer time and storage requirements. A general 

matrix solution algorithm is used to simultaneously solve 

the resultant branch-transformation and boundarycondition equations. The implicit solution method is 

employed because of its inherent efficiency and superior 

stability properties. An optional iteration procedure, controllable by user-defined tolerance specifications, is additionally provided to permit improving the accuracy of the 

computed unknowns.

FINITE-DIFFERENCE TECHNIQUE

The space-time grid system shown in figure 1 depicts 

the region in which solution of the flow equations is 

sought. The symbols 0 and ^ represent weighting factors 

used to specify the time and location, respectively, within 

the Atj time increment and Ax, distance increment at 

which derivative and functional quantities are to be 

evaluated. The temporal and spatial derivatives of the 

functional value, /(/), that denotes the dependent 

variables stage (water-surface elevation) and 

discharge are discretized, respectively, as follows:

dt

and

(6)

t

1

V V

 ^~1 (

Atj

' ' 

 - g-1 t

^

°1

i > 

c ^ >i

> 

> -

f

a-^i

- i

Ax,

"

> ^^~

> ^ 

"

X] X, X, +1 X m

IE 1.- Space-time grid system for finite-difference approximation.

in which 

BAXi

and

and the equatic 

2 

in which

rfStL. 2A2Jj4'3e

.- 2^ .

P* ~ '

dx AXi AXi

Usually 6 is assigned in the range 0.5<0<1. A value 

of 0.5 yields the fully centered scheme used by Preissmann 

(1960) and by Amein and Fang (1970), whereas a value 

of 1.0 yields the fully forward scheme presented by 

Baltzer and Lai (1968).

In a manner similar to treatment of the spatial 

derivatives, the cross-sectional area, top width, hydraulic 

radius, and discharges in nonderivative form in the equation of motion, denoted/(/), are discretized as follows:

(8)

Thus, these functional values can be represented on the 

same time level as the spatial derivatives or at any other 

different level within the time increment. The weighting 

factor x may be assigned in the range 0<x^lCOEFFICIENT MATRIX FORMULATION

The partial-differential flow equations 1 and 2 are 

transformed into finite-difference expressions by application of the operators defined in equations 6-8 (Schaffranek and others, 1981). Using tilde (7) notation to 

signify quantities taken as local constants, updated 

through iteration in the computation process, the equation of continuity can be reduced to

-Qr1 -*, (9)

(10)
FLOW MODEL FOR OPEN-CHANNEL REACH OR NETWORK

and

e= (1-0)

-V- (1-0)

gA38

z

V-1 Ua cos a. gAe

Equations 9 and 10, which define the flow in the Ax, segment, can then be expressed in the following matrix form:

1 1 i

7 -

-1 (11)

EQUATION TRANSFORMATION PROCEDURE

Equation 11 can be applied to all A#, segments within 

the network and the resultant equation set solved directly using appropriate boundary conditions and initial 

values. In the branch-network model, however, transformation equations are developed from the segment flow 

equations to correlate the unknowns at the ends of the 

branches, that is, at the junctions.

From a two-component vector of state for the ith cross 

section,

Qf

the following transformation equation for the ith segment 

can be written

S-+l = U(i)S- +«(,->, (12)

in which S£ is the vector of state for the (i + l)th cross 

section. The transformation matrices of the ith segment, 

U(i) and «< , in which the subscript (i) denotes the segment, follow from the previously defined coefficient 

matrices:

7(0 

1

-7(o

and

 (0

Successive application of the segment-transformation 

equation 12 to all segments contained in a branch results 

in an expression that relates the unknowns at the end 

cross sections 1 and ra of the nth branch,

s;j i =unsr+ Un . as)

The transformation matrices of the nth branch, Un and 

« , in which the subscript n denotes the branch, are obtained through successive substitution of the segmenttransformation equation from the (ra- l)th segment down 

to the first segment. These branch-transformation 

matrices,

and

(14)

(15)

describe the relationship between the vectors of state, SC1 

and S£l , at the end cross sections of the branch, that is, 

at the junctions.

After applicable boundary-condition equations are formulated, the resultant equations are solved simultaneously, yielding stages and discharges at the termini of the 

branches (at the junction cross sections). Intermediate 

values of the unknowns at the internal segment ends (at 

cross sections between junctions) are subsequently determined through successive solution of the segment-transformation equation 12. This transformation procedure effects significant reductions in the model's requirements 

for computer memory and execution time. For example, 

if segment flow equations are used, a network consisting 

of AT sequentially connected branches, each composed of 

MI segments, would form a coefficient matrix of minimum 

order 2M+2, where M is the total number of segments 

in the network, that is, the sum of the Af/s for the Nbranch system. By combining segments into branches and 

using branch-transformation equations instead of segment 

flow equations, the size of the coefficient matrix can be 

reduced to order 4AT.

BOUNDARY CONDITIONS

To solve the branch-transformation equations implicitly, boundary conditions must be specified at internal junctions located at branch confluences within the network 

as well as at external junctions located at the extremities 

of branches, for example, where branches physically terminate or are delimited for modeling purposes. Equations 

describing the boundary conditions at internal junctions 

are automatically generated by the model, whereas 

boundary-condition equations for external junctions are 

formulated by the model from user-supplied time-series 

data or from user-specified functions.

Discharge and stage compatibility conditions can be expressed for internal junctions by neglecting velocity-head 

differences and turbulent energy losses. At a junction of 

n branches, discharge continuity requires that

(16)
MODEL APPLICATIONS

where Wk is zero or some user-specified external flow (inflow or outflow) at junction k, and stage compatibility requires that

Zm =Zm+l, m=l, 2, (17)

Various combinations of boundary conditions can be 

specified for external junctions. A null discharge condition (as, for example, at a dead-end channel), known stage 

or discharge as a function of time, or a known, unique 

stage-discharge relationship can be prescribed.

Together, the internal and external boundary conditions 

provide a sufficient number of additional equations to 

satisfy requirements of the solution technique.

SOLUTION METHOD

The solution process begins at time t0 by use of specified 

initial conditions and proceeds in Ai time increments to 

the end of the simulation at time tn . Gauss elimination using maximum pivot strategy is employed to solve the 

system of equations. Iteration within a time step is performed to provide results within user-specified tolerances. 

The primary effect of iteration is to improve on the quantities taken as local constants within the time step, which 

in turn increases the accuracy of the computed unknowns. 

User-defined accuracy requirements are typically 

achieved in two or fewer iterations per time step.

MODEL APPLICATIONS

The thoroughness of the equation formulation on which 

a model is based largely governs the range of complexity 

of flows it can accommodate. The choice of numerical computation scheme primarily determines whether or not the 

model will be stable, convergent, accurate, and computationally efficient given that it is correctly and precisely 

implemented. However, for any model to be useful it must 

be subsequently transformed into a functional useroriented simulation system, and its accuracy, reliability, 

and versatility must be adequately proved and demonstrated.

The branch-network model is being used to simulate the 

time-varying flows of several coastal and upland water 

bodies, as identified in table 1. These represent a broad 

spectrum of hydrologic field conditions, depicting such 

diverse hydraulic and field situations as hydropower-plantregulated flows in a single upland-river reach, tide-induced 

flows in riverine and estuarine reaches and networks, 

unsteady flow in a residential canal system, and 

meteorologically generated seiches and wind tides in a 

multiply connected network of channels joining two large 

lakes.

Four types of model application are identified in table 

1. The simplest of these is the single-branch type, which

TABLE I.-Applications of the branch-network flow model

State Water body location Application type

Alabama __Coosa River near Childersburg.

Alabama River near 

Montgomery.

Alaska___Knik/Matanuska River Delta 

near Palmer.

California __Sacramento River from Sacramento to Freeport.

Sacramento River from Sacramento to Hood.

Sacramento Delta between Sacramento and Rio Vista.

Threemile Slough near Rio 

Vista.

Connecticut _ Connecticut River near Middletown.

Connecticut River downstream 

from Hartford.

Florida ___Cape Coral residential canal 

system.

Peace River from Arcadia to

Fort Ogden. 

Peace River from Fort Ogden

to Harbour Heights.

Idaho____Kootenai River near Porthill. 

Kentucky _ 

Louisiana _

Ohio River downstream from 

Greenup Dam.

Atchafalaya River near Morgan

City.

Wax Lake Outlet near Calumet.

Calcasieu River between Lake 

Charles and Moss Lake.

Quachita River from Monroe to 

Columbia.

Vermillion River from Lafayette 

to Perry.

Loggy Bayou near Ninock.

Mermentan River from

Mermentan to Lake Arthur. 

Maryland __Potomac River near 

Washington, D.C.

35.2-km multiple 

branch.

21-branch multiconnected 

network.

20-branch multiconnected 

network.

17.4-km single 

branch.

34.3-km multiple 

branch.

24-branch multiconnected 

network.

5.2-km single 

branch.

9.8-km single 

branch.

41.2-km multiple 

branch.

16-branch multiconnected 

network.

30-km multiple 

branch.

21-branch multiconnected 

network.

54.8-km multiple 

branch.

21.7-km single 

branch.

8-branch multiconnected 

network.

15-branch multiconnected 

network.

13-branch multiconnected 

network.

78.9-km multiple 

branch.

48.3-km multiple 

branch.

9.2-km single 

branch.

25.7-km multiple 

branch.

25-branch multiconnected 

network.
FLOW MODEL FOR OPEN-CHANNEL REACH OR NETWORK

TABLE I.-Applications of the branch-network flow model- Continued

State Water body location Application type

Michigan__Detroit River near Detroit.

Saginaw River near Saginaw.

Missouri __Osage River near Schell City. 

New York_ 

N. Dakota__ 

S. Carolina.

S. Dakota __. 

Washington . 

Wisconsin _.

Hudson River from Albany to

Poughkeepsie. 

Red River of the North at

Grand Forks. 

Intracoastal Waterway near

Myrtle Beach. 

Cooper River at Diversion

Canal. 

Cooper River at Lake Moultrie

Tailrace. 

Back River near Cooper River

confluence. 

James River near Hecla.

.Columbia River downstream

from Rocky Reach Dam. 

Pheasant Branch near

Middleton. 

Menomonee River near

Milwaukee. 

Milwaukee Harbor at

Milwaukee.

12-branch multiconnected

network. 

14-branch dendritic

network. 

2.6-km single

branch. 

9-branch dendritic

network. 

1.3-km single

branch. 

36.7-km multiple

branch. 

6.3-km single

branch. 

1.5-km single

branch.

2.2-km single

branch. 

8.5-km multiple

branch. 

3.1-km single

branch. 

0.27-km single

branch. 

0.61-km single

branch.

12-branch multiconnected

network.

is an application to a single reach of channel delimited by 

a pair of external boundary conditions. The multiplebranch type is an application to a channel, again delimited 

by a pair of external boundary conditions, but schematized 

as a series of sequentially connected reaches. The dendritic-network type is an application to a channel system 

composed of branches connected in treelike fashion. The 

multiply connected network type is likewise an application to a channel system, but one in which the branches 

are interconnected, thereby permitting multiple flow 

paths between certain locations in the system.

To illustrate the diverse capabilities of the model, two 

applications identified in table 1 are discussed briefly 

herein. These particular applications were selected to 

demonstrate the flexibility of the model in accommodating 

a wide range of hydrologic conditions and field situations.

PHEASANT BRANCH NEAR MIDDLETON, WIS.

Pheasant Branch is a tributary to Lake Mendota near 

the city of Middleton in Dane County, Wis. A 5-year study

has been conducted by the U.S. Geological Survey, in 

cooperation with the city of Middleton and the Wisconsin Geological and Natural History Survey, to determine 

the sediment transport, streamflow characteristics, and 

stream-channel morphology in the Pheasant Branch 

drainage basin. In support of this effort, a short reach of 

Pheasant Branch was modeled to provide data on streamflow. Backwater effects from the lake and stormgenerated transient flows necessitate use of an unsteady 

flow model.

The Pheasant Branch reach, which begins in marshland 

and ends 274 m downstream at Lake Mendota, is treated 

as a single segment in the model. Under typical flow conditions, the channel is on the order of 6.5 m wide, with 

a maximum depth of 1.5 m.

Water-surface elevations used as boundary conditions 

(identified as station 05-4279.52 at the upstream end and 

05-4279.53 at the downstream end) for simulating flow 

in Pheasant Branch during June 15-21,1978, are shown 

in the upper part of figure 2. Discharges computed by the 

model at the upstream end of the reach are illustrated in 

the lower part of figure 2. Some rapid oscillations in the 

boundary-value data, identified as noise caused by windgenerated waves, are discernible, particularly in the stage 

hydrograph recorded at the downstream end of the reach 

near Lake Mendota. These oscillations in the boundaryvalue data are reflected and accentuated in the hydrograph of computed discharges.

The Pheasant Branch model results plotted in figure 2 

were computed using a 15-minute time step. The 

weighting factors 6 and x were set at 0.8. A value of 

0.0385 was used for 77, and the momentum coefficient, 13, 

was assigned a value of 1.0. These parameter values were 

determined in model calibration tests conducted using 

discharges measured at the mouth of Pheasant Branch 

during other flow periods.

The June 15-21 simulation required 1.6 CPU seconds 

to complete on an Amdahl 470/V71 computer. The model 

required less than one (0.9) iteration per time step, on the 

average, during the simulation.

The computed results indicate that this stream is extremely responsive and sensitive to changing climatological conditions; therefore, these factors must be accurately represented by the model.

POTOMAC RIVER NEAR WASHINGTON, D.C.

In October 1977, the Water Resources Division of the 

U.S. Geological Survey instituted a 5-year interdisciplinary study of the tidal Potomac River and Estuary 

(Callender and others, 1984). The research areas undertaken in this investigation included historical geologic

1 Use of firm or trade names in this report is for identification purposes only and does not 

constitute endorsement by the U.S. Geological Survey.
MODEL APPLICATIONS

175 

to

DC

LU

| 150

H 

Z

LU 

C_>

Sl25h

LU

(3

<

k

100 -

8 -

6-

4 

1 I 

Pheasant Branch at upstream end 

(5-4279.52)

Pheasant Branch at downstream end 

(5-4279.53)

-Pheasant Branch at upstream end _ 

(5-4279.52)

J___]_

June 15, I 16 

1978

17 I 18 I 19 

TIME, IN DAYS

20 I June 21, 

1978

FIGURE 2.-Measured water-surface elevations and computed 

discharges for Pheasant Branch near Middleton, Wis.

studies, geochemistry of bottom sediments, nutrient 

cycling, sediment transport and tributary loading, wetland studies, benthic ecology, and hydrodynamics. The objective of the hydrodynamics project was to devise, implement, calibrate, and verify a series of numerical 

flow/transport simulation models in support of the other 

research efforts. To quantify the hydrodynamics of the 

tidal river, the branch-network model was applied to the 

50-km segment of the Potomac, including its major 

tributaries and inlets from the head of tide at the fall line 

in the northwest quadrant of Washington, D.C., to Indian 

Head, Md., as shown in figure 3.

The Potomac River downstream from Chain Bridge is 

confined for a short distance (approximately 5 km) to a 

narrow, deep, but gradually expanding channel bounded 

by steep rocky banks and high bluffs. Farther downstream

the river consists of a broad, shallow, and rapidly expanding channel confined between banks of low to moderate 

relief. Seven cross-sectional profiles illustrating the channel geometry are plotted in figure 3. The cross-sectional 

area and corresponding channel width expand more than 

fortyfold between Chain Bridge and Indian Head. In 

general, the depth varies from about 9 m at Chain Bridge 

to about 12 m at Indian Head.

Flow in the upstream portion of the tidal river is typically unidirectional and pulsating; bidirectional flow occurs 

in the broader downstream portion. The location of the 

transition from one flow pattern to the other varies, 

primarily in response to changing inflow at the head of 

tide but also to changing tidal and meteorological 

conditions.

The tidal river system is schematized as shown in figure 

4. The network is composed of 25 branches (identified by 

roman numerals) that join or terminate at 25 junction locations (identified by numbered boxes). Junctions that do 

not constitute tributary or inlet locations in figure 4 were 

included in the network schematization to accommodate 

potential nodal flows (point source inflows or outflows 

such as sewage treatment outfalls or pump withdrawals) 

or to account for abrupt changes in channel characteristics.

A total of 66 cross sections were used to depict the channel geometry in 52 flow segments. Whereas the coefficient matrix of segment flow equations would require 

15,376 computer words, use of branch-transformation 

equations reduces the matrix size to 10,000 words. The 

computational effort required to effect a solution is also 

proportionally reduced.

In the tidal Potomac River model, flow discharges derived at a rated gaging station (01-6465.00) 1.9 km 

upstream from Chain Bridge are used as boundary values 

at junction 1. Water-surface elevations recorded at a gaging station (01-6554.80) at Indian Head are used as the 

downstream boundary values at junction 19. All other external boundary conditions are fulfilled by specifying that 

zero discharge conditions prevail at the upstream tidal extent of the particular channel or embayment.

Water-surface elevations recorded near Key Bridge 

(station 01-6476.00), near Wilson Bridge (station 

01-6525.88), and near Hains Point (station 01-6521.00) 

were used to calibrate and verify the model. (See figs. 3 

and 4.) Model-computed discharges were also compared 

with discharges measured for complete tidal cycles at 

Daingerfield Island, Broad Creek, and Indian Head.

In the model calibration process, values of 0.6 and 0.5 

were assigned to weighting factors 6 and x, respectively. 

Eta values in the calibrated model range between 0.0275 

at Chain Bridge and 0.019 at Indian Head. A value of 1.06 

was used for the momentum coefficient, /3, and a 

15-minute time step was used in the simulations. Using
77° 15'

39°00'

FLOW MODEL FOR OPEN-CHANNEL REACH OR NETWORK

77°00'

38°45'

Mount

Accotink \ Vemon 

Bay

MARYLAND

5 MILES

I i I

I I l I I I

0 5 KILOMETERS

Head

01-6554.80

FIGURE 3.-Potomac River near Washington, D.C.
MODEL APPLICATIONS

,01-6465.00

Chain Bridge

Key Bridge

01-6476.00

Roosevelt Is. 

Channel

XIX

Memorial Bridge

12| 

r01-6521.00

Wilson Bridge 7 A 01-6525.88

XII

XIV

Piscataway Creek

Dogue Creek

Accotink

Pohick

Indian Head 19 A<»-6554.80

FIGURE 4.-Schematization of the tidal Potomac River 

system for the branch-network flow model.

these parameter assignments, the model has satisfied convergence criteria, set at 0.46 cm and 3.54 m3/s for watersurface elevations and flow discharges, respectively, in 

fewer than two iterations per time step.

In figure 5, model-computed discharges are plotted 

against discharges measured at Indian Head from 2015 

hours on June 3 to 0830 hours on June 4, 1981. This 

15-hour simulation, from 1900 hours on June 3 to 1000 

hours on June 4, required 10.3 CPU seconds on an Amdahl 470/V7 computer. On the average, 1.4 iterations were 

required per time step. As is evident from the plot, there 

is excellent agreement between computed and measured

Q

Z

o 

o

01 

co 2

ul -2 

O 

cc 

<

I 

o

CO

Q -3

1 1

-5 

18 20

June 3, 

1981

I . I

22 24

TIME, IN HOURS

10 12 

June 4, 

1981

FIGURE 5.-Model-generated plot of computed versus measured 

discharges for the Potomac River at Indian Head, Md., on June 3-4, 

1981.
10 FLOW MODEL FOR OPEN-CHANNEL REACH OR NETWORK

discharges. Computed and measured ebb and flood volume fluxes compare within +0.6 and -2.3 percent, 

respectively.

This application of the model clearly demonstrates its 

adaptability to the simulation of unsteady flow in a network of interconnected channels.

MODEL USE

Mathematical/numerical models can address a variety 

of practical hydrologic field problems. They can be used, 

for example, to provide flow information for complex 

interdisciplinary riverine and estuarine investigations, to 

appraise hydraulic project-design alternatives, and to support environmental-impact assessments. Such varied uses 

emphasize the need, however, for models and (or) their 

supporting data base systems to provide efficient means 

of inputting and managing the required data and of 

analyzing and displaying computed flow information in 

a variety of graphical, pictorial, and alphanumeric forms.

To satisfy this need, the branch-network model has a 

wide range of graphical-display capabilities for computer 

generation of line-printer-drawn, mechanically drafted, 

or optically produced plots. These graphical capabilities 

not only help expedite model calibration and verification, 

but also provide a unique, rapid, and economical 

mechanism for portraying the flow and transport information required for various water-resources investigations. As an example, in figure 6 the particle-tracking 

capability of the model is illustrated in a model-derived 

plot. The time-of-travel graph of figure 6 depicts the 

movement of seven simultaneously injected index particles (labeled A through G) along the main Potomac River 

channel. The inflow-discharge hydrograph representing 

the upstream boundary condition is plotted above the 

time-of-travel graph. The stage hydrograph representing 

the downstream boundary condition is plotted below the 

time-of-travel graph. Model output such as this can be 

used to gain insight into the tidal-cycle variability in the 

concentration and dispersal of nutrients and sediments, 

as these are alternately or concurrently influenced by 

freshwater inflow conditions, meteorological effects, and 

tidal fluctuations.

SUMMARY AND CONCLUSIONS

An operationally oriented, usable model has been 

developed to compute flow and transport information for 

a single open-channel reach or an interconnected network 

of open channels. The branch-network flow model, along 

with its supporting operational data systems (Schaffranek 

and Baltzer, 1978; Regan and Schaffranek, 1985), constitutes a complete one-dimensional numerical-simulation 

system. Based on a comprehensive set of unsteady flow 

equations and structured to accommodate a diversity of

2.4

1.8 _ £g£

u - z - % en " oc

F? < Q £

tj i ^ i_

u ^ C 1.2

i i r

30 60 90 120 150 180 

TIME, IN HOURS

FIGURE 6.-Time-of-travel plot of injected particles for the Potomac 

River from midnight of November 30, 1980, to noon of December 8, 

1980.

complex open-channel configurations, the model has a 

wide range of utility, as is exemplified by the numerous 

applications cited. Two specific applications are illustrated. The model includes numerous graphical display 

capabilities that provide both model engineers and water 

managers with flow information compiled and condensed into easily comprehensible formats tailored to suit their 

specific requirements. One specific output type designed 

to depict the transport properties and flushing capacity 

of a riverine network is illustrated herein; others have 

been reported elsewhere (Lai and others, 1978; Schaffranek and Baltzer, 1978; Lai and others, 1980; Schaffranek and others, 1981; Schaffranek, 1982).

ACKNOWLEDGMENTS

Development of the branch-network flow model and its 

supporting operational data systems is being conducted
REFERENCES 11

in the research program of the Water Resources Division 

of the U.S. Geological Survey. The author is grateful to 

Survey colleagues who have contributed to this effort - 

especially to Robert A. Baltzer and Chintu Lai, who have 

consulted and in some instances collaborated in the 

development of the concepts and techniques expounded 

herein and to cooperating Federal, State, and local agencies that have contributed financially to the collection of 

the data used in this report.

REFERENCES

Amein, Michael, and Fang, C.S., 1970, Implicit flood routing in natural 

channels: American Society of Civil Engineers Proceedings, Journal of the Hydraulics Division, v. 96, no. HY12, p. 2481-2500.

Baltzer, R.A., and Lai, Chintu, 1968, Computer simulation of unsteady 

flows in waterways: American Society of Civil Engineers Proceedings, Journal of the Hydraulics Division, v. 94, no. HY4, p. 

1083-1117.

Callender, Edward, Carter, Virginia, Hahl, D.C., Hitt, Kerie, and 

Schultz, B.I., eds., 1984, A water-quality study of the tidal Potomac 

River and estuary An overview: U.S. Geological Survey WaterSupply Paper 2233, 46 p.

Cunge, J.A., Holly, F.M., Jr., and Verwey, Adri, 1980, Practical aspects 

of computational river hydraulics: Marshfield, Mass., Pitman, 420 p.

Dronkers, J.J., 1969, Tidal computations for rivers, coastal areas, and 

seas: American Society of Civil Engineers Proceedings, Journal of 

the Hydraulics Division, v. 95, no. HY1, p. 29-77.

Fread, D.L., 1974, Numerical properties of implicit four-point finitedifference equations of unsteady flow: National Oceanic and Atmospheric Administration Technical Memorandum, NWS, 

HYDRO-18, 38 p.

Lai, Chintu, Baltzer, R.A., and Schaffranek, R.W., 1980, Techniques 

and experiences in the utilization of unsteady open-channel flow

models: American Society of Civil Engineers, Specialty Conference 

on Computer and Physical Modeling in Hydraulic Engineering, 

Chicago, HI., August 6-8, 1980, Proceedings, p. 177-191.

Lai, Chintu, Schaffranek, R.W., and Baltzer, R.A., 1978, An operational 

system for implementing simulation models - A case study: American Society of Civil Engineers, Specialty Conference on Verification of Mathematical and Physical Models in Hydraulic Engineering, College Park, Md., August 9-12,1978, Proceedings, p. 415-454.

Preissmann, Alexander, 1960, Propagation des intumescences dans les 

canaux et les riviers [Propagation of translator^ waves in channels 

and rivers]: ler Congre's ^association Frangaise de calcul [First Congress of the French Association for Computation], Grenoble, France, 

p. 433-442.

Regan, R.S., and Schaffranek, R.W., 1985, A computer program for 

analyzing channel geometry: U.S. Geological Survey WaterResources Investigations Report 85-4335, 49 p.

Schaffranek, R.W., 1982, A flow model for assessing the tidal Potomac 

River: American Society of Civil Engineers, Specialty Conference 

on Applying Research to Hydraulic Practice, Jackson, Miss., August 

17-20, 1982, Proceedings, p. 521-545.

Schaffranek, R.W., and Baltzer, R.A., 1978, Fulfilling model timedependent data requirements: American Society of Civil Engineers, 

Symposium on Technical, Environmental, Socioeconomic and 

Regulatory Aspects of Coastal Zone Management, San Francisco, 

Calif., March 14-16, 1978, Proceedings, p. 2062-2084.

Schaffranek, R.W., Baltzer, R.A., and Goldberg, D.E., 1981, A model 

for simulation of flow in singular and interconnected channels: U.S. 

Geological Survey Techniques of Water-Resources Investigations, 

Book 7, Chap. C3, 110 p.

Strelkoff, Theodor, 1969, One-dimensional equations of open-channel 

flow: American Society of Civil Engineers Proceedings, Journal of 

the Hydraulics Division, v. 95, no. HY3, p. 861-876.

Yen, Ben Chie, 1973, Open-channel flow equations revisited: American 

Society of Civil Engineers Proceedings, Journal of the Engineering Mechanics Division, v. 91, no. EM5, p. 979-1009.