Depth-Discharge Relations 

of Alluvial Streams 

Discontinuous Rating Curves

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1498-C

** *

Depth-Discharge Relations 

of Alluvial Streams 

Discontinuous Rating Curves

By DAVID R. DAWDY

STUDIES OF FLOW IN ALLUVIAL CHANNELS

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1498-C

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1961
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

REPRINTED

1965

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington, D.G., 20402 - Price 15 cents (paper cover)
CONTENTS

	Pag« 

Definitions___________________________________________________ iv

Symbols. _____________________________________________________ iv

Abstract . ____________________________________._ C-1

The problem of rating alluvial streams_______________________________ 1

An approach to the solution of the problem_______--_-___-____________ 5

Relation of roughness to size of bed material________________________ 11

Other variables considered________________________________________ 16

Conclusions_ ________-_-_____-_____________-_--____---___--_-____ 16

References cited.__________________________________________________ 16

ILLUSTRATIONS

Page 

FIGURE 1. Idealized diagram of bed and surface configuration of alluvial

streams with various regimes of flow----_---_--__-__ .___ C-3

2. Variation of velocity with VJRS> 0.45 mm sand in laboratory. 4

3. Stage-discharge relation for Huerfano River near Undercliffe,

Colo______._______-_____-___-__--_-_._-__. 5

4. Relation of velocity to hydraulic radius for Huerfano River

near Undercliffe, Colo_______________________________ 6

5. Relation of velocity to hydraulic radius for Pigeon Roost

Creek near Byhalia, Miss____________________________ 7

6. Relation of velocity to hydraulic radius for Republican River

at Stratton, Nebr__________________________________ 8

7. Relation of velocity to hydraulic radius for Cheyenne River

near Spencer, Wyo______________-__-_____-_-_-__ __ 9

8. Relation of velocity to hydraulic radius for Middle Loup

River at St. Paul, Nebr.____________________ 10

9. Relation of velocity to hydraulic radius for Rio Grande near

BernaliUo, N. Mex.________________________ 11

10. Relation of velocity to hydraulic radius for South Fork

Powder River near Kaycee, Wyo______-______-________- 12

11. Relation of Chezy C to median diameter of bed material.___ 14

TABLE

Page 

TABLE 1. Variables measured._________--_-----__--_-_-_-_----__-_ C-15

m
DEFINITIONS

Median diameter of bed material. That size for which 50 percent 

of bed material is finer. Size is determined either by sieve or 

visual accumulation-tube analysis.

Sand-channel stream. A stream which has an unlimited source of 

sand of a given size available to the channel. The topmost bed 

layer consists statistically of grains of the same size.

Standard deviation of bed material. Obtained from formula 

%[CZW#5o)+.ZW#ie)] where Z>16, Z>60, and Z>84 denote size 

of bed material of which 16 percent, 50 percent, and 84 percent, 

respectively, are progressively finer in a given sample or combination of samples.

SYMBOLS

A area of cross section of stream

C Chezy roughness parameter; C=V/-<JlRS

F Froude number; F = V/^gR

g gravity constant

k intercept constant in relation V k R** at each station

n Manning roughness parameter; w=1.5 R* S*jV

R hydraulic radius, approximated by R=Aj \W-\-2, ^\

8 stream-channel slope, in feet per foot, generally abstracted from

topographic maps 

V mean velocity of the stream 

W top width for individual measurement 

<r standard deviation of bed material

IV
STUDIES OF FLOW IN ALLUVIAL CHANNELS

DEPTH-DISCHARGE RELATIONS OF ALLUVIAL 

STREAMS DISCONTINUOUS RATING CURVES

By DAVID R. DAWDY

ABSTRACT

A discontinuity occurs in the depth-discharge relation of many alluvial streams. 

For the higher part of the relation, after the discontinuity, Froude number and 

Chezy C remain constant For gaged sites, Froude number may be used for the 

extension of the relation. For ungaged sites, Chezy C may be estimated on the 

basis of bed-material properties.

THE PROBLEM OP RATING ALLUVIAL STREAMS

The development of methods for determining the discharge of alluvial streams has been a problem since the beginning of systematic 

stream gaging in this country. The usual approach is to relate discharge to stage and thus to determine discharge from a record of 

stage. Inspection of the stage-discharge relation shows that the relation is poor for many streams. On some streams, and particularly 

on many alluvial streams, an abrupt discontinuity occurs.

The Geological Survey began laboratory studies of the mechanics 

of flow in alluvial streams at Colorado State University in 1956. These 

studies are conducted by D. B. Simons and E. V. Richardson. One 

of the primary contributions of the laboratory study has been the 

definition of the various regimes of configuration of the sand-bed 

channel. These are described below.

Regime of flow Description* 

Plane bed _ _____ For flow prior to movement 

Ripples Small, uniform sand waves, with little sediment

movement

Dunes Much larger, more irregular sand waves, with a

great deal of turbulence.

C-l
C-2 STUDIES OF FLOW IN ALLUVIAL CHANNELS

Regime of How Descriptions 

Plane-bed transition-_____ Dunes are smoothed out Both the water surface and the bed are plane, with little turbulence.

Standing waves _______ Both the water surface and the bed are characterized by standing waves, often termed "sand 

waves."

Antidunes _ The "sand waves" move upstream, until at some

critical point they break, then reform.
DEPTH-DISCHARGE RELATIONS, DISCONTINUOUS RATING CURVES C-3

Idealized diagrams of these various regimes of flow are shown in 

figure 1.

PLANE-BED REGIME PRIOR TO MOVEMENT

RIPPLE REGIME

DUNE REGIME

PLANE-BED TRANSITION REGIME

STANDING-WAVE REGIME

ANTIDUNE REGIME

FIGURE 1. Idealized diagram of bed and surface configuration of 

alluvial streams with various regimes of flow.
C-4 STUDIES OF FLOW IN ALLUVIAL CHANNELS

For the laboratory data there is a distinct discontinuity in the 

relation of V oc -JRS, where V is the mean velocity, R is the hydraulic radius, and S is the channel slope which is assumed to equal 

the energy slope. The first three regimes, blend into each other in 

such a relation; the last three also blend together. The discontinuity 

occurs between the dune and plane-bed transition regimes. This is 

shown graphically by the flume data for a 0.45-mm sand in figure 2.

0.08

0.06

0.04

0.02

0.01

0.008

0.006

ft

*

0.6 0.8 I 2 468 

MEAN VELOCITY, IN FEET PER SECOND

FIGURE 2. Variation of velocity with JRS, 0.46 mm sand In laboratory.

The average Chezy C for this sand for the lower regimes is about 65; 

for the upper regimes it is about 110.
DEPTH-DISCHARGE RELATIONS, DISCONTINUOUS RATING CURVES C-5 

AN APPROACH TO THE SOLUTION OF THE PROBLEM

The various regimes of flow and bed configuration observed in the 

flume have been seen in sand-channel streams by most stream gagers, 

but their significance to depth-discharge relations has been realized 

only recently (Colby, 1960). The method of determining discharges 

by a stage-diseharge relation often obscures any underlying relation. 

This is because in alluvial streams neither the bottom nor the sides of 

the channel are fixed. Figure 3 shows as a typical example the stagedischarge plot for Huerfano River near ITndercliffe, Colo., for 1941 

and 1942.

7

5

hUJ

£ 3

z

K-~

X 

CO

UJ 1

X

UJ 

CO

CO

0

-1

-2 

-3

-4

*

4

1 .

 .

I

 v

ooo o oooo o

N CO O O OOOO O

 CM sr co co o q

DISCHARGE, IN CUBIC FEET PER SECOND

FIGURE 3, Stage-discharge relation foe Huerfano River near Undercllffe, Colo.
C-6 STUDIES OF FLOW IN ALLUVIAL CHANNELS

The underlying relation may be revealed by a change of variables. 

The effect of variation in bottom elevation is eliminated by replacing 

stage by mean depth or hydraulic radius. The effect of variation in 

width is eliminated by using mean velocity. Figure 4 shows the 

same measurements for Huerfano River, as were plotted in figure 3, replotted on the basis of velocity and hydraulic radius. Figures 5-9 show 

a similar plot for five other streams. Each graph shows a similar 

pattern with a trend for the lower and a trend for the higher measurements, and an amorphous range of discontinuity. On each graph the 

trend for the upper measurement is represented by a curve of relation. 

Each of these curves can be expressed as

where k represents a constant in the relation. According to this relation, the Froude number, F , remains constant, because the other 

term, </, in the formula

UJ 

UJ

to

5 I

cc 

o

-J

o:

*0.5

2 4 

VELOCITY, IN FEET PER SECOND

10

FIGTIKE 4. Relation of velocity to hydraulic radius for Huerfano River near Undercllffe,

Colo.
DEPTH-DISCHARGE RELATIONS, DISCONTINUOUS RATING CURVES C-7

is also a constant. If the energy slope remains constant at a given 

stream site, each curve also represents a constant value of the Chezy 

roughness coefficient, #, for

contains only constant terms.

\-

UJ

tu 

u.

CC 

O

cc 

o

i I

0.5

l_88>

o

o o o 

o

0<p

oo 

o o

00

2 4 

VELOCITY, IN FEET PER SECOND

10

5. Relation of velocity to hydraulic radius for Pigeon Roost Creek near Byhalla,

Miss.
0-8 STUDIES OF FLOW IN ALLUVIAL CHANNELS

UJ

CL 

O

DC 

Q

>.

248

VELOCITY, IN FEET PER SECOND

FIODBB 6. Relation of velocity to hydraulic radius for Republican River at Stratton,

Nebr,
<

rn 

ro 

o

««

Q, __

,3 2

rn 

m

9 *

5 (/>

g m * o

g o

00

HYDRAULIC RADIUS, 

1 T 1
HYDRAULIC RADIUS, I

I1 m

£ r

3 o ro

9 2

 «

 "s 

a

m 

m

H

M, T>

g m S1 o

M °

? z

s o
DEPTH-DISCHARGE RELATIONS, DISCONTINUOUS RATING CURVES C-ll

LU 4

LU

U_

V)

ID

o

o: 

o

EXPLANATION

Standing waves or antidunes

248 

I I I

VELOCITY, IN FEET PER SECOND

FIGURE 9. -Relation of velocity to hydraulic radius for Rio Grande near Bernalillo, N. Mex.

Chezy C Is not constant for all stations, however. The parameters 

most likely to explain the variation between stations would be the 

channel slope and the characteristics of the bed material. This is assuming all sands are of about the same specific weight, and that temperature variations are averaged out in the relation at each station. 

If the slope remains virtually constant at a station, as one normally 

would expect in alluvial channels, then the Chezy C is constant. The 

hypothesis easily is made that the magnitude of this constant Chezy C 

should be related to properties of the bed material.

RELATION OF ROUGHNESS TO SIZE OF BED MATERIAL

To test the hypothesis that a general relation for the upper curve 

existed which could be related to slope and bed material, a study was 

undertaken and records were obtained and studied of the relation be-
C-12 STUDIES OF FLOW IN ALLUVIAL CHANNELS

LJ 

UJ

O

o: 

o

248

VELOCITY, IN FEET PER SECOND

FIODBH 10. Relation of velocity to hydraulic radius for South Fork Powder River near

Kaycee, Wyo.
DEPTH-DISCHARGE RELATIONS, DISCONTINUOUS RATING CURVES C-13

tween hydraulic radius and velocity for stream-gaging stations having 

sand channels.

For many stations no discontinuity in the rating was found. The 

slope and depth of some streams apparently were too small to create 

sufficient shear to wash out the dunes. The gaging station sometimes 

was established at a constriction or above a diversion dam or outcrop 

of bedrock. At places, attempts had been made to stabilize the ratings 

by dumping rock in the stream or by building low-control structures. 

Quite often, for convenience in servicing, gaging stations were established on bridges. All of these conditions tend to complicate the 

hydraulics, inducing variable acceleration and backwater, and no 

break in the rating could be defined. In addition, many streams have 

high flows of such short duration that measurements were difficult to 

obtain, and enough were not available to define the upper curve. For 

other streams, notably those draining the Sand Hills of Nebraska, the 

flow is so uniform that a sufficient range in discharge was not experienced to define any relation.

The stations showing a discontinuity in rating were chosen for this 

study. Generally they were stations in relatively typical reaches 

with no abrupt constrictions. Measurements of high water were made 

either from a cableway or from a single-span bridge which did not 

obstruct the flow. The streams had slopes of 0.0004 feet per foot or 

greater, and had a wide range in discharge.

The relation R « V was determined by plotting the data on logarithmic scales and fitting by eye a straight line to the data. For most 

of the ratings studied, the slope of one-half (abscissa to ordinate) 

represented the line of best fit For some ratings, however, the range 

in velocity in the upper regime was not sufficient to determine the 

slope, and for them a slope of one-half was fitted arbitrarily. For one 

station, South Fork Powder Eiver near Kaycee, Wyo., shown on 

figure 10, the measurements define a slope of two-thirds rather than 

one-half, indicating a constant Manning's n rather than a constant 

Chezy C. The scatter about the line fitted for each station includes 

error in measured discharge and the effect of variations in slope, temperature, or sediment properties which may alter the relation. The 

curve represents a constant Froude number or Chezy G. The scatter 

about the curve partly is natural scatter introduced by measuring 

error, and partly is due to minor variations in the relation. Thus, for 

all but one of the sites studied, rating extensions could be made by 

using a constant Froude number for higher flows. For ungaged sites, 

a constant Chezy G could be used and acceptable results be obtained 

in a great majority of cases.
G-14 STUDIES OF FLOW IN ALLUVIAL CHANNELS

In order to determine the Chezy C at each station, the slope of the 

stream, was determined, generally from topographic maps. It was 

assumed that the energy slope could be approximated by the average 

bed slope through a long reach. The median diameter and the standard deviation of the bed material were abstracted, wherever possible, 

from published reports. In many cases, however, no information was 

available, and single samples of bed material were obtained in the field. 

Some of the samples were taken during periods of no flow, so that both 

median diameter and standard deviation are subject to considerable 

error. These data are shown in table 1.

Despite the possibility of error in both variables, a plot of the Chezy 

G against median diameter shows a trend in the expected direction. 

The larger the bed material the greater the resistance to flow (fig. 

11). A refinement in the measurement of both slope and bed material 

probably would reduce the scatter somewhat. The standard error 

about the line of relation is +25 percent and 20 percent.

200

UJ

x 100

50

©

EXPLANATION

Maximum concentration 

greater. than 80,000 

parts per million

Maximum concentration 

less than 20,000 parts 

per million

°®

®

O.I 0.2 0.4 I 

MEDIAN DIAMETER OF BED MATERIAL, IN MILLIMETERS

FIQDBB 11. Relation of Chezy C to median diameter of bed material.
DEPTH-DISCHARGE RELATIONS, DISCONTINUOUS RATING CURVES C-15

TABLE 1. Variables measured

Station

Bighorn River at Manderson, Wyo _ ___________

Bighorn River near Manderson, Wyo___ _________

Browns Wash near Green 

River, Utah __________ _

Canadian River at Logan, 

N. Mex__________._____

Cherry Creek near Frankto. m, Colo___ _______ _

Cherry Creek near Melvin, 

Colo-__-_-_-------____.

Cheyenne River near Spencer, Wyo ______________

Cuffawa Creek at Chulahoma, Miss____.________

Elkhorn River at Waterloo, 

Nebr_---__-_-_-_ __ ._

Galisteo Creek at Domingo, 

N. Mex____-__.-_______

Huerfano River near Undercliffe, Colo___________

Lance Creek at Spencer, 

Wvo___________________

Middle Loup River at St. 

Paul, Nebr_--_-___--___

Muddy Creek near Shoshoni, Wvo __-__-.-____

Pigeon Roost Creek near

Powder River, South Fork, 

near Kaycee, Wvo __ ___

Republican River at 

Stratton, Nebr _ _________

Rio Grande at Bernalillo, 

N. Mex.____. _______ __

Rio Grand near Belen. 

N Me*

Rio Grande at San Antonio, 

N. Mex________ _______

Rio Grande at San Marcial, 

N. Mex. _______________

Rio Puerco near Bernardo, 

N. Mex____ ___________

San Francisco Riverside 

Drain near Bernardo, 

N. Mex___________.____

San Juan River at Shiprock, 

N. Mex._______________

Walkers Bottom Creek, near 

Hollv Springs, Miss _ ___

White "River, South Fork, 

below White River, S. 

Dak. ___________ __ ___

Willie Wilkins Creek near

Intercept

2.3 

2.6

4.6 

4.1 

4.9 

5.7 

2.85 

4. 1 

2.5 

5.6 

5.2 

3.3 

3.7 

4.7 

3.2 

3.6 

4.0 

3.5 

3.6 

3.5 

3.0 

4.35

2.75 

3.5 

5.1

3.25 

3.8

Slope 

(feet per foot)

0. 000947

.0025 

.00057 

.0074 

.0051 

.0018 

.002 

.0004 

.0055 

.0113 

. 00073 

.001 

.0044 

.0009 

.0016 

.0017 

. 00095 

. 00082 

. 00055 

. 00041 

.001

. 00075 

.0019 

.0125

.0015 

.0058

Chezy C

75

84 

92 

172 

57 

80 

67 

92 

125 

75 

49 

122 

117 

71 

107 

1 0.0165 

97 

114 

126 

149 

148 

138

101 

80 

46

90 

50

Bed material

Median 

diameter 

(mm)

0.34 

.27 

. 18 

.95 

.40 

.8 

.39 

.24 

.40

.3

. 26 

.25 

.40 

.5 

.32 

.30 

.24 

.20 

. 14 

.24

. 26 

.35 

.40

.34 

.51

Standard 

deviation

2.3 

2.1 

1.43 

2.75 

1.96 

5 

1.45 

1. 68 

2.35

1.53 

3.75 

1.4 

9.4 

1. 62 

1.60 

1.50 

1.51 

1.46 

2.42

1.42 

1.58 

1.49

1.54 

1.49

i Manning's n.
C-16 STUDIES OF FLOW IN ALLUVIAL CHANNELS 

OTHER VARIABLES CONSIDERED

The use of the standard deviation of the bed material as a third 

variable indicated a trend that one might predict, but the results were 

not significant. The standard deviation was negatively correlated 

with the Chezy <7, but it reduced the variance by about 3 percent only, 

being significant at about the 40-percent level. The standard deviation, however, probably is the least well defined of all the variables 

measured. The relation may be determined better by refinement of 

the measurements of the variables.

It has been suggested (Brooks, 1958; Vanoni and Nomicos, 1959) 

that suspended sediment is one of the independent variables which 

determines the velocity of the stream* This was not found to be true 

once the discontinuity occurred, although it may influence the point 

of discontinuity. At each of the stations used in this study, there is a 

range of suspended loads in the regimes of flow beyond the discontinuity, yet Chezy C remains virtually constant at each station. There 

also is a wide variation in the rates of sediment transport among the 

various stations. On figure 11, 3 stations for which the maximum 

measured suspended-sediment concentration is less than 20,000 ppm 

are compared with 4 stations for which the maximum is greater than 

80,000 ppm. The differences in the value of Chezy C are not explained by the gross differences in sediment transport.

CONCLUSIONS

Sand-channel streams with wide ranges in discharge often have a 

discontinuous rating. The upper part of the rating represents planebed, standing-wave, and antidune regimes. For this upper part, the 

Froude number and the Chezy G are constant at a site, and the variation of C from site to site appears to be related primarily to grain 

size. The relation of roughness to bed-material size should prove 

valuable in extending rating curves above the highest measured discharge and in determining peak discharge of ungaged streams from 

floodmarks.

REFERENCES CITED

Brooks, N. H., 1958, Mechanics of streams with movable beds of fine sand: Am.

Soe. Civil Engineers Trans. paper 2931, v. 123, p. 526-594. 

Colby, B. R., 1960, Discontinuous rating curves for Pigeon Roost and Cuffawa

Creeks in northern Mississippi: U.S. Dept Agriculture, Agr. Research

Service 41-36,31 p. 

Simons, D. B., Richardson, E. V., and Albertson, M. L., 1961, Flume studies using

median sand (0.446 mm) : U.S. Geol. Survey Water-Supply Paper 149S-A. 

Vanomi, V. A., and Nomicos, G. N., 1959, Resistance properties of sediment-laden

streams: Am. Soc. Civil Engineers Proc., paper 2020, p. 77-107.

 & U.S. GOVERNMENT PRINTING OFFICE: 1965 O 767-089