Effect of Depth of 

Flow on Discharge of 

Bed Material

By BRUCE R. COLBY

STUDIES OF FLOW IN ALLUVIAL CHANNELS

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1498-D

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

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

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

Washington 25, D.C. - Price 15 cents (paper cover)
CONTENTS

 Page

Abstract_______________________________.__________., D-1

Introduction. ________________________________________ 1

Definitions_________________________________________________ 2

Depth effect computed from the Einstein procedure__________________ 3

Depth effect experimentally determined____________________________ 5

General reason for the depth effect________________________________ 7

Depth effect for shear velocity._____________________________________ 9

Depth effect for stream power_____________________________________ 9

Conclusions___________________________________________________ 11

References__________________________________________________ 12

ILLUSTRATIONS

FIGURE 1. Qualitative effect of depth on the relationship between mean 

velocity and discharge of bed material as computed from a 

procedure somewhat revised from Einstein's ______________ D-4

2. Effect of depth OB the relationship between mean velocity 

and empirically determined discharges of bed material at 

60° T______.._______________________ 6

3. Schematic diagram showing the general reasons for variability 

of effect of depth on the relationship of discharge of material to mean velocity...___________________________ 8

4. Effect of depth on the relationship between discharge of bedmaterial at 60° F and shear velocity with respect to the 

particles____________________________________________ 10

m

STUDIES OF FLOW IN ALLUVIAL CHANNELS

EFFECT OF DEPTH OF FLOW ON DISCHARGE OF BED

MATERIAL

By BEUCE R. COLBY

ABSTRACT

Computations with the Einstein procedure and an empirical analysis of data 

from flumes and natural streams agree qualitatively in showing a complex effect 

of depth of flow on the relationship of discharge of bed material to mean velocity. 

The effect of increasing the depth is in the opposite direction at low mean velocities 

from what it is at high mean velocities and is almost negligible at some intermediate 

velocities. Depth of flow has an equally complex and varied effect on the relationship between discharge of bed material and stream power and has, except at low 

shears, a large but simpler effect on the discharge of bed material as related to 

shear velocity with respect to the sediment particles.

INTRODUCTION

In most flume experiments, the range of depth is relatively small 

and the discharge of bedload frequently is on the order of magnitude 

of the discharge of suspended bed material. Hence the variable 

effect of depth of flow on the relationship between discharge of bed 

material, which at high velocities may be mostly in suspension, and 

parameters of flow is sometimes overlooked or misunderstood, 

especially for natural streams.

The purpose of this paper is to give at least a rough idea of the 

effect of depth on relationships between the discharge of bed material 

and such flow parameters as shear velocity with respect to the particles, 

mean velocity, and stream power. The theoretical and empirical 

derivation of graphs of discharge of bed material per foot of stream 

width as a function of depth and mean velocity for flow over sand 

beds is described first. The effect of depth is then explained in general 

terms on the basis of these graphs and other broad relationships.

This paper is based on computations and studies that were made 

partly while the writer was working for the Agricultural Kesearch 

Service, U.S. Department of Agriculture. Information from the files 

of that agency and the Geological Survey was used to define figures 

2 and 4.

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

DEFINITIONS

A few of the terms used in this discussion are defined as follows:

Bedload is the sediment that moves by sliding, rolling, or skipping 

on or very near the streambed and is supported mainly by the bed 

rather than by the turbulence of the flow.

Bed material is a size classification and includes the sizes of sediment 

particles that are found in appreciable quantity near the surface of 

the streambed, but the bed material may be transported anywhere 

between the bed and the water surface.

Discharge, as applied to sediment movement, is a time rate of 

transportation of dry weight of sediment through a cross section.

Measured discharge of bed material is the discharge of bed material 

that can be determined from streamflow and depth-integrated sediment samples, which generally are obtained by sampling the flow front 

the water surface to within 0.3 or 0.4 foot of the streambed.

Shear velocity with respect to the grains or particles is Einstein's 

concept (1950) and may be defined as the shear velocity that can be 

computed from the energy gradient and a particular hydraulic radius 

R'. The square root of the ratio of R' to the hydraulic radius R 

equals the ratio of the actual mean velocity to the mean velocity that 

would exist if all resistance were due to the particles on the bed and 

none to the configuration of the bed and banks. The Configuration 

of the bed and banks of sand-bed streams causes resistance to flow, 

not only because of general changes in channel form but also because 

of ripples and dunes on the bed. The resistance caused by ripples 

and dunes is usually large and may change rapidly at times.

Total discharge of bed material is the discharge of all the bed 

material that passes through a stream cross section.

Stream power per unit area of the bed is the product of shear and 

mean velocity or y dSu, in which 7 is the weight of the water-sediment 

mixture per unit volume, d is the depth of flow, S is the energy gradient, and u is the mean velocity.

Unmeasured sediment discharge is the difference between total 

sediment discharge and measured sediment discharge. It is nearly 

all discharge of bed material, and hence is about equivalent to the 

difference between total and measured discharge of bed material.

The symbol "2" is an exponential measure of the vertical distribution of concentration of sediment of uniform size or of a narrow size 

range. Under the assumptions for which it was theoretically derived, 

it equals vs/(k^gRS}, in which v, is the fall velocity, g is the gravity 

constant, and k is the turbulence constant.
EFFECT OF DEPTH OF FLOW ON DISCHARGE OF BED MATERIAL D~3

DEPTH EFFECT COMPUTED FROM THE EINSTEIN 

rPROCEDURE

One approach to understanding the relationship of discharge of 

bed material to characteristics of the cross section and flow is to make 

many computations with some formula or procedure for computing 

bed-material discharges. The procedure described by Einstein (1950) 

is one of the most comprehensive and detailed methods for computing 

discharges of bed material by at least partly theoretical relationships 

and was, therefore, selected for the computations. The sediment of 

the streambed was assumed to be cohesionless sand of which 35, 50, 

and 65 percent by weight was finer than 0.35, 0.40, and 0.45 millimeter, respectively.

Discharges of bed material per foot of stream width were computed, 

according to a somewhat revised Einstein procedure, for water temperatures of 40°, 60°, and 80° F and for wide ranges of depth and velocity. 

The revisions of the procedure consisted mainly of computing shear 

velocities with respect to the particles from mean velocities and 

deterinining z's partly from experimental data, rather than from 

theory only. The revised z's for flow over dune beds were more 

realistic than those that might be computed as Einstein suggested, 

but they were too small for flow over plane or antidune beds and hence 

gave too large computed discharges of bed material for high mean 

velocities. However, the computed results show qualitatively (fig. 1) 

the relative effects of velocity, depth, and water temperature on the 

discharge of the assumed bed material.

Mean velocity is obviously the major factor in determining the discharge of the assumed bed material according to figure 1. (However, 

its apparent effect is overstated at mean velocities of 5 fps or more.) 

The computed effect of temperature is reasonably consistent although 

the effect increases in general as depth of flow increases. The effect 

of depth is variable. At a mean velocity of 5 feet per second, depth 

has little computed effect; about as much bed material would be discharged per foot of width at a depth of 1 foot as at a depth of 100 feet. 

As the depth increases, the discharge of bed material increases at high 

constant mean velocities and decreases at low constant mean velocities.

If computations similar to those that were used to define figure 1 

had been made for a coarser bed material, the mean velocity at which 

depth has little effect would have been greater than 5 feet per second; 

if they had been made for finer bed material, the mean velocity at 

which depth has little effect would have been less than 5 feet per 

second.

If computed concentrations had been plotted instead of discharge 

of bed material, the effect of depth would be large except for high mean
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EFFECT OF DEPTH OF FLOW ON DISCHARGE OF BED MATERIAL D~5

velocities and small enough sediment particles so that the vertical distribution of suspended sediment would be almost uniform.

DEPTH EFFECT EXPERIMENTALLY DETERMINED

Of course, a figure similar to figure 1 can be defined empirically if 

enough information is available for wide ranges of depth and velocity. 

Hence, many determinations of total discharge of bed material per 

foot of width and concurrent mean velocities were obtained and 

plotted. These determinations were mostly for flume studies, although some field determinations of total discharge of bed material 

were available particularly for Middle Loup River near Dunning, 

Nebr. (Hubbell and Matejka, 1959), and Niobrara River near Cody, 

Nebr. (Colby and Hembree, 1955). These field determinations of 

total discharge of bed material were based on measurements at cross 

sections where velocity and turbulence were high enough to suspend 

practically the entire sediment discharge of the stream. Widths and 

mean velocities were computed at nearby cross sections where the flow 

was over beds of sand whose median diameter was determined by 

sampling.

At cross sections of some natural streams that have beds of sand, 

the total discharges of bed material usually were not determined, and 

only the measured discharges were known. If the stream widths, 

mean velocities, and other necessary information were observed as 

the sediment samples were taken the total discharge of bed material 

could be computed by using the modified Einstein procedure (Colby 

and Hembree, 1955) or from the relation of unmeasured sediment discharge to mean velocity (Colby, 1957). Although accurate field determinations of total discharge of bed material are preferable to partly 

computed discharges, the computed discharges were used for the Rio 

Grande, the Mississippi River, and some other streams for which 

better information was not available.

The total discharges of bed material were first roughly adjusted to 

a water temperature of 60° F and were then used to define individual 

graphs empirically for median particle sizes of bed material of 0.1, 0.2, 

0.3, and 0.4 millimeter. Obviously, most median particle sizes and 

mean velocities were not exactly equal to desired quantities such as 

0.20 millimeter and 4.0 feet per second but might be, for example, 

0.24 millimeter and 3.7 feet per second. Hence, a complex system of 

judgment and graphic analysis was used to interpolate and extrapolate the basic information until a fairly consistent set of smooth 

graphs was obtained. The graphs, of which figure 2 is one, are somewhat inexact and conventionalized. They are intended to apply to 

fields streams rather than to laboratory flumes, which usually have 

more uniform lateral distributions of depth and velocity than field
D-6 STUDIES OF FLOW IN ALLUVIAL CHANNELS

1000

100

10

1.0

- O.I

O.I 1.0 10

DEPTH, IN FEE'T

100

FIGURE 2. Effect of depth on the relationship between mean velocity and empirically determined discharges of bed material (0.3-mm median diameter) at 60° F.

streams. Also, the median size reported for sediment in a flume investigation is generally for the material that is placed hi the flume 

rather than for the sediment that is actually resting at the surface of 

the bed during flow, when some of the finest sediment may be in suspension. In experiments such as those by Gilbert (1914), some selective sorting must occur during a series of runs even though the range 

of particle sizes is small.

The graph for bed material whose median diameter is 0.3 millimeter 

(fig. 2) is probably as well defined as any of the four empirically
EFFECT OF DEPTH OF FLOW ON DISCHARGE OF BED MATERIAL D~7

determined graphs and is, therefore, used as a basis for consideration 

of the effect of depth. However, the effect of depth on the relation 

between discharge of bed material and mean velocity varies not only 

with velocity but also with size of the bed material when the mean 

velocity is consant. For example, the mean velocities at which depth 

has little apparent effect according to the empirically defined graphs 

are about 1.6, 2.0, and 4.0 feet per second for bed materials of 0.1-, 

0.2-, and 0.4-miUimeter median diameters as compared to 3.0 feet per 

second for bed material of 0.3-millimeter median diameter. Of 

course, the line for a velocity of 3.0 feet per second on figure 2, if moreaccurately defined, might not be exactly horizontal through part or 

any of its length.

Qualitatively, the effects of depth on the relation between dischargeof bed material and mean velocity are roughly comparable (figs. 1 

and 2) whether determined from the Einstein procedure or defined 

empirically, even though the sizes of bed material differ somewhat. 

That is, the indicated effect of depth is insignificant for some intermediate mean velocity, whereas at higher constant mean velocities 

the discharge of bed material increases as depth increases and at 

lower constant mean velocities the discharge decreases as depth 

increases. The effect of depth is small at those velocities and depths 

that characterize many sand-bed streams when they are dischargingmuch of their bed-material loads. Hence, a sand-bed stream at a 

particular cross section generally has a reasonably good relationshipbetween discharge of bed material per foot of stream width and mean 

velocity when water temperature is constant, because the variation 

in depth and in lateral distribution of velocity at a given mean velocity 

is usually rather small. If cross sections of the same or different 

streams have comparable bed material, the relationship for one crosssection may also hold reasonably well for others especially at intermediate mean velocities but less well, perhaps, at high and low velocities for which depth effects are larger.

Although factors other than velocity, depth, and water temperaturemay affect the discharge of bed material, a group of graphs each 

similar to that on figure 2 but for a different size of bed material 

gives a good practical basis for estimating discharge of bed material. 

However, the uses and adjustments of such graphs for computing or 

estimating the discharge of bed material in sand-bed streams ara 

beyond the scope of this paper.

GENERAL REASON FOR THE DEPTH EFFECT

The variable effect of depth on the discharges of bed material iseasily understood in broad outline, although the actual computation 

of such discharges for different depths is inexact and complicated-
D-8 STUDIES OF FLOW IN ALLUVIAL CHANNELS

If mean velocity is low say 2 feet per second and water flows in a 

shallow stream over a sand bed whose median diameter is 0.3 or 0.4 

millimeter, a certain amount of bed material, as indicated by the 

-curves of velocity and concentration on figure 3, will be discharged

VELOCITY AND CONC EN T RA t ION'

3. Schematic diagram showing the general reasons for variability of effect of depth on the 

relationship of bed-material discharge to mean velocity.

mostly near the streambed. If the depth of flow is greatly increased 

and the mean velocity is kept constant, the velocity near the streambed will decrease and the discharge of bed material both at the bed 

and throughout the original shallow depth of flow will decrease. 

Only a little bed material will be discharged in suspension in the 

added depth of flow. Hence, the total discharge of bed material is 

likely to be less for a given low mean velocity when the flow is deep 

than when it is shallow. On the other hand, if the mean velocity is 

high say 6 feet per second the discharge of bed material in a shallow 

flow is greater than that in a layer of equal depth at the bottom of a 

deep flow because of the difference in the velocity near the streambed. However, the difference in discharge of bed material near the 

bed is more than balanced by the discharge in the superposed layer 

of the deeper flow. (See fig. 3.)

To a considerable extent, the effect of increasing the mean velocity 

over a bed of constant particle size is equivalent to decreasing the 

particle size at constant velocity. Therefore, the effect of depth for 

fine bed material at moderately low mean velocity may be about 

equivalent to that for coarser bed material at some higher, mean 

velocity.
EFFECT OF DEPTH OF FLOW ON DISCHARGE OF BED MATERIAL D~9

DEPTH EFFECT FOB SHEAR VELOCITY

If Chezy's C were constant, the same sort of effect of depth would 

apply to relationships between discharge of bed material and shear or 

shear velocity as applies to the relationships of discharge of bed material 

to mean velocity. However, Chezy's C usually varies with depth and 

bed configuration. Also, as has been shown by Brooks (1958), shear 

is a poor measure of discharge of either water or sediment for certain 

conditions of flow and sand beds. Hence, the data of figure 2 were 

transposed to show the effect of depth on the relationship between discharge of bed material and shear with respect to the particles.

Shear velocity with respect to the particles was computed from a 

velocity equation given by Keulegan and used by Einstein (1950, 

p. 10):

u=5.75 V log (12.27 dx/kt) 

in which 

u is the mean velocity 

u* r is the shear velocity with respect to the particles and equals

d is the depth of flow

x is a dimensionless parameter for the transition between hydraulically smooth and hydraulically rough boundaries and is a 

function of particle size, water temperature, and u* f

k, is the grain roughness of the bed and was assumed to be 0.00115 

foot (0.35 mm) for a median diameter of 0.30 millimeter

The equation was solved graphically by trial and error for values of 

u* r for different depths and mean velocities.

When the information on figure 2 was replotted on figure 4 in 

terms of u* r rather than mean velocity, a different pattern of depth 

effect appeared. The indicated effect of depth is small at low shear 

velocities with respect to the particles but increases rapidly as these 

velocities increase. Thus, figure 4 indicates that the effect of depth 

can be neglected at very low shear velocities, but large adjustments 

for this effect are required at nearly all flows that have practical 

significance in natural streams. However, this depth effect is always 

in the same direction.

DEPTH EFFECT FOB STREAM POWER

Bagnold (1960) has recently suggested the use of stream power as 

a measure of sediment discharge. In more familiar units and terminology than his, stream power of a volume of water in a uniform 

reach of river, L units long and W units wide, at steady flow equals 

the product of the weight of the water in the reach and the distance 

that the average elevation of the water decreases in unit time. Hence, 

the stream power for low concentrations of sediment and dissolved
D-10 STUDIES OF FLOW IN ALLUVIAL CHANNELS

O.I 1.0 10 

DEPTH, IN FEET

100

FIGUBE 4. Effect of depth on the relationship between discharge of bed material at 60° F and shear 

velocity with respect to the particles.

solids and in pound-foot-second units is Q2ALWduS. For unit 

width and length of channel the stream power is QZAdSii or is the 

product of shear and mean velocity.

If Chezy's 0 is constant and a change in depth is compensated by 

an inverse change in energy gradient, the effect of depth on the relation of discharge of bed material to stream power is like that indicated by figure 2. That is, if the mean velocity is 3 feet per second, 

median diameter is 0.3 millimeter, and shear is constant; the indicated
EFFECT OF DEPTH OF FLOW ON DISCHARGE OF BED MATERIAL D-ll

discharge of bed material remains unchanged as the depth increases. 

For a lower mean velocity such as 1.5 feet per second, the discharge 

of bed material decreases rapidly as the depth increases at constant 

shear; for mean velocities higher than 3 feet per second, the discharge 

of bed material increases as depth increases at constant shear.

If Chezy's O varies, as it does, for example, when the bed configuration changes, mean velocity will not be proportional to the shear, 

and the effect of depth on the relationship of bed material 

discharge to stream power becomes even more complex. In other 

words, a given percentage change in any one of the variables u, d, or 

S is not necessarily equivalent to an equal percentage change in either 

of the other two.

If all resistance to flow is caused by particle roughness, the shear 

is proportional to the square of shear velocity with respect to the 

particles and the stream power is proportional to the product of (u*') 2 

and u. The effect of depth under this assumption is a combination 

of effects that are indicated by figures 2 and 4. At constant low 

velocities and shears, the discharge of bed material decreases as depth 

increases; whereas at constant high velocities and shears, the discharge of bed material increases as depth increases.

CONCLUSIONS

The relation of discharge of bed material per foot of stream width 

to mean velocity, to shear velocity with respect to the particles, or 

to stream power vary, sometimes widely and complexly, with changes 

in depth. At constant low mean velocity an increase in depth reduces 

the discharge of bed material, but at constant high mean velocity 

the effect of depth is reversed. At some intermediate velocities, 

which are frequently the velocities at which most of the bed material 

is transported in natural streams, the effect of depth is usually small.

Depth has a somewhat similar but more complex effect on the relation of discharge of bed material to stream power, because stream 

power is a compound parameter that equals the product of shear and 

mean velocity.

Discharge of bed material at constant shear velocity with respect to 

the particles increases slowly with depth at low shears but increases 

rapidly with depth at higher shears. Practically, this effect of depth 

has the advantage of being always in the same direction as compared 

to the variable effect of depth on the relations that involve mean 

velocity or stream power. It has the disadvantage of being large 

throughout the usual range of significant flows in natural streams.

The effect of depth varies with particle size at constant mean 

velocity, constant shear velocity with respect to the particles, or constant stream power. For example, the effect of depth is small or
D-12 STUDIES OF FLOW IN ALLUVIAL CHANNELS

negligible at mean velocities of about 1.6, 2.0, 3.0, and 4.0 feet per 

second for bed materials whose median sizes are about 0.1, 0.2, 0.3, 

and 0.4 millimeter, respectively.

REFERENCES

Bagnold, R. A., 1960, Sediment discharge and stream power a preliminary 

announcement: U.S. Geol. Survey Circ. 421, 23 p.

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

Soc. Civil Engineers Trans., v. 120, p. 526-594.

Colby, B. R., 1957, Relationship of unmeasured sediment discharge to mean 

velocity: Am. Geophys. Union. Trans., v. 38, p. 708-717.

Colby, B. R., and Hembree, C. H., 1955, Computations of total sediment discharge, Niobrara River near Cody, Nebraska: U.S. Geol. Survey WaterSupply Paper 1357, 187 p.

Einstein, H. A., 1950, The bed-load function for sediment transportation in open 

channel flows: U.S. Dept. Agr. Tech. Bull. 1026, 71 p.

Gilbert, G. K., 1914, The transportation of debris by running water: U.S. Geol. 

Survey Prof. Paper 86, 263 p.

Hubbell, D. W., and Matejka, D. Q., 1959, Investigations of sediment transportation, Middle Loup River at Dunning, Nebraska: U.S. Geol. Survey WaterSupply Paper 1476, 123 p.