Sediment Transport 

in Alluvial Channels 

1963-65

GEOLOGICAL SURVEY PROFESSIONAL PAPER 462

This volume was published 

as separate chapters A I
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEW ART L. UDALL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director
CONTENTS

[The letters designate separately published chapters]

(A) Discharge of sands and mean-velocity relationships in sand-bed streams, by Bruce R. Colby.

(B) Vertical distribution of velocity and suspended sediment, Middle Rio Grande, New Mexico, by Carl F. Nordin, Jr., and 

George R. Dempster, Jr.

(C) A preliminary study of sediment transport parameters, Rio Puerco near Bernardo, New Mexico, by Carl F. Nordin, Jr.

(D) Scour and fill in sand-bed streams, by Bruce R. Colby.

(E) An analysis of some storm-period variables affecting stream sediment transport, by H. P. Guy.

(F) Sediment transport in the Rio Grande, New Mexico, by Carl F. Nordin, Jr., and Joseph P. Beverage.

(G) Effects of water temperature on the discharge of bed material, by B. R. Colby and C. H. Scott.

(H) Bedload equation for ripples and dunes, by D. B. Simons, E. V. Richardson, and C. F. Nordin, Jr.

(I) Summary of alluvial channel data from flume experiments, 1956-61, by H. P. Guy, D. B. Simons, and E. V. Richardson.

Discharge of Sands and 

Mean-Velocity Relationships 

in Sand-Bed Streams

By BRUCE R. COLBY

SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 462-A

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1964
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEW ART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

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

Washington, D.C. 20402 -
CONTENTS

Definitions_ ______________________________________

Symbols ..__ .____.--_-._--_______.____.__..

Abstract_ _ _-____-__-___-_____-_._-____-_-____.__.

Introduction _______________________________________

Concept of sediment transportation ___________________

Complexity of sediment and streamflow relationships__ ______________________________________

Simplifying assumptions.________________________

Bedload discharge ______________________________

Meyer-Peter and Miiller equation___________

Einstein equations __________________________

Bagnold equation.__________________________

Major variables__--____--__-___-_____._____

Suspended-sediment discharge.___________________

Concentration immediately above the bed layer _

Vertical distribution of sediment concentration. _

The classical equation for sediment of one

fall velocity._________________________

Theoretical and observed distributions. 

Equation for flow over dune beds _________

Total sediment discharge..______________________

General methods of computation___________

Fine sediment._____________________________

Page 

v

VI

Al

1

2

2

3

4

4

5

6

7

7

7

8

9

11

13

13

13

	Page 

Effects of major factors. --------------------------- A15

Mean velocity________--_________--____----.__ 15

Natural streams.._____.______-_--____--__-_ 15

Flumes-__________.______-_-__ _-. 18

Shear._________.._.--_--_----_..--_.---_--__ 20

Shear velocity computed from mean velocity.--.--. 24

Stream power._________________________________ 24

Depth..,_________________________ 25

Viscosity. ____-_-___--_-----_--_------_-_-----_ 29

Water temperature_______________________ 29

Concentration of fine sediment_---___._______ 32

Graphs for computing discharge of sands.__--___-_--__ 34

Basic information_____________________________ 34

General method of analysis._____________________ 34

Determining the discharges of sands from the graphs. 37

Probable accuracy of the graphs._________________ 37

Experimental errors in basic information ______ 37

Data included in the study.________________ 38

Stream reaches and flume investigations not

included in the study____________________ 40

Applications. _________--_____--__-___-_--_---__ 43

Deficiencies of basic information._________________ 44

Conclusions._______________________________________ 45

References.._____-_____-_-_-__--_____--______.__--_ 46

ILLUSTRATIONS

Page 

FIGURE 1. Bedload discharges, computed with the Meyer-Peter and Muller equation, as functions of RS and mean velocity._ A5

2. Relationship of $* to ** for individual size ranges--___-___---_________---_-----_-------_-------------- 6

3. zi plotted against zm _-________________---__-______-______________________--_-_---_----------------_- 10

4. z\ plotted against &i.for shallow streams in Nebraska-.-----.-----------.---.---------------------------- 12

5. ziu-jf. plotted against k\ for shallow streams in Nebraska__________-_________----____-.-___-----_-.------ 12

6. Distribution of concentrations of different sediment sizes at a stream vertical_____-__----__-----_--___---- 14

7. Discharge of sands plotted against mean velocity for Niobrara River near Cody, Nebr___-_____---._-_-_____ 16

8. Discharge of sands plotted against mean velocity for Middle Loup River at Dunning, Nebr__________________ 16

9. Curves of relationship between discharge of sands and mean velocity for stations on Pigeon Roost Creek in

Mississippi__ _ __________________________________________________________________________________ 17

10. Discharge of sands plotted against mean velocity for the Rio Grande in New Mexico and for some diversions

from it__________________________________________________________________________________________ 18

11. Discharge of sands plotted against mean velocity for Rio Puerco near Bernardo, N. Mex____________._----__ 19

12. Discharge of sands plotted against mean velocity for Mississippi River at St. Louis, Mo_______--_____------ 19

13. Curves showing relationship between discharge of sands and mean velocity for five sand-bed streams at average

temperatures of about 60° F_______________________________________________________________________ 20

14. Curves showing relationship between discharge of sands and mean velocity for some flume experiments with two

sands---__-_---__________________________-__________________________---____----_____-_____--_--- 21

15. Curves showing relationship between discharges of sands and mean velocity for some flume experiments.-----. 22

16. Discharges of sands plotted against parameters that are related to the force which moves sediment particles.._ 23

17. Effect of depth and water temperature on the relationship between discharge of sands and mean velocity, as.

computed from a somewhat revised Einstein procedure.____________-________-____----_--_----_-------- 26

18. Schematic diagram showing the general reasons for variability of effect of depth on the relationship of bedmaterial discharge to velocity____-_--_-__----_____------_---__-__-----------------_---------------- 27

ill
IV CONTENTS

Page 

FIGURE 19. Empirically determined graph of the effect of depth on the relationship between discharge of sands and mean

velocity.--_..___-._.___.________________________..__.____________________ A28

20. Effect of depth on the relationship between discharge of sands and shear velocity computed from mean velocity ___ 28

21. Effect of depth on the relationship of concentration of sands to mean velocity___-___-_----_-------------__- 29

22. Adjustment for effect of water temperature on the discharge of sands, Niobrara River near Cody, Nebr_______ 30

23. Effect of water temperature on the discharge of sands in four natural streams.-____-_-_-________-________-. 30

24. Approximate effect of water temperature and concentration of fine sediment on the relationship of discharge of

sands to mean velocity.___________________________________________________________________________ 31

25. Comparison of computed and observed discharges of sands for Rio Puerco near Bernardo, N. Mex____________ 33

26. Relationship of discharge of sands to mean velocity for six median sizes of bed sands, four depths of flow, and a

water temperature of 60°F___________________________-______---____--_-_____---___--_____--_-____ 36

27. Comparison of computed and observed discharges of sands for natural streams that were included in the study.. _ 39

28. Comparison of computed and observed discharges of sands for flume data that were included in the study.._____ 41

29. Comparison of computed and observed discharges of sands for stream reaches that were not included in the

study..-_.....___._._....._..._____.__.._____.___.__..________.__._.__.__....-_.___._-__.-_.._._ 42

30. Comparison of computed and observed discharges of sands for flume data that were not included in the study__ 43
DEFINITIONS

Antidunes are roughly symmetrical sand waves that 

are about in phase with similar water waves. This 

definition includes as antidunes the roughly symmetrical sand waves that may exist below relatively 

quiet and symmetrical standing waves on the water 

surface.

Bars are extensive raised areas or ridges, larger than 

dunes in the same flow, on the bed of a stream whose 

bottom shifts readily, at least at high flow.

Bed layer is the layer through which the bedload moves. 

It is usually considered to be only a few grain diameters thick.

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 the sediment of those particle sizes 

that are found in appreciable quantity in the streambed. It is classified according to particle size rather 

than mode of transportation or location in the cross 

section.

Bed sediment is the sediment on the streambed.

Bed sand is bed sediment within the size range of sand.

Concentration is a ratio of the weight of sediment to the 

weight of the water-sediment mixture. In this paper 

it is the weight ratio of the sediment that is transported through a stream cross section or part of a 

cross section to the water-sediment mixture that 

moves through the same area during the same time 

interval.

Dunes are somewhat irregularly spaced mounds of 

loose sand. Their upstream faces are usually gently 

inclined as compared with their much steeper downstream faces.

Fall velocity of a sediment particle is the rate at which 

the suspended particle settles with respect to the 

surrounding fluid.

Fine sediment as the term is used in this paper is the 

sediment that is finer than 0.062 mm.

Froude number is the ratio of the velocity of flow to the 

square root of the product of the gravity constant 

and the depth of flow.

Grain roughness is the boundary roughness that results 

solely from the sizes of the stationary particles

without any effect from the configuration of the bed 

or banks.

Mean velocity is the velocity that is obtained by dividing the flow by the perpendicular area of section 

through which the flow passes. Usually, the mean 

velocity is computed for the whole cross section of a 

stream, but sometimes a mean velocity is computed 

for part of a flow.

Measured discharge of sediment or of sands is the 

discharge of sediment or sands that is usually determined directly from streamflow and depthintegrated sediment samples, which generally are 

obtained by sampling the flow from the water surface 

to within 0.3 or 0.4 foot of the streambed.

Median particle size is the size for which 50 percent 

of the sediment by weight is finer.

Plane bed is a relatively smooth and firm sand bed 

that is free of ripples, dunes, and antidunes and, 

consequently, has low resistance to flow.

Resistance to flow is an inclusive term for the total 

resistance to flow in a channel.

Roughness of bed or banks is a boundary irregularity 

and is usually expressed as a linear measure of the 

effective size of the irregularities of bed or banks.

Sand is sediment particles that have diameters between 

0.062 and 2.0 mm.

Sand-bed stream is a reach of stream whose bed is 

composed almost entirely of cohesionless, shifting 

sand and whose channel is not confined by walls 

of rough rock or other material that would cause 

unusually high turbulence throughout much of the 

flow.

Sediment discharge is the rate at which weight of 

sediment moves through a cross section perpendicular to the direction of flow.

Shear per unit area of the streambed is the product 

of the specific weight of water-sediment mixture, 

the depth of flow, and the energy gradient.

Shear velocity is the square root of the product of 

the gravity constant, the depth of flow, and the 

energy gradient.

Shear velocity with respect to the grains or particles is 

a concept used by Einstein (1950) and is the shear 

velocity that can be computed from the energy 

gradient and a particular hydraulic radius R''.
VI SYMBOLS

The square root of the ratio of E' to the hydraulic 

radius R equals the ratio of the actual mean velocity 

to the mean velocity that would exist if all resistance to flow were due to the stationary particles on 

the bed.

Shear velocity computed from mean velocity is the 

shear velocity that can be computed from a known 

mean velocity with the use of an equation such as 

the one Einstein (1950) derived from work by 

Keulegan (1938).

Size class or size range is a definitely limited range of 

sediment particles that are grouped together because their behavior can be reasonably well represented by some one particle size such as the

geometric mean size, which is the square root of 

the product of the upper and lower limiting sizes.

Suspended sediment (suspended load) or suspended 

sand is sediment or sand that is supported by the 

upward components of turbulent currents or by 

colloidal suspension if the sediment particles are 

very small.

Standard fall velocity is the average rate at which a 

particle would eventually settle if falling alone in 

quiescent distilled water of infinite extent and at a 

temperature of 24° C.

Unmeasured sediment discharge is the difference 

between total sediment discharge and measured 

sediment discharge.

A and B

A,

a 

C 

c

D

D

d

9

it>

KT

SYMBOLS

Dimensionless constants used as 0.047 

and 0.25, respectively, by Meyer-Peter 

and Muller (1948).

Dimensionless constant used by Bagnold 

(1956). It may be about 9.

Dimensionless constant used by Bagnold. 

It varies slowly with particle size.

Distance above the streambed. __

Chezy C, which is equal to u/^RS.

Concentration of sediment. Concentration at a is denoted by ca and at y by c y.

Particle size expressed as a diameter or the 

geometric mean diameter of a size 

range.

Effective diameter of the bed-material 

mixture (the particle size of uniform 

sediment) and equal to SZX6 for bed 

sediment of a range of sizes.

Particle size for which 35 percent of the 

bed material by weight is finer. Subscripts 50, 65, and 90 have comparable 

meanings. Hence, D& is the median 

particle size.

Depth of flow.

The gravity constant.

Fraction by weight of bed sediment in a 

size range.

A measure of resistance to flow and equal 

to the mean velocity u divided by the 

quantity (#2/3S1/2).

A comparable measure of the flow resistance due only to the resistance of the 

stationary grains, and for metric units 

it can be computed from 26/(-D90) 1/6 in

R

R'

which Dgo is the particle size, in meters, 

for which 90 percent of the bed sediment 

by weight is finer.

Constants.

The turbulence constant, which is about 

0.40 for clear-water flow and certain 

types of boundary roughness but may 

vary widely for sediment-laden flow and 

dune roughness.

A linear measure of roughness that may 

be assumed to equal D65 if the bed 

roughness is due wholly to the roughness 

of the stationary grains.

The form of k that is computed from the 

vertical distribution of velocity over a 

dune bed or other rough bed.

Length of channel reach.

The mixing length.

Bedload discharge per foot of width in 

terms of weight of sediment per unit 

time.

Bedload discharge per unit width in terms 

of underwater weight of sediment per 

unit time.

The hydraulic radius, which equals average depth of flow when the bank resistance is negligible.

The hydraulic radius with respect to the 

grains or particles. It is the hydraulic 

radius that could be used to compute 

velocity if all the resistance to flow 

were due to the stationary sediment 

particles.
SYMBOLS VII

S 

u

u*

V

W

w

X

The energy gradient.

Time-averaged velocity at a point. A

subscript may be used to indicate distance of the point above the streambed. 

Mean velocity, which is computed by

dividing the streamflow by the area of

the cross section. 

Shear velocity computed from mean velocity and equal to 0.40w/[2.30 log (12.27

__

Shear velocity and equal to -JgRS. 

Shear velocity with respect to the sediment

particles and equal to -JgR'S. 

Average vertical velocity of the upwardor downward-moving water in turbulent

flow.

Channel width. 

Fall velocity of a sediment particle or

particles. 

A dimensionless parameter to cover the

transition from hydraulically smooth to

hydraulically rough boundaries. 

Distance above the streambed. 

Theoretical exponent for vertical distribution of sediment of one fall velocity or a

narrow range of fall velocity and equal

to w/(0.40iO for uniform flow over a

plane bed. 

The form of z that can be computed from

w/(0.40«m). 

The form of z for flow over dunes or other

rough beds and equal to

log

7*

ep

V

Pf

Ps

Exponential measure of the actual distribution of sediment with depth as determined from a logarithmic plot of concentration for a size range against

(d y)ly.

Logarithm to the base 10.

Logarithm to the base e.

Declination of the bed surface below the 

horizontal; cos /8 generally can be considered to equal unity and be disregarded.

Specific weight of water-sediment mixture.

Specific weight of the fluid.

Specific weight of the sediment particles.

A dimensionless measure of the apparent 

tangential stress due to action of external forces on the fluid and equal to 

\Pfl(p.-Pf)](BS/D).

That value of 6F (determined by extrapolation) at which sediment particles begin 

to move on a rippled sand bed.

Kinematic viscosity.

Density of the fluid.

Density of the sediment particles.

Intensity of bedload transportation for 

uniform bed sediment.

Intensity of bedload transportation for a 

size range of nonuniform bed sediment.

Function for correlating effect of flow with 

intensity of sediment transportation.

Function ^ as adjusted by Einstein (1950, 

p. 37) for an individual size range.

SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

DISCHARGE OF SANDS AND MEAN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS

By BRUCB R. COLBY

ABSTRACT

Theories of sediment movement and general observation 

provide the basis for a practical concept of sediment transportation in sand-bed streams. This concept covers bedload 

discharge, the concentrations of sediment of different size 

ranges immediately above the bed layer, and the vertical 

distribution of suspended sediment of different size ranges 

within the flow. According to the concept, the bedload discharge per unit width depends mostly on the velocity near 

the streambed, the fall velocities of the bed sediment, and 

the depth of flow. Immediately above the bed layer, the 

concentrations of sands in the different size ranges are 

determined mainly by the bedload discharge of the different 

size ranges, the depth of the bed layer, the velocity of flow 

through the bed layer, and the probable continuity of the 

concentrations from the bed layer to the suspended sediment immediately above the bed layer. The concentrations 

of suspended sands within the different size ranges at any 

level in the flow depend on the concentrations immediately 

above the bed layer and the rate at which the concentrations 

decrease with distance above the bed layer. The rate of 

decrease of sediment concentration in each size range can 

be computed, but only roughly, from the ratio of the fall 

velocity to the shear velocity.

An analysis of the relationships of discharge of sands per foot 

of width to the controlling factors that were indicated or suggested by the general concept of sediment transportation showed 

approximate major effects of several variables as follows:

1. Mean velocity or, as an alternative factor, shear velocity 

computed from mean velocity has a dominant effect on the 

discharge of sands in most sand-bed streams. At a cross 

section of a sand-bed stream a reasonably close relationship usually exists between discharge of sands and mean 

velocity unless water temperature changes greatly or unless 

the concentration of fine sediment is high and variable. 

The curve of relationship for one sand-bed stream may 

approximately apply for some other sand-bed streams.

2. Shear and stream power each has a dominant effect, somewhat similar to that of mean velocity, on the discharge of 

sands except for the uncertainty of the relationship when 

the resistance to flow is variable.

3. For moderate to high velocities and shallow flows, discharge 

of sands per foot of width varies somewhat but not greatly 

for changes in median diameter of the bed material from 

0.30 to 0.80 mm. However, the effect of particle size on 

the discharge of sands may be large at low velocities in any 

depth of flow or at high velocities in deep flows.

4. Depth sometimes has a large effect on the relationship between discharge of sands per foot of width and any one of 

the four measures of the force or the energy that causes 

sediment movement, and the effect varies complexly with 

changes in velocity and particle size.

5. If other factors remain constant, an increase in viscosity as 

a result of a decrease in water temperature causes an increase in discharge of sands because the change in viscosity 

causes a decrease in the fall velocity of the sand grains. 

In much the same way, a high concentration of fine sediment may increase the apparent viscosity of the water and 

thus decrease the fall velocities of the sand grains. 

Graphs based on the empirical relationships of discharge 

of sands to major variables were prepared for a wide range 

of velocity, depth, water temperature, and concentration of 

fine sediment; they provide a method for quickly approximating the discharge of sands in sand-bed streams. In spite 

of many inaccuracies in the available data and uncertainties 

in the graphs, about 75 percent of the sand discharges that 

were used to define the relationships were less than twice 

or more than half of the discharges that were computed 

from the graphs of average relationship.

The agreement of computed and observed discharges of 

sands for sediment stations whose records were not used to 

define the graphs seemed to be about as good as that for 

stations whose records were used.

INTRODUCTION

Two practical objectives of sedimentation research are to understand the effects of major factors 

on sediment discharge in streams and to develop 

methods for computing the sediment discharge. An 

understanding of the effects of the major factors is 

essential to evaluate correctly the somewhat incomplete and inexact information available on sediment 

discharge and on the behavior of sediment-laden 

streams. Methods for computing sediment discharge 

are needed in studies and computations relating to 

channel design and maintenance, sediment yields 

from drainage basins, rates of scour or fill in natural 

or artificial channels, and depletion of reservoir 

storage by sediment accretion.

Because sediment transportation varies with 

complex characteristics of sediment and flow, an

Al

727-985 O - 64 - 2
A2 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

accurate and always applicable method for determining sediment discharge from characteristics of 

drainage basins, transported sediment, and streamflow may never be developed. If such a method is 

ever devised for all types of sediment, drainage 

areas, channels, and flows, it is likely to be unmanageably complex and generally impracticable 

because of the large amount of field information 

and the complicated office procedures that would be 

necessary. However, the relating of sediment discharge to characteristics of streamflow is much 

simplified (although it still remains difficult) if the 

sediment discharge is computed for only those 

particle sizes generally present in the streambed in 

appreciable quantities and if only streams with beds 

of sand are considered.

The main purpose of this paper is to indicate 

in as simple, accurate, and practical a form as possible 

the major relationships between the discharge of sands 

in sand-bed streams and the characteristics of flow 

and sediment. Secondary purposes are to provide a 

rapid method for computing the discharge of sands 

in sand-bed streams and to show deficiencies in the 

information that is necessary to define more accurately 

the relationships between discharge of sands and 

characteristics of flow and sediment.

Because the relationships are complex and are 

inexactly defined, they have been expressed in graphs 

rather than in equations. The graphs are based on 

laboratory and field information and on a general concept of the theories and equations of sediment transportation. They express the discharge of sands for 

flows as shallow as 0.1 foot and as deep as 100 feet, 

although for the deepest flows the graphs are poorly 

defined extrapolations. Bed-material sizes on the 

graphs are shown only as median diameters and range 

from 0.1 to 0.8 mm. The relationships for the largest 

and the smallest sizes are based to a considerable 

degree on extrapolation.

First, a concept of sediment transportation is given 

on the basis of recognized theories and personal 

observations of water and sediment discharge. It 

provides a background for understanding the major 

factors that can be expected to affect the discharge of 

sands in sand-bed streams. Then, the quantitative 

effects of the major factors as empirically defined by 

available information are individually shown and 

discussed. Finally, graphs are given for computing 

the approximate discharge of sands in sand-bed 

streams; and the probable accuracy, limitations, and 

possible improvements of the graphs and relationships 

are suggested.

Studies on which this paper is based were made 

over a period of several years; some were made while 

the writer was working for the Agricultural Kesearch 

Service, U.S. Department of Agriculture.

Unpublished data for Pigeon Boost Creek in northern Mississippi and its tributaries were used in some 

of the studies and were from the files of the Agricultural Eesearch Service, University, Miss. Much unpublished information collected by the Geological Survey 

was made available by: D. M. Culbertson, district 

engineer, for streams in Nebraska and for the Mississippi River at St. Louis, Mo.; J. M. Stow, district 

chemist, for streams in New Mexico and for the 

Colorado River; D. B. Simons, project chief, for 

flume investigations at Fort Collins, Colo.; and T. F. 

Hanly, district engineer, for streams in the Bighorn 

River basin in Wyoming.

CONCEPT OF SEDIMENT TRANSPORTATION

An understanding of the causes of sediment movement by streamflow and the general pattern of the 

movement is necessary for developing sound relationships for computing sediment discharge. Such a concept cannot be precise, however, partly because our 

knowledge is not complete and partly because our 

minds cannot wholly grasp the complexities of all the 

complete and detailed relationships involved in sediment transportation. Fortunately, a concept of sediment movement in flowing water may be adequate for 

many studies if it includes approximately correct 

interrelationships of major factors and characteristics 

of the flow, the sediment particles, and the sediment 

discharge. An attempt will be made to state a useful 

concept in terms of theories, equations, and observations of streamflow and sediment movement.

The statement of this concept is divided into several 

parts. The complexity of sediment transportation is 

explained briefly; then, the assumptions made to simplify and to limit the scope of the studies and the 

relationships for this paper are stated. Next, each 

of the two general modes of sediment discharge 

bedload discharge and suspended-sediment discharge 

are discussed separately; and finally, the total discharge 

of sediment at a cross section is considered.

COMPLEXITY OF SEDIMENT AND STREAMFLOW 

RELATIONSHIPS

The relationships of sediment discharge to characteristics of sediment, drainage basin, and streamflow 

are complex because of the large number of variables 

involved, the problems of expressing some variables 

simply, and the complicated relationships among the
DISCHARGE OF SANDS AND ME AN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS A3

variables. At a cross section of a stream, the sediment 

discharge may be considered to depend: on depth, 

width, velocity, energy gradient, temperature, and 

turbulence of the flowing water; on size, density, shape, 

and cohesiveness of particles in the banks and bed at 

the cross section and in upstream channels; and on 

the geology, meteorology, topography, soils, subsoils, 

and vegetal cover of the drainage area. Obviously, 

simple and satisfactory mathematical expressions for 

such factors as turbulence, size and shape of the 

sediment particles in the streambed, topography of the 

drainage basin, and rate, amount, and distribution of 

precipitation are very difficult, if not impossible, to 

obtain.

The concentration of the moving sediment is generally nonuniform both laterally and vertically in 

the cross section, and each sediment size (more 

precisely, sediment of each fall velocity) will have 

a vertical and lateral distribution that differs from 

the distribution of sediment of any other size. 

Similarly, the depth, velocity, turbulence, shear, and 

other characteristics of flow vary, sometimes widely, 

across a stream. Thus, a single value such as an 

average depth, a mean velocity, or a median particle 

size may be a somewhat unsatisfactory measure of 

depth, velocity, or particle size for a cross section. 

Furthermore, the lateral and vertical distributions of 

sediment and flow depend partly on nearby cross 

sections and on channel alinement. The flow may be 

over a relatively firm and smooth bed or over bars, 

dunes, or antidunes; and the bed configuration may 

vary laterally, longitudinally, and with time. In addition, sediment particles interfere with each other at 

high concentrations, and the chemical composition of 

the water may affect the degree of flocculation of 

fine sediment. Because the flow pulsates and the coarse 

sediment tends to discharge erratically both in time 

and in lateral position in the flow, observations of 

velocity, discharge of coarse sediment, and slopes of 

the water surface must be made carefully over a considerable period of time to obtain satisfactory averages. 

An adequate knowledge of these difficulties indicates 

that (a) sediment discharge cannot, in general, be 

precisely computed from characteristics of the flow, 

the sediment, and the drainage basin and (b) some 

assumptions are necessary to reduce the problem of 

sediment transportation to manageable proportions.

SIMPLIFYING ASSUMPTIONS

The assumptions made for this study should be 

well understood because they limit significantly the 

scope and applicability of the discussions and results.

The derived relationships should only be applied to 

compute discharge of sands for streams and cross 

sections that have characteristics reasonably consistent 

with the assumptions.

When one cross section is compared with others, 

sediment discharge is assumed to be proportional 

to width if other major variables are constant from 

section to section. Thus, sediment discharge is computed and discussed in terms of discharge per unit 

width. The assumption does not imply a uniform 

distribution of sediment discharge across a section. 

It implies only that the sediment discharge through 

two sections should be about proportional to width 

if the sections have the same mean velocity, average 

depth, bed sediment, water temperature, and concentration of fine sediment.

Mean velocity, average depth, and average shear 

at a cross section are assumed to be acceptable measures of the actual nonuniform velocities, depths, and 

shears across the section. Obviously, this assumption 

will not always be satisfactory. Some irregular cross 

sections should be subdivided into parts, and 

separate computations of sediment discharge should 

be made for each part. However, the error in using 

a mean velocity for an entire nonuniform cross section may be partly compensated, or even overcompensated, for by the assumption that sediment discharge is proportional to stream width.

For example, if 30 feet of the width of a 100-foot 

section is shallow and has a very low velocity, almost 

the entire discharge of sands may be within the other 

70 feet of width. The mean velocity may be only a few 

percent lower for the whole section than for the 70- 

foot width (mean velocity is weighted with width 

and depth, not with width alone). Hence, even though 

the discharge of sands may vary as the third or fourth 

power of the mean velocity, the computed decrease in 

discharge of sands per foot of width for the 100-foot 

width as compared with the 70-foot width will be 

partly compensated, or even overcompensated, for 

when multiplication is by 100 feet rather than by 

70 feet.

Cross sections for determining sediment discharges 

are assumed to be in reasonably straight reaches of 

channel.

The bed sediment is considered to be adequately 

represented by the median particle size, although for 

certain somewhat extreme conditions of flow, such as 

the beginning of sediment movement, low discharges 

of sands, or deep flows, the assumption is not generally 

satisfactory.

The discharge of fine sediment is assumed to be 

more closely related to supply of fine sediment in the
A4 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

drainage basin than to conditions of flow at a cross 

section, and hence no attempt is made to compute the 

discharge of fine sediment from characteristics of the 

flow at a cross section. (However, some attention is 

given to the effect of high concentrations of clay on 

the discharge of sands.) This assumption is in line 

with findings of Einstein and Chien (1953) that the 

discharge of fine sediment theoretically follows the 

same laws as the discharge of sands but that limitations on sampling the fine bed sediment and on applying the laws make the computation of the discharge 

of fine sediment impracticable and unreliable. Theoretically, the dividing size between sediment too fine 

to have its discharge related satisfactorily to characteristics of flow at a cross section and sediment whose 

discharge can be so related should vary with mean 

velocity and other factors. For this paper the dividing 

size, which is arbitrary because no sharp division 

exists, has been selected as 0.062 mm., the lower limit 

of sand sizes.

Only stream reaches whose beds are composed almost 

entirely of nearly cohesionless sand and whose channels are not confined between rough rock walls are 

considered. Such walls may cause unnaturally high 

turbulence for a given mean velocity and depth.

BEDLOAD DISCHARGE

When water flows very slowly over a bed of sand, 

none of the sand grains may move. If velocity near 

the bed is slowly increased, a critical velocity will be 

reached at which some sand grains occasionally move 

along the bed for short distances and then stop. 

Obviously, this critical velocity is somewhat indefinite 

because initial movement of the grains depends on the 

arrangement of the grains and on local variations of 

velocity. Hence, Einstein (1950) related the picking 

up of the grains from the bed and the beginning of 

movement to the probability of individual grains to 

move. This reasonable concept of Einstein indicates 

that any critical velocity can only be an inexact and 

partly arbitrary measure of the beginning of significant bedload movement.

If the velocity near the bed is greater than critical, 

sand grains move intermittently by rolling, sliding, 

or skipping along the bed. The movement is within 

a very thin layer, called the bed layer, a few grain 

diameters thick. (The bed-layer concept is, of course, 

inexact, especially for a dune bed and a wide range of 

grain sizes in the bed sediment.) The grains which 

thus move in the bed layer and which are supported 

mainly by contact with the streambed compose the 

bedload. In general, the rate of travel of these grains 

while in motion and the frequency with which the

grains begin intermittent movement depend on the 

velocity of flow near the bed. Obviously, the particle 

sizes of bed sediment and the difference in density 

between the sediment and the water are also significant 

factors that affect bedload discharge. The viscosity of 

the fluid probably also has an appreciable effect. There 

are three generally known procedures by different 

investigators for computing bedload discharge.

MBYER-PBTER AND MtTLLER EQUATION

A reasonably simple and evidently fairly accurate 

formula for bedload discharge was developed by MeyerPeter and Muller (1948). For channels having negligible bank friction, the formula may be written 

yf(Kr/K/r*RS=A(vs -yf)Dm +B(yf/9y>*(qB ')2 '3 , (D 

in which

yf and ys are the specific weights, respectively, of

the fluid and the sediment particles; 

Kr is a measure of resistance to flow and equals the

mean velocity u divided by (#2/3£1/2 ) ; 

Kr is a comparable measure of the flow resistance 

due only to the resistance of the stationary grains 

and for metric units can be computed from 26/ 

(D90 ) 1/6 . DM is the particle size, in meters, for 

which 90 percent of the bed sediment by weight 

is finer;

R is the hydraulic radius, which equals average 

depth of flow when the bank resistance is 

negligible;

S is the energy gradient; 

A and B are dimensionless constants that may be

used as 0.047 and 0.25, respectively; 

Dm is the particle size of uniform sediment and 

equals 2Z%6 for bed sediment of a range of sizes; 

it, is the fraction by weight of sediment of a size 

D in the streambed ; 

g is the gravity constant; and 

qB ' is the bedload discharge per unit width by

weight under water.

Equation 1 for bedload discharge, in foot-poundsecond units, when 7^ is equal to 165 pounds per cubic 

foot, can be written as

For convenience, the bedload discharge per foot of 

width can be expressed in terms of the dry weight per 

foot of width (qB ) through the relationship

and equation 1 becomes

(2)
DISCHARGE OF SANDS AND MEAN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS A5

According to equation 2, no bedload moves until the 

left-hand side exceeds O.Q77Dm. For high velocities 

and small particle sizes, Q.Q77Dm becomes relatively 

small. If it is disregarded and (KT/K/) is constant, 

the computed bedload discharge varies as (RS)3/2 or as 

the third power of the mean velocity if the Chezy C is 

constant.

The ratio Kr/K/ in equation 2 equals the ratio of 

the actual mean velocity to the mean velocity if all 

resistance to flow were due to the bed roughness that 

the grains would cause if stationary on a plane streambed. (The equation is based on assumed negligible 

bank friction.) A convenient way to evaluate Kr/K/ 

in foot-pound-second units is to compute the ratio of 

the actual velocity to the velocity that can be determined from an equation of the type given by Keulegan 

(1938) and from a roughness equal to the Z)65 size of 

the bed material as suggested by Einstein (1950). In 

whatever way the ratio is computed, KT/K/ is an adjustment that makes the left side of equation 2 vary 

much more closely with mean velocity than with RS 

or with shear with respect to the particles rather than 

with yRS.

The close relationship between bedload discharges 

computed from the Meyer-Peter and Miiller equation 

and mean velocity can readily be shown by an example 

for a cross section of Pigeon Roost Creek near Byhalia, 

Miss. Within a narrow range of depths and shears, 

two considerably different flows may occur at a particular depth at this cross section because of differences 

in configuration of the streambed. The bed sediment 

at the cross section is cohesionless sand that has a 

median diameter of 0.40 mm. Dm in equation 2 is 

0.45 mm, and Dgo is 0.66 mm. Bedload discharges 

were computed from equation 2 for five velocities and 

depths of flow; the three lowest velocities were over 

dune beds, and the two highest were over plane beds 

or a combination of plane bed and antidune bed. The 

computed bedload discharges increase rapidly and 

consistently as mean velocity increases, but a break 

exists in their relationship to RS. (See fig. 1.) Evidently, RS may be a poor measure of the bedload 

discharge unless RS is adjusted by some factor, such 

as KJK/, that corrects for the effect of changes in 

resistance to flow that are due to changes in bed 

roughness.

EINSTEIN EQUATIONS

Einstein (1950) suggested a comprehensive procedure for the computation of bed-material discharge 

from the characteristics of flow and bed sediment. 

In this procedure, the part of the bed-material discharge 

that is bedload was computed on the basis of the

o.oi i 10

RS, IN FEET MEAN VELOCITY, IN FEET

PER SECOND

FIGURE 1. Bedload discharges, computed with the Meyer-Peter and Miiller 

equation, as functions of RS and mean velocity.

probability of movement of particles at the bed surface 

as related to intensity of flow. Einstein's bedload 

equations for uniform sediment can be written

psg \PS 

in which pf and ps are densities of the fluid and the 

sediment particles, respectively. Also,

P»-P D

pf

R'S

and

*=/(*).

For constant densities and for one consistent system 

of units of measurement these three equations 

Einstein's become

of

(3)

and KI and K2 are constants. R' is the hydraulic 

radius that could be used to compute velocity if all 

the resistance to flow were due to the roughness that 

might be caused by the sediment particles if stationary 

on the bed.
A6 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

Einstein used R' in equations based on work by 

Keulegan (1938), but R' can be used equally well in a 

Manning equation; that is, R' can be stated in terms 

of R, the actual hydraulic radius (or average depth of 

flow for negligible bank friction), as

hence

=K/(R')2'3 S1/2 ',

R'=R(Kr/K/y'*.

Einstein's equation for qB then may be written, in 

which Kz is a constant, as

D

Thus, Einstein's parameter for the effect of flow on 

bedload discharge of uniform particle size is like that 

used in the Meyer-Peter and Miiller equation (see eq. 

2). Einstein's graph of the function (fig. 2) sjhows, of 

course, that $* and, hence, qB increase as D/[(Kr/K/)3/2 

RS] decreases.

Einstein stressed the need for the adjustment from 

R to R', an adjustment that ties the bedload discharge 

more closely to mean velocity than to RS. However, 

he computed the relationship between R and R' through 

the use of a curve (Einstein, 1950, fig. 5) that may be 

a good average but sometimes does not apply well 

(Brooks, 1958, p. 588-589).

If the bed sediment is not uniform in size, the Einstein procedure requires considerable adjustment of $ 

and *. The adjusted quantities for individual size 

classes or size fractions of bed sediment are <J>* and **. 

Einstein's graph of <J>* as a function of ** (fig. 2) shows 

a rapid decrease of bedload discharge when the velocity, 

as measured by R'S, becomes low in relation to the 

particle size; that is, when V* becomes large. The

100

10

o.i

o.ooi o.oi o.i 10 100

bedload discharge of a size fraction is directly proportional to the percentage by weight of that size fraction 

in the bed sediment. In other words, if the percentage 

of bed sediment in a size range is doubled but $* remains constant, the bedload discharge of that size 

range is also doubled. The statement of proportionality does not mean that the bedload discharges for 

different size fractions are in direct proportion to the 

relative percentages of the size fraction in the streambed.

BAGNOL.D EQUATION

Using a different approach from that of Einstein, 

Bagnold (1956) derived a formula for computing 

bedload discharge similar to Einstein's. For bed sediment of uniform size, his equation (Bagnold, 1956, 

eq 42c) is

(4)

in which

FIGURE 2. Relationship of ** to ** for individual size ranges. From Einstein 

(1950, fig. 11).

AI is a dimensionless constant that may be

about 9, 

BI is a dimensionless constant that varies slowly

with particle size, 

KI is the same constant that was used in

equation 3, 

0F is a dimensionless measure of the apparent

tangential stress due to action of external

forces on the fluid and equals [pf/(ps pf)](RS/D), 

8t is that value of 6F (determined by extrapolation)

at which sediment particles begin to move on

a rippled sand bed, and 

/3 is declination of the bed surface below the

horizontal, and cos /3 generally can be considered

to equal unity and be disregarded.

Except for the use of R instead of Rf , Bagnold's 

6f is the reciprocal of Einstein's ¥. Of course, the 

bedload discharge increases as 6F increases or as * 

decreases. Also, for a particular grain size, for constant 

densities, and for moderately high velocities for which 

6t becomes relatively insignificant the bedload discharge according to equation 4 is about proportional 

to (RS)m. For these stated conditions, the bedload 

discharge is also about proportional to the third power 

of the mean velocity if the Chezy C is constant.

A major difference between the Bagnold equation 

for computing bedload discharges and the equations 

of Einstein or Meyer-Peter and Muller is that Bagnold 

made no direct provision for the effect of changing 

resistance caused by changing bed configuration (if 

6t is intended to be such an adjustment, it is inadequate 

for most natural streams); that is, Bagnold used
DISCHARGE OF SANDS AND MEAN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS A7

neither an R' instead of R nor a KT/KT ' ratio. If RS 

is as inadequate a measure of bedload discharge as 

figure 1 indicates that it may sometimes be, Bagnold's 

equation may generally not be as applicable as the 

equations of Einstein and Meyer-Peter and Miiller.

MAJOR VARIABLES

The general concept of bedload discharge and certain mathematical statements of the concept indicate 

that no sediment may move as bedload at very low 

velocities. At relatively high velocities, the bedload 

discharge may vary roughly with the third 

power of the mean velocity. Hence, mean velocity 

or more exactly, velocity near the streambed must 

be a highly significant variable in the computation of 

bedload discharge. Shear (or /?$) is an alternate 

major variable and might be used rather than velocity, 

but an adjustment of the shear from total shear to 

shear with respect to the particles evidently is required 

for changing bed roughness. If mean velocity is used 

as a parameter of bedload discharge, depth can be 

expected to affect bedload discharge because for a 

given velocity near the streambed the mean velocity 

usually increases as the depth of flow increases.

The emphasis here placed on mean velocity as a 

measure of bedload discharge is wholly consistent 

with the usual concept that shear with respect to the 

particles is a major parameter of bedload discharge. 

Unfortunately, shear with respect to the particles can 

at present be obtained only indirectly and often inexactly except for flow over plane beds. Usually it can 

be computed as accurately from mean velocity and 

depth as in any other way. Thus, mean velocity is a 

practical and convenient measure of bedload discharge 

to substitute for shear with respect to the particles.

Another significant variable is the bed sediment. 

The size of the bed sediment, as measured by the 

median diameter and the size distribution, has three 

noteworthy effects on bedload discharge. One effect 

is that particle size largely determines the velocity 

that is necessary near the bed to start appreciable 

sediment movement. Particle size also partly determines resistance to flow because it directly determines grain roughness and affects bed configuration. 

A third effect is that particles of different sizes discharge at different rates even though the velocity 

near the bed is constant.

The properties of the fluid may also affect the 

discharge of sands as bedload. The density of the 

fluid is included in each of the three bedload equations that were discussed, but the practical effect of 

changes in the different densities of fluid and sediment is usually small because these densities are

generally about constant in sand-bed streams. Changes 

in viscosity of the fluid or of the fluid-sediment mixture 

seem to have an effect on the relationship of shear 

to mean velocity because these changes may modify 

the bed configuration.

Thus for sand-bed streams, the discharge per foot 

of width of sands as bedload can be expected to vary 

mainly with mean velocity, depth of flow, median 

particle size or median fall velocity of bed sediment, 

and size distribution of the bed sediment. If the 

velocity near the bed could be satisfactorily determined, perhaps the only other major parameter that 

would usually have to be considered in determining 

bedload discharge in sand-bed streams would be the 

sizes of the bed sediment.

SUSPENDED-SEDIMENT DISCHARGE

Although in sand-bed streams much more sand is 

generally discharged as suspended sediment than as 

bedload, bedload discharge was discussed first because 

it forms a base to which the suspended sediment 

adjusts itself. The most direct relationship of suspended sediment to bedload results from continuity 

of the concentrations of suspended sediment of each 

size range at the upper surface of the bed layer; 

that is, the concentrations of suspended sediment 

immediately above the bed layer depend on the 

concentrations within the bed layer. Suspendedsediment concentrations at higher levels in the flow 

depend, in turn, on the concentrations immediately 

above the bed layer and on the vertical attenuation of 

the concentrations of sediment of the different sizes 

from the bed layer upward.

CONCENTRATION IMMEDIATELY ABOVE THE BED LAYER

If the velocity near the streambed is great enough, 

some sand grains, perhaps usually the lightest or the 

fastest moving ones, escape from the bed layer by 

hydraulic lift, aerodynamic lift (E. E. Glover, written 

communication, 1960), or other mechanical or hydraulic 

action. Grains that rise above the bed layer may be 

supported by the turbulence of the flow and thus 

become suspended sediment rather than bedload. Of 

course, if the flow contains suspended sediment, some 

particles of the suspended sediment are continually 

settling into the bed layer. The ready interchange 

of particles from bedload to suspended load or of 

suspended load to bedload means that the suspendedsediment concentration just above the bed layer adjusts 

itself to changes in discharge of bedload.

Einstein (1950) used the assumption that the concentration of each particle size of suspended sediment 

at the upper surface of its own 2-grain-diameter bed
A8 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

layer was equal to the concentration of that particle 

size in the bed layer. This assumption may be only 

approximate, and the idea of separate thicknesses of 

bed layer for each particle size may be somewhat 

unrealistic. However, changes in the concentration of 

suspended sediment of each size class at the upper 

surface of the bed layer should be roughly proportional 

to changes in concentration of particles of that size 

class within the bed layer. If the thickness of the bed 

layer and the average velocity in the bed layer remain 

constant, changes in the concentration of sediment of 

a given size class in the bed layer will be proportional 

to changes in bedload discharge of that size class of 

sediment. Hence, the same variables that determine 

the bedload discharge will mainly determine the 

suspended-sediment concentrations at the upper surface 

of the bed layer and immediately above it. Thus, 

the major variables that affect concentration of suspended sediment immediately above the bed layer 

are mean velocity, depth of flow, and particle sizes 

or fall velocities of the bed material.

VERTICAL, DISTRIBUTION OF SEDIMENT CONCENTRATION

A detailed discussion of the concepts that relate 

to the vertical distribution of suspended sediment 

seems to be necessary for two reasons. The first 

reason is that much more sand is transported in 

most streams as suspended sediment than as bedload. 

The second reason is that the factors affecting the 

vertical distribution of sediment are incompletely 

defined and their relationships to the discharge of 

sands are complex. Available experimental data are 

generally inadequate to determine the factors and their 

relationships except, perhaps, for narrow ranges of 

the factors. Therefore, empirical analysis of the data 

should be based on as adequate and practical an 

understanding of the basic concepts as can be obtained.

By definition, suspended sediment is supported 

by the vertical currents in turbulent flow. These 

currents exist because of vertical eddies or random 

vertical movements of some parts of the flow relative to other parts of the flow. The amount of the 

vertical movement in either direction is largely a 

function of the time-averaged horizontal velocity, 

although roughness of the channel boundary and 

other factors have considerable effect. Hence, the 

mean velocity of a stream is a rough measure of 

the turbulent currents that suspend sediment. If the 

water surface and the streambed are plane and parallel, any upward movment of part of the flow must be 

balanced by a downward movement of another part 

of the flow. If the sediment particles were uniformly

distributed in the turbulent flow and did not settle 

with respect to it, the rate of upward sediment movement, like the rate of upward water movement, would 

on the average equal the rate of downward sediment 

movement through any area of plane parallel to the 

water surface.

Because sediment particles settle with respect to 

the surrounding water, vertical equilibrium requires 

that the number of sediment particles raised by the 

upward components of flow must exceed the number 

lowered by the downward components of flow. Thus, 

equilibrium may be reached if the upward movement 

of water starts at a lower altitude where the concentration is comparatively high and if the downward 

movement of water starts from a higher altitude where 

the concentration is comparatively low. The sediment 

in both upward and downward components of the flow 

is assumed to move an appreciable distance before mixing with other water. If this vertical distance of 

movement before mixing is the same for all particle 

sizes, the ratio of the number of particles that starts 

moving upward to the number that starts moving 

downward is directly related to the fall velocity; that 

is, in a particular flow the fall velocity is a measure 

of the rate of change of concentration with distance 

above the streambed. For a constant fall velocity, an 

increase in turbulence means an increase in the vertical 

movements of the flow and results in a lower rate of 

change of concentration with distance above the 

streambed. Thus, the major parameters that determine vertical variation of concentration of sediment 

in a flow are fall velocity and turbulence. Mean 

velocity is probably the best available measure of 

turbulence, although relative roughness of the channel 

boundary is likely to affect the relationship between 

mean velocity and turbulence.

THE CLASSICAL EQUATION FOR SEDIMENT OF ONE FALL VELOCITY

The theoretical equation for the vertical distribution 

of concentration of suspended sediment of a particular 

fall velocity was given by Rouse (1937) and was 

restated by Einstein (1950, p. 17) in the familiar form

Cyfd y a V 

ca \ y d aj

and

_ ~

w

(5)

(6)

in which

cy is the concentration at a distance y above the

streambed, 

c0 is the concentration at a distance a above the

streambed,
DISCHARGE OF SANDS AND MEAN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS A9

d is the depth of flow,

w is the fall velocity of the particles,

k is the turbulence constant, and __

u* is the shear velocity and is equal to -\gRS

The exponent z is the theoretical slope of a log-log 

graph of concentration of sediment of one fall velocity 

against (d y}/y and increases as the rate of change 

of the concentration with depth increases. When z 

is zero, the vertical distribution of concentration is 

uniform.

Equation 5, in which z is defined as in equation 6, is 

the classical equation for the vertical distribution of 

suspended sediment. Theoretically, it can be used to 

compute the vertical distribution of concentration of 

suspended sediment of any particular fall velocity in 

turbulent flow over a plane bed that is parallel to the 

water surface if the suspended sediment concentration 

is known at any one distance above the streambed.

THEORETICAL AND OBSERVED DISTRIBUTIONS

Observed vertical distributions of sediment concentration in a natural stream (Anderson, 1942) and in a 

laboratory flume (Vanoni, 1946) checked equation 5 

to the extent that the log-log graphs of concentration 

against (d y)/y defined approximately straight lines. 

However, the slope z\ determined from a line was 

usually different from the theoretical slope z computed 

from equation 6. Vanoni (1946) and Ismail (1952) 

considered that the coefficient of sediment transfer 

is not the same as the coefficient of momentum transfer. 

Einstein and Chien (1954) suggested other changes 

in the relatively simple assumptions on which equations 

5 and 6 are based in order to make z and z\ agree better. 

However, they apparently restricted their explanations 

to flows over plane beds, whereas equation 6 is likely 

to be less accurate for flows over dune beds than for 

flows over plane beds. Their suggestions merit careful 

study but are generally too complicated and uncertain 

for use at the present time.

Because the vertical distribution of sediment concentration is an important phase of sediment transportation particularly in deep flows, equation 6 is here 

examined to show some possible uncertainties and 

inaccuracies th^___ -involved in its use. A few of 

the uncertainties and inaccuracies result directly from 

the assumptions on which the equation is based. 

Each of the variables w, k, and u^ is considered 

individually.

The variation of Zi with fall velocity can be defined 

experimentally from the concentrations and size 

analyses of sediment samples that are collected at 

several points on a vertical during a short period of

time when the turbulent flow over a sand bed is reasonably constant. A value of z\ for each size range at 

a vertical can readily be determined by plotting. 

Because ku% is the same for all size ranges, the values 

of z\ for the different size ranges should vary in proportion to the fall velocities of representative particles 

from each size range, if the numerator of the right 

side of equation 6 is correct. However, the values of 

Zi thus determined for sands in the Enoree River 

(Anderson, 1942) did not increase as rapidly as the 

values of z and were generally smaller for the coarser 

particles than the corresponding values of z computed 

from equation 6. Similarly, Colby and Hembree 

(1955) found that the values of z\ for different ranges 

of sand sizes in several natural streams varied roughly 

with about the 0.7 power of the fall velocity as computed from equations given by Rubey (1933). The 

relationship of Zi to fall velocity was based on the 

slopes of lines that might be drawn through points on 

figure 3 for a given time and cross section. The first 

power of the fall velocity from the Rubey equations 

evidently is an inadequate measure, at least for sands, 

of the change in z^ with change in particle size within 

a given flow. The vertical distributions of concentration of the coarse sizes of sediment are more uniform 

than equations 5 and 6 indicate that they should be.

The first power of the mean velocity may be an 

inadequate measure of the change of Zi with particle 

size for any one of several reasons or for a combination 

of reasons. These reasons include the possible effects 

of using an average vertical velocity rather than a 

range of vertical velocities and of disregarding higher 

order terms for the concentration gradients of coarse 

sediments (Einstein and Chien, 1954). Other possible 

reasons are mass movement of water and sediment 

from the bed upward, inapplicability of a laboratory 

determination of fall velocity of a representative 

particle of a size range as a measure of the effective fall 

velocity in the turbulent flow of a natural stream, and 

inadequacy of the assumptions regarding the mixing 

length.

On figure 3 the wide vertical scatter, which is relatively consistent from one size range to another, indicates that 0.40wm does not satisfactorily show the effect 

that the characteristics of the flow have on z^ at least 

for these shallow flows that were mostly over dune beds. 

Colby and Hembree (1955, p. 75) suggested that large 

variations in k might explain much of this vertical 

scatter. Hence, the significance and variability of k, 

especially for flow over dune beds, may help to explain 

differences between z and z\.

727-985 O - 64 - 3
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DISCHARGE OF SANDS AND MEAN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS All

EQUATION FOB FLOW OVER DUNE BEDS

The equations of the type given by Keulegan (1938) 

for mean velocity u and for time-averaged velocity at a 

point uv may be written, respectively,

«/2.30=^[1.09+log (xdlk.)] (7)

(8)

in which

a; is a dimensionless parameter to cover the transition from hydraulically smooth to hydraulically rough boundaries, and

ks is a linear measure of roughness and may be taken 

equal to Z?e6 (the size for which 65 percent of the 

bed material by weight is finer) if the roughness 

is due wholly to grain roughness.

Usually, kg is large and is not known for flow over dune 

beds, and therefore these equations cannot then be used 

to compute velocities directly. Equation 8, however, 

shows that the difference in point velocity at distances 

Wy and y above the streambed is 2.3Qu*/k, which is the 

same as the difference for flow over a plane bed provided 

that equations 7 and 8 apply; that is, the vertical 

distribution of velocity depends on the total shear over 

a dune bed the same as it does over a plane bed, if 

equation 8 is applicable.

Three types of information indicate that equations (7) 

and (8) do not necessarily apply to shallow flows over 

dune beds. The first type shows that the verticalvelocity distribution changes (and hence k changes) as 

the measuring vertical is shifted with respect to a dune. 

For example, in the Middle Loup River at Dunning, 

Nebr., two values of k computed from vertical distributions of velocity at the crest of a dune were 0.32 and 

0.62 as compared to k values of 0.11 and 0.19 for verticals 

in a trough downstream from the crest. A second type 

of information shows that some k's as computed from 

equation (8) and from observed vertical distributions of 

velocity are much higher than 0.40 for shallow flows 

over dune beds (Hubbell and Matejka, 1959, p. 71), 

whereas, in general, k values for flows that transport 

bed material in appreciable quantity probably should 

be somewhat lower than 0.40 (Einstein and Chien, 

1954). The third type of information, obtained by 

Sayre and Albertson (1959) for clear-water flow, rigid 

boundaries, and fixed baffles, shows that the k computed from an equation similar to equation (7) differs 

from the k computed from an equation similar to equation (8) for the same flow if the baffles are spaced too 

widely apart. In other words, the vertical distribution

of velocity is determined partly by position of the 

vertical with respect to a baffle if the spacing of baffles 

is wide. The variability of vertical velocity distribution and k from place to place in the clear-water flow is 

consistent to that for the Middle Loup River at Dunning. On the basis of these three types of information, 

the k that is computed from the vertical distribution 

of velocity in flow over a dune bed or other rough bed 

is here called "ki" because it may not be a true turbulence constant.

Although ki may not be a true turbulence constant, 

it seems to have a relationship as it logically should 

to the vertical distribution of sediment concentration 

as was shown by Hubbell and Matejka (1959, p. 71 

and fig. 36). They used values of ki (k in their terminology) that were computed by averaging the values 

of ki for individual verticals in the cross section and 

found that Zi for shallow flows in some sand-bed 

streams in Nebraska varied inversely with about the 

0.6 power of ki. For this paper, the k^ values for the 

cross sections were recomputed from a vertical-velocity 

difference and a shear velocity that were averages of 

figures for the individual verticals. The recomputed 

ki values give less weight to a high k^ at an individual 

vertical than did the k values that were used for a 

cross section by Hubbell and Matejka. In general, 

the Zi values vary inversely with about the 0.8 power 

of the recomputed k^ values. (See fig. 4.) The quantity ZiU% also varies inversely with about the 0.8 power 

of ki (fig. 5), whereas equation (6) indicates that it 

should vary inversely with the 1.0 power of k^ The 

difference may result from the fact that an abnormally 

high ki is usually higher than the ki for the surrounding flows; that is, the vertical distribution of sediment 

is probably partly determined by the ki at nearby verticals; hence, the average or effective ki may often be 

lower than an abnormally high observed ki even though 

the observed ki is itself an average for perhaps three to 

five verticals.

The use of w* in equation 6 was not accepted by 

Einstein (1950), although he obtained u* in his derivation. He computed z from the equation

3=i0/(0.40«*')- (9)

In this equation, u*' or the shear velocity with respect 

to the particles is the shear velocity that can be computed from the energy gradient and the part of the 

hydraulic radius that relates to the movement of 

sediment particles on the bed; that is, Einstein used 

the mean velocity equation

log (12.27*57^5). (10)
A12 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

EXPLANATION

O 0.062 to 0.125 mm

x 0.125 to 0.250 mm

 0.250 to 0.500 mm

0.05

dcv_ wddy

dy

By differentiation of equation 8 with ki rather than k

Hence,

For convenience define z0 by the equation

dcy_ w ddy 

cy kiii* y(d y)

(11)

(12)

Equation 11 can now be integrated from a distance a to 

a distance y above the streambed to obtain

Cyfd-y a V 

ca \ y d a/

which is the same as the classical equation 5 except for 

the change from k to ki in the exponent. If equation 8 

applies, ki equals k, and the classical equation should 

apply to flow over a dune bed provided that the equation for momentum exchange also applies to flow over 

a dune bed. Of course, the equation for momentum 

exchange is somewhat questionable for flow over a 

dune bed, especially if the dune height is relatively 

large in comparison to the depth of flow.

FIGURE 4. z/ plotted against kt for shallow streams in Nebraska.

As Z>65 is only a small fraction of the total roughness of 

a dune bed, a comparison of equations 7 and 10 shows 

that u* '/0.40 must be much smaller for flow over dune 

beds than u*/ki (ki rather than k because the constant 

may not be a true turbulence constant). Whether 

ki is about equal to 0.40 or is larger than 0.40, u* must 

be considerably larger than u*' for flow over dune beds, 

and for such flow the two kinds of shear velocity are 

by no means equivalent.

A reexammation of Einstein's development (1950, 

p. 14-17) of equation 6 does not seem to indicate that 

u*' should be substituted for u* in this equation. 

Einstein used the equation for the vertical distribution 

of suspended sediment at equilibrium in the form

dcv_ cvw ~~

dy vle/2

He then evaluated the product of the average vertical 

water movement v and the mixing length le from the 

equation for momentum exchange to obtain

EXPLANATION 

O 0.062 to 0.125 mm

X 0.125 to 0.250 mm 

 0.250 to 0.500 mm

0.02

FIGUEE 5. ziw* plotted against fc, for shallow streams in Nebraska.
DISCHARGE OF SANDS AND MEAN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS A13

The difference between z0 from equation 12 and z 

from equation 9 is usually large for flow over dune beds. 

If ki differs appreciably from k, it is usually larger than 

0.40. Also, w* for flow over dune beds is usually much 

larger than u* f . Hence, the product kiU* is usually 

much larger than 0.40 u*', and the computed vertical 

distributions of sediment concentration that are based 

on kiU* are much more uniform than those that are 

based (as Einstein suggested) on 0.40w*'.

In summary, the vertical distribution of sediment 

concentration of a particular size range becomes more 

uniform as the shear velocity and k or ^ increase and as 

the fall velocity decreases. However, the classical equation for the vertical distribution holds only inexactly 

and is especially questionable for flow over dune beds. 

Hence, a procedure like Einstein's (1950) for computing 

the discharge of bed material, although based on 

generally sound concepts, is unlikely to give consistently accurate computations of suspended-sediment 

discharge particularly for flow over dune beds.

TOTAL SEDIMENT DISCHARGE

Some general relationships involved in the computation of the total sediment discharge at a cross section 

can be visualized from figure 6, which is consistent with 

the preceding explanations of a concept of sediment 

transportation. The figure was prepared for flow at 

a depth of 5.0 feet and at a mean velocity of 3.5 feet per 

second over a dune bed and for a bed material in which 

DS5 and D65 are 0.0007 and 0.001 foot, respectively. 

(The same depth and energy gradient might be associated with a plane bed, but the velocity would then be 

much greater and the vertical distributions of concentration would be different.) The vertical distribution of 

sediment concentration varies widely from one particle 

size to another.

The complicated interrelationships indicated by the 

general concept of sediment transportation and partly 

illustrated by figure 6 preclude any simple modeling in 

a flume of the sediment transportation in a river. 

Although many phases of sediment transportation can 

well be studied in the laboratory, flume investigations, 

in the absence of satisfactory model laws, are basically 

studies of sediment transportation in small channels 

rather than model studies of the sediment transportation in natural streams.

The complexity of sediment and flow relationships 

and the difficulties of modeling sediment transportation suggest extreme caution in the use of compound 

parameters, whether dimensionless or not, in studies 

of sediment transportation. Even the simple ratio d 

to D can be highly misleading if used as a parameter,

because the effect of doubling the depth d is often 

much different from the effect of dividing the particle 

size D by 2.

GENERAL METHODS OF COMPUTATION

The total sediment discharge at a cross section of 

a stream is the sum of the bedload discharges and the 

suspended-sediment discharges of all particle sizes. 

The bedload discharge either can be computed directly 

as a whole or can be obtained by adding the computed 

bedload discharges for the different size ranges. It 

must be computed by individual size ranges if the 

concentrations of suspended sediment by size ranges 

are to be based on the bedload discharges. The computation of suspended-sediment discharge can be made 

from defined distributions, such as those shown on 

figure 6, by multiplying average velocity and average 

concentration within each of several layers of flow, 

adding the products for all the layers of flow, and 

converting the sum of the products to a convenient 

rate, such as tons per day. Thin layers of flow should 

be selected at depths where the concentration and 

velocity change rapidly as the distance above the bed 

changes.

FINE SEDIMENT

The discharge of the fine sediment can seldom be 

computed from the characteristics of the bed sediment 

and of the flow at a cross section by the general procedure that can be used, though inexactly, for coarser 

sediment such as sand. As Einstein and Chien (1953) 

have pointed out, the two main reasons preventing 

this type of computation for fine sediments are that 

(a) a representative sample of the fine sediment in 

the upper layer of the streambed can seldom be obtained especially under high flows and (b) the relative 

amount of the fine sediment in this upper layer 

changes rapidly. (Silt and clay together are roughly 

equivalent to "wash load" as the term was used by 

Einstein and Chien.) Thus, the relative amount of 

fine sediment at the surface of the streambed is seldom 

known; hence, the bedload discharge of the fine sediment usually cannot be computed with any reasonable 

accuracy. The bedload discharge of the fine sediment 

is so small as compared to the discharge of suspended 

fine sediment that it could be disregarded, except that 

the only theoretical way of computing the concentration and discharge of the suspended fine sediment 

is based directly on the bedload discharge. As figure 

6 indicates, a small percentage of fine sediment at the 

streambed may provide a base for much suspended 

fine sediment.
VELOCITY, IN FEET PER SECOND

Assumed about constant c

in bed layer

2000 4000 6000 

CONCENTRATION, IN PARTS PER MILLIO

400 800 1200 1600 2000 2400

CONCENTRATION, IN PARTS PER MILLION 

FIGURE 6. Distribution of concentrations of different sediment sizes at a stream vertical.

2800 320
DISCHARGE OF SANDS AND MEAN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS A15

High concentrations of fine sediment in a flow 

increase the discharge of the coarser sediment because 

they decrease the fall velocity of the suspended 

sediment and may modify the bed configuration 

(D. B. Simons, E. V. Richardsons, and W. L. 

Haushild, written comm.). The fine sediment seems 

to affect sediment transportation by changing the 

apparent viscosity of the mixture of water and sediment. The effect of the fine sediment should be 

comparable to the effect of a decrease in water 

temperature, but the effect of high concentrations of 

fine sediment may be much greater than the effect 

of usual changes of the temperature of natural streams.

EFFECTS OF MAJOR FACTORS

The preceding discussion of the concept of sediment 

transportation indicates the major variables that 

probably control the discharge of sands in sandbed 

streams through the effect of these variables on bedload discharges, suspended-sediment concentrations 

immediately above the bed layer, and vertical distributions of suspended-sediment concentrations. The quantitative effect of individual variables can be approximately defined, at least for some flows and streams 

cross sections, by a type of graphical analysis. An 

important part of the analysis is to determine which 

of four possibly interchangeable variables namely, 

mean velocity, shear, shear velocity computed from 

mean velocity, or stream power shows the closest and 

most useful relationship to the discharge of sands. Of 

course, shear velocity computed from mean velocity is 

the -type of shear velocity with respect to the particles 

that is computed from a known mean velocity with the 

use of an equation such as equation 10.

Three general assumptions or limitations should be 

kept in mind in the following discussions. One is 

that the assumed bed material of the streams is relatively cohesionless sands. Another is that discharge of sands in these streams is equivalent to the 

sum of the bedload discharge and the suspended discharge of bed material. The third is that discharge 

of sands is proportional to the width of stream; this 

assumption, in effect, excludes some highly irregular 

cross sections from the studies but does permit the use 

of discharge of sands per foot of width as the usual 

unit for expressing the rate of transportation of sands 

through a cross section.

MEAN VELOCITY

Probably the simplest measure of the force that 

moves sediment in a stream is the mean velocity. 

Studies by Gilbert (1914), Colby (1957), and 

Rottner (1959) have shown an approximate relationship between the discharge of bed material in shallow 

flows or near the streambed and roughly the third 

power of the mean velocity, except for low velocities. 

Hence, the relationship of discharge of sands per unit 

width of stream to mean velocity was investigated 

in detail for natural streams and less extensively for 

flumes.

NATURAL STREAMS

Total discharges of sands for Niobrara River near 

Cody, Nebr., show a close relationship to mean velocity. 

(See fig. 7.) (For this and other curves that involve 

a relationship of discharge of sediment to mean 

velocity or to other variables, the sediment discharge 

is considered to be the statistically dependent variable; 

that is, each curve is drawn to show, as well as possible, 

the average sediment discharge for each of several 

narrow ranges of the independent variable.) These 

total discharges of sands were measured at a contracted 

section 1,900 feet downstream from the gaging-station 

section where velocities, depths, and widths of flow and 

particle sizes of the bed sediment were determined 

(Colby and Hembree, 1955). The median size of the 

bed sediment was about 0.28 mm. The total discharges of sands have the usual inaccuracies inherent 

in sampling suspended sands at a somewhat unsatisfactory cross section. Also, at times some appreciable 

gain or loss of sands must have occurred between the 

two sections. No adjustment was applied for change 

in water temperature. In spite of the inaccuracies, 

the scatter of points from the average curves of figure 

7 is not excessive.

An average curve of discharge of sands per foot 

of width against mean velocity was also defined (fig. 

8) from information given by Hubbell and Matejka 

(1959) for Middle Loup River at Dunning, Nebr. 

Sections A, B, O, and E, at which velocities, depths, 

and widths of flow and particle sizes of the bed sediment were determined, have average widths of roughly 

280, 70, 80, and 150 feet, respectively. Section A is 

about 1.4 miles upstream from the turbulence flume 

where total sediment discharges were measured (Benedict, Albertson, and Matejka, 1955), and section E is 

about 700 feet downstream from the flume. Section B 

is about 1,200 feet and section C is less than 100 feet 

upstream from the flume. Median particle size of the 

bed material at the four sections is about 0.32 mm. 

Rough adjustments for changes in water temperature 

were applied before the total discharges of sands were 

 plotted on figure 8. A single curve represents the 

average relationship fairly well for all four sections. 

Total discharges of sands (data from the files of

the Agricultural Research Service, U.S. Dept. of
A16 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

200

100

MEAN VELOCITY, IN FEET PER SECOND

FIGURE 7. Discharge of sands plotted against mean velocity for Niobrara Elver 

near Cody, Nebr. (Data unadjusted for water temperature.)

Agriculture, University, Miss.) for several stations 

on Pigeon Eoost Creek or its tributaries in Mississippi were approximately determined by adding 

estimates of unmeasured sediment discharges to 

measured discharges of sands. These total discharges were plotted against mean velocity to define 

the six average curves that are indicated by broken

EXPLANATION

o 

Section A

A 

Section B

MEAN VELOCITY, IN FEET PER SECOND

FIGURE 8. Discharge of sands plotted against mean velocity for Middle Loup 

Eiver at Dunning, Nebr. (Data roughly adjusted for water temperature.)

lines on figure 9. Each of five of the broken curves 

is for an individual sediment station, and the sixth 

is a composite curve that is based on only a few 

determinations for each of several sediment stations. 

Median diameter of the bed sands is about 0.40 mm. 

at all stations. The unmeasured sediment discharges, 

mostly of sands, were estimated from a graphical
DISCHARGE OF SANDS AND MEAN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS A17

600

=: 100

MEAN VELOCITY, IN FEET PER SECOND

FIGURE 9. Curves of relationship between discharge of sands and mean velocity 

for stations on Pigeon Roost Creek in Mississippi. (Data unadjusted for water 

temperature.)

727-985 O - 64 - 4

relationship to mean velocity (Colby, 1957). Hence, 

the unmeasured portion of the total discharge of 

sands was based partly on mean velocity. Also, the 

mean velocities for the high flows at several of the 

stations were determined by measuring from bridges 

where the streambed scoured locally. Most measurements of high flows were made on falling stages when 

the mean velocity under a bridge may have been 

appreciably lower than the mean velocity away from 

the bridge. In general, the six curves are reasonably 

consistent and may be roughly expressed by the single 

continuous curve of figure 9.

Total discharges of sands for stations on the Rio 

Grande or diversions from it in New Mexico were 

obtained from unpublished computations that were 

made by J. K. Culbertson and D. Q. Matejka according to the modified Einstein procedure (Colby and 

Hembree, 1955) and were used to define a curve of 

relationship to mean velocity. (See fig. 10.) Two 

total discharges of sands in the Rio Grande near 

Socorro, N. Mex., and one near San Antonio, N. Mex., 

were made when most of the flow originated in the 

drainage basin of the Rio Puerco and contained very 

high concentrations of clay and silt. The plotted points 

for the three determinations when the concentration 

of fine sediment was very high are circled on figure 

10. The circled points indicate discharges of sands 

that are roughly three to eight times greater than 

the discharges of sands from the average curve for 

times when the concentrations of fine sediment were 

low.

Total discharges of sands for Rio Puerco near 

Bernardo, N. Mex., where concentrations of silt plus 

clay are on the order of 100,000 ppm (parts per 

million) or more on many rises, have only a rough 

relationship to mean velocity (fig. 11). They are 

generally very high as compared with total discharges 

indicated by an estimated curve for which factors 

such as velocities, depths, and bed-materials sizes are 

similar but the concentration of fine sediment is low. 

The effect of the high concentrations of fine sediment 

apparently is very large. It is reasonably consistent 

with the laboratory findings of D. B. Simons, 

E. V. Richardson, and W. L. Haushild (written 

communication).

For a large stream, Mississippi River at St. Louis, 

Mo., the total discharges of sands plot fairly close 

to an average curve of relationship to mean velocity. 

(See fig. 12.) They were based on measured discharges of sands plus unmeasured sediment discharges 

that were estimated from the same graphs used for the 

unmeasured sediment discharges of Pigeon Roost
A18 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

1000

Very high concentration 

of fine sediment

Generally low concentration of fine sediment

MEAN VELOCITY, IN FEET PER SECOND

FIGURE 10. Discharge of sands plotted against mean velocity for the Rio 

Qrande in New Mexico and for some diversions from it. (Data unadjusted 

for water temperature.)

Creek. Hence, these unmeasured sediment discharges 

are related to mean velocity. However, the relationship 

of measured discharge of sands to mean velocity (P. R. 

Jordan, written communication) seems to be equally 

as consistent as the relationship of figure 12.

Scatter of total discharges of sands from an average 

relationship such as that of figures 7, 8, 10, or 12 is 

generally less for a single and reasonably uniform 

cross section for example, the cross section of Niobrara River near Cody, Nebr. than for a group of 

cross sections or for an irregular cross section.

Curves showing relationship between discharge of 

sands per foot of stream width and mean velocity for 

the Niobrara River, the Middle Loup River, Pigeon 

Roost Creek, the Rio Grande, and the Mississippi 

River are rather surprisingly consistent (fig. 13) and 

can be used to estimate quickly and roughly the discharge of sands in many sand-bed streams. The average water temperature is about 60° F. for each curve. 

Of course, the somewhat steeper slope of the curve 

for the Mississippi River is due to the fact that any 

particular mean velocity occurs at greater depths in 

the Mississippi River than in the shallower streams. 

However, in general, a sand-bed stream seems to have 

not only a reasonable close relationship between discharge of sands per foot of width and mean velocity 

at a cross section, but the relationship defined for one 

cross section may apply approximately for other sandbed streams if bed-material sizes and depths are not 

greatly different and if the concentrations of silt plus 

clay are not unusually high.

FLUMES

The discharge of sands per foot of width for a 

series of flume experiments with the same bed sediment 

and constant water temperature usually shows a fairly 

close relationship to mean velocity. (See fig. 14). 

However, in a flume whose slope can be adjusted, a 

particular mean velocity may occur over a wide percentage range of depths. Thus, some scattering of 

points on figure 14 is caused by changes in depth, but 

some is caused by changes in water temperature, by 

unidentified factors, or by experimental error. Except 

for the effect of depth, the scatter of the observed 

discharges of sands about an average curve would probably be considerably less for most flume experiments 

in which a constant bed material is used that it would 

be for similar mean velocities at a cross section of a 

natural stream where the determinations are likely to 

be less accurate and where the lateral distributions of 

depth, velocity, and sediment discharge generally are 

less uniform. If the mean velocities in a flume are 

much lower than in a natural stream, they may be in
DISCHARGE OF SANDS, IN TONS PER DAY PER FOOT OF WIDTH

IfC

6 o

IE.

g

s « 

I?

is

s- a

CD « 

sr& 

g^

DISCHARGE OF SANDS, IN TONS PER DAY PER FOOT OF WIDTH

i " 

a 8
A20 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

500

100

10

EXPLANATION

Niobrara River 0.28

Middle Loup River 0.32

Mississippi River 0.2 to 0.7

Rio Grande 0.2 to 0.3

Pigeon Roost Creek 0.40

10

MEAN VELOCITY, IN FEET PER SECOND

FIGURE 13. Curves showing relationship between discharge of sands and mean 

velocity for five sand-bed streams at average temperatures of about 60° F.

the range for which the discharge of sands is very 

sensitive to small changes in velocity, depth, and size 

distribution of the bed sediment.

The main reason for the curvature of the lines 

showing average relationship on figure 14 is that the 

discharges of sands are plotted against mean velocity 

rather than against mean velocity minus the critical 

velocity at which movement of significant amounts of 

sand begins. If 0.8 and 0.7 foot per second are subtracted from the mean velocities for the left and right 

curves of figure 14, respectively, straight lines can be 

substituted for the curves.

The average relationships of discharges of sands 

to mean velocity for flume experiments differ considerably among themselves and especially for shallow flows are rather inconsistent with the relationship for Niobrara River near Cody, Nebr. (See fig. 

15.) Much of the disagreement is due to differences 

in sizes of bed sediment or in depth. Because of 

the divergence among the curves for flumes, the 

comparative consistency of the curves for natural 

streams (fig. 13) probably would not have been predicted from flume experiments.

SHEAR

Shear or tractive force is an obvious measure of 

the force that may cause sediment to move and has 

been widely used as a parameter of sediment discharge. Shear in the form of a shear velocity is 

directly proportional to mean velocity if the Chezy 

C is constant. Thus, if resistance to flow as expressed 

by C were always constant, mean velocity and shear 

velocity could be used interchangeably as parameters 

of discharge of sands in sand-bed streams. The 

choice between those two parameters should be made, 

therefore, mainly on the basis of which one has the 

more consistent relationship to the discharge of sands 

when resistance to flow is variable.

Other factors should also be considered in making 

a choice between mean velocity and shear as a measure of the discharge of sands. Shear can be used 

directly in the design of a channel, whereas mean 

velocity for channel design normally must be computed from the shear. However, if the discharge 

of sands is more closely related to mean velocity 

than to shear, the problem of the relationship of 

mean velocity to shear cannot be avoided. Also, 

canals usually are designed for velocities that are 

slow enough to be in the range of flow over dune 

beds, and in this range the major change of roughness at the transition from dunes to a smoother bed 

does not enter the computations. In a natural stream
DISCHARGE OF SANDS AND MEAN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS A21

100

Median size 0.45 mm (Simons 

and others, 1961)

Median size 0.18 mm (Barton 

and Lin, 1955)

O 

0.01

0.001

0.1 1 

MEAN VELOCITY, IN FEET PER SECOND

FIGTJEE 14. Curves showing relationship between discharge of sands and mean velocity for some flume experiments with two 

sands. (Data unadjusted for water temperature and depth.)
A22 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

200

100

O 

MEAN VELOCITY, IN FEET PER SECOND

Approx

Dso depth

(mm) (ft)

a.......... NiobraraKiver..___ _ 0.28... ...... 1.0-6.0

6...- -- Brooks (1958)-._ .088-...- .- 0.15-0.3

c ..... Barton and Lin (1955) . -- .18-.. --- 0.3-1.4

d........-- Brooks (1958)--. .145-- - -- 0.15-0.3

e- ------ Simons and others (1961).---.---------- .45.-------------- 0.25-1.0

FIGURE 15. Curves showing relationship between discharges of sands and mean 

velocity for some flume experiments. (Data unadjusted for water temperature 

and depth.)

at a particular time, the mean velocity usually can 

be determined experimentally more easily than the 

shear. For channels that are not uniform laterally, 

the average depth of flow ordinarily used to compute 

shear is weighted with lateral distance in contrast 

to mean velocity, which is weighted with the product of lateral distance and depth; that is, average 

depth is equal to the area divided by the width, but 

mean velocity is equal to the streamflow divided by 

the area, which is the product of the average depth 

and the width. Thus, the areas of highest velocity 

in a cross section usually are given more weight in 

computing the mean velocity than in computing the 

average depth because the velocities generally are 

highest where the flow is deepest.

The discharges of sands reported by Simons, 

Kichardson, and Albertson (1961) and plotted 

against mean velocity on figure 14 indicate an 

approximate relationship (fig. 16) to shear. For easy 

comparison with stream power, the discharges of sands 

are replotted against the 3/2 power of the shear 

and the cube of the mean velocity. At a shear of 

about 0.07 to 0.10 pound per square foot (a shear 

to the 3/2 power of about 0.018 to 0.030), the range 

in discharge of sands at a particular shear is large. 

From independent experiments that were made mostly 

within a similar range of uncertain relationship, 

Brooks (1958) concluded that neither the velocity 

nor the sediment discharge is a single-valued function 

of the shear. Although the conclusion applies to 

only a narrow range of shears in most natural streams 

if the bed material, water temperature, and energy 

gradient are constant, within this range the shear is 

not an acceptable measure of the mean velocity or of 

the discharge of sands. The range of uncertain relationship between discharge of sands and shear differs 

considerably from one stream to another and cannot 

be acccurately predicted at the present time from characteristics of the channel. Hence, the shear is at times 

an undependable measure of the discharge of sands.

A conclusion by Barton and Lin (1955, p. 21, 37) 

that the bed-material concentration may be a function 

only of resistance to flow is a further indication of 

the relatively poor relationship of discharge of sands 

to shear. Of course, the reason for some relationship 

between bed-material discharge and resistance to flow 

is that mean velocity depends both on resistance to 

flow and on shear, and the relationship either to 

resistance to flow or to shear is an incomplete part of 

the more basic relationship to mean velocity. In other 

words, shear at constant resistance to flow is a measure
DISCHARGE OF SANDS AND MEAN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS A23

Q .  -

°

Q. UJ

I CE t~ <

go rt "-W 6

z

o:

^H LU°-

_. UJ

HiQIM JO iOOd d3d AVQ d3d SNOi Nl 'SQNVS JO 39dVHOSia
A24 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

of mean velocity and discharge of bed material, and 

resistance to flow at constant shear is also a measure 

of mean velocity and discharge of bed material.

SHEAR VELOCITY COMPUTED FROM MEAN 

VELOCITY

Einstein (1950, p. 9-10) used a shear velocity with 

respect to the grains, which is the shear velocity that 

can be computed from the hydraulic radius with respect 

to the grains. This hydraulic radius with respect to 

the grains or particles is the hydraulic radius that can 

be used with the energy gradient to compute mean 

velocity from equation 10. However, the use of this 

hydraulic radius, R'', under the logarithm sign in the 

equation is questionable. The shear velocities used in 

this paper were computed from known mean velocities 

and from equation 10 except that d or R was substituted 

for Rf under the logarithm sign.

The same discharges of sands that were plotted 

against the mean velocity cubed and against the 3/2 

power of the shear show no discontinuity in their 

relationship to the shear velocities computed from 

mean velocity. Hence, this type of shear velocity has 

a major advantage over shear as a measure of the 

discharge of bed material. The scatter about an average curve is, of course, approximately the same whether 

mean velocity or shear velocity computed from mean 

velocity is used as the independent variable. However, 

as will be shown later, the effect of depth on the 

relationship between discharge of sands and shear 

velocity computed from mean velocity is somewhat 

different from that on the relationship between discharge of sands and mean velocity. Except for the 

difference in effect of depth, the choice between mean 

velocity and shear velocity computed from mean 

velocity is based mostly on convenience and simplicity

of use.

STREAM POWER

"Stream power" is a term for the rate at which a 

stream loses energy. It has been discussed as a measure 

of the discharge of sediment by Cook (1935) and 

Bagnold (1960), but its relationship to sediment discharge is not simple, as Cook indicated. Specifically, 

the stream power of a volume of water in a reach of 

river L units long and W units wide is equal to the 

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

loss of energy head per unit time. Hence, the stream 

power for low concentrations of sediment and of dissolved solids and in pound-foot-second units is 62.4 

LWduS. For unit width and length of channel the 

stream power is 62.4 dSu or the product of shear and 

mean velocity.

If the Chezy C were constant, the stream power 

would be directly proportional to the third power of

the mean velocity and would be neither a better nor 

a worse parameter than mean velocity. However, when 

C varies, the shear is a somewhat inadequate measure 

of the discharge of sands. Hence, the stream power 

also is a poor measure of the discharge of sands when 

C varies widely, as it does within the range of shear 

at which a dune bed changes to a plane or antidune 

bed. In other words, the discontinuity in the relationship between discharge of sands and shear at about a 

shear of 0.07 to 0.10 pound per square foot also exists 

in the relationship between discharge of sands and 

stream power but is partly obscured by the inclusion 

in stream power of mean velocity.

Any continuous curve that might be drawn to 

represent the experimental relationship between discharge of sands and stream power on figure 16 

obviously cannot be as smooth and consistent as the 

curve for the relationship to the cube of the mean 

velocity. A comparison of the two curves of figure 

16 suggests some conclusions with respect to stream 

power.

At velocities high enough to cause plane or antidune beds, the resistance to flow should be reasonably 

constant; hence, all curves showing relationship 

between discharge of sands and one of the independent 

variables of figure 16 should have about the same 

slope. Curves showing relationship of discharge of 

sands to mean velocity or to shear velocity computed 

from mean velocity are smooth and continuous and 

merge gradually into the slope for high velocity. In 

fact, as was shown (fig. 14) for two sands, the relationship between discharge of sands per foot of width 

and mean velocity minus the velocity at which sediment 

begins to move significantly is about a straight-line 

relationship. No constant quantity can be subtracted 

from the cube of the mean velocity to obtain a straightline relationship to the discharge of sands. Similarly, 

no constant stream power can be subtracted from the 

stream power on figure 16 to define a straight-line 

relationship between discharge of sands and stream 

power. This fact and objections based on theoretical 

physics make very questionable the usefulness of 

Bagnold's suggestion (1960, p. 4) that a power 

threshold should be used for the beginning of sediment 

movement.

The relationship between discharge of sands and 

stream power is poor within the range of stream 

power from about 0.10 to 0.30 foot-pound per second 

per square foot, because within this range the shear 

has little relationship either to mean velocity or to 

discharge of sands. Above and below this range, the 

curve showing the relationship of discharge of sands 

to stream power is much like that to the cube of the
DISCHARGE OF SANDS AND MEAN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS A25

mean velocity. The break in the relationship occurs 

at the transition from dunes below this range to a 

plane bed or antidunes above this range and is caused 

by the unsatisfactory independent variable. It can be 

eliminated by using mean velocity or shear velocity 

computed from mean velocity as the independent 

variable.

At high discharges of sands, the scatter of points 

about a curve showing average relationship is less 

from stream power than for any of the other three 

independent variables that were used on figure 16. 

This relatively good agreement for one set of experiments at high discharges of sands may or may not be 

characteristic.

A reasonably constant concentration of sands is indicated by the approximately 45° slope of the curve 

showing the relationship between the discharge of 

sands and stream power at high flows and at high discharges of sands. At a particular reach of a sand-bed 

stream, the slope of the energy gradient usually varies 

comparatively little with changes in depth. Hence, if 

the depth is quadrupled, the mean velocity should double provided that the Chezy C remains constant, and 

the flow per foot of width should be eight times 

greater for the deeper flow. For the concentration of 

sands to remain constant, the discharge of sands should 

increase eightfold or in proportion to the cube of the 

mean velocity.

DEPTH

The effect of depth on the relationship of discharge 

of sands to any one of the four independent variables 

 namely, mean velocity, shear, shear velocity computed from mean velocity, and stream power is 

different from the effect of depth on the relationship 

of discharge to any of the other three variables. It 

can be defined more accurately for the relationships 

of discharge of sands to mean velocity or to shear 

velocity computed from mean velocity than for the 

relationships to shear or to stream power.

The complicated effect of depth of flow on the 

relationship of discharge of sands to mean velocity at 

different water temperatures can be shown qualitatively 

by the results of many computations of discharge of 

sands. The computations were made according to a 

procedure that was based mostly on the work of 

Einstein (1950) and for a natural bed of unconsolidated 

sand whose Z>35 , D50 , and D65 sizes were about 0.35, 

0.40, and 0.45 mm, respectively. They were made for 

many different combinations of depth and mean 

velocity and usually for water temperatures of 40°, 60°, 

and 80 °F. The main departures from the Einstein 

procedure were the use of shear velocities computed 

from the mean velocities and of some assumptions with

regard to the vertical distribution of sediment concentrations. These assumptions were probably fairly 

accurate for flow over dune beds but resulted in too 

great discharges of sands at high velocities. A qualitative graph based on these computations (fig. 17) 

indicates the dominant general relationship between 

discharge of sands and mean velocity. It also indicates 

a complex effect of depth on this dominant relationship. 

At constant low velocities, an increase in depth is 

accompanied by a decrease in discharge of sands; but 

at constant high velocities, an increase in depth is 

accompanied by an increase in discharge of sands. At 

some intermediate constant velocities, depth has little 

effect on the relationship between discharge of sands 

and mean velocity. At constant mean velocity, the 

effect of depth varies somewhat, but not greatly, as the 

water temperature changes.

The general reason for the variable effect is easily 

understood in broad outline, although the actual computation of bed-material discharges for different 

depths is inexact and complicated. If mean velocity 

is low perhaps 2 feet per second and if water 

flows in a shallow stream over a sand bed whose 

median diameter is 0.3 or 0.4 mm, a certain amount 

of bed material (as generally indicated by the curves 

of velocity and concentration on figure 18) will be 

discharged, mostly near the streambed. If the mean 

velocity is constant at 2 feet per second and if the 

flow is deep, the velocity near the streambed will be 

lower and the amount of bed material discharged will 

be less, both at the bed and throughout the original 

shallow depth of flow, than they were in the shallow 

stream. Only a little bed material will be discharged 

in suspension in the additional depth of flow. Hence, 

the total bed-material discharge 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 perhaps 6 feet per second the bedmaterial discharge is greater in a shallow flow than in 

a layer of equal depth at the bottom of a deep flow 

because of the difference in velocity near the streambed. 

However, the difference in the bed-material discharge 

in this layer near the bed is more than balanced by 

the discharge of bed material in the superposed layer 

of the deeper flow. (See fig. 18.)

If enough information is available for one bed 

material and one water temperature and for a wide 

range of depths and mean velocities, the effects of 

mean velocity and depth on the discharge of bed 

material can be determined by the empirical definition 

of a graph similar to figure 17. Even information for 

different water temperatures and different bed materials can be analyzed by trial-and-error procedures to
A26

100

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10

1.0

0.1 

0

SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

100,000 I

3 fps

2 fps

10,000

1000

1.0 10

100

7 fps

5 fps 40°F

100 1.0 

DEPTH,IN FEET

10 100

FIGUHE 17. Effect of depth and water temperature on the relationship between discharge of sands (0.40 mm median diameter of the bed sands) and mean velocity,

computed from a somewhat revised Einstein procedure. From Colby (1961, fig. 1).

define a graph of empirical relationship of discharge 

of sands to mean velocity and to depth. (See p. 34- 

35.) Such a relationship for a water temperature at 

60° F and bed sediment having a 0.30-mm median 

diameter is shown on figure 19. Qualitatively, the relationships of figures 17 and 19 are similar in spite of 

the difference in particle size and in method of determination ; but quantitatively, figure 19 is much more 

accurate than figure 17. The fundamental soundness of 

the Einstein procedure is well shown, at least for 

the assumed bed sand, by the general qualitative 

agreement of the computed effects with the empirically 

defined effects of mean velocity, depth, and water 

temperature.

For a median particle size of 0.30 mm and a water 

temperature of 60°F, about the same discharge of 

sands is indicated for a mean velocity of 3.0 feet per 

second whether the depth is 1.0 foot or 100 feet. 

Of course, if the line on figure 19 for a mean velocity 

of 3.0 feet per second were defined more accurately, 

it might have a somewhat different shape and position.

The effect of depth varies somewhat with changes 

in particle size. The mean velocities at which depth 

has little apparent effect on the relationship between 

discharge of sands and mean velocity are about 1.6, 

2.2, and 3.5 feet per second for bed sediments of 0.10-, 

0.20-, and 0.40-mm median diameters, respectively,

according to empirically defined graphs similar to 

figure 19. To some extent an increase in mean velocity 

at constant particle size has about the same effect as a 

decrease of particle size at constant mean velocity.

Only a small apparent effect of depth on discharge 

of sands in natural streams was indicated by the 

relative consistency of the relationship between discharge of sands and mean velocity for different 

streams. (See fig. 13.) These streams generally have 

mean velocities, depths, and sizes of bed sediments 

for which changes in depth should have comparatively little effect, according to figure 19. However, 

compared with shallower streams, the deeper flows 

of the Mississippi River at St. Louis result in lower 

discharges of sands at low velocities and higher discharges of sands at high velocities.

Some inconsistencies of the relationships between 

discharge of sands in flumes and mean velocity (fig. 

15) are explainable on the basis of the effect of 

depth and particle size. Not only are the depths 

in flume experiments usually shallow, but they may 

decrease as the velocities increase. At a high velocity 

the discharge of sands, especially fine sands, is likely 

to be far lower in a shallow flow in a flume than at 

the same mean velocity in a deep river. In contrast, 

at a low velocity the discharge of sands is likely to 

be higher in a shallow flow in a flume than at the
DISCHARGE OF SANDS AND MEAN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS

Water surface of deep flow

A27

Water surface of shallow flow

VELOCITY AND CONCENTRATION

FIGURE 18. Schematic diagram showing the general reasons for variability of effect of depth on the relationship of bed-material discharge to velocity. From Colby

(1961, fig. 3).

same mean velocity in a deep river. Thus, the rate 

at which the discharge of sands increases with respect 

to the mean velocity is usually greater in a natural 

stream than in a flume and in a deep river than in 

a shallow stream. Also, the depth effect becomes 

increasingly large at low velocities, and the velocities 

in flumes particularly for a given type of bed configuration are generally low as compared with those 

in rivers.

The effect of depth on the relationship between 

discharge of sands and shear velocity computed from 

mean velocity differs somewhat from the effect on the 

relationship between discharge of sands and mean 

velocity but can be computed readily from figure 19 

for a median diameter of 0.30 mm and a water temperature of 60°F. The computations require the solving 

of equation 10 by graphic or algebraic trial and error 

to obtain values of the shear velocity computed from 

the mean velocity for many different combinations of 

depth and velocity. The discharges of sands corresponding to these combinations can then be plotted 

against depth and shear velocity computed from mean 

velocity to obtain a reasonably simple pattern of depth 

effect. (See fig. 20.) One advantage of this depth 

effect is that it is always in the same direction; that is, 

at constant shear velocity computed from mean 

velocity, the discharge of sands increases as the depth 

increases if there is any appreciable discharge of sands. 

However, the rate of increase in the discharge of sands 

is slow at low shear velocities computed from mean 

velocities, but the rate becomes progressively greater 

as the shear velocity computed from the mean velocity 

increases. In many flume experiments, the effect of 

depth on the relationship between discharge of sands 

and shear velocities computed from mean velocities 

would be simpler to use than the effect of depth on the 

relationship between discharge of sands and mean 

velocities. In computations for natural streams, the 

latter effect of depth is smaller and, hence, is generally 

easier to apply with reasonable accuracy.
A28 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

The effect of depth on the relationship between 

discharge of sands and shear is somewhat indefinite 

except when shear is about proportional to mean 

velocity or to shear velocity computed from mean 

velocity. When shear has this relationship, the effect 

of depth on the relationship of discharge of sands to 

mean velocity or to shear velocity computed from mean 

velocity can be applied.

The effect of depth on the relationship between 

discharge of sands and stream power is a combination of the effects on the relationships to mean 

velocity and to shear. It is, therefore, both complex 

and sometimes indfinite. If the Chezy C is constant 

and if a change in hydraulic radius is exactly compensated by an inverse change in energy gradient 

so that. R8 remains constant, the depth effect on 

the relationship of discharge of sands to stream 

power can be determined from figure 19 for a 

median particle size of 0.30 mm and a water temperature of 60°F. If the mean velocity is 3.0 feet per 

second, median diameter is 0.30 mm, and shear and 

water temperature are constant, the indicated discharge of sands remains nearly unchanged as the 

depth increases from about 1 foot to 100 feet. For a

1000

1000

100

1 10 

DEPTH, IN FEET

100

FIGURE 19. Empirically determined graph of the effect of depth on the relationship between discharge of sands and mean velocity. (Water temperature 60 °F, 

and median diameter of bed sands 0.30 mm.)

FIGUBE 20. Effect of depth on the relationship between discharge of sands and 

shear velocity computed from mean velocity. (Water temperature 60°F, and 

median diameter of bed sands 0.30 mm.)

lower mean velocity such as 1.5 feet per second, 

the discharge of sands decreases rapidly as the depth 

increases at constant shear. If the Chezy C varies, 

as it does when the bed configuration changes, the 

effect of depth on the relationship between discharge 

of sands and stream power becomes more complex 

and is somewhat indefinite. It is indefinite because the 

compound parameter stream power is also somewhat 

indefinite in the sense that doubling the shear may 

cause a different effect than doubling the mean velocity. 

The effect of depth on relationships that involve 

concentrations of sands should also be understood 

because it seriously limits the usefulness of concentration of sands as a measure of the rate of sediment movement. The effect of depth on the relationship between concentration of sands and mean velocity 

for flow over a sand bed whose median particle size 

is 0.30 mm is large and variable. (See fig. 21, which is 

plotted directly from the information on fig. 19.) 

Also, the effect of depth on the relationship between 

the concentration of sands and mean velocity obviously 

will differ considerably from one particle size to 

another. The effect of depth on the relationship of 

concentration of sands to mean velocity is large and
DISCHARGE OF SANDS AND ME AN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS A29

100,000

10,000

1000

100

10

0.1 1 10 

DEPTH,IN FEET

100

FIGURE 21. Effect of depth on the relationship of concentration of sands to mean 

velocity. (Water temperature 60°F, and median diameter of bed sands 0.30 

mm.)

variable; therefore, the concentration of sands is a 

relatively meaningless parameter of rates of sediment 

transportation as compared to discharge of sands 

per foot of width, unless both particle size and depth 

of flowT are stated or are kept within narrow ranges. 

Hence, the concentration of sands may be significant 

in a flume study of about equal depths of flow over a 

particular bed of sand, but any attempts to relate concentrations of sands at a cross section in a flume to 

concentrations at a cross section in a natural stream 

are likely to be unfruitful. The effect of depth is 

also large and variable on the relationships of concentration of sands to shear, to shear velocity computed from mean velocity, or to stream power.

VISCOSITY

According to the concept of sediment transportation, 

viscosity has an effect on the thickness of the laminar 

sublayer, but its major effect on the relationship between discharge of sands and mean velocity is through 

its influence on the fall velocities of the particles 

and thus on the vertical distributions of sediment of 

different sizes. The percentage effects of changes in 

water temperature on the discharge of sands obviously 

varies with velocity, depth, and particle size. If the 

concentrations of the different particle sizes and the 

vertical distributions of the concentrations could be 

computed accurately, the effect of a change in viscosity 

could also be determined accurately, although the computations would be rather complex and tedious. Unfortunately, the vertical distributions of concentration cannot now be accurately computed, especially for 

flows over dune beds. Until vertical distributions of 

sands are more accurately predictable, somewhat oversimplified adjustments based mainly on empirical 

analysis of available field and flume information may 

be more practical measures of the effect of viscosity 

on the discharge of sands than are more theoretical 

adjustments.

The viscosity of water or the apparent viscosity 

of a water-sediment mixture can be changed appreciably by variations in water temperature or in concentrations of fine sediment. The relationship between 

the viscosity of distilled water and water temperature 

is accurately known. In contrast, high concentrations 

of fine sediment have a large and less readily defined 

effect on the apparent viscosity of the water-sediment 

mixture (D. B. Simons, E. V. Kichardson, and W. L. 

Haushild, written commun.; J. C. Mundorff, written 

commun.). The effect varies not only with concentration of the fine sediment but with characteristics of the 

fine sediment. The effects of changes in water temperature and in concentration of fine sediment on discharge 

of sands are sometimes too large to be disregarded.

"WATER TEMPERATURE

The effect of water temperature at a particular 

cross section can be approximated with trial-and-error 

multiple correlation. Usually, the size and the composition of the bed sediment at a particular cross section 

of a sand-bed stream are fairly constant with respect 

to time and flow, and either the relationship between 

depth and velocity is reasonably constant or the velocity is in the range for which depth has little effect 

on the relationship between discharge of sands and 

mean velocity. Hence, the effect of changes in depth at 

a particular mean velocity is likely to be small, and the 

discharge of sands at the cross section is controlled 

mainly by mean velocity and water temperature if 

concentrations of fine sediment are not high. An 

average curve drawn through the observed discharges 

of sands plotted against mean velocities for Niobrara
A30 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

Eiver near Cody, Nebr. (fig. 7), gives an approximation of the effect that a change in mean velocity 

has on the average discharge of sands. The ratio 

of each observed discharge of sands to the average 

discharge of sands from the curve of figure 7 at that 

mean velocity can be computed and then plotted against 

water temperature. (See fig. 22.) Although the points

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of sands in Niobarara Eiver near Cody, the second 

approximation for the effect of temperature changes 

did not seem to differ significantly in slope from the 

curve of figure 22.

The effect of water temperature on discharge of 

sands, defined by trial-and-error multiple correlation, 

for three sediment stations is surprisingly consistent. 

(See fig. 23.) It also agrees approximately with the 

computed effect of temperature (fig. 17) for usual 

combinations of depth and velocity (for example, 10 

feet and 7.0 feet per second or 2.0 feet and 3.0 feet 

per second) in natural flows over a bed material whose 

median diameter is 0.40 mm. However, the percentage 

effect shown on figure 17 is generally a little smaller 

for the 0.40-mm bed sand than for the somewhat finer 

bed sands at the three sediment stations for which 

the temperature effect was empirically defined.

A complete expression of the temperature effect 

might be made in terms of several graphs, each for 

a different median size of bed material and each 

similar to figure 17. However, presently available 

information is inadequate to define such graphs accurately either empirically or theoretically.

In the absence of such detailed and accurate graphs, 

an oversimplified relationship indicated by the dashed 

curves on figure 24 is suggested as a practical measure 

of the effect of water temperature on the discharge 

of sands in sand-bed streams. It is based on figures 18 

and 23 and on general considerations of the effect of 

viscosity on the discharge of sands. It may be satisFIGURE 22. Adjustment for effect of water temperature on the discharge of 

sands, Niobrara River near Cody, Nebr.

scatter widely, in general the discharge of sands 

decreases as the temperature increases.

The curves of figure 7 and 22 are only first 

approximations of the relationships that they represent. Second approximations can be obtained by 

using the curve of figure 22 to adjust the observed 

discharges of sands to equivalent sand discharges 

for a constant temperature of, for example, 60°F. 

The temperature-adjusted discharges of sands can 

then be plotted to obtain a second approximation of 

an average curve of relationship of discharge of sands 

to mean velocity. This second approximation is, of 

course, for a temperature of 60°F. Next, the ratio of 

each observed discharge of sands (unadjusted) to the 

average discharge of sands from the curve of second 

approximation can be plotted against water temperature to obtain the second approximation of the effect 

of changes in water temperature. For the discharge

0.4

EXPLANATION

Niobrara River

Middle Loup River 

Mississippi River

'From fig 17 (computed for 

depth of 10 ft. and velocity of 7.0 fps)

________I I

20 100

WATER TEMPERATURE, IN °F

FIGURE 23. Effect of water temperature on the discharge of sands in four natural

streams.
DISCHARGE OF SANDS AND MEAN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS A31

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A32 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

factorily accurate for most natural streams whose bed 

sands have median diameters of 0.20 to 0.40 mm but 

may be very inaccurate for some other particle sizes, 

for very low velocities, or for very deep flows. A 

theoretically accurate temperature correction would be 

different for each particle size of sand in suspension 

and presumably cannot be made without a more exact 

knowledge of the vertical distribution of sediment 

than is now available. Fortunately, the temperature 

correction generally is rather small, and a large percentage error in the correction usually causes only a 

moderate percentage error in discharge of sands.

A curve for making rough estimates of adjustments 

for different particle sizes is also given on figure 24, 

but it may be very inexact for some particle sizes. 

This curve and its usage will be discussed in more 

detail later.

CONCENTRATION OF FINE SEDIMENT

According to D. B. Simons, E. V. Kichardson, and 

W. L. Haushild (written commun.), a high concentration of fine sediment has an effect on the fall velocity 

of sands that is explainable in terms of the change 

that the fine sediment causes in the apparent viscosity 

of the water-sediment mixture. Their laboratory 

investigations showed that the median fall diameter of 

each of three sands decreased as the concentration of 

suspended fine sediment increased. Roughly four times 

as high a concentration of kaolin as of "bentonite" was 

required to decrease the median fall diameter by equal 

amounts. (However, the fine material that Simons, 

Kichardson, and Haushild called "bentonite" contained 

about 25 percent of particles coarser than 0.004 mm 

and about 3 percent coarser than 0.062 mm. In this 

paper the term "bentonite" when inclosed in quotation 

marks refers specifically to the fine sediment that 

they used.) Although these laboratory findings show 

the great effect that high concentrations of fine sediment have on the discharge of sands, they cannot be 

used directly to indicate the effect in natural streams.

To some extent the effect of an increase in concentration of fine sediment is equivalent to a decrease of 

water temperature. Median fall diameters of 0.19 and 

0.29 mm were decreased about as much by an increase 

of 40,000 ppm of "bentonite" as by a change in water 

temperature from 100° to 32°F. The same general 

equivalence can be established between the change in 

concentration of "bentonite" that will cause a given 

change in apparent viscosity of the water-sediment 

mixture and the change in water temperature that 

will cause an equal change in the viscosity of distilled 

water.

For concentrations of "bentonite" on the order of 

50,000 ppm or less, an adjustment cofficient for their 

effect on the discharge of sands can be estimated on 

the basis of the indicated equivalent effect of a particular concentration of "bentonite" and a given 

change in water temperature; that is, if the temperature and the concentration of fine sediment affect the 

relationship of discharge of sands to mean velocity 

only through changes in viscosity or apparent viscosity, the adjustment coefficient can be approximated 

on figure 24 on the basis of the curves already drawn 

on that figure for water temperature. However, the 

extension of the graph to high concentrations of 

"bentonite" would obviously be very questionable if 

based only on the relationship to water temperature.

A crude idea of the adjustment coefficients for 

high concentrations of fine sediment can be obtained 

from information for Rio Puerco near Bernardo, 

N". Mex., whose sediment concentrations frequently 

are very high. Usually, the discharge of sands at a 

given mean velocity, especially a high mean velocity, 

is far greater (fig. 11) than that for flows over a 

bed of roughly similar sand in the Rio Grande or 

other streams. However, discharges of sands may 

be unusually high (fig. 10) in the Rio Grande at 

infrequent times when the concentrations of fine sediment are high. The clay carried by the Rio Puerco 

and the Rio Grande is probably bentonite or similar 

fine material, but appreciable quantities of the fine 

sediment are silt. Because 25 percent of the "bentonite" 

used in the laboratory studies consisted of particles 

coarser than clay, the fine material of the Rio Puerco 

and the Rio Grande probably is similar to the "bentonite." The ratios of observed discharges of sands 

(roughly adjusted to a water temperature of 60°F) 

for Rio Puerco near Bernardo to discharges of sands 

from the estimated curve for low concentrations of fine 

sediment (fig. 11) were plotted on a draft of figure 24. 

The ratios together with extrapolations from the curves 

for lower concentrations of fine sediment on figure 24 

were used to define the adjustment coefficients for high 

concentrations of "bentonite" or of the fine sediment that is transported by the Rio Puerco.

Discharges of sands computed from figure 24 

average roughly the same as the observed discharges 

of sands for the Rio Puerco, but the observed discharges tend to be somewhat higher at high flows 

and lower at low flows. (See fig. 25.) Although the 

percentage differences between computed and observed 

discharges of sands are rather large, the adjustments 

of figure 24 can be used to remove most of the wide

O

departures of the observed discharges of sands in the 

Rio Puerco from the approximate curve (fig. 11) for
DISCHARGE OF SANDS AND ME AN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS A33

low concentrations of fine sediment. Of course, the 

adjustments of figure 24 are unlikely to apply as well 

to other streams as to the Kio Puerco for which they 

were defined.

Presumably, the adjustment coefficients for water 

temperature and for concentration of fine sediment 

can be multiplied together to obtain a single adjustment coefficient for the combined effects of water 

temperature and concentration of fine sediment on the 

relationship between discharge of sands and mean 

velocity.

Adjustment coefficients similar to those on figure 

24 will certainly vary with changes in bed-material 

size, but the relative effects of the different factors 

that affect the coefficients have not been satisfactorily 

evaluated. Hence, the right-hand diagram of figure 24 

is only a guess at the relative effect of concentration 

of fines or of water temperature for different median 

sizes of bed sediment. The effect is stated in,percent, 

and the adjustments on the main diagram of figure 24 

are assumed to be for a median size of bed sediment 

of 0.20 to 0.30 mm and to represent 100 percent, Of 

course, the adjustment coefficients from the main 

diagram of figure 24 are not to be directly multiplied 

by percentages from the right-hand diagram; rather, 

the adjustment coefficients minus 1.00 are to be multiplied by the percentages. For example, if an adjustment coefficient from the main diagram is 1.50 and 

the median diameter of the bed sediment is 0.50 mm, 

the adjustment for the effect of viscosity would be

10,000

1000

100

10 100 1000 10,000 

COMPUTED DISCHARGE OF SANDS, IN TONS PER DAY PER FOOT OF WIDTH

FIGURE 25. Comparison of computed and observed discharges of sands for Rio 

Puerco near Bernardo, N. Mex.

60 percent of 0.50 or 30 percent. The final adjustment 

coefficient would be 1.30.

If the fine sediment transported by a stream is 

mostly kaolin, the adjustments of figure 24 might 

be used as a very rough estimate after the concentrations of fines is divided by four (D. B. Simons, E. V. 

Richardson, and W. L. Haushild, written commun.). 

The effects of high concentrations of other fine sediments or of mixtures of fine sediments are scarcely 

known at all. Also, the chemical character and the 

concentration of the dissolved solids in the flow probably control to an appreciable degree the effect of 

any particular high concentration of fine sediment.

At times, the fine sediment may deposit on or in 

the top layers of the streambed if the concentrations 

are very high and if the velocities near the bed are 

relatively low. The presence of fine sediment at the 

surface of the bed may reduce the discharge of sands. 

If the free movement of the sand particles at the surface of the bed is known to be inhibited by the fine 

sediment, the discharge of sands may be relatively 

low, and the adjustments of figure 24 should not be 

applied.

The adjustment for the effect of high concentrations 

of fine sediment on the discharge of sands could be 

based on the assumption that the fall velocity of a sand 

particle is the only measure of size that has an appreciable effect on the discharge of the sand. In other 

words, a sand bed whose median particle size is 0.40 

mm might be assumed to have the same discharge of 

sands for a given condition of flow as a bed whose 

median particle size was 0.20 mm if the concentration 

of fine sediment was high enough to reduce the fall 

velocity of the 0.40 mm sand to that of a 0.20 mm sand. 

This type of adjustment was not made for two main 

reasons. First, in shallow flows the discharge of sands 

seems to increase appreciably if the concentration of 

fine sediment is greatly increased even for ranges of 

depth and velocity at which a decrease in particle 

size of the bed sands apparently causes little increase 

in the discharge of sands. Second, knowledge of the 

relationship of the discharge of sands to mean velocity 

and depth is very inexact for sands that have as low 

fall velocities as those of 0.25 to 0.40 mm sands when 

settling through water that contains very high concentrations of fine sediment.

In general, the effect of fine sediment on the 

relationship between discharge of sands and mean 

velocity is highly complex and poorly understood. 

The effect, although sometimes very large in natural 

streams, has usually been disregarded in the computation of discharges of sands or bed material. The 

relatively simple adjustments suggested here are
A34 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

intended to give only a rough idea of (a) the actual 

relationships and (b) the possibly large effect of 

concentration of fine sediment on the discharge of 

sands. Computed discharges of sands that are based 

on these adjustments for high concentrations of fine 

sediment are likely to be very inaccurate. However, 

the comparatively minor adjustments for concentrations of fine sediment lower than perhaps 50,000 ppm 

may usually be realistic for flows in most natural 

streams whose bed materials have median diameters of 

0.20 to 0.40 mm.

GRAPHS FOR COMPUTING DISCHARGE OF SANDS

The complicated relationships involving major 

factors that control the discharge of sands in sandbed streams can probably be better understood from 

graphs than from mathematical equations. Also, the 

discharges of sands can be computed more readily 

from graphs than from equations. Hence, graphs were 

prepared to indicate the relationships of discharge of 

sands to the major factors that affect the discharge 

of sands. The general concept of sediment transportation gives an idea of qualitative relationships to be 

expressed by the graphs; but until sediment transportation is better understood, the quantitative definition 

of the graphs must depend mainly on field and laboratory information.

BASIC INFORMATION

A graph such as the one (fig. 19) for a bed sand 

whose median diameter is 0.30 mm could be defined 

readily if enough accurate determinations of discharge of sands were available for a particular size 

of bed sand, for low concentrations of fine sediment, 

and for one water temperature. Enough determinations 

would require discharges of sands for a wide range 

of depths and velocities. In the absence of enough 

accurate information, a few graphs, each similar to 

that of figure 19 and each for a different size of bed 

sand, were approximated from somewhat random and 

sometimes questionably accurate data. None of the 

flume or field information was collected specifically 

for this analysis of the discharge of sands.

Total discharges of bed material over beds of sand 

were tabulated for many flume investigations, but 

no attempt was made to investigate all flume studies 

of bed-material discharge because the relationships 

were to be defined mainly for field streams. Particularly helpful were published data from flume experiments that were reported by Gilbert (1914), Barton 

and Lin (1955), Simons, Richardson, and Albertson 

(1961), and Brooks (1958).

Field data on the discharge of sands for several 

streams in different parts of the United States were 

tabulated and studied. Sand discharges for Middle 

Loup Kiver near Dunning, Nebr. (Hubbell and 

Matejka, 1959), and for Niobrara River near Cody, 

Nebr. (Colby and Hembree, 1955), were approximately total discharges. They were based on measurements of bed-material discharge 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 measured at nearby 

cross sections where the flow was over beds of sand 

whose median diameter was determined by sampling. 

For most other sand-bed streams in this study, the 

total discharges of sands were not so directly determined, and only the measured discharges of sands 

were known. For these streams, the total discharges 

of sands usually had been computed in local offices 

from the modified Einstein procedure (Colby and 

Hembree, 1955), but some were computed locally or 

by the writer from measured discharges of sands and 

from the relationship of unmeasured sediment discharge to mean velocity (Colby, 1957). Thus, total 

discharges of sands based partly on indirect computations were used for Elkhorn River near Waterloo, 

Nebr. (Beckman and Furness, 1962), the lower Colorado River (U.S. Bureau of Reclamation, 1958), 

Pigeon Roost Creek and tributaries in northern Mississippi (unpublished information, Agricultural Research Service, U.S. Dept. of Agriculture, University, 

Miss.), Mississippi River at St. Louis, Mo., the Cedar, 

Little Blue, North Loup, and South Loup Rivers in 

Nebraska, and the Rio Grande and the Rio Puerco in 

New Mexico (unpublished information, U.S. Geol. 

Survey, Lincoln, Nebr., or Albuquerque, N. Mex.).

GENERAL METHOD OF ANALYSIS

Graphical analysis of the -information on the discharge of sands consisted in eliminating progressively 

the approximate effect of one variable after another. 

On the basis of the assumption that discharge of 

sands is proportional to stream width, all discharges 

of sands were expressed in tons per day per foot of 

stream width. These discharges for a single cross 

section or sometimes for a combination of cross sections were then usually plotted against mean velocity 

to define a relationship that applied specifically to 

each section or combination of sections. For some sections, the ratios of observed discharges of sands to 

the discharges from the average curve were plotted 

against water temperature to obtain the approximate 

effect of water temperature (fig. 22, for example) on 

the relationship between discharge of sands and mean
DISCHARGE OF SANDS AND MEAN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS A35

velocity. After the effect of water temperatures had 

been defined for a few cross sections (fig. 23), the discharges of sands per foot of width were roughly 

adjusted to a water temperature of 60°F, if the water 

temperature were known.

The next major step in the graphical analysis was 

to define the separate graphs showing the effect of 

depth (such as fig. 19) on bed sands of different 

median diameters. Each temperature-adjusted discharge of sands for a median diameter of bed sands 

between about 0.06 and 0.90 mm was plotted against 

depth on the graph sheet for the selected median 

diameter (0.10, 0.20, 0.30, 0.40, 0.60, or 0.80 mm) 

that was closest to the reported median diameter. 

The mean velocity was noted beside each plotted 

point. Some discharges of sand over beds whose 

median diameter was, for example, about 0.25 mm 

were plotted on graphs for both the nearest smaller 

and the nearest larger median diameter. Lines for 

different mean velocities were drawn on each of the 

six graphs on the basis of the plotted points and 

judgment. The general form of the graph of figure 

19 was used as a guide. On parts of a graph for 

which data were insufficient, somewhat contradictory, 

or entirely lacking, rough estimates were made by 

comparison among the six graphs or by consideration 

of the general concept of sediment transportation. 

Then the six graphs were redrawn as relationships 

between discharge of sands and mean velocity for each 

of four depths of flow; the median size of the bed 

sands was used as a second independent variable. The 

smoothing of the curves for these relationships, which 

were first approximations of the graphs of figure 26, 

removed any large inconsistencies from one median 

size of bed sand to another. Smoothing of the graphs 

was continued by replotting alternately in the form of 

figure 19 and in the form of figure 26 until a reasonably 

consistent set of curves (fig. 26) was obtained. The 

form of figure 26 was selected rather than that of 

figure 19, because the interpolation for the effect 

of depth can usually be made more conveniently and 

accurately than the interpolation for the effect of mean 

velocity.

The derivation of figure 24 for the effect of changing 

water temperature and for the effect of high concentartion of fine sediment has already been discussed. 

The effect of bed configuration on the relationship 

between mean velocity and discharge of sands may 

sometimes be large and is likely to be most evident 

in natural streams at the cross sections that are 

uniform laterally. One of the dashed curves of figure 

9, the curve for Pigeon Roost Creek near Byhalia,

Miss., has a flatter slope at mean velocities of 3.0 to 

5.0 feet per second than at either lower or higher mean 

velocities, and the flatter slope coincides with the transition from a dune bed to a plane bed. A distinguishing 

characteristic of the channel at this station is lateral 

uniformity, which results in a change of bed configuration at about the same time across most of the channel 

width. In natural streams, changes in bed configuration 

usually occur gradually; that is, a change usually 

affects only a small part or strip of the streambed at 

first and then spreads over more of the bed. Thus, at 

any one time only part of the width of the bed is 

likely to be plane unless the channel is unusually uniform laterally. According to D. B Simons (written 

commun.), flume experiments have shown considerably 

lower discharges of sands for a given mean velocity 

over a plane bed than over an antidune bed. Such a 

relationship might be explainable, at least in part, by 

the fact that the turbulence for a given mean velocity, 

depth, viscosity, and particle size should be less over 

a plane bed than over a dune or antidune bed. Although 

no adjustments have been applied in this paper for 

differences in bed configuration and none may be 

generally required for most natural streams, the discharge of sands should be expected to be considerably 

lower over a plane bed than for the same velocities 

and depths over irregular sand beds.

The available information was inadequate for determining the usually minor effects of such factors as 

the variations or scattering of particle size, velocity, 

or depth about the median or mean. Obviously, the 

variation of particle size of the bed material about 

the median diameter may sometimes appreciably affect 

the discharge of sands, perhaps especially at low flows. 

A nonuniform lateral distribution of velocity and 

depth may also have a large effect on the discharge of 

sands at some irregular cross sections. If an appreciable part of a cross section is shallow and has a low 

velocity as compared with the rest of the section, the 

discharge of sands may well be computed separately 

for each of two or more parts of the cross section.

The graphical analysis of the data, which is largely 

based on judgment and evaluation of complex factors, 

would not be exactly duplicated if the study were repeated. Obviously, the adequacy and accuracy of the 

analysis cannot be directly stated but can be crudely 

indicated by comparison of observed discharges of 

sands with discharges computed from the results of 

the graphical analysis. This comparison and an evaluation will be made as a part of the discussion of 

probable accuracy of sand discharges that are determined from the final graphs.
A36 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

10,000

1000

100

MEAN VELOCITY, IN FEET PER SECOND 

FIGURE 26. Relationship of discharge of sands to mean velocity for six median sizes of bed sands, four depths of flow, and a water temperature of 60° F.
DISCHARGE OF SANDS AND MEAN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS A37

DETERMINING THE DISCHARGES OF SANDS FROM 

THE GRAPHS

The discharge of sands in a sand-bed stream can 

be computed from the empirical graphs according 

to one or more of three methods. The choice of a 

method depends mainly on the number of variables 

that need to be considered in the computation. If 

the velocity, depth, bed-material sizes, water temperature, and concentration of fine sediment are 

within the general range for which the curves of 

figure 13 were defined, the effect of variables other 

than mean velocity may perhaps be disregarded, and 

the discharge of sands per foot of width may be 

taken with reasonable accuracy from one of the 

curves of figure 13. However, if the depth effect is 

likely to be appreciable, the discharge of sands 

should be determined from figure 26. If concentrations of fine sediment are high and water temperatures 

differ considerably from 60°F, the discharge of sands 

from figure 26 should be adjusted according to figure 

24. Examples show the computation of the discharge 

of sands according to each of the three methods.

First method discharge of sands based on figure 13

Selected data for example 1:

Mean velocity__________-_-_________ft per sec__ 2. 70

Depth___.______.._-___._..__________ft__ 3. 5

Median size of bed sediment______________mm_i 0. 37

Water temperature_______________________° F__ 65

Concentration of fine sediment.____________ppm__ 1, 500

Summary of the computations: Because the size of bed material 

is about that for Pigeon Roost Creek, the discharge of sands 

per foot of stream width can be taken directly as 11 tons per 

day per foot of width from the curve for Pigeon Roost Creek 

on figure 13. (If computed by the second method, the result 

is about 8.5 tons per day per foot.)

Second method discharge of sands based on figure 26

Selected data for example 2:

Mean velocity______________________ft per sec__ 6. 2

Depth-_--_-------------------------_------ft-. 15. 5

Median size of bed sediment______________mm__ 0. 26

Water temperature_______________________°F__ 57

Concentration of fine sediment ____________ _ppm_ _ 650

Summary of the computations: From figure 26 the indicated 

discharges of sands are by interpolation for median size about 

188 and 400 tons per day per foot for depths of 10 and 100 

feet, respectively. Interpolation for the depth of 15.5 feet 

can be made by guess, by use of a simple diagram, or with a 

straightedge on a sheet of log-log paper to obtain 210 tons per 

day per foot. This discharge of sands can be quickly checked 

against a discharge of 220 tons per day per foot that might be 

picked directly from the curve (fig. 13) for the Mississippi 

River at St. Louis, Mo. Thus, the first method can usually 

be used as an easy check for major errors in computations by 

the second method.

Third method discharge of sands based on figures 24 and 26

Selected data for example 3:

Mean velocity..-.- _____--__-_--- ft per sec__ 6. 5

Depth______------------_------------------ft-- 4. 8

Median size of bed sediment._______-__--__mm__ 0. 43

Water temperature________--__-__------- 0 F__ 75

Concentration of fine sediment, mostly bentonite

ppm__ 33,000

Summary of the computations: From figure 26 the discharges 

of sands are about 92 and 150 tons per day per foot of width 

for depths of 1.0 and 10 feet, respectively. Hence, about 

130 tons per day per foot of width can be interpolated for the 

depth of 4.8 feet. The adjustment coefficient from figure 24 

for 75°F and a depth of 4.8 feet is 0.86 for a median diameter 

of 0.20 or 0.30 mm. Also, the adjustment coefficient for 33,000 

ppm of fine sediment, mostly bentonite, is 1.92 for a median 

diameter of 0.20 to 0.30 mm, and the total adjustment coefficient is 1.92X0.86 or 1.65. According to the right-hand 

graph of figure 24, the effect of a change in viscosity or 

apparent viscosity is only 78 percent as large for the median 

diameter of 0.43 mm as for a median diameter of 0.20 or 

0.30 mm. Therefore, 78 percent of (1. 65 1.00) or 0.51 is 

added to 1.00 to obtain the estimated adjustment coefficient 

for the median diameter of 0.43 mm. The 130 tons per day 

per foot multiplied by 1.51 gives 196 tons per day per foot, 

which could well be rounded to 200 tons per day per foot 

because the discharge of sands ordinarily should not be 

determined to more than two significant figures. The discharge of sands from the curve of figure 13 for Pigeon Roost 

Creek is about 150 tons per day per foot.

PROBABLE ACCURACY OF THE GRAPHS

Three phases of the probable accuracy of the graphs 

for computing the discharge of sands need consideration. In order of their discussion, these phases are the 

experimental errors inherent in the basic field and 

laboratory information, the accuracy of the definition 

of the graphs for the data that were included in the 

study, and the applicability of the graphs to discharges 

of sands for streams that were not included in the 

study.

EXPERIMENTAL ERRORS IN BASIC INFORMATION

A major difficulty in an evaluation of the probable 

accuracy of the graphs is that the amount of experimental error in individual determinations of discharge 

of sands is seldom known satisfactorily. However, the 

fact of appreciable experimental error can be easily 

established. For example, rather wide variations of 

concentration of suspended sediment (mostly but not 

all sands) were measured at a turbulence flume in 

Middle Loup River at Dunning, Nebr., by Benedict, 

Albertson, and Matejka (1955). Also, C. F. Nordin 

and J. K. Culbertson (written commun.) reported 

two sets of suspended-sediment samples at about the 

same times and at nearly steady flow at each of seven 

sections in Rio Grande near Barnalillo, N. Mex.; 

these samples, although probably collected more care-
A38 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

fully than most samples, showed considerable percentage differences in the concentrations of sands. To the 

nearest whole percent, the differences between the two 

concentrations of silt plus clay for a cross section ranged 

from 0 to 8 percent and averaged 3 percent for the 

seven cross sections. The differences between the two 

concentrations of suspended sands ranged from 3 to 33 

percent and averaged 20 percent. Such differences 

as these (which probably are reasonably typical) 

between individual determinations of average concentration of suspended sands at a cross section indicate 

proportional errors in the measured discharges of 

the suspended sands. (Actually, these differences 

are, of course, measures of consistency rather than of 

absolute accuracy.) In addition to differences in carefully determined discharges of sands at a cross section, 

wide differences in measured discharges of sands may 

occasionally result from the collection in a suspendedsediment sample of sands from the bed layer or the 

streambed unless the excess sand is noted immediately 

and the sampling is repeated. Most experienced field 

men consider that sampled concentrations of suspended 

sands are likely to be much too large at times but 

are unlikely to be much too small. Usually several 

individual determinations of the discharge of sands 

may be required to establish an accurate average discharge of sands for a particular time and flow. Even 

in a flume experiment for which conditions of flow and 

sediment discharge are likely to be more constant 

than in field investigations, the sampling of sands 

may have to be continued over a period of a few hours 

in order to establish an accurate average discharge 

of sands.

Because the average discharge of sands during 

a period of steady flow is the discharge that should 

correlate with characteristics of the flow and the bed 

sand, departures from this average are considered to 

be experimental errors. Thus, the experimental error 

is a measure of the difference between an experimentally determined or computed discharge of sands 

and the average discharge of sands in the same flow 

at a steady state. It is not the difference between the 

experimentally determined or computed discharge of 

sands and the actual instantaneous discharge of sands 

at that time. Therefore, because of short-time fluctuations in the discharge of sands, the experimental error 

may be large even though the determination of the 

instantaneous discharge of sands was accurate.

At many of the high flows and sometimes at 

rather low flows per foot of width, the mean velocities 

used in this study were determined from bridges 

where the sand bed is affected by local scour and where 

piers may cause turbulence that is relatively high as

compared to the mean velocity. In general, the discharges of sands per foot of width at bridge sections 

are likely to be larger than they would be for the 

same velocities in representative reaches of channel 

away from the bridges.

Lateral distributions of depth and velocity are sometimes so far from uniform that the reported means 

are not good measures of the discharge of sands. 

However, inaccuracies that are caused in the relationships of discharge of sands to mean velocity and to 

depth by poor lateral distributions are more strictly 

failures in analysis of the data than experimental 

errors.

If the slope of the water surface is used as a parameter of the discharge of sands, considerable experimental error should be expected in the determinations 

of slope, unless the total fall through the reach is 

large enough to be determined accurately and unless 

several observations of the slope are made and 

averaged.

Total discharges of sands in field streams are 

generally measurable only at very constricted sections or at structures that have been designed for 

the determination of the total sand discharges of 

the stream. The widths, depths, and velocities at 

such places are not useful measures of the discharge 

of sands in sand-bed streams. If useful relationships 

are to be obtained, the discharges of sands at these 

places must be correlated with widths, depths, and 

velocities at nearby reaches of channel where the 

streambed is mostly cohesionless sand. Obviously, 

temporary net scour or net fill and the short-time 

fluctuations in discharge of sands even at constant 

flow make uncertain any individual determinations 

of total discharge of sands in terms of conditions 

of flow at another cross section.

In most streams, only part of the discharge of sands 

is measured, and the rest is computed somewhat indirectly. The computation of part of the discharge 

of sands introduces error because the procedures for 

computing the unmeasured discharges of sands are by 

no means exact and because some of the procedures 

depend on the measured concentrations of sand, which 

are themselves inexact. The probable inaccuracy of 

the computation of the discharge of unmeasured sands 

is known only roughly.

DATA INCLUDED IN THE STUDY

The many and complicated steps involved in the 

graphical analysis cannot readily be reproduced or 

explained in detail in this paper and were partly 

based on judgment. Hence, the probable accuracy 

of the graphs on figures 24 and 26 is difficult to
DISCHARGE OF SANDS AND MEAN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS A39

evaluate or to state. A practical measure of the probable accuracy for data that were included in the study 

can be obtained by comparing the discharges of sands 

computed from figures 24 and 26 with the original 

observed discharges of sands. (The term "observed 

discharges" is used for simplicity even though for 

many field streams the "observed discharges" of sands 

are partly based on indirect computations of the unmeasured discharges of sands.) Because many observations were used in the graphical analysis, an arbitrary

1000

selection was made for the test of accuracy. Within 

each set of field or flume data, the first, sixth, eleventh, 

sixteenth, etc., determinations were selected. Thus, at 

least one observation was included from each set of 

data.

Although the difference between the computed and 

the observed discharge of sands sometimes is large, 

especially for low velocities and shallow depths, the 

general relationship for field sand-bed streams is 

reasonably good. (See fig. 27.) Of about 200 indi0.1

1 10 100 

COMPUTED DISCHARGE OF SANDS, IN TONS PER DAY PER FOOT OF WIDTH

1000

FIGUBE 27. Comparison of computed and observed discharges of sands for natural streams that were included in the study.
A40 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

vidual comparisons, about three-fourths of the observed 

discharges of sands are within limits that are 100 

percent larger and 50 percent smaller than the computed discharges of sands. The widest percentage 

differences and proportionately the fewest close agreements are for shallow depths and low mean velocities. 

Usually in field sand-bed streams the lateral distributions of depth and velocity are more uneven for shallow 

flows, especially perhaps in very wide channels, than 

for deep flows. Also, at low velocities the percentage 

effects of small changes in particle size or in mean 

velocity may be large. For example, at a depth of 1 

foot and for a median particle size of 0.30 mm, a 

change in mean velocity from 1.50 to 1.70 feet per 

second slightly more than doubles the discharge of 

sands, according to figure 26. The differences in tonnage of the computed and the observed discharges of 

sands at these low velocities and shallow depths are 

usually insignificant in terms of the annual discharge 

of sands for creeks and rivers even though the percentage difference is large.

At low depths and mean velocities, a somewhat 

better average agreement between computed and 

observed discharges of sands in natural streams could 

be obtained by revising the curves of figures 26, but 

such a change is probably not worthwhile until the 

reasons for the differences at low flows are better 

understood. Perhaps some differences are due to nonuniformity of the cross sections and could be partly 

removed by subdividing the cross sections into parts 

and then computing the discharge of sands separately 

for each of the parts. Disagreement may also be caused 

by different bed configurations at the same mean 

velocity.

The agreement between the computed and the 

observed discharges of sands is a little better for flows 

over sand beds in flumes (fig. 28) than in natural 

streams. Any change in figure 26 that would make 

the computed discharges of sands agree better with 

the observed discharges of sands from 1 to 10 tons 

per day per foot of width in natural streams would 

make the agreement poorer for the discharges of sands 

in flumes. Obviously, separate graphs like those of 

figure 26 could be prepared for natural streams and 

for flumes, or an arbitrary adjustment could be 

applied to the computed discharges of sands for low 

flows in natural streams. However, such procedures 

may not be desirable at this time.

STREAM REACHES AND FLUME INVESTIGATIONS NOT 

INCLUDED IN THE STUDY

Of course, really significant tests of the accuracy 

of the derived graphs should be based on the total 

discharge of sands at stream reaches for which information was not used to define the graphs. Some 

unpublished information for such reaches of natural 

streams was available in the files of the Geological 

Survey at Lincoln, Nebr., and Worland, Wyo. Also, 

two computations of total discharges of sands at 

individual verticals in the Missouri River at Omaha, 

Nebr., were furnished by Lloyd C. Fowler, U.S. 

Army Corps of Engineers, Omaha, Nebr. The observed discharges of sands in natural streams were 

determined by the modified Einstein procedure. The 

flume investigations by the U.S. Waterways Experiment Station (1935) were not used to define the 

graphs, so about every fifth run that had an observed 

discharge of sands greater than 0.1 ton per day per 

foot of width was selected as a check on the probable 

accuracy of the graphs for flume flows.

The comparison between the computed discharges 

of sands and the observed discharges of sands for 

cross sections that were not used to define the 

graphs is generally good for the cross sections that 

are relatively uniform and moderately deep and that 

have dependable basic information. The comparison 

was relatively much poorer for some of the other cross 

sections. (See fig. 29.) The computed and observed 

discharges of sands agree rather closely for the individual verticals of Missouri River at Omaha and for 

the cross sections of Niobrara River near Valentine, 

Nebr., Snake River near Burge, Nebr., and Bighorn 

River at Thermopolis, Wyo. The Popo Agie River 

near Riverton, Wyo., is not a sand-bed stream, 

although the beds of many pools are mostly sand. 

Also, the samples on which the size of the bed material 

for the Popo Agie River is based were collected more 

than a year after the dates for which the discharges 

of sands were observed. Probably because of the 

incomplete sand cover, the observed discharges of 

sands are somewhat lower than 50 percent of the 

computed discharges of sands. In general, shallow 

cross sections, such as those of Fivemile, Muddy, and 

Badwater Creeks and Pavillion drain, have considerably greater observed discharges of sands than 

computed discharges of sands. Beaver Creek near 

Arapahoe, Wyo., although its flows were rather 

shallow, does have a close relationship between observed and computed discharges of sands. The cross 

sections of Badwater Creek at Bonneville, Wyo., in 

addition to being shallow, are nonuniform laterally at 

low flow and have very high concentrations of fine 

sediment. In general, the graphs of figures 24 and 26 

seem to represent the discharges of sands as well for 

these stream reaches that were not included in their 

definition as for the stream reaches that were included.
DISCHARGE OF SANDS AND MEAN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS A41

100 

1 10 

COMPUTED DISCHARGE OF SANDS, IN TONS PER DAY PER FOOT OF WIDTH

FIGURE 28. Comparison of computed and observed discharges of sands for flume data that were included in the study.

100

For flume investigations, the graphs of figures 

24 and 26 do not apply as well to the discharges of 

sands that were reported by the U.S. Waterways 

Experiment Station (fig. 30) as they did to the discharges of sands that were used to define the graphs. 

Perhaps the observed discharges of sands from the 

Waterways Experiment Station studies were lower 

than those from other flume studies because of relatively more coarse particles in several of their sands. 

The inconsistencies may be due, at least partly, to 

differences in bed configuration at about the same 

shallow depths and velocities. Sand 6 (U.S. Waterways Experiment Station, 1935), for example, had

very little sand that was appreciably finer than the 

median particle size but had considerable sand that 

was considerably coarser than the median particle 

size. Observed discharges of sand 6 were very low 

as compared with the computed discharges of sands. 

On the basis of figures 27 through 30, the graphs 

of figures 24 and 26 probably provide a means of 

computing fairly consistent discharges of sands at 

a particular cross section. However, especially for 

shallow flows and low velocities, the effects of some 

factors that vary from one stream or flume to another 

or from one sand to another are not yet satisfactorily 

 represented by the graphs. Although the graphs
A42

10,000

SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

1000

o

 

0

X

a 

D

A 

A 

M

EXPLANATION

Badwater Creek at Bonneville, Wyo 

Beaver Creek near Arapahoe, Wyo 

Bighorn River at Thermopolis, Wyo 

Fivemile Creek near Shoshoni, Wyo 

Fivemile Creek near Riverton, Wyo 

Muddy Creek near Shoshoni, Wyo 

Pavillion drain near Pavillion, Wyo 

Popo Agie River near Riverton, Wyo 

Missouri River at Omaha, Nebr 

Niobrara River near Valentine, Nebr 

Snake River near Burge, Nebr

0.1

1 10 100 

COMPUTED DISCHARGE OF SANDS, IN TONS PER DAY PER FOOT OF WIDTH

1000

PIGUEE 29. Comparison of computed and observed discharges of sands for stream reaches that were not included In the study.
DISCHARGE OF SANDS AND MEAN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS A43

0.1 1 10

COMPUTED DISCHARGE OF SANDS, IN TONS PER DAY

PER FOOT OF WIDTH

FIGUBE 30. Comparison of computed and observed discharges of sands for flume 

data (U.S. Waterways Experiment Station, 1935) that were not included in the 

study.

cannot be used to obtain highly accurate discharges of 

sands in sand-bed streams, they provide a quick and 

reasonably accurate procedure for computing the 

average discharge of sands for many sand-bed streams. 

The graphs are applicable only to sand-bed streams 

that do not have unusually rough confining walls or 

distinct pools and riffles (except, perhaps, at very low 

flow). Even though the bed of a stream is composed 

of cohesionless sand, the graphical relationships may 

not apply well at narrow sections that are confined 

between rough rock walls. Rough rock walls, such 

as those at the cableway on the Colorado River near 

Grand Canyon, Ariz., may cause considerably more 

turbulence than would normally exist for a combination of velocity and depth; as was previously pointed 

out, the vertical distribution of the sands is determined 

by the turbulence, which seems to be at least partly 

dependent on the total shear rather than on the shear 

with respect to the sediment particles.

APPLICATIONS

The empirically defined graphs have two obvious 

general uses. One is to show the effect on the discharge of sands of a change in any major variable. 

The other is to compute the approximate discharge 

of sands in flows over sand beds.

In addition to the two general uses, more specialized 

applications of the graphs also can be made. For 

example, the average relationship at a particular

cross section between discharge of sands and mean 

velocity for a water temperature of about 60°F can be 

readily approximated from the graphs of figure 26 if 

the median diameter of the bed sands and the widths, 

depths, and mean velocities are known for one or two 

low, medium, and high streamflows.

The change is velocity from a wide to a narrow 

cross section along a reach of stable channel can also 

be approximated on the basis of continuity of flow 

and of discharge of sands. Because a large change 

in discharge of sands usually accompanies a small 

change in mean velocity, the mean velocity must 

change comparatively little from one cross section 

to another along a stable reach of channel. This 

relative constancy of mean velocity has sometimes 

been noted in terms of a tendency for the area of 

cross section to remain about constant along a reach 

of sand-bed channel at steady flow. According to 

tht curves of figure 13, an increase of 100 percent 

in width might be compensated (a constant discharge 

of sands might be maintained from one cross section 

to the other) by a 25- or 30-percent decrease in mean 

velocity.

The large changes in discharge of sands for small 

changes in mean velocity help to explain channel 

configuration. For example, a particular flow per 

foot of width can occur at a wide range of depths 

and mean velocities; but, if that flow transports a 

given discharge of sands per foot of width, the 

velocity and depth become fixed within narrow limits 

for a given water temperature and for low concentrations of fine sediment. In broad terms, a particular 

discharge of sands per foot of width largely determines the mean velocity that must exist in a stable 

reach of a sand-bed stream, which has a particular 

size of bed sands. The mean velocity and the rate 

of flow per foot of width largely determine the depth 

of flow and hence the slope if the resistance to flow 

is known. (However, within certain ranges of depth, 

velocity, particle size, and water temperature, the 

resistance to flow may change widely and be somewhat 

indeterminate.)

In a channel reach that is not at equilibrium, the 

probable rate of net scour or net fill may be approximated from the graphs, provided that the flowduration curves, the channel widths, and the velocities for given rates of streamflow can be predicted 

for each end of the reach.

An understanding of the effect of a few major 

factors on the discharge of sands may aid in understanding the effect of other factors. For example, 

the effect of high concentrations of fine sediment on 

the discharge of sands in natural streams was not
A44 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

generally recognized for many years because the 

relationship of the discharge of sands per foot of 

width to mean velocity had not been defined for low 

concentrations of fine sediment. As soon as the 

dominant relationship to mean velocity was known, 

the effect of the high concentrations of fine sediment 

became obvious for such a stream as the Eio Puerco.

DEFICIENCIES OF BASIC INFORMATION

Because of the number of variables, the wide ranges 

in value of the variables, and the complex interrelationships among the variables, the empirical determination of the major graphs in this paper should have 

been based on a large amount of accurate information 

that was collected specifically for this study. Unfortunately, the available field and flume data were all 

collected for purposes other than this study and are 

deficient in coverage of the ranges of the variables, in 

applicability to the specific relationships to be defined, 

and to some extent in accuracy. Lack of accuracy was 

due partly to the necessity for computing the unmeasured part of the discharge of sands for most sediment 

stations but probably more to differences between 

measured discharge of sands and the correct time 

average for the cross section when samples were 

collected. A correct time average of discharge of 

sands is an average over a long enough period of time 

to eliminate both random and short-term cyclical 

variations in discharge of sands.

Adequate field information for a particular median 

diameter of bed sand should cover at least the usual 

range and combinations of depths and velocities in 

sand-bed streams in enough detail to define a graph 

like that of figure 19. The information is definitely 

inadequate if significant parts of such a figure must 

be based on estimates, interpolations, or extrapolations 

as was necessary to some extent for all the different 

median sizes of bed sands. For most particle sizes, 

available information is inadequate or entirely lacking 

especially for high velocities and for deep flows. 

Therefore, discharges of sands for Mississippi Eiver 

at St. Louis are particularly helpful because they are 

for deep flows and some are for moderately high 

velocities.

Field information on total discharges of sands is 

wholly inadequate to define a graph like that of 

figure 19 for median diameters of bed sands smaller 

than 0.10 mm or larger than 0.50 mm for most 

combinations of depth and mean velocity.

The effect of water temperature on the relationship 

of discharge of sands to mean velocity has been defined 

empirically for only a few streams and then only for 

the usual combinations of depths and mean velocity in

those streams. The somewhat similar but sometimes 

much larger effect of concentration of fine sediment on 

the discharge of sands is based mostly on a few laboratory studies and on the discharges of sands in Rio 

Puerco near Bernardo. More experimental data on 

the effect of water temperature and of high concentrations of fine sediment in field streams are badly needed.

Information applicable and sufficiently accurate to 

define possible effects of such usually minor factors 

as the lateral variation of velocity at a given mean 

velocity is almost entirely lacking. Also, the effects 

of deviation of sizes of the bed sands from the median 

size have been obscured by the scarcity of precise 

and pertinent information. The bed configuration is 

seldom known from available information for field 

streams; hence, the effect of bed configuration on the 

relationship between discharge of sands and mean 

velocity is difficult to establish.

The empirical determination of the relationships of 

discharges of sands in sand-bed streams to the major 

factors that control such discharges could be made more 

exact and dependable by a planned program for obtaining the needed coverage and accuracy of field information. Such a program should include at least the following procedures:

1. Determine the probable accuracy of the basic information on the discharge of sands in natural 

streams either by studying the accuracy of sediment discharges in general or by repeating individual field observations, particularly of concentration of sands, often enough to obtain a practical 

measure of the consistency of measured discharges 

of sands.

2. Determine by measurement the total discharges of 

sands in streams other than the Niobrara and the 

Middle Loup Rivers; that is, obtain more determinations of the discharge of sands at cross sections where all or nearly all the sands are in 

suspension and can be sampled and then relate 

these total discharges of sands to characteristics 

of the flow and the bed sands at nearby cross sections where the stream has a sand bed. These 

other streams should preferably have other bedmaterial sizes and combinations of depth and 

mean velocity than the Niobrara and the Middle 

Loup Rivers.

3. Use the total discharges of sands from procedure 2 

to confirm or to revise the present methods of 

computation so that discharges of unmeasured 

sands can be determined with acceptable accuracy 

in any sand-bed stream for which suitable information on flow and measured sediment discharge is 

available.
DISCHARGE OF SANDS AND MEAN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS A45

4. Use the total discharges of sands from procedure 2 

as part of the basis for determining or improving 

individual graphs similar to the graph of figure 19 

for bed sands whose median diameters are 0.10, 

0.20, 0.30, 0.40, 0.60, 0.80, and 1.00 mm.

5. Complete the definition of these graphs. Obtain 

measured discharges of sands for particle sizes, 

depths, and mean velocities for which total discharges of sands could not be measured; add to the 

measured discharges the computed unmeasured 

. discharges of sands; and define the graphs from the 

computed total discharges of sands.

6. Obtain accurate determinations of discharges of total 

sands (either measured or partly unmeasured) at 

the same cross sections and flows but at widely 

different water temperatures to define the effect of 

water temperature through the range of usual 

combinations of depth, velocity, and bed-material 

size.

7. Investigate in the laboratory the fall velocities of 

sand grains in water that contains high concentrations of different combinations of clays and 

silts of usual mineralogical compositions. Also, 

study the effect of changes in the dissolved solids 

in the flow on the effect of high concentrations of 

different fine sediments on the discharge of sands.

8. Check the effect of fine sediments as indicated by the 

laboratory studies against the apparent effect on 

the discharge of sands in natural streams.

After the major effects have been clearly defined, 

develop a program for the determination of the effects 

of usually minor variables on the discharge of sands. 

Presumably, such a program may require the careful 

control and evaluation of all major factors in order that 

relatively minor effects can be discerned.

CONCLUSIONS

The discharge of sands in a sand-bed stream depends 

on many variables whose effects are complicated and 

interrelated. Hence, no precise relationship between 

discharge of sands and characterics of the flow and 

sediment can be expected.

Flume investigations can provide much helpful 

information on sediment transportation; but, until 

scale effects are understood more completely, flume 

investigations of the discharge of sands are not 

model studies of the discharge of sands in field 

streams.

The use of discharge of sands per foot of width 

as a measure of the rate of transportation of sands 

is based directly on the assumption that, if other 

factors remain constant, the discharge of sands is 

in proportion to stream width. Although this assumption is not precisely correct, it seemed to afford a 

reasonable basis for comparison of sand discharges in 

streams of greatly different widths.

Concentration of sands is a poor basis for comparison of discharge of sands with the factors that 

control that discharge in natural streams because 

the concentration varies greatly with depth of flow. 

However, for a narrow range of depths in a set of 

flume experiments, concentration of sands may be 

a fairly satisfactory measure of transportation of 

sands.

For the velocities and depths at which most of the 

sand is transported in many natural streams, the discharge of sands per foot of width does not vary greatly 

with changes in the depth of flow.

Shear, which has been widely used as a measure 

of the discharge of sediment, is sometimes a poor 

measure. When the roughness of the channel changes, 

the discharge of the coarse sediment generally is more 

closely related to the mean velocity than to the shear. 

However, the vertical distribution of sands may vary 

more nearly with total shear than with shear velocity 

with respect to the particles.

When discharge of sands per foot of width at one 

cross section is plotted against mean velocity, a 

reasonably good relationship is usually defined for 

that cross section, unless the concentration of fine 

sediment is high and variable.

The relationships between discharge of sands per 

foot of width and mean velocity may be much the 

same for one cross section as for another, provided 

that differences in depth, water temperature, concentration of fine sediment, or size of bed sand are 

not excessively large. The agreement in relationships 

indicates that mean velocity is a dominant measure of 

the discharge of sands per foot of width in many 

sand-bed streams.

The relationship to mean velocity implies that mean 

velocity must be relatively constant from one cross 

section to another along a reach of sand-bed channel. 

If mean velocity were to change appreciably, the discharge of sands per foot of width would change much 

more than the mean velocity. Such a velocity change 

would require very large percentage differences in 

width. Otherwise more sands would be discharged 

at one section that the other, and the channel would 

aggrade or degrade.

Shear velocity computed from mean velocity correlates with discharge of sands about as well as mean 

velocity. The effect of depth on the relationship 

between discharge of sands and shear velocity computed from mean velocity is, however, somewhat 

different than on the relationship between discharge
A46 SEDIMENT TRANSPORT IN ALLUVIAL CHANNELS

of sands and mean velocity. The effect of depth 

indicates that for flume investigations shear velocity 

computed from mean velocity may be a more convenient parameter (because of a smaller and more 

consistent depth effect) than mean velocity and that 

for field streams mean velocity is the more convenient 

parameter because of the smaller effect of depth for 

many combinations of depth and mean velocity.

Stream power, the product of shear and mean 

velocity, may be a good measure of discharge of sands 

when roughness is about constant, but it is unsatisfactory for variable roughness.

The effect of depth on the relationships between 

discharge of sands per foot of width and mean 

velocity, shear, shear velocity computed from mean 

velocity, or stream power is complex and variable 

and may be large. The effect of depth is usually easier 

to apply to the relationship of discharge of sands 

to mean velocity or to shear velocity computed from 

mean velocity than to the relationship of discharge 

of sands to shear or to stream power. The effect of 

depth also largely determines the use of discharge 

of sands per foot of width rather than concentration 

as a measure of the discharge of sands.

Qualitatively, the effects of mean velocity, depth, 

and water temperature on the discharge of sands 

can be computed, for some streams at least, from the 

Einstein procedure.

Median particle size of the bed sands has an appreciable effect on the discharge of sands especially at low 

velocities in shallow flows and at either low or high 

velocities in deep flows.

The discharge of sands increases as the water temperature decreases, but the change is usually small as 

compared with changes that may be caused by variations in mean velocity or in depth.

High concentrations of fine sediment, especially of 

bentonite, may greatly increase the discharge of sands 

per foot of width even though velocity, depth, and 

water temperature are kept constant.

The method of analysis that is suggested for determining empirically the effects of major factors on 

the discharge of sands is limited by the available 

information, but it gives relationships that should 

be helpful in understanding the relative effects of 

different major factors.

The empirically defined graphs provide a basis for 

rapid estimation of the discharge of sands in sandbed streams. The accuracy of the estimated discharges 

should be reasonably satisfactory for many sand-bed 

streams. Accuracy, however, may be poor for bed 

sands whose median diameters are outside the range 

of 0.15 to 0.50 mm, for unusually low or high velocities,

for deep flows, for turbulence that is unusually high 

in comparison to mean velocity, or for high concentrations of fine sediment.

The graphs are applicable only to streams whose 

beds are composed of nearly cohesionless sands and 

whose banks do not have the unusually high roughness that might be caused by rock walls that constrict the flow.

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DISCHARGE OF SANDS AND MEAN-VELOCITY RELATIONSHIPS IN SAND-BED STREAMS A47

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o