Aspects of Flow Resistance 

and Sediment Transport 

Rio Grande Near Bernalillo 

New Mexico

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1498-H

Aspects of Flow Resistance 

and Sediment Transport 

Rio Grande Near Bernalillo 

New Mexico

By CARL F. NORDIN, JR.

STUDIES OF FLOW IN ALLUVIAL CHANNELS

GEOLOGICAL SURVEY WATER-SUPPLY PAPER 1498-H

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

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

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

Washington. D.C., 20402
CONTENTS

Page

Glossary of terms________________________________________________ v

List of symbols__________________________________________________ v

Abstract__ ______________________________________________________ Hi

Introduction. _____________________________________________________ 1

Purpose and scope.____________________________________________ 2

Observed data.___________________________________________________ 2

Flow regime and bed configuration_________________________________ 7

Resistance to flow.________________________________________________ 14

Chezy equation.______________________________________________ 14

Manning equation.____________________________________________ 14

Friction factor ratio._________________________________________ 17

Bar resistance.._________-_____________--_________-_--___-_--_- 17

Transition flow.-__________________________________________________ 22

Sediment transport________________________________________________ 27

Conclusions..-_--_________-______________-_-_____-__--_---_-_---_- 37

References cited.__________________________________________________ 40

ILLUSTRATIONS

Page 

FIGURE 1. Sketch map of the study reach_______.___________________ H3

2. Average size distribution of bed material-__-_----____----_ 8

3. Velocity-depth relation________________________________ 10

4. Wave length-velocity relation________________-____-___-_- 12

5. Wave length-depth relation_________________--____-___.__ 13

6. Relation of n to discharge______-_-___-_-----_--_-_--_-_- 15

7. Relation of///'to discharge____________________________ 18

8. Bar resistance for sections A-2 and F. ____________________ 20

9. Bar resistance for the reach____________________________ 21

10. Stage-discharge relations, sections A-2 and F- _____________ 23

11. Vari ations in the velocity-depth relations __________________ 25

12. Mean daily discharge and concentration, April-July, 1958___ 28

13. Variation of discharge and concentration, June 12-19, 1958. _ 31

14. Discharge rates of bed material, sections A-2 and F, using an

average bed-material size distribution.__________________ 32

15. Discharge rates of bed material, sections A-2 and F, using

individual bed-material size distributions ________________ 33

16. Total transport rates of bed material, modified Einstein

method and the Einstein bed-load function______________ 34

17. Total transport rates of bed material, Laursen method._____ 35

18. Comparison of measured and computed discharge rates of bed

material, modified Einstein method._-_____-_---_----_-_ 36

m
IV CONTENTS

Page

19. Comparison of measured and computed discharge rates of bed

material, Laursen method_____________________________ H37

20. Comparison of water discharge and concentration of fine material for sections A-2 and F___________________________ 39

TABLES

Page 

TABLE 1. Observed data and computed parameters for section A-2,..... __ H4

2. Observed data and computed parameters for section F. _______ 5

3. Particle-size analysis of suspended-sediment samples, section

A-2.................................................. 6

4. Particle-size analysis of suspended-sediment samples, section F- 7

5. Particle-size analysis of bed-material samples, section A-2..... 8

6. Particle-size analysis of bed-material samples, section F....... 9

7. Observed data for standing waves.________________________ 12

8. Observed data and computed parameters for flow with high

concentrations of suspended sediment, section A-2........ 16

9. Observed data and computed parameters for the average of 13

cross sections, 1952____________________________________ 22

10. Particle-size analysis of bed-material samples for the reach,

1952_____.__.__._.._.____._._._____________._________ 22

11. A comparison of total transport rates of bed material for

sections A-2 and F, April 6 and June 22, I960.___________ 27

12. Total transport rates of bed material._____________________ 38
GLOSSARY OF TERMS

Bed material: The material which composes the channel bed. For 

this study, bed material was considered to be all sediment coarser 

than 0.062 mm.

Fine material: Sediment finer than 0.062 mm.

Sand: Sediment between 0.062 mm and 2.0 mm in size, regardless 

of minerological composition. The terms "sand", "sand sizes" 

and "bed material" are used interchangeably.

Sediment: Fragmental material that originates from weathering 

rocks and is transported by, suspended in, or deposited by water.

Wash load: Material finer than any size of sediment found in appreciable quantities in the stream bed. The terms "wash load" 

and "fine material" are used interchangeably.

LIST OF SYMBOLS

Symbol Description Units

A Area of cross section of stream._____________ ft

B Channel width.____________________________ ft

C Chezy discharge coefficient______---___------ ft1/2 per sec

C/^g Dimensionless Chezy coefficient

Cm Measured suspended sediment concentration. __ ppm

D Depth at a vertical.________________________ ft

D Mean depth at a cross section______________ ft

d Median diameter of bed material.____________ mm

di6 Diameter of bed material for which 16 per cent,

by weight, is finer._______________________ mm

das Diameter of bed material for which 35 per cent,

by weight, is finer._______________________ mm

rfes Diameter of bed material for which 65 per cent,

by weight, is finer.______________________ mm

dsi Diameter of bed material for which 84 per cent,

by weight, is finer________________________ mm

F Froude number

/ Friction factor

/' Friction factor from a pipe-friction diagram

///' Relative friction factor

g Acceleration due to gravity __________________ ft per sec2

h Height of standing waves, crest to trough. _ _ _ _ ft

kf Representative grain roughness, d65 ___________ ft

v
VI GLOSSARY

L Wave length of standing waves._______________ ft

n Manning resistance coefficient______________ fty6

Pw Wetted perimeter.________________________ ft

Q Water discharge.___________________________ ft3 per sec

Qs Discharge rate of measured suspended sand____ tons per day

Qt Discharge rate of bed material.______________ torn; per day

R Hydraulic radius (R ~D) ____________________ ft

R' Hydraulic radius with respect to the sand

grains. __________________________________ ft

R" Hydraulic radius with respect to dunes and

bars.___________________________________ ft

S Water surface slope

Se Energy gradient slope

T Temperature-------____-------------------- °F

V^ Shear velocity with respect to the sand grains,

 \/gR'Se _______________________-----___-- ft per sec

V? Shear velocity with respect to the bars and

dunes, -JgR" 6V-------------------------- ft per sec

V Mean velocity at a vertical._________________ ft per sec

V Mean velocity at a cross section,_____________ ft per sec

x A correction factor for transition from smooth

	to rough boundary 

7 The unit weight of water.___________________ Ibs per ft3

5 Theoretical thickness of the laminar sublayer__ ft

v Kinematic viscosity_______________________ ft2 per sec

TT 3.1416

pf Density of the fluid_______________________ slugs per ft3

ps Density of the sediment.____________________ slugs per ft3

a Measure of gradation of the bed material

T Shear stress or tractive force on the bed, yRSe - _ Ibs per ft2

T' Shear stress resisted by the sand grains, yR'Se - - - Ibs per ft2

T" Shear stress resisted by the bars, dunes, and

	other bed features, yR"Se . _______________ Ibs per ft2

\f/ f A flow intensity parameter.
W* 1+9 *

STUDIES OF FLOW IN ALLUVIAL CHANNELS

ASPECTS OF FLOW RESISTANCE AND SEDIMENT

TRANSPORT, RIO GRANDE NEAR BERNALILLO

NEW MEXICO

By CARL F. NORDIN, JR.

ABSTRACT

Twenty-one roughly concurrent observations of hydraulic variables at the 

partially confined section A-2 and the wider section F for a reach of the Rio 

Grande near Bernalillo, N. Mex., form the basis for this study of flow resistance 

and sediment transport. In addition, the averages of the hydraulic properties of 

13 cross sections for the reach are used to determine bar resistance and bedmaterial transport rates by the Einstein procedure. Some miscellaneous observations on standing waves and on flows with high sediment concentrations are given.

Because bed configurations change, flow resistance for the Rio Grande varies 

widely. For lower regime flow, the bed configuration is dunes, and the flow 

resistance is high. Values of the dimensionless Chezy coefficient, C/Vff, range 

from about 9 to 12. For upper regime flow, the bed is relatively plane, the flow 

resistance is low, and Cfi/g varies approximately from 19 to 24. Flow in the 

transition region between lower and upper regime is characterized by variable bed 

configurations, flow resistance, velocity and depth, both laterally at a cross section 

and from section to section along the reach. Width adjustments are effected by 

channelization. Both width adjustments and minor variations in characteristics 

of the bed material reflect a tendency for the channel to maintain continuity of 

bed-material transport.

Flow resistance is consistently greater at the narrow section A-2 than at section 

F. The Einstein bar-resistance curve does not adequately describe the flow 

resistance for either of these cross sections. However, data for the average of 13 

cross sections in the reach fit the bar-resistance curve reasonably well.

High concentrations of suspended sediment have more effect on the values of 

overall resistance coefficients for lower regime flow and flow in the transition 

region than for upper regime flow.

Transport rates of bed material are estimated by several methods and compared 

with the transport rates of measured suspended sand.

High concentrations of bed material accompany high velocity and low flow 

resistance. The interrelation of sediment transport and flow resistance is complicated by the extreme variability of water and sediment discharge and by the 

time-dependency of sediment concentration during flood events.

INTRODUCTION

Studies of flow phenomena in alluvial channels have been conducted 

mostly in carefully controlled laboratory experiments. It is sometimes difficult to extend the results of such studies to natural channels

HI
H2 STUDIES OF FLOW IN ALLUVIAL CHANNELS

because field data often are inadequate or incomplete. Comparable 

field studies are difficult to conduct, and the interpretation of field 

data is complicated by the indeterminacy introduced from nonuniform 

flow conditions and from the inherent inaccuracies in measuring, 

estimating, or sampling some of the factors involved.

Nonetheless, it is of interest to consider field data in the light of 

carefully controlled laboratory investigation and, with due consideration of the limitations of field measurements, to examine some of the 

observed similarities and differences in the experimental and natural 

conditions.

PURPOSE AND SCOPE

This paper presents the results of a series of observations of ike 

Rio Grande near Bernalillo, N. Mex. The characteristics of flow 

resistance are described qualitatively in terms of flow regime and 

bed configuratiqn, and quantitatively in terms of standard resistance 

coefficients and of the Einstein bar-resistance curve (Einstein and 

Barbarossa, 1952). Some limited data for flows with high concentration of suspended fine material are given. The magnitude of the 

resistance coefficients, the regimes of flow, and the general forms of 

the bed configurations are compared to those reported in several flume 

investigations.

Sediment transport rates are computed by the modified Einstein 

method (Colby and Hubbell, 1961) and by the method proposed by 

E. M. Laursen (1958). For several observations, transport rates 

are estimated using the Einstein bed-load function (Einstein, 1950).

OBSERVED DATA

The analysis which follows is based principally upon data from a 

reach of the Rio Grande near Bernalillo, N. Mex. (fig. 1). The right 

bank of the upstream part of the reach is a high bluff composed of 

calcareous sandstone, which partially confines the flow at section 

A-2. Throughout the rest of the reach, the banks are composed of 

a mixture of sand, silt, and clay, and are partially stabilized by vegetation. Section A-2 is the site of a U.S. Geological Survey gaging 

station. Permanent cables are installed at sections A-2 and F. 

Other sections are equipped to accommodate temporary cables for 

boat measurements. The stationing shown in figure 1 represents 

the distance, in feet, along the center of the channel.

Tables 1 and 2 present observed data and computed parameters 

for 21 observations at sections A-2 and F. The observations were 

approximately concurrent, except in 1960, when water discharge 

was measured only at section A-2. For these dates, the crosssectional area at section F was determined by sounding, and the
FLOW RESISTANCE AND SEDIMENT TRANSPORT, RIO GRANDE H3

Section A-3

A-2 (Cableway)

A-l

F (Cableway)

Index map of New Mexico

15920

M

2500 2500 FEET

FIGURE 1. Sketch map of the study reach. 

716-825 O 64 2
TABLE 1. Observed data and computed parameters 

Date

4-25-52.. 

5-12-52. 

6-17-52. -

6-20-52.. 

6-26-52.-- _

7-24-52 

6- 8-58 

5-13-58--. 

5-21-58-. 

6-27-58 

6-4 -58-- 

6-10-58... 

6-13-58 

6-18-58- -

6-25-58. 

4-6 -60 

5-24-60.. 

6-22-60. 

4-27-61_. 

5-3-61-. 

5-19-61__._ 

Q

(cfe)

2,730 

6,490 

6,140 

4,830 

2,760 

2,060

6,860 

8,320 

8,680 

10,100 

8,160 

5,800 

4,340 

4,000 

6,040

2,100 

1,240 

2,030

2,230 

3,360 

2,260

B

(ft)

272 

272 

272 

272 

269 

270

270 

271 

270 

273 

272 

270 

266 

267 

273

269 

133 

268

267 

270 

268

V 

(ft per

sec)

4.06 

6.57 

5.96 

5.09 

3.71 

2.84

6.91 

6.88 

7.82 

7.71 

6.92 

6.27 

6.10 

5.06 

6.50

3.04 

2.71 

2.89

3.16 

3.99 

3.62

D

(ft)

2.47 

3.63 

3.79 

3.49 

2.76 

2.69

3.68 

4.46 

4.11 

4.80 

4.34 

3.43 

2.67 

2.96 

3.40

2.56 

3.44 

2.93

2.64 

3.12 

2.33

T

(°F)

58 

62 

70 

72 

70 

75

58 

60 

67 

74 

67 

65 

63 

67 

73

59 

65 

74

57 

66 

63

Watersurface 

elevation 

(ft above

MSL)

5,034.64 

5,035.07 

5, 034. 95 

5,034.38 

5,033.88 

5, 034. 00

5,034.52 

5,034.81 

5,034.89 

5, 035. 10 

5,034.55 

5,033.86 

5,033.55 

5, 033. 49 

5,034.09

5, 034. 00 

5,033.64 

5,034.11

5, 034. 12 

5,034.50 

5,034.33

S. 

(ft per ft)

0.00089 

.00084 

.00083 

.00079 

.00076 

.00080

.00080 

.00080 

.00079 

.00080 

.00083 

.00074 

.00076 

.00076 

.00080

.00083 

.00083 

.00082

.00083 

.00083 

.00085

Bed material

d 

(mm)

0.23 

.39 

.28 

.32 

.30 

.27

.32

.29 

.37 

.29 

.28 

.34 

.35 

.27 

.35

.22 

.23 

.22

a

1.65 

1.65 

1.53 

1.55 

1.57 

1.66

1.89 

1.73 

1.48 

1.54 

1.42 

1.49 

1.67 

1.88 

1.44

1.56 

1.53 

1.46

F

0.455 

.606 

.539 

.480 

.394 

.306

.635 

.576 

.680 

.619 

.584 

.596 

.658 

.519 

.621

.335 

.258 

.297

.343 

.398 

.419

V, 

(ft per

sec)

0.266 

.314 

.318 

.298 

.260 

.263

.308 

.339 

.323 

.352 

.341 

.290 

.256 

.269 

.296

.262 

.303

.278

.266 

.289 

.253

n 

(ft»/«

0.020

.015

.017

.018

.021

.028

.014

.016

.013

.015

.016

.014

.012

.016

.014

.026

.036

.030

.025

.023

.021

1 Average bed-material size distribution from figure 2 used in calculations.
TABLE 2. Observed data and computed parameters 

Date

4-25-52. 

5-12-52. 

6-17-52_-. 

6-20-52. 

6-26-52. 

7-24-52.- 

6-8-58---. 

6-13-58--- 

5-21-58.... 

5-27-58-- 

6-4 -58.- ..

6-10-58... ....

6-13-58. . 

6-18-58.- 

6-25-58.. 

4-6 -60-... 

K_94_CA

6-22-60.-. 

4-27-61.. 

6-3 -61.. 

6-19-61.. 

Q

(cfe)

2,910 

6,390 

6,100 

4,720 

2,850 

2,030

6,730 

8,310 

8,700 

9,970 

7,790 

5,450 

4,380 

4,730 

5,960

2,100 

1,240 

2,030

2,280 

3,890 

2,260

B

(ft)

640 

570 

489 

350 

370 

580

557 

541 

645 

637 

508 

520 

498 

518 

532

632 

459 

357

552 

638 

335

V 

(ft per

sec)

3.86 

5.42 

5.73 

5.50 

3.20 

2.20

5.61 

6.02 

6.17 

5.93 

5.64 

4.91 

4.60 

4.01 

5.36

3.03 

2.05 

2.23

3.02 

3.49 

3.60

D

(ft)

1.18 

2.07 

2.18 

2.45 

2.41 

1.58

2.15 

2.55 

2.19 

2.64 

2.72 

2.13 

1.91 

2.28 

2.09

1.09 

1.32 

1.44

1.37 

1.75 

1.87

T

(°F)

64 

65 

74 

73 

70 

80

61 

63

68 

68 

71 

69 

66 

72 

77

62 

65 

73

58 

65 

66

Watersurface 

elevation 

(ft above

MSL)

5, 027. 31 

5,028.32 

5,028.17 

5,027.84 

5,027.64 

5, 027. 44

5,028.2 

5, 028. 4 

5, 028. 7 

5,028.9 

5,028.0 

5,028.0 

5,027.5 

5,027.4 

5, 027. 7

5,027.17 

5, 026. 86 

5,027.41

5,027.26 

5,027.75 

5, 027. 31

S. 

(ft per ft)

0.00089 

.00084 

.00083 

.00079 

.00076 

.00080

.00080 

.00080 

.00079 

.00080 

.00083 

.00074 

.00076 

.00076 

.00080

.00083 

.00083 

.00082

.00083 

.00083 

.00085

Bed material

d 

(mm)

0.29 

.34 

.31 

.33 

.31 

.33

.24 

.28 

.22 

.25 

.30 

.32 

.45 

.29 

.32

.30

.19 

.22 

.21

a

1.61 

1.73 

1.53 

1.72 

1.52 

1.60

1.53 

1.56

2.35 

1.67 

1.51 

2.25 

3.34 

1.68

1.52

1.58 

1.51 

1.49

F

0.630 

.663 

.682 

.617 

.364 

.308

.673 

.663 

.735 

.645 

.603 

.592 

.587 

.468 

.651

.514 

.314 

.327

.455 

.466 

.463

Vf 

(ft per 

sec)

0.183 

.237 

.242 

.250 

.243 

.202

.236 

.257 

.236 

.261 

.270 

.225 

.216 

.236 

.232

.170 

.188 

.195

.191 

.216 

.226

n 

(fti/«

0.012

.012

.012

.013

.023

.026

.012

.013

.011

.013

.014

.013

.013

.017

.012

.015

.025

.024

.017

.017

.018

1 Average bed material size distribution from figure 2 used in calculations.
H6 STUDIES OF FLOW IN ALLUVIAL CHANNELS

area so determined was used with the discharge from section A~2 

to obtain mean velocity.

Water-surface elevations were obtained from staff gages referred 

to a datum of mean sea level. Slopes of the energy gradients were 

determined as the drop in elevation of the energy gradients divided 

by the distance, 8,240 feet, between sections A-2 and F, and were 

assumed applicable to both sections.

Suspended-sediment samples were collected with a US D-49 

sampler. These samples were analyzed for particle-size distribution 

by the pipette method for sizes smaller than 0.062mm (Kilmer and 

Alexander, 1949) and by sieving or by the visual accumulation method 

(U.S. Inter-Agency Kept. 11, 1957) for sand sizes. The D-49 suspended-sediment sampler collects samples from the water surface 

to about 0.3 foot above the stream bed. The measured suspendedsediment concentrations, Cm , and the particle-size distributions for 

samples at sections A-2 and F are given in tables 3 and 4. All concentrations are given in parts per million (ppm) by weight.

The particle-size distributions of bed-material samples are given 

in tables 5 and 6. Usually, 3 to 15 samples of bed material were 

collected from the top inch or two of the bed. The samples were 

analyzed separately by sieves, and the arithmetic average of the 

separate samples was plotted on log-probability paper to define the 

median diameter, d, and the measure of the gradation, or, shown in

TABLE 3. Particle-size analysis of suspended-sediment samples, section A-#

[Methods of analysis: C, chemically dispersed; M, mechanically dispersed; P, pipet; S, sieve; V, visual 

accumulation tube; W, in distilled water!

Date

4-25-52 

5-12-52 

6-17-52 

6-20-52 

6-26-52 

7-24-52 

5- 8-58 

5-13-58 

5-21-58 

5-27-58 

6- 4-58 

6-10-58 

6-13-58 

6-18-58 

6-25-58 

4- 6-60 

5-24-60 

6-22-60 

4-27-61 

5- 3-61 

5-19-61 

Water discharge 

(cfs)

2,730 

6,490 

6,140 

4,830 

2,760 

2,060

6,860 

8,320 

8,680 

10, 100 

8,160 

5,800 

4,340 

4,000 

6,040

2,100 

1,240 

2,030

2,230 

3,360 

2,260

Concentration of 

sample 

(ppm)

3,320 

3,160 

1,990 

1,690 

1,040 

2,490

4,420 

4,740 

3,400 

3,040 

2,580 

3,480 

2,080 

5,980 

5,410

1,900 

1,200 

781

1,790 

2,670 

1,250

Percent finer than indicated size (mm)

0.004

14 

12 

7 

5 

6 

16

15 

13 

13 

10 

8 

4 

6 

25 

8

20 

18

15 

18 

13

0.016

20 

19 

9 

10 

9 

22

23 

19 

18 

16 

12 

7 

6 

38 

13

27 

24

22 

28 

20

0.062

44 

33 

22 

22 

18 

65

44 

33 

34 

32

28 

18 

18 

51 

27

50 

36 

33

36 

51 

39

0.125

78 

58 

35 

46 

44 

82

61 

58 

58 

54 

50 

34 

45 

63 

30

83

55 

64

68 

74 

66

0.250

97 

88 

68 

85 

81 

95

92 

92 

87 

88 

84 

64 

91 

78 

44

97 

89 

94

92 

97 

96

0.500

100 

99 

98 

99 

99 

99

99 

100 

96 

100 

99 

92 

99 

96 

89

100 

100 

100

100 

100 

100

1.000

100 

100 

100 

100 

100

100

166

100 

100 

100 

100 

100

Metnods 

of 

analysis

SPWCM 

SPWCM 

SPWCM 

SPWCM 

SPWCM 

SPWCM

VPWCM 

VPWCM 

VPWCM 

VPWCM 

VPWOM 

VPWCM 

VPWCM 

VPWCM 

VPWCM

VPWCM 

VPWCM 

V

VPWCM 

VPWCM 

VPWCM
FLOW RESISTANCE AND SEDIMENT TRANSPORT, RIO GRANDE H7

TABLE 4. Particle-size analysis of suspended-sediment samples, section F

[Methods of analysis: C, chemically dispersed; M, mechanically dispersed; P, pipet; S, sieve; V, visual 

accumulation tube; W, in distilled waterl

Date

4-26-52. ........

5-12-52 

6-17-52 

6-20-52 

6-26-52 

7-24-52 .

5- 8-58 

5-13-58 

5-21-58 

5-27-58 

6- 4-58 

6-10-58 

6-13-58 

6-18-58 

6-25-58 

4- 6-60 

5-24-60 

6-22-60 

4-27-61 

5- 3-61 

5-19-61 

Water discharge 

(cfs)

2,910 

6,390 

6,100 

4,720 

2,850 

2,030

6,730 

8,310 

8,700 

9,970 

7,790 

6,450 

4,380 

4,730 

6,960

2,100 

1,240 

2,030

2,280 

3,890 

2,260

Concentration of 

sample 

(ppm)

2,820 

2,690 

1,500 

1,460 

852 

1,740

4,700 

4,220 

3,350 

3,060 

2,500 

2,070 

1,990 

5,420 

2,800

1,870 

828 

1,090

1,540 

2,690 

1,140

Percent finer than indicated size (mm)

0.004

15 

14 

8 

5 

8 

18

16 

14 

13

12 

8 

8

13 

18

22 

29

18 

19

0.016

21 

21 

14 

10 

12 

27

23 

21 

19 

18 

13 

13

22

28

29 

38

26 

29

0.062

45 

43 

25 

22 

26 

67

43 

39 

36 

34 

30 

30 

16 

40 

55

55 

55 

23

46 

52 

43

0.125

79 

67 

51 

46 

57 

86

66 

69 

63 

60 

53 

53 

37 

56 

64

80 

72 

41

72

77 

67

0.250

94 

95 

88 

83 

89 

97

95 

94 

95 

94 

80 

88 

80 

77 

88

96 

93 

67

96 

97 

93

0.500

99 

100 

100 

99 

100 

100

99 

100 

100 

100 

96 

97 

99 

90 

100

100 

100 

94

100 

100 

100

1.000

100

100

100

100 

100 

100 

100

98

Methods 

of 

analysis

SPWCM 

SPWCM 

SPWCM 

SPWCM 

SPWCM 

SPWCM

VPWCM 

VPWCM 

VPWCM 

VPWCM 

VPWCM 

VPWCM 

V 

VPWCM 

VPWCM

VPWCM 

VPWCM 

V

VPWCM 

VPWCM 

V

tables 1 and 2. The measure of gradation, or, was calculated from 

the equation

ff=l/2(d/du+dsi/d) (1) 

in which

d is the median diameter

die is the size, by weight, for which 16 percent is finer

ds4 is the size for which 84 percent is finer.

For some of the computations, the average size distribution of bed 

material shown in figure 2 was assumed applicable. This distribution 

is based upon a previous analysis of 139 observations, wherein it was 

concluded that particle-size distribution of bed material for the Rio 

Grande near Bernalillo is approximately constant (Nordin and 

Culbertson, 196 la).

The data presented for the period from 1952 through 1958 were 

collected as part of a cooperative program between the U.S. Geological 

Survey and the U.S. Bureau of Reclamation.

FLOW REGIME AND BED CONFIGURATION

Flow in the Rio Grande may be classified into lower and upper 

flow regime, with a transition zone between. This classification is 

described in detail by Simons and Richardson (1961) and is based 

upon magnitude of resistance to flow, bed and water surface configurations, and mode and magnitude of sediment transport.

Lower regime is characterized by a large resistance to flow and a 

low transport rate of bed material. The bed roughness is ripples or
H8 STUDIES OF FLOW IN ALLUVIAL CHANNELS

99 

98

95 

90

LJ

z

PARTICLES OFPERCENT F

O1!- 

oi\> eni-* 

X

x

/

/

/

/

/

/

/

/

/

/

/

'

'

|X^

^

d35=0.24 mm

c/50=0.2 

d65 =0.3

9 mm 

5 mm

0.1 1.0 

DIAMETER, IN MILLIMETERS

FIGTJEE 2. Average size distribution of bed material.

TABLE 5. Particle-size analysis of bed-material samples, section ADate

4-25-52. _ __

5-12-52......

6-17-52...... 

6-20-52......

6-26-52......

7-24-52......

5- 8-58...... 

5-13-58.. .

5-21-58-. 

5-27-58. ..... 

6- 4-58.- 

6-10-58.- 

6-13-58- 

6-18-58-- 

6-25-58-- 

4-6-60-. 

5-24-60.. 

6-22-60-- 

4-27-61.. 

5- 3-61...... 

5-19-61. _ 

Number 

of 

samples

3 

3 

6 

3 

3 

3

4 

3 

3 

3 

3 

3 

3 

3 

3

»18 

29

312

6 

6 

6

Percent finer than indicated size, (mm)

0.062

2.1 

0.2 

0.4 

0.4 

0.3 

1.0

0.7 

0.8 

0.2 

0.3 

0.4 

0.2 

0.3 

0.9 

0.3

0.5 

0.3 

0.2

0.4 

0.4 

0.2

0.125

10.8 

1.5 

3.0 

2.8 

2.8 

5.6

3.2 

6.1 

1.3 

2.4 

3.3 

2.2 

2.4 

7.3 

1.4

3.7 

3.3 

4.3

9.8 

6.4 

4.1

0.250

56.5 

15.2 

37.4 

26.7 

32.7 

45.5

34.2 

39.0 

12.8 

38.0 

33.4 

20.9 

22.4 

45.2 

16.1

32.2 

35.6 

42.6

61.1 

58.8 

67.2

0.500

93.4 

70.6 

93.0 

86.4 

87.4 

86.9

77.9 

83.0 

78.9 

88.6 

97.3 

86.1 

78.0 

80.5 

83.1

84.7 

89.4 

86.8

97.4 

96.6 

97.8

1.000

98.2 

93.7 

98.9 

95.3 

97.7 

98.2

91.4 

97.0 

98.5 

97.1 

99.9 

94.6 

93.9 

94.8 

98.1

95.6 

96.9 

96.9

99.6 

99.8 

99.9

2.000

98.8 

97.8 

99.4 

67.3 

99.2 

99.5

94.5 

98.4 

99.6 

98.8 

100 

96.7 

97.9 

98.1 

99.2

97.4 

98.4 

98.8

100 

100 

100

4.000

99.2 

99.1 

99.5 

98.2 

99.6 

99.7

95.7 

98.8 

99.8 

99.6

97.7 

98.9 

99.1 

99.4

98.2 

99.4 

99.4

8.000

99.2 

99.3 

99.6 

99.2 

100 

100

96.3 

99.2 

99.8 

99.7

98.6 

99.7 

99.6 

99.7

99.0 

99.9 

99.8

16.00

100 

100 

100 

99.9

96.6 

100 

100 

100

100 

100 

100 

100

100 

100 

100

32.00

......

100

- 

1 Average of samples from sections A, B, C, D, E, F.

2 Average of samples from section A, C, E. 

s Average of samples from sections A, C, E, G.
FLOW RESISTANCE AND SEDIMENT TRANSPORT, RIO GRANDE H9 

TABLE 6. Particle-size analysis of bed-material samples, section F

Date.

4-26-52......

5-12-52.......

6-17-52.. ....

6-20-52......

6-26-52.. _ 

7-24-52-. _ .

5- 8-58.. 

5-13-58.. 

5-21-58. _ ..

5-27-58.. 

6- 4-58-.... .

6-10-58-- 

6-13-58-. 

6-18-58......

6-25-58.. 

4- 6-60.. 

5-24-60..- 

6-22-60-.- 

4-27-61.- 

5-3-61.. 

5-19-61.. 

Number 

of

samples

3 

3 

3 

3 

3 

3

4 

4 

4 

4 

3 

3 

S 

3 

3

118 

»9

812

7 

7 

4

Percent finer than Indicated size (mm)

0.125

1.1 

0.4 

0.3 

0.3 

0.4 

0.7

1.2 

0.8 

22.6 

10.2 

1.0 

0.2 

0.1 

0.7 

0.3

0.5 

0.3 

0.2

2.2 

0.4 

0.6

0.125

6.5 

2.8 

2.0 

1.4 

3.0 

2.6

8.4 

5.7 

25.6 

29.4 

8.0 

1.6 

0.9 

5.2 

2.4

3.7 

3.3 

4.3

16.9 

7.0

7.8

0.250

34.2 

26.6 

31.0 

30.6 

29.4 

25.9

52.8 

39.4 

54.7 

49.5 

34.6 

27.8 

11.3 

42.2 

31.1

32.2 

35.6 

42.6

72.2 

62.9 

69.2

0.500

90.5 

76.7 

87.3 

79.3 

89.2 

82.0

96.5 

93.0 

93.4 

93.4

87.7 

88.2 

59.4 

74.6 

81.9

84.7 

89.4 

86.8

98.7 

96.6 

95.4

1.000

99.7 

95.7 

97.1 

94.5 

96.8 

95.4

99.8 

98.6 

98.7 

99.3 

97.5 

94.7 

79.3 

82.6 

91.9

95.6 

96.9 

96.9

99.8 

100 

99.8

2.000

99.0 

98.1 

98.7 

97.1 

97.3 

97.5

99.9 

99.2 

99.4 

99.8 

99.1 

96.3 

90.6 

85.1 

95.0

97.4 

98.4 

98.8

100

100

4.000

99.6 

98.5 

99.2 

98.2 

97.5 

98.7

100 

99.5 

99.8 

100 

99.7 

97.3 

94.6 

86.7 

96.8

98.2 

99.4 

99.4

8.000

99.8 

98.6 

99.4 

99.1 

98.4 

99.5

99.8 

99.9

100

98.2 

96.6 

88.5 

98.2

99.0 

99.9 

99.8

16.00

100 

100 

100 

100 

100 

100

100 

100

100 

99.3 

90.1 

99.4

100 

100 

100

32.00

100 

100 

100

1 Average of samples from sections A, B, C, D, E, F. 

3 Average of samples from sections A, C, E. 

8 Average of samples from sections A, C, E, 0.

dunes or both, and the water surface typically shows boils downstream from the crests of the dunes or in the vicinity of potholes. 

For upper regime flow, the bed is relatively flat, the flow resistance is 

low, and the transport rate of bed material is high. The water 

surface generally is quite smooth; under some conditions standing 

waves or antidunes may be present.

Figure 3 shows a simple but convenient method for classifying flow 

regime and bed configuration for the Rio Grande near Bernalillo. 

It has been determined from field observations that flow with values 

of mean velocity, V, and mean depth, D^R, which plot along the 

lower regime line, is over a bed that has a dune configuration, whereas 

flow along the upper regime line is over a bed that is relatively plane. 

In the transition zone, between the two regimes of flow, the bed configuration is unpredictable, but it is generally a composite of dunes 

over portions of the bed and of plane bed or washed-out dunes and 

bars over other portions. The importance of this transition zone will 

be discussed subsequently. Bed configurations shown in tables 1 and 

2 were classified into dune, transition, and plane bed by use of field 

notes and the relation in figure 3. These classifications are rough and 

necessarily overlapping at times, but they appear to give a fair 

qualitative indication of the bed condition.

Details of the geometry of the bed configurations for lower regime 

flow have not been observed for the Rio Grande, but some general 

features have been determined. The major form of bed roughness is
H10 STUDIES OF FLOW IN ALLUVIAL CHANNELS

10

</)

5

<

C£

O

_J

<

tr

o

i

i.o

0.5

O Standing waves or

antidunes

Lower regime(dunes)

Transition zone

A

Upper regime 

(Plane bed, standing 

waves, antidunes)

1.0

VELOCITY, IN FEET PER SECOND

FIGURE 3. Relation of hydraulic radius to velocity for Rio Grande near Bemalillo, N. Mex.

10

dunes. Dunes with ripples superposed have been observed in some 

reaches of the Rio Grande upstream from Bernalillo, where the sediment transport rate is extremely low, but there is no evidence to 

indicate that this form of bed roughness is present in the Bernalillo 

reach. Ripples are not especially important and generally occur at 

very shallow depths.
FLOW RESISTANCE AND SEDIMENT TRANSPORT, RIO GRANDE Hll

The approximate ratios of the length of the dune, from crest to 

crest, to the height of the dune range from 10 to 24, and average 

about 16. This compares with approximate ratios from 12 to 20 in 

irrigation canals (Bender, 1956), and ratios from 14 to 28 reported by 

Simons and others (1961) for flume studies with a sand of 0.45 mm 

median diameter. E. M. Laursen (1958) reported ratios ranging 

approximately from 4 to 7 for flume studies with 0.1 mm sand. The 

shorter dune length may be related in some way to the scale of the 

flume system or the size distribution of the bed material.

The height of dunes for the Rio Grande varies widely, and has 

been observed in many cases to exceed the mean depth of flow. Dunes 

in the Rio Grande appear to grow to greater height, relative to the 

mean flow depth, than do the dunes in channels of small width-depth 

ratios. R. A. Bagnold (1956) has pointed out that in flow of 

unrestricted width, the fluid can divert and rejoin around a bed 

feature, while in channels that are narrow relative to the scale of the 

dune, the entire flow must pass over the bed feature. Because the 

fluid boundary drag on the upstream face of the dune is decreased 

in the three-dimensional case, the dunes grow steeper and higher 

than hi the case of restricted width.

For upper regime flow, the bed is usually firm; under some conditions, it may become so hard packed that sampling of the bed 

material by conventional techniques is difficult or impossible. 

Standing waves may be present. Trains of waves will build and die 

out in a regular cyclic pattern. Relative to a fixed point in the 

channel, the trams of waves may be stationary, or they may move 

upstream or (rarely) downstream. Violently breaking waves are not 

uncommon.

Generally, the standing waves have a wave length L from crest 

to crest of 3 to 6 times the mean flow depth, and the width of the 

waves, measured perpendicularly to the direction of flow, averages 

about 1% times the wave length and has never been observed to exceed 

3 tunes the wave length.

Some quantitative information for standing waves in the Rio 

Grande and several tributaries is given in table 7. The velocities 

and depths listed in the table are local values which were observed 

in the regions where the standing waves occurred. Most of the 

observations were made from boats, and the data are admittedly 

crude. Figure 4 shows wave length, L, plotted against velocity, V. 

The solid line represents the wave length criterion established by 

Kennedy (196la), namely:

V=T/gL/2ir. (2) 

71&-825
H12 STUDIES OF FLOW IN ALLUVIAL CHANNELS

10 I I I I I I I I I 

I 1 I [ I 1 J 11

10 

WAVE LENGTH (I)

FIGURE 4. Relation of velocity to the wave length of standing waves.

30

The field data indicate generally greater wave lengths for a given 

velocity than equation 2 would predict. When the difficulties of 

accurately measuring velocity and wave length under field conditions 

are considered, the scatter from the predicted relation is not excessive. 

Figure 5 shows wave length plotted against depth, D, where the 

solid line represents the equation:

L=4.2 D. (3)

Kennedy concluded that the wave length, L, was independent of flow 

depth. Figure 5 does not reject this contention, but merely indicates 

that the velocity-depth relation was reasonably stable, for the range

TABLE 7. Observed data for standing waves

Location

Qalisteo Creek ......

Date

8-17-54 

8-17-54 

8-17-54 

5-27-58 

6-25-58 

4- 6-60 

5-10-61 

6-10-61 

5-11-61

8-16-60 

8-16-60

9-11-61 

9-20-61 

9-20-61

V

(ft per 

sec)

7.4 

8.3 

8.8 

7.7 

6.5 

3.5 

6.7 

9.5 

10.0

2.9 

4.5

7.2 

8.4 

8.4

D

(ft)

3.2 

2.9 

3.6 

4.6 

3.3 

1.2 

2.5 

4.5 

5.4

0.5 

1.0

2.8 

3.8 

3.7

L

(ft)

15.0 

15.0 

16.0 

16.0 

13.0 

3.5 

13.0 

15.0 

17.0

3.0 

5.0

15.0 

15.4 

14.0

h

(ft)

2.0 

1.8 

2.0

1.6 

0.5 

1.5 

2.5 

2.0

0.3 

0.4

2.5 

2.8 

2.8

Bed material

d (mm)

0.28 

0.32 

0.18 

0.24 

0.35 

0.30

0.29 

0.54

0.23 

0.51 

0.20

(T

1.43 

1.51 

1.78 

1.64 

1.24 

1.52

1.90 

1.71

1.86 

2.21 

2.44

Cm 

(pm)

114,000 

129,000 

115,000 

3,040 

2,080 

1,870

47,000 

47,000

175,000 

240,000 

240,000

Remarks

Breaking. 

Do. 

Do.

Breaking. 

Do.

Breaking. 

Do. 

Do.
FLOW RESISTANCE AND SEDIMENT TRANSPORT, RIO GRANDE H13

I I I I I I I

1 10 

WAVE LENGTH (L)

FIOTTBE 5. Relation of depth to wave length of standing waves.

of conditions in table 7, and was similar to the relation for upper 

regime flow shown in figure 3. From equations (2) and (3),

(4) 

and

V= V4. 2Dg(2w = 4.64Z>1/2. (5)

The upper regime line in figure 3 may be represented by the equation "

The ratio of the wave height h from crest to trough to the wave 

length L for the breaking waves varied from 0.12 to 0.20 and averaged 

about 0.15. The values agree reasonably well with the theoretical 

value of 0.142 for deep water waves and with the observed values 

given by Kennedy (1961a).

The presence or absence of standing waves makes no impression 

on the overall flow resistance in the Rio Grande near Bernalillo, 

N. Mex.

Several criteria for predicting bed configurations in alluvial channels 

have been advanced (Liu and Hwang, 1959; and Garde, 1960). It 

has been shown (Culbertson and Dawdy, 1964) that these criteria 

do not adequately define the bed configuration for the Rio Grande.
H14 STUDIES OF FLOW IN ALLUVIAL CHANNELS 

RESISTANCE TO FLOW 

CHEZY EQUATION

In many natural channels it is sufficiently accurate for preliminary 

analyses to assume that the slope of the energy gradient, Se , equals 

the water-surface slope, S, and is constant (Dawdy, 1961). If this 

assumption is introduced in the interpretation of figure 3, the lines 

of lower and upper regime represent the Chezy equation with constant 

coefficients:

V=C^/lfS. (6)

Values of C/^/9, the dimensionless Chezy coefficient, range approximately from 9 to 12 for lower regime flow and from 19 to 24 for upper 

regime flow. These values are in reasonable agreement with values 

computed from data presented by Simons and Richardson (1961), 

wherein C/^/9 ranged from 10.1 to 11.4 for flow over a dune bed of 

0.28 mm median diameter sand, and from 17.4 to 18.1 for flow over 

a plane bed.

The lower and upper regime lines of figure 3 also represent lines of 

constant Froude number, \F=V/-\gD, where ~D^R. It is significant 

that the limiting Froude number for dunes is about 0.3, the upper 

limit for channel stability given by Simons and Albertson (1960). 

The limiting Froude number for upper regime flow is about 0.6. 

Leopold and others (1960) have suggested that the upper limit for 

Froude number in a river is controlled by the onset of spill resistance, 

which represents a localized and concentrated dissipation of energy 

near the banks. Field observations show that spill resistance occurs 

in the Rio Grande with upper regime flow, and the presence of standing waves and antidunes also represents a concentrated dissipation of 

energy.

During sustained upper regime flow, general degradation occurs 

through the reach. Leopold and Wolman (1956) reported that the 

bed elevation of the entire reach was lowered by scour during spring 

floods. The decreasing bed elevation and velocity-depth ratio for the 

1952 spring flow is evident from the data in tables 1 and 2. It is 

concluded that along with spill resistance, the tendency to scour is an 

important factor controlling the upper limit of the velocity-depth 

ratio in the Rio Grande.

MANNING EQUATION

Probably the most common approach to flow resistance in natural 

channels is the Manning equation:

 i 1 4.Q V=±^ #2/35.1/2. (7)
FLOW RESISTANCE AND SEDIMENT TRANSPORT, RIO GRANDE H15

Values of Manning's n computed using D=R are plotted against 

discharge in figure 6. The n values in the figure represent the data 

in tables 1 and 2, and some additional observations at section A-2 

(table 8) where the concentration of suspended sediment was relatively high in the range of 10,000 to 75,000 ppm.

For upper regime flow, which occurs above 4,000 cubic feet per 

second, the n values range approximately from 0.012 to 0.018, while 

for flows over a dune bed, the values range from 0.024 to 0.036.

The resistance to flow at section A-2 is generally greater than the 

resistance at section F for either lower or upper regime of flow. Part 

of this difference is due to a side-wall effect at section A-2; but a 

side-wall correction, as outlined by H. A. Einstein (1942) with an 

assumed n value of 0.05 for the banks, is not sufficient to bring the n 

values for section A-2 down to the values for section F.

For lower regime flow, dunes persist on the channel bed during 

greater discharges at section A-2 than at section F, and the dunes 

are of greater height at section A-2. The height of dunes appears 

to be dependent on the depth of flow, and the higher dunes account 

for the greater flow resistance at section A-2.

For upper regime flow, where the bed is plane, the flow resistance 

should depend upon the grain roughness and upon the depth of flow, 

and the overall flow resistance should be greater at the wide shallow

0.05

EXPLANATION

0.04

c 0.03

CO

0.02

0.01

O 

Section A-2

X

Section A-2

o (high concen 

0 trations)

° o Section F

V

o

Upper regime 

i i i i i

1000 10,000 20,000 

DISCHARGE, IN CUBIC FEET PER SECOND

FIGURE 6. Relation of Manning's n to discharge.
H16 STUDIES OF FLOW IN ALLUVIAL CHANNELS

section than at the deeper section because of relative roughness 

effects. However, figure 6 indicates a higher resistance at section 

A-2, and it must be concluded that additional frictional losses at 

section A~2 are related in some way to local conditions which have 

not been determined. Probably, the changing cross section of the 

channel in the approach reach to section A-2 creates a form of spill 

resistance or an effect of sinuosity similar to that found by Leopold 

and others (1960) in a rigid-boundary flume study.

Figure 6 also shows values of n for section A-2 for flow transporting 

high concentrations of fine material in suspension. Most of the 

values were presented by Culbertson and Dawdy (1964). The 

measured data upon which the calculations were based are shown in 

table 8. Concentrations of suspended sediments range upward to 

about 75,000 ppm. Slopes were not observed, and an assumed 

average value of 0.00095 is probably on the conservative side, as most 

of the observations were on the falling side of the hydrograph.

The resistance for high concentration flows over a plane bed 

(table 8) is about the same as resistance for lower concentration 

flows. There is no consistent trend toward either an increase or 

decrease in resistance due to suspended-sediment concentrations. 

If a dampening of turbulence occurred because of the presence of 

suspended sediment, the effects on flow resistance probably are 

negligible for practical purposes. Vanoni and Nomicos (1960) 

reported a reduction in n of 13 percent due to sediment load. Changes 

in n of this magnitude could not be detected over the gross scatter 

in figure 6.

Simons and Richardson (1960) indicated that concentrations of 

fine sediment on the order of 40,000 ppm reduced resistance to 

flow by as much as 40 percent in the lower flow regime. This decrease 

in resistance was explained as due to the increased mass density and 

kinematic viscosity of the fluid, with a resulting decrease in fall velocity 

of the bed material. The decrease in fall velocity presumably allowed

TABLE 8. Observed data and computed parameters for How with high concentrations of suspended sediment, section A-#

Date

7-18-53. ........ .....

7-30-53... .............

7-15-52................

8-27-55................

8-30-57 . . 

8-23-61 --..-. ...

8-27-55................

7-18-53................

8-12-52 . - 

9-25-55... .............

(cfs)

1,370

1,430

4,280

4,300

5,000

6,340

7,360

8,020

8,200

13,000

V 

(ft per 

sec)

3.18

3.31

6.11

5.80

6.48

6.82

7.34

7.46

7.52

8.50

D

(ft)

1.60

1.60

2.60

2.75

2.86

3.43

3.70

3.95

4.01

Cm

(ppm)

42, 400

29,200

10 700

59,500

20 400

51,600

48, 100

74, 700

63,400

n fifift

T(°F)

73

66

63

63

70

70

63

70

64

61

n

(ft 1/6)

0. 0197

.0190

.0142

.0155

.0143

.0145

.0150

.0154

.0154

.0170

/

0.0388

.0356

.0171

.0200

.0167

.0180

.0168

.0174

.0173

.0190

///'

3.0

2.7

1.4

1.8

1.5

1.6

1.5

1.6

1.6

1.9

Flow regime

Do.

Do.

Do.

Do.

Do.

Do.

Do.

Do
FLOW RESISTANCE AND SEDIMENT TRANSPORT, RIO GRANDE H17

the bed material to be transported at greater rates that is, the bed 

becomes plane at lower flows than for conditions of lower concentrations. The data in table 8 for July 18 and 30, 1953 would tend to 

confirm the observations of Simons and Richardson.

In terms of overall resistance coefficients, high-concentration 

flow has more effect on lower regime flow or flow in the transition 

region than on flow over a plane bed. The changes in bed configuration occur at lower discharges for high-concentration flow than for 

the low-concentration flow. This interpretation appears to be in 

conflict with the findings of Vanoni and Nomicos (1960), who concluded that "the reduction in friction factor due to the suspended 

sediment is of importance only for streams carrying high suspended 

load over flat beds, and is of minor importance where there are dunes 

on the bed." However, in their flume study, Vanoni and Nomicos 

solidified the flume bed, and thus could not detect the interrelation 

of concentration to bed configuration. Their conclusion that changes 

in the friction factor due to variations in bed configuration are much 

larger than those due to the suspended load is well supported by the 

data in tables 1 through 4.

FRICTION FACTOR EATIO

Taylor and Brooks (1961) proposed that the ratio ///' be used 

as a measure of bed configuration, where

SgDS, f~ f ^- (8)

and /' is determined from a standard Moody pipe-friction diagram 

by use of 4Z5 in place of pipe diameter and use of the median diameter 

of the bed material for the equivalent sand roughness. This value, 

///', represents the ratio of the actual overall resistance as measured 

by / to the resistance /7 which would result from clear flow over a 

flat bed with grain roughness only, and thus should serve as an index 

to the energy dissipation due to form roughness of dunes, bars, and 

other irregularities on the bed. Taylor and Brooks found that an 

///' value of 2 separated the flow over a dune bed configuration from 

flow over a plane bed. Values of///7 for section A~2 and F are plotted 

against discharge in figure 7. It may be seen that all the values of 

///' for upper regime flow are less than 2.

BAB RESISTANCE

Einstein and Barbarossa (1952) introduced an approach to the 

determination of flow resistance wherein the total resistance was 

divided into two parts, one due to the sand-grain roughness, and the
HIS STUDIES OF FLOW IN ALLUVIAL CHANNELS 

10 

9 

8

6

EXPLANATION

o 

Section A-2

X

Section A-2 

(high concentrations)

5

_ Section F

4

 o

X

»

3

2

1

i

Upper regime 

0

1000 10,000 20,000 

DISCHARGE, IN CUBIC FEET PER SECOND

FIGURE 7. Relation of ///' to discharge.

other due to bars, dunes, and the general irregularities of the channel 

bed. The total shear stress on the bed, r=yRSe , was divided into 

two parts:

r = r' + r" (9)

where

T' yR'fic is the shear stress resisted by the sand grains, 

r" = yR"Se is the shear stress resisted by the form drag of the dunes and

bars, 

7= the unit weight of water,

For a given velocity, R' may be determined from the logarithmic 

velocity equation

'* = 5.75 logio (12.27 R'xfk.) (10) 

where

g is the acceleration due to gravity,

ks is the representative grain roughness, d®,

a; is a correction factor for transition from smooth to rough boundary, is

given as a function of k,/Sf, and is applicable where fc s/8'<5, 

5' is the theoretical thickness of the laminar sublayer, which is equal to

ll.fo/FV 

v is the kinematic viscosity.
FLOW RESISTANCE AND SEDIMENT TRANSPORT, RIO GRANDE H19

Reasoning that the shape, size, and frequency of bed irregularities 

should be a function of the bed-load transport rate, which is itself a 

function of flow intensity, ^', Einstein and Barbarossa plotted a 

measure of the bar resistance, the dimensionless resistance factor 

V/V"* , against flow intensity \f/', where

(11) 

e

pt is the density of the sediment, 

Pf is the density of the fluid,

^35 is the representative grain diameter of the bed material and is the size 

for which 35 percent by weight is finer.

The functional relationship of V/V"* to \ff' was defined from the average relation for reaches in several natural channels.

From the manner in which the bar-resistance curve was derived, it 

is apparent that it was not intended to be applicable to individual 

cross sections (Einstein, 1950). However, the application of the 

method to individual sections does give a means of comparing qualitatively the behavior of one section against another

Values of V/V"* are plotted against $' for sections A-2 and F in 

figure 8. In these calculations, d35 and d65 were determined from 

figure 2, and the slopes used were values of Se shown in tables 1 and 2. 

The behavior at section F, in spite of considerable scatter, is about as 

predicted by the curve, while for section A-2, the plotted points fall 

below the curve, and indicate that the dunes persist to greater shear 

stresses (low \ff' values) than predicted by the curve. Corrections for 

side-wall effects at section A-2, with n values of 0.05, do not appreciably change the relationship shown, and it is concluded that the 

difference in resistance at the two sections is not due to bank effects.

It is of interest to note that the behavior at section A-2, which is of 

almost constant width, is different from the behavior of many flumes, 

which are also of constant width. Flume data generally plot above 

the curve, which indicates that dunes will disappear from the bed at 

lower shear stresses than predicted by the curve, whereas for section 

A-2 the dunes persist to greater shear stresses. One reason for the 

differences might be in the effect of flow depth upon the height to 

which dunes will grow. Another reason probably lies in the method 

of operation of flume experiments that is, a range of shear stresses 

in a flume requires a wide range of slope with a narrow range of depth, 

while in the field case, for a given reach, changes in shear stress result 

from wide variations in depth with a narrow range in slope. There 

exist some basic differences in behavior for the two cases.

Vanoni and Brooks (1957) have previously discussed the 1952 

observations for sections A-2 and F in terms of the bar-resistance

716-825 O 64 4
H20 STUDIES OF FLOW IN ALLUVIAL CHANNELS

100

EXPLANATION

O 

Section A-2

 

Section F

~v

8

10

10

FIGURE 8. Bar resistance for sections 4-2 and F.

curve. In their analysis, they used the individual measurements for 

d35 and das, as well as the local water-surface slope for each section. 

Their values of T/F"* and \ff' show the same general trend shown in 

figure 8, but with considerably more scatter and with a definite time 

sequence. They concluded that the 1952 Kio Grande data do not 

"lend confidence to the bar-resistance curve."

As indicated above, application of the curve to individual cross 

sections probably is not justified. In fact, when average conditions 

through the Bernalillo reach are applied to the bar-resistance curve 

(fig. 9), the computed values of "V/V" and \l/f show excellent agreement 

with the predicted relation (Nordin and Dempster, 1963). The 1952 

Rio Grande data would not detract from the bar-resistance curve but 

would tend to confirm the relation.

Basic data for figure 9 are given in table 9. The areas and wetted 

perimeters are the average of 13 cross sections through the reach, and 

water surface slopes were determined by fitting a line by the method
FLOW RESISTANCE AND SEDIMENT TRANSPORT, RIO GRANDE H21

100

\

10

10

. Bar resistance tor the average of 13 cross sections.

of least squares through the plotted water-surface profiles. Bedmaterial samples were collected at each cross section and were sieved 

to determine particle-size distributions. Values of c?3S and d^ in 

table 9 were from the average size distributions given in table 10.

Figures 8 and 9 show that (1) the flow resistance for the reach is 

reasonably well described by the bar-resistance curve; (2) for a given 

flow intensity, there is an appreciable difference in bar resistance from 

one cross section to another; and (3) the bar resistance and hydraulic 

characteristics of an individual cross section, and especially a confined 

or partially confined section, are not indicative of the bar resistance 

and hydraulic characteristics of the reach.

In summary, the data for the Rio Grande, near Bernalillo, N. Mex., 

indicate that the general characteristics of the flow regime and forms 

of bed roughness and the values of the resistance coefficients compare 

reasonably well with the characteristics and values reported for 

several flume studies. Flow resistance for the average cross section
H22 STUDIES OF FLOW IN ALLUVIAL CHANNELS

TABLE 9. Observed data and computed parameters for the average of IS cross

sections, 1952

Date

4-25-52..

5-12-52. . 

6-17^52. _ 

6-20-52. - 

6-26-52..

7-24-52..

A

2,820

6,440 

6,120

4,775 

2,800

2,030

A 

(ft*)

736

1,140 

1,151 

938

784

826

Pw

(ft)

454.0

485.2 

477.3 

429.5 

384.3

424.4

R

(ft)

1.63

2.35 

2.41 

2.20 

2.04

1.95

B

(ft)

452

477 

475 

427 

382

422

V 

(ft per 

sec)

3.83

5.65 

5.32 

5.09 

3.57

2.46

D

(ft)

1.63

2.39 

2.42 

2.20 

2.05

1.96

T

(°F)

61

64 

72 

72 

70

78

S 

(ft per 

ft)

0.00090

.00086 

.00086 

.00086 

.00085

.00088

Bed 

material

d*t 

(mm)

0.230

.278 

.270 

.290 

.260

.283

dm 

(mm)

0.330

.406 

.385 

.420 

.375

.405

*'

1.61

.96 

1.04 

1.16 

1.99

3.94

VI V*

25.7

47.1 

36.4 

38.9 

19.5

11.9

Flow 

regime

Transition. 

Upper.

Do! Transition. 

Lower.

TABLE 10. Particle-size analysis of bed-material samples for the reach, 1958

Date

4-25^52. 

5-12-52-.... 

6-17-51. . 

6-20-52-. 

6-26-52. 

7-24-52.. 

Number

of cross 

sections

13

12 

14 

13

13

13

Total 

number of 

samples

42

41 

46 

40

39

39

0.062

1.5

.5 

.3 

.3

.3

1.1

0.125

7 a

3.4 

2.3

O 1

3.2

4.3

Percei

0.250

41.1

27.9 

29.8

OA Q

QI Q

25.8

it finei

0.500

91 1

79.3 

83.0

7fi Q

QQ Q

79.9

than!

1.0

Q7 Q

93.8

95.1 

92 5

Q4. *>

95.1

Indicat

2.0

QQ Q

96.6 

97.2 

96 1

95.7

97.4

ed size

4.0

QQ 9

97.8 

98.1 

97 6

96.5

98.4

(mm)

8.0

QQ A

98.0 

98.7

QQ 7

97.5

99.2

16

QQ 7

100 

99.3

QQ 8

100

100

32

100 165"

100

 --

d

(mm)

0.28

.33 

.32 

.35

.31

.34

<r

1 64

1.73 

1.61 

1.76

1.62

1.68

of the reach is described reasonably well by the bar-resistance curve, 

but the flow resistance may vary widely, for a given discharge, from 

one section to the next. Most of this variation in flow resistance is 

associated with flow in the transition zone between lower and upper 

regime flow. In the following section, some aspects of flow resistance 

in this transition region will be considered in detail.

TRANSITION FLOW

Flow in the transition region between upper and lower^ regimes 

will be called simply "transition flow". It has been shown in laboratory investigations (Brooks, 1958) and confirmed in the field 

(Colby, 1960) that for transition flow neither the velocity nor the 

sediment transport rate can be expressed as a single-valued function 

of the bed shear stress or any combination of the depth and slope or 

bed hydraulic radius and slope.

For the Rio Grande near Bernalillo, transition flow occurs frequently 

and generally occurs between discharges of 1,000 and 4,000 cfs. In 

terms of frequency, these flows are equaled or exceeded about 28 and 

6 percent of the time respectively. Thus, flow may be in the transition between lower and upper regime about one day out of five. 

On the basis of the period 1942-59, it is estimated that transition
FLOW RESISTANCE AND SEDIMENT TRANSPORT, RIO GRANDE H23

flow accounts for about 34 percent of the total volume of flow and 

upper regime flow accounts for about 39 percent of the volume.

Flow conditions in the transition region are complicated by rather 

extreme changes in bed elevation. Figure 10, previously presented

UJ UJ

U. UJ

tE >

D O

(/> CD

tr <

LU (-

h- UJ

< UJ

5035 

5034 

5033 

5029 

5028 

5027 

5026

Section A-2

 

U*

 * 

»

<

^~

 

 

(

Section F

».* 

 

 

l

* 

>

1

 

4

1000 4000

DISCHARGE, IN CUBIC FEET PER SECOND

10,000

FIGUBE 10. Relation of water-surface elevation to discharge, sections A-2 and F.

by the writer and J. K. Culbertson (Nordin and Culbertson, 1961b) 

shows discharge plotted against water-surface elevation for sections 

A-2 and F. Above 4,000 cfs, flow resistance is approximately constant, and variations in the stage-discharge relation reflect changes 

in bed elevation due to scour and fill. Below 4,000 cfs, however, 

discontinuities in the stage-discharge relations reflect both changing 

bed elevations and changing bed configurations, and generally it 

is impossible to distinguish between the two effects except by frequent 

discharge measurements.

In terms of average flow conditions through the reach, the relation 

of flow resistance to discharge probably is best represented by some 

sort of gradual transition, such as the transition shown by the barresistance curve (fig. 9). Abrupt changes in the bed configurations 

and in the flow resistance through the entire reach are obvious only 

when the change in stage is very rapid.

A gradual transition through the reach is conditioned by two factors: 

(1) lateral variations in bed configuration and flow resistance at the 

individual cross sections, and (2) variations in flow resistance from 

section to section along the reach.
H24 STUDIES OF FLOW IN ALLUVIAL CHANNELS

An abrupt change from a dune bed configuration to a plane bed 

configuration requires, among other things, that the cross section be 

relatively uniform (Colby, 1960). Where lateral nonuniformity of 

depth and velocity exists, as it does in most natural streams, multiple 

forms of bed roughness may exist (Simons and Richardson, 1962) 

and an appreciable change in discharge may be required to shift the 

entire cross section into a single form of bed configuration.

The multiple forms of bed roughness most commonly observed 

during transition flow in the Rio Grande consist of dunes and washedout dunes across some part of the section, with a relatively plane bed 

across the remainder of the section.

The variations in flow resistance from section to section along the 

reach are shown in terms of the velocity-depth relations in figure 11. 

Basic data for the figure was provided by J. P. Beverage, U.S. Geological Survey (written communication, 1962). The letters beside 

the plotted points designate the cross sections shown in figure 1. 

Because the slope was about constant, the lines representing lower 

and upper flow regime are shown as line of constant C/Vg, and are 

identical to the lines shown on figure 3.

For the observation of April 6, 1960 (fig. HA), the velocity was 

remarkably constant through the reach. Five of the sections snowed 

about the same velocity, and the other two varied by about 16 percent 

from the average. The Chezy coefficient varied by a factor of about 

2, and bed configurations ranged from dunes across the entire bed at 

section E to a flat bed across most of the cross section F. For the 

May 24th observation (fig. HB), the flow had receded and the general 

flow characteristics throughout the reach were those of lower regime, 

except at section B, where multiple bed forms persisted. Flow had 

channelized to about two-thirds of the bankful width at most of the 

sections.

On June 22, 1960 (fig. 11(7), the water discharge was comparable to 

the discharge for the April 6th observation. Five of the eight sections 

had dune bed configurations, whereas multiple forms of bed roughness 

were observed at sections A, D, and Q. Values of Cj-^g ranged from 

10 to 18, about the same range as for the April observations. Sections 

B and F, which had the lowest flow resistance for the April observation, showed a higher resistance typical of flow over a dune bed.

Although complicated in detail, the general pattern of channel 

adjustments through a spring runoff event is consistent from year to 

year. Leopold and Wolman (1956) have shown that the bed elevation 

of the reach (sections A-3 through M) is lowered by scour during a 

flood event, and that during the late summer and winter flows, the 

bed returns to about the same elevation as before. On the rising stage 

of a spring flood, the channel bed elevation is higher than later in the
FLOW RESISTANCE AND SEDIMENT TRANSPORT, RIO GRANDE H25

10 i r

I I

i r

i_i

10

I r

LJ

>

10

133d Nl 'Hld3Q
H26 STUDIES OF FLOW IN ALLUVIAL CHANNELS

spring, the cross sections are reasonably uniform, and flow will occupy 

the entire width of the channel between the active banks. Bed configurations adjust quite rapidly to accommodate the imposed sediment 

and water discharge, and the velocity is nearly constant through the 

reach, regardless of the width of the cross section. If the discharge is 

great enough, upper regime flow will occur through the entire reach, 

together with a continued scour to lower the bed. During sustained 

transition flow or on receding stages, considerable channelization 

occurs with a concurrent decrease in the average width of the channel 

(Nordin and Culbertson, 196Ib). This tendency toward a decrease in 

width is seen clearly in the data for the average flow conditions of the 

reach in 1952, table 9. The average width was 452 feet for a discharge 

of 2820 cfs on the rising stage (April 25, 1952) and 382 feet for a discharge of 2800 cfs on the falling stage (June 26, 1952).

Changes in width during transition flow occur quite rapidly and 

generally are somewhat variable and unstable. These changes 

should not be confused with the adjustments in bankfull width, 

which are features of the general regimen of the river, and which are 

related primarily to the bankfull discharge or to some dominant discharge (Blench, 1957) and to the characteristics of the bed and bank 

material (Schumm, 1960).

The most rapid channelization occurs in the transition region of 

velocities of about 3 fps. Qualitatively, this width development 

may be explained by considering the relation of velocity to the discharge rate of bed material, given by B. R. Colby (1961). Colby 

determined from empirical relations that, for a mean velocity of 3 

feet per second and a median diameter of bed material of 0.3 mm, the 

discharge rate of bed material was independent of flow depth. For 

the flow conditions shown in figure 11 (mean velocity approximately 

3 feet per second), the transport rate of bed material would vary 

directly with the width of the channel. For the channel to maintain 

continuity of sediment transport, channel adjustments must be in 

the direction required for equilibrium conditions to be established 

a decrease in width or velocity at the wide sections, or an increase in 

velocity at the narrow sections, or both.

Table 11 presents a re"sum6 of the velocities, channel widths, and 

bed-material transport rates at sections A-2 and F for the observations on April 6, 1960, and June 22, 1960. All sediment coarser than 

0.062 mm was considered bed material, and total transport rates 

were computed both by the modified Einstein method (Colby and 

Hubbell, 1961) and by the method proposed by Laursen (1958). 

For April 6, the velocity was about the same at both sections, and the 

transport rate at section F, estimated by either method, was considerably greater than the transport rate at section A-2. For June 22,
FLOW RESISTANCE AND SEDIMENT TRANSPORT, RIO GRANDE H27

after section F had channelized to about two-thirds of bankfull width, 

the transport rates at the two sections were approximately the same. 

These figures support the idea that width development below bankfull stage is primarily in response to a tendency for the channel to 

maintain continuity of sediment transport.

The width adjustments of a channel, like the changes in bed configuration, flow resistance, velocity, and depth, are related in some 

manner to the sediment transport rates. The tendency for low flow 

resistance to accompany high concent ations of bed material is obvious 

from the data in tables 1 and 2. It is beyond the scope of this study 

to treat sediment transport in detail, but several general aspects 

will be considered.

TABLE 11. A comparison of total transport rates of bed material for sections A-# 

and F, April 6 and June 22, 1960

Date

April 6, I960..... ......................

June 22, 1960 . .. .-... ...... .

Section

A-2 

F A-2 

F

Mean 

velocity 

(ft per 

sec)

3.04 

3.03 

2.89 

2.23

Width 

(ft)

269 

632 

268 

357

Total transport rate of bed 

material (tons per day)

Modified 

Einstein 

method

8,400 

12.000 

4,900 

5,400

Laursen 

method

3,400 

4,700 

1,800 

1,500

SEDIMENT TRANSPORT

Figure 12 shows daily mean water discharge and sediment concentration for the Rio Grande near Bernalillo, April-July 1958. The 

maximum concentration preceded the maximum discharge by a period 

of about six weeks. Thereafter, the concentration decreased regularly, more or less independently of the water discharge.

The concentration curve in figure 12 was defined from suspendedsediment samples which contained both fine material (or wash load) 

and sand-size material (or bed-material load). Particle-size distributions of the samples (tables 3 and 4) show that much of the material 

transported on the rising spring flow was wash load. However, for 

the data presented herein, neither the size distributions of the samples 

nor the concentrations of bed material computed by various totalload formulas are sufficient to indicate whether the transport rate of 

bed material, for a given discharge, is greater on the rising stage or on 

the falling stage.

Usually, in a natural channel where the concentration of bed 

material is higher on rising stages, the increased concentration may 

be accounted for, at least in part, by the lower temperatures and increased viscosities, as clearly shown by L. G. Straub and his coworkers
H28 STUDIES OF FLOW IN ALLUVIAL CHANNELS

14,000 -

12,000 -

^ 10,000 -

o

8000

6000

4000

2000

APRIL MAY JUNE JULY 

FIGURE 12. Mean daily water discharge and sediment concentration, April-July, 1958,

(Straub and others, 1958). Straub found, in flume studies, that at 

a constant discharge, a decrease in temperature caused an increase in 

sediment transport and an increase hi the energy slope, while the 

depth and the velocity remained about constant. In a natural 

channel with relatively slow changes in stage, the slope must conform 

to the natural topography and therefore will vary conservatively, 

while the velocity and depth change appreciably.

In the case of tributary inflow from storm runoff, where the temperature may be about constant but the change in stage is rapid, the 

increase in bed-material concentration on the rising stage is associated 

with an increased slope and with the higher viscosity induced by the 

high concentrations of fine material.
PLOW RESISTANCE AND SEDIMENT TRANSPORT, RIO GRANDE H29

Leopold and Maddock (1953) have shown that for a given width 

and discharge, an increase in suspended sediment requires an increase 

in velocity and a reduction in depth. They concluded, from observations of channel adjustments during flood events, that the observed 

changes in the stream bed resulted from changes in the sediment load 

brought into the measuring reach from upstream. No evaluation 

was given of temperature effects or of the influence of fine sediments, 

and it must be assumed that some part of the observed channel adjustments may be attributed to these factors. However, the results 

of flume experiments, conducted at constant temperature, have supported the findings of Leopold and Maddock (Vanoni and Brooks, 

1957; Kennedy, 1961b). Accordingly, several investigators have 

suggested that the bed-material load is an imposed variable, and that 

for a given discharge, a stream will adjust its bed form, friction factor, 

velocity, and depth to enable it to transport the load (Vanoni and 

Nomicos, 1960; Taylor and Brooks, 1961).

While the fine material certainly must be treated as an independent 

variable, there is little field data to support the idea that the bedmaterial concentration should be considered an imposed variable. 

This is especially true in respect to spring flood events, when there is a 

considerable change in temperature and fine material concentration 

between a given discharge on the rising stage and the same discharge 

on the falling stage.

In the case of tributary inflow, the bed-material concentration 

attributable to the inflow must be considered an imposed variable. 

However, the adjustments in the main channel which accompany 

tributary inflow are quite complex, and the observed behavior of a 

channel usually does not provide much insight into the relation between sediment load and flow resistance.

For example, consider the data in tables 1 and 2 for June 13 and 

June 18, 1958. The average of the measured discharges on both days 

was 4360 cfs. Slopes were the same for the two observations. On 

June 13 the stage was about constant, but on June 18 the discharge 

was increasing during the observations. The total concentration and 

the concentration of bed material was greater on June 18 than on 

June 13. Associated with the higher concentration were a lower 

velocity, greater depth, greater flow resistance, and higher temperature. Between June 13 and June 18, the median diameter of the bed 

material decreased, and the measure of sorting, <r, increased.

The difference in the observed behavior of the channel for the two 

dates was due to inflow on June 17,1958, of a relatively heavy sediment 

load from Galisteo Creek, 23 miles upstream from the Bernalillo reach. 

The peak concentration from the inflow passed the Bernalillo reach 

about 12 hours before the June 18th observation. After the peak, the
H30 STUDIES OF FLOW IN ALLUVIAL CHANNELS

total concentration and the concentration of any particular size class 

decreased exponentially with time (fig. 13).

Relative to the Rio Grande, the Galisteo Creek transported higher 

concentrations and bed material of coarser grade. One might expect 

the bed material at Bernalillo to become coarser with inflow from the 

Galisteo, but such was not the case. In general, the channel adjustments which occur with tributary inflow, especially the minor changes 

in the characteristics of the bed material, are highly unpredictable.

From the data in tables 1 and 2, it is evident that minor changes 

in the characteristics of the bed material are not reflected in the measures of flow resistance.

The effects of changes in the bed-material size distribution on sediment transport rates are aot obvious. However, there are some indirect indications that the minor variations of the bed material are 

important, as was suggested by Kennedy (1961b). Figure 14 shows 

the transport rates of bed material at section A-2 plotted against the 

transport rates at section F. The rates were computed using the 

modified Einstein method (Colby and Hubbell, 1961), the data in 

tables 1 through 4, and the average bed material size distribution of 

figure 2. There is a systematic deviation toward higher transport 

rates at section F. In figure 15, the transport rates were computed 

using the individual bed-material size distributions in tables 5 and 6. 

The plotted points show only random scatter about a line of perfect 

agreement, the relation which would be expected if there is continuity 

of sediment transport through the reach. In fact, it seems reasonable 

to assume that the minor variations in size distributions of bed material 

at different sections along a reach, like the width adjustments during 

transition flow, are manifestations of a tendency for the channel to 

maintain continuity of sediment transport hence the random scatter 

in figure 15.

For the Rio Grande, as for most natural channels, total transport 

rates of bed material cannot be measured directly but must be computed by one of the various transport formulae. Figure 16 shows 

the transport rates of bed material for sections A-2 and F, computed 

by the modified Einstein method, and the transport rates for the 

average cross section of the reach in 1952 (tables 9 and 10), computed 

with the Einstein bed-load function. Figure 17 shows transport 

rates for sections A-2 aad F computed by the Laursen method. It is 

obvious that for a given water discharge, there are considerable 

differences in the transport rates computed by the various methods.

The total discharge rates of bed material, QT, are compared with the 

discharge rate of measured suspended sand, Qs, in figures 18 and 19. 

The discharge rates of measured suspended sand were determined
100,000

12 

FIGUBE 13. Variation of discharge and concentration with t
H32 STUDIES OF FLOW IN ALLUVIAL CHANNELS

- 100,000

10,000

1000 I I I l I

FIGURE 14.-

1000 10,000 100,000 

DISCHARGE RATE OF BED MATERIAL, IN TONS PER DAY, SECTION A-2

-A comparison of discharge rates of bed material at sections A-2 and F, computed using an 

average size distribution of bed material.

from water discharges and concentrations of sand in suspendedsediment samples (tables 3 and 4).

Figure 18 shows that, as an average condition, the total transport 

rate of bed material by the modified Einstein method, for a given 

cross section, is equal to the discharge rate of measured suspended 

sand multiplied by a constant. Culbertson and Dawdy (1964) determined the Bagnold transport function for the Rio Grande near 

Bernalillo to be a constant, 1.46, times the discharge rate of measured 

suspended sands. A line with the equation QT=IAQ Qs would fall 

below most of the points on figure 18.

In figure 19, most of the plotted points fall below the line QT= 

Qs ; thus the total transport rates by the Laursen equation are less 

than the transport rates determined from suspended sediment samples.

A summary of the transport rates of bed material computed by 

the modified Einstein and Laursen methods for sections A-2 and F 

and by the Einstein bed-load function for the reach is given in table 12.
FLOW RESISTANCE AND SEDIMENT TRANSPORT, RIO GRANDE H33

- 100,000 -

10,000 -

1000

1000 10,000 100,000 

DISCHARGE RATE OF BED MATERIAL, IN TONS PER DAY, SECTION A-2

FIGURE 15. A comparison of discharge rates of bed material at sections A-1 and F, computed using individual size distributions of bed material.

The modified Einstein method utilized the size distribution of suspended-sediment samples. Both the modified Einstein and the 

Laursen method require values of mean velocity and depth; so the 

methods have limited application in design where it is necessary to 

estimate depth and velocity for a specific range of discharge. The 

bar-resistance curve provides a method of estimating mean velocity 

but from figure 16 and table 9, it is seen that the relation of water 

discharge to bed-material discharge is poorly defined. Some of the 

scatter in the relation determined by the Einstein bed-load function 

is attributable to changes in slope, temperature, and bed-material 

size distributions, but as noted by Brooks (1958), the major cause of 

the scatter is that hydraulic radius and velocity are not unique functions of discharge.

Although the data for the Bernalillo reach fit the the bar-resistance 

curve better than most field data, velocities estimated from the curve
H34 STUDIES OF FLOW IN ALLUVIAL CHANNELS

100,000

10,000

1000

IT

EXPLANATION

O 

Section A-2

 

Section F

Reach, 1952, average 

cross section

O

A A

I_I l I 1 I I M

1000 10,000 

DISCHARGE, IN CUBIC FEET PER SECOND

FIGTTKE 16. Total transport rates of bed material, computed by the modified Einstein method tor sections 

A-2 and F, and by the Einstein bed-load function for the reach.

vary by as much as 20 percent from the observed values given in table 

9. The bar-resistance curve is not a sensitive measure of the flow 

resistance due to forms of bed roughness.

Colby (1961) has shown that mean velocity is the major factor 

determining the discharge of bed material. Although a discontinuity 

exists in the relation of bed shear stress to bed-material discharge for 

several stations of the Rio Grande in New Mexico, there is no discontinuity, for a given cross section, in the relation of velocity to 

transport rate. It is apparent that refinements in the methods of 

estimating the transport rates of bed material are contingent upon 

more reliable expressions for the interrelation of bed configuration, 

flow resistance, velocity, and depth. Of necessity, such expressions 

must be developed in carefully controlled laboratory experiments.
FLOW RESISTANCE AND SEDIMENT TRANSPORT, RIO GRANDE H35

100,000

UJ

C5 01

I 

O 

<2 10,000

a

1000

1 I I I 1 1 1

EXPLANATION

O 

Section A-2

 

Section F

 O'

O ^0 

O O

<D

i i i i i

1000 10,000 

DISCHARGE, IN CUBIC FEET PER SECOND

FIGURE 17. Total transport rates of bed material, computed by the Laursen method.

The applicability and the limitations of experimental results, however, must be determined in the field.

Precise relations between bed configurations, flow resistance, 

hydraulic variables, and transport parameters probably can never be 

defined for many field conditions. Several of the more important 

reasons for this lack of precision in the Rio Grande data are as follows :

1. Both the streamflow and the sediment discharge rates are 

extremely variable, as illustrated hi figure 12. Uniform flow and 

uniform transport rates are probably never realized. There should 

be a tendency for the stream to adjust toward equilibrium conditions, 

but over a short time base such as the time required for a single 

observation there is no reason to assume that equilibrium exists.
H36 STUDIES OF FLOW IN ALLUVIAL CHANNELS

100,000

O

10,000

1000

I I I I I I I I

EXPLANATION

O 

Section A-2

Section F

I i I I i i

1000 10,000 100,000 

DISCHARGE RATE OF MEASURED SUSPENDED SAND (Q5 ), IN TONS PER DAY

FIGURE 18. A comparison of the discharge rate of measured suspended sand with the total discharge rate 

of bed material computed by the modified Einstein method.

2. It is difficult to estimate, measure, or sample some of the variables 

involved. In figure 20, the water discharge and the concentration 

of fine material for section A-2 are plotted against the discharge 

and concentration for section F. Because the observations were 

only approximately concurrent, the scatter in the relations reflects 

both errors inherent in the measuring and sampling techniques and 

minor short-term fluctuations in the actual discharges and concentrations. Fine material is uniformly distributed in a cross section, 

so sampling errors should be minimized. The variations in concentration in figure 20 probably are representative of the actual fluctuations of concentrations of fine material in the stream.

Generally, the fluctuations in the concentrations for sand sizes 

are more pronounced than the fluctuations for the fine material. The 

concentration of sand may be affected by local factors. For example,
FLOW RESISTANCE AND SEDIMENT TRANSPORT, RIO GRANDE H37

100,000

^ o o 

<: °

g t

5 s

S z

_ UJ a co

UJ Q£

oo =

o 3. 10,000

UJ ^

i 2 01 as

LU UJ O °-

2c *"

o P

1000

EXPLANATION

O 

Section A-2

Section F

1000 10,000 100,000 

DISCHARGE RATE OF MEASURED SUSPENDED SAND (Q5 ), IN TONS PER DAY

FIGURE 19. A comparison of the discharge rate of measured suspended sand with the total discharge rate 

of bed material computed by the Laursen method.

the high concentration of sand on June 25, 1958, at section A-2 was 

caused by antidunes upstream from the section, a local influence 

which was not reflected in the concurrent sample at section F.

3. There exists a time-dependency of sediment concentration 

during flood events, both for spring floods extending over several 

months (fig. 12) and for minor flood events from tributary inflow 

(fig. 13). During such flood events, the concentration varies as a 

function of time, apparently independent of the hydraulics of the 

system. Thus, for any single observation, antecedent conditions 

may be of prime importance.

CONCLUSIONS

The principle conclusions of this study may be summarized as 

follows:

1. Flow resistance for the Rio Grande near Bernalillo, N. Mex., 

varies widely due to changing bed configurations.
H38 STUDIES OF FLOW IN ALLUVIAL CHANNELS 

TABLE 12. Total transport rates of bed material

Date

4-25-52.........

5-12-52.........

6-17-52.........

6-20-52.........

6-26-52.........

7-24-52.........

5- 8-58.........

5-13-58.........

5-21-58.........

5-27-58.........

6- 4-58.........

6-10-58.........

6-13-58.........

6-18-58.........

6-25-58.........

4- 6-60.........

5-24-60.........

6-22-60.........

4-27-61. __ ....

5- 3-61.........

5-19-61.........

Section A-2

Q (cfs)

2,730 

6,490 

6,140 

4,830 

2,760 

2,060

6,860 

8,320 

8,680 

10,100 

8,160 

5,800 

4,340 

4,000 

6,040

2,100 

1,240 

2,030

2,230 

3,360 

2,260

<?r (tons 

per 

day)i

21, 700 

58,000 

37,000 

28,900 

10,700 

8,330

74,000 

108,000 

77,900 

85,000 

59,900 

52,300 

30, 100 

36,300 

88,200

7,900 

4,050 

5,230

10,800 

19,700 

9,820

QT (tons 

per

day)'

23,100 

75,900 

35,700 

30,500 

11,300 

7,000

72,700 

101,000 

119,000 

80,100 

61,600 

61,800 

42,300 

47,400 

93,900

8,400 

4,300 

4,900

9,900 

18,500 

5,700

<?r(lons 

per 

day)'

9,800 

45,400 

27,700 

16,000 

4,900 

1,800

63,700 

63,600 

80,400 

59,200 

50,200 

30,100 

25,100 

13,500 

28,400

3,400 

1,400 

1,800

4,600 

7,400 

7,400

Section F

Q(cfe)

2,910 

6,390 

6,100 

4,720 

2,850 

2,030

6,730 

8,310 

8,700 

9,970 

7,790 

5,450 

4,380 

4,730 

5,960

2,100 

1,240 

2,030

2,280 

3,890 

2,260

QT (tons 

per 

day)i

26,700 

64,600 

50,000 

36,700 

12,000 

6,100

90,300 

123,100 

101,000 

106,000 

58,500 

43,600 

38,800 

51,900 

44,600

12,300 

2,280 

5,790

12,000 

31,700 

9,200

QT (tons 

per 

day)»

22,800 

56,200 

43,300 

29,200 

8,700 

5,600

82,100 

96,000 

99,800 

107,000 

59,800 

43,700 

54,500 

61,000 

44,500

12,000 

2,500 

5,400

9,900 

23,000 

7,900

Or (tons 

per

day)'

13,700 

40,500 

22,100 

16,600 

4,100 

1,300

39,600 

51,200 

50,700 

57,100 

28,400 

17,800 

15,000 

9,600 

23,000

4,700 

1,000 

1,500

5,300 

8,800 

5,200

Reach

Q(cfs)

2,820 

6,440 

6,120 

4,780 

2,800 

2,030

Or (tons 

per 

day)*

3,570 

47,800 

62,000 

8,980 

10,100 

3,480

i Computed by the modified Einstein method, using average bed-material size distribution of figure 2. 

»Computed by the modified Einstein method, using individual bed-material size distributions of tables 

5 and 6.

> Computed by the Laursen equation, using average bed-material size distributions of figure 2. 

* Computed by the Einstein bed-load function, using the bed-material size distribution in table 10.

2. Flow in the Rio Grande may be classified as lower regime flow 

over a dune bed and upper regime flow over a plane bed, with a 

transition zone between the two regimes.

3. Flow resistance is consistently greater at the partially confined 

section A-2 than at the wider section F.

4. The Einstein bar-resistance curve does not adequately describe 

the flow resistance for either section A-2 or F, but it does describe 

the flow resistance for the average cross section of the reach.

5. High concentrations of suspended sediment have more effect 

on the values of overall resistance coefficients for lower regime flow 

and flow hi the transition zone than for upper regime flow.

6. Flow in the transition region is characterized by variable bed 

configuration, flow resistance velocity, and depth. These variations 

occur laterally at a cross section and from section to section along 

the reach.

7. Adjustments hi the flow width and minor variations hi characteristics of the bed material reflect the tendency for the channel to 

maintain continuity of sediment transport.

8. Transport rates of bed material, computed by .the Laursen 

method, are less than the transport rates of measured suspended 

sands. For a given cross section, transport rates of bed material
MATERIAL FINE OF CONCENTRATION 

F SECTION AT 

§ o 

o o 

8

!

3 

s

o

o 

I

o 

o

o m

o SS Z 3

5$

3 Z

2 > 

I I_______I I I I I I I I 1 

F SECTION AT DISCHARGE 

g

8 

> 

E

m

i 1 i_ I I I I I I I _____I 

6 H
H40 STUDIES OF FLOW IN ALLUVIAL CHANNELS

computed by the modified Einstein method bear a constant ratio to 

the transport rates of measured suspended sands.

9. The interrelation of flow resistance and sediment transport is 

complicated by the extreme variability of water and sediment discharge, and by the time-dependency of sediment concentration during 

flood events.

REFERENCES CITED

Bagnold, R. A , 1956, The flow of cohesionless grains in fluids: Royal Soc. [London]

Philos. Trans., Ser. A, v. 249, p. 235-297. 

Bender, D. L., 1956, Suspended sediment transport in alluvial irrigation channels:

Fort Collins, Colo., Colorado State Univ., Dept. Civil Engineering, CER

no. 56DLB7. 

Blench, T., 1957, Regime behavior of canals and rivers: London, Butterworths

Scientific Publications. 

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

Soc. Civil Engineers Trans., v. 123, p. 526-549. 

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

Creeks in Northern Mississippi: U.S. Dept. Agriculture, ARS 41-36.

 1961, Effects of depth of flow on discharge of bed material: U.S. Geol.

Survey Water-Supply Paper 1498-D. 

Colby, B. R., and Hubbell, D. W., 1961, Simplified methods for computing total

sediment discharge with the modified Einstein procedure: U.S. Geol. Survey

Water-Supply Paper 1593. 

Culbertson, J. K., and Dawdy, D. R., 1964, A study of fluvial characteristics and

hydraulic variables, Middle Rio Grande, New Mexico: U.S. Geological Survey Water-Supply Paper 1498-F. 

Dawdy, D. R., 1961, Depth-discharge relations in alluvial streams discontinuous

rating curves: U.S. Geol. Survey Water-Supply Paper 1498-C. 

Einstein, H. A., 1942, Formulas for the transportation of bed load: Am. Soc. Civil

Engineers Trans., v. 107, p. 561-597.

 1950, The bed-load function for sediment transport in open-channel flows:

U.S. Dept. Agriculture Tech. Bull. 1026. 

Einstein, H. A., and Barbarossa, N. L., 1952, River channel roughness: Am. Soc.

Civil Engineers Trans., v. 117, p. 1121-1132. 

Garde, R. J., 1960, Total sediment transport in alluvial channels, Ph. D.

Dissertation, Colo. State Univ., Ft. Collins, Colo. 

Kennedy, J. F., 1961a, Stationary waves and antidunes in alluvial channels:

Pasadena, Calif., California Inst. Technology, W. M. Keck Lab. Hydraulics

and Water Resources, Rept. KH-R-2.

 1961b, Further laboratory studies of roughness and suspended load of 

alluvial streams: Pasadena, Calif., California Inst. Technology, W. M. Keck 

Lab. Hydraulics and Water Resources, Rept. KH-R-3.

Kilmer, V. J., and Alexander, L. T., 1949, Methods of making mechanical analysis 

of soils: Soil Science, v. 68, p. 15-24.

Laursen, E. M., 1958, The total sediment load of streams: Am. Soc. Civil Engineers Proc., v. 84, no. HY 1.

Leopold, L. B., and Maddock, T., Jr. 1953, The hydraulic geometry of stream 

channels and some physiographic implications: U.S. Geol. Survey Prof. 

Paper 252.
FLOW RESISTANCE AND SEDIMENT TRANSPORT, RIO GRANDE H41

Leopold, L. B., and Wolman, M. G., 1956, Floods in relation to river channels:

Dijon, France, Internat. Assoc. Hydrology, Pub. 42. 

Leopold, L. B., Bagnold, R. A., Wolman, M. G., and Brush, L. M., Jr., 1960,

Flow resistance in sinuous or irregular channels: U.S. Geol. Survey Prof.

Paper 282-D. 

Liu, H. K., and Hwang, S. Y., 1959, Discharge formula for straight alluvial

channels: Am. Soc. Civil Engineers Proc., v. 85, no. HY 11. 

Nordin, C. F., Jr., and Culbertson, J. K., 1961a, Particle-size distribution of

stream-bed material in the Middle Rio Grande Basin, New Mexico: U.S.

Geol. Survey Prof. Paper 424-C, Article 265.

 1961b, Discussion of forms of bed roughness in alluvial channels: Am.

Soc. Civil Engineers Proc., v. 87, no. HY 6. 

Nordin, C. F., Jr., and Dempster, G. R., Jr., 1963, Vertical distribution of velocity

and suspended-sediment concentration, Middle Rio Grande, New Mexico:

U.S. Geol. Survey Prof. Paper 462-B. 

Schumm, S. A., 1960, The shape of alluvial channels in relation to sediment type:

U.S. Geol. Survey Prof. Paper 352-B. 

Simons, D. B., and Albertson, M. L., 1960, Uniform water conveyance channels

in alluvial materal: Am. Soc. Civil Engineers Proc., v. 86, no. HY 5. 

Simons, D. B., and Richardson, E. V., 1960, Discussion of resistance properties

of sediment laden streams: Am. Soc. Civil Engineers Trans., v. 125. p.

1170-1172.

 1961, Forms of bed roughness in alluvial channels: Am. Soc. Civil 

Engineers Proc., v. 87, no. HY 3.

1962, The effect of bed roughness on depth-discharge relations in alluvial

channels: U.S. Geol. Survey Water-Supply Paper 1498-E. 

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

using medium sand (0.45 mm): U.S. Geol. Survey Water-Supply Paper

1498-A. 

Straub, L. G., Anderson, A. G., and Flammer, G. H., 1958, Experiments on the

influence of temperature on sediment load: U.S. Dept. Army, Missouri

River Div. Sediment Series no. 10. 

Taylor, R. H., Jr., and Brooks, N. H., 1961, Discussion of resistance to flow in

alluvial channels: Am. Soc. Civil Engineers Proc., v. 87, no. HY 1. 

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