
1

MOVEMENT OF GRADED SEDIMENT WITH
DIFFERENT SIZE RANGES
Jau

Yau LU
1
and I

Yu WU
2
ABSTRACT
Bagnold’s (1980) empirical correlation is one of the most applicable bed load relationships
in the literature. However, few drawbacks still exist in Bagnold’s corr
elation. In this study,
a modified Bagnold’s (1980) formula was proposed and tested with both reliable laboratory
and field data under a wide range of sediment and flow conditions. In general, it was found
that the modified Bagnold’s formula provided bet
ter results on the prediction of sediment
transport rates for most of the conditions tested. The phenomenon of selective transport for
different sediment sizes for both the unimodal and bimodal bed materials were also analyzed
based on the reliable flume
and river data.
Key Words:
Sediment transport, Nonuniform material, Selective transport, U
nimodal
and b
imoda
l bed
materials
,
Critical shear stress.
I. INTRODUCTION
Numerous sediment transport equations have been proposed in the past few decades. The
app
licability of each of the sediment transport equation proposed is restricted by the data
range from which the equation was developed.
Bagnold (1980) proposed
an
empirical
bed
load equation
u
sing a stream
p
ower
as
the
dominant independent variable (see Appe
ndix
Ⅲ
for details). A significant amount of data
collected from both natural rivers and small laboratory flumes was used to test the proposed
correlation, and reasonably good results were obtained.
Few drawbacks, however, still exist in Bagnold’s (1980) correl
ation.
First, a constant
value of 0.04 for the dimensionless Shields parameter was adopted in the calculation of the
threshold stream power. Second, the mode size (or median size if mode size is unknown )
was selected as the representative particle size
for a unimodal bed,
and these modes were
used as the representative sizes for a bimodal bed in calculating the sediment transport rates
with Bagnold’s (1980) equation. In the case of bimodal rivers, where the behavior of the
two size groups
of
sediment ca
nnot be independent of one another, a common threshold
stream power,
o
2
0
1
0
was assumed. This value was adopted, except where the
data demands a smaller value, to avoid a negative
0
. However, t
he selection of “a
smaller value” was somewhat subjective.

2

1
Professor, Department of Civil Engineering, National Chung

Hsing University, Taichung 402, Taiwan, China.
2
Ph.D, Department of Civil Engineering, National Chung

Hs
ing University, Taichung 402, Taiwan, China.
2
Service Committee, Taiwan Association of Hydraulic Engineer.
The process of initiation of sediment motion is statistical in nature.
Shields’(1936)
curve or the revised curve (Miller et al., 1977 ; Vanoni, 1977
) were usually adopted to
estimate the critical shear stress. However, these curves were developed based on nearly
unisize sediment. Also, the dimensionless Shields parameter approaches a constant value
only for a high boundary particle Reynolds number (
500
/
d
U
R
s
*
*
e
; Vanoni, 1977). The
incipient motion for grains in a mixture of sizes has
also been studied for the past
fifteen years (White and Day, 1982; Wiberg and Smith, 1987; Wilcock and Southard, 1988;
Kuhnle, 1993). Bed shear stress at i
ncipient motion for a mixture can be defined following
the technique of Parker et al. (1982). In this work, the bed shear stress and sediment
transport rate were nondimensionalized as follows:
i
s
*
i
gD
0
(1)
i
s
si
s
*
i
f
U
gq
W
3
1
(2)
where
0
bed shear stress;
s
density of sediment;
i
D
diameter of the
i
th size
fraction;
si
q
unit sediment transport rate for the
i
th size fraction;
2
1
0
/
U
;
i
f
fraction of the
i
th size contained in the bed material. The bed shear stress at incipient motion (or reference
shear stress,
*
ri
) was
defined as
*
i
*
ri
when
002
0
.
W
*
i
. This s
hear stress
corresponds to a very low rate of sediment transport.
The main objectives of this study were
(1)to modify Bagnold’s (1980) empirical
equation for the estimation of sediment transport rates for nonuniform sediment; (2)to collect
additional flume
data with
graded sediment
and
steep gradients for testing the modified
Bagnold’s (1980) equation; (3)to analyze the interaction of the movement of different
sediment particles in a sediment mixture.
II. LABORATORY EXPERIMENT
Mountain streams are character
ized by steep slopes and large roughness elements.
Reliable flume
data with
graded gravel collected under steep gradient conditions are still
very limited in the literature. In this study, a series of laboratory experiments were
conducted using a steep r
ecirculating flume to collect bedload data for testing the sediment
transport equations in the following section (section 3.3).
The laboratory flume used for the experiment was 0.6 m wide, 0.6 m high,
and
11.1
m
long. The adjustable range of the flume slo
pe was 0

15 %, and the maximum discharge was

3

0.15 m
2
/s. The recirculating flume and the other related research apparatus are shown in
Fig.1.
A specially designed sediment
feeding system
with a maximum feeding rate of
approximately 700 kg/min was designed
in this study. The feeding rate of the sediment was
controlled by the rotational speed of a belt. The relationship between the sediment feeding
rate and the rotational speed is shown in Fig.2.
Fig. 1
Schematic diagram of recirculating flume
Fig. 2
Relationship between sediment feeding rate and rotational speed of belt
The natural river sediment particles
used for
this experiment
were sieved from an
11.1 m
60 cm
15 cm
2.0 m
2.0 m
4.5 m
5.0 m
60 cm
sediment settling basin
stilling basin
water
storage
tank
steel pipe
screwthread jack
guide plate
tailwater gate control
tailwater gate
trash rack
fixed bed
fixed bed
automatic sediment feeder
moveable bed
head tank
drainage pipe
flow regulation
plate
folw control valve
motor
conveyer
motor
flow control valve
pumping system
supporting frame
air valve
pump
concrete block
drainage pipe
triangular weir
drainage pipe
honeycomb
bypass
guide
plate
plastic pipe
concret block
1.0 m
compound weir
1.35 m
1.24 m
2.80 m
flow
collecting basket
motor
turbine
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
Rotational speed , N
c
(revolutions/min)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
S
e
d
i
m
e
n
t
f
e
e
d
i
n
g
r
a
t
e
,
Q
s
(
K
g
/
s
e
c
)
C.E.III Q
s
= 0.1050 N
c
+0.0167
R = 0.998
2

4

aggregate supply company located in the central region of Taiwan. The nonuniform mate
rial
varied from 1.18 mm to 50.8 mm with a median size of 7.5 mm. The standard deviation of
the sediment
σ
16
84
D
/
D
was 3.0. The dry specific gravity of the material was about
2.65. The grain size distribution of the material is shown in Fig.
4
.
The major independent variables chosen in the experimental design of this study
were the
bed slope and the flow discharge. Three levels for the initial bed slope, 2%, 5%, and 8%,
and three levels for the unit flow discharge 0.045, 0.090, and 0.135 m
2
/s were selected in the
experimental program. All of the values of the Froude numb
er for the current experimental
runs were greater than one. In other words, all of them belong to the supercritical flow.
The detailed procedures of this experiment are given in Su (1995) and Huang (1998). The
data collected are summarized in Table 1. Th
e average water t
emperature
was about 20
℃
(
±
2
℃
).
Table 1
Summary of experimental conditions (C.E.
Ⅲ
)
Run
Number
(1)
Flow
Discharge
(m
3
/s)
(2)
Flow
Depth
(m)
(3)
Final Water
Surface
Slope
(%)
(4)
Sediment
Transport
Rate
)
s
m
/
kg
(
(5)
R311
.1
0.027
0.0639
2.18
0.3366
R311.2
0.027
0.0672
2.68
0.3975
R312.1
0.027
0.0589
5.24
3.9114
R312.2
0.027
0.0556
6.23
3.9326
R313.1
0.027
0.0550
8.81
7.5976
R313.2
0.027
0.0550
9.80
8.1753
R313.1
0.054
0.0861
2.18
0.5592
R321.2
0.054
0.
0888
2.34
0.5698
R322.1
0.054
0.0800
5.09
6.1454
R322.2
0.054
0.0850
5.19
6.2010
R323.1
0.054
0.0825
6.97
11.2628
R323.2
0.054
0.0800
7.03
11.1965
R331.1
0.081
0.1143
1.87
1.3913
R331.2
0.081
0.1083
2.03
1.3065
R332.1
0.081
0.0950
4.8
2
7.0543
R332.2
0.081
0.0950
4.82
7.4545
R333.1
0.081
0.0850
8.01
14.4611
R333.2
0.081
0.0850
7.17
14.0702

5

III. MODIFICATIONS OF BAGNOLD’S FORMULA
3.1 Necessities of modification
–
representative sizes and threshold criterion
Different representa
tive particle sizes have been chosen in the calculation of the sediment
transport rates for nonuniform sediment. For example,
35
D
,
50
D
and
m
D
were selected by
Einstein (1950), Colby (1964) and M
eyer

Peter and M
ü
ller (1948), respectively in their
sediment transport relationships.
For unimodal bed materials, Bagnold (1980) suggested that the mode or “ peak “ size be
used. However, where detailed size analyses are not given, the median size
50
D
must be
accepted. The threshold stream power
0
in Bagnold’s (1980) formula was defined in
terms of Shields’ threshold criterion
D
/
c
c
, where
c
was assumed to have a constant
va
lue of 0.04. In this study, the value of
c
*
was calculated from the modified Shields
diagram (Miller et al., 1977).
For bimodal bed materials with two mode sizes
1
D
and
2
D
, Bagnold (1980) made
separate computations, one for each value of
D
.
A common threshold power
o
(geometric mean
o
2
0
1
0
) was adopted except where the data demands a
“somewhat smaller value” to
avoid a negative
0
. However, as mentioned earlier, the
selection of the “smaller value” is somewhat subjective. To minimize this drawback, in this
study, the representative size was assumed to be the weighted average value
2
2
1
1
D
w
D
w
D
,
where w
1
and w
2
are the weighting factors (or the fractions ) corresponding to the bed
material with modes
1
D
and
2
D
respectively. The threshold stream power
0
was
assumed to
be
2
1
2
1
w
w
o
[ Note that when
w
1
=
w
2
=50 %, this equation is identical to
Bagnold’s (1980) original relationship, i,
e.
o
2
0
1
0
].
Similar to the case
with
unimodal bed material, the threshold stream powe
rs (
0
)
1
and (
0
)
2
corresponding to
1
D
and
2
D
were calculated using Shields diagram (Miller et al., 1977).
3.2 Available data
To test the modified Bagnold’s (1980) formula
as proposed in Section 3.1, both the
laboratory and the field data with 114 sets of unimodal and bimodal bed materials in the
literature were selected. For the laboratory data, Wilcock’s (1987) data with unimodal bed
material, Kuhnle’s (1994) data with bo
th unimodal and bimodal bed materials, and Wu’s
(1998) data with weak bimodal bed material (see Section II) were chosen. For the field data,
Hollingshed’s (1972) Elbow River data with unimodal bed material (1967

1969) and
Emmett’s (1979) Tanaca River data
with bimodal bed material (1977

1978) were included in
the analysis.
Fig. 3 gives the particle size distributions of the unimodal bed materials, including both
nearly uniform and log

normal distributions, with particle sizes ranging from 0.2 mm to 100
mm.
Fig. 4 shows the particle size distributions of the bimodal bed materials with particles

6

ranging from 0.08 mm to 50 mm. The properties of the grain size distributions and the
hydraulic properties of the selected data are given in Table 2 and Table 3, re
spectively. The
flow discharge ranged from 0.0104 to 1,678 m
3
/s, the channel width ranged from 0.356 to 424
m, the flow depth varied from 0.06 to 2.83 m, and the water surface slope varied from 0.745
% to 9.8 %.
Fig. 3
Particle size distributi
ons of the unimodal
bed materials
Fig. 4
Particle size distributions of the bimodal
bed materirals
Table
2
Properties of grain

size distributions
Type
of
Data
References
Type of Mixture
D
m
(mm)
D
16
(mm)
D
50
(mm)
D
84
(mm)
D
mode
(mm)
D
s
(mm)
D
G
(mm)
σ
(geom.)
Flume
Data
Kuhnle (1994) Sand
100%
Unisize
0.49
0.32
0.476
0.67
0.48
―
―
ㄮ㐵
䭵桮汥
ㄹ 㐩4
SG10
Strongly bimodal
1.06
0.32
0.48
1.14
―
〮㐳
㐮㜰
ㄮ㠹
䭵桮汥
ㄹ 㐩4
SG25
Strongly bimodal
1.88
0.35
0.57
5.05
―
〮㔱
㔮㈰
㌮㠰
䭵桮汥
㤹㐩9
SG45
Strongly bimodal
2.93
0.37
0.94
6.22
―
〮㔰
㔮
㐮
䭵桮汥
ㄹ 㐩4
Gravel 100%
Unisize
5.85
4.50
5.579
7.60
5.00
―
―
ㄮ㌰
Wilcock (1987) MUNI
≒
Unisize
1.87
1.63
1.86
2.16
1.87
―
―
ㄮㄵ
Wilcock (1987) MIT 1/2
Φ
Log

normal
1.83
1.25
1
.82
2.49
1.84
―
―
ㄮ㐱
Wilcock (1987) MIT 1
Φ
Log

normal
1.85
0.89
1.83
3.53
1.85
―
―
ㄮ㤹
Wu (1998) C.E.
Ⅲ
Weakly bimodal
10.4
2.80
7.50
26.0
―
㈮㈰
ㄳ⸵
㌮〰
Field
Data
Emmett (1979) Tanaca River
Strongly bimodal
Strongly bimodal
2.49
8.13
0.16
0.22
2.80
7.0
5.0
15.6
―
―
〮㌲
〮㈵
ㄸ⸰
ㄳ⸰
㔮㔹
㠮㐱
Hollingshead (1972) Elbow
River
Log

normal
32.44
13.5
25.0
58.4
20.0
―
―
㈮〸
Table
3
Hydraulic properties of experimental data for testing the modified formula
0.1
1
10
100
1000
D (mm)
5
15
25
35
45
55
65
75
85
95
0
10
20
30
40
50
60
70
80
90
100
P
e
r
c
e
n
t
f
i
n
e
r
Kuhnle(1994),Sand 100 %
Kuhnle(1994),Gravel 100%
Wilcock(1987),MUNI
Wilcock(1987),MIT 1/2
Wilcock(1987),MIT 1
Hollingshead(1972),Elbow River
£X
£X
0.01
0.1
1
10
100
D (mm)
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
P
e
r
c
e
n
t
f
i
n
e
r
Kuhnle(1994),S90G10
Kuhnle(1994),S75G25
Kuhnle(1994),S55G45
Tanaca River(1977),S33G67
Tanaca River(1978),S86G14
Wu(1998),S54G46

7

Type
of
Data
Reference
s
No
of
Data
(1)
Flow
Discharge
(m
3
/s)
(2)
Flume
Width
(m)
(3)
Flow
Depth
(m)
(4)
Final Water
Surface
Slope
(%)
(5)
Sediment
Transport
Rate
)
s
m
/
kg
(
(6)
Flume
Data
Kuhnle (1994) Sand 100%
4
0.0105

0.0168
0.356
0.1039

0.1073
0.038

0.208
0.0000065

0.02062
Kuhnle (1994) SG10
6
0.0104

0.0295
0.356
0.1049

0.1073
0.038

0.373
0.000045

0.1530
Kuhnle (1994) SG25
6
0.0104

0.0293
0.356
0.1013

0.1054
0.0407

0.470
0.0000040

0.1690
Kuhnle (1994) SG45
6
0.0120

0.0287
0.356
0.1012

0.1070
0.091

0.
418
0.000076

0.1422
Kuhnle (1994) Gravel 100%
4
0.0256

0.0302
0.356
0.1026

0.1052
0.428

0.514
0.00021

0.02406
Wilcock (1987) MUNI
6
0.0306

0.0557
0.60
0.1140

0.1280
0.096

0.293
0.00000185

0.0464
Wilcock (1987) MIT 1/2
Φ
7
0.0278

0.0549
0.60
0.1100

0.1
170
0.100

0.490
0.0000144

0.0970
Wilcock (1987) MIT 1
Φ
11
0.0286

0.0377
0.60
0.1090

0.1130
0.104

0.330
0.0000159

0.0594
Wu (1998) C.E.
Ⅲ
18
0.0270

0.0810
0.60
0.060

0.110
1.87

9.80
0.3366

14.4611
Field
Data
Emmett (1979) Tanaca River
21
410

1678
98

424
2.26

2.83
0.44

0.53
0.016

0.151
Hollingshead (1972) Elbow
River
25
35.4

109.0
6.1

24.4
0.6092

0.8830
0.745
0.002

2.80
To t a l
1 1 4
0.0 1 0 4

1678
0.356

424
0.060

2.830
0.745

9.80
0.00000185

14.4611
3.3 Tested results
From the probability theory it
follows (see e.g. Mood et al., 1974) that when the coefficient
of skewness
C
s
>0, the particle size distribution is skewed to the right (i.e. positively skewed),
and that
D
mode
<
D
50
<
D
m
, (where
D
mode
=mode size,
D
50
=median size, and
D
m
=mean size).
In contr
ast to this, if
C
s
<0, the particle size distribution is skewed to the left (negatively
skewed), and
D
mode
>
D
50
>
D
m
.
Fig.5 shows the calculated values of the “reference shear stress”
r
(Wilcock, 1987;
Kuhnle, 1994) based on
D
mode
(A),
D
50
(B), and
D
m
(C) for the unimodal bed materials. The
Shields curve proposed by Miller (1977), the incipient criterion
c
=0.04 as suggested by
Bagnold (1980), and a dashed curve (
s
W
U
) representing the critical co
ndition of initial
sediment suspension (Bagnold, 1966) are also plotted in this figure for comparison. Fig.5
indicates that, for the laboratory data, the results based on
D
mode
,
D
50
and
D
m
were fairly close
to each other and they were also very close to t
he Shields curve proposed by Miller (1977,
solid curve). However, for the field data (Elbow River), the differences due to
D
mode
,
D
50
and
D
m
were detectable, and the deviations between these data points and the Shields curve
proposed by Miller (1977) were
significant.

8

Fig. 5
Calculated reference shear stress vs. particle size for unimodal bed materials
(A=mode size,
D
mode
; B=median,
D
50
; C=mean size,
D
m
)
There were at least two possible explanations for the results of the Elbow river. First,
for mou
ntain rivers or rivers with steep gradients, the sediment load is usually limited by the
supply rate (detachment limiting) from the source area rather than the sediment transport
capacity (Simons & Sentürk, 1992; Bathurst, 1978). Second, based on the limi
ted data, the
dimensionless critical shear stress
c
for high particle Reynolds number
e
R
(
/
D
U
) was
found to vary from 0.04 to 0.06 (Vanoni, 1977). The value of
c
for high
e
R
was very
close to 0.04 for Miller’s (1977) Shields curve. For comparison, a line corresponding to
06
0
.
c
for high
e
R
(
500
e
R
) was also plotted in Fig. 5.
For the bimodal bed mat
erial, as mentioned in Section 3.1, a weighted average particle
size
2
2
1
1
D
w
D
w
D
was suggested to be the representative size for calculating the sediment
transport rate in the modified Bagnold’s formula. When either
1
w
(%
of sand) or
2
w
(% of
gravel) is zero, the representative size becomes the mode size of the unimodal bed material.
Fig. 6 shows the calculated values for the “reference shear stress”
r
(Wilcocks, 1987;
Kuhnle, 1994)
for the bimodal bed materials. The Shields curve proposed by Miller (1977),
the incipient criterion
04
0
.
c
as suggested by Bagnold (1980), a dash line for
06
0
.
c
for
e
R
>500, and a dash curve (
s
W
U
) representing the critical condition of initial sediment
suspension (Bagnold, 1966) are also plotted in the figure for comparison. In addition, for
each bed material, the mode size of sand (
S
D
, hallow symbol), the mode
size of gravel (
G
D
,
solid symbol), and the weighted representative particle size (semi

solid symbol, with
weighted
r
value) are also plotted in the figure.
In Fig. 6, the semi

solid symbols (weighted
D
vs. weighted
ri
values) fell slightly
below theShields curve (Miller, 1977) for
e
R
<500, and they fell above the Shields curve
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
0.1
1
10
100
D (mm)
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
0.1
1
10
100
r
i
(
N
/
m
)
Suspension Transport
Fully Mobilized Transport
Partial Transport
Shields curve (Muller et al., 1977)
Bagnold,
c
=0.04
c
=0.06, (R
e
>500)
U*=W
Kuhnle(1994),Sand 100 %
Kuhnle(1994),Graveld 100 %
Wilcock(1987),MUNI
Wilcock(1987),MIT 1/2
Wilcock(1987),MIT 1
Hollingshead(1972),Elbow river
No Motion
£X
£X
A
B
C
ABC
Sand
Gravel
Cobble
ABC
ABC
*
s
2
*
*
R
e
>
5
0
0
R
e
<
5
0
0
*
*

9

(close to the dashed line
04
0
.
c
) and were close to the dash
ed line
06
0
.
c
for
e
R
>500.
Shen and Lu (1983) found that the turbulence level, particle protrusion, and the gradation of
the particle sizes had a significant effect on the critical shear stress. Quantitative
modifica
tion of the Shields diagram for nonuniform sediment sizes with bimodal distribution
is not yet available. In this study, the critical shear stress was calculated with a Miller’s
(1977) Shields curve based on the weighted mode size.
Fig. 6
C
alculated reference shear stress vs. particle size for
bimodal bed materials
Table 4
Comparison of computed and measured bed

load
Type
of
Data
References
Type of Mixture
% of Predicted Bed Load in
Discrepancy Rang
e
1/2
≦δ≦
2
1/3
≦δ≦
3
Bagnold
Wu
Bagnold
Wu
Flume
Data
Kuhnle (1994) Sand 100%
Unisize
50
75
75
75
Kuhnle (1994) SG10
Strongly bimodal
83
83
83
83
Kuhnle (1994) SG25
Strongly bimodal
50
83
67
83
Kuhnle (1994) SG45
Strongly bimodal
33
67
50
83
Kuhnle (1994) Gravel 100%
Unisize
75
100
75
100
Wilcock (1987) MUNI
≒
Unisize
17
67
50
100
Wilcock (1987) MIT 1/2
Φ
Log

normal
29
86
57
86
Wilcock (1987) MIT 1
Φ
Log

normal
9
64
55
82
Wu (1998) C.E.
Ⅲ
Weakly Bimodal
56
89
100
100
Field
Data
Emmett (1979) Tanaca River
Strongly bimodal
52
62
81
90
Hollingshead (1972)
Elbow River
Log

normal
50
58
73
81
Percentage
46 %
76 %
70 %
88 %
Discrepancy ratio
δ
=Computed i
b
/ Measured i
b
Table 4 gives a comparison of the measured and computed bed load for both Bagnold’s
(1980) and the modified Bagnold’s (designated as “Wu”) met
hods. The parameter
in
Table 4 is the discrepancy ratio, i.e. the ratio of the computed bed load to the measured bed
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
2
3
0.1
1
10
100
D (mm)
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
0.1
1
10
100
r
i
(
N
/
m
)
Suspension Transport
Fully Mobilized Transport
Partial Transport
No Motion
Shields curve(Muller,1977)
Bagnold,
c
=0.04
c
=0.06, (R
e
>500)
U*=W
Kuhnle(1994),S90G10
Kuhnle(1994),S75G25
Kuhnle(1994),S55G45
Emmett(1979),Tanaca River
Kuhnle(1994),Goodwin Creek
Sand
Gravel
Cobble
R
e
>
5
0
0
D
s
D
G
sand mode
gravel mode
*
s
Weighted
r
& mode
*
(hallow)
(solid)
(semisolid)
2
R
e
<
5
0
0
*
*
*

10

load. The percentages falling within the ranges 1/2
≦
≦
2.0, and 1/3
≦
≦
3.0 for both
methods are listed in the table. As can be seen from the table, in general, the percentage
using the modified method (“Wu”) was higher than the corresponding value using the original
Bagnold’s (1980) method.
Fig. 7 shows the relationships bet
ween the measured and predicted bed load using both the
original Bagnold’s (1980) formula and the modified formula (“Wu”). Three straight lines
corresponding to the discrepancy ratios of 2.0, 1.0 and 0.5 are also shown in the figure. As
can be seen in th
e figure, for high bed load rates (
b
i
greater than about
01
.
0
s
m
/
kg
),
both methods gave fairly good predictions. For median and low bed load rates (
b
i
less than
about 0.01
s
m
/
kg
), the modified method provided better predictions. In Fig. 7, for the
convenience of presentation, the predicted
b
i
value was assumed to be a very small value of
10

8
s
m
/
kg
if
0
. Based on the actual calculations, it was found that there were six
sets and one set of data with
0
for the original Bagnold’s (1980) formula and the
modified formula (“Wu”), respectively. Although the sediment transport rate f
or most of
these data were small (<10

4
s
m
/
kg
), it still implies that one needs to further investigate
the criterion of incipient motion for nonuniform sediment under different flow conditions.
Fig. 7
Measured vs. pred
icted bed load using both the original Bagnold’s
formula and
the modified formula (Wu)
；
δ
=discrepancy ratio
1E8
1E7
1E6
1E5
1E4
0.001
0.01
0.1
1
10
100
Measured i
b
( kg/m s)
1E8
1E7
1E6
1E5
1E4
0.001
0.01
0.1
1
10
100
P
r
e
d
i
c
t
e
d
i
b
(
k
g
/
m
s
)
Bagnold, (Kuhnle,Sand 100 %)
Wu, (Kuhnle,Sand 100 %)
Bagnold, (Kuhnle,Gravel 100 %)
Wu, (Kuhnle,Gravel 100 %)
Bagnold, (Wilcock,MUNI)
Wu, (Wilcock,MUNI)
Bagnold, (Wilcock,MIT 1/2 )
Wu, (Wilcock,MIT 1/2 )
Bagnold, (Wilcock,MIT 1 )
Wu, (Wilcock,MIT 1 )
Bagnold, (Hollingshead,Elbow River)
Wu, (Hollingshead,Elbow River)
Bagnold,(Kuhnle,SG10)
Wu,(Kuhnle,SG10)
Bagnold,(Kuhnle,SG25)
Wu,(Kuhnle,SG25)
Bagnold,(Kuhnle,SG45)
Wu,(Kuhnle,SG45)
Bagnold,(Tanaca River)
Wu,(Tanaca River)
Bagnold,(CE III)
Wu,(CE III)
f
f
f
f
P
e
r
f
e
c
t
A
g
r
e
e
m
e
n
t
L
i
n
e
.
.
.
2.0
0.5

11

IV. HIDING EFFECT AND SELECTIVE TRANSPORT
4.1 Derivation of hiding function
&
sediment
transport relationship
The bed material of a gravel river bed
usually contains a broad range of grain sizes, at
times from fine sand to large boulders. Therefore, the movement of grains in a gravel bed
is much more selective than that in a sandy river. In this section, Diplas’ (1987) theory
with a hiding function
is adopted to analyze the selective transport of sediment particles for
both the unimodal and the bimodal bed materials.
With consideration of the assumptions and the data needs of Diplas’ (1987) theory, the
laboratorydata with unimodal (MIT, 1
) collected by Wilcock (1987) and the field data
with bimodal distribution (Goodwin Creek) collected by Kuhnle (personal communication)
were selected for the analysis. The derivations of the hiding functions and the sediment
transport relations
hips are briefly summarized as follows (see Diplas 1987 for detailed
procedure) :
Fig. 8
i
W
versus
i
for five size ranges of
Fig. 9
i
W
versus
i
for eight si
ze ranges of
Wilcock’s MIT 1
Φ
摡ta
the data for Goodwin Creek
1.
The bed materials were divided into five and eight grain

size ranges, respectively for
Wilcock and Kuhnle’s data. The dimensionless bedload for each size range,
i
W
, and the
Shields stress based on the mean particle diameter of that range,
i
were plotted on
log

log scales (Figs. 8 and 9). For each size range, a relation of the form
mi
i
i
i
W
was obtained based on t
he least

squares log

log regression and shown in Fig. 8 or 9.
0.01
0.05
0.1
i
0.0001
0.001
0.01
0.1
1
10
100
W
i
*
D=6.17 mm
D=3.67 mm
D=2.18 mm
D=1.30 mm
D=0.77 mm
W
ri
*
*
= 0.002
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
0.01
0.1
1
10
i
0.0001
0.001
0.01
0.1
1
10
W
i
*
W
ri
45.25 mm
22.63 mm
11.31 mm
5.66 mm
2.83 mm
1.41 mm
0.71 mm
0.35 mm
*
*
= 0.002

12

2.
The relation for the reference Shields stresses
i
r
, which were obtained from Figs. 8 and 9
for
W
i
*
=0.002, and
D
i
/
D
50
is expressed by
b
i
r
r
D
D
i
50
50
(3)
where
50
r
is the reference Shields stress that corresponds to the median size
50
D
.
Based on log

log regressions, it was found that
50
r
=0.0356
&
970
.
0
b
for Wilcock’s
data (unimodal); and
50
r
=0.0856
&
805
.
0
b
for Kuhnle’s data (bimodal).
3.
The dimensionless sediment transport rate
i
W
[defined by Eq.(2) ] can be expressed as
d
c
D
D
i
ri
i
W
W
50
1
(4)
where
i
i
/
ri
, and
ri
W
=0.002. Using the regression analysis it has been found that
c=0.0892
&
d=6.134 with correlation coefficient r=0.91 for Wilcock’s data; and c=0.1723
&
d=6.390 with r=0.89 for Kuhnle’s da
ta.
4.
In the case of a sediment mixture, the coarser particles project into the flow further than
the finer ones. As a result, the mobility of the larger (smaller) particles increased
(decreased) in comparison to their mobility as part of a uniform material
of the same size.
The dimensionless bed load rate can be expressed as
50
50
50
1
D
D
,
h
fct
W
i
i
(5)
where
50
50
1
D
/
D
,
h
i
is called the reduced hiding function, which can be derived from
Eqs. (3) and (4). Based on the actual
measurements, the following forms of the reduced
hiding functions for Wilcock’s (unimodal) and Kuhnle’s (bimodal) data were derived by
the authors
unimodal:
0892
0
50
0892
0
50
030
0
50
1
50
0
50
1
.
i
.
i
D
D
.
i
D
D
i
D
D
D
D
,
h
f
f
(6)
bimodal:
1723
0
50
1723
0
50
195
0
50
1
50
50
50
1
.
i
.
i
D
D
.
i
D
D
i
D
D
D
D
,
h
f
f
(7)
5.
Using Eq.(2), the dimensionle
ss sediment transport relation, Eq.(4) can be converted into
a dimensional form which can be utilized for practical applications. Thus, the bedload
formulae for Wilcock and Kuhnle’s data can be converted into the following forms:

13

unimodal
f
i
.
D
D
.
i
i
s
.
ri
.
i
B
D
D
f
YS
g
W
q
134
6
030
0
50
50
5
1
0892
0
50
1
(8)
bimodal
f
i
.
D
D
.
i
i
s
.
ri
..
i
B
D
D
f
YS
g
W
q
390
6
195
0
50
50
5
1
1723
0
50
1
(9)
Fig. 1
0
Comparison of reduced hiding fun

Fig. 1
1
C o mp a r i s o n o f r e d u c e d h i d i n g f u n

c t i o n
1
h
, wi t h d i m e n s i o n l e s s p a r t i c l e s i z e
c t i o n
1
h
, wi t h d i me n s i o n l e s s p a r t i c l e s i z e
D
i
/
D
50
& the flow intensity parameter
50
f
for
D
i
/
D
50
& the flow intensity parameter
50
f
for
the laboratory data of Wilcock 1
Φ
⡵湩浯摡氩l
瑨攠t楥汤i瑡t 䝯G
摷d渠n牥e欠⡢業潤k氩
1.0
D
i
/D
50
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
h
1
0.75
3.35
3.34
2.86
2.23
1.89
1.29
1.05
Equal mobility (h
1
2.0
f
50
Absence of hiding
=1.0 )
D
i
/D
50
0.1
0.2
0.3
0.4
0.6
0.7
0.8
0.9
1.1
1.2
1.3
1.4
1.6
1.7
1.8
1.9
0.0
0.5
1.0
1.5
2.0
h
1
f
50
2.22
2.01
0.89
4.81
0.12
1.0
4.34
3.49
1.76
1.06
Equal mobility
Absence of hiding

14

Fig. 12
Particle size distributions of bed load
Fig. 13
Measured vs. predicted bed load
samples and bed subsurface for
[
Eqs. (8) & (9)
]
Goodwin Creek
4.2 Discus
sions
For the convenience of discussion, the reduced hiding functions Eqs. (6) and (7) are plotted
in Figs.10 and 11, respectively. The standard deviation
σ
of the bed material for Wilcock’s
data (unimodal) was 1.99; and
σ
=5.4 and 6.9 for two sets of data (station 2) from Goodwin
Creek (bimodal). In Fig. 10, it can be seen that the range of
1
h
was approximately
0.98~1.05 for the experimen
tal conditions considered (
.
e
.
i
0.75
≦
D
i
/
D
50
≧
3.35; and 1.05
≦
Φ
50
≧
3.34). The fact that the value of
1
h
was very close to 1.0 implies that the condition
of equal mobility for all grain sizes was nearly true for the unim
odal bed material with small
σ
value (
σ
2.0 in Fig. 3 ).
Fig. 11 shows the variation of the reduced hiding function
1
h
with dimensionless particle
size
D
i
/
D
50
and the flow intensity parameter
Φ
50
for the field data of Goodwin Creek
(bim
odal). As can be seen in the figure, for
Φ
50
values larger than approximately 2.0, the
coarser particles become more mobile than the finer ones. The actual measured particle size
distributions for the subsurface material and the bed load with different f
low intensities (
Φ
50
=0.89~4.34) are plotted in Fig. 12. As shown in the figure, in general, the median size of
the bed load increased with an increase in the value of the flow intensity parameter
Φ
50
.
Fig. 13 shows a comparison of the measured and the pred
icted bed load rates based on Eqs.
0
0
1
10
100
D (mm)
0
10
20
30
40
50
60
70
80
90
100
P
e
r
c
e
n
t
f
i
n
e
r
bed suface
bed subsuface
bed load( )
bed load( )
bed load( )
bed load( )
bed load( )
bed load( )
bed load( )
f 4.34
f 3.49
f 2.22
f 2.01
f 1.76
f 1.06
f 0.89
50
50
50
50
50
50
50
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
1E5
1E4
1E3
0.01
0.1
1
10
Measured i
b
(kg/m s)
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
1E5
1E4
1E3
0.01
0.1
1
10
P
r
e
d
i
c
t
e
d
i
b
(
k
g
/
m
s
)
Wilcock(1988), MIT 1 (Unimodal Bed)
Kuhnle(1992), Goodwin Creek (Bimodal bed)
£X
P
e
r
f
e
c
t
A
g
r
e
e
m
e
n
t
L
i
n
e
1
.
0
0
.
5
2
.
0
.
.

15

8
& 9.
Three straight lines corresponding to the discrepancy ratios of 2.0, 1.0 and 0.5 are also
plotted in the figure. It was found that 65 % and 83 % of the data fell within the discrepancy
intervals of 1/2
≦δ≦
2.0 and 1
/3
≦δ≦
3.0, respectively.
V. SUMMARY AND CONCLUSIONS
Based on the laboratory experiments and the data analysis performed in this study, the
following conclusions can be drawn:
1.
A series of experiments for a weakly bimodal bed material with steep slope gradie
nts were
conducted. The data collected were useful to widen the applicability of the sediment
transport relationships for gravel bed rivers.
2.
A modified Bagnold’s (1980) formula was proposed and tested with both reliable
laboratory and field data under a w
ide range of sediment (unimodal and bimodal) and flow
conditions. Two of the major modifications on Bagnold’s (1980) formula were:
(a)A weighted mode size was used as a representative size to calculate the sediment
transport rate for the bimodal bed mater
ial (instead of a geometric mean of the modes
for sand and gravel).
(b)Instead of calculating critical shear stress based on a constant value of dimensionless
critical shear stress
c
=0.04, the critical shear stress was calculated usin
g the weighted
mode size and the Shields diagram (Miller, 1977).
In general, the modified Bagnold’s formula provided better results in the prediction of
sediment transport rates for most of the conditions tested.
3.
The phenomenon of selective transport fo
r different sizes of sediment particles in a
nonuniform mixture was analyzed using Diplas’ theory with a hiding function. It was
found that for the unimodal bed material, all grain sizes had approximately “equal
mobility”. For bimodal bed material with a
wide gradation in Goodwin Creek, the size of
the bed load varied with the normalized Shields stress
Φ
50
.
4.
The estimation of the critical shear stress for nonuniform sediment under different flow
conditions is an important and difficult task. More laboratory and field tests are still
needed to increase our understanding of this problem.
Appendix I. REFER
ENCES
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“
An approach to the sediment transport problem from general physics.
”
U.S. Geol. Surv. Prof. Pap. 422

J, 1966.
Bagnold, R. A.,
“
Bed load transport by natural rivers.
”
Water. Resour. Res., pp.303

312,
1977.
Bagnold, R. A.,
“
An empirica
l correlation of bedload transport rates in flumes and natural
rivers.
”
Proc. R. Soc. London., pp.453

473, 1980.

16

Bagnold, R. A.,
“
Transport of solids by natural water flow: evidence for a worldwide
correlation.
”
Proc. R. Soc. London., A 405, pp. 369

374, 1
986.
Bathurst, J. C., “Flow resistance of large

scale roughness.” J. Hydraul. Div., ASCE, 104(12),
pp. 1587

1603, 1978.
Church, M., Wolcott, J. F., and Fletcher, W. K.,
“
A test of equal mobility in fluvial sediment
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”
Water Resour. Res., 27, pp.2941

2951, 1991.
Colby, B. R., “Discharge of sands and mean velocity relationships in sand

bed streams.” U.S.
Geol. Surv. Prof. Pap. 462

A, 1964.
Diplas, P.,
“
Bedload transport in gravel

bed streams.
”
J. Hydraul. Engrg., ASCE,
113(3), pp.
277

292, 1987.
Einstein, H. A., “The bed

load function for sediment transport in open channel
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Emmett, W. W., et al. U.S. Geol. Surv. Open File Rep. 153
9, 1979.
Hollingshead, A. B.,
“
Sediment transport measurements in gravel river.
”
J. Hydraul. Engrg.,
ASCE, pp. 1817

1834, 1972.
Huang, T. C.
, “Transport mechanism of nonuinform gravels in a steep flume.” M.S. Thesis,
Dept. of Civil Engrg., National Chung

H
sing University (in Chinese), 1998.
Kuhnle, R. A.,
“
Fractional transport rates of bedload on Goodwin Creek.
”
Dynamic of
gravel

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and Sons, Ltd., Chichester, U.K., pp. 141

155, 1992.
Kuhnle, R. A.,
“
Incipient motion of sand

gravel sediment mixtures.
”
J. Hydraul. Engrg.,
ASCE, 119(12), pp. 1400

1415, 199
3
.
Meyer

Peter, E., and M
űller, R., “Formulae for bedload transport.” Trans., Intern. Assoc.
Hyd. Res., 2nd. Meeting, Stockholm, pp. 39

65, 1948.
Milhous, R. T.,
“
Sediment transport in a gravel

bottomed stream.
”
Ph. D. Thesis, Oregon
State University, Corvallis, Oregon, 1973.
Mill
er, M. C., McCave, I. N., and Komar, P. D.,
“
Threshold of sediment motion under
unidirectional currents.
”
Sedimentology, 24, pp. 507

527, 1977.
Parker, G., and Klingeman, P. C.,
“
On why gravel bed streams are paved.
”
Water Resour.
Res., 18(5), pp. 1409

142
3, 1982.
Parker, G., Klingeman., P. C., and McLean, D. G.,
“
Bedload and size distribution in paved
gravel

bed streams.
”
J. Hydraul. Div., ASCE, 108(4), pp. 544

571, 1982.
Shen, H. W., and Lu, J. Y., “Development and predicition of bed armoring.” J. Hydraul
.
Engrg., ASCE, 109(4), pp. 611

629,1983.
Shields, A.,
“
Application of similarity principles and turbulence research to bedload
movement.
”
Hydrodynamic Lab. Rep. 167, California Institute of Technology, pasadena,
Calif., 1936.

17

Simons, D. B.
, and Sentürk, F
., “Sediment transport technology.” Water Resources
Publications, Fort Collins, Colorado, 1992.
Su, T. C.
, “Transport rates for gravels with different ranges of grain sizes.” M.S. Thesis,
Dept. of Civil Engrg., National Chung

Hsing University (in Chinese),
1995.
Vanoni, V. A.
, “Sedimentation engineering.” ASCE Manual and reports on engineering
practice, No. 54, 1977.
Wiberg, P. L., and Smith, J. D.,
“
Calculations of the critical shear stress for motion of
uniform and heterogeneous sediments.
”
Water Resour.
Res., 23, pp.1471

1480, 1987.
Wilcock, P. R.,
“
Bed

load transport of mixed

size sediment.
”
In:
Dynamics of gravel

bed
rivers
, Billi, P., Hey, R. D., Thorne, C. R., and Tacconi, P., eds., John Wiley and Sons, Ltd.,
Chichester, U.K., pp. 109

131, 1992.
Wilc
ock, P. R.,
“
Critical shear stress of natural sediments.
”
J. Hydraul. Engrg., ASCE,
119(4), pp. 491

505, 1993.
Wilcock. P. R., and McArdell, B. W.,
“
Surface

based fractional transport rates: mobilization
thresholds and partial transport of a sand

gravel s
ediment.
”
Water Resour. Res., 29(4), pp.
1297

1312, 1993.
Wilcock, P. R., and Southard, J. B.,
“
Experimental study of incipient motion in mixed

size
sediment.
”
Water Resour. Res., 24, pp. 1137

1151, 1988.
Williams, G. P.,
“
Flume
width and water depth effec
ts in sediment transport experiments.
”
U.S. Geol. Surv. Prof. Pap. 562

H, 1970.
Appendix II. NOTATION
The following symbols are used in this paper:
b
=
Exponent of Eq.(3);
s
C
=
the coefficient of skewness;
D
=
2
2
1
1
D
w
D
w
, the weighted average value;
1
D
=
Mode size for sand;
2
D
=
Mode size for gravel;
16
D
=
size for which 16 % by weight of sediment is finer;
50
D
=
Median size;
84
D
=
size for which 84 % by weight of sediment is finer;
G
D
=
Mode
size of gravel;
i
D
=
Diameter of the
i
th size fraction;
m
D
=
Mean s
ize;
e
mod
D
=
Mode size;
S
D
=
Mode size of sand;
d
=
Exponent of Eq.(4);
s
d
=
grain size;
i
f
=
Fraction of
i
th size contained in bed material;
g
=
Acceleration of gravity;

18

1
h
=
(
Φ
50
, D
i
/D
50
), reduced hiding function;
b
i
=
Transport rate of bedload
)
s
m
/
kg
(
;
i
m
=
Exponent of
mi
i
i
i
W
;
N
c
=
Rotational speed (revolutions/min);
s
Q
=
Sediment feeding rate (kg/sec);
B
q
=
unit sediment transport rate for the total bedload (m
2
/s);
si
q
=
unit sediment transport rate for the
i
th siz
e fraction (m
2
/s);
R
=
Hydraulic radius = flow cross

sectional area/wetted perimeter;
*
e
R
=
/
d
U
s
*
, Reynolds number;
r
=
Correlation coefficient;
S
=
slope of energy gra
de line;
U
=
2
1
0
/
= bed shear velocity;
*
i
W
=
Dimensionless bed load in
i
th size range;
*
r
W
=
Reference value for dimensionless sediment transport rate (
*
i
W
=
*
ri
W
=
0.002);
s
W
=
fall velocity;
1
w
=
the weighted factor (% of sand);
2
w
=
the weighted factor (% of gravel);
Y
=
mean flow depth ;
50
f
=
50
/
50
r
= normalized Shields stress;
i
f
=
i
/
ri
= normalized Shields stress;
i
=
Coefficient of
mi
i
i
i
W
;
=
Measured
i
b
/
computed
i
b
=
discrepancy ratio
;
=
density of water;
s
=
density of sediment;
=
16
84
D
/
D
, geometric standard deviation of sediment size distri
bution;
0
=
gRS
= bed shear stress;
c
=
critical shear stress from modified Shields diagram (Miller,1977);
*
i
=
)
)
/((
i
s
o
gD
= Shields stress for
D
i
;
r
=
Reference shear stress;
ri
=
Reference critical shear stress (N/m
2
);
*
ri
=
Reference value of
*
i
at which
*
i
W
*
ri
W
0.002;
50
r
*
=
the reference Shield stress that corresponds to the subsurface
D
50
;
=
/
=
kinematic viscosity
;
=
stream power per unit bed area
)
s
m
/
kg
(
;
o
=
Nominal threshold value of
at which bed movement starts
)
s
m
/
kg
(
;
o
=
2
0
1
0
=
geometric mean;
1
o
=
stream power for D
1
;
2
o
=
strea
m power for D
2
;

19

Appendix III. Bagnold’s (1980) empirical correlation of b
edload transport rates
Bagnold’s (1980) proposed an empirical expression for the prediction of bedload transport
rates as follows:
2
1
3
2
2
3
0
0
*
*
*
*
b
b
D
D
Y
Y
i
i
(i)
where t he starr
ed values
refer
to any single point on a reliable
experiment al pl ot.
Bagnold (1980) chose the following reference values based on Williams’ (1970) data :
*
b
i
=
0.1
s
m
/
kg
, (

0
)
*
=0.5
s
m
/
kg
,
Y
*
=0.1 m,
*
D
=
1.1
×
10

3
m,
The definitions for other variables in Eq(i) are :
Y
=
mean depth of flow (m)
D
=
mode size of bed material (m)
b
i
=
transport rate of bedload
by immersed weight per unit width
)
s
m
/
kg
(
=
stream power per unit bed area
)
s
m
/
kg
(
0
=
nominal threshold value of
at which bed movement starts
)
s
m
/
kg
(
0
290
2
3
D
log (12
D
Y
)
(Assume Shields’ threshold criterion
=
D
/
o
0.04
; and
= average sediment excess
density in water
1600 kg/m
3
).
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