NOTES ON THE DIVISION BETWEEN BED MATERIAL LOAD AND WASH
LOAD IN SAND-BED STREAMS
Gary Parker
August 22, 2000
Caution: these notes are preliminary and subject to change
References on wash load cutoff here
Introduction
Consider a sand-bed river at a morphologically significant (flood) flow. Let c
denote the total volume concentration of suspended sediment (bed material load
plus wash load). That is,
c
Qs
Q  Qs
(1)
where Q denotes the water discharge and Qs denotes the volume discharge of
suspended sediment.
Furthermore, let Fs(D) denote the fraction of the
suspended load that is finer than size D. There presumably exists some cutoff
size Dc such that the sediment finer than this can be considered to be wash load.
Here a mechanistic model is developed for this cutoff size.
Fall velocity
Here fall velocity is denoted as vs. The relation of Dietrich (1982) for fall velocity
can be reduced to the following functional form;
R f  R f (R ep )
(2)
where Rf is a dimensionless fall velocity and Rep is a dimensionless explicit
particle Reynolds number, given respectively by
Rf 
vs
RgD
R ep 
RgDD

(3)
where g denotes the acceleration of gravity and  denotes the kinematic viscosity
of water. In addition R = (s/) – 1, where s denotes the density of sediment and
 denotes the density of water. The precise form of the functional relation can be
written in the form
1/ 3
 10 y 

Rf  
 Re 
 p
y  -3.76715 + 1.92944 x - 0.09815 x 2 - 0.00575 x 3 + 0.00056 x 4
(4)
x  og10 (Re )
2
p
Condition for near-equilibrium suspension
For a flow that is not too far from equilibrium the rate of entrainment of a given
size into suspension from the bed should be nearly equal to the rate of deposition
of the same size from suspension onto the bed. Let the grain size distribution be
divided into N ranges, each with size Di, fraction Fbi in the bed and near-bed
volume concentration cbi. The fall velocity associated with size Di is denoted as
vsi. The following condition, then, should be satisfied for the rate of deposition to
balance the rate of entrainment:
v sic bi  v siFbiEi
(5)
where Ei denotes a normalized dimensionless entrainment rate. According to the
relation of Garcia and Parker (1993),
aZ i5
Ei 
a 5
1
Zi
0 .3
(6)
where
Z i  Re
0.6
pi
us
v si
a  1.3x10 7
0.2
RgDi Di
 Di 


Repi 

 D 50 
  1  0.288 
(7)
In the above relations us denotes the part of the shear velocity due to skin
friction, D50 denotes the median size of the bed material and  denotes the
arithmetic standard deviation of the bed mateial on the phi scale.
Condition for cutoff size
Let Dc denote the cutoff size between bed material load and wash load. Two
assumptions are made regarding this size: a) the vertical distribution of
suspended sediment at and below the cutoff size can be approximated as
uniform (a condition that must be verified a posteriori) and b) the fraction content
of bed material Fbw consisting of grains no larger than the cutoff size is an
appropriately selected very small number, e.g. 0.005 or a half of a percent. The
choice of Fbw is somewhat arbitrary, but a good model should not be overly
sensitive to the value chosen.
The cutoff size between bed material load and bed load is here defined to be the
size Dc such that if all the material in suspension finer than D c were replaced with
material of size Dc, the content of this material in the bed would be just equal to
the cutoff bed fraction content Fbc. In accordance with the above-stated
assumption that the vertical size distribution of sediment of size D c should be
nearly uniform in the vertical, (5) reduces to the condition
cFsc  Fbc E( Z c )
(8)
or
D
Z c  Re  c
 D 50
0 .6
pc



0 .2
u s
cF
 E 1 ( sc )
v sc
Fbc
(8)
where E-1 denotes a functional inversion, such that
cFsc




 cF 
Fbc

E 1  sc   
cF
F


sc
bc


 a(1  0.3 F ) 
sb 

1/ 5
(9)
In the above relations Fsc denotes the fraction of the suspended load that is finer
than Dc, vsc denotes the fall velocity of size Dc and Repc denotes the value of Rep
for size Dc.
Equation (8) can be manipulated into the form
0.6
Repc
 s  D c

R f c  (1  0.288  )
1 cFsc  D 50
E (
)
Fbc



0.3
(10)
where Rfc denotes the value of Rf for size Dc and s denotes the part of the
Shields stress due to skin friction, given by the relation
 s 
u2s
RgD 50
Description of the flow
(11)
The following simple formulation for water mass and momentum balance is used
for a long, quasi-equilibrium reach of a sand-bed stream with water discharge per
unit width q that is held constant downstream (no tributaries in the reach) and
constant bed friction coefficient Cf;
q  UH
u2  C f U2  gHS
(12a,b)
where U denotes cross-sectionally averaged flow velocity, H denotes crosssectionally averaged depth, u denotes shear velocity, S denotes down-channel
bed slope (averaged over meandering etc.) and Cf is a friction coefficient, here
approximated as constant . Solving for H, it is found that
 C q2 
H   f 
 gS 
1/ 3
(13)
Defining the Shields stress  as
u2
 
RgD 50

(14)
it is found upon manipulation that
 
q̂ 2 / 3 S 2 / 3 C1f/ 3
R
(15)
where
q̂ 
q
(16)
gD 50 D 50
The parameter s in (10) can be estimated from  using the relation of Engelund
and Hansen (1967);
 s  0.06  0.4 (   ) 2
Solution for cutoff size
With the above relations (10) can be reduced to the form
(17)

 q̂2 / 3 C1f/ 3 S 2 / 3
0.6
0.06  0.4
(1  0.288  )Repc
R


Rfc 
1/ 5
cFsc




Fbc


cFsc 

 a(1  0.3 F ) 
sb 





2



1/ 2
 Dc

 D 50



0.3
(18)
The solution for cutoff size Dc is iterative. In the above equation the values of a,
Dr, c, q̂ , Cf, S, and R must be specified. For each estimated value Dc the
parameter Repc and the fraction Fsc of suspended load that is finer than Dc can
be computed. Once this is done, the value of Rfc can be computed from (18). In
order for the estimated value of Dc to be the correct one, the value of Rfc
computed from (18) must agree with the value computed from the fall velocity
relation (2).
Sample calculation
This example is loosely based on the middle Fly River, Papua New Guinea. The
reach in question has a length L = 500 km long. The bankfull water discharge
per unit width q is assumed to be 30 m 2/s and the friction coefficient Cf is
assumed to be 0.0012. Bed slope S declines exponentially from an upstream
value Su = 8x10-5 to a downstream value Sd = 2x10-5, so that
x

S  S u exp    
L

  1.386
(19a,b)
Median bed grain size is also assumed to decline exponentially, such that
 2 x
D 50  D 50 u exp    
 3 L
(20)
where D50u = 0.3 mm (The meaning of the choice of exponent of (2/3)  will
become apparent below.) It can then be deduced from the above relation that
D50d = 0.119 mm at the downstream end of the reach. In all calculations the
value of the arithmetic standard deviation of the grain size distribution  of the
bed material is taken to be constant at 0.7.
The particular choice of the exponent in (20) is such that the Shields stress  is
rendered constant downstream. That is, substituting (19) and (20) into (15) and
reducing, it is found that the Shields stress is given by the relation
q̂u2 / 3 S u2 / 3 C1f/ 3
 
R

(21)
where
q̂u 
q
(22)
gD 50 uD 50 u
independently of x.
Because the slope of the reach is declining downstream, it can be assumed that
a grain size that is washload at any point x was also washload farther upstream.
With this in mind, the concentration of particles participating in washload at some
point x where the cutoff size is Dc is given as cuFscu, where cu denotes the volume
concentration of suspended load at the upstream end of the reach and and Fsuc
denotes the fraction of the incoming suspended sediment at the upstream end of
the reach that is finer than Dc. That is, the transformation cFsc  cuFsuc can be
made in (18).
Here cu is assumed to take the values 2x10-4, 4x10-4 and 6x10-4, corresponding
to total suspended solids concentrations of 530, 1060 and 1590 mg/l for the case
R = 1.65. The assumed grain sizes Di and fractions finer at the upstream and of
the reach Fsui are given in Table 1 below.
Table 1
Di mm
0.5
0.25
0.125
0.0625
0.03125
0.015625
0.007813
0.003906
0.001953
Fsui
1
0.97
0.93
0.85
0.66
0.44
0.31
0.24
0.18
The above grain size distribution is given in graphical form in Figure 1.
1
Fsu
0.8
0.6
0.4
0.2
0
0.001
0.01
0.1
1
D mm
Figure 1
Values of Fbc of 0.001, 0.002 and 0.003 are adopted for the calculations,
corresponding to cases A and B. In Figure 2 the assumed variation of D 50 and
the predicted variations of Dc are shown for the three cases (Fbc, cu) = (0.002,
0.0002). (0.002, 0.0004), and (0.002. 0.0006). In Figure 3 the assumed variation
of D50 and the predicted variations of Dc are shown for the three cases (Fbc, cu) =
(0.001, 0.0004), (0.002, 0.0004) and (0.003, 0.0004).
D50 and Dc versus x: Dc for three
concentrations c with Fbc = 0.002
D, Dc (mm)
1
D50
c = 0.0002
c = 0.0004
0.1
c = 0.0006
0.01
0
100
200
300
x km
Figure 2
400
500
D50 and Dc versus x: Dc for three
values of Fbc with c = 0.004
D50, Dc (mm)
1
D50
Fbc = 0.001
Fbc = 0.002
Fbc = 0.003
0.1
0.01
0
100
200
300
400
500
x km
Figure 3
Figure 2 indicates that for a specified value of Fbc the cutoff size Dc drops with
increasing suspended sediment concentration. Figure 3 indicates that for a
specified suspended sediment concentration Dc increases with increasing Fbc.
References
Dietrich, W. E. 1982 Settling velocity of natural particles. Water Resources
Research, 18(6), 1615-1626.
Engelund, F. and Hansen, E. 1967 A monograph on sediment transport in
alluvial streams. Teknisk Forlag, Copenhagen, Denmark, 62 p.
Garcia, M. and Parker, G. 1991 Entrainment of bed sediment into suspension.
J. Hydraul. Engrg., ASCE, 117(4), 414-435.
REFERENCES ON WASHLOAD CUTOFF.
ASCE and Colby give traditional 62 micron value.
method similar to the present one.
Einstein and Chien give
ASCE, 1975. Sedimentation Engineering. Edited by Vanoni, V. A., V.A., New
York, 745 p.
Colby, B. R., 1957. “Relationship of unmeasured sediment discharge to mean
velocity.” Trans. American Geophysical Union, 38(5), 708-717.
Einstein, H. A. and Chien, N. 1953. “Can the rate of wash load be predicted
from the bed-load function?” Trans. American Geophysical Union, 34(6), 876882.
ADDENDUM
Chris Paola and Gary Parker
10/28/00
The basic transport relation is
qui Fi  qT pi
(1)
where Fi denotes the fractions in the active layer of the bed (active layer), pi
denotes the fractions in the load (bedload + suspended load), q T is the total load
and qui is a unit transport rate specified in the following form;
Rgqui
 b





3/2
 D 
 G  b , i 
 D 
g 

(2)
In the above relation R = (s/) –1, s denotes sediment density,  denotes water
density, g denotes the acceration of gravity, b denotes boundary shear stress, Di
denotes the ith grain size, Dg denotes the geometric mean size of the bed (active
layer), b denotes a Shields stress based on b and Dg given by the relation
 b 
b
RgD g
(3)
corresponding to bankfull flow and G is a functional form that must be specified.
In addition, where
Di  2 i
Dg  2 
(4)
then
N
   Fi  i
(5)
i1
where N denotes the number of grain size ranges. Note: the above treatment
could be formulated in terms of D50 as well as Dg.
Here it is envisaged that b is a specified parameter. Suppose that qT and
pi are known as well. Solving for Fi using (1) and (2),
Fi 
qT p i
Rg
 b





3/2
1
 D 
G  b , i 
 D 
g 

(6)
Now (6), (3) and the relation
N
n 2 (D g )   Fi  i
(7)
i1
obtained from (4) and (5) define N+2 relations in N+2 unknowns F i, b and Dg.
Wash load is provisionally defined as that fine fraction of the bed material
that could fit comfortably within the pore space of its coarser brethren. With this
in mind, it is assumed that the following relation exists between porosity p and
the grain size distribution;
 p   po f  (,...)
(8)
where po denotes the porosity associated with uniform material with the
geometric mean size of the mixture,  denotes the arithmetic standard deviation
of the mixture, given as
 2  ( i   )2 Fi
(9)
and the ellipsis denotes the possibility of higher moments participating in the
relation.. It is expected that f decreases as  increases, as a wider range of
sizes allows for fine particles to fit within the pore space between coarse
particles.
Now let i be ordered from coarsest (i = 1) to finest (i = N). The cutoff size
index for bed material load is denoted as Nc. Let c be defined as
Nc
 c   ( i   )2 Fi
(10)
 pc   po f  ( c ,...)
(11)
2
i1
and let
Then the following equation must be solved for the washload cutoff N c, and thus
Dc = D iNc ;
N
 F  
iNc 1
i
dc
(12)
where  is an order-one coefficient, probably less than one, that accounts for the
fact that the fine material stuffed into the pores of the coarser material has its
own pore space as well. The parameter  may also need to account for the
possibility that some of the grains that ought to fit into the pore space may not
find individual pores that are big enough to fit in.