1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHAPTER 7:
RELATIONS FOR 1D BEDLOAD TRANSPORT
Let qb denote the volume bedload transport rate per unit width (sliding, rolling,
saltating). It is reasonable to assume that qb increases with a measure of flow
strength, such as depth-averaged flow velocity U or boundary shear stress b.
A dimensionless Einstein bedload number q* can be defined as follows:
qb
q 
RgD D

b
A common and useful approach to the quantification of bedload transport is to
empirically relate qb* with either the Shields stress * or the excess of the Shields
stress * above some appropriately defined “critical” Shields stress c*. As pointed
out in the last chapter, c* can be defined appropriately so as to a) fit the data and
b) provide a useful demarcation of a range below which the bedload transport rate
is too low to be of interest.
The functional relation sought is thus of the form
qb  qb (  ) or
qb  qb (   c )
1
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
BEDLOAD TRANSPORT RELATION OF MEYER-PETER AND MÜLLER
All the bedload relations in this chapter pertain to a flow condition known as “planebed” transport, i.e. transport in the absence of significant bedforms. The influence
of bedforms on bedload transport rate will be considered in a later chapter.
The “mother of all modern bedload transport relations” is that due to Meyer-Peter
and Müller (1948) (MPM). It takes the form
qb  8(   c )3 / 2
, c  0.047
The relation was derived using flume data pertaining to well-sorted sediment in the
gravel sizes.
Recently Wong (2003) and Wong and Parker (submitted) found an error in the
analysis of MPM. A re-analysis of the all the data pertaining to plane-bed transport
used by MPM resulted in the corrected relation
qb  4.93 (   c )1.6
, c  0.047
If the exponent of 1.5 is retained, the best-fit relation is
qb  3.97 (   c )3 / 2
, c  0.0495
2
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
Bedload Relation: Modified MPM
1.0E+00
1.0E-01
qb* = 3.97 (* - c*)1.50
c* = 0.0495
qb*
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-02
Data of Meyer-Peter and Muller (5.21 mm,
28.65 mm) and Gilbert (3.17 mm, 4.94 mm,
7.01 mm)
1.0E-01
*
1.0E+00
3
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
LIMITATIONS OF MPM
The “critical Shields stress” c* of either 0.047 or 0.0495 in either the original or
corrected MPM relation(s) must be considered as only a matter of convenience for
correlating the data. This can be demonstrated as follows.
Consider bankfull flow in a river. The bed shear stress at bankfull flow bbf can be
estimated from the depth-slope product rule of normal flow:
bbf  gHbf S
The corresponding Shields stress bf50* at bankfull flow is then estimated as
bf 50 
Hbf S
RD s50
where Ds50 denotes a surface median size. For the gravel-bed rivers presented in
Chapter 3, however, the average value of bf50* was found to be about 0.05 (next
page).
According to MPM, then, these rivers can barely move sediment of the surface
median size Ds50 at bankfull flow. Yet most such streams do move this size at
bankfull flow, and often in significant quantities.
4
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
LIMITATIONS OF MPM contd.
1.E+01
1.E+00
bf 50
gravel-bed streams
Grav Brit
Grav Alta
Sand Mult
Sand Sing
Grav Ida
1.E-01
1.E-02
1.E-03
1.E+02
1.E+04
1.E+06
1.E+08
1.E+10
1.E+12
1.E+14
Q̂
There is nothing intrinsically “wrong” with MPM. In a dimensionless sense,
however, the flume data used to define it correspond to the very high end of the
transport events that normally occur during floods in alluvial gravel-bed streams.
While the relation is important in a historical sense, it is not the best relation to use
with gravel-bed streams.
5
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
A SMORGASBORD OF BEDLOAD TRANSPORT RELATIONS FOR UNIFORM
SEDIMENT
Some commonly-quoted bedload transport relations with good data bases are given
below.
43.5qb
1 ( 0.143 /  )2 t 2
1
e dt 

( 0.143 /  ) 2
1  43.5qb


qb  17   c



  c ,
qb  18.74   c


b
q  5.7   

b

 1.5
c

 
q  11.2 
 1.5
c  0.05


  0.7 c ,
c  0.05
, c  0.037 ~ 0.0455
 
1 
 


c





Einstein (1950)
Ashida & Michiue (1972)
Engelund & Fredsoe (1976)
Fernandez Luque & van Beek (1976)
4 .5
,
c  0.03
Parker (1979) fit to
Einstein (1950)
6
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
PLOTS OF BEDLOAD TRANSPORT RELATIONS
1.E+02
1.E+01
1.E+00
1.E-01
E
AM
EF
FLBSand
P approx E
FLBGrav
qb *
1.E-02
1.E-03
1.E-04
1.E-05
1.E-06
1.E-07
1.E-08
1.E-09
0.01
E = Einstein
AM = Ashida-Michiue
EF = Engelund-Fredsoe
P approx E = Parker approx of Einstein
FLBSand = Fernandez Luque-van
Beek, c* = 0.038
FLBGrav = Fernandez Luque-van
Beek, c* = 0.0455
0.1
*
1
7
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
NOTES ON THE BEDLOAD TRANSPORT RELATIONS
 The bedload relation of Einstein (1950) contains no critical Shields number. This
reflects his probabilistic philosophy.
 All of the relations except that of Einstein correspond to a relation of the form
qb ~ ( )3 / 2
In the limit of high Shields number. In dimensioned form this becomes
 u 
qb

 K 
RgD D
 RgD 
2

3/2
or
Rgq b
u
3

K
where K is a constant; for example in the case of Ashida-Michiue, K = 17. Note
that in this limit the bedload transport rate becomes independent of grain size!!
 Some of the scatter between the relations is due to the face that c* should be a
function of Rep. This is reflected in the discussion of the Fernandez Luque-van
Beek relation in the next slide. (Recall that Rep  RgD D / .)
 Some of the scatter is also due to the fact that several of the relations have been
plotted well outside of the data used to derive them. For example, in data used
to derive Fernandez Luque-van Beek, * never exceeded 0.11, whereas
8
the plot extends to * = 1.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NOTES ON THE RELATION OF FERNANDEZ LUQUE AND VAN BEEK
In the experiments on which the relation is based;
a) Streamwise bed slope angle  varied from near 0 to 22.
b) The material tested included five grain sizes and three specific gravities, as
given below; also listed are the values of Rep and the critical Shields number co*
determined empirically at near vanishing bed slope angle.
Material
Walnut shells
Sand1
Sand2
Gravel
Magnetite
D mm
1.5
0.9
1.8
3.3
1.8
R
0.34
1.64
1.64
1.64
3.58
Rep
106
108
306
760
453
co*
0.038
0.038
0.037
0.0455
0.042
c) It is thus possible to check the effect of Rep and  on the transport relation of
Fernandez Luque and van Beek (FLvB).
9
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
CRITICAL SHIELDS NUMBER IN THE RELATION OF FLvB
The experimental values of co* generally track the modified Shields relation of
Chapter 6, but are high by a factor ~ 2. This reflects the fact that they correspond to
a condition of “very small” transport determined in a consistent way (see original
reference).
0.05
0.045
.
0.04
0.035
co*
0.03
Data of FLvB
Modified Shields Curve
0.025
0.02
0.015
( 7.7 Rep0.6 )
0.01
c  0.5 [0.22 Rep0.6  0.06  10
]
0.005
0
10
100
Rep
1000
10
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
CRITICAL SHIELDS NUMBER IN THE RELATION OF FLvB contd.
The ratio c*/co* decreases with streamwise angle  as predicted by the relation of
Chapter 6, but to obtain good agreement r must be set to the rather high value of
47 (c = 1.07).
1.1
1
.
c*/co*
0.9
Observed
Predicted
0.8
0.7
c
tan 

cos

(
1

)
co
c
0.6
0.5
0.4
0
5
10
15

20
25
30
11
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
SHEET FLOW
• For values of * < a threshold value sheet*, bedload is localized in terms of
rolling, sliding and saltating grains that exchange only with the immediate bed
surface.
• When s* > sheet* the bedload layer devolves into a sliding layer of grains
that can be several grains thick. Sheet flows occur in unidirectional river flows
as well as bidirectional flows in the surf zone.
• Values of sheet* have been variously estimated as 0.5 ~ 1.5. (Horikawa,
1988, Fredsoe and Diegaard, 1994, Dohmen-Jannsen, 1999; Gao, 2003). The
parameter sheet* appears to decrease with increasing Froude number.
• Wilson (1966) has estimated the bedload transport rate in the sheet flow
regime as obeying a relation of the form
qb  K( )3 / 2 , K  12
All the previously presented bedload relations except that of Einstein also
devolve to a relation of the above form for large *, with K varying between
3.97 and 18.74.
12
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
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© Gary Parker November, 2004
A VIEW OF SHEET FLOW TRANSPORT
Double-click on the image to run the video.
D = 0.116 mm
S = 0.035
U = 1.05 m/s
Fr = 1.85
sheet* = 0.51
rte-booksheetpeng.mpg: to run without relinking, download to same folder as PowerPoint presentations.
13
Video clip courtesy P. Gao and A. Abrahams; Gao (2003)
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MECHANISTIC DERIVATIONS OF BEDLOAD TRANSPORT RELATIONS
A number of mechanistic derivations of bedload transport relations are available.
These are basically of two types, based on two forms for bedload continuity. Let bl =
the volume of sediment in bedload transport per unit area, ubl = the mean velocity of
bedload particles, Ebl = the volume rate of entrainment of bed particles into bedload
movement (rolling, sliding or saltation, not suspension) and Lsbl = the average step
length of a bedload particle (from entrainment to deposition, usually including many
saltations). The following continuity relations then hold:
qb  blubl
, qb  EblL sbl
In a Bagnoldean approach, separate predictors are developed for ubl and bl, the latter
determined from the Bagnold constraint (Bagnold, 1956). Models of this type include
the macroscopic models of Ashida and Michiue (1972) and Engelund and Fredsoe
(1976), and the saltation models of Wiberg and Smith (1985, 1989), Sekine and
Kikkawa (1992), and Nino and Garcia (1994a,b). Recently, however, Seminara et al.
(2003) have shown that the Bagnold constraint is not generally correct.
In the Einsteinean approach the goal is to develop predictors for Ebl and Lsbl. It is the
former relation that is particularly difficult to achieve. Models of this type include 14
the Einstein (1950), Tsujimoto (e.g. 1991) and Parker et al. (2003).
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CALCULATIONS WITH BEDLOAD TRANSPORT RELATIONS
To perform calculations with any of the previous bedload transport relations, it is
necessary to specify:
1) the submerged specific gravity R of the sediment;
2) a representative grain size exposed on the bed surface, e.g. surface geometric
mean size Dsg or surface median size Ds50, to be used as the characteristic size
D in the relation;
3) and a value for the shear velocity of the flow u* (and thus b).
Once these parameters are specified, * = (u*)2/(RgD) is computed, qb* is calculated
from the bedload transport relation, and the volume bedload transport rate per
unit width is computed as qb = (RgD)1/2Dqb*.
The shear velocity u* is computed from the flow field using the techniques of Chapter
5. For example, in the case of normal flow satisfying the Manning-Strickler
resistance relation,
k q
u  
2

r

2

1/ 3
s
2
w



3 / 10
g7 / 10 S7 / 10
1/ 3 2

k
q
   s 2 w
 r g



3 / 10
S7 / 10
RD
15
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
ALTERNATIVE DIMENSIONLESS BEDLOAD TRANSPORT
Again, the case under consideration is plane-bed bedload transport (no bedforms).
As a preliminary, define a dimensionless sediment transport rate W* as
Rgq
qb
W   3/2  3 b
( )
u

Now all previously presented bedload transport rates for uniform sediment can be
rewritten in terms of W* as a function of *:
1 ( 0.143 /  )2 t 2
43.5 (  )3 / 2 W 
1
e dt 

( 0.143 /  )2
1  43.5 (  )3 / 2 W 

 c  
c 
W  171    1   , c  0.05
 
   
 c  
c 

W  18.741    1  0.7  , c  0.05
 
   
1 .5




W   5.71  c  , c  0.037 ~ 0.0455
  

 c 

W  11.2 1   
 s 
4 .5
,
c  0.03
Einstein (1950)
Ashida & Michiue (1972)
Engelund & Fredsoe (1976)
Fernandez Luque & van Beek (1976)
Parker (1979) fit to
Einstein (1950)
16
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
SURFACE-BASED BEDLOAD TRANSPORT FORMULATION FOR MIXTURES
Consider the bedload transport of a mixture of sizes. The thickness La of the active
(surface) layer of the bed with which bedload particles exchange is given by as
La  naDs90
where Ds90 is the size in the surface (active) layer such that 90 percent of the
material is finer, and na is an order-one dimensionless constant (in the range 1 ~ 2).
Divide the bed material into N grain size ranges, each with characteristic size Di, and
let Fi denote the fraction of material in the surface (active) layer in the ith size range.
The volume bedload transport rate per unit width of sediment in the ith grain size
range is denoted as qbi. The total volume bedload transport rate per unit width is
denoted as qbT, and the fraction of bedload in the ith grain size range is pbi, where
N
qbT   qbi
i1
qbi
, pbi 
qbT
Now in analogy to *, q* and W*, define the dimensionless grain size specific
Shields number i*, grain size specific Einstein number qi* and dimensionless grain
size specific bedload transport rate Wi* as
2

u
b
i 
 
RgD i RgD i
qbi
, qbi 
RgD i D i Fi

Rgq bi
q

bi
, Wi   3 / 2 
( i )
(u )3 Fi
17
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
SURFACE-BASED BEDLOAD TRANSPORT FORMULATION contd.
It is now assumed that a functional relation exists between qi* (Wi*) and i*, so that
qbi
q 
 fq ( i ) or
RgDi Di Fi

bi

i
W 
Rgq bi
(u )3 Fi
 fW ( i )
The bedload transport rate of sediment in the ith grain size range is thus given as
qbi  Fi RgDi Di fq ( i ) or
u3
qbi  Fi
fW (i )
Rg
qbi
According to this formulation, if the grain
size range is not represented in the
surface (active) layer, it will not be
represented in the bedload transport.
qbi
La

x
z'
18
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
BEDLOAD RELATION FOR MIXTURES DUE TO ASHIDA AND MICHIUE (1972)

qbi  17 i  ci

i  ci

Basic transport relation

ci

scg


1



 0.843 Di  for Di  0.4
D 
Dsg

 sg 

2


 

 log(19) 
Di
for
 0 .4
 

Dsg
 log19 Di  
  Dsg  

 
Modified version of Egiazaroff
(1965) hiding function
Dsg  2
s
N
,
 s    iFi
i1
scg  0.05
Effective critical Shields stress for
surface geometric mean size
Note This relation has been modified
slightly from the original formulation.
Here the relation specifically uses
surface fractions Fi, and surface
geometric mean size Dsg is specified in
preference to the original arithmetic
mean size Dsm = DiFi.
19
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
BEDLOAD RELATION FOR MIXTURES DUE TO PARKER (1990a,b)
This relation is appropriate only for the computation of gravel bedload transport rates
in gravel-bed streams. In computing Wi*, Fi must be renormalized so that the sand is
removed, and the remaining gravel fractions sum to unity, Fi = 1. The method is
based on surface geometric size Dsg and surface arithmetic standard deviation s on
the  scale, both computed from the renormalized fractions Fi.
Wi  0.00218 Gi 
D 
i  sgo  i 
D 
 sg 
0.0951
,
sgo 
sg


ssrg
u2
 
,
RgDsg

sg
,
4 .5

 0.853 
 for   1.59
54741 






G()  exp 14.2(  1)  9.28(  1)2 for 1    1.59

14.2 for   1


s
  1
O (sgo )  1
Dsg  2 s ,
O (sgo )

ssrg  0.0386



N
s    iFi
i 1
N
,
    i  s  Fi
2
s
2
i1
In the above O and O are set functions of sgospecified in the next slide.
20
1D SEDIMENT TRANSPORT MORPHODYNAMICS
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© Gary Parker November, 2004
BEDLOAD RELATION FOR MIXTURES DUE TO PARKER (1990a,b) contd.
1.6
1.4
1.2
1
omegaO
o
o
sigmaO
 O,  O 0.8
0.6
0.4
0.2
0
0.1
1
10
100
1000
 sgo
It is not necessary to use the above chart. The calculations can be
performed using the Visual Basic programs in RTe-bookAcronym1.xls
21
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BEDLOAD RELATION FOR MIXTURES DUE TO PARKER (1990a,b) contd.
An example of the renormalization to remove sand is given below.
Surface grain size fractions
Raw Values
Range
Range mm
no
1 0.25 ~ 0.5
2 0.5 ~ 1
3 1~2
4 2~4
5 4~8
6 8 ~ 16
7 16 ~ 32
8 32 ~ 64
9 64 ~ 128
10 128 ~ 256
Fraction sand
Fraction gravel
Di mm
Fi
0.35
0.71
1.41
2.83
5.66
11.31
22.63
45.25
90.51
181.02
0.05
0.14
0.07
0.03
0.06
0.12
0.25
0.18
0.07
0.03
Renormalized Values
Range no
Range
mm
Di mm
1
2
3
4
5
6
7
2.83
5.66
11.31
22.63
45.25
90.51
181.02
2~4
4~8
8 ~ 16
16 ~ 32
32 ~ 64
64 ~ 128
128 ~ 256
Fi
0.041
0.081
0.162
0.338
0.243
0.095
0.041
(=0.03/0.74)
(=0.06/0.74)
(=0.12/0.74)
(=0.25/0.74)
(=0.18/0.74)
(=0.07/0.74)
(=0.03/0.74)
0.26
0.74
Sum
1
Sum
1
22
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
PROGRAMMING IN VISUAL BASIC FOR APPLICATIONS
The Microsoft Excel workbook RTe-bookAcronym1.xls is an example of a
workbook in this e-book that uses code written in Visual Basic for Applications
(VBA). VBA is built directly into Excel, so that anyone who has Excel (versions
2000 or later) can execute the code directly from the relevant worksheet in the
workbook RTe-bookAcronym1.xls. The relevant code is contained in three
modules, Module1, Module2 and Module3 of the workbook, which may be
accessed from the Visual Basic Editor.
All the code in this e-book is written in VBA in Excel modules. A self-teaching
tutorial in VBA is contained in the workbook RTe-bookIntroVBA.xls. Going though
the tutorial will not only help the reader understand the material in this e-book, but
also allow the reader to write, execute and distribute similar code.
To take the tutorial, open RTe-bookIntroVBA.xls and follow the instructions.
23
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NOTES ON Rte-bookAcronym1.xls
The workbook RTe-bookAcronym1.xls provides interfaces for three different
implementations.
In the implementation of “Acronym1” the user inputs the specific gravity of the
sediment R+1, the shear velocity of the flow and the grain size distribution of the
bed material. The code computes the magnitude and size distribution of the
bedload transport.
In the implementation of “Acronym1_R” the user inputs the specific gravity of the
sediment R+1, the flow discharge Q, the bed slope S, the channel width B, the
parameter nk relating the roughness height ks to the surface size Ds90 and the
grain size distribution of the bed material. The code computes the shear velocity
using a Manning-Strickler resistance formulation and the assumption of normal
flow, and then computes the magnitude and size distribution of the bedload
transport.
The implementation of “Acronym1_D” uses the same formulation as
“Acronym1_R”, but allows specification of a flow duration curve so that average24
annual gravel transport rate and grain size distribution can be computed.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EXAMPLE: INTERFACE FOR “Acronym1” IN Rte-bookAcronym1.xls
25
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BEDLOAD RELATION FOR MIXTURES DUE TO WILCOCK AND CROWE (2003)
The sand is not excluded in the fractions Fi used to compute Wi*. The method is based
on the surface geometric mean size Dsg and fraction sand in the surface layer Fs.
Wi*  Gi 
 0.0027.5
4.5

G    0.894 
141  0.5 

 
for   1.35
for   1.35
b
  Di 


i 

  Dsg 
ssrg  0.021  0.015 exp( 20Fs )

sg

ssrg
b
0.67
1  exp(1.5  Di / Dsg )
2
u

sg 
RgDsg
26
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
EFFECT OF SAND CONTENT IN THE SURFACE LAYER ON GRAVEL MOBILITY
IN A GRAVEL-BED STREAM
Wilcock and Crowe (2003) have shown that increasing sand content in the bed surface
layer of a gravel-bed stream renders the surface gravel more mobile. This effect is
captured in their relationship between the reference Shields number for the surface
geometric mean size ssrg (a surrogate for a critical Shields number) and the fraction
sand Fs in the surface layer:
ssrg  0.021 0.015 exp( 20Fs )

ssrg
Note how  decreases
as Fs increases. The
surface layers of gravelbed streams rarely contain
more than 30% sand;
ssrg
beyond this point the
gravel tends to be buried
under pure sand.
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0
0.05
0.1
0.15
Fs
0.2
0.25
0.3
27
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BEDLOAD RELATION FOR MIXTURES DUE TO POWELL, REID AND LARONNE
(2001)
The sand is excluded in the fractions Fi used to compute Wi*. The method is
based on the surface median size Ds50, computed after excluding sand..
 1
Wi  11.21  
 
i
 
sci

sci

sc 50
 Di 

 
 Ds50 
4.5
0.74
,
sc 50  0.03
More information about bedload transport relations for mixtures can be
28
found in Parker (in press; downloadable from http://www.ce.umn.edu/~parker/).
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATIONS OF BEDLOAD TRANSPORT RATE OF MIXTURES
Assumed grain size distribution of the bed surface: Dsg = surface geometric mean,
sg = surface geometric standard deviation, Fs = fraction sand in surface layer.
i 
7.5
6.5
5.5
4.5
3.5
2.5
1.5
0.5
-0.5
-1.5
Raw Surface Grain Size Distribution
Db,I, mm
Di, mm
% Finer
256
100
181.02
128
97
90.51
64
71
45.25
32
45
22.63
16
26
11.31
8
20
5.66
4
18
2.83
2
16
1.41
1
7
0.71
0.5
2
0.35
0.25
0
Fi
0.03
i 
0.26
7.5
0.26
6.5
0.19
5.5
0.06
4.5
0.02
0.02
3.5
2.5
0.09
1.5
0.05
0.02
Dsg = 22.3 mm, sg = 4.93, Fs = 0.16
Surface Grain Size Distribution Gravel
Only
Db,I, mm
Di, mm % Finer
Fi
256
100.00
181.02
0.036
128
96.43
90.51
0.310
64
65.48
45.25
0.310
32
34.52
22.63
0.226
16
11.90
11.31
0.071
8
4.76
5.66
0.024
4
2.38
2.83
0.024
2
0.00
Dsg = 40.7 mm, sg = 2.36
(sand excluded)
29
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATIONS, MIXTURES contd.
Grain Size Distributions for Bedload Calculations
100
Percent Finer
80
60
Surface
Surface Gravel
40
20
0
0.1
1
10
D (mm)
100
1000
30
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATIONS, MIXTURES contd.
Other input parameters:
R = 1.65
u* = 0.15 to 0.40 m/s
Relations used:
A-M = Ashida and Michiue (1972), sand not excluded
P = Parker (1990), sand excluded
P-R-L = Powell et al. (2001), sand excluded
W-C = Wilcock and Crowe (2003), sand not excluded
Output parameters
qG = volume gravel bedload transport rate per unit width
DGg = geometric mean size of the gravel portion of the transport
Gg = geometric standard deviation of the gravel portion of the bedload
pG = fraction gravel in the bedload transport (only for A-M and W-C):
fraction sand = 1 - pG
31
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATIONS, MIXTURES contd.
Gravel Bedload Transport Rate
0.1
0.01
qG (m2/s)
0.001
0.0001
W-C
A-M
0.00001
P-R-L
0.000001
P
0.0000001
0.1
1
u* (m/s)
32
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATIONS, MIXTURES contd.
Geometric Mean Size of Gravel Bedload
100
DGg (mm)
W-C
A-M
P
10
P-R-L
1
0.1
1
u* (m/s)
33
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATIONS, MIXTURES contd.
Geometric Standard Deviation of Gravel Bedload
3.5
P
3
P-R-L
Gg
2.5
W-C
2
A-M
1.5
1
0.1
1
u* (m/s)
34
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SAMPLE CALCULATIONS, MIXTURES contd.
Fraction Gravel in Bedload
1
A-M
pG
W-C
0.5
0
0.1
1
u* (m/s)
35
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
COMPUTATIONS OF ANNUAL BEDLOAD YIELD
It is necessary to have a flow duration curve to perform the calculation. The flow duration
curve specifies the fraction of time a given water discharge is exceeded, as a function of
water discharge.
This curve is divided into M bins k = 1 to M, such that pQk specifies the fraction of time the
flow is in range k with characteristic discharge Qk. The value u*k must be computed for each
range. For example, in the case of normal flow with constant width B, the Manning-Strickler
resistance relation from Chapter 5 yields
u2,k
 k1s/ 3Qk2 
  2 2 
 B r 
3 / 10
g7 / 10 S7 / 10
The grain size distribution of the bed material must be specified; ks can be computed as 2
Ds90 (for a plane bed). Once u*,k is computed, either qb,k (material approximated as uniform)
or qbi,k (mixtures) is computed for each flow range, and the annual sediment yield qba or total
yield qbTa and grain size fractions of the yield pai are given as
M
M
qba   qb,kpQk
k 1
M

    qbi,kpQk  , pai 
i1  k 1

q
N
or
qbTa
k 1
M
bi,k
pQk


q
p


  bi,k Qk 
i1  k 1

N
Expect the flood flows to contribute disproportionately to the annual sediment yield.
An implementation is given in “Acronym1_D” of Rte-bookAcronym1.xls .
36
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REFERENCES FOR CHAPTER 7
Ashida, K. and M. Michiue, 1972, Study on hydraulic resistance and bedload transport rate in
alluvial streams, Transactions, Japan Society of Civil Engineering, 206: 59-69 (in Japanese).
Bagnold, R. A., 1956, The flow of cohesionless grains in fluids, Philos Trans. R. Soc. London A,
249, 235-297.
Dohmen-Janssen, M., 1999, Grain size influence on sediment transport in oscillatory sheet flow,
Ph.D. thesis, Technical University of Delft, the Netherlands, 246 p.
Einstein, H. A., 1950, The Bed-load Function for Sediment Transportation in Open Channel
Flows, Technical Bulletin 1026, U.S. Dept. of the Army, Soil Conservation Service.
Egiazaroff, I. V., 1965, Calculation of nonuniform sediment concentrations, Journal of Hydraulic
Engineering, 91(4), 225-247.
Engelund, F. and J. Fredsoe, 1976, A sediment transport model for straight alluvial channels,
Nordic Hydrology, 7, 293-306.
Fernandez Luque, R. and R. van Beek, 1976, Erosion and transport of bedload sediment,
Journal of Hydraulic Research, 14(2): 127-144.
Fredsoe, J. and Deigaard, R., 1994, Mechanics of Coastal Sediment Transport, World Scientific,
ISBN 9810208405, 369 p.
Gao, P., 2003, Mechanics of bedload transport in the saltation and sheetflow regimes, Ph.D.
thesis, Department of Geography, University of Buffalo, State University of New York
Horikawa, K., 1988, Nearshore Dynamics and Coastal Processes, University of Tokyo Press, 522
p.
37
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REFERENCES FOR CHAPTER 7 contd.
Meyer-Peter, E. and Müller, R., 1948, Formulas for Bed-Load Transport, Proceedings, 2nd
Congress, International Association of Hydraulic Research, Stockholm: 39-64.
Nino, Y. and Garcia, M., 1994a, Gravel saltation, 1, Experiments, Water Resour. Res., 30(6),
1907-1914.
Nino, Y. and Garcia, M., 1994b, Gravel saltation, 2, Modelling, Water Resour. Res., 30(6), 19151924.
Parker, G., 1979, Hydraulic geometry of active gravel rivers, Journal of Hydraulic Engineering,
105(9), 1185-1201.
Parker, G., 1990a, Surface-based bedload transport relation for gravel rivers, Journal of
Hydraulic Research, 28(4): 417-436.
Parker, G., 1990b, The ACRONYM Series of PASCAL Programs for Computing Bedload
Transport in Gravel Rivers, External Memorandum M-200, St. Anthony Falls Laboratory,
University of Minnesota, Minneapolis, Minnesota USA.
Parker, G., Solari, L. and Seminara, G., 2003, Bedload at low Shields stress on arbitrarily sloping
beds: alternative entrainment formulation, Water Resources Research, 39(7), 1183,
doi:10.1029/2001WR001253, 2003.
Parker, G., in press, Transport of gravel and sediment mixtures, ASCE Manual 54, Sediment
Engineering, ASCE, Chapter 3, downloadable at
http://cee.uiuc.edu/people/parkerg/manual_54.htm .
Powell, D. M., Reid, I. and Laronne, J. B., 2001, Evolution of bedload grain-size distribution with
increasing flow strength and the effect of flow duration on the caliber of bedload sediment
38
yield in ephemeral gravel-bed rivers, Water Resources Research, 37(5), 1463-1474.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REFERENCES FOR CHAPTER 7 contd.
Sekine, M. and Kikkawa, H., 1992, Mechanics of saltating grains, J. Hydraul. Eng., 118(4), 536558.
Seminara, G., Solari, L. and Parker, G., 2002, Bedload at low Shields stress on arbitrarily sloping
beds: failure of the Bagnold hypothesis, Water Resources Research, 38(11), 1249,
doi:10.1029/2001WR000681.
Tsujimoto, T., 1991, Mechanics of Sediment Transport of Graded Materials and Fluvial Sorting,
Report, Faculty of Engineering, Kanazawa University, Japan (in Japanese and English).
Wiberg, P. L. and Smith, J. D., 1985, A theoretical model for saltating grains in water, J.
Geophys. Res., 90(C4), 7341-7354.
Wiberg, P. L. and Smith, J. D., 1989, Model for calculating bedload transport of sediment, J.
Hydraul. Eng, 115(1), 101-123.
Wilcock, P. R., and Crowe, J. C., 2003, Surface-based transport model for mixed-size sediment,
Journal of Hydraulic Engineering, 129(2), 120-128.
Wilson, K. C., 1966, Bed load transport at high shear stresses, Journal of Hydraulic Engineering,
92(6), 49-59.
Wong, M., 2003, Does the bedload equation of Meyer-Peter and Müller fit its own data?,
Proceedings, 30th Congress, International Association of Hydraulic Research, Thessaloniki,
J.F.K. Competition Volume: 73-80.
Wong, M. and Parker, G., submitted, The bedload transport relation of Meyer-Peter and Müller
overpredicts by a factor of two, Journal of Hydraulic Engineering, downloadable
at
39
http://cee.uiuc.edu/people/parkerg/preprints.htm .