Sediment in mountain rivers tends to be poorly sorted, including a wide range of grain size from san

1
1D MORPHODYNAMICS OF MOUNTAIN RIVERS SEDIMENT
MIXTURES
Sediment in mountain rivers tends to be poorly
sorted, including a wide range of grain size from
sand to gravel and coarser. The bed and bedload
should be characterized in terms of a grain size
distribution rather than a single grain size. In
characterizing grain size distributions, grain
size is often specified in terms of a base-2
logarithmic scale (phi scale or psi scale).
These are defined as follows where D is given in
mm,
Gravel and sand in cut bank, Las Vegas Wash,
Arizona, USA
2
SAMPLE EVALUATIONS OF ? AND ? SEDIMENT SIZE
RANGES
3
SEDIMENT GRAIN SIZE DISTRIBUTIONS
The grain size distribution is characterized in
terms of N1 sizes Db,i such that ff,i denotes
the mass fraction in the sample that is finer
than size Db,i. In the example below N 7.
Note the use of a logarithmic scale for grain
size.
4
SEDIMENT GRAIN SIZE DISTRIBUTIONS contd.
In the grain size distribution of the last slide,
the finest size (0.03125 mm) was such that 2
percent, not 0 percent was finer. If the finest
size does not correspond to 0 percent content, or
the coarsest size to 100 percent content, it is
often useful to use linear extrapolation on the
psi scale to determine the missing values.
Note that the addition of the extra point has
increased N from 7 to 8 (there are N1 points).

5
SEDIMENT GRAIN SIZE DISTRIBUTIONS contd.
The grain size distribution after extrapolation
is shown below.
6
CHARACTERISTIC SIZES BASED ON PERCENT FINER
Dx is size such that x percent of the sample is
finer than Dx Examples D50 median size D90
roughness height
To find Dx (e.g. D50) find i such that
Then interpolate for ?x
and back-calculate Dx in mm
7
STATISTICAL CHARACTERISTICS OF SIZE DISTRIBUTION
N1 bounds defines N grain size ranges. The ith
grain size range is defined by (Db,i, Db,i1) and
(ff,i, ff,i1)
?i (Di) characteristic size of ith grain size
range fi fraction of sample in ith grain size
range
8
STATISTICAL CHARACTERISTICS OF SIZE DISTRIBUTION
contd.
mean grain size on psi scale ? standard
deviation on psi scale
Dg geometric mean size ?g geometric standard
deviation ( ? 1) Sediment is well sorted if ?g lt
1.6
Dg 0.273 mm, ?g 2.17
9
GRAIN SIZE DISTRIBUTION CALCULATOR
Workbook RTe-bookGSDCalculator.xls computes the
statistics of a grain size distribution input by
the user, including Dg, ?g, and Dx where x is a
specified number between 0 and 100 (e.g. the
median size D50 for x 50). It uses code in VBA
(macros) to perform the calculations. You will
not be able to use macros if the security level
in Excel is set to High. To set the security
level to a value that allows you to use macros,
first open Excel. Then click Tools, Macro,
Security and then in Security Level check
Medium. This will allow you to use macros.
10
GRAIN SIZE DISTRIBUTION CALCULATOR contd.
When you open the workbook RTe-bookGSDCalculator.x
ls, click Enable Macros. The GUI is contained
in the worksheet Calculator. Now to access the
code, from any worksheet in the workbook click
Tools, Macro, Visual Basic Editor. In the
Project window to the left you will see the
line VBA Project (FDe-bookGSDCalculator.xls).
Underneath this you will see Module1.
Double-click on Module1 to see the code in the
Code window to the right. These actions allow
you to see the code, but not necessarily to
understand it. In order to understand this
course, you need to learn how to program in VBA.
Please work through the tutorial contained in the
workbook RTe-bookIntroVBA.xls. It is not very
difficult! All the input are specified in the
worksheet Calculator. First input the number
of pairs npp of grain sizes and percents finer
(npp N1 in the notation of the previous
slides) and click the appropriate button to set
up a table for inputting each pair (grain size in
mm, percent finer) in order of ascending size.
Once this data is input, click the appropriate
button to compute Dg and ?g. To calculate any
size Dx where x denotes the percent finer, input
x into the indicated box and click the
appropriate button. To calculate Dx for a
different value of x, just put in the new value
and click the button again.
11
GRAIN SIZE DISTRIBUTION CALCULATOR contd.
This is what the GUI in worksheet Calculator
looks like.
12
GRAIN SIZE DISTRIBUTION CALCULATOR contd.
If the finest size in the grain size distribution
you input does not correspond to 0 percent finer,
or if the coarsest size does not correspond to
100 percent finer, the code will extrapolate for
these missing sizes and modify the grain size
distribution accordingly. The units of the code
are Subs (subroutines). An example is given
below.
Sub fraction(xpf, xp) 'computes fractions
from finer Dim jj As Integer
For jj 1 To np xp(jj) (xpf(jj) -
xpf(jj 1)) / 100 Next jj End Sub
In this Sub, xpf denotes a dummy array containing
the percents finer, and xp denotes a dummy array
containing the fractions in each grain size
range. The Sub computes the fractions from the
percents finer. Suppose in another Sub you know
the percents finer Ff(i), I 1..npp and wish to
compute the fraction in each grain size range
F(i), i 1..np (where np npp 1). The
calculation is performed by the
statement fraction Ff, f
13
WHY CHARACTERIZE GRAIN SIZE DISTRIBUTIONS IN
TERMS OF A LOGARITHMIC GRAIN SIZE?
Consider a sediment sample that is half sand,
half gravel (here loosely interpreted as material
coarser than 2 mm), ranging uniformly from 0.0625
mm to 64 mm. Plotted with a logarithmic grain
size scale, the sample is correctly seen to be
half sand, half gravel. Plotted using a linear
grain size scale, all the information about the
sand half of the sample is squeezed into a tiny
zone on the left-hand side of the diagram.
Logarithmic scale for grain size
Linear scale for grain size
14
UNIMODAL AND BIMODAL GRAIN SIZE DISTRIBUTIONS
The fractions fi(?i) represent a discretized
version of the continuous function f(?), f
denoting the mass fraction of a sample that is
finer than size ?. The probability density pf of
size ? is thus given as p df/d?.
The example to the left corresponds to a Gaussian
(normal) distribution with -1 (Dg 0.5
mm) and ? 0.8 (?g 1.74)
The grain size distribution is called unimodel
because the function p(?) has a single mode, or
peak.
The following approximations are valid for a
Gaussian distribution
15
UNIMODAL AND BIMODAL GRAIN SIZE DISTRIBUTIONS
contd.
A sand-bed river has a characteristic size of bed
surface sediment (D50 or Dg) that is in the sand
range. A gravel-bed river has a characteristic
bed size that is in the range of gravel or
coarser material. The grain size distributions
of most sand-bed streams are unimodal, and can
often be approximated with a Gaussian
function. Many gravel-bed river, however, show
bimodal grain size distributions, as shown to the
upper right. Such streams show a sand mode and a
gravel mode, often with a paucity of sediment in
the pea-gravel size (2 8 mm).
Plateau
Gravel mode
Sand mode
A bimodal (multimodal) distribution can be
recognized in a plot of f versus ? in terms of a
plateau (multiple plateaus) where f does not
increase strongly with ?.
16
UNIMODAL AND BIMODAL GRAIN SIZE DISTRIBUTIONS
contd.
The grain size distributions to the left are all
from 177 samples from various river reaches in
Alberta, Canada (Shaw and Kellerhals, 1982). The
samples from sand-bed reaches are all unimodal.
The great majority of the samples from gravel-bed
reaches show varying degrees of
bimodality. Note geographers often reverse the
direction of the grain size scale, as seen to the
left.
Figure adapted from Shaw and Kellerhals (1982)
17
VERTICAL SORTING OF SEDIMENT
Gravel-bed rivers such as the River Wharfe often
display a coarse surface armor or pavement.
Sand-bed streams with dunes such as the one
modeled experimentally below often place their
coarsest sediment in a layer corresponding to the
base of the dunes.
River Wharfe, U.K. Image courtesy D. Powell.
Sediment sorting in a laboratory flume. Image
courtesy A. Blom.
18
EXNER EQUATION OF CONSERVATION OF BED SEDIMENT
FOR SIZE MIXTURES MOVING AS BEDLOAD
fi'(z', x, t) fractions at elevation z' in ith
grain size range above datum in bed 1. Note
that over all N grain size ranges qbi(x, t)
volume bedload transport rate of sediment in the
ith grain size range L2/T
Or thus
19
ACTIVE LAYER CONCEPT
The active, exchange or surface layer
approximation (Hirano, 1972) Sediment grains in
active layer extending from ? - La lt z lt ? have
a constant, finite probability per unit time of
being entrained into bedload. Sediment grains
below the active layer have zero probability of
entrainment.
20
REDUCTION OF SEDIMENT CONSERVATION RELATION USING
THE ACTIVE LAYER CONCEPT
Fractions Fi in the active layer have no vertical
structure. Fractions fi in the substrate do not
vary in time.
Thus
where the interfacial exchange fractions fIi
defined as
describe how sediment is exchanged between the
active, or surface layer and the substrate as the
bed aggrades or degrades.
21
REDUCTION OF SEDIMENT CONSERVATION RELATION USING
THE ACTIVE LAYER CONCEPT contd.
Between
and
it is found that
(Parker, 1991).
22
REDUCTION contd.
The total bedload transport rate summed over all
grain sizes qbT and the fraction pbi of bedload
in the ith grain size range can be defined as
The conservation relation can thus also be
written as
Summing over all grain sizes, the following
equation describing the evolution of bed
elevation is obtained
Between the above two relations, the following
equation describing the evolution of the grain
size distribution of the active layer is obtained
23
EXCHANGE FRACTIONS
where 0 ? ? ? 1 (Hoey and Ferguson, 1994
Toro-Escobar et al., 1996). In the above
relations Fi, pbi and fi denote fractions in the
surface layer, bedload and substrate,
respectively. That is The substrate is mined as
the bed degrades. A mixture of surface and
bedload material is transferred to the substrate
as the bed aggrades, making stratigraphy. Stratig
raphy (vertical variation of the grain size
distribution of the substrate) needs to be
stored in memory as bed aggrades in order to
compute subsequent degradation.
24
ALTERNATIVE DIMENSIONLESS BEDLOAD TRANSPORT
The generalized bedload transport relation of the
type of Meyer-Peter and Müller (1948) was written
in the form where Recalling that ?b ?u2,
the relation can be written in the alternative
form where (Parker et al., 1982). The
form W versus ? is often used as the basis for
generalizing to sediment mixtures.
25
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 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 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
26
SURFACE-BASED BEDLOAD TRANSPORT FORMULATION contd.
It is now assumed that a functional relation
exists between qi (Wi) and ?i, so that The
bedload transport rate of sediment in the ith
grain size range is thus given as
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.
27
BEDLOAD RELATION FOR MIXTURES DUE TO PARKER (1990)
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.
In the above ?O and ?O are set functions of
?sgospecified in the next slide.
28
BEDLOAD RELATION FOR MIXTURES DUE TO PARKER
(1990) contd.
?o ?o
It is not necessary to use the above chart. The
calculations can be performed using the Visual
Basic programs in RTe-bookAcronym1.xls
29
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.
30
AGGRADATION AND DEGRADATION OF RIVERS
TRANSPORTING GRAVEL MIXTURES
Results of a flood in the gravel-bed Salmon
River, Idaho. Photo by author
31
MODELING AGGRADATION AND DEGRADATION IN
GRAVEL-BED RIVERS CARRYING SEDIMENT MIXTURES
Gravel-bed rivers tend to be steep enough to
allow the use of the normal (steady, uniform)
flow approximation. Here this analysis is
applied using a Manning-Strickler formulation
such that roughness height ks is given as where
Ds90 is the size of the surface material such
that 90 is finer and nk is an order-one
dimensionless number (1.5 3 the work of
Kamphuis, 1974 suggests a value of 2). No
attempt is made here to decompose bed resistance
into skin friction and form drag. The reach is
divided into M intervals bounded by M 1 nodes.
In addition, sediment is introduced at a ghost
node at the upstream end. Since the index i
has been used for grain size ranges, the index
k is used here for spatial nodes.
32
COMPUTATION OF BED SLOPE AND BOUNDARY SHEAR STRESS
At any given time t in the calculation, the bed
elevation ?k and surface fractions Fi,k must be
known at every node k. The roughness height ks,k
and thickness of the surface layer La,k are
computed from the relations where nk and na are
specified order-one dimensionless constants.
(Beware in the equation for roughness height the
k in nk is not an index for spatial node.)
Using the normal flow approximation, the boundary
shear stress ?b,k at the kth node is given from
Chapter 5 as where u?,k denotes the shear
velocity and bed slope Sk is computed as Bed
slope need not be computed at k M 1, where
bed elevation is specified as a boundary
condition.
33
COMPUTATION OF BEDLOAD TRANSPORT
Once Fi,k and ?b,k are known, the bedload
transport rates qbi, and thus qbT and pi can be
computed at any node. An example is given here
in terms of the Wilcock-Crowe (2003) formulation.
The surface geometric mean size Dsg,k is
calculated at every node as where ?i
ln2(Di). The Shields number and shear velocity
based on the surface geometric mean size are then
given as The same fractions Fi,k allow the
computation of the fraction sand Fs,k in the
surface layer at node k. This parameter is
needed in the formulation of Wilcock and Crowe
(2003).
34
COMPUTATION OF BEDLOAD TRANSPORT contd.
It follows that the volume bedload transport rate
per unit width in the ith grain size range is
given as where in the case of the relation of
Wilcock and Crowe (2003),
35
MODELING AGGRADATION AND DEGRADATION IN
GRAVEL-BED RIVERS CARRYING SEDIMENT MIXTURES
contd.
The discretized versions of the Exner relations
are where fIi,k is evaluated from a
relation of the type given in Slide 4 In
the above relation fs,i,int,k denotes the
fractions of the substrate just below the surface
layer at node k and ? is a user-specified
parameter between 0 and 1.
36
MODELING AGGRADATION AND DEGRADATION IN
GRAVEL-BED RIVERS CARRYING SEDIMENT MIXTURES
contd.
The spatial derivatives of the sediment transport
rates are computed as where au is a
upwinding coefficient equal to 0.5 for a central
difference scheme. When k 1, the node k 1
refers to the ghost node, where qbi, and thus qbT
and pi are specified as feed parameters. The
term ?La,k/?t ?t is not a particularly important
one, and can be approximated as where La,k,old
is the value of La,k from the previous time step.
In the case of the first time step, La,k,old may
be set equal to 0.
37
BOUNDARY CONDITIONS, INITIAL CONDITIONS AND FLOW
OF THE COMPUTATION

• The boundary conditions are
• Specified values of qb,i (and thus qbT and pbi)
  at the upstream ghost node
• Specified bed elevation ? at node k M1.
• The initial conditions are
• Specified initial bed elevations ? at every
  node (here simplified to a specified initial bed
  slope Sfbl
• Specified surface and substrate grain size
  distributions Fi and fs,i at every node (here
  taken to be the same at every node).
• At any given time fractions Fi and elevation ?
  are known at every node. The values Fi are used
  to compute Ds90 Dsg, Ds50, ks, La and other
  parameters (e.g. Fs) at every node. The values of
  ? are used to compute slopes S and combined with
  the computed values of ks to determine the shear
  stress ?b at every node except M1, where the
  information is not needed. The resulting
  parameters are used to compute qbi, qbT and pbi
  at all nodes except M1. The Exner relations are
  then solved to determine bed elevations ? and
  surface fractions Fi at all nodes. At node M1
  only the change in grain size distribution is
  evaluated.

38
INTRODUCTION TO RTe-bookAgDegNormGravMixPW.xls
The workbook is a descendant of the PASCAL code
ACRONYM3 of Parker (1990a,b). It allows the user
to choose from two surface-based bedload
transport formulations those of Parker (1990)
and Wilcock and Crowe (2003). In the relation of
Parker (1990) the surface grain size
distributions need to be renormalized to exclude
sand before specification as input to the
program. This step is neither necessary nor
desirable in the case of the relation of Wilcock
and Crowe (2003), where the sand plays an
important role in mediating the gravel bedload
transport. The basic input parameters are the
water discharge per unit width qw, flood
intermittency If, gravel input rate during floods
qbTf, reach length L, initial bed slope SfbI,
number of spatial intervals M, time step ?t,
fractions pbf,i of the gravel feed, fractions
FI,i of the initial surface layer (assumed the
same at every node) and fractions fsI,I of the
substrate (assumed to be uniform in the vertical
and the same at every node). The parameters
Mprint and Mtoprint control output. Auxiliary
parameters include nk for roughness height, na
for active layer thickness, ?r of the
Manning-Strickler relation, submerged specific
gravity R of the sediment, bed porosity ?p,
upwinding coefficient au and interfacial transfer
coefficient ?.
39
INTRODUCTION TO RTe-bookAgDegNormGravMixPW.xls
contd.
One interesting problem of sediment mixtures is
when the river first aggrades, creating its own
substrate with a vertical structure in the
process, and then degrades into it. The code in
the workbook is not set up to handle this. The
necessary extension is trivial in theory but
tedious in practice the vertical structure of
the newly-created substrate must be stored in
memory as the calculation proceeds.
A gravel-bed reach of the Las Vegas Wash, USA,
where the river is degrading into its own
deposits.
Some calculations with the code follow.
40
CALCULATIONS WITH RTe-bookAgDegNormGravMixPW.xls
The calculations are performed with the Parker
(1990) bedload transport relation. The grain
size distributions of the feed sediment, initial
surface sediment and substrate sediment are all
taken to be identical, as given below. Note that
sand has been removed from the grain size
distributions.
41
CALCULATIONS WITH RTe-bookAgDegNormGravMixPW.xls
contd.
A case is chosen for which the bed must aggrade
from a very low slope. Calculations are
performed for 60 years, 600 years and 6000 years
in order to study the evolution of the profile.
The software produces graphical output for the
time development of the long profiles of a) bed
elevation ?, b) surface geometric mean size Dsg
and c) volume gravel bedload transport rate per
unit width qbT.
42
Parker relation After 60 years
43
Parker relation After 60 years
44
Downstream variation of qbT/qbTf, where qbT
Bedload Transport Rate and qbTf Upstream
Bedload Feed Rate
Parker relation After 60 years
qbT/qbTf
45
Parker relation After 600 years
46
Parker relation After 600 years
47
Downstream variation of qbT/qbTf, where qbT
Bedload Transport Rate and qbTf Upstream
Bedload Feed Rate
Parker relation After 600 years
qbT/qbTf
48
Parker relation After 6000 years
49
Parker relation After 6000 years
50
Downstream variation of qbT/qbTf, where qbT
Bedload Transport Rate and qbTf Upstream
Bedload Feed Rate
Parker relation After 6000 years
qbT/qbTf
51
CALCULATIONS WITH RTe-bookAgDegNormGravMixPW.xls
contd.
The next case is one for which the bed which the
bed must degrade to a new equilibrium. The input
grain size distributions are the same as the
previous case. Again, the Parker (1990) relation
is used. The input parameters are given below.
The calculation shown is over a duration of 240
years.
52
Parker relation After 240 years
53
Parker relation After 240 years
54
Downstream variation of qbT/qbTf, where qbT
Bedload Transport Rate and qbTf Upstream
Bedload Feed Rate
qbT/qbTf
Parker relation After 240 years
55
CALCULATIONS WITH RTe-bookAgDegNormGravMixPW.xls
contd.
Sand is excluded from the input grain size
distributions when using the Parker (1990)
relation. The Wilcock-Crowe (2003) relation
explicitly includes the sand. Two calculations
follow. In the first of them, the input data are
exactly the same as that for the calculations
using Parker (1990) of Slides 51-54 (degradation
to a new equilibrium). In particular, sand is
excluded from the input grain size distributions.
In the second of them, 25 sand is added to the
grain size distribution. The Wilcock-Crowe
(2003) relation predicts that the addition of
sand makes the gravel more mobile. It will be
seen that the bed elevation at the end of the
240-year calculation is predicted to be
significantly lower when sand is included than
when it is excluded.
56
Wilcock-Crowe relation Sand excluded After 240
years
57
Wilcock-Crowe relation Sand excluded After 240
years
58
Downstream variation of qbT/qbTf, where qbT
Bedload Transport Rate and qbTf Upstream
Bedload Feed Rate
qbT/qbTf
Wilcock-Crowe relation Sand excluded After 240
years
59
Wilcock-Crowe relation Sand included After 240
years
60
Wilcock-Crowe relation Sand included After 240
years
61
Downstream variation of qbT/qbTf, where qbT
Bedload Transport Rate and qbTf Upstream
Bedload Feed Rate
qbT/qbTf
Wilcock-Crowe relation Sand included After 240
years
62
NOTES ON THE EFFECT OF SAND IN THE GRAVEL
Comparing Slides 56 and 59, it is seen that the
upstream end of the reach has degraded
considerably more in the case of Slide 56, i.e.
when sand is included in the Wilcock-Crowe (2003)
calculation. Comparing Slides 52 and 59, it is
seen that the bed profile at the end of the
calculation using Wilcock-Crowe (2003) with sand
included is almost the same as the corresponding
profile using Parker (1990), in which sand is
automatically excluded. The correspondence is
not an accident. The field data used to develop
the Parker (1990) relation did indeed include
sand in the bed and load sand was excluded in
the development of the relation because of
uncertainty as to how much might go into
suspension. So the Parker (1990) relation
implicitly includes a set fraction of sand in the
bed. This notwithstanding, the Wilcock-Crowe
(2003) relation has the considerable advantage
that the quantity of sand in the feed sediment
and substrate can be varied. As the calculations
show, for all other factors equal the relation
predicts that an increased sand content can
significantly increase the mobility of the
gravel.
63
REFERENCES
Hirano, M., 1971, On riverbed variation with
armoring, Proceedings, Japan Society of Civil
Engineering, 195 55-65 (in Japanese). Hoey, T.
B., and R. I. Ferguson, 1994, Numerical
simulation of downstream fining by selective
transport in gravel bed rivers Model development
and illustration, Water Resources Research, 30,
2251-2260. Meyer-Peter, E. and Müller, R., 1948,
Formulas for Bed-Load Transport, Proceedings, 2nd
Congress, International Association of Hydraulic
Research, Stockholm 39-64. Parker, G., 1990,
Surface-based bedload transport relation for
gravel rivers, Journal of Hydraulic
Research, 28(4) 417-436. Parker, G., Klingeman,
P. and McLean, D., 1982, Bedload and size
distribution in natural paved gravel bed streams,
Journal of Hydraulic Engineering, 108(4),
544-571. Shaw, J. and R. Kellerhals, 1982, The
Composition of Recent Alluvial Gravels in Alberta
River Beds, Bulletin 41, Alberta Research
Council, Edmonton, Alberta, Canada. Wilcock, P.
R., and Crowe, J. C., 2003, Surface-based
transport model for mixed-size sediment, Journal
of Hydraulic Engineering, 129(2), 120-128.
For more information see Gary Parkers e-book 1D
Morphodynamics of Rivers and Turbidity Currents
http//cee.uiuc.edu/people/parkerg/morphodynamics
_e-book.htm
64
REFERENCES FOR CHAPTER 17
Parker, G., 1990a, Surface-based bedload
transport relation for gravel rivers, Journal of
Hydraulic Research, 28(4) 417-436. 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
. Toro-Escobar, C. M., G. Parker and C. Paola,
1996, Transfer function for the deposition of
poorly sorted gravel in response to streambed
aggradation, Journal of Hydraulic Research,
34(1) 35-53. Wilcock, P. R., and Crowe, J. C.,
2003, Surface-based transport model for
mixed-size sediment, Journal of Hydraulic
Engineering, 129(2), 120-128.