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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

SAMPLE EVALUATIONS OF ? AND ? SEDIMENT SIZE

RANGES

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.

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).

SEDIMENT GRAIN SIZE DISTRIBUTIONS contd.

The grain size distribution after extrapolation

is shown below.

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

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

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

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.

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.

GRAIN SIZE DISTRIBUTION CALCULATOR contd.

This is what the GUI in worksheet Calculator

looks like.

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

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

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

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 ?.

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)

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.

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

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.

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.

REDUCTION OF SEDIMENT CONSERVATION RELATION USING

THE ACTIVE LAYER CONCEPT contd.

Between

and

it is found that

(Parker, 1991).

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

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.

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.

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

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.

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.

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

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.

AGGRADATION AND DEGRADATION OF RIVERS

TRANSPORTING GRAVEL MIXTURES

Results of a flood in the gravel-bed Salmon

River, Idaho. Photo by author

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.

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.

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).

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),

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.

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.

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.

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 ?.

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.

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.

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.

Parker relation After 60 years

Parker relation After 60 years

Downstream variation of qbT/qbTf, where qbT

Bedload Transport Rate and qbTf Upstream

Bedload Feed Rate

Parker relation After 60 years

qbT/qbTf

Parker relation After 600 years

Parker relation After 600 years

Downstream variation of qbT/qbTf, where qbT

Bedload Transport Rate and qbTf Upstream

Bedload Feed Rate

Parker relation After 600 years

qbT/qbTf

Parker relation After 6000 years

Parker relation After 6000 years

Downstream variation of qbT/qbTf, where qbT

Bedload Transport Rate and qbTf Upstream

Bedload Feed Rate

Parker relation After 6000 years

qbT/qbTf

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.

Parker relation After 240 years

Parker relation After 240 years

Downstream variation of qbT/qbTf, where qbT

Bedload Transport Rate and qbTf Upstream

Bedload Feed Rate

qbT/qbTf

Parker relation After 240 years

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.

Wilcock-Crowe relation Sand excluded After 240

years

Wilcock-Crowe relation Sand excluded After 240

years

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

Wilcock-Crowe relation Sand included After 240

years

Wilcock-Crowe relation Sand included After 240

years

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

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.

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

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.