AGGRADATION AND DEGRADATION OF RIVERS TRANSPORTING GRAVEL MIXTURES

1
CHAPTER 17 AGGRADATION AND DEGRADATION OF RIVERS
TRANSPORTING GRAVEL MIXTURES
Results of a flood in the gravel-bed Salmon
River, Idaho. Photo by author
2
REVIEW SEDIMENT CONSERVATION OF GRAVEL MIXTURES

• Gravel-bed rivers tend to be poorly-sorted.
  During floods, bed material load consists almost
  exclusively of bedload. (Sand is often
  transported in copious quantities as washload
  during floods.) The surface material (armor or
  pavement) tends to be coarser than the substrate.
  By definition the median size Dsub50 or
  geometric mean size Dsubg of the substrate is in
  the gravel range, but the substrate may contain
  up to 30 sand in the interstices of an otherwise
  clast-supported deposit.
• Material from Chapters 4 and 7 is used
  extensively in this chapter, and is reviewed in
  the next few slides. Definitions follow below.
• Fi fraction of material in the surface layer
  in the ith grain size range, i 1..N
• Di characteristic size of the ith grain size
  range
• La thickness of the surface (active,
  exchange) layer
• fIi fraction in the ith grain size range of
  material exchanged between the surface and the
  substrate as the bed aggrades or degrades
• qbi volume bedload transport rate per unit
  width of material in the ith grain size range
• qbT Sqbi total volume bedload transport
  rate per unit width
• pi qbi/qbT fraction of bedload material in
  the ith grain size range

3
REVIEW FROM CHAPTER 4 EXNER RELATIONS FOR
MIXTURES
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 Exner equation of sediment conservation of
Chapter 4, here generalized to include the flood
intermittency If of Chapter 14, can thus 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
4
REVIEW FROM CHAPTER 4 EXCHANGE FRACTIONS
where 0 ? ? ? 1 (Hoey and Ferguson, 1994
Toro-Escobar et al., 1996). In the above
relations Fi, pi 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 if subsequent
degradation into the deposit is to be computed.
5
REVIEW FROM CHAPTER 7 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
6
REVIEW FROM CHAPTER 7 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.
7
REVIEW 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, i.e. ?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.
8
REVIEW BEDLOAD RELATION FOR MIXTURES DUE TO
PARKER (1990a,b) 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
9
REVIEW 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.
10
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.
11
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.
12
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.
Using the relations of Chapter 2, 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).
13
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),
14
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.
15
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.
16
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.

17
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 (1990a)
and Wilcock and Crowe (2003). In the relation of
Parker (1990a) 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 ?.
18
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.
19
CALCULATIONS WITH RTe-bookAgDegNormGravMixPW.xls
The calculations are performed with the Parker
(1990a,b) 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.
20
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.
21
Parker relation After 60 years
22
Parker relation After 60 years
23
Downstream variation of qbT/qbTf, where qbT
Bedload Transport Rate and qbTf Upstream
Bedload Feed Rate
Parker relation After 60 years
qbT/qbTf
24
Parker relation After 600 years
25
Parker relation After 600 years
26
Downstream variation of qbT/qbTf, where qbT
Bedload Transport Rate and qbTf Upstream
Bedload Feed Rate
Parker relation After 600 years
qbT/qbTf
27
Parker relation After 6000 years
28
Parker relation After 6000 years
29
Downstream variation of qbT/qbTf, where qbT
Bedload Transport Rate and qbTf Upstream
Bedload Feed Rate
Parker relation After 6000 years
qbT/qbTf
30
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 (1990a,b)
relation is used. The input parameters are given
below. The calculation shown is over a duration
of 240 years.
31
Parker relation After 240 years
32
Parker relation After 240 years
33
Downstream variation of qbT/qbTf, where qbT
Bedload Transport Rate and qbTf Upstream
Bedload Feed Rate
qbT/qbTf
Parker relation After 240 years
34
CALCULATIONS WITH RTe-bookAgDegNormGravMixPW.xls
contd.
Sand is excluded from the input grain size
distributions when using the Parker (1990a,b)
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 (1990a,b) of Slides 30-33
(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.
35
Wilcock-Crowe relation Sand excluded After 240
years
36
Wilcock-Crowe relation Sand excluded After 240
years
37
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
38
Wilcock-Crowe relation Sand included After 240
years
39
Wilcock-Crowe relation Sand included After 240
years
40
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
41
NOTES ON THE EFFECT OF SAND IN THE GRAVEL
Comparing Slides 35 and 38, it is seen that the
upstream end of the reach has degraded
considerably more in the case of Slide 38, i.e.
when sand is included in the Wilcock-Crowe (2003)
calculation. Comparing Slides 31 and 38, 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 (1990a,b), in which sand is
automatically excluded. The correspondence is
not an accident. The field data used to develop
the Parker (1990a,b) 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 (1990a,b) 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.
42
REFERENCES FOR CHAPTER 17
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. Kamphuis, J. W., 1974, Determination
of sand roughness for fixed beds, Journal of
Hydraulic Research, 12(2) 193-202. 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/park
erg/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.