1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHAPTER 17:
AGGRADATION AND DEGRADATION OF RIVERS TRANSPORTING GRAVEL
MIXTURES
Results of a flood in the gravel-bed Salmon River, Idaho.
Photo by author
1
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
2
• pi = qbi/qbT = fraction of bedload material in the ith grain size range
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
N
q
qbT   qbi , pbi  bi
qbT
i1
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

q p
 

(1  p )fIi (  La )  FiLa   If bT bi
t
x
 t

Summing over all grain sizes, the following equation describing the evolution of
bed elevation is obtained:
(1  p )

q
 If bT
t
x
Between the above two relations, the following equation describing the evolution
of the grain size distribution of the active layer is obtained:
La 
qbTpbi
qbT
 Fi
(1  p )La
 Fi  fIi 
 If
 If fIi

t 
x
x
 t
3
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REVIEW FROM CHAPTER 4: EXCHANGE FRACTIONS


fi zL ,
0

a
t
fIi  

Fi  (1  )pbi ,
0
t

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.
Stratigraphy (vertical variation of the grain size distribution of the substrate)
needs to be stored in memory as bed aggrades if subsequent
4
degradation into the deposit is to be computed.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
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 , pbi 
i1
qbi
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 
 
RgDi RgDi
qbi
, qbi 
RgDi Di Fi

Rgq bi
q

bi
, Wi   3 / 2 
( i )
(u )3 Fi
5
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
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'
6
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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. SFi = 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.
7
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REVIEW: BEDLOAD RELATION FOR MIXTURES DUE TO PARKER (1990a,b)
contd.
1.6
1.4
1.2
1
o
omegaO
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
8
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
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( 14Fs )

sg

ssrg
b
0.69
1  exp(1.5  Di / Dsg )
2
u

sg 
RgDsg
9
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
k s  nkDs90
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.
x
ghost
k=1
2
3
M -1
L
M
k = M+1
M+1
10
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
k s,k  nkDs90,k
L a,k  naDs90,k
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
3 / 10
given from Chapter 5 as
1/ 3 2


b,k  u
2
,k
k s,k qw

 
2


r


g7 / 10Sk7 / 10
where u,k denotes the shear velocity and bed slope Sk is computed as
 1  2
, k 1

x
Sk  
  k 1
 k 1
, k  2..M
 2x
Bed slope need not be computed at k = M + 1, where bed elevation is specified as a
11
boundary condition.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
N
k   iFi,k
, Dsg,k  2k
i1
where i = ln2(Di). The Shields number and shear velocity based on the surface
geometric mean size are then given as

sg,k

k q
 
 g
1/ 3 2
s,k w
2
r




3 / 10
Sk7 / 10
RD sg,k
u,k
k q
 
 
1/ 3 2
s,k w
2
r




3 / 20
g7 / 20Sk7 / 20
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
12
(2003).
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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
qbi,k
u3,k 
 Fi,k
Wi,k
Rg
N
qbT,k   qbi,k
, pbi,k
i1
qbi,k

qbT,k
where in the case of the relation of Wilcock and Crowe (2003),
 0.002i7,k.5

4.5

Wi,k    0.894 
14 1  0.5 
i,k

 
for i,k  1.35
for i,k  1.35
bi ,k
sg,k  Di 


i,k  

ssrg,k  Dsg,k 
ssrg,k  0.021  0.015 exp( 20Fs,k )
bi,k 
0.67
1  exp(1.5  Di / Dsg,k )
13
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MODELING AGGRADATION AND DEGRADATION IN GRAVEL-BED RIVERS
CARRYING SEDIMENT MIXTURES contd.
The discretized versions of the Exner relations are:
k t  t
Fi,k tt
If qbT
 k 
t
1  p x k
La,k
 qbTpbi
1
If
qbT
 
Fi,k  fIi,k  t 
 Fi,k 
 fIi,k
La,k
t
La,k (1  p ) 
x k
x

t
k
where fIi,k is evaluated from a relation of the type given in Slide 4:
fIi,k


fs,i,int,k ,
0

t k

Fi,k  (1  )pbi,k ,   0

t k
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.
14
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MODELING AGGRADATION AND DEGRADATION IN GRAVEL-BED RIVERS
CARRYING SEDIMENT MIXTURES contd.
The spatial derivatives of the sediment transport rates are computed as
qbT,k  qbT,k1
qbT,k1  qbT,k
qbT
 au
 (1  au )
x k
x
x
qbT,kpbi,k  qbT,k1pbi,k 1
qbT,k1pbi,k1  qbT,kpbi,k
qbTpi
 au
 (1  au )
x k
x
x
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
La,k
t  La,k  La,k,old
t
where La,k,old is the value of La,k from the previous time step. In the case of the first
15
time step, La,k,old may be set equal to 0.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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 = M+1.
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 M+1,
where the information is not needed. The resulting parameters are used to compute
qbi, qbT and pbi at all nodes except M+1. The Exner relations are then solved to
determine bed elevations  and surface fractions Fi at all nodes. At node M+1 only
16
the change in grain size distribution is evaluated.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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 .
17
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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 newlycreated 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.
18
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
Grain Size Distributions
100
90
Percent Finer
80
70
60
Feed
Initial Surface
Substrate
50
40
30
20
10
0
1
10
100
Size mm
Dd,i mm Feed
256
128
64
32
16
8
4
2
1
0.5
0.25
0.125
100
95
80
50
25
10
5
0
0
0
0
0
Initial
Surface Substrate
100
100
95
95
80
80
50
50
25
25
10
10
5
5
0
0
0
0
0
0
0
0
0
0
1000
19
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
Input parameters
qw
6
qbTo
1.00E-04
Inter
0.05
etadI
3
SfbI
1.00E-04
L
20000
dt
73.05
M
25
Mtoprint
50
Mprint
6
60
The input cells are in gold
water discharge/width, m^2/s
These cells contain useful information
Input "1" for Parker (1990) relation, "2" for
gravel input rate, m^2/s
Intermittency
Wilcock-Crowe (2003) relation:
initial base level, m
1 Input here to choose
initial bed slope
reach length,m
Click to Run Program
time step, days
no. of intervals
no. of steps until a printout of results is made
no. of printouts after the initial one
dt
6311520 sec
years calculation time
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.
20
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream Variation in Bed Elevation
16
14
Parker relation
After 60 years
Elevation m
12
0 yr
10 yr
20 yr
30 yr
40 yr
50 yr
60 yr
final w.s.
10
8
6
4
2
0
0
5000
10000
Distance m
15000
20000
21
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream Variation in Surface Geometric Mean Size
Surface Geometric Mean Size
mm
100
Parker relation
After 60 years
0 yr
10 yr
20 yr
30 yr
40 yr
50 yr
60 yr
10
1
0
5000
10000
Distance m
15000
20000
22
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DownstreamVariation
variationofofqbT/qbTo,
qbT/qbTf,where
where
qbT
Downstream
qbT
==
Bedload
Bedload
BedloadTransport
TransportRate
Rateand
andqbTo
qbTf==Upstream
Upstream
Bedload
Feed
FeedRate
Rate
100
Parker relation
After 60 years
qbT/qbTf
qbT/qbTo
10
0 yr
10 yr
20 yr
30 yr
40 yr
50 yr
60 yr
1
0.1
0.01
0.001
0
5000
10000
Distance m
15000
20000
23
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream Variation in Bed Elevation
35
30
Parker relation
After 600 years
Elevation m
25
0 yr
100 yr
200 yr
300 yr
400 yr
500 yr
600 yr
final w.s.
20
15
10
5
0
0
5000
10000
Distance m
15000
20000
24
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream Variation in Surface Geometric Mean Size
Surface Geometric Mean Size
mm
100
Parker relation
After 600 years
0 yr
100 yr
200 yr
300 yr
400 yr
500 yr
600 yr
10
1
0
5000
10000
Distance m
15000
20000
25
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream
qbT
==
DownstreamVariation
variationofofqbT/qbTo,
qbT/qbTf,where
where
qbT
Bedload
Bedload
BedloadTransport
TransportRate
Rateand
andqbTo
qbTf==Upstream
Upstream
Bedload
Feed
FeedRate
Rate
100
Parker relation
After 600 years
qbT/qbTf
qbT/qbTo
10
0 yr
100 yr
200 yr
300 yr
400 yr
500 yr
600 yr
1
0.1
0.01
0.001
0
5000
10000
Distance m
15000
20000
26
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream Variation in Bed Elevation
70
60
Parker relation
After 6000 years
Elevation m
50
0 yr
1000 yr
2000 yr
3000 yr
4000 yr
5000 yr
6000 yr
final w.s.
40
30
20
10
0
0
5000
10000
Distance m
15000
20000
27
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream Variation in Surface Geometric Mean Size
Surface Geometric Mean Size
mm
100
0 yr
1000 yr
2000 yr
3000 yr
4000 yr
5000 yr
6000 yr
Parker relation
After 6000 years
10
1
0
5000
10000
Distance m
15000
20000
28
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream
qbT
==
DownstreamVariation
variationofofqbT/qbTo,
qbT/qbTf,where
where
qbT
Bedload
Bedload
BedloadTransport
TransportRate
Rateand
andqbTo
qbTf==Upstream
Upstream
Bedload
Feed
FeedRate
Rate
100
Parker relation
After 6000 years
qbT/qbTf
qbT/qbTo
10
0 yr
1000 yr
2000 yr
3000 yr
4000 yr
5000 yr
6000 yr
1
0.1
0.01
0.001
0
5000
10000
Distance m
15000
20000
29
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
30
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream Variation in Bed Elevation
120
Elevation m
100
Parker relation
After 240 years
0 yr
40 yr
80 yr
120 yr
160 yr
200 yr
240 yr
final w.s.
80
60
40
20
0
0
5000
10000
Distance m
15000
20000
31
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream Variation in Surface Geometric Mean Size
Surface Geometric Mean Size
mm
100
0 yr
40 yr
80 yr
120 yr
160 yr
200 yr
240 yr
Parker relation
After 240 years
10
1
0
5000
10000
Distance m
15000
20000
32
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DownstreamVariation
variationofofqbT/qbTo,
qbT/qbTf,where
where
qbT
Downstream
qbT
==
Bedload
Bedload
BedloadTransport
TransportRate
Rateand
andqbTo
qbTf==Upstream
Upstream
Bedload
Feed
FeedRate
Rate
100
qbT/qbTf
qbT/qbTo
10
0 yr
40 yr
80 yr
120 yr
160 yr
200 yr
240 yr
1
Parker relation
After 240 years
0.1
0.01
0.001
0
5000
10000
Distance m
15000
20000
33
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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 WilcockCrowe (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.
Input parameters
qw
6
qbTo
1.00E-04
Inter
0.05
etadI
3
SfbI
5.00E-03
L
20000
dt
7.305
M
25
Mtoprint
2000
Mprint
6
240
The input cells are in gold
water discharge/width, m^2/s
These cells contain useful information compu
Input "1" for Parker (1990) relation, "2" for
gravel input rate, m^2/s
Intermittency
Wilcock-Crowe (2003) relation:
initial base level, m
2 Input here to choose relation
initial bed slope
reach length,m
Click to Run Program
time step, days
no. of intervals
no. of steps until a printout of results is made
no. of printouts after the initial one
dt
631152 sec
34
years calculation time
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream Variation in Bed Elevation
120
Wilcock-Crowe relation
Sand excluded
After 240 years
Elevation m
100
80
60
40
0 yr
40 yr
80 yr
120 yr
160 yr
200 yr
240 yr
final w.s.
20
0
0
5000
10000
Distance m
15000
20000
35
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream Variation in Surface Geometric Mean Size
Surface Geometric Mean Size
mm
100
0 yr
40 yr
80 yr
120 yr
160 yr
200 yr
240 yr
10
Wilcock-Crowe relation
Sand excluded
After 240 years
1
0
5000
10000
Distance m
15000
20000
36
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DownstreamVariation
variationofofqbT/qbTo,
qbT/qbTf,where
where
qbT
Downstream
qbT
==
Bedload
Bedload
BedloadTransport
TransportRate
Rateand
andqbTo
qbTf==Upstream
Upstream
Bedload
Feed
FeedRate
Rate
100
qbT/qbTf
qbT/qbTo
10
0 yr
40 yr
80 yr
120 yr
160 yr
200 yr
240 yr
1
0.1
Wilcock-Crowe relation
Sand excluded
After 240 years
0.01
0.001
0
5000
10000
Distance m
15000
20000
37
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream Variation in Bed Elevation
120
Wilcock-Crowe relation
Sand included
After 240 years
Elevation m
100
80
60
40
0 yr
40 yr
80 yr
120 yr
160 yr
200 yr
240 yr
final w.s.
20
0
0
5000
10000
Distance m
15000
20000
38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream Variation in Surface Geometric Mean Size
Surface Geometric Mean Size
mm
100
0 yr
40 yr
80 yr
120 yr
160 yr
200 yr
240 yr
10
Wilcock-Crowe relation
Sand included
After 240 years
1
0
5000
10000
Distance m
15000
20000
39
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
Downstream
qbT
==
DownstreamVariation
variationofofqbT/qbTo,
qbT/qbTf,where
where
qbT
Bedload
Bedload
BedloadTransport
TransportRate
Rateand
andqbTo
qbTf==Upstream
Upstream
Bedload
Feed
FeedRate
Rate
100
qbT/qbTf
qbT/qbTo
10
0 yr
40 yr
80 yr
120 yr
160 yr
200 yr
240 yr
1
0.1
Wilcock-Crowe relation
Sand included
After 240 years
0.01
0.001
0
5000
10000
Distance m
15000
20000
40
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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.
41
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
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/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.
42