1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHAPTER 14:
1D AGGRADATION AND DEGRADATION OF RIVERS: NORMAL FLOW
ASSUMPTION
The disposal of large amounts of waste sediment
from the Ok Tedi Copper Mine, Papua New Guinea,
has caused significant aggradation, or bed level rise,
in the Ok Tedi (“Ok” means “river”) and Fly Rivers.
The mine
Aggradation in gravel-bed reaches of Ok Tedi
5 m aggradation at bridge
Aggradation
where the Ok
Ma joins the Ok
Tedi
1
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
CHANNEL AGGRADATION AND FLOODPLAIN DEPOSITION OF OK
TEDI AT GRAVEL-SAND TRANSITION
River slope drops by an
order of magnitude in
the transition zone
from braided gravelbed to meandering
sand-bed stream,
leading to massive
deposition of sand.
Sediment depositing on the
floodplain has destroyed the forest.
Sand is dredged from the river to
ameliorate the deposition.
2
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RESPONSE OF A RIVER TO SUDDEN VERTICAL FAULTING CAUSED
BY AN EARTHQUAKE
View in November, 1999, shortly
after the earthquake caused a sharp
3 m elevation drop at a fault.
View in May, 2000 after aggradation
and degradation have smoothed out
the elevation drop.
The above images of the Deresuyu River, Turkey, are courtesy
of Patrick Lawrence and François Métivier (Lawrence, 2003)
3
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RESPONSE OF A RIVER TO SUDDEN VERTICAL FAULTING CAUSED
BY AN EARTHQUAKE contd.
Inferred initial profile
immediately after faulting in
November, 1999
Profile in May, 2001
Upstream degradation (bed level lowering) and downstream aggradation
(bed level increase) are realized as the river responds to the knickpoint
created by the earthquake (Lawrence, 2003)
4
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
BACKGROUND AND ASSUMPTIONS
Change in channel bed level (aggradation or degradation) can occur in response to:
• increase or decrease in upstream sediment supply;
• change in hydrologic regime (water diversion or climate change);
• change in river slope (e.g. channel straightening, as outlined in Chapter 2);
• increased or decreased sediment supply from tributaries;
• sudden inputs of sediment from debris flows or landslides;
• faulting due to earthquakes or other tectonic effects such as tilting along the reach,
and;
• changing base level at the downstream end of the reach of interest.
Here “base level” loosely means a
controlling elevation at the downstream
end of the reach of interest. It means
water surface elevation if the river flows
into a lake or the ocean, or a
downstream bed elevation controlled
by e.g. tectonic uplift or subsidence at
a point where the river is not flowing
into standing water.
Base level of this reach
of the Eau Claire river,
Wisconsin, USA is
controlled by a reservoir,
Lake Altoona
5
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE EQUILIBRIUM STATE contd.
Rivers are different in many ways from laboratory flumes. It nevertheless helps to
conceptualize rivers in terms of a long, straight, wide, rectangular flume with high
sidewalls (no floodplain), constant width and a bed covered with alluvium. Such a
“river” has a simple mobile-bed equilibrium (graded) state at which flow depth H, bed
slope S, water discharge per unit width qw and bed material load per unit width qt
remain constant in time t and in the streamwise direction x. A recirculating flume
(with both water and sediment recirculated) at equilibrium is illustrated below.
water
sediment
pump
6
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE EQUILIBRIUM STATE contd.
The hydraulics of the equilibrium state are those of normal flow. Here the case of a
plane bed (no bedforms) is considered as an example. The bed consists of uniform
material with size D. The governing equations are (Chapter 5):
Water conservation:
qw  UH
Momentum conservation:
Friction relations:
b  Cf U2
b  gHS
1/ 6
C f  const (Chezy )
or
C f 1/ 2
H
  r  
 kc 
(Manning  Strickler )
where kc is a composite bed roughness which may include the effect of bedforms
(if present).
Generic transport relation of the form of Meyer-Peter and Müller for total bed
material load: where t and nt are dimensionless constants:
 

qt
  t  b  c 
RgD D
 RgD

nt
7
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE EQUILIBRIUM STATE contd.
In the case of the Chezy resistance relation, the equations governing the
normal state reduce to:
 C f q2w
H  
 gS
1/ 3



 C q2
qt  RgD D  t  f w
 g
1/ 3



S   
  c 
RD 



nt
2/3
In the case of the Manning-Stickler resistance relation, the equations
governing the normal state reduce with to:
 k1c/ 3 q2w
H   2
  r gS



3 / 10
1/ 3 2

 k c qw
qt  RgD D  t  2
r g








3 / 10

nt


S

  c 
RD 



7 / 10
Let D, kc and R be given. In either case above, there are two equations for four
parameters at equilibrium; water discharge per unit width qw, volume sediment
discharge per unit width qt, bed slope S and flow depth H. If any two of the set (qw,
qt, S and H) are specified, the other two can be computed. In a sediment-feed
flume, qw and qt are set, and equilibrium S and H can be computed from either of the
above pair. In a recirculating flume, qw and
8
H are set (total water mass in flume is conserved), and qt and S can be computed.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE EQUILIBRIUM STATE contd.
The basic nature of the arguments of the previous slide do not change if a) total bed
material transport is divided into bedload and suspended load components, each
with its own predictor, b) bed shear stress is divided into skin friction and form drag,
each with its own predictor, and c) transport/entrainment relations for uniform
material are replaced with relations for sediment mixtures. Each new variable is
accompanied by one new constraint (governing equation). For example, consider
the case of gravel transport in the absence of bedforms. Using the gravel bedload
transport relation of Powell et al. (2001) as an example, and setting kc = nkDs90 (no
form drag), the problem reduces with the relations of Chapters 5 and 7 to
 n1k/ 3D1s/903 q2w
H  
2
  r gS



3 / 10
g1/ 20 S21 / 20
qbi  Fi
R
 n1k/ 3D1s/903 q2w

r2




9 / 20
0.74


 Di 



0.03 


Ds50 

1 

3 / 10
1/ 3 1/ 3 2
7 / 10 

 nk Ds90 qw 


S






r2g 
RDi  
 

4.5
Recalling that qbT = qbi and bedload fractions pi =
qbi/qbT, if any two of the set (H, S, qw, qT) and either the bed surface fractions Fi or
the bedload fractions pi are specified, the equilibrium values of the other parameters
can be computed from the above equations. For example, if S, qw and Fi (from
which Ds50 can be computed according to the relations of Chapter 2) are
9
specified, H, qbT and pi can be computed directly from the above relations.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SIMPLIFICATIONS
The concepts of aggradation and degradation are best illustrated by using simplified
relations for hydraulic resistance and sediment transport. Here the following
simplifications are made in addition to the assumptions of constant width and
the absence of a floodplain:
1.
2.
3.
The case of a Manning-Strickler formulation with constant composite roughness
kc is considered;
Bed material is taken to be uniform with size D;
The Exner equation of sediment conservation is based on a computation of total
bed material load, which is computed via the generic equation
 

qt
  t  s b  c 
RgD D
 RgD

nt
where s  1 is a constant to convert total boundary shear stress to that due to
skin friction (if necessary). For example, to recover the corrected version of
Meyer-Peter and Müller (1948) relation of Wong and Parker (submitted) for
gravel transport, set t = 3.97 , nt = 1.5, c* = 0.0495 and s = 1. For the bed
material load relation of Engelund and Hansen (1967) for sand transport, which
uses total boundary shear stress, not that due to skin friction,
10
t = 0.05/Cf, nt = 2.5, c* = 0 and s = 1.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
4.
SIMPLIFICATIONS contd.
The full flood hydrograph or flow duration curve of discharge variation is
replaced by a flood intermittency factor If, so that the river is assumed to be at
low flow (and not transporting significant amounts of sediment) for time fraction
1 – If, and is in flood at constant discharge Q, and thus constant discharge per
unit width qw = Q/B for time fraction If (Paola et al., 1992). The implied
hydrograph takes the conceptual form below:
flood
Q
low flow
t
In the long term, then, the relation between actual time t and time that the river has
been in flood tf is given as
t f  If t
Let the value of the total bed material load at flood flow qt be computed in m2/s.
Then the total mean annual sediment load Gt in million tons per year is given as
Gt  s qtBIf t a /(1x106 )
11
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SIMPLIFICATIONS contd.
There are many reasonable ways to compute the intermittency factor If. One
reasonable way to do so is to:
a) compute the volume bed material transport rate Qtbf at bankfull flow;
b) use the full flow duration curve to compute the mean annual volume bed
material transport rate Qtanav as
Qtan av   Qt,kpk
c)
where qt,k denotes the value of qt in the kth discharge range, and pk denotes the
fraction of time the flow is in this range, and
Compute the flood value of
qt and If as
flood
Q tbf
qt 
Bbf
Q tan av
If 
Q tbf
Q
low flow
t
In this way If denotes the fraction of time per year that continuous
bankfull flow would yield the annual sediment yield.
12
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
SIMPLIFICATIONS contd.
It is important to realize that none of these simplifications are necessary. Once
methods for computing aggradation and degradation are developed using the above
simplifications, however, the analysis easily generalizes to cases with mixed grain
sizes, a distinction between bedload and suspended bed material load, a
computation of both form drag and skin friction, and computations using the full flow
hydrograph or flow duration curve.
The Minnesota River, USA
near Le Sueur during the
Generalization to the case of varying width is
flood of record in 1965
also rather straightforward, and is implemented
in future chapters of this e-book. Including the
floodplain, however, is more difficult, especially
in the case of meandering rivers. This is
because when the floodplain is inundated and
the floodplain depth is substantial, the thread of
high velocity may no longer completely follow
the river channel. The simplest reasonable
assumption is that the bed material load at
above-bankfull flows is equal to that at bankfull
flow.
13
channel
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
AGGRADATION AND DEGRADATION AS TRANSIENT RESPONSES TO
IMPOSED DISEQIUILBRIUM CONDITIONS
Aggradation or degradation of a river reach can be considered to be a response to
disequilibrium conditions, by which the river tries to reach a new equilibrium. For
example, if a river reach has attained an equilibrium with a given sediment supply
from upstream, and that sediment supply is suddenly increased at t = 0, the river
can be expected to aggrade toward a new equilibrium.
final equilibrium bed profile in
balance with load qt > qta
transient aggradational profile
sediment supply
increases from qta
to qt at t = 0
h
antecedent equilibrium bed profile
established with load qta
14
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NORMAL FLOW FORMULATION OF MORPHODYNAMICS: GOVERNING
EQUATIONS
In this chapter the flow is calculated by approximating it with the normal flow
formulation, even if the profile itself is in disequilibrium. The approximation is of loose
validity in most cases of interest, and becomes more rigorously valid with increasing
Froude number. Gradually varied flow is considered in Chapter 20. Using the Exner
formulation of Chapter 2 and the Manning-Strickler formulation for flow resistance,
the morphodynamic problem has the following character:
(1   p )
q
h
 -If t
t
x
1/ 3 2

  k c qw
qt  RgD D  t s  2
 g

  r



3 / 10
nt
S7 / 10   

  c 
RD 



,
S
h
x
In the above relations t denotes real time (as opposed to flood time) and the
intermittency factor If accounts for the fact that the river is only occasionally in flood
(and thus morphologically active).
15
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
THE NORMAL FLOW MORPHODYNAMIC FORMULATION AS A
NONLINEAR DIFFUSION PROBLEM
The previous formulation can be rewritten as:
h  
h 

 d (S ) 

t x 
x 
where d is a kinematic “diffusivity” of sediment (dimensions of L2/T) given by the
relation
1/ 3 2
If RgD D 
  k c qw
d 
 t s  2
(1   p ) S    r g
 



3 / 10
nt


S

  c 
RD 



7 / 10
The top equation is a diffusion equation. In the bottom equation, it is seen that d is
dependent on S = - h/x, so that the diffusion formulation is nonlinear.
The problem is second-order in x and first order in t, so that one initial
condition and two boundary conditions are required for solution.
16
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
INITIAL AND BOUNDARY CONDITIONS
The reach over which morphodynamic evolution is to be described must have a
finite length L. Here it extends from x = 0 to x = L.
The initial condition is that of a specified bed profile;
h( x, t ) t 0  hI ( x )
The simplest example of this is a profile with specified initial downstream elevation
hId at x = L and constant initial slope SI;
h( x, t ) t 0  hId  SI (L  x )
The upstream boundary condition can be specified in terms of given sediment
supply, or feed rate qtf, which may vary in time;
qt ( x, t ) x 0  qtf ( t )
The simplest case is that of a constant value of sediment feed.
The downstream boundary condition can be one of prescribed base level in terms of
bed elevation;
h( x, t ) xL  hd (t )
Again the simplest case is a constant value, e.g. hd = 0.
17
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
NOTES ON THE DOWNSTREAM BOUNDARY CONDITION
In principle the best place to locate the downstream boundary condition is at a
bedrock exposure, as illustrated below. In most alluvial streams, however, such
points may not be available. Three alternatives are possible:
a) Set the boundary condition at a point so far downstream that no effect of e.g.
changed sediment feed rate is felt during the time span of interest;
b) Set the boundary condition where the river joins a much larger river; or
c) Set the boundary condition at a point of known water surface elevation, such as
a lake (see Chapter 20 and the use of the gradually varied flow model).
Alluvial Kaiya River, Papua New Guinea, and downstream bedrock exposure
Bedrock
makes a
good
downstream
b.c.
18
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DISCRETIZATION FOR NUMERICAL SOLUTION
The morphodynamic problem is nonlinear and requires a numerical solution. This
may be done by dividing the domain from x = 0 to x = L into M subreaches bounded
by M + 1 nodes. The step length x is then given as L/M. Sediment is fed in at an
extra “ghost” node one step upstream of the first node.
x 
Feed sediment here!
ghost
i=1
2
L
M
xi  (i  1)x , i  1..M  1
x
3
M -1
L
Bed slope can be computed by the
relations to the right. Once the
slope Si is computed the sediment
transport rate qt,i can be computed
at every node. At the ghost node,
qt,g = qtf.
M
i = M+1
 h1  h2
,i 1

x
 h  h
i1
Si   i1
, i  2..M
 2x
 hM  hM1 , i  M  1
 x
19
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DISCRETIZATION OF THE EXNER EQUATION
Let t denote the time step. Then the Exner equation discretizes to
hi t  t
1 qt,i
 hi t 
If t , i  1..M
1   p x
where
qt,i
qt,i  qt,i1
qt,i1  qt,i
 au
 (1  au )
x
x
x
and au is an upwinding coefficient. In a pure upwinding scheme, au = 1. In a
central difference scheme, au = 0.5. A central difference scheme generally works
well when the normal flow formulation is used.
At the ghost node, qt,g = qtf. In computing qt,i/x at i = 1, the node at i – 1 (= 0) is
the ghost node. At node M+1, the Exner equation is not implemented because
bed elevation is specified as hM+1 = hd.
20
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
INTRODUCTION TO RTe-bookAgDegNormal.xls
The basic program in Visual Basic for Applications is contained in Module 1, and is
run from worksheet “Calculator”.
The program is designed to compute a) an ambient mobile-bed equilibrium, and b)
the response of a reach to changed sediment input rate at the upstream end of the
reach starting from t = 0.
The first set of required input includes: flood discharge Q, intermittency If, channel
(bankfull) width B, grain size D, bed porosity p, composite roughness height kc
and ambient bed slope S (before increase in sediment supply). Composite
roughness height kc should be equal to ks = nkD, where nk is in the range 2 – 4, in
the absence of bedforms. When bedforms are expected kc should be estimated at
bankfull flow using the techniques of Chapter 9 and 10 (compute Cz from
hydraulic resistance formulation; kc = (11 H)/exp(Cz)).
Various parameters of the ambient flow, including the ambient annual bed material
transport rate Gt in tons per year, are then computed directly on worksheet
“Calculator”.
21
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
INTRODUCTION TO RTe-bookAgDegNormal.xls contd.
The next required input is the annual average bed material feed rate Gtf imposed
after t > 0. If this is the same as the ambient rate Gt then nothing should happen;
if Gtf > Gt then the bed should aggrade, and if Gtf < Gt then it should degrade.
The final set of input includes the reach length L, the number of intervals M into
which the reach is divided (so that x = L/M), the time step t, the upwinding
coefficient au (use 0.5 for a central difference scheme), and two parameters
controlling output, the number of time steps to printout Ntoprint and the number of
printouts (in addition to the initial ambient state) Nprint.
The downstream bed elevation hd is automatically set equal to zero in the
program.
Auxiliary parameters, including r (coefficient in Manning-Strickler), t and nt
(coefficient and exponent in load relation), c* (critical Shields stress), s (fraction
of boundary shear stress that is skin friction) and R (sediment submerged specific
gravity) are specified in the worksheet “Auxiliary Parameters”.
22
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
INTRODUCTION TO RTe-bookAgDegNormal.xls contd.
The parameter s estimating the fraction of boundary shear stress that is skin
friction, should either be set equal to 1 or estimated using the techniques of
Chapter 9.
In any given case it will be necessary to play with the parameters M (which sets
x) and t in order to obtain good results. For any given x, it is appropriate to
find the largest value of t that does not lead to numerical instability.
The program is executed by clicking the button “Do a Calculation” from the
worksheet “Calculator”. Output for bed elevation is given in terms of numbers in
worksheet “ResultsofCalc” and in terms of plots in worksheet “PlottheData”
The formulation is given in more detail in the worksheet “Formulation”, which is
also available as a stand-alone document, Rte-bookAgDegNormalFormul.doc.
23
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MODULE 1 Sub Main
This is the master subroutine that controls the Visual Basic program.
Sub Main()
Clear_Old_Output
Get_Auxiliary_Data
Get_Data
Compute_Ambient_and_Final_Equilibria
Set_Initial_Bed_and_time
Send_Output
j=0
For j = 1 To Nprint
For w = 1 To Ntoprint
Find_Slope_and_Load
Find_New_eta
Next w
More_Output
Next j
End Sub
24
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MODULE 1 Sub Set_Initial_Bed_and_time
This subroutine sets the initial ambient bed profile.
Sub Set_Initial_Bed_and_time()
For i = 1 To N + 1
x(i) = dx * (i - 1)
eta(i) = Sa * L - Sa * dx * (i - 1)
Next i
time = 0
End Sub
25
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MODULE 1 Sub Find_Slope_and_Load
This subroutine computes the load at every node.
Sub Find_Slope_and_Load()
Dim i As Integer
Dim taux As Double: Dim qstarx As Double: Dim Hx As Double
Sl(1) = (eta(1) - eta(2)) / dx
Sl(M + 1) = (eta(M) - eta(M + 1)) / dx
For i = 2 To M
Sl(i) = (eta(i - 1) - eta(i + 1)) / (2 * dx)
Next i
For i = 1 To M + 1
Hx = ((Qf ^ 2) * (kc ^ (1 / 3)) / (alr ^ 2) / (B ^ 2) / g / Sl(i)) ^ (3 / 10)
taux = Hx * Sl(i) / Rr / D
If fis * taux <= tausc Then
qstarx = 0
Else
qstarx = alt * (fis * taux - tausc) ^ nt
End If
qt(i) = ((Rr * g * D) ^ 0.5) * D * qstarx
Next i
End Sub
26
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
MODULE 1 Sub Find_New_eta
This subroutine implements the Exner equation to find the bed one time step later.
Sub Find_New_eta()
Dim i As Integer
Dim qtback As Double: Dim qtit As Double: Dim qtfrnt As Double: Dim qtdif
As Double
For i = 1 To M
If i = 1 Then
qtback = qqtf
Else
qtback = qt(i - 1)
End If
qtit = qt(i)
qtfrnt = qt(i + 1)
qtdif = au * (qtback - qtit) + (1 - au) * (qtit - qtfrnt)
eta(i) = eta(i) + dt / (1 - lamp) / dx * qtdif * Inter
Next i
time = time + dt
27
End Sub
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
A SAMPLE COMPUTATION
Calculation of River Bed Elevation Variation with Normal Flow Assumption
(Qf)
(Inter)
(B)
(D)
(lamp)
Calculation of ambient river conditions (before imposed change)
Assumed parameters
Q
70 m^3/s
Flood discharge
If
0.03
Intermittency
The colored boxes:
B
25 m
Channel Width
indicate the parameters you must specify.
D
30 mm
Grain Size
The rest are computed for you.
p
0.35
Bed Porosity
(kc)
kc
(S)
S
75 mm
0.008
The ambient sediment transport
rate is 305,000 tons/year. At time
t = 0 this is increased to 700,000
tons per year. The bed must
aggrade in response.
Roughness Height
If bedforms are absent, set kc = ks, where ks = nk D and nk is an order-one factor (e.g. 3).
Ambient Bed Slope
Otherwise set kc = an appropriate value including the effects of bedforms.
Computed parameters at ambient conditions
H
0.875553 m
Flow depth (at flood)
*
0.141503
Shields number (at flood)
q*
0.232414
Einstein number (at flood)
qt
0.004859 m^2/s
Volume sediment transport rate per unit width (at flood)
Gt
3.05E+05 tons/a
Ambient annual sediment transport rate in tons per annum (averaged over entire year)
Calculation of ultimate conditions imposed by a modified rate of sediment input
Gtf
7.00E+05 tons/a
Imposed annual sediment transport rate fed in from upstream (which must all be carried during floods)
qtf
0.011161 m^2/s
Upstream imposed volume sediment transport rate per unit width (at flood)
ult
0.211523
Ultimate equilibrium Shields number (at flood)
Sult
0.014207
Ultimate slope to which the bed must aggrade
Hult
0.736984 m
Ultimate flow depth (at flood)
Click the button to perform a calculation
Calculation of time evolution toward this ultimate state
L
qt,g
x
t
10000
0.011161
1.67E+02
0.01
m
m^2/s
m
year
length of reach
Ntoprint
sediment feed rate (during floods) at ghost node
Nprint
spatial step
M
u
time step
Duration of calculation
200
5
60
0.5
10
Number of time steps to printout
Number of printouts
Intervals
28
Here 1 = full upwind, 0.5 = central difference
years
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RESULTS OF SAMPLE COMPUTATION
Bed evolution
160
140
Elevation in m
120
0 yr
2 yr
4 yr
6 yr
8 yr
10 yr
Ultimate
100
80
60
40
20
0
0
2000
4000
6000
Distance in m
8000
10000
29
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
INTERPRETATION
The long profile of a river is a plot of bed elevation h versus down-channel
distance x. The long profile of a river is called upward concave if slope S = h/x is decreasing in the streamwise direction; otherwise it is called upward
convex. That is, a long profile is upward concave if
S
 2h
 2 0
x
x
upward-convex
h
upward-concave
Aggrading reaches often show
transient upward concave
profiles. This is because the
deposition of sediment causes the
sediment load to decrease in the
downstream direction. The
decreased load can be carried with
a decreased Shields number *,
and thus according to the normalflow formulation of the present
chapter, a decreased slope:
k q
  
 g

x
1/ 3 2
c
w
2
r



3 / 10
S7 / 10
RD
30
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
INTERPRETATION contd.
The transient long profile of Slide 29 is upward concave because the river is
aggrading toward a new mobile-bed equilibrium with a higher slope. Once the
new equilibrium is reached, the river will have a constant slope (vanishing
concavity). This process is outlined in the next slide (Slide 32), in which all the
input parameters are the same as in Slide 28 except Ntoprint, which is varied so
that the duration of calculation ranges from 1 year (far from final equilibrium) to
250 years (final equilibrium essentially reached).
Slide 33 shows a case where the profile degrades to a new mobile-bed
equilibrium. During the transient process of degradation the long profile of the bed
is downward concave, or upward convex. This is because the erosion which
drives degradation causes the load, and thus the slope to increase in the
downstream direction. The input conditions for Slide 32 are the same as that of
Slide 28, except that the sediment feed rate Gtf is dropped to 70,000 tons per year.
This value is well below the ambient value of 305,000 tons per year (see Slide 28),
forcing degradation and transient downward concavity. In addition, Ntoprint is
varied so that the duration of calculation varies from 1 year to 250 years.
It will be seen in Chapter 25 that factors such as subsidence or sea level rise
can drive equilibrium long profiles which are upward concave.
31
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
AGGRADATION TO A NEW MOBILE-BED EQUILIBRIUM
Bed evolution
Bed evolution
160
160
0 yr
0.2 yr
0.4 yr
0.6 yr
0.8 yr
1 yr
Ultimate
Elevation in m
120
100
80
0 yr
2 yr
4 yr
6 yr
8 yr
10 yr
Ultimate
140
120
Elevation in m
140
60
40
100
80
60
40
20
20
0
0
0
2000
4000
6000
8000
10000
0
2000
Distance in m
6000
8000
10000
Distance in m
Bed evolution
Bed evolution
160
160
100
80
120
Elevation in m
120
0 yr
50 yr
100 yr
150 yr
200 yr
250 yr
Ultimate
140
0 yr
5 yr
10 yr
15 yr
20 yr
25 yr
Ultimate
140
Elevation in m
4000
60
100
80
60
40
40
20
20
0
0
0
2000
4000
6000
Distance in m
8000
10000
0
2000
4000
6000
Distance in m
8000
10000
32
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
DEGRADATION TO A NEW MOBILE-BED EQUILIBRIUM
Bed evolution
Bed evolution
90
90
80
60
50
40
0 yr
2 yr
4 yr
6 yr
8 yr
10 yr
Ultimate
70
Elevation in m
70
Elevation in m
80
0 yr
0.2 yr
0.4 yr
0.6 yr
0.8 yr
1 yr
Ultimate
30
60
50
40
30
20
20
10
10
0
0
0
2000
4000
6000
8000
10000
0
2000
Distance in m
6000
8000
10000
Distance in m
Bed evolution
Bed evolution
90
90
80
80
0 yr
20 yr
40 yr
60 yr
80 yr
100 yr
Ultimate
60
50
40
0 yr
50 yr
100 yr
150 yr
200 yr
250 yr
Ultimate
70
Elevation in m
70
Elevation in m
4000
30
60
50
40
30
20
20
10
10
0
0
0
2000
4000
6000
Distance in m
8000
10000
0
2000
4000
6000
Distance in m
8000
10000
33
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
ADJUSTING THE NUMBER M OF SPATIAL INTERVALS
AND THE TIME STEP t
The calculation becomes unstable, and the program crashes if the time step t is
too long. The above example resulted in a crash when t was increased from the
value of 0.01 years in Slide 29 to 0.05 years. The larger the value M of spatial
intervals is, the smaller is the maximum value of t to avoid numerical
34
instability. Acceptable values of M and t can be found by trial and error.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
AN EXTENSION:
RESPONSE OF AN ALLUVIAL RIVER TO VERTICAL FAULTING DUE TO AN
EARTHQUAKE
The code in RTe-bookAgDegNormal.xls represents a plain vanilla version of a
formulation that is easily extended to a variety of other cases. The spreadsheet
RTe-bookAgDegNormalFault.xls contains an extension of the formulation for
sudden vertical faulting of the bed. The bed downstream of the point x = rfL (0 < rf
< 1) is suddenly faulted downward by an amount hf at time tf. The eventual
smearing out of the long profile is then computed.
h
35
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RESULTS OF SAMPLE CALCULATION WITH FAULTING
Bed evolution
12
10
Elevation in m
8
0 yr
0.05 yr
0.1 yr
0.15 yr
0.2 yr
0.25 yr
6
4
2
0
-2
-4
-6
0
2000
4000
6000
Distance in m
8000
10000
36
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
RESULTS OF SAMPLE CALCULATION WITH FAULTING contd.
In time the fault is erased by degradation upstream and aggradation downstream,
and a new mobile-bed equilibrium is reached.
Bed evolution
12
12
10
10
8
8
0 yr
0.001 yr
0.002 yr
0.003 yr
0.004 yr
0.005 yr
6
4
2
0
Elevation in m
Elevation in m
Bed evolution
0 yr
0.025 yr
0.05 yr
0.075 yr
0.1 yr
0.125 yr
6
4
2
0
-2
-2
-4
-4
-6
-6
0
2000
4000
6000
8000
0
10000
2000
6000
8000
Bed evolution
Bed evolution
12
10
10
8
8
0 yr
0.5 yr
1 yr
1.5 yr
2 yr
2.5 yr
6
4
2
0
Elevation in m
12
4
2
0
-2
-4
-4
-6
0 yr
5 yr
10 yr
15 yr
20 yr
25 yr
6
-2
37
-6
0
2000
4000
6000
Distance in m
10000
Distance in m
Distance in m
Elevation in m
4000
8000
10000
0
2000
4000
6000
Distance in m
8000
10000
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
© Gary Parker November, 2004
REFERENCES FOR CHAPTER 14
Engelund, F. and E. Hansen, 1967, A Monograph on Sediment Transport in Alluvial Streams,
Technisk Vorlag, Copenhagen, Denmark.
Lawrence, P., 2003, Bank Erosion and Sediment Transport in a Microscale Straight River, Ph.D.
thesis, University of Paris 7 – Denis Diderot, 167 p.
Meyer-Peter, E. and Müller, R., 1948, Formulas for Bed-Load Transport, Proceedings, 2nd
Congress, International Association of Hydraulic Research, Stockholm: 39-64.
Paola, C., Heller, P. L. & Angevine, C. L., 1992, The large-scale dynamics of grain-size variation
in alluvial basins. I: Theory, Basin Research, 4, 73-90.
Powell, D. M., Reid, I. and Laronne, J. B., 2001, Evolution of bedload grain-size distribution with
increasing flow strength and the effect of flow duration on the caliber of bedload sediment
yield in ephemeral gravel-bed rivers, Water Resources Research, 37(5), 1463-1474.
Wong, M. and Parker, G., submitted, The bedload transport relation of Meyer-Peter and Müller
overpredicts by a factor of two, Journal of Hydraulic Engineering, downloadable at
http://cee.uiuc.edu/people/parkerg/preprints.htm .
38