MORPHODYNAMICS OF RIVERS ENDING IN 1D DELTAS

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CHAPTER 34 MORPHODYNAMICS OF RIVERS ENDING IN 1D
DELTAS
When rivers flow into bodies of standing water
such as lakes or reservoirs, they typically form
fan-deltas that spread out laterally as they
prograde in the streamwise direction. If the
river is confined by a narrow canyon, however,
the installation of a dam can lead to a nearly 1D
delta that progrades downstream. An example is
shown on the next page.
Fan-delta at the upstream and of Mills Lake, a
reservoir on the Elwha River, Washington,
USA. Image courtesy Y. Cui.
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AN EXAMPLE OF A 1D DELTA
Hoover Dam was closed in 1936. Backwater from
the dam created Lake Mead. Initially backwater
extended well into the Grand Canyon. For much of
the history of Lake Mead, the delta at the
upstream end has been so confined by the canyon
that is has propagated downstream as a 1D delta.
As is seen in the image, the delta is now
spreading laterally into Lake Mead, forming a 2D
fan-delta.
View of the Colorado River at the upstream end of
Lake Mead. Image from NASA https//zulu.ssc.nasa.g
ov/mrsid/mrsid.pl
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HISTORY OF SEDIMENTATION IN LAKE MEAD, 1936 - 1948
Image based on an original from Grover and Howard
(1937)
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STRUCTURE OF A DELTA TOPSET, FORESET AND
BOTTOMSET
A typical delta deposit can be divided into a
topset, foreset and bottomset. The topset is
coarse-grained (sand or sand and gravel), and is
emplaced by fluvial deposition. The foreset is
also coarse-grained, and is emplaced by
avalanching. The bottomset is fine-grained (mud,
e.g. silt and clay) and is emplaced by either
plunging turbidity currents are rain from
sediment-laden surface plumes.
standing water
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SIMPLIFICATION TOPSET AND FORESET ONLY
Here the problem is simplified by considering a
topset and foreset only. That is, the effect of
the mud is ignored. Mud is included in a later
chapter.
standing water
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MOVING BOUNDARIES
The problem is charactized by two moving
boundaries. The point x ss(t) corresponds to
the topset-foreset break, and the point x sb(t)
corresponds to the foreset-bottomset break. Bed
elevation is denoted as ?(x, t) the antecedent
bed profile over which the delta progrades is
denoted as ?base(x). Bed elevation at the
topset-foreset break is given as ?s(t) ?ss(t),
t. Bed elevation at the foreset-bottomset break
is given as ?b(t) ?sb(t), t.
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EXNER EQUATION OF SEDIMENT CONSERVATION
The Exner equation of sediment conservation takes
the form An appropriate boundary condition at
the upstream end is where qtf denotes a
sediment feed rate
qtf
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LINEAR BED PROFILE ON FORESET (DELTA FRONT)
Bed elevation on the fluvial region, i.e. 0 ? x ?
ss(t) is denoted as ?f(x,t). The delta front is
assumed to have a specified constant angle of
avalanching Sa. This angle is usually less,
often much less than the angle of repose of the
sediment (Kostic and Parker, 20xx). The profile
on the foreset is thus given as
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SHOCK CONDITION ON FORESET (DELTA FRONT)
The Exner equation of sediment continuity can
be integrated across the delta front to yield the
shock condition (Swenson et al., 2000 Kostic
and Parker, 2003a,b). Now it is assumed that no
(coarse-grained) sediment escapes the toe of the
delta front, so that qtsb(t), t 0. In
addition, the bed profile across the foreset,
can be used to calculate ??/?t across it.
Remembering to compute the material derivative of
?fss(t), t, it is found that
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SHOCK CONDITION ON FORESET (DELTA FRONT) contd.
Denoting The condition at the bottom of the
last slide reduces to Where Ss denotes the bed
slope of the fluvial region at the topset-foreset
break. Now substituting the above relation
into and integrating, it is found that
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SHOCK CONDITION ON FORESET (DELTA FRONT) contd.
Upon reduction, the shock condition can be
reduced to the following form where qts
qtss(t), t denotes the volume rate per unit
width of delivery of bed material load to the
brink of the foreset. This condition can be
understood in a simple way. Suppose that Ss ltlt
Sa and that the term can be
neglected. Further noting that the height of the
delta front ?? is equal to Sa (sb ss), it is
seen that the condition reduces to
That is, all the sediment delivered to the
topset-foreset break is consumed in prograding
the delta forward at speed .
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CONTINUITY CONDITION AT FORESET-BOTTOMSET BREAK
The foreset elevation profile must match
continuously with the antecedent bed profile
?base(x) at the point x sb(t), i.e. the
foreset-bottomset break. Here ?base is a
specified profile. Thus (Kostic and Parker,
2003a,b)
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CONTINUITY CONDITION AT FORESET-BOTTOMSET BREAK
contd.
Taking the derivative with respect to time of the
equation below yields the result Denoting
the basement slope at the foreset-bottomset break
as Sb, where the condition reduces to Thus
if Ss and Sb are small compared to Sa and the
time derivate in ?f can be neglected, the
condition reduces simply to
so that the foreset-bottomset break progrades at
the same speed as the topset-foreset break.
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MOVING BOUNDARY FORMULATION
The Exner equation on the fluvial zone is
transformed using the following moving-boundary
coordinates Note that the fluvial zone is now
traversed from sediment feed point to
topset-foreset break as . The
derivatives transform according to the chain rule
as
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EXNER EQUATION, SHOCK CONDITION AND CONTINUITY
CONDITION IN MOVING-BOUNDARY COORDINATES
The Exner equation transforms to The shock
condition transforms to The continuity
condition transforms to
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IMPLEMENTATION SEDIMENT TRANSPORT RELATION
The calculation is implemented using a generic
relation for total bed material
transport where ?t, nt and ?c must be
specified. The flow field is computed using a
constant Chezy coefficient Cz for simplicity. As
was seen in Chapters 19 and 21, the
generalization to a) a sediment transport
formulation that divides bed material load into
bed load and suspended load components, b) a
resistance predictor that divides resistance into
skin friction and form drag and c) a flow
hydrograph is relatively straightforward.
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IMPLEMENTATION BACKWATER FORMULATION
Deltas are zones where rivers meet standing
water. As a result, it can be expected that
backwater effects are important in deltas. The
backwater formulation was given in Chapter 5. In
moving-boundary coordinates, it takes the
form with the boundary condition on H
of where ?s denotes the elevation of standing
water. Note that the elevation of standing water
?s can in general be a function of time. It is
variation in this parameter that allows the
determination of the effect of base level change
on delta morphodynamics.
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IMPLEMENTATION BACKWATER FORMULATION contd.
Once the depth H is computed everywhere at a
given time from the backwater formulation, the
Shields number ? is evaluated from the
relation This in turn allows evaluation of
qt from the total bed material load equation
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IMPLEMENTATION NORMAL FLOW APPROXIMATION
In some cases it is desirable to use the normal
flow approximation, even with its disadvantages
in the vicinity of a delta. In such case the
backwater formulation is replaced with the
computation
Thus if qw, R, D, Cz and the bed profile ? are
specified at any time, ? and qt can be computed
everywhere from the above relations and the load
equation In the backwater formulation,
changing base level is mediated by a time-varying
elevation of standing water ?d, where the
function must be specified. In a
backwater formulation, changing base level is
approximated in terms of a time-varying
downstream bed elevation. That is, where
must be specified. This condition places some
limits on the results of the analysis, as
outlined in subsequent slides.
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SPECIFICATION OF THE ANTECEDENT BED PROFILE
?base(x), THE INITIAL BED CONFIGURATION AND THE
UPSTREAM BOUNDARY CONDITION
Although in general the antecedent bed profile
can be specified arbitrarily, here it is assumed
for simplicity that this profile has constant bed
slope - ??base/?x Sbase. The upstream end of
the fluvial zone is always located at x 0. The
initial length of the fluvial zone is ssI and the
initial bed slope is SfI. The initial elevation
of the topset-foreset break is ?sI, and the
initial elevation of the bottom of the
topset-foreset break is ?bI. The initial bed
profile is thus given as The antecedent bed
profile is a straight line with slope Sbase and
passing though elevation ?bI at x ssI (?sI -
?bI)/Sa. In any calculation, Sb, ssI, SfI, ?sI,
?bI and foreset slope Sa must be specified by the
user. The upstream boundary condition is
specified in terms of a sediment feed rate qtf,
which in general may vary in time.
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FLOW OF THE CALCULATION BACKWATER FORMULATION

• For any given time
• Specify the elevation of downstream standing
  water ?d.
• Calculate the backwater curve upstream from
  .
• Use this to evaluate qt everywhere, including qts
  at .
• Implement the shock condition to find .
  This shock condition requires knowledge of the
  term
• It is sufficient to evaluate this term using the
  current bed profile and that obtained one step
  earlier. At this term can be
  ignored.
• Solve Exner everywhere to find new bed elevations
  at time later.
• Use continuity condition to find .
• Update boundaries

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FLOW OF THE CALCULATION NORMAL FLOW FORMULATION

• For any given time
• Specify the downstream bed elevation ?d.
• Calculate the bed slope S everywhere, and use
  this to find H everywhere.
• Use this to evaluate qt everywhere, including qts
  at .
• Implement the shock condition to find .
  This shock condition requires knowledge of the
  term
• This term can be directly computed in terms of
  the imposed function .
• Solve Exner everywhere except the last node
  (where bed elevation is specified) to find new
  bed elevations at time later.
• Use continuity condition to find .
• Update boundaries

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DISCRETIZATION AND SPECIFICATION OF INITIAL BED
Initial bed
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CALCULATION OF DERIVATIVES
Two Excel workbooks with code are presented here.
The backwater formulation is given in
RTe-book1DDeltaBW.xls, and the normal flow
formulation is given in Rte-book1DDeltaNorm.xls.
In both cases the spatial derivative of ?f
appearing in the transformed Exner relation
is computed using a central difference scheme
except at end nodes. That is, In the
case of the normal flow formulation, it is not
necessary to compute the above derivate at node
M 1, because bed elevation ?d is specified
there.
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CALCULATION OF DERIVATIVES contd.
Spatial derivatives in qt should be based on an
upwinding scheme for a backwater formulation. In
the code presented here, a pure upwinding scheme
is used for all nodes, i.e.
A partial upwinding scheme will also work. In
the case of a normal flow formulation, spatial
derivatives in qt can be based on a central
difference scheme for internal nodes Again,
in the normal flow formulation the above
derivative need not be computed at node M 1.
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INTRODUCTION TO Rte-book1DDeltaBW.xls
The code in this spreadsheet uses a backwater
formulation. Input variables include Flood
discharge/width qw Time step ?t Flood
intermittency If No. of spatial
intervals M Chezy resistance coefficient Cz
Steps to printout Mtoprint Grain size D
Printouts beyond initial Mprint Sed submerged
specific gravity R Deposit porosity lp Volum
e bed material feed rate/width qtf Coefficient
in bed material transport relation at Exponent in
bed material transport relation nt Critical
Shields number in transport relation ?c Elevation
of downstream standing water ?d Initial
elevation of topset-foreset break ?sI Initial
elevation of foreset-bottomset break ?bI Initial
bed slope of fluvial zone SfI Bed slope of
subaqueous basement Sb Initial length of fluvial
zone sfI Maximum length of fluvial
zone sfmax Slope of avalanching on foreset Sa
The code uses a fixed elevation ?d of standing
water at the downstream end. The code can be
easily modified to handle the case of
time-varying base level.
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CALCULATIONS WITH Rte-book1DDeltaBW.xls
Calculations are presented for the indicated
input parameters, except for Mtoprint, which is
varied to capture the time evolution of the river
profile.
28
Note how the backwater formulation captures the
new delta front as it progrades over the
antecedent bed.
Mtoprint 20
29
Mtoprint 100
30
The incipient delta merges with the antecedent
one, which then progrades outward vigorously.
Mtoprint 200
31
Progradation with a nice upward-concave long
profile.
Mtoprint 5000
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The calculation craps out for Mtoprint
10000 (final time 30 years).
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Not to worry. Just increase the number of
spatial intervals M from 20 to 40 and it works
fine for Mtoprint 10000 (final time 30
years)!
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The conditions for this run are identical to
those of the previous slide, except that basement
slope Sb has been changed from 0 to 0.0003. Note
that the long profiles are less concave, and that
the progradation rate is reduced.
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WORKSHEET InData OF Rte-book1DDeltaBW.xls
PROVIDES USEFUL GUIDANCE IN REGARD TO INPUT
PARAMETERS

• Make sure that ?d gt ?sI!
• If initial depth Hi ?d - ?sI is less than
  Hcrit then the flow will be locally supercritical
  and the calculation will fail!
• If Hi lt Hni, where Hni normal depth for the
  initial bed, then the initial flow will follow an
  M2 curve, which requires a very dense spatial
  grid to capture.
• Certain combinations of ?sI, ?bI and ?d will
  cause the calculation to fail as the height of
  the foreset falls to zero. Increasing ?sI
  usually fixes the problem.
• As always, it is necessary to tinker with ?t
  and M to obtain numerical stability.

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AN EXAMPLE OF A CALCULATION FAILURE
All parameters are the same as those of slides 27
and 31 except that ?d has been lowered from 8.5 m
to 7 m. The calculation fails for the very
physical reason that the delta front is prograded
out of existence.
To get the calculation to work, either raise ?d
or lower ?bI.
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EXAMPLE OF A HIGH BASE LEVEL
This case is the same as that of slide 30
(Mtoprint 200), except that ?d is increased
from 8.5 m to 20 m. Note how this retards delta
progradation.
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INTRODUCTION TO Rte-book1DDeltaNorm.xls
The code in this spreadsheet uses a normal flow
formulation. Input variables include Flood
discharge/width qw Time step ?t Flood
intermittency If No. of spatial
intervals M Chezy resistance coefficient Cz
Steps to printout Mtoprint Grain size D
Printouts beyond initial Mprint Sed submerged
specific gravity R Deposit porosity lp Volum
e bed material feed rate/width qtf Coefficient
in bed material transport relation at Exponent in
bed material transport relation nt Critical
Shields number in transport relation ?c Elevation
of topset-foreset break ?d Initial elevation of
foreset-bottomset break ?bI Initial bed slope of
fluvial zone SfI Bed slope of subaqueous
basement Sb Initial length of fluvial
zone sfI Slope of avalanching on foreset Sa
The code uses a fixed bed elevation ?d at the
downstream end. The code can be easily modified
to handle the case of time-varying base level,
but if ?d rises too rapidly in time the delta
goes into autoretreat, requiring special
treatment. See Chapter 38.
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CALCULATIONS WITH Rte-book1DDeltaNorm.xls
The input conditions are identical to slide 27
(backwater formulation), except that base level
is specified in terms of a fixed downstream bed
level ?d. Calculations are presented for the
indicated input parameters, except for Mtoprint,
which is varied to capture the time evolution of
the river profile.
40
This case (normal flow) corresponds to slide 31
(backwater). By this time the results are not
much different between the two.
41
The two cases (backwater versus normal) are
compared at the end time (15 years) of slides 31
and 40. The profile for the backwater
calculation has prograded less, because the
specified downstream water surface elevation of
8.5 m has led to a computed downstream bed
elevation that is higher than the specified value
of 3 m for the normal flow calculation.
42
This case (normal flow) corresponds to slide 29
(backwater). At this earlier time, the
difference between the two is fairly strong.
43
In the case of the backwater calculation, the
antecedent delta is still inactive by 0.3 years
as the deposit struggles to overcome backwater
upstream. In the case of the normal flow
calculation, the antecedent delta is forced to
prograde immediately, as there is no backwater.
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NOTES IN CLOSING

• ? This formulation uses a constant Chezy
  resistance coefficient.
• Could you modify it for Manning-Strickler? For
  Wright-Parker?
• This formulation uses a relation for total bed
  material load
• ? Could you modify it to treat bed load and
  suspended load separately?
• ? This formulation assumes a uniform sediment
  size
• ? Could you modify it to treat sediment
  mixtures?
• ? This formulation assumes constant base level
  at the downstream end.
• ? Could you modify it to treat time-varying base
  level?
• ? This formulation assumes an intermittency and
  a constant flood flow.
• ? Could you modify it to treat hydrographs?
• The purpose of this e-book is not to work out
  each one of these permutations. Rather the
  purpose is to provide the reader with the
  necessary tools to allow adaptation to specific
  research and engineering problems of interest.

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REFERENCES FOR CHAPTER 34
Grover, N.C., and Howard, C.L., 1937, The passage
of turbid water through Lake Mead, Transactions,
American Society of Civil Engineers, 103,
720-732. Kostic, S. and Parker, G., 2003a,
Progradational sand-mud deltas in lakes and
reservoirs. Part 1. Theory and numerical
modeling, Journal of Hydraulic Research, 41(2),
127-140. Kostic, S. and Parker, G., 2003b,
Progradational sand-mud deltas in lakes and
reservoirs. Part 2. Experiment and numerical
simulation, Journal of Hydraulic Research, 41(2),
141-152 Swenson, J. B., Voller, V. R., Paola, C.,
Parker G. and Marr J., 2000, Fluvio-deltaic
sedimentation a generalized Stefan problem,
European Journal of Applied Math., 11, 433-452.