MORPHODYNAMICS OF 1D SUBMARINE/SUBLACUSTRINE FANS

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CEE 598, GEOL 593 TURBIDITY CURRENTS
MORPHODYNAMICS AND DEPOSITS
LECTURE 12 MORPHODYNAMICS OF 1D
SUBMARINE/SUBLACUSTRINE FANS
As the Colorado River flows into Lake Mead, USA,
it forms a delta.
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STRUCTURE OF A DELTAIC DEPOSIT
Rivers flowing into lakes and reservoirs deposit
their sediment in deltas. In a delta, the
coarser material (e.g. sand and coarser)
material is fluvially emplaced in a topset
deposit and emplaced by avalanching in a foreset
deposit. Finer material (e.g. mud) is
preferentially emplaced in deep water beyond the
toe of the foreset. A major mechanism for this
deep-water emplacement consists of turbidity
currents associated with plunging.
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PLUNGING IN LAKE MEAD
plunge line
Logjam near plunge point
Image from USBR
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DEPOSITION WITHIN LAKE MEAD, COLORADO RIVER, USA
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SOME DETAIL OF THE DEPOSIT
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MOST DELTAS SPREAD OUT LATERALLY TO MAKE
FAN-DELTAS
Delta of the Selenga River as it flows into Lake
Baikal, Russia
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WHEN LAKE MEAD BEGAN TO FILL, HOWEVER, THE DELTA
WAS CONFINED TO A NARROW CANYON FOR MANY YEARS
So the somewhat abstract case of a 1D delta
prograding into a zone of constant width is not
entirely without field analogs. Besides, if we
can understand 1D delta, the treatment can then
be generalized to 2D deltas.
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HERE WE SIMPLIFY THE PROBLEM TO A 1D SUBAQUEOUS
FAN
The upstream feed point of the 1D subaqueous fan
is set at x 0. The vertical position of the
feed point is, however, allowed to adjust with
the evolution of the bed.
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FURTHER SIMPLIFICATIONS

• The flow is assumed to be continuous.
• Only a single grain size is considered.
• The flow is assumed to be supercritical
  everywhere, so that the backwater equations for
  the flow can be integrated downstream from the
  feed point.
• Erosion is neglected in a first formulation, so
  that the turbidity current is treated as purely
  depositional.
• All of these assumptions can be relaxed in a more
  elaborate model.

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THE GOVERNING EQUATIONS
Here we use a 3-equation formulation for the
flow. Since the flow is assumed to be dilute and
continuous, the relations
can be simplified to
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PARAMETERS AND BOUNDARY CONDITIONS
In the model presented here, ro, Cf, vs, R (and
of course g, because we just might not be working
on Earth) must be specified. We are assuming
supercritical flow, i.e. Ri lt 1, so that the
governing equations are integrated in
the downstream direction. Upstream boundary
conditions for C, U and H must be specified at x
0. Here we specifiy the upstream water
discharge per unit width qwo UoHo, the
suspended sediment discharge per unit width qso
UoHoCo and the upstream Richardson number Rio
(RgCoHo/Uo2) (Rgqso/Uo3). Thus
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BED EVOLUTION
The Exner equation of bed sediment continuity
takes the form That is, here we neglect both
sediment entrainment and bedload
transport An initial condition must be
specified for the bed profile To simplify
things here, we assume that the initial bed has a
constant slope SbI and an initial bed elevation
of 0 at x 0.
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FLOW OF THE CALCULATION
At any given time t, solve the equations below
over the existing bed from x 0 to x L, where
L is some domain length.
Use the results of this solution to find the bed
some time t ?t later
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SPATIAL DISCRETIZATION
The problem is solved over a domain extending
from x 0 to x L, where L is a specified
parameter. This domain is discretized to M
intervals of length ?x bounded by M 1 points
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SOLUTION FOR THE FLOW
The simplest method you can use to solve for the
flow is the Euler Step Method. (Yes, you can use
a more accurate method if you know it). We
rewrite the equations of the flow as
where for example
The equations discretize to where for
example Hi denotes the value of H at xi. For any
given bed profile (which specifies S), the
equations can be solved stepwise downstream from
i 1 to i M 1.
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SOLUTION FOR THE BED EVOLUTION
Solve for the new bed elevations from
Exner and obtain the new bed slopes Si
as And with these new bed slopes, it is
possible to solve for the flow over the new bed!
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WELCOME TO THE EXCEL WORKBOOK WITH IMBEDDED CODE
IN VISUAL BASIC FOR APPLICAITIONS Rte-book1DSubaq
ueousFan.xls
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NOTES ON THE PARAMETERS
If the flow were morphodynamically active for
only fraction I of time (i.e. a constant flow
that is maitained for only a few days per year),
the Exner equation must be modified to The
code allows for this possibility through the
parameter I Inter. The coefficient of bed
friction Cf is related to Czs via the
relation The fall velocity vs is computed from
the Dietrich (1982) relation introduced earlier.
The water entrainment relation used is that of
Parker et al. (1987)
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THE CODE
The code is found in the Visual Basic Editor.
From the Excel menu, select Tools, Macro, Visual
Basic Editor If the code is (macros are) not
enabled when you open the Excel file, you will
have to go to the Excel Menu, select Tools, Macro,
Security and set Security no higher than
medium. You then have to close and open
Excel in order to have the code enabled when you
open the file.
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SAMPLE CALCULATION BASE CASE
These are the base input parameters. Note that D
0.02 mm (mud) SbI initial bed slope 0.05
Simulation time 0.6 years of continuous flow
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D 0.02 mm, SbI initial bed slope 0.05
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D 0.005 mm, SbI initial bed slope 0.05
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D 0.05 mm, SbI initial bed slope 0.05
Bed gets too steep because the flow is purely
depositional!
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D 0.02 mm, SbI initial bed slope 0.05
(again)
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D 0.02 mm, SbI initial bed slope 0.02
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D 0.02 mm, SbI initial bed slope 0.065
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MATERIAL TO BE ADDED EROSIONAL CASE
NEED TO ADD. RTe-book1DSubaqueousFanWErosTRY.xls
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REFERENCES
Ashida, K. and M. Michiue, 1972, Study on
hydraulic resistance and bedload transport rate
in alluvial streams, Transactions, Japan Society
of Civil Engineering, 206 59-69 (in
Japanese). UNDER CONSTRUCTION