Mathematical Models of Sediment Transport Systems

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Mathematical Models of Sediment Transport
Systems Vaughan R.
Voller
Tetsuji Muto, Wonsuck Kim, Chris Paola, Gary
Parker, John Swenson, Jorge Lorenzo Trueba, Man
Liang Matt Wolinsky, Colin Stark, Andrew Fowler,
Doug Jerolmack
1m
10km
Anomalous Diffusion at Experimental Scales
A Model of Delta Growth
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Bangladesh
Katrina
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The Disappearing Mississippi DeltaMotivation
Provided by Wonsuck Kim et al, EOS Aug 2009
Due to Upstream Damming (limiting sediment
supply) Artificial Channelization
of the river (limiting flooding) Increased
subsidence (?) creating off shore space that
needs to be filled
Birds foot
Each year Louisiana loses 44 sq k of costal
wetlands
Loss of a buffer that could protect inland
infrastructure
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A plan is on the table to reverse this trend is
to create breaks in the levees to allow for
flooding, sediment deposit, and land growth
Costly and Risky Is there enough sediment?
Will it be sustainable
? How long will it take ?
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A lucky accidental natural experiment
Some 100 k or so to the West of New-Orleans ,in
the 1970s a navigation channel was created on a
tributary of the Mississippi. This resulted in a
massive sediment diversion and over the next 30
years the building of an delta 20K in dimension

20k
Wax-Lake Delta
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Can the experience of Wax Lake be transported to
the Birds Foot? Sediment Delta Growth Models
developed can be validated with Wax Lake data?
Graphic by Wonsuck Kim, UAT
Building Delta Models is achieved by appealing to
heat and mass transfer analogies
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Examples of Sediment Deltas
Water and sediment input
Sediment Fans
1km
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The delta shoreline is a moving boundary Advanced
in time due to sediment input
sediment flux
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A One D Experiment mimicking building of delta
profile, Tetsuji Muto and Wonsuck Kim Sediment
and Water Mix introduced into a slot flume (2cm
thick) with a fixed Sloping bottom and fixed
water depth
shore-line moves in response to
sediment input
Maintains a constant submarine slope
Can we construct a model for this ?
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In a Laboratory setting with constant flow
discharge and shallow depth
d(epth)
Momentum Balance
Drag

And when coupled to the Sediment Transport Law
(assuming bed shear gtgt Sheilds stress)
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The Swenson AnalogyMelting and Shoreline Movement
Stefan Melting Problem
T
Shore-line condition
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Apply this analogy to experiments
JORGE LORENZO-TRUEBA1, VAUGHAN R. VOLLER, TETSUJI
MUTO ,WONSUCK KIM, CHRIS PAOLA AND JOHN B. SWENSON
J. Fluid Mech. (2009), vol. 628, pp. 427443
Provide
sediment line-flux mm2/s
water line-discharge mm2/s
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Governing Equations
fixed basement
Note Two moving boundaries moving in opposite
directions. (1) shoreline, (2) bed-rock/alluvial
transition (point on basement where sediment
first deposits )
Four Boundary Conditions Are Needed
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A closed form similarity solution for tracking
fronts is found
Slope Ratio
Where the lambdas are functions of the
dimensionless variables the slope ratio R and
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Slope Ratio
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Experiment vs. Analytical VALIDATION
J. Fluid Mech. (2009), vol. 628, pp. 427443
experimental
analytical
Get fit by choosing diffusivity from Geometric
measurements From one exp. snap-shot
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In field setting
Value of slope ratio R controls sensitivity of
fronts
J. Fluid Mech. (2009), vol. 628, pp. 427443
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Common Field observation
Lower than expected curvature for fluvial surface
experimental
analytical
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In a Laboratory setting with constant flow
discharge and shallow depth
d(epth)
Momentum Balance
Drag
And when coupled to the sediment transport law
(assuming bed shear gtgt Sheilds stress)
Suggests a non-linear diffusive model
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Non-Linear diffusion model
J. Lorenzo-Trueba, V.R. Voller
J. Math. Anal. Appl. 366 (2010) 538549
also has sim. sol but requires numerical solution
Closed form only when
geometric wedge
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Linear
Geometric
J. Lorenzo-Trueba, V.R. Voller
J. Math. Anal. Appl. 366 (2010) 538549
Not until you reach high values of R do you see
any real difference
R
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Back to lack of curvature in Experiments
Jurasic Tank Experiment at close to steady
state
Diffusion solution too-curved
subsidence
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Is this equation valid
Clear separation between scale of heterogeneity
and domain. An REV can be identified
Not a slot
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Model
x
Transport controlled by Non-local events
suggesting --- path-dependence described
through hereditary integrals Non-Gaussian
behaviors with thick power-law tails
allowing for occurrence of extreme events
Through use of volume averaging generic
Advection-Diffusion transport equation will have
form
Processes that can be embodied into a fractional
Advection-Diffusion Equation (fADE)
fractional flux depends on weighted average of
non-local slopes (up and down stream)
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First we take a pragmatic approach and
investigate what happens if we replace the
diffusion flux with a fractional flux
Will this reduce curvature ?
A toy problem is introduced
area/time
solution
length/s
Piston subsidence of base
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First we will just blindly try a pragmatic
approach where we will write down a Fractional
derivative from of our test problem, solve it
and compare the curvatures.

Our first attempt is based on the left hand
Caputo derivative
With
LOOKS UPSTREAM
The divergence of a non-local fractional flux
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Clearly Not a good solution
expected
predicted
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Our second attempt is based on the right hand
Caputo derivative

With
LOOKS DOWN-STREAM
Note
Solution
On 0,1
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Right-Hand Caputo
Looks like this Has correct behavior
When we scale to The experimental setup We get a
good match
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And when a fraction flux is used it can match the
observed lack of curvature
Voller and Paola JGR (to appear)
Right
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But the question remains
Is this physically meaningful ?
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A simple minded model Down stream conditions
influence upstream transport
Imagine that particles transport through system
as chains The lengths of the chains vary and can
take values up to the length of the system
So at a given cross section x we can write down
a the flux as a weighted average of the
down-stream slopes
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Basic diffusion models can lead to interesting
math and reproduce experiments
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Thanks
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Shown How classic numerical heat transfer
(enthalpy method) can be used to model
key geoscince problem
Illustrated how a Monte-Carlo Solution based on
a Levy PDF
Can solve fractional BVP
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CLAIM If steps are chosen from a Levy
distribution
Maximum negative skew,
This numerical approach will also recover
Solutions to
Suggest that Monte Carlo Associated with a
PDF Could resolve multiple situations
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A Monte Carlo Solution
Well know (and somewhat trivial) that a Monte
Carlo simulation originating from a point and
using steps from a normal distribution will after
multiple realizations recover the temperature at
the point
Nright
Nleft
Tpoint fraction of walks that exit on Left
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As a demonstration of one-way we may go-about
solving such systems let us Consider the example
fractional BVP
This is a steady state problem in which the left
hand side represents a Local balance of a
Non-Local flux
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On using results from fractal methods a scale
independent model can be posed in terms of a
fractional derivative
Related to a Levy PDF distribution It has Fat
Tails Extreme events have finite probability
Such considerations could be important in
micro-scale heat transfer-where the required
resolution is close the scale of the mechanisms
in the heat conduction Process.
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Monte Carlo Calculation of Fractional Heat
Conduction
As noted above the transport of sediment (flux
volume/area-time) can be described by A
diffusion like law
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BUT On a land surface, spatial and temporal
variations are at an observable scale This is
similar to situation in a porous mediawhere it
is known that
length scale of resolution
where the hydraulic Conductivity has a power
law dependence with the scale at which it is
resolved.
Modeling a reservoir at scale
using a hydraulic conductivity determined at a
scale
Will result in under prediction of transport
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Errors appears when slope ratio is high A thin
wedge at on-lap
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The Modeling Paradigm
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