1. 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

2. Bangladesh
Katrina

3. The Disappearing Mississippi Delta—Motivation 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
New Orleans
Bird’s foot
Each year Louisiana loses ~44 sq k of costal wetlands
Loss of a buffer that could protect inland infrastructure

4. 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 ?

5. A lucky accidental natural experiment
Some 100 k or so to the West of New-Orleans ,in the 1970’s 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
New Orleans
~20k
Wax-Lake Delta

6. Can the experience of Wax Lake be transported to the Bird’s 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

7. Examples of Sediment Deltas
Water and
sediment input
Sediment Fans
1km

8. The delta shoreline is a moving boundary
Advanced in time due to sediment input
sediment flux
Land
Water
advancing
shore-line
profile view
water
land

9. 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 ?

10. In a Laboratory setting with constant flow discharge and shallow depth
Momentum Balance +
Drag
And when coupled to the Sediment Transport Law
(assuming bed shear >> Sheild’s stress)

11. The Swenson Analogy—Melting and Shoreline Movement
Stefan Melting Problem T
Water and Sediment
line discharges
Shore-line Advance
no subsidence or
sea-level change
Latent heat increases
in space
Shore-line condition

12. 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. 427–443
Provide
sediment line-flux mm2/s
water line-discharge mm2/s

13. Governing Equations At capacity transport 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

14. 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

15. Slope Ratio

16. Experiment vs. Analytical: VALIDATION
J. Fluid Mech. (2009), vol. 628, pp. 427–443
predicted fluvial
surface
experimental
analytical
Get fit by choosing
diffusivity from
Geometric measurements
From one exp.
snap-shot

17. In field setting
Value of slope ratio R controls “sensitivity” of fronts
High R
lower R
J. Fluid Mech. (2009), vol. 628, pp. 427–443

18. Common Field observation
Lower than expected curvature
for fluvial surface
experimental
analytical 18

19. Suggests a non-linear diffusive model
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 >> Sheild’s stress)
Suggests a non-linear diffusive model 19

20. Non-Linear diffusion model
J. Lorenzo-Trueba, V.R. Voller
J. Math. Anal. Appl. 366 (2010) 538–549
also has sim. sol but requires
numerical solution
Closed form only when
geometric wedge

21. Not until you reach high values of R do you see any real difference
Linear
Geometric
J. Lorenzo-Trueba, V.R. Voller
J. Math. Anal. Appl. 366 (2010) 538–549
Not until you reach high values of R
do you see any real difference R

22. Back to lack of curvature in Experiments
“Jurasic Tank” Experiment at close to steady state
Diffusion solution “too-curved”
subsidence 22

23. Is this equation valid ~3m Not a slot Model Exp.
Heterogeneity occurs at all scales
Up to an including the domain.
REV can not be identified
Exp.
~3m
Clear separation between scale of
heterogeneity and domain.
An REV can be identified
Not a slot
Volume over which average properties
can be applied globally.

24. x Model Exp 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)

25. 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 25

26. Our first attempt is based on the left hand Caputo derivative
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
Note
The divergence of a non-local fractional flux
Solution 26

27. Clearly Not a good solution
expected
predicted 27

28. Our second attempt is based on the right hand Caputo derivative
With
LOOKS DOWN-STREAM
Note
Solution
On [0,1] 28

29. Has “correct behavior”
Right-Hand Caputo
Looks like this
Has “correct behavior”
When we scale to
The experimental setup
We get a good match 29

30. And when a fraction flux is used it can match the observed lack of curvature
Voller and Paola
JGR
(to appear)
Right 30

31. But the question remains
Is this physically meaningful ? 31

32. 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
plan view x 1
side view
The movement of the red particle
is controlled by the movement of
the green particle at the
chain head –a movement controlled
by the slope at
the green particle
So at a given cross section x we can write
down a the flux as a weighted average of
the down-stream slopes

33. If we choose power law-weights
And take limit as
With change in variable
With simple mined particle chain model Flux ix given by the Right-Hand Caputo

34. Basic diffusion models can lead to
interesting math and reproduce
experiments
Fractional diffusion can predict
observed low curvature
A simple minded model
can provide a physical rational
for fractional model based on
down stream control of flux

35. Thanks

36. 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
Nleft
Nright
Can solve fractional BVP

37. CLAIM: If steps are chosen from a
Levy distribution
Maximum negative skew,
This numerical approach will also recover Solutions to
Comparison of Monte-Carlo
and analytical
Suggest that Monte Carlo
Associated with a PDF
Could resolve multiple situations

38. 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
CLAIM: If steps are chosen from a
Levy distribution
Maximum negative skew,
This numerical approach will also recover Solutions to

39. Local balance of a Non-Local flux
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
If the fractional derivative is identified as a Caputo derivative
(there are a number of reasonable definitions) then the closed solution is

40. 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
time scale for such a process
Anomalous: Super-Diffusion
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.

41. 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
BUT On a land surface, spatial
and temporal variations
are at an “observable scale”
–at or close
to the scale of resolution
Observed
-Holdup-release events
-History dependent fluxes
~3m

42. BUT On a land surface, spatial and temporal variations
are at an “observable scale”
This is similar to situation in
a porous media—where 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

45. Errors appears when slope ratio is high A thin wedge at on-lap
observed
range R
For field condition then plot
of solution vs. R=

46. The Modeling Paradigm Approximation Assumptions Numerical Solution
Verification: Comparison
of numerical and analytical
predictions VERIFY
Numerical Approach
Phenomenological
Assumptions
Model
Limit Case
Assumptions
Analytical Solution
Physical
Process
Validation: If assumptions for
Analytical solution are consistent with
Physical assumptions In experiment
Can VALIDATE phenomenological assumptions
Isolate Key
Phenomena
Experiment