Mathematical Models of Sediment Transport Systems Vaughan R. Voller

Tetsuji Muto, Wonsuck Kim, Chris Paola, Gary Parker, John Swenson, Jorge Lorenzo Trueba, Man LiangMatt Wolinsky, Colin Stark, Andrew Fowler, Doug Jerolmack

Anomalous Diffusion at Experimental ScalesA Model of Delta Growth

10km

1m

Katrina

Bangladesh

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

Each year Louisiana loses ~44 sq k of costal wetlands

Loss of a buffer that could protect inland infrastructure

Bird’s foot

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 ?

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 ~20Kin dimension

~20k

Wax-Lake Delta

New Orleans

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?

Building Delta Models is achieved by appealing to heat and masstransfer analogies

Graphic byWonsuck Kim, UAT

1km

Sediment Fans

Examples of Sediment Deltas

Water and sediment input

The delta shoreline is a moving boundaryAdvanced in time due to sediment input

sediment flux

Land

Wateradvancingshore-line

water

land

profile view

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 fixedSloping bottom and fixed water depth

0q

Can we construct a model for this ?

shore-line moves in response to sediment input

Maintains a constantsubmarine slope

h

wq

x

holpeS

)(

d(epth)

In a Laboratory setting with constant flow discharge and shallow depth

Momentum Balance

dS~

Drag2

2~

d

qCUC w

ff

size)grain andnumber

Reynolds offunction (a

constant fC

And when coupled to the Sediment Transport Law (assuming bed shear >> Sheild’s stress)

SCq fw ][~ 21

23

23

~

sqx

h

x

hqs

32

~

2~0

+

T

2

2

x

T

t

T

0q

)t(s

dt

dsLqin

Stefan Melting Problem

The Swenson Analogy—Melting and Shoreline Movement

h

0q

2

2

x

h

t

h

dt

dssqin

Shore-line Advanceno subsidence or sea-level change

Latent heat increases in space

ssL

Water and Sediment line discharges

Shore-line condition

Apply this analogy to experiments

sediment line-flux mm2/sos

qq0

owq water line-discharge mm2/s

Provide

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

)t(sx)t(s,x

h

t

hbash

2

2

dt

dss

x

hiii sh

2

0h)iv(

0qx

h)i(

bash)ii(

Governing Equations

tssh tsba

fixed basement

At capacity transport

Four Boundary Conditions Are Needed

Note: Two moving boundaries moving in opposite directions. (1) shoreline,(2) bed-rock/alluvial transition (point on basement where sediment first deposits )

A closed form similarity solution for tracking fronts is found

21

21

tts shsh 21

21

tts baba

Where the lambdas are functions of the dimensionless variables

the slope ratio R and

s

Slope Ratio

0q

x

h

10 0 q

R

)( fieldin 1

1/

2

212

21

2 sh

shbashsh

shbash Rerferfe

erferfR

Rerferfe

eR

sh

ba

shbashsh

ba

1212

2

)( fieldin 1

21

21

tts baba

21

21

tts shsh

10 0 q

R

Slope Ratio

0q

x

h

Experiment vs. Analytical: VALIDATION

predicted fluvial surface

experimental

analytical

Get fit by choosing diffusivity fromGeometric measurementsFrom one exp. snap-shot

J. Fluid Mech. (2009), vol. 628, pp. 427–443

shs abs

High R

shs abs

lower R

In field setting 1 Value of slope ratio R controls “sensitivity” of fronts

J. Fluid Mech. (2009), vol. 628, pp. 427–443

experimental

analytical

Common Field observation

Lower than expected curvaturefor fluvial surface

h

wq

x

holpeS

)(

d(epth)

In a Laboratory setting with constant flow discharge and shallow depth

Momentum Balance

dS~

Drag2

2~

d

qCUC w

ff

size)grain andnumber

Reynolds offunction (a

constant fC

And when coupled to the sediment transport law (assuming bed shear >> Sheild’s stress)

SCq fw ][~ 21

23

23

~

sq

x

h

x

hqs

32

~0Suggests a non-linear diffusive model

Non-Linear diffusion model

J. Math. Anal. Appl. 366 (2010) 538–549

J. Lorenzo-Trueba, V.R. Voller

also has sim. sol but requiresnumerical solution

Closed form only when

largeor (linear) 0

geometric wedge

R

Linear

Geometric

Not until you reach high values of Rdo you see any real difference

2

10

J. Math. Anal. Appl. 366 (2010) 538–549

J. Lorenzo-Trueba, V.R. Voller

Diffusion solution “too-curved”

~3m

“Jurasic Tank” Experiment at close to steady state

Back to lack of curvature in Experiments

Lxdx

dhK

dh

d 0,

subsidence

~3m

Clear separation between scale of heterogeneity and domain. An REV can be identified

Volume over which average properties can be applied globally.

Heterogeneity occurs at all scalesUp to an including the domain.REV can not be identified

Exp.

Lxdx

dhK

dh

d 0,Is this equation valid

Model

Not a slot

Through use of volume averaginggeneric Advection-Diffusion transport equation will have form Processes that can be embodied into a

fractional Advection-Diffusion Equation (fADE)

Model

x

2

2

x

hK

x

hu

t

h

Exp

Transport controlled byNon-local “events” suggesting ---

path-dependence described throughhereditary integrals

Non-Gaussian behaviors with“thick” power-law tails allowingfor occurrence of extreme events

h

hK

xx

hu

t

h

fractional flux depends on weighted averageof non-local slopes (up and down stream)

1,0

A toy problem is introduced

First we take a pragmatic approach and investigate what happens if

we replace the diffusion flux

with a fractional flux

10,22

2

xdx

hd

[area/time]

length/s]

Piston subsidence of base

dx

dhq 2

2

0h2)1(12 xxxh

solution

dx

dhq

10,

x

hq Will this reduce curvature ?

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.

10,10,2)(0 xhDD xC

0)1(,2)0(0 hqhDxC

With

Our first attempt is based on the left hand Caputo derivative

dx

d

dfd

dxxfD

x

xCo

01

1

The divergence of a non-local fractional flux

Solution

)1(

)1(2

)2(

)1(2)(

1

xx

xh

xxDC 21

1 xDC

0cDC

Note 10

LOOKS UPSTREAM

10,10,2)(0 xhDD xC

0)1(,2)0(0 hqhDxC

)1(

)1(2

)2(

)1(2)(

1

xx

xh

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2

Clearly Not a good solution

5.0

predicted

expected

10,10,2)( 1 xhDD Cx

0)1(,2)0(1 hqhDCx

With

Our second attempt is based on the right hand Caputo derivative

NoteSolution

dy

d

dxxyD

x

Cx )()(

)1(

1)(

1

1

)()( 101 xyDxyD xCC

x

On [0,1] 10

1)1()2(

2)(

xxh

LOOKS DOWN-STREAM

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1

10,10,2)( 1 xhDD Cx

0)1(,2)0(1 hqhDCx

1)1()2(

2)(

xxh

Looks like this Has “correct behavior”

When we scale toThe experimental setupWe get a good match

Right-Hand Caputo

5.01.0

1

5.0

1

Right

And when a fraction flux is used it can match the observed lack of curvature

Voller and PaolaJGR(to appear)

But the question remains

Is this physically meaningful ?

A simple minded model: Down stream conditions influence upstream transport

Imagine that particles transport through system as chainsThe 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

n

jj n

xjxhWq

0

1),('

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

j

n

jj n

xjxhWq

0

1),('

j

If we choose power law-weights

jW j )1(

1

And take limit as n

x

dxhq1

0

)(')1(

1

With change in variable x

)()()()1(

11

1

xhDdhd

dxq C

xx

With simple mined particle chain model Flux ix given by the Right-Hand Caputo

Basic diffusion models can lead to interesting math and reproduceexperiments

Fractional diffusion can predict observed low curvature

A simple minded model can provide a physical rationalfor fractional model based ondown stream control of flux

n

jj n

xjxhWq

0

1),('

Thanks

Shown How classic numerical heat transfer (enthalpy method) can be used to model keygeoscince problem

Illustrated how a Monte-Carlo Solutionbased on a Levy PDF

NleftNright

0)1(,1)0(

10,0

hh

dx

hd

dx

d

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1

Can solve fractional BVP

CLAIM: If steps are chosen from aLevy distribution

Maximum negative skew, 1

This numerical approach will also recover Solutions to

0)1(,1)0(

10,0

hh

dx

hd

dx

d

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1

25.0

5.0

1

xh 1

Comparison of Monte-Carloand analytical

Suggest that Monte CarloAssociated with a PDFCould resolve multiple situations

A Monte Carlo Solution

NleftNright

Tpoint = fraction of walks that exit on Left

Well know (and somewhat trivial) that a Monte Carlo simulation originating froma ‘point’ and using steps from a normal distribution will after multiple realizationsrecover the temperature at the ‘point’

CLAIM: If steps are chosen from aLevy distribution

Maximum negative skew, 25.0,1

This numerical approach will also recover Solutions to

0)1(,1)0(

10,0

hh

dx

hd

dx

d

0)1(,1)0(

0

TT

dx

dT

dx

d

As a demonstration of one-way we may go-about solving such systems let usConsider 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

0)1(,1)0(

10,0

hh

dx

hd

dx

d

If the fractional derivative is identified as a Caputo derivative(there are a number of reasonable definitions) then the closed solution is xh 1

0

0.2

0.4

0.6

0.8

1

0 0.5 1

125.0

~3m

On using results from “fractal” methods a scale independent model can be posed in termsof a fractional derivative

10,

x

hKqx

Such considerations could be important in micro-scale heat transfer-wherethe required resolution is close the scale of the mechanisms in the heat conductionProcess.

Related to a Levy PDF distributionIt has “Fat Tails” Extreme events have finite probability

time scale for such a process

21

1,1

1

t

Anomalous: Super-Diffusion

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

~3m

x

hKqx

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

BUT On a land surface, spatial and temporal variations are at an “observable scale”

This is similar to situation ina porous media—where it is known that

x

hKqx

length scale of resolution

K where the hydraulic Conductivity has a power law dependence with the scale at which it is resolved.

Modeling a reservoir at scale eLarg

using a hydraulic conductivity determined at a scale eLsmall arg

Will result in under predictionof transport

Errors appears when slope ratio is highA thin wedge at on-lap

10 0

qR

0q

For field condition then plot

of solution vs. R=

1

0q 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

observed range

sh

ba

R

0q

21

21

tts shsh

21

21

tts baba

Physical Process

Isolate Key

Phenomena

Experiment

Pheno

men

ologic

al

Assum

ption

s 2

2

x

h

t

h

Model

Approximation

Assumptions

NumericalSolution

Validation: If assumptions forAnalytical solution are consistent withPhysical assumptions In experimentCan VALIDATE phenomenological assumptions

Verification: Comparisonof numerical and analytical predictions VERIFY Numerical Approach

The Modeling Paradigm

Limit CaseAssumptions

01

12

212

21

2

shsh

shshsh

erfe

erfsh

Analytical Solution