An attempt to provide a physical interpretation of fractional transport in heterogeneous domains

Vaughan Voller

Department of Civil Engineering and NCED University of Minnesota

With Key inputs from Chris Paola, Dan Zielinski, and Liz Hajek

Themes:

Heterogeneity can lead to interesting non-local effects that Confound our basic models

Some of these non-local effects can be successfully modeled with fractional calculus

Here: I will show Two geological examples and try and develop an “Intuitive” physical links between the mathematical and statistical nature of fractional derivatives and field and experimental observations

Example 1: Models of Fluvial Profiles in an Experimental Earth Scape Facility

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0 500 1000 1500 2000 2500 3000

inq --flux)(mmh

)/( smm

sediment deposit

subsidence

)(mmx x

In long cross-section, through sediment deposit Our aim is to redict steady state shape and height of sediment surface above sea level for given sediment flux and subsidence

~3m

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0 500 1000 1500 2000 2500 3000

inq)(mmh

)/( smm

sediment deposit

subsidence

0)(, with

,

0

hqdxdhk

dxdhk

dxdq

dxd

in

)(mmx x

One model is to assume that transport of sediment at a point is proportional to local slope -- a diffusion model

dxdhkq

In Exnerbalance

This predicts a surfacewith a significant amount of curvature

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00

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0 500 1000 1500 2000 2500 3000

)(mmh

)/( smm

--fluxinq

x)(mmx

BUT -- experimental slopestend to be much “flatter” thanthose predicted with a diffusion model

Hypothesis:

The curvature anomaly isdue to

“Non-Locality”

Referred to as “Curvature Anomaly ”

ponded water

local property

infil

trati

on ra

te

time

Theory

Reality—After Logsdon, Soil Science, 162, 233-241, 1997

small is when ,~0.5)(,~ 1 ttsti

small is when ,~~ 5.5. ttsti

)(0 tszzhq

Example 2: The Green-Ampt Infiltration Model

soil

ponded water

zhq

Why ?

Heterogeneitiesfissures, lenses, worms

Such a system could exhibit non-local control of flux If length scales of heterogeneities are

power law distributed

10,

zhq

A fractional derivative rep. of flux may be appropriate

Probable Cause: is heterogeneity in the soil

Possible Solution is Fractional Calculus

The 1-alpha fractional integral of the first derivative of h

10,11

0

dddhx

xhq

x

xxx 1

1 01 10 For real or

10,1

1)(

1

dddhx

xhq

x

Also (on interval ) can define the right hand Caputo as 10 x

Non-localityValue depends on“upstream values”

Non-localityValue depends on“downstream values”

Left-hand Caputo

)1()( xf

xf

Note:

A probabilistic definition

The zero drift Fokker-Planck equation describes the time evolution (spreading) of a Gaussian distribution

exponential decaying tail

A fractional form of this equation

Describes the spreading of an -stable Lévy distribution

exponential decaying tail

power law thick tailUpstream points have finite influence over longdistances-- Non-local

Note this distribution is associatedwith the left-hand Caputo

–if maximally skewed to the right

Y

Y

A discrete non-local conceptual model

Assumption flux across a given part of Y—YIs “controlled” by slope up-streamat channel head

--a NON-LOCAL MODEL

Surface made up of “channels” representation of heterogeneity

Xflux across a small sectioncontrolled by slopeat channel head

~3m

Motivated by “Jurassic Tank” Experiments

X

Can model global advance of shoreline with a one-d diffusion equation withAn “average” diffusive transport in x-direction—see Swenson et al Eur. J. App. Math 2000

But at LOCAL time and space scales –transport is clearly “channelized” and NON-LOCAL

Y

Y

A discrete non-local conceptual model IT is just a Conceptual Model

Assumption flux across a given part of Y—YIs “controlled” by slope up-streamat channel head

--a NON-LOCAL MODEL

Surface made up of “channels” representation of heterogeneity

n

j

upjjx sWkq

1

Flux across Y—Y is then a weighed sum of up-stream slopes

X

Unroll

Y

YxGives more weight to channelHeads closer to x

flux across a small sectioncontrolled by slopeat channel head

n

j

upjjx sWkq

1

Represent by a finite –difference scheme

A discrete non-local conceptual model -- continued

n

j

jIjIjx x

hhWkq

1

1

1 xn

x

ii-1i-2i+1-n i+2-n

Scaled max. heterogeneity length scale

x

One possible choice is the power-law

xxjW j ]))[(1(

1)1(101

where

0

0.1

0.2

0.3

-10 -8 -6 -4 -2 0

0

0.1

0.2

0.3

-7 -6 -5 -4 -3 -2 -1

xxhh

xjqn

j

jIjIx

1

1])[()1(

1

A discrete non-local conceptual model -- link to Caputo

)1()1(1

k

If

10,11lim

10

dddhxq

x

xx

xn

A left hand Caputo

If the right hand is treated as a Riemann sum we arrive at

dxhqnx

1

0lim)(

)1(1

xWith transform

An illustration of the link between Math, Probability and Discrete non-local model

Consider “trivial” steady sate equation

0)1(1)0( hh

xxx 1

1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

left, alpha=.5h = 1 - x.5

alpha=1h = 1 - x

right, alpha=.5h = (1 - x).5

Math Solution

1)0( h

Probability Solution

0)1( h

x

A Monte-Carlo “Race” between two particles starting random walks from boundaries

Each Step of the walk is chosen from the appropriate Lévy distribution.

The race ends when one particle reaches or moves past the target point x— a win tallied for the color of that particle.

1)0( h 0)1( h

x

Probability Solution

00.10.20.30.40.50.60.70.80.9

1

0 0.5 1

1

125.0,5.0,1

Lines: math analytical solutions

n

j

jIjIjx x

hhWkq

1

1

xxjW j ]))[(1(

Discrete Numerical Non-Local Model -- Daniel Zielinski

A flux balance in each volume. Simply truncate sums “lumping” weights of heterogeneitiesthat extend beyond

0

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1

0 0.2 0.4 0.6 0.8 1

x

T

0.7, 1

0.3, 1

0.5, 0

h = 1 h = 0

0.5

1

1.5

2

2.5

0 0.5 1

infil

tratio

n ra

te

timeb

A predicted infiltration rate

Non-monotonic

Results calculated through to

max length of het.

What happens onceInfiltration exceeds heterogeneity Length scale ???

Do we revert to homogeneous behavior?

0.2

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0 2500 5000

infil

tratio

n ra

te

time (s)a

0.5

0.9

1.3

1.7

2.1

2.5

0 2500 5000in

filtra

tion

rate

time (s)b

0.25

0.29

0.33

0.37

0.41

0.45

0 2500 5000

infil

tratio

n ra

te

time (t)c

Compare with Field Data

data

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1

1.5

2

2.5

0 0.5 1

infil

tratio

n ra

te

timeb

0.75

0.95

1.15

1.35

0 0.5 1

infil

tratio

n ra

tetime c

1.5

2

2.5

3

3.5

0 0.2 0.4

infil

traio

n ra

te

timea

Fractional Green-Ampt

Beyond het. Length scale ??

So with Math

10,11

0

dddhx

xhq

x

Long finite influence

Probability

j

n

jj xhW

xh

1

Discrete Physical Analogy

~hereditary integral

I have tried to show how fractional derivativesCan be related to descriptions of transport in

heterogeneous domains

through the non-local quantities

Non-local values

0.5

1

1.5

2

2.5

0 0.5 1

infil

tratio

n ra

te

timeb

soil

ponded water

)(0,0 tszzh

z

0)(0)0( 0

ssshhh

szzh

dtds

Based on this it has been shown that a “FRACTIONAL Green-Ampt model can match “Anomalous” Field infiltration behaviors attributed to soil heterogeneity

0.5

0.9

1.3

1.7

2.1

2.5

0 2500 5000

infil

tratio

n ra

te

time (s)b

But what about the Fluvial Surface problem

~3m

0)1(,2)0(

10,

hq

xdxdh

dxd

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Solution too-curved

10,

x

dxhd

dxd

BUT Left hand DOES NOT WORK—predicetd fluvial surface dips below horizon (z=0)

~3m

YY

In experiment surface made up oftransient channels with a wide range of length scales Assumption flux in any channel (j) crossing Y—Y

Is “controlled” by slope at down-stream channel head

x

Use an alternative conceptual model

Y

Y

downjj sq

x

max channel length

0)(,)(

with

,)(

0

*

*

hqxdhdk

xdhdk

dxd

in

Results in RIGHT-HAND fractional model

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0

0)(,)(

with

,)(

0

*

*

hqxdhdk

xdhdk

dxd

in

So with a small value of alpha (non-locality) we reduce curvature and get closer to theexperiment observation

Use numerical solution of

0

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0 500 1000 1500 2000 2500 3000

=1

=0.25

XES10

NOTE change of sign in curvature

So Non-local fractional model is also successful in modeling curvature abnormality

0.5

1

1.5

2

2.5

0 0.5 1

infil

tratio

n ra

te

timeb

0.5

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1.3

1.7

2.1

2.5

0 2500 5000

infil

tratio

n ra

te

time (s)b

Inconsistent measurement data is a modler's dream---

But fair to say that here I have demonstrated a consistency between a scheme(fractional derivative) to describe transport in heterogeneous systems and some field and experimental observations

Any model works on a selection of the data

Concluding Comments

Whereas this does not result in a predictive model it does begin to provide an understanding of the non-local physical features that may control infiltration in heterogeneous soils and fluvial sediment transport.

[1] Metzler R, Klafter J. The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Physics Reports 2000; 339 1-77. [2] Schumer R, Meerschaert MM, Baeumer B. Fractional advection-dispersion equations for modeling transport at the Earth surface. Journal Geophysical Research 2009; 114. doi:10.1029/2008jf001246. [3] Voller VR, Paola C. Can anomalous diffusion describe depositional fluvial profiles? Journal. Of Geophysical Research 2010; 115. doi:10.1029/2009jf001278. [4] Voller VR. An exact solution of a limit case Stefan problem governed by a fractional diffusion equation. International Journal of Heat and Mass Transfer 2010; 53: 5622-25.[5] Podlubny I. Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of their Solution and Some of their Applications. San Diego, Academic Press, 1998.

http://en.wikipedia.org/wiki/Fractional_calculusA bibilography

Thank You