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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: - PowerPoint PPT Presentation
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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
-0.08
-0.07
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-0.02
-0.01
00
20
40
60
80
100
120
140
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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00
20
40
60
80
100
120
140
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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-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
00
20
40
60
80
100
120
140
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
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
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
0.3
0.4
0.5
0.6
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
0.5
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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-0.02
-0.01
00
20
40
60
80
100
120
140
0 500 1000 1500 2000 2500 3000
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
-0.08
-0.07
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-0.05
-0.04
-0.03
-0.02
-0.01
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
20
40
60
80
100
120
140
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
0.9
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