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 --flux sediment deposit subsidence 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

sediment deposit subsidence One model is to assume that transport of sediment at a point is proportional to local slope -- a diffusion model In Exner balance This predicts a surface with a significant amount of curvature

--flux BUT -- experimental slopes tend to be much “flatter” than those predicted with a diffusion model Hypothesis: The curvature anomaly is due to “Non-Locality” Referred to as “Curvature Anomaly ”

ponded water local property Theory Reality—After Logsdon, Soil Science, 162, , 1997 Example 2: The Green-Ampt Infiltration Model

soil ponded water Why ? Heterogeneities fissures, lenses, worms Such a system could exhibit non-local control of flux If length scales of heterogeneities are power law distributed 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 For real or Also (on interval ) can define the right hand Caputo as Non-locality Value depends on “upstream values” Non-locality Value depends on “downstream values” Left-hand Caputo Note:

A probabilistic definition The zero drift Fokker-Plan c k 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 tail Upstream points have finite influence over long distances-- Non-local Note this distribution is associated with 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—Y Is “controlled” by slope up-stream at channel head --a NON-LOCAL MODEL Surface made up of “channels” representation of heterogeneity X flux across a small section controlled by slope at channel head

~3m Motivated by “Jurassic Tank” Experiments X Can model global advance of shoreline with a one-d diffusion equation with An “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—Y Is “controlled” by slope up-stream at channel head --a NON-LOCAL MODEL Surface made up of “channels” representation of heterogeneity Flux across Y—Y is then a weighed sum of up-stream slopes X Unroll Y Yx Gives more weight to channel Heads closer to x flux across a small section controlled by slope at channel head

Represent by a finite –difference scheme A discrete non-local conceptual model -- continued i i-1i-2 i+1-n i+2-n Scaled max. heterogeneity length scale x One possible choice is the power-law where

A discrete non-local conceptual model -- link to Caputo If A left hand Caputo If the right hand is treated as a Riemann sum we arrive at With transform

An illustration of the link between Math, Probability and Discrete non-local model Consider “trivial” steady sate equation Math Solution

Probability Solution 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.

x Probability Solution Lines: math analytical solutions

Discrete Numerical Non-Local Model --Daniel Zielinski A flux balance in each volume. Simply truncate sums “lumping” weights of heterogeneities that extend beyond h = 1 h = 0

A predicted infiltration rate Non-monotonic Not -0.5 Results calculated through to max length of het. What happens once Infiltration exceeds heterogeneity Length scale ??? Do we revert to homogeneous behavior?

Compare with Field Data data Fractional Green-Ampt Beyond het. Length scale ??

So with Math Long finite influence Probability Discrete Physical Analogy ~hereditary integral I have tried to show how fractional derivatives Can be related to descriptions of transport in heterogeneous domains through the non-local quantities Non-local values

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

But what about the Fluvial Surface problem ~3m Solution too-curved BUT Left hand DOES NOT WORK—predicetd fluvial surface dips below horizon (z=0)

~3m YY In experiment surface made up of transient 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 Use an alternative conceptua l model Y Y max channel length Results in RIGHT-HAND fractional model

So with a small value of alpha (non-locality) we reduce curvature and get closer to the experiment observation Use numerical solution of NOTE change of sign in curvature So Non-local fractional model is also successful in modeling curvature abnormality

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; [2] Schumer R, Meerschaert MM, Baeumer B. Fractional advection-dispersion equations for modeling transport at the Earth surface. Journal Geophysical Research 2009; 114. doi: /2008jf [3] Voller VR, Paola C. Can anomalous diffusion describe depositional fluvial profiles? Journal. Of Geophysical Research 2010; 115. doi: /2009jf [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: [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, bibilography

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