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

1. 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

2. Example 1: Models of Fluvial Profiles in an Experimental Earth Scape Facility ~3m --flux 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 sediment deposit subsidence

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

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

5. Example 2: The Green-Ampt Infiltration Model Theory ponded water local property Reality—After Logsdon, Soil Science, 162, 233-241, 1997

6. Why ? ponded water Heterogeneities fissures, lenses, worms soil 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

7. Probable Cause: is heterogeneity in the soil Possible Solution is Fractional Calculus Non-locality Value depends on “upstream values” Non-locality Value depends on “downstream values” Left-hand Caputo The 1-alpha fractional integral of the first derivative of h For real or Also (on interval ) can define the right hand Caputo as Note:

8. 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 –if maximally skewed to the right Describes the spreading of an a-stable Lévy distribution power law thick tail Upstream points have finite influence over long distances-- Non-local exponential decaying tail Note this distribution is associated with the left-hand Caputo

9. A discrete non-local conceptual model Surface made up of “channels” representation of heterogeneity Assumption flux across a given part of Y—Y Is “controlled” by slope up-stream at channel head --a NON-LOCAL MODEL Y flux across a small section controlled by slope at channel head X Y

10. ~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

11. A discrete non-local conceptual model IT is just a Conceptual Model Surface made up of “channels” representation of heterogeneity Assumption flux across a given part of Y—Y Is “controlled” by slope up-stream at channel head --a NON-LOCAL MODEL Y X Unroll Y Flux across Y—Y is then a weighed sum of up-stream slopes Y flux across a small section controlled by slope at channel head Gives more weight to channel Heads closer to x x Y

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

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

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

15. 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.

16. Probability Solution x Lines: math analytical solutions

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

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

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

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

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

22. 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)

23. ~3m Use an alternative conceptual model Y Y Y Y max channel length 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 Results in RIGHT-HAND fractional model

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

25. Concluding Comments Inconsistent measurement data is a modler's dream--- Any model works on a selection of the data 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 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.

26. A bibilography http://en.wikipedia.org/wiki/Fractional_calculus [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.

27. Thank You