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

Séminaire de Potamologie Résumé: Vaughan R Voller, Civil Engineering, University of Minnesota To first order, transport in many natural systems can be adequately described by diffusion processes. When heterogeneities are present in the system, however, a simple linear, integer diffusion treatment may not be sufficient to account for experimental and filed observations. For example in sedimentary deposition experiments, the observed fluvial surfaces are much “flatter” than those predicted with a linear diffusion model. If the length scales of heterogeneities in the system are power law distributed and occur up to the scale of the system itself, then a case can be made to model the transport with a fractional diffusion equation. A significant part of this talk some time will be taken in explaining what fractional diffusion is, why it is a good model in heterogeneous systems, and how it represents non-local behavior. Particular emphasis is placed on developing discrete analog models (inspired by observation) that provide a physical interpretation of fractional diffusion. Following these developments a simple steady state diffusion problem is used to illustrate the connections between a mathematical, probabilistic, and discrete non-local treatment of a fractional derivative. These ideas are then carried forward into to the development of a fractional diffusion models that are able to describe observations taken from experimental scale sediment transport systems and field moisture infiltration measurements.