On upscaling operator-stable Lévy motions in fractal porous media

Abstract

The dynamics of motile particles, such as microbes, in random porous media are modeled with a hierarchical set of stochastic differential equations which correspond to micro, meso and macro scales. On the microscale the motile particle is modeled as an operator stable Lévy process with stationary, ergodic, Markov drift. The micro to meso and meso to macro scale homogenization is handled with generalized central limit theorems. On the mesoscale the Lagrangian drift (or the Lagrangian acceleration) is assumed Lévy to account for the fractal character of many natural porous systems. Diffusion on the mesoscale is a result of the microscale asymptotics while diffusion on the macroscale results from the mesoscale asymptotics. Renormalized Fokker–Planck equations with time dependent dispersion tensors and fractional derivatives are presented at the macro scale.