Scaling laws and dispersion equations for Lévy particles in one-dimensional fractal porous media

Abstract

Over some range of scales natural porous media often display a fractal Eulerian velocity or conductivity field. If one assumes the fractal conductivity gives rise to fractal drift velocities, then particle paths may be studied in the framework of stochastic differential equations (SODEs). On the microscale, trajectories are modeled as solutions to a SODE with Markovian, stationary, ergodic drift subject to a fluctuating Lévy force. The Lévy force allows for self-motile particles such as flagellated microbes. On the mesoscale the trajectories are modeled as solutions to a SODE with Lévy (fractal) drift and diffusion arising from the microscale asymptotics. On the macroscale the process is driven by the asymptotics of the mesoscale drift without diffusion. Asymptotic scaling laws and dispersion equations are presented.