Non Fickian flux for advection-dispersion with immobile periods

The fractal mobile-immobile model (MIM) is intermediate between advection dispersion (ADE) and fractal Fokker-Planck (FFKPE) equations. It involves two time derivatives, whose orders are 1 and gamma ( between 0 and 1) on the left-hand side, whereas all mentioned equations have identical right-hand sides. The fractal MIM model accounts for non-Fickian effects that occur when tracers spread in media because of through-flow, and can get trapped by immobile sites. The solid matrix of a porous material may contain such sites, so that non-Fickian spread is actually observed. Within the context of the fractal MIM model, we present a mapping that allows the computation of fluxes on the basis of the density of spreading particles. The mapping behaves as Fickian flux at early times, and tends to a fractional derivative at late times. By means of this mapping, we recast the fractal MIM model into conservative form, which is suitable to deal with sources and bounded domains. Mathematical proofs are illustrated by comparing the discretized fractal p.d.e. with Monte Carlo simulations.