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 γ (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.
Lévy flights, which are Markovian continuous time random walks possibly accounting for extreme events, serve frequently as small-scale models for the spreading of matter in heterogeneous media. Among them, Brownian motion is a particular case where Fick's law holds: for a cloud of walkers, the flux is proportional to the gradient of the probability density of finding a particle at some place. Lévy flights resemble Brownian motion, except that jump lengths are distributed according to an α-stable Lévy law, possibly showing heavy tails and skewness. For α between 1 and 2, a fractional form of Fick's law is known to hold in infinite media: that the flux is proportional to a combination of fractional derivatives or the order of α − 1 of the density of walkers was obtained as a consequence of a fractional dispersion equation. We present a direct and natural proof of this result, based upon a novel definition of usual fractional derivatives, involving a convolution and a limiting process. Taking account of the thus obtained fractional Fick's law yields fractional dispersion equation for smooth densities. The method adapts to domains, limited by boundaries possibly implying non-trivial modifications to this equation.
We address the description of solutes flow with trapping processes in porous media. Starting from a small-scale model for tracer particle trajectories, we derive the corresponding governing equations for the concentration of the mobile and immobile phases within a fractal mobile-immobile model approach. We show that this formulation is fairly general and can easily take into account nonconstant coefficients and in particular space-dependent sorption rates. The transport equations are solved numerically and a comparison with Monte Carlo particle-tracking simulations of spatial contaminant profiles and breakthrough curves is proposed, so as to illustrate the obtained results.
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