Non Fickian flux for advection–dispersion with immobile periods

Abstract

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.

Mass transport with sorption in porous media

“…In [9], following [6,13,22] we proved that the mapping Id + λI 1−γ 0,+ has an inverse, the proof is completely similar when h depends on x instead of being equal to 1, so that the above results imply P m = (Id + λI…”

“…(1) was proposed at first by Schumer et al [21], who stressed the interesting asymptotic behavior of the solutions. Then, it was proved [2,9,23] to represent the density P(x, t) of a Brownian Motion subordinated to a time process accounting for immobile periods, distributed by a maximally skewed stable law of exponent γ. Similar equations, with constant coefficients and only the I 1−γ 0,+ ∂ t P(x, t) on the left hand side, were also proved to represent the evolution of the density of random walks.…”

“…Such approaches do not adapt to coefficients, that depend on the space variable. For (1), the density of the mobile and immobile tracer fractions P m and P i was found by [9] to satisfy…”

“…Bounded domains with "natural" boundary conditions were addressed in [9,12], with uniform diffusivity. In this purpose, we have to set some physical ideas that are at the basis of the reasoning leading to (4).…”

“…The idea of an immobile phase for a tracer [5], results in the Mobile-Immobile Model (MIM), assuming exchanges with the mobile fraction, described by first order kinetics. Experimental data [3] showed Break-Through-Curves with very marked heavy tails at the outlet of unsaturated columns with average pore velocity v, better described by the fractal variant of the MIM [2,9,21,23] which is (Id + λI 1−γ 0,+ )∂ t P(x, t) = ∇ · (∇DP − vP)(x, t), (1) with γ between 0 and 1, I 1−γ 0,+ being defined as follows [8,14,16].…”