Mass transport subject to time-dependent flow with nonuniform sorption in porous media

Abstract

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.

Mass transport with sorption in porous media

“…The mapping R τ also involves the (time) convolution of kernel τ −1 Ψ γ = τ −1 Ψ (./τ 1/γ ). That E τ → 0 and R τ → h(x)λI 1−γ 0,+ when τ → 0 was proved in [12] in this context. The theorem and Proposition 1 adapt this result to the case where D is a function of space, satisfying Hypothesis H 2 .…”

“…The interest of such a reasoning is that it applies to more general situations than that of the above mentioned results, including non-uniformly distributed trapping sites or non-uniform diffusivity. The case where the velocity depends on time whereas the diffusivity is constant and uniform was addressed in [12], where (2) was proved.The case where the diffusivity depends on the space variable x was addressed in [24], on the basis of numerical comparisons. Here, we present a proof of a more general version of (1) and (2), valid in this case.…”

“…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).…”

From particles scale to anomalous or classical convection-diffusion models with path integrals

“…For the time spent during displacements (mobile phase), several sceneries may be conceived, depending on the time of occurrence of the diffusive jump within a mobile period [t − τ, t]. Since we assume here stochastic independence between waiting times and jumps, they are equivalent in the diffusive limit [37,54]. For the sake of simplicity, we thus assume that walkers perform a single instantaneous diffusive jump at the end of each mobile period.…”

“…in [56,3,35,29,28], using mainly continuous time random walks and Laplace transforms. In the present paper, we choose primarily the MIM approach that seems better suited for modeling mass transport in porous media ( [49,4,37]). Indeed, fMIM's solutions behave as FFPE's ones at late times and as ones of the advection-dispersion equation at early times, in agreement with many heavy-tailed data.…”