A mathematical model for colloids deposition in porous media combined with a moving boundary at the microscale: Solvability and numerical simulation

Abstract

We study a reaction-diffusion model posed on two distinct spatial scales that accounts for diffusion, aggregation, fragmentation, and deposition of populations of colloidal particles within a porous material. In this model, the macroscopic transport of the particles is described by an effective equation whose transport coefficients are determined by cell problems posed on the underlying pore scale. The internal pore geometry can change over time due to deposition or detachment of colloidal particles. We represent the evolving microstructure as solid cores whose phase boundaries can grow or shrink over time. As deposition progresses, neighbouring growing cores may come into contact, leading to local clogging of the pore space. We investigate how such evolving microstructures influence the effective transport and storage properties of porous layers. We establish basic analytical results concerning the weak solvability of the resulting multiscale evolution problem, which takes the form of a strongly non-linear parabolic system, in the non-clogging regime. For the numerical approximation of weak solutions we propose a two-scale finite element discretization. Numerical experiments illustrate how local clogging affects the effective dispersion tensor and quantify the resulting trade-off between transport efficiency and storage capacity.