A Homogenized Model for Dispersive Transport and Sorption in a Heterogeneous Porous Medium
Lucy C. Auton, Mohit P. Dalwadi, Ian M. Griffiths · SIAM Journal on Applied Mathematics · 2025
Abstract. When a fluid carrying a passive solute flows quickly through porous media, three key macroscale transport mechanisms occur. These mechanisms are diffusion, advection, and dispersion, all of which depend on the microstructure of the porous medium; however, this dependence remains poorly understood. For idealized microstructures, one can use the mathematical framework of homogenization to examine this dependence, but heterogeneous materials are more challenging. Here, we consider sorption of a solute by a two-dimensional microstructure comprising an array of adsorbent obstacles of smooth but arbitrary shape, the size and spacing of which can vary along the length of the porous medium. We use homogenization via the method of multiple scales to systematically upscale a microscale problem involving nonperiodic cells of varying area to obtain effective continuum equations for macroscale transport and sorption. The equations are characterized by the local porosity, an effective local adsorption rate, and an effective local anisotropic solute diffusivity. All of these macroscale properties depend nontrivially on the two degrees of microstructural geometric freedom in our problem; obstacle size, and obstacle spacing. Further, the coefficient of effective diffusivity comprises the molecular diffusivity, the suppressive effect of the presence of obstacles, and the enhancing effect of dispersion. To illustrate the mathematical model, we focus on a simple example geometry comprising circular obstacles on a hexagonal lattice, for which we numerically determine the macroscale permeability and effective diffusivity. We find a power law for the dispersive component of solute transport, consistent with classical Taylor dispersion.