A homogenised model for dispersive transport and sorption in a heterogeneous porous medium

When a fluid carrying a passive solute flows quickly through porous media, three distinct mechanisms of transport 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 idealised microstructures, we can use the mathematical framework of homogenisation theory to examine this dependence. Here, we consider a two-dimensional microstructure comprising an array of obstacles of smooth but arbitrary shape, the size and spacing of which can vary along the length of the porous medium. We use homogenisation via the method of multiple scales to systematically upscale a novel problem involving cells of varying area to obtain effective continuum equations for macroscale flow, transport and sorption. The equations are characterized by the local porosity, a local anisotropic flow permeability, 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 promotive effect of dispersion. To illustrate the matematical model, we focus on a simple example geometry comprising circular obstacles on a hexagonal lattice, for which we numerically determine the macroscale permeability and coefficient of effective diffusivity.