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

TL;DR

This paper develops a homogenised model for dispersive transport and sorption in heterogeneous porous media, accounting for microstructural variations like obstacle size and spacing, and derives effective macroscale equations.

Contribution

It introduces a systematic homogenisation approach for non-periodic microstructures with variable obstacle size and spacing, linking microstructure to macroscale transport properties.

Findings

Effective diffusivity depends on obstacle geometry and dispersion effects.

Macroscale equations incorporate local porosity, adsorption, and anisotropic diffusivity.

Numerical example confirms power-law dispersive behavior consistent with Taylor dispersion.

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 idealised microstructures, one can use the mathematical framework of homogenisation to examine this dependence, but strongly heterogeneous materials are more challenging. 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 microscale problem involving non-periodic cells of varying area to obtain effective continuum equations for macroscale transport and sorption. The equations are characterised 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.