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

A major challenge in flow through porous media is to better understand the
link between pore-scale microstructure and macroscale flow and transport. For
idealised microstructures, the mathematical framework of homogenisation theory
can be used for this purpose. Here, we consider a two-dimensional
microstructure comprising an array of circular obstacles, the size and spacing
of which can vary along the length of the porous medium.We use homogenisation
via the method of multiple scale to systematically upscale a novel problem that
involves cells of varying area to obtain effective continuum equations for
macroscale flow and transport. The equations are characterized by the local
porosity, an effective local anisotropic flow permeability, and an effective
local anisotropic solute diffusivity. These macroscale properties depend
non-trivially on both degrees of microstructural geometric freedom (obstacle
size and spacing). We take advantage of this dependence to compare scenarios
where the same porosity field is constructed with different combinations of
obstacle size and spacing. For example, we consider scenarios where the
porosity is spatially uniform but the permeability and diffusivity are not. Our
results may be useful in the design of filters, or for studying the impact of
deformation on transport in soft porous media.