  A major challenge in flow through porous media is to better understand the
link between 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 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 and transport. The equations are characterised by
the local porosity, a local anisotropic flow permeability, an effective local
anisotropic solute diffusivity, and an effective local adsorption rate. These
macroscale properties depend nontrivially on the two degrees of microstructural
geometric freedom in our problem: obstacle size, and obstacle spacing. We
exploit this dependence to construct and compare scenarios where the same
porosity profile results from different combinations of obstacle size and
spacing. We focus on a simple example geometry comprising circular obstacles on
a rectangular lattice, for which we numerically determine the macroscale
permeability and effective diffusivity. We investigate 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.
