A volume averaging approach for asymmetric diffusion in porous media

Asymmetric diffusion has been observed in different contexts, from transport in stratified and fractured porous media to diffusion of ions and macromolecular solutes through channels in biological membranes. Experimental and numerical observations have shown that diffusion is facilitated in the direction of positive void fraction (i.e., porosity) gradients. This work uses the method of volume averaging in order to obtain effective medium equations for systems with void fraction gradients for passive and diffusive mass transport processes. The effective diffusivity is computed from the solution of an associated closure problem in representative unit cells that allow considering porosity gradients. In this way, the results in this work corroborate previous findings showing that the effective diffusivity exhibits important directional asymmetries for geometries with void fraction gradients. Numerical examples for simple geometries (a section with an obstacle and a channel with varying cross section) show that the diffusion asymmetry depends strongly on the system configuration. The magnitude of this dependence can be quantified from the results in this work.