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In this work, a two-dimensional time-fractional subdiffusion model isdeveloped to investigate the underlying transport phenomena evolving in abinary medium comprised of two sub-domains occupied by homogeneous material. Weutilise an unstructured mesh control volume method to validate the modelagainst a derived semi-analytical solution for a class of two-layered problems.This generalised transport model is then used to perform computationalhomogenisation on various two-dimensional heterogenous porous media. A keycontribution of our work is to extend the classical homogenisation theory toaccommodate the new framework and show that the effective diffusivity tensorcan be computed once the cell problems reach steady state at the microscopicscale. We verify the theory for binary media via a series of well-known testproblems and then investigate media having inclusions that exhibit a molecularrelaxation (memory) effect. Finally, we apply the generalised transport modelto estimate the bound water diffusivity tensor on cellular structures obtainedfrom environmental scanning electron microscope (ESEM) images for Spruce woodand Australian hardwood. A highlight of our work is that the computeddiffusivity for the heterogeneous media with molecular relaxation is quitedifferent from the classical diffusion cases, being dominated at steady-stateby the material with memory effects.

The use of a time-fractional transport model for performing computational homogenisation of 2D heterogeneous media exhibiting memory effects