  In this work, a two-dimensional time-fractional subdiffusion model is
developed to investigate the underlying transport phenomena evolving in a
binary medium comprised of two sub-domains occupied by homogeneous material. We
utilise an unstructured mesh control volume method to validate the model
against a derived semi-analytical solution for a class of two-layered problems.
This generalised transport model is then used to perform computational
homogenisation on various two-dimensional heterogenous porous media. A key
contribution of our work is to extend the classical homogenisation theory to
accommodate the new framework and show that the effective diffusivity tensor
can be computed once the cell problems reach steady state at the microscopic
scale. We verify the theory for binary media via a series of well-known test
problems and then investigate media having inclusions that exhibit a molecular
relaxation (memory) effect. Finally, we apply the generalised transport model
to estimate the bound water diffusivity tensor on cellular structures obtained
from environmental scanning electron microscope (ESEM) images for Spruce wood
and Australian hardwood. A highlight of our work is that the computed
diffusivity for the heterogeneous media with molecular relaxation is quite
different from the classical diffusion cases, being dominated at steady-state
by the material with memory effects.
