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

Abstract

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.

Analytical and numerical investigation on the tempered time-fractional operator with application to the Bloch equation and the two-layered problem

“…To date, Zeng et al [36] used a discrete least squares collocation method to deal with a coupled system of time-fractional diffusion equations with different fractional indices in an irregularly shaped region. Feng et al [21,37] proposed a two-dimensional time-fractional model to study the underlying transport phenomena in a binary medium based on the Riemann-Liouville fractional derivative, in which it showed that the generalised transport model could exhibit the correct physical solution behaviour and produce a more accurate overall mass balance. In this section, we focus on a two-layered problem with a tempered operator (see Fig.…”

Analytical and numerical investigation on the tempered time-fractional operator with application to the Bloch equation and the two-layered problem

“…To date, Zeng et al [34] used a discrete least squares collocation method to deal with a coupled system of time-fractional diffusion with different fractional indices in an irregularly shaped region. Feng et al [19,35] proposed a 2D time-fractional model to study the underlying transport phenomena in a binary medium based on the Riemann-Liouville fractional derivative, in which it showed that the generalised transport model could exhibit the correct physical solution behaviour and produce a more accurate overall mass balance. In this section, we focus on a two-layered problem with a tempered operator (see Figure 7):…”