Effective medium equations for fractional Fickʹs law in porous media

This paper studies reaction–diffusion phenomena in disordered porous media with non-Fickian diffusion effects. The aim is to obtain an effective medium equation of the concentration dynamics having a fractional Fickʹs law description for the particles flux. Since the methodology is based on a volume averaging approach, a fractional spatial averaging theorem is developed to interchange averaging integration and fractional differentiation. Model structure simplifications are made on the basis of an order of magnitude analysis from physical insights. The closure problem associated with the effective diffusivity definition is also developed, showing that the macroscale diffusion parameter is affected by (i) the scaling from mesoscales to macroscales, and (ii) by the disordered structure of the porous medium.