A fractional-order Darcyʹs law

By using spatial averaging methods, in this work we derive a Darcyʹs-type law from a fractional Newtonʹs law of viscosity, which is intended to describe shear stress phenomena in non-homogeneous porous media. As a prerequisite towards this end, we derive an extension of the spatial averaging theorem for fractional-order gradients. The usage of this tool for averaging continuity and momentum equations yields a Darcyʹs law with three contributions: (i) similar to the classical Darcyʹs law, a term depending on macroscopic pressure gradients and gravitational forces; (ii) a fractional convective term induced by spatial porosity gradients; and (iii) a fractional Brinkman-type correction. In the three cases, the corresponding permeability tensors should be computed from a fractional boundary-value problem within a representative cell. Consistency of the resulting Darcyʹs-type law is demonstrated by showing that it is reduced to the classical one in the case of integer-order velocity gradients and homogeneous porous media.