Journal
PHYSICA A-STATISTICAL MECHANICS AND ITS APPLICATIONS
Volume 374, Issue 1, Pages 1-14Publisher
ELSEVIER
DOI: 10.1016/j.physa.2006.07.033
Keywords
Darcy's law; fractional constitutive equation; volume averaging
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By using spatial averaging methods, in this work we derive a Darcy's-type law from a fractional Newton's 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 Darcy's law with three contributions: (i) similar to the classical Darcy's 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 Darcy's-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. (c) 2006 Published by Elsevier B.V.
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