A fractional-order Darcy's law |
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Authors: | J. Alberto Ochoa-Tapia Francisco J. Valdes-ParadaJose Alvarez-Ramirez |
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Affiliation: | División de Ciencias Basicas e Ingenieria, Universidad Autonoma Metropolitana-Iztapalapa, Apartado Postal 55-534, Mexico D.F. 09340, Mexico |
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Abstract: | 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. |
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Keywords: | Darcy's law Fractional constitutive equation Volume averaging |
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