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A fractional-order Darcy's law

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  • Ochoa-Tapia, J. Alberto
  • Valdes-Parada, Francisco J.
  • Alvarez-Ramirez, Jose

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.

Suggested Citation

  • Ochoa-Tapia, J. Alberto & Valdes-Parada, Francisco J. & Alvarez-Ramirez, Jose, 2007. "A fractional-order Darcy's law," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 374(1), pages 1-14.
    • Handle: RePEc:eee:phsmap:v:374:y:2007:i:1:p:1-14
    DOI: 10.1016/j.physa.2006.07.033
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    References listed on IDEAS

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    1. Meerschaert, Mark M. & Mortensen, Jeff & Wheatcraft, Stephen W., 2006. "Fractional vector calculus for fractional advection–dispersion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 367(C), pages 181-190.
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