Physica A 373 (2007) 339–353 Effective medium equations for fractional Fick’s law in porous media Francisco J. Valdes-Parada à , J. Alberto Ochoa-Tapia, Jose Alvarez-Ramirez Departamento de Ingenierı´a de Procesos e Hidrau´lica, Universidad Auto´noma Metropolitana-Iztapalapa, Aparatado Postal 55-534, Mexico D.F. 09340, Mexico Received 8 February 2006; received in revised form 26 May 2006 Available online 10 July 2006 Abstract 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 Fick’s 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. r 2006 Elsevier B.V. All rights reserved. Keywords: Non-Fickian diffusion; Fractional calculus; Reaction–diffusion; Porous media; Effective medium equations; Pseudo- homogeneous equations 1. Introduction Chemical reaction is commonly coupled to transport phenomena in many natural and industrial systems. At certain length scales, an equation widely used to describe the process is the traditional reaction–diffusion equation section qc qt ¼ D= 2 c þ RðcÞ, where c is a given particle concentration, D is the diffusion coefficient, and RðcÞ is a reaction rate. The underlying hypothesis behind this reaction–diffusion model is that the transport mechanisms are constitutive properties equivalent to the average of unmeasurable transport properties at arbitrarily small scales, and so the parameters of the model are scale independent. That is, the transport mechanisms are invariant under spatial scaling, implying that the model is able to describe the transport process at any spatial and time scales. This corresponds to Fickian behavior, which is valid when the particle jump size (i) is uncorrelated in time, ARTICLE IN PRESS www.elsevier.com/locate/physa 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.06.007 à Corresponding author. Tel.: +52 585 8044648; fax: +52 55 58044900. E-mail address: iqfv@xanum.uam.mx (F.J. Valdes-Parada).