par De Wit, Anne
Référence Physics of fluids, 7, 11, page (2553-2562)
Publication Publié, 1995
Article révisé par les pairs
Résumé : A theory is given in which the effective permeability tensor K eff of heterogeneous porous media is derived by a perturbation expansion of Darcy's law in the variance σ2 of the log-permeability ln[K(r)]. The only assumption is that the spatially varying permeability K(r) is a stationary random function of position. The effective permeability obtained is expressed in terms of the moments of the distribution of ln[K(r)], i.e. Keff can formally be computed for any given distribution of the fluctuations of the log-permeability. The explicit dependence of Keff on multi-point statistics is given for non-gaussian log-permeability fluctuations up to order σ6. As a special case of the theory, we examine Keff for a normal distribution function for both isotropic and anisotropic media. In the case of three-dimensional isotropic porous media, a conjecture has been made in the past according to which the scalar effective permeability Keff = K Gexp[σ2/6] where KG is the geometric mean of the log-permeability. It is shown here that this conjecture is incorrect as the σ6-order term of Keff contains additional terms than those corresponding to the development of the above formula. Moreover, these additional terms depend on the structure of the two-point correlation function of ln[K]. The resulting Keff computed for both Gaussian and exponentially decaying covariances lies below the exponential formula. This result might suggest the exponential formula as being an upper bound for K eff. For anisotropic systems, Keff is given up to the σ4-order for the general case where the mean flow is arbitrarily oriented with regard to the axes of stratification.