Correlation structure dependence of the effective permeability of heterogeneous porous media

A theory is given in which the effective permeability tensor Keff of heterogeneous porous media is derived by a perturbation expansion of Darcy’s law in the variance σ2 of the log‐permeability ln[κ(ub;;‐45rubx)]. The only assumption is that the spatially varying permeability κ(ub;;‐45rubx) is a expressed in terms of the moments of the distribution of ln[κ(ub;;‐45rubx)], 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 κeff=KGexp[σ2/6] where KG is the geometric mean of the log‐permeability. It is shown here that this conjecture is incorrect as the σ6‐o...

[1]  Andrew F. B. Tompson,et al.  Numerical simulation of solute transport in three-dimensional, randomly heterogeneous porous media , 1990 .

[2]  Peter K. Kitanidis,et al.  Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach: 1. Method , 1992 .

[3]  Allan L. Gutjahr,et al.  Stochastic analysis of spatial variability in subsurface flows: 1. Comparison of one‐ and three‐dimensional flows , 1978 .

[4]  Gedeon Dagan,et al.  A Note on Higher‐Order Corrections of the Head Covariances in Steady Aquifer Flow , 1985 .

[5]  B. Noetinger,et al.  Experimental Tests of a Simple Permeability Composition Formula , 1991 .

[6]  S. P. Neuman,et al.  Prediction of steady state flow in nonuniform geologic media by conditional moments: Exact nonlocal , 1993 .

[7]  P. Indelman,et al.  Effective permittivity of log-normal isotropic random media , 1995 .

[8]  Gedeon Dagan,et al.  Higher-order correction of effective permeability of heterogeneous isotropic formations of lognormal conductivity distribution , 1993 .

[9]  J. R. Macmillan,et al.  Stochastic analysis of spatial variability in subsurface flows: 2. Evaluation and application , 1978 .

[10]  P. Indelman,et al.  A higher‐order approximation to effective conductivity in media of anisotropic random structure , 1994 .

[11]  Peter K. Kitanidis,et al.  Analysis of the Spatial Structure of Properties of Selected Aquifers , 1985 .

[12]  E. M. Lifshitz,et al.  Electrodynamics of continuous media , 1961 .

[13]  J. Bear.  Dynamics of Fluids in Porous Media , 1975 .

[14]  R. Freeze.  A stochastic‐conceptual analysis of one‐dimensional groundwater flow in nonuniform homogeneous media , 1975 .

[15]  G. Dagan.  Flow and transport in porous formations , 1989 .

[16]  C. Axness,et al.  Three‐dimensional stochastic analysis of macrodispersion in aquifers , 1983 .

[17]  J. C. Griffiths,et al.  A Stochastic Model for Predicting Variations in Reservoir Rock Properties , 1966 .

[18]  E. Sudicky.  A natural gradient experiment on solute transport in a sand aquifer: Spatial variability of hydraulic conductivity and its role in the dispersion process , 1986 .