CORRELATION STRUCTURE DEPENDENCE OF THE EFFECTIVE PERMEABILITY OF HETEROGENEOUS POROUS-MEDIA

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 sigma(2) of the log-permeability In[kappa(($) under bar r)]. The only assumption is that the spatially varying permeability kappa(($) under bar r) is a stationary random function of position. The effective permeability obtained is expressed in terms of the moments of the distribution of In[kappa(($) under bar r)], i.e. K-eff can formally be computed for any given distribution of the fluctuations of the log-permeability. The explicit dependence of K-eff on multi-point statistics is given for non-gaussian log-permeability fluctuations up to order sigma(6). As a special case of the theory, we examine K-eff 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 kappa(eff) = K(G)exp[sigma(2)/6] where K-G is the geometric mean of the log-permeability. It is shown here that this conjecture is incorrect as the sigma(6)-order term of kappa(eff) 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 In[kappa]. The resulting kappa(eff) 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 kappa(eff). For anisotropic systems, K-eff is given up to the sigma(4)-order for the general case where the mean flow is arbitrarily oriented with regard to the axes of stratification. (C) 1995 American Institute of Physics.