Nonlinear spectral analysis of flow through porous media with isotropic lognormal hydraulic conductivity

摘要

We extend previous work on flow through D-dimensional random porous media with isotropic, lognormal and multifractal hydraulic conductivity K to include all (not necessarily multifractal) isotropic lognormal K fields. We use an approximate nonlinear analysis method to obtain the marginal distributions of the hydraulic gradient delH and the specific flow (q) under bar, the effective conductivity K-eff, and the spectral density tensors of the delH and (q) under bar fields. We find that the amplitudes of delH and (q) under bar have lognormal distribution and their common orientation is distributed like Brownian motion on the unit sphere in R-D, at a 'time' that depends on the variance of ln(K) and the space dimension D. Under ergodicity conditions, we obtain K-eff = E[K] expt{-(1/D)sigma(ln(K))(2)}, which is the formula conjectured by Matheron [Elements Pour Une Theorie Des Millieux Poreaux, 166 pp]. Our spectral density tensors of delH and (q) under bar differ from the classical linear perturbation results in that they become more isotropic as the wavenumber amplitude k = (k) under bar increases. A second difference, which becomes noticeable when log(K) has a broad-band spectrum and high variance, is that the nonlinear spectra decay more slowly with k. The theoretical results are confirmed through two-dimensional simulations. (C) 2004 Elsevier B.V. All rights reserved.