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

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 ∇H and the specific flow q̄, the effective conductivity Keff, and the spectral density tensors of the ∇H and q̱ fields. We find that the amplitudes of ∇H and q̱ have lognormal distribution and their common orientation is distributed like Brownian motion on the unit sphere in RD, at a ‘time’ that depends on the variance of ln(K) and the space dimension D. Under ergodicity conditions, we obtain Keff=E[K]exp{−(1/D)σln(K)2}, which is the formula conjectured by Matheron [Elements Pour Une Theorie Des Millieux Poreaux, 166 pp]. Our spectral density tensors of ∇H and q̱ differ from the classical linear perturbation results in that they become more isotropic as the wavenumber amplitude k=|k̄| 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.