| Popis: |
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 q , the effective conductivity K eff , and the spectral density tensors of the ∇ H and q fields. We find that the amplitudes of ∇ H and q 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 ]exp{−(1/ D ) σ ln( 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 q differ from the classical linear perturbation results in that they become more isotropic as the wavenumber amplitude k=| k | 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. |
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