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Flow through porous media with multifractal hydraulic conductivity
Author(s) -
Veneziano Daniele,
Essiam Albert K.
Publication year - 2003
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/2001wr001018
Subject(s) - multifractal system , isotropy , scaling , random field , anisotropy , mathematics , hydraulic conductivity , porous medium , mathematical analysis , statistical physics , physics , geometry , fractal , statistics , geology , porosity , geotechnical engineering , optics , soil science , soil water
We make a nonlinear analysis of flow through saturated porous media when the hydraulic conductivity K is an isotropic lognormal field with multifractal scale invariance. In this case, logK is isotropic Gaussian with spectral density , where D is the space dimension. Our main result is that the hydraulic gradient ∇H and specific flow q are also multifractal fields, whose renormalization under space contraction involves random rotation of the field and random scaling of its amplitude. The scaling properties and marginal distributions of ∇H and q are obtained analytically as functions of the space dimension D and a multifractal parameter of K (the codimension C K ). The fields ∇H and q are anisotropic at large scales but approach isotropy at very small scales. Using scaling arguments, we obtain the effective conductivity of the medium K eff as an explicit function of D, C K , and the scaling range of K.