Flow through porous media with multifractal hydraulic conductivity

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.