Flow through porous media with multifractal hydraulic conductivity

Abstract

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 SlogK(k) ∝ |k|-D, 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 CK). 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 Keff as an explicit function of D, CK, and the scaling range of K.