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 ?, where D is the space dimension. Our main result is that the hydraulic gradient ∇H and specific flow <span cssStyle="textunderlinestyle:single">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 <span cssStyle="textunderlinestyle:single">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 <span cssStyle="textunderlinestyle:single">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.
 Publication:

Water Resources Research
 Pub Date:
 June 2003
 DOI:
 10.1029/2001WR001018
 Bibcode:
 2003WRR....39.1166V
 Keywords:

 Hydrology: Groundwater hydrology;
 Hydrology: Instruments and techniques;
 subsurface flow;
 multifractality;
 scale invariance;
 random fields;
 stochastic hydrology;
 effective conductivity;
 Hydrology: Groundwater hydrology;
 Hydrology: Instruments and techniques