Cross-property relations for momentum and diffusional transport in porous media

Cross-property relations linking the fluid permeability k associated with viscous flow through a porous medium to effective diffusion properties of the medium have recently been derived. Torquato [Phys. Rev. Lett. 64, 2644 (1990)] found that k≤Dφ₁τ, where τ is the "mean survival time" associated with steady-state diffusion of "reactants" in the fluid region of diffusion coefficient D and porosity φ₁ of a porous medium containing absorbing walls (i.e., trap boundaries). Subsequently, Avellaneda and Torquato [Phys. Fluids A 3, 2529 (1991)] related k to the electrical formation factor F (inverse of the dimensionless effective electrical conductivity) and the principal (largest) diffusion relaxation time T ₁ associated with the time-dependent trapping problem, namely, k≤DT₁/F. In this study, we compute the aforementioned bounds, using an efficient first-passage-time algorithm, for grain-consolidation models of porous media and compare them to exact results for these models. We also conjecture a new relation connecting k to τ and F for a wide class of porous media, namely, k≤Dτ/F, and show that it gives the sharpest permeability estimate among the existing bounds. The importance of this relation lies not only in its usefulness as an estimator of the permeability but that it involves the diffusional parameters τ and F which can be measured in situ.