Rigorous link between fluid permeability, electrical conductivity, and relaxation times for transport in porous media

A rigorous expression is derived that relates exactly the static fluid permeability k for flow through porous media to the electrical formation factor F (inverse of the dimensionless effective conductivity) and an effective length parameter L, i.e., k=L2/8F. This length parameter involves a certain average of the eigenvalues of the Stokes operator and reflects information about electrical and momentum transport. From the exact relation for k, a rigorous upper bound follows in terms of the principal viscous relation time Θ1 (proportional to the inverse of the smallest eigenvalue): k≤νΘ1/F, where ν is the kinematic viscosity. It is also demonstrated that νΘ1≤DT1, where T1 is the diffusion relaxation time for the analogous scalar diffusion problem and D is the diffusion coefficient. Therefore, one also has the alternative bound k≤DT1/F. The latter expression relates the fluid permeability on the one hand to purely diffusional parameters on the other. Finally, using the exact relation for the permeability, a derivation of the approximate relation k≂Λ2/8F postulated by Johnson et al. [Phys. Rev. Lett. 57, 2564 (1986)] is given.