Length scales relating the fluid permeability and electrical conductivity in random two-dimensional model porous media.

Abstract

We present results of a study testing proposed length scales l relating the bulk electrical conductivity \ensuremath{\sigma} of a fluid-saturated porous medium to its permeability k, via the relation k\ensuremath{\propto}${\mathit{l}}^{2}$\ensuremath{\sigma}/${\mathrm{\ensuremath{\sigma}}}_{0}$, where ${\mathrm{\ensuremath{\sigma}}}_{0}$ is the fluid conductivity. For a class of two-dimensional model random porous media, we compute \ensuremath{\sigma}, k, and the following three length scales: h, the ratio of pore volume to pore surface area; \ensuremath{\Lambda}, as defined by electrical-conductivity measurements; and ${\mathit{d}}_{\mathit{c}}$, as defined by a nonwetting-fluid-injection experiment. Over a range of two and half decades in k, we find that both \ensuremath{\Lambda} and ${\mathit{d}}_{\mathit{c}}$ are reasonably good predictors of k, whereas h clearly fails. We also examine differences between the electric fields and the fluid-flow fields for a given pore structure by comparing their respective two-point correlation functions. The length scale \ensuremath{\Lambda} is analytically related to an electric-field correlation length, and is found, to a good approximation, to be proportional to a fluid-velocity correlation length. The results of this paper demonstrate the important effect that spatial randomness in the pore space has on flow problems. In a random pore structure, with a distribution of pore sizes, the flow will tend to go more through the largest pore necks, decreasing the importance of the narrowest necks that tend to dominate the behavior of periodic models.