Dependence of the electrical conductivity on saturation in real porous media

Archie's Law for the porosity and saturation dependence of electrical conductivity, sigma(theta) = a sigma(m)(b phi)(theta/phi)(n) (where sigma is electrical conductivity, the subscript b denotes the brine or bulk solution, phi is porosity, theta is volume wetness, and a, m, and n are fitting parameters) was recently derived by applying continuum percolation theory to fractal porous media. We have recast Archie's Law in terms of saturation alone to obtain sigma(theta) = sigma(0) (theta - theta(c))(mu), where theta(c) is the critical volume fraction for percolation, and sigma(0) = a sigma(b)/(1 - theta(c))(mu). The value of the exponent, mu = 2.0 for three-dimensional systems and 1.28 for two, is consistent with theory and simulations. We examined the universality of the exponent's value, and the range of validity of our expression. Drawing on published data, we compared predicted and measured values of sigma(theta) across the full range of saturation, and found that the newly derived expression provides good predictions, is robust with respect to secondary effects such as residual salinity and contact resistance, and yields meaningful physical parameters.