Numerical estimation of electrical conductivity in saturated porous media with a 2-D lattice gas

A 2-D lattice gas is used to calculate the effective electrical conductivity of saturated porous media as a function of porosity and conductivity ratio Rc. between the pore-filling fluid and the solid matrix for various microscopic structures of the pore space. The way the solid phase is introduced allows the porosity ϕ to take any value between 0 and 1 and the geometry of the pore structure to be as complex as desired. The results are presented in terms of the formation factor F = σw/σr, with σr the effective conductivity of the saturated rock and σw the conductivity of the fluid. It is shown that the formation factor F as a function of the porosity ϕ follows a power law F = aϕ−m, equivalent to the empirical Archie's law. The exponent m varies with the microgeometry of the pore space and could therefore reflect the microstructure at the macroscopic scale. The prefactor a of the power law, however, is close to 1 regardless of the microstructure. For a given microgeometry of the pore space, the variation of the residual electrical conductivity of the solid matrix induced by a finite conductivity ratio Rc does not significantly influence the variation of the effective conductivity of the fluid-solid binary mixture unless the porosity is low.