A physical basis for the Cole-Cole description of electrical conductivity of mineralized porous media

We found a physical interpretation of the Cole-Cole relation for the conductivity of fluid-saturated, mineralized rocks. The first step was to rederive a diffusive version of the Cole-Cole relation from a pore-scale description that couples diffusion and electrochemistry in a simplified way. We wanted to give the parameters of the Cole-Cole relation, the chargeability and relaxation time, a microscopic interpretation. In this treatment, the exponent known as the frequency dependence was fixed to either c = 1/2 or c = 1 depending on the ratio of the diffusion length and effective pore length. However, many field observations give 0.1 < c < 0.5. The second step was to calculate the conductivity of a network in which the individual bonds are described by the Cole-Cole conductivity function with c = 1/2. These network simulations, which did not rely on effective medium approximations, turned out to exhibit Cole-Cole behavior as well with c taking on a continuous range of values, provided the pore-scale parameters are picked from sufficiently wide distributions. These network results agreed with observations based on field data inversion.