Predicting permeability from the characteristic relaxation time and intrinsic formation factor of complex conductivity spectra

Low-frequency quadrature conductivity spectra of siliclastic materials exhibit typically a characteristic relaxation time, which either corresponds to the peak frequency of the phase or the quadrature conductivity or a typical corner frequency, at which the quadrature conductivity starts to decrease rapidly toward lower frequencies. This characteristic relaxation time can be combined with the (intrinsic) formation factor and a diffusion coefficient to predict the permeability to flow of porous materials at saturation. The intrinsic formation factor can either be determined at several salinities using an electrical conductivity model or at a single salinity using a relationship between the surface and quadrature conductivities. The diffusion coefficient entering into the relationship between the permeability, the characteristic relaxation time, and the formation factor takes only two distinct values for isothermal conditions. For pure silica, the diffusion coefficient of cations, like sodium or potassium, in the Stern layer is equal to the diffusion coefficient of these ions in the bulk pore water, indicating weak sorption of these couterions. For clayey materials and clean sands and sandstones whose surface have been exposed to alumina (possibly iron), the diffusion coefficient of the cations in the Stern layer appears to be 350 times smaller than the diffusion coefficient of the same cations in the pore water. These values are consistent with the values of the ionic mobilities used to determine the amplitude of the low and high-frequency quadrature conductivities and surface conductivity. The database used to test the model comprises a total of 202 samples. Our analysis reveals that permeability prediction with the proposed model is usually within an order of magnitude from the measured value above 0.1 mD. We also discuss the relationship between the different time constants that have been considered in previous works as characteristic relaxation time, including the mean relaxation time obtained from a Debye decomposition of the spectra and the Cole-Cole time constant.