Long and short narrow pore models for membrane polarization

Abstract

Many recent approaches to derive hydraulic soil and rock parameters from spectral induced polarization data are based on the determination of the characteristic time of measured impedance spectra. The characteristic time, sometimes also termed relaxation time, depends on the characteristic length scale of the pore geometry. One commonly used theory describing the complex electrical impedance of porous media is the membrane polarization. The pore space is modeled as a sequence of wide and narrow pores with different anion and cation mobilities. A question of particular interest is the dependence of the characteristic time on the pore lengths. The dominance of diffusion processes leads to a quadratic dependence of the characteristic time on the length scale of the pore system. However, analytical and numerical studies revealed an inconsistency concerning the pore length that determines the characteristic time. Whereas the analytical short narrow pore (SNP) model predicts a dependence on the length of the narrow pore, the length of the wide pore dominates in various numerical studies. We propose a new SNP model, which involves fewer assumptions concerning the ionic mobilities. Our new SNP model is a special case of the Cole-Cole impedance. Its characteristic time is controlled by the length of the narrow pore. A second model is derived for the case of long narrow pores and termed the long narrow pore model. The resulting impedance compares, but is not equal to a special case of the Cole-Davidson impedance and its characteristic time varies with the length of the wide pore. Both new models are special cases of the Marshall-Madden impedance for membrane polarization. The limitation of their respective validity with respect to certain combinations of mobilities and pore lengths, allows for the coexistence of both behaviors and resolves the apparent contradiction between theoretical and numerical studies.