Geometrical constraints for membrane polarization
Membrane polarization is one process that might describe the causes of induced polarization in sediments at the pore scale. Here, we investigate the practical relevance of one particular model, which consists of two cylindrical pores with different radii r and lengths L, for which the impedance can be calculated analytically. We derive approximate equations that relate polarizabilities and relaxation times directly to r and L. Based on these relations and on a systematic exploration of the parameter space, we investigate under which conditions membrane polarization is relevant in the sense that it produces measurable phase shifts in the frequency range typically observed in the laboratory or at the field scale. In principle, a wide range of spectra can be obtained. Maximum phase shifts up to hundreds of milliradian can be simulated, and the characteristic time scales cover the entire range typically measured in the laboratory. We discuss some specific constraints in the context of results from mercury injection porosimetry and recently published laboratory data and show that the required geometries are not unrealistic, even if a moderate ratio between pore length and width is included as an additional condition. We conclude that membrane polarization as a possible mechanism is not limited to a particular frequency range. We also provide evidence that the pore length of the wide pore is likely to control the measured relaxation times in practical situations. The results encourage further attempts to combine impedances of two-pore systems to approach the simulation of real sediments, with the aim to extract pore space parameters from measured data.