Processes related to the membrane-like behaviour of pore throats are generally considered relevant for the induced polarization of granular media. A model to describe the impedance of membrane polarization analytically was suggested previously for a 1-D system, consisting of a sequence of narrow and wide pores of different lengths. In that model, the membrane effect is caused by different mobilities of the ion species, resulting in different ion transference numbers in the two pore types. We extend the model by explicitly including the properties of the electrical double layer and the pore radii into the impedance calculation. Our basic assumption is that in a cylindrical pore system, the ion concentrations in the direction perpendicular and parallel to the pore are independent of each other. We then obtain concentrations averaged over the pore section from known theories of the charge distribution in a cylindrical pore. We also include the effect of the Stern layer in form of a partition coefficient. Our extension enables us to use the analytical expression of the impedance, but with an extended parametrization of the transference numbers. We demonstrate the validity of our approach by comparing the results of our calculations with previously published numerical simulations for a particular model. The usefulness of our method is illustrated with parameter studies that illustrate the possible dependence of relaxation times on pore radii.

Water‐saturated porous media exhibit a low‐frequency (<1 MHz) dispersion of the electrical conductivity caused by the polarization of the electrical double layer (EDL) coating the charged solid‐liquid interface. We develop a mathematical framework describing the polarization caused by field‐induced perturbations of the ion densities in the Stern and the diffuse layer of the EDL for two different geometrical configurations of solid and liquid phase. For spherical grains immersed in an electrolyte we derive an improved analytical description by combining suitable models for diffuse‐ and Stern‐layer polarization. The selected models differ from those usually used in geophysical literature and improve the agreement with the corresponding finite‐element solution significantly. We then employ the validated finite‐element model to examine the EDL in a pore‐constriction geometry, which is often used to study membrane polarization. Here, a suitable analytical model can only be set up for a pure diffuse‐layer polarization. The results for the coupled Stern‐ and diffuse‐layer polarization in both geometries indicate that (1) the polarization of the Stern layer is much stronger than the polarization of the diffuse layer as long as the EDL is not connected at the system scale; (2) this dominance of the Stern‐layer polarization can be observed in both geometries, but (3) the contribution of the diffuse layer increases with increasing compaction as represented by the pore‐constriction geometry; and (4) the contributions of both parts of the EDL reach similar levels, when the EDLs on different surfaces are interconnected at the system scale.

We have developed an extension of the theory and mathematical description of the commonly used Wong model of electrode polarization. The full solution of the Poisson-Nernst-Planck system is provided, including three additional unknown coefficients, which describe the microscale polarization response at the surface of a perfectly conducting particle. The model involves two simultaneously acting polarization mechanisms. The first is related to the dynamic charging of diffuse layers induced over the poles of the particles, which relaxes on a time scale proportional to the particle radius and Debye length of the electrolyte solution. The second is a volume-diffusion mechanism, which is activated by reaction currents through the solid-liquid interface, i.e., faradaic currents due to charge-transfer reactions. The relaxation time of the resulting concentration polarization around the particle increases quadratically with the particle radius and the fraction of electroactive cations in the electrolyte. Although diffuse-layer polarization dominates the effective response for small particles and in the absence of reactive cations, volume-diffusion polarization only affects the macroscopic behavior if the particle size exceeds a critical value. From the closed-form analytic expressions for both relaxation times that we used, we derive a critical particle radius that depends on Debye length and reactive cation concentration. Based on the improved understanding of the underlying polarization processes, we also review some recent conceptual models of metallic polarization and correct several physical misconceptions inherent to these reinterpretations of the classic theory.

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.

In floodplain environments, a naturally reduced zone (NRZ) is considered to be a common biogeochemical hot spot, having distinct microbial and geochemical characteristics. Although important for understanding their role in mediating floodplain biogeochemical processes, mapping the subsurface distribution of NRZs over the dimensions of a floodplain is challenging, as conventional wellbore data are typically spatially limited and the distribution of NRZs is heterogeneous. In this study, we present an innovative methodology for the probabilistic mapping of NRZs within a three‐dimensional (3‐D) subsurface domain using induced polarization imaging, which is a noninvasive geophysical technique. Measurements consist of surface geophysical surveys and drilling‐recovered sediments at the U.S. Department of Energy field site near Rifle, CO (USA). Inversion of surface time domain‐induced polarization (TDIP) data yielded 3‐D images of the complex electrical resistivity, in terms of magnitude and phase, which are associated with mineral precipitation and other lithological properties. By extracting the TDIP data values colocated with wellbore lithological logs, we found that the NRZs have a different distribution of resistivity and polarization from the other aquifer sediments. To estimate the spatial distribution of NRZs, we developed a Bayesian hierarchical model to integrate the geophysical and wellbore data. In addition, the resistivity images were used to estimate hydrostratigraphic interfaces under the floodplain. Validation results showed that the integration of electrical imaging and wellbore data using a Bayesian hierarchical model was capable of mapping spatially heterogeneous interfaces and NRZ distributions thereby providing a minimally invasive means to parameterize a hydrobiogeochemical model of the floodplain.