[1] A model combining low‐frequency complex conductivity and high‐frequency permittivity is developed in the frequency range from 1 mHz to 1 GHz. The low‐frequency conductivity depends on pore water and surface conductivities. Surface conductivity is controlled by the electrical diffuse layer, the outer component of the electrical double layer coating the surface of the minerals. The frequency dependence of the effective quadrature conductivity shows three domains. Below a critical frequency f_p, which depends on the dynamic pore throat size Λ, the quadrature conductivity is frequency dependent. Between f_p and a second critical frequency f_d, the quadrature conductivity is generally well described by a plateau when clay minerals are present in the material. Clay‐free porous materials with a narrow grain size distribution are described by a Cole‐Cole model. The characteristic frequency f_d controls the transition between double layer polarization and the effect of the high‐frequency permittivity of the material. The Maxwell‐Wagner polarization is found to be relatively negligible. For a broad range of frequencies below 1 MHz, the effective permittivity exhibits a strong dependence with the cation exchange capacity and the specific surface area. At high frequency, above the critical frequency f_d, the effective permittivity reaches a high‐frequency asymptotic limit that is controlled by the two Archie's exponents m and n like the low‐frequency electrical conductivity. The unified model is compared with various data sets from the literature and is able to explain fairly well a broad number of observations with a very small number of textural and electrochemical parameters. It could be therefore used to interpret induced polarization, induction‐based electromagnetic methods, and ground penetrating radar data to characterize the vadose zone.

I developed a new model named POLARIS describing the complex conductivity of (pyrite‐free) shaly poorly sorted sands. This model is based on the solution given by the effective medium theory for grains coated by an electrical double layer and immersed in a background electrolyte. The electrical double layer comprises the Stern layer and the diffuse layer. Both layers play very distinct roles in the in‐phase and quadrature conductivities. The polarization of the shaly sands is mainly controlled by the polarization of the Stern layer (except at very high salinities) with a very small mobility of the counterions contained in this layer. The in‐phase component of the conductivity is controlled by the conductivity of the pore water with a contribution associated with the diffuse layer (the contribution of the Stern layer seems negligible). The fraction of counterions in the Stern layer is computed from a simple sorption isotherm and is used to infer the quadrature conductivity. The quadrature conductivity is assumed to be frequency independent, which is a reasonable approximation in clayey sands and sandstones, in agreement with observations. The polarization model is also based on the assumption that the Stern layer is discontinuous between grains, an assumption that is consistent with recent models of ionic transport in clayey sands. POLARIS explains the dependence of the quadrature conductivity on the salinity, cation exchange capacity, specific surface area (or specific surface per unit pore volume), and temperature. It can be used to predict the saturation and the permeability (inside 1 order of magnitude).

Summary. The surface conductivity of porous rocks has two contributions: the first is associated with the diffuse layer coating the grains and is frequency-ind

Water‐saturated clay‐rich media exhibit low‐frequency (1 Hz to 1 MHz) effective conductivity and effective permittivity dispersions that are the consequence of both the polarization of the mineral/water interface coating the surface of the grains and the Maxwell‐Wagner polarization. These low‐frequency properties are modeled by combining (1) a complexation model of the surface properties of clay minerals (kaolinite, illite, and smectite), (2) a polarization model of the Stern layer (the inner portion of the electrical double layer coating the surface of the minerals), and (3) a macroscopic model comprising the electrochemical polarization of the grains and the contribution of the Maxwell‐Wagner effect. The macroscopic model is based on the differential effective medium theory. It includes a convolution product with the grain size distribution. For kaolinite, the diffuse layer occupies a small fraction of the pore space and is considered as part of the surface of the grains. This is due to the low specific surface area of kaolinite. In the case of illite and smectite, the situation is different. Because of the high specific surface areas of these minerals, the diffuse layer occupies a large fraction of the pore space and is considered as part of the pore space and is described using a Donnan equilibrium model. We obtain excellent comparisons between various experimental data reported in the literature and our model. Then, we considered low‐porosity (compacted or cemented) clay rocks and shales. Here too, we obtained a good agreement between the data and the predictions of a model based on a volume‐averaging approach. We also note that at very low frequencies (<1 Hz), another polarization mechanism exists that is not reproduced by our model. We believe that this polarization corresponds to a nonlinear membrane polarization contribution.

An algorithm is developed to interpret self‐potential (SP) data in terms of distribution of Darcy velocity of the ground water. The model is based on the proportionality existing between the streaming current density and the Darcy velocity. Because the inverse problem of current density determination from SP data is underdetermined, we use Tikhonov regularization with a smoothness constraint based on the differential Laplacian operator and a prior model. The regularization parameter is determined by the L‐shape method. The distribution of the Darcy velocity depends on the localization and number of non‐polarizing electrodes and information relative to the distribution of the electrical resistivity of the ground. A priori hydraulic information can be introduced in the inverse problem. This approach is tested on two synthetic cases and on real SP data resulting from infiltration of water from a ditch.