A. Revil

Electrical conductivity in shaly sands with geophysical applications

We develop a new electrical conductivity equation based on Bussians model and accounting for the different behavior of ions in the pore space. The tortuosity of the transport of anions is independent of the salinity and corresponds to the bulk tortuosity of the pore space which is given by the product of the electrical formation factor F and the porosity ϕ. For the cations, the situation is different. At high salinities, the dominant paths for the electromigration of the cations are located in the interconnected pore space, and the tortuosity for the transport of cations is therefore the bulk tortuosity. As the salinity decreases, the dominant paths for transport of the cations shift from the pore space to the mineral water interface and consequently are subject to different tortuosities. This shift occurs at salinities corresponding to ξ/t(+)f ∼ 1, where ξ is the ratio between the surface conductivity of the grains and the electrolyte conductivity, and t(+)f is the Hittorf transport number for cations in the electrolyte. The electrical conductivity of granular porous media is determined as a function of pore fluid salinity, temperature, water and gas saturations, shale content, and porosity. The model provides a very good explanation for the variation of electrical conductivity with these parameters. Surface conduction at the mineral water interface is described with the Stern theory of the electrical double layer and is shown to be independent of the salinity in shaly sands above 10−3 mol L−1. The model is applied to in situ salinity determination in the Gulf Coast, and it provides realistic salinity profiles in agreement with sampled pore water. The results clearly demonstrate the applicability of the equations to well log interpretation of shaly sands.

Effective conductivity and permittivity of unsaturated porous materials in the frequency range 1 mHz–1GHz

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 fp, which depends on the dynamic pore throat size Λ, the quadrature conductivity is frequency dependent. Between fp and a second critical frequency fd, 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 fd 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 fd, the effective permittivity reaches a high-frequency asymptotic limit that is controlled by the two Archies 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.

Pore‐scale heterogeneity, energy dissipation and the transport properties of rocks

The pore structure of rocks is highly complex, with wide variations in pore size and shape. In this work, pore-scale heterogeneity was simulated by distributing spheres, tubes and cracks with variable dimensions on a square lattice. The transport properties of 100 such network realizations, covering 11 orders of magnitude in permeability, were calculated. Seeking the appropriate averaging procedure to calculate the permeability and electrical conductivity from the local pore parameters, we computed the energy locally dissipated in each bond during fluid or electric flow and the energy globally dissipated in the whole network. By equating the latter to the sum of the former, we obtained averaging expressions exactly predicting the transport properties of the network realizations. Since these relations hold on a wide variety of heterogeneous networks covering a broad range of permeabilities and electrical conductivities, we propose that they should also be valid on rocks. We can thus gain insights into how pore-scale heterogeneity affects the transport properties of rocks.

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.

Influence of the electrical diffuse layer and microgeometry on the effective ionic diffusion coefficient in porous media

This study models the effect of the electrical diffuse layer at the pore matrix interface on ionic diffusion processes in a porous medium in presence of a concentration gradient. We find that the equation D = Df/Fo usually used to compute in steady state conditions the effective diffusion coefficient, D, of a porous medium as a function of the salt diffusion coefficient, Df, the electrical formation factor, F, and the porosity, o, should be replaced by: D = ( Df/Fo)ξ, in the presence of an electrical diffuse layer at the pore-matrix interface. The correction factor, ξ arises from the cross-coupling of ionic fluxes due to electroneutrality requirement.