Key Points
A new model for the effective conductivity and permittivity
It explains an important number of data in a broad frequency range
It accounts for clay content, mineralogy and saturation
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 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.
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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).
Key Points
Induced polarization of shaly sands is controlled by the Stern layer
The quadrature conductivity depends on salinity and CEC
The model accounts for temperature and saturation
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•The Nernst–Planck and Stokes equations are upscaled with volume-averaging.•The electrokinetic properties are modeled with an effective charge density.•The constitutive equations include filtration ...effects in unsaturated conditions.•Comparison with experimental data is successful.
I developed a model of cross-coupled flow in partially saturated porous media based on electrokinetic coupling including the effect of ion filtration (normal and reverse osmosis) and the multi-component nature of the pore water (wetting) phase. The model also handles diffusion and membrane polarization but is valid only for saturations above the irreducible water saturation. I start with the local Nernst–Planck and Stokes equations and I use a volume-averaging procedure to obtain the generalized Ohm, Fick, and Darcy equations with cross-coupling terms at the scale of a representative elementary volume of the porous rock. These coupling terms obey Onsager's reciprocity, which is a required condition, at the macroscale, to keep the total dissipation function of the system positive. Rather than writing the electrokinetic terms in terms of zeta potential (the double layer electrical potential on the slipping plane located in the pore water), I developed the model in terms of an effective charge density dragged by the flow of the pore water. This effective charge density is found to be strongly controlled by the permeability and the water saturation. I also developed an electrical conductivity equation including the effect of saturation on both bulk and surface conductivities, the surface conductivity being associated with electromigration in the electrical diffuse layer coating the grains. This surface conductivity depends on the CEC of the porous material.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK, ZRSKP
The surface conductivity of porous rocks has two contributions: the first is associated with the diffuse layer coating the grains and is frequency-independent as long as the diffuse layer is above a ...percolation threshold. The second contribution is associated with the Stern layer of weakly sorbed counterions on the mineral surface and is frequency-dependent if the Stern layer is discontinuous at the scale of the representative elementary volume. In the frequency range 1 mHz–100 Hz, this second contribution is also associated with the main polarization mechanism observed by the spectral induced polarization method in granular media (neglecting the contribution of other polarization processes like those associated with redox processes and membrane polarization). At the macroscale, we connect the Stern layer contribution to the complex conductivity and to the expectation of the probability distribution of the inverse of the grain size. This is done by performing a convolution between the probability distribution of the inverse of the grain size and the surface conductivity response obtained when all the grains have the same size. Surface conductivity at the macroscopic scale is also connected to an effective pore size used to characterize permeability. From these relationships, a new equation is derived connecting this effective pore size, the electrical formation factor, and the expected value of the probability distribution for the inverse of the grain size, which is in turn related to the distribution of the relaxation times. These new relationships are consistent with various formula derived in the literature in the limit where the grain size distribution is given by the delta function or a log normal distribution and agree fairly well with various experimental data showing also some limitations of the induced polarization method to infer permeability. One of these limitations is the difficulty to detect the relaxation, in the phase, associated with the smaller grains, as this polarization may be hidden by the Maxwell—Wagner polarization at relatively high frequencies (>100 Hz). Also, cemented aggregates of grains can behave as coarser grains.
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•A model describing coupled transport properties in partially saturated clayey materials.•The model predicts correctly the osmotic coefficient.•The model predicts correctly the diffusion coefficient.
...A new set of constitutive equations describing the transport of the ions and water through charged porous media and considering the effect of ion filtration is applied to the problem of reverse osmosis and diffusion of a salt. Starting with the constitutive equations derived in Paper 1, I first determine specific formula for the osmotic coefficient and effective diffusion coefficient of a binary symmetric 1:1 salt (such as KCl or NaCl) as a function of a dimensionless number Θ corresponding to the ratio between the cation exchange capacity (CEC) and the salinity. The modeling is first carried with the Donnan model used to describe the concentrations of the charge carriers in the pore water phase. Then a new model is developed in the thin double layer approximation to determine these concentrations. These models provide explicit relationships between the concentration of the ionic species in the pore space and those in a neutral reservoir in local equilibrium with the pore space and the CEC. The case of reverse osmosis and diffusion coefficient are analyzed in details for the case of saturated and partially saturated porous materials. Comparisons are done with experimental data from the literature obtained on bentonite. The model predicts correctly the influence of salinity (including membrane behavior at high salinities), porosity, cation type (K+ versus Na+), and water saturation on the osmotic coefficient. It also correctly predicts the dependence of the diffusion coefficient of the salt with the salinity.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UL, UM, UPCLJ, UPUK, ZRSKP
Recent developments in induced polarization allow for the characterization of alteration halos within hydrothermal systems. We explore the possibility of using electrical conductivity and normalized ...chargeability tomograms in concert to image hydrothermal conduits thanks to their high cation exchange capacities associated with alteration. The hydrothermal plumbing system of the late‐stage rift area of Lake Abhé (Republic of Djibouti) is used to test the ability of induced polarization to reveal such plumbing system associated with the prominent hydrothermal chimneys serving as vents for the hydrothermal fluids. We show that induced polarization can be used to provide an alteration tomogram, which highlights the flow path toward 2 chimneys in the top 70 m below the surface of the sediments at Lake Abhé.
Plain Language Summary
We are currently lacking tools to visualize the plumbing system of hydrothermal bodies. Because of its unique sensitivity to alteration, induced polarization represents a welcome and useful non‐intrusive technique to image hydrothermal conduits including at large depths (kilometer scale). We develop a methodology to image the hydrothermal flow paths associated with the chimneys observed at Lake Abhé, Republic of Djibouti, located at the triple junction between the Red Sea, Gulf of Aden, and main Ethiopian rift trends. These vents are associated with hydrothermal sublacustrine springs. The hydrothermal pathways are characterized by intense hydrothermal alteration.
Key Points
Induced polarization is sensitive to hydrothermal alteration
The hydrothermal vents at Lake Abhé are highly altered
Induced polarization images the conduits associated with two vents
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FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SAZU, SBCE, SBMB, UL, UM, UPUK
We provide a unified model for the soil‐water retention function, including the effect of bound and capillary waters for all types of soils, including clayey media. The model combines a ...CEC‐normalized isotherm describing the sorption of the bound water (and the filling of the trapped porosity) and the van Genuchten model to describe the capillary water sorption retention but ignore capillary condensation. For the CEC‐normalized isotherm, we tested both the BET and Freundlich isotherms, and we found that the Freundlich is more suitable than the BET isotherm in fitting the data. It is also easier to combine the Freundlich isotherm with the van Genuchten model. The new model accounts for (1) the different types of clay minerals, (2) the different types of ions sorbed in the Stern layer and on the basal planes of 2:1 clays, and (3) the pore size distribution. The model is validated with different data sets, including mixtures of kaolinite and bentonite. The model parameters include two exponents (the pore size exponent of the van Genuchten model and the exponent of the Freundlich isotherm), the capillary entry pressure, and two critical water contents. The first critical water content is the water content at saturation (porosity), and the second is the maximum water content associated with adsorption forces, including the trapped nonbound water.
Key Points
A new CEC- normalized Freundlich isotherm
Unified Freundlich and van Genuchten model for water retention
Validation made with a large set of experimental data
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Key Points
First model built for unsaturated materials
Basis for new methods to investigate the vadose zone
All the material properties are related to measurable parameters
A theory of cross‐coupled ...flow equations in unsaturated soils is necessary to predict (1) electroosmotic flow with application to electroremediation and agriculture, (2) the electroseismic and the seismoelectric effects to develop new geophysical methods to characterize the vadose zone, and (3) the streaming current, which can be used to investigate remotely ground water flow in unsaturated conditions in the capillary water regime. To develop such a theory, the cross‐coupled generalized Darcy and Ohm constitutive equations of transport are extended to unsaturated conditions. This model accounts for inertial effects and for the polarization of porous materials. Rather than using the zeta potential, like in conventional theories for the saturated case, the key parameter used here is the quasi‐static volumetric charge density of the pore space, which can be directly computed from the quasi‐static permeability. The apparent permeability entering Darcy's law is also frequency dependent with a critical relaxation time that is, in turn, dependent on saturation. A decrease of saturation increases the associated relaxation frequency. The final form of the equations couples the Maxwell equations and a simplified form of two‐fluid phases Biot theory accounting for water saturation. A generalized expression of the Richard equation is derived, accounting for the effect of the vibration of the skeleton during the passage of seismic waves and the electrical field. A new expression is obtained for the effective stress tensor. The model is tested against experimental data regarding the saturation and frequency dependence of the streaming potential coupling coefficient. The model is also adapted for two‐phase flow conditions and a numerical application is shown for water flooding of a nonaqueous phase liquid (NAPL, oil) contaminated aquifer. Seismoelectric conversions are mostly taking place at the NAPL (oil)/water encroachment front and can be therefore used to remotely track the position of this front. This is not the case for other geophysical methods.
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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.
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.
Key Points:
Permeability is related to a relaxation time and the formation factor
Time constant and formation factor are obtained from induced polarization
The relationship is tested against a broad database of experimental data
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