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 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.

Coupled hydromechanical and electromagnetic disturbances in unsaturated porous materials.

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.

Time-domain induced polarization as a tool to image clogging in treatment wetlands.

During the last decade, treatment (artificial) wetlands have flourished all over Europe for the treatment of sewages from small communities thanks to their low cost of operation. The clogging of the filter of these wetlands is an issue affecting their efficiency and considered as their main operational problem. The present work shows the results of the application of a geophysical method called time-domain induced polarization. It is used to non-intrusively image, in 3D, the clogging of the gravel filters in a quick and efficient way. Induced polarization characterizes the ability of a porous material to reversibly store electrical charges when submitted to an electrical field. The material property characterizing this ability is called normalized chargeability. A set of laboratory experiments allows to determine an empirical relationship between the normalized chargeability and the weight amount of clogging. Induced polarization measurements have been performed in the field over a treatment wetland to get a 3D reconstructed image (tomography) of the normalized chargeability. From this tomography and the previously defined relationship, we are able to image in 3D the distribution of clogging and where it is concentrated in the filter. We can therefore identify the areas requiring preventive measures to minimize this clogging issue.

Diffusion of ions in unsaturated porous materials.

In a salinity gradient, the diffusion of ions through the connected porosity of a porous and charged material is influenced by the charged nature of the interface between the pore water and the solid. This influence is exerted through the generation of a macroscopic electrical field termed the diffusion or membrane potential. This electrical field depends on the excess of counterions located in the pore space counterbalancing the charge density of the surface of the solid. In unsaturated porous materials, we have to consider (1) the effect of the charged nature of the air/water interface, (2) the increase of the counterion density as the counterions are packed in a smaller volume when the saturation of the nonwetting phase (air) increases, and (3) the influence of the water saturation upon the tortuosity of the water phase. The volume average of the Nernst-Planck equation is used to determine the constitutive equations for the coupled diffusion flux and current density of a multicomponent electrolyte in unsaturated conditions. We assume that water is the wetting phase for the solid phase. We neglect the electro-osmotic flow in the coupled constitutive equations and the deformation of the medium (the medium is assumed to be both isotropic and rigid). This model explains well the observed tendency of strong decreases of the apparent diffusion coefficient of ions with the decrease of the saturation of the water phase under steady-state conditions. This decrease is mainly due to the influence of the saturation upon the tortuosity of the water phase.

Complex conductivity of water-saturated packs of glass beads.

The low-frequency conductivity response of water-saturated packs of glass beads reflects a combination of two processes. One process corresponds to the polarization of the mineral/water interface coating the surface of the grains. The other process corresponds to the Maxwell-Wagner polarization associated with accumulation of the electrical charges in the pore space of the composite medium. A model of low-frequency conductivity dispersion is proposed. This model is connected to a triple-layer model of electrochemical processes occurring at the surface of silica. This model accounts for the partition of the counterions between the Stern and the diffuse layers. The polarization of the mineral/water interface is modeled by the electrochemical polarization model of Schurr for a spherical grain. We take into account also the DC surface conductivity contribution of protons of the sorbed water and the contribution of the diffuse layer. At the scale of a macroscopic representative elementary volume of the porous material, the electrochemical polarization of a single grain is convoluted with the grain size distribution of the porous material. Finally, the Maxwell-Wagner polarization is modeled using the complex conductivity of a granular porous medium obtained from the differential effective medium theory. The predictions of this model agree well with experimental data of spectral induced polarization. Two peaks are observed at low frequencies in the spectrum of the phase. The first peak corresponds to the distribution of the size of the beads and the second peak is due to the roughness of the grains.

Thermodynamics of ions and water transport in porous media.

The thermodynamic framework of Prigogine, de Groot, and Mazur is extended to study the transport of ions and water in thermoporoelastic materials assuming infinitesimal deformations. New expressions are developed for the first and second principles of nonequilibrium thermodynamics of multicomponent systems and a generalized power balance equation is derived. For porous materials, all the components cannot be treated on a symmetric basis. A Lagrangian framework associated with deformation of the solid phase is introduced and, in this framework, Curie's principle is used to set up the form of the linear constitutive equations describing the transport of ions, water, and heat through the pore network. The material properties entering these equations were recently obtained by Revil and Linde [J. Colloid Interface Science 302 (2006) 682-694] using a volume-averaging approach based in the Nernst-Planck and Stokes equations. This provides a way to relate the material properties entering the constitutive equations to two textural parameters characterizing the topology of the pore space of the material (namely the tortuosity of the pore space and the permeability). The generalized power balance equation is used to derive the linear poroelastic constitutive equations (including the osmotic pressure) to describe the reversible contribution of deformation of the medium in response to ions and water transport through the connected porosity.

Introducing interacting diffuse layers in TLM calculations: a reappraisal of the influence of the pore size on the swelling pressure and the osmotic efficiency of compacted bentonites.

The truncation of the Gouy-Chapman diffuse part in compacted clay-rocks and bentonite is introduced into the electrical triple-layer model (TLM) recently developed by P. Leroy and A. Revil [J. Colloid Interface Sci. 270 (2004) 371]. The new model is used to explain the dependence of the osmotic efficiency and the swelling pressure as functions of the mean pore size of the medium, determined from the porosity and the specific surface. The truncation of the diffuse layer introduces a new variable in the system of equations to be solved, the electrical potential at the midplane between adjacent charged surfaces. This new variable is evaluated through a Taylor expansion of the electrical potential. The present model is able to capture the variation of the osmotic efficiency and the swelling pressure with the mean pore size. The partition of counterions between the Stern layer and the diffuse layer as a function of the pore size calculated by the TLM also shows a good consistency with the model. This implies that more than 90% of the counterions are located in the Stern layer.

Influence of the Dukhin and Reynolds numbers on the apparent zeta potential of granular porous media.

The Helmholtz-Smoluchowski (HS) equation is widely used to determine the apparent zeta potential of porous materials using the streaming potential method. We present a model able to correct this apparent zeta potential of granular media of the influence of the Dukhin and Reynolds numbers. The Dukhin number represents the ratio between the surface conductivity (mainly occurring in the Stern layer) and the pore water conductivity. The Reynolds number represents the ratio between inertial and viscous forces in the Navier-Stokes equation. We show here that the HS equation can lead to serious errors if it is used to predict the dependence of zeta potential on flow in the inertial laminar flow regime without taking into account these corrections. For indifferent 1:1 electrolytes (such as sodium chloride), we derived two simple scaling laws for the dependence of the streaming potential coupling coefficient (or the apparent zeta potential) on the Dukhin and Reynolds numbers. Our model is compared with a new set of experimental data obtained on glass bead packs saturated with NaCl solutions at different salinities and pH. We find fairly good agreement between the model and these experimental data.

Electrokinetic coupling in unsaturated porous media.

We consider a charged porous material that is saturated by two fluid phases that are immiscible and continuous on the scale of a representative elementary volume. The wetting phase for the grains is water and the nonwetting phase is assumed to be an electrically insulating viscous fluid. We use a volume-averaging approach to derive the linear constitutive equations for the electrical current density as well as the seepage velocities of the wetting and nonwetting phases on the scale of a representative elementary volume. These macroscopic constitutive equations are obtained by volume-averaging Ampère's law together with the Nernst-Planck equation and the Stokes equations. The material properties entering the macroscopic constitutive equations are explicitly described as functions of the saturation of the water phase, the electrical formation factor, and parameters that describe the capillary pressure function, the relative permeability functions, and the variation of electrical conductivity with saturation. New equations are derived for the streaming potential and electro-osmosis coupling coefficients. A primary drainage and imbibition experiment is simulated numerically to demonstrate that the relative streaming potential coupling coefficient depends not only on the water saturation, but also on the material properties of the sample, as well as the saturation history. We also compare the predicted streaming potential coupling coefficients with experimental data from four dolomite core samples. Measurements on these samples include electrical conductivity, capillary pressure, the streaming potential coupling coefficient at various levels of saturation, and the permeability at saturation of the rock samples. We found very good agreement between these experimental data and the model predictions.

Non-intrusive characterization of the redox potential of landfill leachate plumes from self-potential data.

Contaminant plumes (e.g., associated with leakages from municipal landfills) provide a source of natural electrical potentials (or "self-potentials") recordable at the Earth's surface. One contribution to these self-potentials is associated with pore water flow (i.e., the "streaming potential"), and the other is related to redox conditions. A contaminant plume can be regarded as a "geobattery": the source current potentially results from the degradation of the organic matter by micro-organisms, which produces electrons. These electrons are then carried by nanowires that connect bacteria and thorough metallic particles that precipitate in areas of strong redox potential gradient. In the case of the Entressen landfill (South of France), reported here, the hydraulic head differences measured in piezometers outside the contaminant plume is strongly linked to the surface self-potential signals, with a correlation coefficient of -0.94. We used a Bayesian method that combines hydraulic head and self-potential data collected outside the contaminated area to estimate the streaming potential component of the collected self-potential data. Once the streaming potential contribution was removed from the measured self-potentials, the correlation coefficient between the residual self-potentials and the measured redox potentials in the aquifer was 0.92. The slope of this regression curve was close to 0.5, which was fairly consistent with both finite element modelling and the proposed geobattery model.

Chemico-electromechanical coupling in microporous media.

We determine the macroscopic transport properties of isotropic microporous media by volume-averaging the local Nernst-Planck and Navier-Stokes equations in nonisothermal conditions. In such media, the excess of charge that counterbalances the charge deficiency of the surface of the minerals is partitioned between the Gouy-Chapman layer and the Stern layer. The Stern layer of sorbed counterions is attached to the solid phase, while the Gouy-Chapman diffuse layer is assumed to have a thickness comparable to the size of the pores. Rather than using Poisson-Boltzmann distributions to describe the ionic concentrations in the pore space of the medium, we rely on Donnan distributions obtained by equating the chemical potentials of the water molecules and ions between a reservoir of ions and the pore space of the medium. The macroscopic Maxwell equations and the macroscopic linear constitutive transport equations are derived in the vicinity of equilibrium, assuming that the porous material is deformable. In the vicinity of thermodynamic equilibrium, the cross-coupling phenomena of the macroscopic constitutive equations of transport follow Onsager reciprocity. In addition, all the material properties entering the constitutive equations depend only on two textural properties, the permeability and the electrical formation factor.

Diffusion of ionic species in bentonite.

We present a macroscopic model of ionic diffusion in bentonites including the effect of the hydraulic-electrical-chemical couplings expected in such charged porous medium. The anomalous concentrations of the ions in the pore water of the bentonite are modeled with a modified Donnan model in which we account for the partition of the counterions between the Stern and Gouy-Chapman layers. This is accomplished using an electric triple layer (TLM) model combined with an explicit complexation model at the mineral/water interface. The porosity entering into the determination of the formation factor of the medium is an effective porosity obtained by removing the fraction of hydration water covering the surface of the clay minerals. We investigate two different cases of diffusion. In the first case, we consider a salinity gradient between two reservoirs in contact with a cylindrical sample of bentonite. The model predicts an increase of the diffusivity of the salt with the salinity of the solution in contact with the bentonite in agreement with experimental data. In the second case, we analyze a self-diffusion experiment of an ionic tracer. The model predicts an increase of the diffusivity of anions with the effective porosity and with the ionic strength. This is also in good agreement with experimental data.

Geophysical investigation of seepage beneath an earthen dam.

A hydrogeophysical survey is performed at small earthen dam that overlies a confined aquifer. The structure of the dam has not shown evidence of anomalous seepage internally or through the foundation prior to the survey. However, the surface topography is mounded in a localized zone 150 m downstream, and groundwater discharges from this zone periodically when the reservoir storage is maximum. We use self-potential and electrical resistivity tomography surveys with seismic refraction tomography to (1) determine what underlying hydrogeologic factors, if any, have contributed to the successful long-term operation of the dam without apparent indicators of anomalous seepage through its core and foundation; and (2) investigate the hydraulic connection between the reservoir and the seepage zone to determine whether there exists a potential for this success to be undermined. Geophysical data are informed by hydraulic and geotechnical borehole data. Seismic refraction tomography is performed to determine the geometry of the phreatic surface. The hydro-stratigraphy is mapped with the resistivity data and groundwater flow patterns are determined with self-potential data. A self-potential model is constructed to represent a perpendicular profile extending out from the maximum cross-section of the dam, and self-potential data are inverted to recover the groundwater velocity field. The groundwater flow pattern through the aquifer is controlled by the bedrock topography and a preferential flow pathway exists beneath the dam. It corresponds to a sandy-gravel layer connecting the reservoir to the downstream seepage zone.

A triple-layer model of the surface electrochemical properties of clay minerals.

We propose an electrical triple-layer model (TLM) to describe the electrochemical properties of clay minerals. This model includes a speciation model of the active crystallographic surface sites plus a classical description of the Stern and diffuse layers. In addition to the surface charges associated with the surface groups (aluminols, silanols, and Al--O--Si sites), the model takes into account the degree of isomorphic substitution rate inside the crystalline network. The model computes both the zeta potential and the low-frequency (few kHz) surface conductivity as these two properties are equally important to interpret electrokinetic properties of colloids and clay-rich porous materials. For surface conductivity, the model comprises two contributions; one is associated with the Stern layer (dynamic Stern layer) and the second is associated with the excess of counterions located in the diffuse Gouy-Chapman layer. The parameters of the TLM model are optimized using the Simplex algorithm. Comparison between the model and the experimental data shows good agreement for a reasonable choice of parameters close to the a priori values determined from quartz and gibbsite. Surface conductivity of smectite appears rather independent of salinity, while for kaolinite, surface conductivity increases with salinity. In both cases, the Stern layer contribution to surface conductivity dominates but the contribution associated with the diffuse layer cannot be neglected.

Characterization of focused seepage through an earthfill dam using geoelectrical methods.

Resistivity and self-potential tomography can be used to investigate anomalous seepage inside heterogeneous earthen dams. The self-potential (SP) signals provide a unique signature to groundwater flow because the source current density responsible for the SP signals is proportional to the Darcy velocity. The distribution of the SP signals is also influenced by the distribution of the resistivity; therefore, resistivity and SP need to be used in concert to elucidate groundwater flow pathways. In this study, a survey is conducted at a small earthen dam in Colorado where anomalous seepage is observed on the downstream face at the dam toe. The data reveal SP and direct current resistivity anomalies that are used to delineate three anomalous seepage zones within the dam and to estimate the source of the localized seepage discharge. The SP data are inverted in two dimensions using the resistivity distribution to determine the distribution of the Darcy velocity responsible for the observed seepage. The inverted Darcy velocity agrees with an estimation of the Darcy velocity from the hydraulic conductivity obtained from a slug test and the observed head gradient.

Low-frequency complex conductivity of sandy and clayey materials.

Low-frequency polarization of sands and sandstones seems to be dominated by the polarization of the Stern layer, the inner part of the electrical double layer coating the surface of the silica grains and clay particles. We investigate a simple model of Stern layer polarization combined with a simple complexation model of the surface of the grains immersed in a 1:1 electrolyte like NaCl. In isothermal conditions, the resulting model can be used to predict the complex conductivity of clayey materials as a function of the porosity, the cation exchange capacity of the clay fraction (alternatively the specific surface area of the material), and the salinity of the pore water. A new set of experimental data is presented. This dataset comprises low-frequency (1 mHz-45 kHz) complex conductivity measurements of saprolites and sandstones that are well characterized in terms of their petrophysical properties (porosity, permeability, specific surface area or CEC, and pore size). This dataset, together with incorporating additional data from the literature, is used to test the Stern layer polarization model. We find an excellent agreement between the predictions of this model and this experimental dataset indicating that the new model can be used to predict the complex conductivity of natural clayey materials and clay-free silica sands.

Changes in induced polarization associated with the sorption of sodium, lead, and zinc on silica sands.

Low-frequency dielectric spectroscopy can be measured in terms of a conductance and a phase lag between the electrical current and the electrical field. This conductance and phase lag can be written as into a complex conductivity with both an in-phase and quadrature components that are frequency dependent. In sands, the low-frequency (10 mHz-40 kHz) spectra of the complex conductivity are dominated by the polarization of the electrical double layer (especially the internal part of the electrical double layer called the Stern layer) and the Maxwell-Wagner polarization (typically above 100 Hz). We present a polarization that is able to explain the complex conductivity spectra including the grain size distribution, the porosity, and the complexation of the mineral surface with the ions of the pore water. To test this model, we investigate the sorption of various cations (Na, Pb, Zn) characterized by different affinities with the surface of silica. Sand column experiments were carried out to see the change in the complex conductivity during the advective/dispersive transport of a lead nitrate solution and a zinc sulfate solution, replacing a sodium chloride solution in the pore space of the sand. The complex conductivity model is able to explain the change of the phase over time.

Ionic contribution to the self-potential signals associated with a redox front.

In contaminant plumes or in the case of ore bodies, a source current density is produced at depth in response to the presence of a gradient of the redox potential. Two charge carriers can exist in such a medium: electrons and ions. Two contributions to the source current density are associated with these charge carriers (i) the gradient of the chemical potential of the ionic species and (ii) the gradient of the chemical potential of the electrons (i.e., the gradient of the redox potential). We ran a set of experiments in which a geobattery is generated using electrolysis reactions of a pore water solution containing iron. A DC power supply is used to impose a difference of electrical potential of 3 V between a working platinum electrode (anode) and an auxiliary platinum electrode (cathode). Both electrodes inserted into a tank filled with a well-calibrated sand infiltrated by a (0.01 mol L(-1) KCl+0.0035 mol L(-)(1) FeSO(4)) solution. After the direct current is turned off, we follow the pH, the redox potential, and the self-potential at several time intervals. The self-potential anomalies amount to a few tens of millivolts after the current is turned off and decreases over time. After several days, all the redox-active compounds produced initially by the electrolysis reactions are consumed through chemical reactions and the self-potential anomalies fall to zero. The resulting self-potential anomalies are shown to be much weaker than the self-potential anomalies observed in the presence of an electronic conductor in the laboratory or in the field. In the presence of a biotic or an abiotic electronic conductor, the self-potential anomalies can amount to a few hundred millivolts. These observations point out indirectly the potential role of bacteria forming biofilms in the transfer of electrons through sharp redox potential gradient in contaminant plumes that are rich in organic matter.

A potential-based inversion of unconfined steady-state hydraulic tomography.

The importance of estimating spatially variable aquifer parameters such as transmissivity is widely recognized for studies in resource evaluation and contaminant transport. A useful approach for mapping such parameters is inverse modeling of data from series of pumping tests, that is, via hydraulic tomography. This inversion of field hydraulic tomographic data requires development of numerical forward models that can accurately represent test conditions while maintaining computational efficiency. One issue this presents is specification of boundary and initial conditions, whose location, type, and value may be poorly constrained. To circumvent this issue when modeling unconfined steady-state pumping tests, we present a strategy that analyzes field data using a potential difference method and that uses dipole pumping tests as the aquifer stimulation. By using our potential difference approach, which is similar to modeling drawdown in confined settings, we remove the need for specifying poorly known boundary condition values and natural source/sink terms within the problem domain. Dipole pumping tests are complementary to this strategy in that they can be more realistically modeled than single-well tests due to their conservative nature, quick achievement of steady state, and the insensitivity of near-field response to far-field boundary conditions. After developing the mathematical theory, our approach is first validated through a synthetic example. We then apply our method to the inversion of data from a field campaign at the Boise Hydrogeophysical Research Site. Results from inversion of nine pumping tests show expected geologic features, and uncertainty bounds indicate that hydraulic conductivity is well constrained within the central site area.

Reconstruction of the water table from self-potential data: a bayesian approach.

Ground water flow associated with pumping and injection tests generates self-potential signals that can be measured at the ground surface and used to estimate the pattern of ground water flow at depth. We propose an inversion of the self-potential signals that accounts for the heterogeneous nature of the aquifer and a relationship between the electrical resistivity and the streaming current coupling coefficient. We recast the inversion of the self-potential data into a Bayesian framework. Synthetic tests are performed showing the advantage in using self-potential signals in addition to in situ measurements of the potentiometric levels to reconstruct the shape of the water table. This methodology is applied to a new data set from a series of coordinated hydraulic tomography, self-potential, and electrical resistivity tomography experiments performed at the Boise Hydrogeophysical Research Site, Idaho. In particular, we examine one of the dipole hydraulic tests and its reciprocal to show the sensitivity of the self-potential signals to variations of the potentiometric levels under steady-state conditions. However, because of the high pumping rate, the response was also influenced by the Reynolds number, especially near the pumping well for a given test. Ground water flow in the inertial laminar flow regime is responsible for nonlinearity that is not yet accounted for in self-potential tomography. Numerical modeling addresses the sensitivity of the self-potential response to this problem.