Induced polarization of clay-rich materials — Part 4: Water content and temperature effects in bentonites

Countries with commercial nuclear power production are planning to isolate their high-level and long-lived radioactive waste in deep geological disposal (DGD) to avoid any discharge of radionuclides in the biosphere (Madsen, 1998; McCombie et al., 2000; Mckinley et al., 2007; Palmu and Torsten, 2010). DGDs involve isolating radioactive waste within a stable geologic environment (typically at a depth of 400–600 m). This is generally achievable by using multiple barriers acting together to provide protection over hundreds of thousands of years (Ewing et al., 2016). These multiple barrier systems include a host rock (such as the Callovo-Oxfordian (COx) formation in France, the Opalinus clay in Switzerland, or the migmatitic gneiss and pegmatitic granite in Olkiluoto, Finland) and an engineered barrier system (EBS). The EBS is composed of a waste form, waste canisters, buffer materials, backfills, seals, and concrete plugs (see OECD, 2003; Mckinley et al., 2007; Sellin and Leupin, 2013, Figure 1).

In the French geological disposal project (called CIGEO), the sealing of the galleries, access ramps, and shafts to prevent potential pathways for water, gas, and radionuclides migration is one of the key points to ensure the long-term safety of the repository (Mokni et al., 2016; Mokni and Barnichon, 2016). Bentonite is generally regarded as the most suitable sealing material (Delage et al., 2010; Tournassat et al., 2015). This is due to its high swelling ability, low permeability, thermal stability, and high retention properties with a large surface area and cation exchange capacity (CEC) (Kanno and Wakamatsu, 1992; Chapman, 2006; Villar and Lloret, 2008; Chen et al., 2014). For applications in geologic disposals, the bentonite is usually installed as compacted blocks of bentonite-sand mixtures or pure bentonite pellets (Cui et al., 2008; Villar et al., 2008; Gens et al., 2009; Ye et al., 2010; Wang et al., 2013; Saba, 2013; Wang et al., 2015). In repository conditions, the compacted sand-bentonite mixtures will be set up in unsaturated conditions. The water content of bentonite is expected to increase through the suction and sorption of groundwater from the surrounding formation. This infiltration would, in turn, trigger the development of swelling pressures (Villar and Lloret, 2008; Dieudonné et al., 2016; Mokni and Barnichon, 2016). One of the major roles of swelling is to ensure that the sealing material closes all the macro-voids in the disposal. It is well documented that swelling pressure and permeability depend principally on the saturation and dry density of the material (e.g., Imbert and Villar, 2006). A small change in density can induce a significant change in swelling pressure and permeability. Such behavior may, in turn, influence the hydromechanical behavior of the sand-bentonite mixture (Villar and Lloret, 2008; Shirazi et al., 2010; Darde et al., 2022).

The addition of sand to bentonite improves certain characteristics, such as increasing the dry density and shear strength and decreasing the shrinkage (Dixon et al., 1985; Srikanth and Mishra, 2016). It presents hydraulic conductivities similar to pure bentonite (Cho et al., 2000). Increasing the sand content within the seal favors gas-pathways within the repository (Smart et al., 2008; Bardelli et al., 2014; Guo et al., 2020). This may help avoid the accumulation of gas (particularly hydrogen) that is expected to be produced by the radiolysis of water due to ionizing radiation and the anaerobic corrosion of steel and reactive metals present in the disposal facility (Guo et al., 2020). The effect of hyperalkaline solution (associated with the presence of cement) is discussed in the next paper of this series (El Alam et al., 2024).

Monitoring parameters such as water content, dry density, saturation, salinity, and permeability through space in a sand-bentonite seal can, therefore, be considered essential to assess and monitor the long-term performance of these seals. The effect of temperature is also of paramount importance in interpreting the in situ experiments in field conditions. Different monitoring technologies sensitive to such a broad range of parameters have been tested to assess their efficiency for DGDs (e.g., Bertrand et al., 2019). The sensors include total pressure cells, pore pressure sensors, thermocouples, time-domain and frequency-domain reflectometry sensors, and fiber optics (see details in Müller et al., 2015; García-Siñeriz et al., 2019; Sakaki et al., 2019). These technologies tend to give accurate readings. That said, they provide only local information, and their setting may offer a preferential pathway for the migration of radionuclides.

In contrast, geophysical methods provide powerful alternatives (or complementary techniques) because (1) they can be designed in a nonintrusive manner, (2) they are sensitive to variations in water and clay content, (3) they allow dynamic phenomena to be monitored over time, and (4) they allow local anomalies to be captured that are potentially missed by localized sensors. A broad range of geophysical methods such as seismic tomography and electrical resistivity tomography (ERT) have already been tested for operational repository monitoring (Biryukov et al., 2016; Wieczorek et al., 2017; De Carvalho Faria Lima Lopes et al., 2019; Maurer et al., 2019; Ducut et al., 2022; Wayal and Sitharam, 2022). However, seismic monitoring experiments have shown that entirely nonintrusive traveltime tomography could be challenging as it can have difficulties in detecting subtle changes within a low-velocity and attenuating environment (Manukyan et al., 2012).

Induced polarization (IP) can be conducted in parallel with ERT surveys. This method yields additional information by exploring the low-frequency polarization mechanisms that occur when an electrical field is applied, either in time or frequency domains (Wait, 1959; Telford et al., 1990; Ward, 1990). Pioneered a century ago by Schlumberger (1920), IP has been applied since to multiple fields and topics, such as the study of the excavation damaged zones (Kruschwitz and Yaramanci, 2004; Okay et al., 2013), ore exploration (Martin et al., 2022), salinity variations in nonconsolidated clays and sandstones (Weller et al., 2011; Mendieta Tenorio, 2021), biogeophysics (Atekwana and Slater, 2009; Martin and Günther, 2013; Mendonça et al., 2015; Song et al., 2022), the study of the critical zone of the earth (Weller et al., 2011, 2013; Jougnot, 2020), the alteration of volcanic rocks (Revil et al., 2021), permafrost (Duvillard et al., 2018; Mudler et al., 2019, 2022), and geoarcheology (Martin et al., 2020).

In IP, conduction and polarization by a primary electrical field are studied in the low-frequency limit (<10 kHz) of the Maxwell equations (Vinegar and Waxman, 1984). The IP of low-porosity, anisotropic clay rocks and illite/pyrites mixes is investigated in the three previous papers of this series (Revil et al., 2023a, 2023b, 2024). However, the IP properties of high CEC, high porosity, and smectite-rich materials need to be investigated as an end-member case of clay-rich materials.

Many works have been published to elucidate the polarization of clay-rich media (e.g., Cosenza et al., 2007; Leroy and Revil, 2009; Jougnot et al., 2010; Breede et al., 2012; Revil et al., 2013, 2017, 2018b; Okay et al. 2013, 2014). In the past, various models have been developed to advance the petrophysics of the IP of clay-rich materials, including the double-layer polarization model of De Lima and Sharma (1992) and Lyklema et al. (1983), the generalized effective-medium theory of IP (GEMTIP) model based on the differential effective medium theory (Zhdanov, 2008; Tong et al., 2023), and the diffuse layer polarization models initially developed for colloidal suspension (e.g., Dukhin and Shilov, 1974; Fixman, 1980; Barreto and Dias, 2014). Other models are based on different physics, such as the membrane polarization model (Marshall and Madden, 1959; Li et al., 2021). The model of Cosenza et al. (2008) is based on the Maxwell-Wagner polarization but cannot explain the low-frequency IP features, such as the relationship between surface conductivity and normalized chargeability. Actually, none of these fundamental models can explain large data sets of experimental data. Indeed, in porous media, at the opposite of dilute colloidal suspensions of clays, the diffuse layers overlap and therefore cannot be easily polarized. As a consequence, early emphasis has been put on the so-called dynamic Stern layer model (see Stern [1924] for the original description; see Rosen and Saville [1991] and Grosse [2009] for a dynamic Stern layer model applied to IP). The GEMTIP model is not based on any physical polarization mechanisms and should be coupled to such mechanisms to be predictive (see the discussion in Revil and Cosenza, 2010). The membrane polarization model is unable to explain experimental data, possibly indicating that membrane polarization may not be the dominant polarization mechanism at play in the conditions used to measure IP phenomena.

Conversely, semiempirical models have been developed by Waxman and Smits (1968) for conduction and Vinegar and Waxman (1984) for polarization of clay-bearing sediments. The model of Vinegar and Waxman (1984) is compatible with a dynamic Stern layer model coupled to a membrane mechanism. The dynamic Stern layer polarization model developed by Revil (2013a, 2013b) is based on volume-averaging upscaling and offers a physics-based model that is entirely compatible with the models of Waxman and Smits (1968) and Vinegar and Waxman (1984). This model, very simple in essence, is therefore used in this paper as a starting basis to explain and elucidate the IP properties of bentonite-sand mixtures.

In this paper, we conducted IP measurements on 42 manufactured samples composed of mixtures of Fontainebleau sand (pure silica, poorly polarizable) and MX80 Wyoming bentonite (highly polarizable). The choice of the frequency range was performed for four reasons: (1) frequencies below 1 Hz were not considered in order to speed up the experimental work (a lot of spectra needed to be done in a reasonable amount of time); (2) smectite particles polarize at high frequencies (>10 Hz); (3) time-domain IP is usually performed at 1 Hz in field conditions, which is inside the frequency range investigated here; and (4) electrode drift can complicate the accuracy of the measurements below 1 Hz.

Our first goal of this study is (1) to establish a database for the complex conductivity spectra of bentonite-sand mixtures at different mass fractions of bentonite and volumetric water contents. In terms of petrophysical parameters, the range of volumetric water contents is 19% to 76%, saturation is 40% to 100%, dry density is 0.98 to 1.61, and the mass fractions of clay are 10% to 90% (100% if we consider the data from El Alam et al., 2023). These conditions are those that may be encountered in the early and late times of the life of a DGD (Gens et al., 2009; Ye et al., 2010; Saba, 2013). (2) Our second goal is to explain the observed trends found between the complex conductivity spectra and mass fraction of bentonite at different saturation states and bentonite contents. (3) We also test a simple linear temperature dependence equation for in-phase and quadrature conductivities and normalized chargeability.

The data set studied in this paper is composed of a total of 42 experiments/samples (see Table 1). The entire data set is made up of mixtures of Wyoming MX80 bentonite, which is composed of crushed pellets at a relatively homogeneous granulometry (Molinero Guerra et al., 2017), and Fontainebleau sand (Figure 2). The MX80 bentonite is mostly composed of Montmorillonite (70% to 90% dry weight) plus quartz, feldspar, muscovite, and calcite. The Fontainebleau sand is a quartz sand (99.98% silica) with a narrow grain size distribution and a mean grain diameter of approximately 250 μm (Bourbié and Zinszner, 1985; Doyen, 1988). The Fontainebleau sand is characterized by a very weak polarization compared with the polarization of bentonite (compare the results of Revil et al. [2014a] for the Fontainebleau sandstone and those of El Alam et al. [2023] for pure bentonite).

The procedure for the creation of the bentonite-sand samples starts by spraying deionized water and crushed pellets of powder on the sand and mixing the three components together (Figure 2). The spraying is done until the desired water content is reached. The hydrated bentonite-sand mixture is then placed in a metallic mold and compacted to a given density using a uniaxial press with uniaxial confining pressure (see Bairlein et al., 2014). Such sample preparation leads to samples that are homogeneous, at least through visual inspection (Figure 2). The compaction states (height, bulk density, dry density, and porosity), as well as the saturations and initial water content, were measured and/or calculated after compaction. The samples have a diameter of 5 cm and heights ranging between 6 and 8 cm (see Figure 2).

Electrocardiogram (ECG) electrodes (pregelled Ag-AgCl) are then placed on the samples, and complex conductivity spectra are obtained using the high-precision impedance meter and a four electrode array, separating the voltage electrodes from the current electrodes (Figure 2). The measurements were done within 10 min after a sample was compacted to minimize its dehydration through drying. The electrodes were placed in a radial configuration, implying four electrodes being placed at 90° from each other and at the center height of each sample (see Jougnot et al., 2010; Okay et al., 2014; and Figure 2). Two current electrodes (A and B) impose the current, and then two potential electrodes (M and N) measure the potential difference.

Transforming the measured impedance (in Ω) into complex conductivity (in $Sm−1$⁠) requires a geometric factor K (in meters). We numerically determined K using COMSOL Multiphysics to be 0.23 and 0.25 m for sample heights of 6 and 8 cm (depending on the degree of compaction), respectively. The procedure to obtain the geometric coefficient is discussed in Jougnot et al. (2010) and is based on numerically solving Ohm’s law with the charge conservation equation, adding the appropriate boundary conditions for the potential and current.

The complex conductivity spectra are obtained using a high-precision MFIA impedance analyzer manufactured by Zurich Instruments (Figure 2a). The total number of measuring points per sweep was set to 100, regardless of the frequency range. Replicates indicate that the in-phase conductivity has an accuracy of 2%, whereas the quadrature conductivity has an accuracy of 5% in the frequency range investigated in this study (1 Hz–45 kHz). Examples of complex conductivity spectra are shown in Figure 3 for different bentonite and water contents and in Figure 4 at different temperatures.

When using deionized water, the true pore water conductivity remains unknown because of the interaction between water and the hydrated ions in the micro- and nanoporosity of the bentonite and possibly salt crystals. Samples E70 to E91 and E153 to E170 were hydrated with deionized water, whereas samples E149 to E152 were saturated with saline solutions (NaCl) of known conductivities. The saline solutions used for samples E149 to E152 correspond to the following four saline solution conductivities $σw$ (NaCl, 25°C) = 0.2, 1.0, 5.0, and 10 $Sm−1$ (or approximately 1.3, 6.5, 32.5, and 65 g/L), respectively. These conductivities were checked through measurements. Samples E149 to E151 all had porosities of approximately 0.58 ± 0.01, whereas E152 had a lower porosity of 0.41. These experiments were performed in order to estimate the pore water conductivity when the samples were mixed with the deionized water after one hour of equilibrium time.

Figure 5 shows the conductivity data for eight fully saturated samples at porosities of approximately 0.57 ± 0.02. These conductivity data confirm a high degree of linearity between the conductivity of bentonite, bentonite-sand mixtures, and the pore water conductivity. All the conductivity data were fitted with the best fit of the classical linear conductivity equation $σ∞=σw/F+σS$ (Waxman and Smits, 1968) to determine the formation factor F and the surface conductivity $σS$ assuming that the salinity dependence of the surface conductivity can be neglected. The salinity and pore water composition dependence of surface conductivity are discussed in Vaudelet et al. (2011a, 2011b) and Weller et al. (2011, 2013) whereas they are neglected in this study (e.g., Revil, 2012).

The volumetric water contents, porosity, and saturation of the 42 experiments are reported in Table 1. We assume no unconnected porosity is present in these granular materials. The CEC of the MX80 used in this study was determined using the cobaltihexamine method (Guillaume, 2002; Bradbury and Baeyens, 2003; Molinero Guerra et al., 2017). We obtained a value of 87.2 ± 0.64 cmol(+)/kg (81.9 meq/100 g) for three replicates. We also measured the hygroscopic water content of MX80 after drying the samples at 105°C for 120 h.

We now use the curve of the (in-phase) conductivity versus the pore water conductivity using four experiments (fully saturated, samples E149 to E152, Table 1) discussed previously to determine the pore water conductivity in the other experiments. We measured the in-phase conductivity of a sample prepared under the same conditions but with deionized water (sample E75, $ϕ$ = 0.54, fully saturated, see Table 1). The conductivity of the pore water is obtained by using the projection of the in-phase conductivity and the curve to obtain the pore water conductivity (see Figure 2). We find a pore water conductivity of 0.28 $Sm−1$ at 25°C. Because we always use the same bentonite powder, the pore water conductivity is assumed to remain the same at a given temperature for all the experiments in Table 1, with the exception of E149 to E152.

The temperature measurements were performed in a temperature-controlled bath using a protocol similar to that of Revil et al. (2018a) (see also Martinez et al., 2012; Hermans et al., 2014; Coperey et al., 2019). The temperatures range from approximately 10°C to 60°C. The maximum temperature of the waste packages in the French concept CIGEO should not exceed 100°C. The French design also implies that the maximum temperature of the surrounding rocks should not exceed 70°C. Consequently, it is highly unlikely that in nominal conditions the bentonite seals will reach temperatures higher than 60°C.

Temperature measurements concern experiments/samples E188, E189, and E190 (see Table 1). The complex conductivity spectra at the different temperatures of sample E188 are shown in Figure 4. We note an increase in the in-phase conductivity and quadrature conductivity with temperature. Because the relaxation times depend on temperature, it is possible that there is a small translation of the spectra with temperature. The samples were indeed wrapped in plastic films. Immediately after compacting the samples, we placed four ECG electrodes on the samples (90° from each other, like in Figure 2 for the measurements at 25°C) that are stable in temperature. We then tightly wrapped each sample and ECG electrodes with a plastic film. We then covered the sample with duct tape to ensure the stability and airtightness of the plastic film around the sample. Finally, we cut the plastic film around the metal ECG snap buttons to allow the electrodes to be connected to the impedance meter and, at the same time, minimize any water loss from the samples. This setup allowed us to connect an alligator clip to each electrode and thus connect each electrode to the impedance meter.

By adding sand grains to bentonite, we add another polarization length scale to the problem. However, because of its size (approximately 250 μm) and charge density, the Fontainebleau sand polarizes weakly at quite low frequencies (<1 Hz, see Revil et al., 2014a), whereas clay aggregates polarize at high frequencies (>10 Hz). In the present study, we consider the complex conductivity spectra above 1 Hz, and the polarization of the sand grains can be safely neglected. That said, it still has a small effect by decreasing the current density associated with the bentonite itself by reducing its volumetric density. This effect is a mere multiplication of the conductivities and normalized chargeability by a factor $φSd1.5$ (see Revil, 2000), in which $φSd$ denotes the volume fraction of sand. This correction factor, being relatively small, will be neglected hereinafter.

Is surface conductivity the dominant conduction mechanism in our data set? To answer this question, in Figure 9, we plotted the normalized chargeability versus the conductivity for the Table 1 data set. We observed that the samples characterized by high fractions of bentonite are usually characterized by high normalized chargeabilities and conductivity values. They are statistically close to the line with a slope of R = 0.10. This implies that for these samples, the surface conductivity has a strong influence on the overall conductivity. Indeed, the ratio between the normalized chargeability and the conductivity is equal to R only if the surface conductivity dominates the conductivity response of the rock. For sandy samples, the data are characterized by lower normalized chargeabilities and conductivities. The data are rather clustered in the domain for which surface conductivity is more and more negligible. This is an intuitive result that can be quantified here, thanks to such a crossplot.

Figure 10 indicates that the in-phase conductivity follows a power law relationship with saturation. These trends for different bentonite contents are used to determine the in-phase conductivity at saturation (see Table 2). The values at saturation are reported as a function of the bentonite content in Figure 11a. The in-phase conductivity increases with the bentonite content because of the increase of the CEC. This is consistent with the theory (equation 3). In Figure 11a, we also report the results of a few experiments performed directly at saturation and the other data points from the literature. They all exhibit the same trend in a consistent way.

In Figure 12, we plot the in-phase conductivity as a function of the water content. We see that the conductivity depends on the water content according to a power law function with an exponent of m = 2.3. Thanks to the dynamic Stern layer model, we can predict the values of the prefactor of the scaling law shown in Figure 12. Taking a grain density of $ρg=2650kgm−3$⁠, a mobility B(Na⁺, 25°C) = 3.1 × 10⁻⁹ m² s⁻¹ V⁻¹, and the value of the CEC of pure bentonite, CEC_B = 90 meq/100 g = 87 × 10³ C kg⁻¹, we have $ρgBCECB=0.75Sm−1$⁠. As the pore water conductivity is $σw=0.21Sm−1$⁠, the prefactor for the in-phase conductivity is $σ′=σ0′θ2.3$ with $σ0′=1.0Sm−1$ for pure bentonite. Figure 12 shows that the prefactor is $σ0′=1.1±0.2Sm−1$ from the experimental data and is, therefore, in close agreement with the predicted value from the dynamic Stern layer model.

In Figure 11b, we plot the quadrature conductivity as a function of the bentonite content. The quadrature conductivity appears mostly independent of the bentonite content, at least for bentonite contents higher than 30%. This means that the presence of the sand grains (with a very weak polarization) does not affect the polarization of the mixture for bentonite content higher than 0.30. For bentonite content lower than 0.30, the quadrature conductivity decreases with the bentonite content.

In Figure 13, we plot the quadrature conductivity as a function of the water content for the complete collection of data reported in Table 1. We observe a clear power law dependence between the two quantities (as predicted by the model) with a power law exponent p = 3.3. This allows fixing the value of this critical parameter for sand-bentonite mixes, which is important for water content monitoring purposes, as discussed subsequently.

Just as for the preceding in-phase conductivity, and thanks to the dynamic Stern layer model, we can predict the values of the prefactor of the scaling law shown in Figure 13. With $α$ = 8 (based on the grain size distribution covering four orders of magnitude as discussed previously) and the value of the dimensionless number R = 0.10, the prefactor of the power law relationship for the quadrature conductivity is $σ0′′=ρgBCECBR/a=0.010Sm−1$⁠. The observed prefactor is $σ′′=σ0′′θ3.3$ with $σ0′′=0.012Sm−1$ (see Figure 13), in close agreement with this prediction. We can perform a consistency test with the data shown in Figure 13 at saturation. At a water content of $θ$ = 0.6, the quadrature conductivity is given by $σ′′=0.00242Sm−1$⁠, which is again consistent with the value determined from the data in Figure 11b (⁠$σ′′=0.00230Sm−1$⁠).

In Figure 16, we plot the quadrature conductivity versus the dry density $ρd$ (at a constant temperature), and we observe a relationship consistent with the form of equation 10 using the same input parameters for the Stern layer as discussed previously. This shows the predictive ability of the dynamic Stern layer model to capture such a data trend.

We now discuss the consistency of the results obtained for bentonite with the other data sets of soils and volcanic rocks that are also rich in smectite. Figure 17 shows Archie’s law (that is, the relationship between the formation factor and the porosity) for a collection of rock samples rich in smectite. We choose mostly volcanic rocks, for which the smectite content is associated with the alteration of the volcanic glasses. As we can see in Figure 17, bentonite data (from Figure 4) are consistent with the large data set shown in this figure with a cementation/porosity exponent m = 2.2 consistent with the trend discussed previously in Figure 12 between the in-phase conductivity and the water content.

In Figure 18, we plot a broad data set of experimental data in terms of surface conductivity versus the reduced CEC, defined as the CEC divided by the bulk tortuosity of the pore space (that is, the product of the formation factor by the porosity; see the last term of equation 3). Here again, the bentonite data (saturated bentonite, see Figure 4) show great consistency with the data set of rocks rich in smectite being, as expected, an end member in the trend.

The result is even more striking in Figure 19, wherein we plot the normalized chargeability versus the surface conductivity. The slope of the trend corresponds to the dimensionless number R discussed previously. The data are consistent with the value R = 0.10, as discussed previously. We can, therefore, conclude that bentonite is really an end member of smectite-rich porous media, and the properties of bentonite are consistent with published data for rocks and soils that are rich in smectite.

There are two questions we want to tackle in this discussion. The first is related to the potential application of IP to monitor bentonite seals in the context of DGDs. We can perform frequency- or time-domain IP measurements to image, in 3D, the conductivity and normalized chargeability at different times (going from one domain to the other is done through a mere Fourier transform). We know the conductivity of the pore water from chemical modeling (possibly including the effects of alkalinity associated with the presence of cement) and the effect of temperature, thanks to the results shown in the present paper. This would require, however, determining the relationship between the normalized chargeability determined for a given period of current injection (by default 1 s) and the normalized chargeability covering the entire spectrum $Mn(∞,0)$⁠, as discussed previously. A correction factor (>1) should be applied to the inverted normalized chargeability data, and we will develop some relationships to determine this correction factor. Then, for each pixel describing a 3D tomogram, we have two dynamic results, namely the conductivity and the normalized chargeability. From these two measurements, we can determine the water content and CEC at each time step. If the clay mineralogy remains the same, only the water content is expected to change over time. However, if the pH changes, we can observe the illitization of the smectite in the presence of potassium, leading, in turn, to a change in the CEC over time (this process is discussed in the next paper of this series; see El Alam et al., 2024).

The second question of interest is what could be done to improve our model. First, we should extend the frequency domain to a higher frequency (perhaps 10 MHz) to be sure we can fit the complex conductivity spectra to account for double-layer polarization and Maxwell-Wagner polarization (see De Lima and Sharma, 1992). The second point would be to modify our model to account for the dual water, such as the bulk pore water and the hydration water, as done by Clavier et al. (1984). The third ingredient would be to consider a surface Archie’s law for the surface conductivity and the quadrature conductivity/normalized chargeability (see preliminary work by Wang and Revil [2020], for surface conductivity in fractal-type porous media). Such an extension will be developed in a future paper of this series (Part 7).

In the previous papers of this series, we have been dealing with the effect of desiccation in anisotropic clay rocks and the influence of pyrite in illite-pyrite mixtures upon the complex conductivity of such clayey materials. In the present paper, we have been dealing with the effect of saturation and the weight fraction of sand in sand-bentonite mixtures. This opportunity offers another way to test the predictions and limits of the dynamic Stern layer model we have developed in this series of papers.

To reach this goal, we performed 42 new experiments to investigate the effect of the water content on the complex conductivity of sand-bentonite mixtures that can be used as seals for the storage of high-level, long-lived radioactive wastes in underground repositories. At a given saturation, the in-phase conductivity linearly depends on the bentonite content because of the increase of the CEC with the amount of smectite. The quadrature conductivity and the normalized chargeability are independent of the mass fraction of bentonite (at least for mass fractions higher than 10%).

The in-phase and quadrature conductivities depend on the water content according to power law relationships, with a power law exponent of m = 2.3 for the in-phase conductivity and p = 3.3 for the quadrature conductivity. The in-phase and quadrature conductivities linearly depend on temperature in the temperature range of 10°C–60°C with a coefficient of 2.1%/°C to 2.8%/°C. When compared with other clay-rich material, the bentonite and bentonite-sand mixtures appear as an end member because of the very high CEC of smectite. This work illustrates the usefulness of the spectral IP method because of its sensitivity to changes in water content in complex media such as sand-bentonite mixtures. The model could be used to interpret field data in a wide variety of contexts, including for volcanoes and landslides.

We thank the editor, M. Sen, the associate editor, and four anonymous reviewers for their constructive comments regarding our manuscript. This work is cofunded by the Institut de Radioprotection et de Sûreté Nucléaire (IRSN) and by the European Union within the MODATS program.

Data associated with this research are confidential and cannot be released.

Biographies and photographs of the authors are not available.