## ABSTRACT

Source rocks for oil and gas are often associated with shales that are rich in pyrite and kerogen. Induced polarization is a suitable tool to characterize these formations. We develop a new experimental database of 43 laboratory experiments using mixtures of clay and pyrite under fully or partially water-saturated conditions. The liquid water saturation is in the range of 20%–100%, whereas the pyrite content is in the range of 0%–17%. Spectral-induced polarization measurements are performed in the frequency range of 0.1 Hz to 45 kHz at room temperature (approximately 25°C ± 1°C). The complex conductivity spectra are fitted with a Cole-Cole model, and the Cole-Cole parameters are determined using a stochastic procedure based on a Markov chain Monte Carlo sampler. The Cole-Cole parameters associated with low-frequency dispersion are then plotted as a function of the (water) saturation and pyrite content (volume fractions). Four predictions of the model are tested against the experimental data. We find that the model is able to explain the results, including (1) the chargeability depends on the pyrite content in a predictable way, (2) the (instantaneous) conductivity depends on the saturation according to Archie’s law, (3) the Cole-Cole exponent does not depend on the saturation and pyrite content (except at very small pyrite content of <1 vol%), and (4) the relaxation time is inversely proportional to the instantaneous conductivity. We also develop a more advanced petrophysical model using a double Cole-Cole distribution, in which one distribution is associated with the clay minerals and the other is associated with the pyrite.

## INTRODUCTION

Induced polarization has a long history in the realm of exploration geophysics (Schlumberger, 1920; Pelton et al., 1978; Kemna et al., 2014). This method is considered a suitable approach to localize ore bodies, including in clay-rich formations (e.g., Hallof and Klein, 1983; Hall and Olhoef, 1986). Source rocks for oil and gas are often rich in pyrite, and induced polarization is a suitable tool to characterize these formations (Clavier et al., 1976; Misra et al., 2016a, 2016b). In addition, this method has also been used in hydrogeophysics to study environmental problems, including contamination plumes (Martin et al., 2020), leakages in dams and embankments (Abdulsamad et al., 2019), clay properties used for the storage of wastes (Revil, 2013a, 2013b; Mendieta et al., 2021, 2023), and in biogeophysics because of its sensitivity to roots (Zanetti et al., 2011) and bacteria (Revil et al., 2012).

A complete model describing the induced polarization properties of ore bodies immersed into porous and polarizable media has been recently described by Revil et al. (2015a, 2015b) (see also Mao and Revil, 2016a; Mao et al., 2016b; Misra et al., 2016a, 2016b; Revil et al., 2017a, 2017c). Other mechanistic models exist, including those based on the redox reactions at the surface of the metallic particles (Wong, 1979) and the polarization of the electrical double layer (Bücker et al., 2018a, 2018b). Furthermore, other models have been developed, including, for instance, empirical and semiempirical models (e.g., Dias, 2000; Gurin et al., 2015, 2019; Gurin and Titov, 2023) or models based on the differential effective medium theory neglecting, however, a precise description of the underlying physicochemistry and fundamental mechanisms for the transport and storage of charge carriers (e.g., Zhdanov et al., 2012; Maineult et al., 2017, 2018; Bücker et al., 2018a, 2018b). Solving the relevant local-based Nernst-Planck equations can also be used to understand the fundamental mechanisms at play, especially the dependence of the relaxation time on the instantaneous conductivity of the material (Gurin et al., 2013, 2015; Abdulsamad et al., 2017; Revil et al., 2018, 2022; Wu et al., 2022). Only the models involving an internal polarization mechanism can explain the dependence of the relaxation time on the instantaneous conductivity of the material.

Most laboratory experiments are usually performed in saturated conditions. Some experiments have been performed with saturated sand and pyrite mixtures (Mahan et al., 1986; Hupfer et al., 2016) as well as with clay-bearing pyrite in desiccation conditions (Tartrat et al., 2019) and sand-pyrite mixtures (Martin et al., 2022). However, an integrated study of the frequency-dependent complex conductivity of a sand-pyrite mixture in an unsaturated conduction has been missing to date. In the two previous papers of this series (Revil et al., 2023a, 2023b), we study the effect of desiccation and anisotropy of mudrocks that were essentially pyrite free. We wish to extend our approach to pyrite-rich anisotropic shales, and, as a first approach, we study here synthetic isotropic mixtures made of illite, pyrite, and water.

Therefore, we propose a new set of 43 laboratory experiments to elucidate the effect of saturation and pyrite content using mixtures of illite and pyrite partially saturated by electrolyte (0.8 S m^{−1}, NaCl, 25°C) mixes, and we test further the formula proposed in the previous papers of this series (Revil et al., 2013a, 2013b). We cover the saturation range of 0.2–1.0 and the pyrite content of 0–0.17. Above a pyrite content of 0.22, the chargeability is close to one (Revil et al., 2015a, 2015b) and stays close to this value, whereas, in the range of 0–0.17, there is a rapid change in the chargeability with the pyrite content. Complex conductivity data are performed with an impedance meter in the frequency range of 0.1 Hz to 45 kHz.

## MODEL

### Single Cole-Cole model

*M*denotes the chargeability (dimensionless),

*c*is the Cole-Cole exponent (dimensionless), and $\tau $ denotes the (Cole-Cole relaxation) time constant (expressed in s). The quantities $\sigma 0$ and $\sigma \u221e$ denote the direct current (DC) and instantaneous conductivity, respectively. For the induced polarization mechanism under consideration, they correspond to the low and high asymptotic behavior of the conductivity. In time-domain induced polarization, the DC conductivity corresponds to the conductivity observed after a long application of the external electrical field and the instantaneous conductivity observed just after the application of the electrical field, respectively.

^{−1}),

*F*denotes the intrinsic formation factor (dimensionless) related to the connected porosity $\varphi $ (dimensionless) by the first Archie’s law $F=\varphi \u2212m$, where $m>1$ is called the porosity or first Archie’s exponent (dimensionless) (Archie, 1942) and $n>1$ denotes the saturation exponent (dimensionless). The mobility $B$ denotes the apparent mobility of the counterions for surface conduction (m

^{2}s

^{−1}V

^{−1}) and $\lambda $ denotes the apparent mobility of the counterions for the polarization associated with the quadrature conductivity (m

^{2}s

^{−1}V

^{−1}). A dimensionless number

*R*has also been introduced previously as $R=\lambda /B$ (for further explanations, see Revil et al., 2017b). From our previous studies (e.g., Ghorbani et al., 2018), we have $\beta $ (Na

^{+}, 25°C) = 3.1 ± 0.3 × 10

^{−9}m

^{−2}s

^{−1}V

^{−1}, $\lambda $ (Na

^{+}, 25°C) = 3.0 ± 0.7 × 10

^{−10}m

^{−2}s

^{−1}V

^{−1}, and the dimensionless number

*R*is approximately 0.10 ± 0.02 (independent of saturation and temperature). It corresponds to an upper limit of chargeability of 100 mV/V in the absence of metallic particles.

*a*larger than 500 μm, the relaxation time is written as

Our goal in this paper is to check the predictive power of the previous set of equations for partially saturated clay-pyrite mixtures. This includes four predictions that are as follows:

The Cole-Cole exponent should be independent of the saturation and pyrite content (except for a very small amount of pyrite for which their polarization is smaller than the polarization associated with clay particles).

The Cole-Cole relaxation time should be inversely proportional to the instantaneous conductivity of the mixture. Martin et al. (2022, their equation 18) also notice that the relaxation time is related to the bulk electrical conductivity of the material. Gurin et al. (2015, their equation 12) also relate the relaxation time to the pore water electrical conductivity, which in turn is related to the bulk conductivity when the bulk conductivity dominates over the surface conductivity, i.e., generally at high salinities or low cation exchange capacity (CEC) materials. The CEC denotes the cation exchange capacity of the material (expressed in C kg

^{−1}and often expressed in meq/100 g with 1 meq/100 g = 963.20 C kg^{−1}).The dependence of the chargeability with the pyrite content should be consistent with equation 7, with the background chargeability being dependent on the saturation according to equation 9.

At high salinity, the (instantaneous) conductivity of the mixture follows Archie’s law with the saturation, and the second Archie’s exponent of the mixture

*n*can be recovered. The dependence of the (instantaneous) conductivity of the mixture on the pyrite content should be predictable as well. At low frequencies, the conductivity of the mixtures of pyrite and illite should be smaller than the conductivity of the illite alone. This is because at low frequencies, the pyrite grains are fully polarized, and they appear as insulators.

### Double Cole-Cole model

## MATERIALS AND METHODS

We performed complex conductivity measurements with a high-precision impedance meter, namely the ZEL-SIP04-V02 (Zimmermann et al., 2007, 2008). The complex conductivity spectra were obtained in the frequency range between 0.1 Hz and 45 kHz at room temperature (approximately 25°C). A sketch of the experimental setup is shown in Figure 3, and we follow the approach discussed in the two previous papers of this series (Revil et al., 2013a, 2013b). The measured impedance was converted into complex conductivity using the geometric factor calculated based on the geometry of the sample holder and the position of the ABMN electrodes (AB denotes the current electrodes, and MN denotes the voltage electrodes).

The (background) clay material is a natural illitic material. Its CEC was determined using the cobalt hexamine chloride method (Guillaume, 2002). We found CEC = 9.74 meq/100 g or 9402 C/kg (two replicates) (Table 1). The complex conductivity spectra of the illite alone were first obtained at the four pore water solutions at 25°C (NaCl). The conductivity (at 1 Hz) is shown in Figure 4a. The formation factor *F* and surface conductivity $\sigma S=\rho gBCEC/F\varphi $ are determined by fitting the conductivity equation for the background to these data, and the values are reported in Table 1. The formation factor *F* is 2.1, and the surface conductivity is 0.021 S m^{−1}. We can check that the value of the surface conductivity is consistent with the value of the CEC and formation factor. Using equation 7, $\sigma S=\rho gBCEC/F\varphi $ and with a grain density $\rho g=2650\u2009\u2009kg\u2009m\u22123$, *F* = 2.1, $\varphi =0.65$, CEC = 9401.5 C kg^{−1}, and $\beta $ (Na^{+}, 25°C) = 3.1 ± 0.3 × 10^{−9} m^{−2}s^{−1} V^{−1}, we obtain a surface conductivity of 5.6 × 10^{−2} S m^{−1}, slightly larger than the measured value of 2.1 × 10^{−2} S m^{−1}.

We also performed complex conductivity experiments to study the saturation behavior of the background properties. The dependence of the background conductivity and background chargeability on the saturation is provided in Table 2 and shown in Figure 4b. The background conductivity follows Archie’s type behavior with a low value of the saturation exponent *n* close to one. The background chargeability seems to increase slightly with the decrease in the saturation but the scatter in the data points does not allow to properly fit these data with the model discussed previously.

This illite is mixed with pyrite cubes of various sizes to obtain the desired metal content $\phi m$ in the range of 0%–17%. The volume of pyrite is first determined by immersion. Then, all of the experiments performed in this paper (see Table 3) were done with a pore water conductivity of 0.8 S m^{−1} (NaCl, 25°C). For the experiments, a homogeneous mixture was prepared with the background clay, the electrolyte at 0.8 S m^{−1}, and the desired saturation and pyrite cubes to mimic the mixture shown in Figure 1. Once prepared, we closed the core sample holders (Figure 2) and let them rest for approximately 10 days. A total of 43 experiments (reported in Table 3) are performed.

A typical set of complex conductivity spectra is shown in Figure 5 at a given saturation (30% in this case). The data show a clear polarization that increases with the increase in the pyrite content, as expected. These data can be fitted by a Cole-Cole model, and we use a Markov chain Monte Carlo (MCMC) sampler to fit the Cole-Cole parameters for each complex conductivity spectrum. The values of the Cole-Cole parameters are reported in Table 3. The fit of the complex conductivity spectra is shown in Figures 6, 7, 8, 9, and 10. We can say that the Cole-Cole model provides a good descriptor of the polarization effect. Then, in the next section, we discuss the dependence of the Cole-Cole parameters on the properties of the mixtures.

## RESULTS

### Single Cole-Cole model

First, in Figure 11, we check that the Cole-Cole exponent *c* is independent of the saturation and pyrite content (except for small pyrite content <1 vol% for which we recover the Cole-Cole exponent that is associated with an illite of approximately 0.50, not shown here). Because the Cole-Cole exponent described the broadness of the polarization length scales, this is not surprising.

In Figure 12, the Cole-Cole relaxation time is plotted as a function of the instantaneous conductivity. We observe, despite the scatter in the data, that the Cole-Cole relaxation time is inversely proportional to the instantaneous conductivity of the material. In Figure 13, we plot the chargeability of the mixture as a function of the pyrite content. We check that the data are grossly consistent with the prediction of the model given in equation 4.

In Figure 14a and 14b, we plot the instantaneous conductivity as a function of the saturation at low and high pyrite contents. We check, in agreement with the model, that the dependence on saturation of the instantaneous conductivity does not depend on the pyrite content. In Figure 15, we check that the increase in the instantaneous conductivity with the pyrite content is grossly consistent with the prediction of the model. Note that the value of the saturation exponent *n* we found is low (*n* = 1.25) but inside the range of realistic values (*n* comprised between 1 and 4).

### Double Cole-Cole model

*n*, and the reduced relaxation time $\tau \u02dc1=\tau 1\sigma \u221e(sw,\phi m)$, so there are only three parameters. The other parameters,

*F*, $\varphi $, CEC, and $\rho g$, are provided in Table 1. Finally, we use $\beta $ (Na

^{+}, 25°C) = 3.1 ± 0.3 × 10

^{−9}m

^{−2}s

^{−1}V

^{−1}and $\lambda $ (Na

^{+}, 25°C) = 3.0 ± 0.7 × 10

^{−10}m

^{−2}s

^{−1}V

^{−1}for the two mobilities and a pore water conductivity $\sigma w=0.8\u2009\u2009S\u2009m\u22121$ (NaCl, 25°C).

A comparison between the double Cole-Cole model and the experimental data for saturation of 20% is shown in Figure 16. The model is able to explain the data, but the fits are not perfect. With a relaxation time of $\tau 2=2\xd710\u22124\u2009\u2009s$ (for the entire data set), we obtain the following values for the three other parameters: $c1$ = 0.7 ± 0.2, *n* = 1.9 ± 0.7, and the reduced relaxation time $\tau \u02dc1=2.2\xb10.9\u2009\u2009S\u2009m\u22121$ for pyrite. The value of $c1$ is what we expect from Figure 11b. The value of *n* conforms to the expected value for the saturation exponent, generally close to the cementation exponent *m*. Finally, the value of $\tau \u02dc1$ is very well determined with a small standard deviation indicating that the relaxation time $\tau 1$ is indeed inversely proportional to the conductivity of the material.

## CONCLUSION

We performed complex conductivity experiments on 43 mixtures of illite/pyrite and an electrolyte to study the effect of the pyrite content and saturation. The complex conductivity spectra can be accurately fitted with a single Cole-Cole model at least above a critical content of pyrite or a double Cole-Cole model accounting for the polarization spectra of the clay particles and pyrite grains. The Cole-Cole parameters are given physical interpretation thanks to a mechanistic model. We check that the Cole-Cole exponents do not depend on the saturation. The instantaneous conductivity depends on the saturation according to a power law relationship with a power law exponent given by the second Archie’s exponent *n*. The chargeability depends linearly on the pyrite content, with a background chargeability that depends on the saturation. Finally, the Cole-Cole relaxation time is inversely proportional to the instantaneous conductivity, which depends on saturation in a predictable way. The proposed double Cole-Cole model represents a step forward in being able to model such complex mixtures accounting for a few textural parameters and state variables.

In the future, we could check if the data obtained in the course of this analysis could be compared with numerical experiments using a network approach. We will also apply our approach to real source rocks that are rich in pyrite and kerogen also accounting for their anisotropy. If we can explain the polarization characteristic of such complex media quantitatively, this means that we could interpret the field data for any complex situation in oil reservoirs and source rocks.

## ACKNOWLEDGMENTS

We thank E. Zimmerman at Forschungszentrum Jülich for developing and building the low-frequency impedance spectrometer used here. We also thank the editor X. Garcia and the three reviewers for their time and useful comments that improved our manuscript. This work is partially funded through the PALLAS project.

## DATA AND MATERIALS AVAILABILITY

Data associated with this research are available and can be obtained by contacting the corresponding author.

Biographies and photographs of the authors are not available.