# Induced polarization of volcanic rocks. 8. The case of intrusive igneous rocks

Loris Piolat, André Revil, Jessy Richard, Ahmad Ghorbani, Pierre Cosme, Yves Géraud, Clément Casotti, Pierre Vaudelet, Marc Diraison, Alexiane Favier

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Loris Piolat, André Revil, Jessy Richard, Ahmad Ghorbani, Pierre Cosme, et al. Induced polarization of volcanic rocks. 8. The case of intrusive igneous rocks. Geophysical Journal International, 2025, 241, pp.1348-1372. 10.1093/gji/ggaf102. hal-05033936

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# Induced polarization of volcanic rocks. 8. The case of intrusive igneous rocks

L. Piolat,<fnref n="1" /> A. Revil,<fnref n="2" /> J. Richard,<fnref n="2" />,<fnref n="3" /> A. Ghorbani,<fnref n="3" /> P. Cosme,<fnref n="1" /> Y. Géraud,<fnref n="1" /> C. Casotti,<fnref n="2" />,<fnref n="4" /> P. Vaudelet,<fnref n="3" /> M. Diraison<fnref n="1" /> and A. Favier<fnref n="1" />

Accepted 2025 March 12. Received 2025 March 4; in original form 2024 December 9

## SUMMARY

In the previous papers of this series, we have developed an in-depth analysis of the low-frequency complex conductivity response of volcanic (extrusive) rocks. We showed that the alteration of these rocks plays a key-role in determining their induced polarization properties, especially regarding the formation of smectite in response to the thermo-activated alteration of the volcanic glasses. We also considered the effects associated with the presence of magnetite and pyrite. In this paper, we look at the induced polarization properties of igneous rocks like granites and granitoids. Usually, the alteration path of these rocks leads to the formation of kaolinite, a clay mineral with a much lower Cation Exchange Capacity (CEC) than smectite. Thirty-three core samples from three sites in France are saturated with NaCl solutions at three salinities (pore water conductivity of 0.1, 1.0 and 10 S m⁻<fnref n="1" />, 25 °C) and their complex conductivity spectra are measured in the frequency range 0.01 Hz–45 kHz. As observed for volcanic rocks, the surface conductivity, normalized chargeability and quadrature conductivity depend strongly on the CEC of the rock, which is independently measured with the cobalt-hexamine method. The (intrinsic) formation factor follows an Archie’s type relationship with the connected porosity with a porosity (cementation) exponent of *m* = 1.70 ± 0.02, much smaller than for volcanic extrusive rocks. Like for volcanic rocks, a dynamic Stern layer model can be used to illustrate the behaviour associated with the clay-minerals (mostly kaolinite). A field investigation is conducted in the Vosges (France) using a deep time-domain induced polarization survey reaching at a depth of investigation ~400 m. We show how the electrical conductivity and the normalized chargeability can be used to image the water content and CEC of the granitic substratum. The conductivity of granite is found to be dominated by surface conductivity rather than by bulk conductivity and therefore Archie’s law cannot be used as a conductivity equation to interpret field data as commonly done in ElectroMagnetic surveys.

**Key words:** Electrical properties; Electrical resistivity tomography (ERT); Induced polarization; Tomography.

## 1 INTRODUCTION

In the previous papers of this series (Revil et al. 2017a, b), we have developed a petrophysical model explaining the induced polarization properties of volcanic rocks from various types of volcanisms. We investigated the case of shield volcanoes (Revil et al. 2021), strato-volcanoes (Zhang et al. 2023) and kimberlites (Titov et al. 2024). Furthermore, we demonstrated how induced polarization can be used to image clay-caps of geothermal systems with an example from Krafla volcano in Iceland (Revil et al. 2019). Our model accounts for the presence of smectite (resulting from the alteration of

the volcanic glasses) and the presence of semiconductors like pyrite and magnetite (see also Gurin et al. 2015; Misra et al. 2016a, b; Emelyanov et al. 2020; Peshtani et al. 2022).

Our approach and model extend previous efforts made in the realms of both ore body exploration and detection as well as in hydrogeophysics to better understand how induced polarization data can be interpreted in field conditions in terms of hydraulic properties of interest (e.g. Schwarz 1962; Macnae 1979; Olhoeft 1981b; Titov et al. 2002; Weller et al. 2013; Glaser 2021; Fu et al. 2024; Revil et al. 2024). For volcanic edifices, understanding the alteration of volcanic rocks is of paramount importance to better comprehend

<footnote n="1">¹Université de Lorraine, GeoRessources, Ecole Nationale Supérieure de Géologie, 54505 Vandoeuvre-lès-Nancy, France</footnote>

<footnote n="2">²Université Grenoble Alpes, Université Savoie Mont-Blanc, CNRS, UMR CNRS 5204, EDYTEM, 73370 Le Bourget du Lac, France.</footnote>

<footnote>E-mail: andre.revil@univ-smb.fr</footnote>

<footnote n="3">³Någa Geophysics, 229 Rue Joseph Fontanet, 73000 Chambéry, France</footnote>

<footnote n="4">⁴GINGER CEBTP, 680 rue Aristide Bergès Parc d'Activités du Pré Millet, 38330 Montbonnot-Saint-Martin, France</footnote>
the mechanical stability of volcanic edifices and the permeability of these rocks, which, in turn, may play a key role in the dissipation of fluid overpressure mechanisms and the collapse of the edifices (Albino et al. 2018).

To date, and to the best of our knowledge, there are no works focusing on understanding the induced polarization properties of granitic rocks or intrusive igneous rocks in general. Intrusive igneous rocks result from the slow crystallization of magma at depth forming intrusions such as batholith, dikes and laccoliths. Few papers have been published regarding the electrical conductivity of granite. Furthermore, published papers have been focusing on the high-temperature (>800 °C) and high-pressure (>100 MPa) dependence of the electrical conductivity of these rocks (e.g. Watanabe 1972; Olhoeff 1981a; Shanov et al. 2000; Dai et al. 2014; Guo et al. 2017). In addition, very few efforts have been done to obtain their formation factor (one of the main ingredients describing the bulk electrical conductivity of these rocks below 200 °C). Even less works have been done concerning the study of their surface conductivity associated with the presence of alumino-silicates with the exception of Le Ber et al. (2022). However, the work by Le Ber et al. (2022) lacks interfacial property characterization such as cation exchange capacity (CEC) or BET (named from Stephen Brunauer, Paul Hugh Emmett, and Edward Teller) specific surface area measurements. Therefore, it cannot be used for a quantitative assessment of the electrical conductivity of granite. Induced polarization data of granitic-type rocks are really scarce (see for instance Duvillard et al. 2018; Coperey et al. 2019, and recently Revil et al. 2024, for measurements and quantitative assessments using few rock samples).

Induced polarization is an extension of the conductivity method to include the polarization (reversible charge storage) of the rock under an applied electrical field (e.g. Olhoeff 1981a, b; Maineult et al. 2017, 2018). Brenner et al. (1992) were probably the first to perform spectral induced polarization measurements on granites and to recognize the role of surface conductivity in the interpretation of well-log resistivity data in the continental deep drilling project (KTB) borehole. Bai et al. (2022) performed spectral conductivity measurements at different water saturations and Gomaa (2022) performed broad-band frequency complex conductivity measurements using eight core samples and made qualitative assessments regarding their results. The alteration of granitic rocks tends to produce kaolinite (Schiavon 2007 and references therein), a clay mineral characterized by a much lower cation exchange capacity than smectite (e.g. Vinegar & Waxman 1984), which was observed as a result of the alteration for volcanic extrusive rocks (e.g. Zhang et al. 2023).

Our goals in this paper are: (1) to develop the first extensive data set of induced polarization data for 33 well-characterized granitic rocks from three sites in France; (2) to see if the results could be explained by the petrophysical model developed in the previous papers of this series; (3) to present a field survey to show how the findings developed on the basis of these laboratory data can be used for the interpretation of time-domain induced polarization data over a granitic basement. The field application concerns an outcropping granitic batholith in the Vosges area (France) and is based on the recently developed FullWaver system from IRIS.

# 2 PETROPHYSICAL STUDY

## 2.1. General considerations

Granites correspond to the main coarse-grained intrusive crystalline rocks located in the continental crust of the Earth as igneous intrusions. They range from dykes with few decimeters thickness to

large-scale batholiths exposed over hundreds of square kilometers. They form from silica-rich magma cooling slowly underground until they crystallize (slower crystallization implies coarser grains). Three minerals dominate their composition: quartz, alkali feldspar and plagioclase. In addition, some mica and/or amphibole minerals plus some secondary minerals like kaolinite can be observed. They may also contain semiconductors like pyrite or magnetite (Fig. 1).

In water-saturated conditions with acidic pore waters, the alteration of granites involves the transformation of primary minerals (mica and alkali feldspars) into secondary clay minerals especially kaolinite Al₄(OH)₈[Si₄O₁₀]. For instance, the alteration of orthoclase (K[AlSi₃O₈], a potassium feldspar) and albite (Na[AlSi₃O₈], a plagioclase) yields (see Stoch & Sikora 1976; Yatsu 1988; Wilson 2004; Nguetnkam et al. 2008):

$$4\text{Me}[\text{AlSi}_3\text{O}_8] + 4\text{H}_2\text{O} + 2\text{CO}_2 \rightleftharpoons \text{Al}_4(\text{OH})_8[\text{Si}_4\text{O}_{10}] + 2\text{MeCO}_3 + 8\text{SiO}_2, \quad (1)$$

where Me = K, Na (potassium and sodium). Usually, the alteration path involves the slow dissolution of alkali feldspars and plagioclase (with a typical timescale of ~100 000 yr) and the precipitation of kaolinite (see Blum 1994). Schiavon (2007) demonstrated that granite weathering leads to the formation of authigenic kaolinite. This process can be accelerated in presence of SO₂ in acidic conditions. Kaolinites, alongside other clayey minerals, can also be indicators of hydrothermalism such as for the quartz monzonite of Mount Princeton (Colorado, see Richards et al. 2010). A weathering classification standard for granite has been developed by Woo et al. (2006) and weathering and alteration can be connected to the seismic velocities of the core samples (Woo et al. 2006; Lin et al. 2014, 2021).

Since kaolinite is characterized by a lower CEC than smectite (3 to 15 meq/100 g for kaolinite versus 80 meq/100 g for smectite, 1 meq/100 g = 963.20 C kg⁻¹ in SI units), but also by a larger particle size typically around one micrometer, the CEC of disaggregated weathered granite is in the range 6–16 meq/100 g (ammonium method) according to Rider et al. (2006). This means that the induced polarization of intrusive rocks is expected to occur at lower frequencies than for extrusive volcanic rocks. Granitic rocks often contain pyrite, which is a semiconductor exhibiting a strong polarization. Magnetite can be also present in granite (see for instance Maulana et al. 2013). Therefore, the composition of the granites in coarse grains, authigenic kaolinite and pyrite (possibly magnetite) is believed to influence their complex conductivity spectra as discussed below in more detail (see Fig. 1). The pore structure of granite is also characterized by a high degree of self-similarity (e.g. Pape & Schopper 1987; Wang et al. 2023), which could translate into flat spectra at low frequencies because of the broad (multiscale) distribution of the associated polarization length scales.

## 2.2. Core samples: origin and characterization

Our 33 core samples are divided into three subsets, each coming from a specific area in France (see Fig. 2). Data set 1 (see Table 1) is composed of 13 cubic core samples (~4 cm in size) from the Middle-Armoricán Batholith in French Brittany (Armoricán Massif, see Fig. 2). This batholith belongs to the Variscan (Hercynian) orogeny. The core samples are leucogranites of age 320 ± 8 Ma (Augier et al. 2015). These tight granites are characterized by coarse-grained facies and a pinkish-grey colour (Fig. 3).

Data set 2 is composed of 10 core samples from the Senones batholith in the Vosgian Mountains (Fig. 2), which are part of the Variscan orogeny (Guillot et al. 2020). These tight granites are characterized by coarse-grained facies and a greyish-pink colour
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Induced polarization of granite. (a) Granite with kaolinite associated with alteration and pyrite. (b) The surface of kaolinite is characterized by the presence of an electrical double layer. The external electrical field $E_0$ polarizes the double layer of the kaolinite particles (modified from Revil et al. 2023). (c) Polarization of pyrite (modified from Revil et al. 2023). The electro-diffusion of the charges carriers inside the semi-conductor (electrons and p-holes) is responsible for the polarization of the grain. The parameter M (dimensionless) corresponds to the chargeability of the material reflecting the ability of the material to reversibly store electrical charges (DL denotes field induced diffuse layers). The chargeability of the granite is a composite of the chargeability associated with the kaolinite (and other non-metallic minerals) and the chargeability associated with the pyrite. The quantities $\sigma_0$ and $\sigma_\infty$ denote the dc—direct current and instantaneous conductivities, respectively. They correspond to the low-frequency and high-frequency asymptotic limit of the induced polarization phenomenon.</figcaption></figure>

(Fig. 3). These samples have been taken from the field case discussed at the end of this work. A complete discussion of the geological context for these granites is given below in Section 4.1.

Data set 3 is composed of eight samples coming from the Black Mountain in the SouthWestern end of the Massif Central, France (Fig. 3). The outcropping granite is 338 ± 2 my, highly fractured and characterized locally by the presence of pyrite (Vallance et al. 2004). Finally, two additional end-members samples are added to the data set, one young poorly altered granite sample collected in the French Alps (Mont Blanc massif, 303 ± 2 my) and a sample of pure kaolinite, sample ABM-ABK (see Tables 1–3). The complete data set comprises therefore 33 samples.

In order to interpret the laboratory measurements, two properties of the core samples need to be independently measured: their (connected) porosity and their cation exchange capacity. The porosity of the core samples (reported in Table 1) was determined using their dry and water-saturated weights as well as the volume of the core samples obtained by immersion. For the dry weights, the core samples were dried in the oven at 58 °C until they reach stable (dry) weights. The porosity of the fresh (unaltered) core samples is typically around 1 to 2 per cent while the porosity of the altered core

sample can be in the range 10–20 per cent, therefore substantially higher.

The CEC measurements were obtained using the cobalt-hexamine technique. Two grams of crushed rocks were used to measure the CEC according to the procedure developed by Aran et al. (2008). The measurements are reported in Table 1. The CEC of kaolinite CEC(K) is typically 3 to 15 meq/100 g and the CEC of the granite is therefore $CEC = \phi_w$CEC(K) where $\phi_w$ (unitless) denotes the weight fraction of kaolinite in granite. The CEC of the pure kaolinite sample is 10.1 meq/100 g. The CEC of the fresh granite core samples are comprised between 0.2 and 2 meq/100 g while the CEC of the altered core samples are in the range 5–14 meq/100 g so in-between the two previous end-members.

The CEC versus porosity data are shown in Fig. 4 for all the core samples. The data follows a clear trend corresponding to an alteration trend associated with the kaolinization process of the fresh granite. Fresh samples are characterized by low porosities (below 1 per cent) with CEC values below 0.4 meq/100 g. The CEC and porosity values of granite increase with alteration and weathering and are somehow correlated to a certain extent.
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Simplified map of the major lithotectonic domains of Western Europe, adapted from Tabaud (2012) and Martínez Catalán *et al.* (2024). Site 1 corresponds to the late-orogenic granite of the Quiberon Peninsula (GPS coordinates: 47.516 900; -3.150 576). Site II corresponds to the Senones granite in the Northern Vosges region (GPS coordinates: 48.389 542; 6.998 542). Site 3 corresponds to the Lampy granite, part of the late-Hercynian (Variscan) batholiths of the Montagne Noire (GPS coordinates: 43.399 1633; 2.168 6731). Position of the three test sites in France with the three corresponding data sets for the core samples. The field case corresponds to the position of Data set 2 (Vosges area).</figcaption></figure>

## 2.3. Formation factor and surface conductivity

We performed frequency-domain induced polarization measurements over the frequency range 10 mHz–45 kHz at three salinities (pore water conductivity of 0.1, 1.0 and 10.0 S m⁻¹, 25 °C, NaCl). The measurements were acquired with the ZELSIP04-V02 impedance meter developed in Germany by Egon Zimmermann (see Zimmermann *et al.* 2008; see also Revil *et al.* 2013; Okay *et al.* 2014). The methodology used to do the measurements has been extensively described in the previous papers of this series and will not be repeated here. As a side note however, the spurious effects of parasitic couplings (e.g. Wang & Slater 2019) are avoided by using fibre optic cables. Furthermore, the sample holder does not contain metallic pieces. Three saline solutions were used corresponding to the following pore water conductivities σ<sub>w</sub>(NaCl, 25 °C) = 0.1, 1.0 and 10 S m⁻¹. Here again, the procedure used for the saturation is extensively described in Zhang *et al.* (2023) and will not be repeated here.

The complex conductivity spectra are shown in Figs 5–7. The in-phase conductivity depends poorly on the frequency in the range

10 mHz–45 kHz. The quadrature conductivity exhibits a strong frequency-dependence at high frequencies showing clearly a plateau at low frequencies (<100 Hz, characterizing the Induced Polarization (IP) response of the core samples) and a polarization increase, which corresponds to the Maxwell–Wagner–Sillars polarization (e.g. Sillars 1937; Glaser *et al.* 2023). We have checked that this high-frequency response is not associated with (inductive or capacitive) coupling effects (see Bore *et al.* 2021, 2024 for broad-range dielectric spectroscopy). The low-frequency plateau in the quadrature conductivity depends on the alteration level of the rock, which can be characterized by the CEC.

In Fig. 8, we show for four selected rock samples, the relationship between the (in-phase) conductivity of the rock σ'(S m⁻¹, measured at 1 Hz) as a function of the pore water conductivity σ<sub>w</sub> (S m⁻¹). The in-phase conductivity has two contributions, one associated with the pore network saturated with the pore water (bulk contribution proportional to the conductivity of the pore water, σ<sub>w</sub>, filling the pore space) and the second associated with the conduction in the electrical double layer coating the surface of the grains (termed surface conductivity σ<sub>s</sub>, see for instance Waxman & Smits 1968).
<table><caption>**Table 1.** Petrophysical properties of the 33 core samples used in this study. Data set 1 (13 core samples) corresponds to Site I (Brittany, France). Data set 2 (nine core samples) corresponds to Site II (Vosges, France). Data set 3 (eight core samples) corresponds to Site III (Montagne Noire, Aude, France). CEC is determined using the cobalt-hexamine method. The CEC is expressed in meq/100 g (1 meq/100 g = 963.20 C kg⁻¹ in SI units). The surface conductivities and formation factors are determined by plotting, for each core sample, the (in-phase) conductivity of the material (at 1 Hz) as a function of the conductivity of the pore water (in a log/log space). For sample RHU225, the analysis of the thin sections shows that 95 per cent of the plagioclase are altered. The in-phase and quadrature conductivities are reported at 10 Hz and for a pore water conductivity between 0.1 and 0.0312 S m⁻¹ at 25 °C.</caption><thead><tr><th>Full name</th><th>Origin</th><th>Porosity φ (−)</th><th>CEC meq/100g</th><th>F (−)</th><th>σ<sub>S</sub> 10<sup>-4</sup> S m<sup>-1</sup></th></tr></thead><tbody><tr><td>BARA203</td><td>Site I</td><td>0.055</td><td>1.35</td><td>49</td><td>22</td></tr><tr><td>CHA201</td><td>Site I</td><td>0.039</td><td>0.26</td><td>187</td><td>4</td></tr><tr><td>RHU227</td><td>Site I</td><td>0.084</td><td>2.40</td><td>58</td><td>35</td></tr><tr><td>RHU228</td><td>Site I</td><td>0.029</td><td>–</td><td>193</td><td>16</td></tr><tr><td>RHU221</td><td>Site I</td><td>0.047</td><td>1.39</td><td>202</td><td>29</td></tr><tr><td>RHU223A</td><td>Site I</td><td>0.044</td><td>1.57</td><td>99</td><td>18</td></tr><tr><td>RHU223B</td><td>Site I</td><td>0.043</td><td>1.13</td><td>119</td><td>4.8</td></tr><tr><td>RHU224</td><td>Site I</td><td>0.040</td><td>0.92</td><td>114</td><td>26</td></tr><tr><td>RHU225</td><td>Site I</td><td>0.151</td><td>4.58</td><td>20</td><td>158</td></tr><tr><td>STJU201</td><td>Site I</td><td>0.048</td><td>1.05</td><td>109</td><td>8.7</td></tr><tr><td>VI208A</td><td>Site I</td><td>0.045</td><td>1.31</td><td>84</td><td>6.1</td></tr><tr><td>VI208B</td><td>Site I</td><td>0.051</td><td>0.96</td><td>74</td><td>2.0</td></tr><tr><td>VI209</td><td>Site I</td><td>0.033</td><td>0.65</td><td>141</td><td>11.6</td></tr><tr><td>GR01</td><td>Site II</td><td>0.019</td><td>1.48</td><td>813</td><td>3.6</td></tr><tr><td>GR02</td><td>Site II</td><td>0.013</td><td>2.48</td><td>1053</td><td>2.8</td></tr><tr><td>GR03</td><td>Site II</td><td>0.012</td><td>1.76</td><td>1648</td><td>0.43</td></tr><tr><td>GR04</td><td>Site II</td><td>0.009</td><td>1.59</td><td>1996</td><td>0.90</td></tr><tr><td>GR05</td><td>Site II</td><td>0.162</td><td>7.53</td><td>44.9</td><td>148</td></tr><tr><td>GR06</td><td>Site II</td><td>0.128</td><td>3.53</td><td>89.6</td><td>26</td></tr><tr><td>GR07</td><td>Site II</td><td>0.045</td><td>1.29</td><td>594</td><td>13</td></tr><tr><td>GR08</td><td>Site II</td><td>0.077</td><td>1.28</td><td>228</td><td>20</td></tr><tr><td>GR09</td><td>Site II</td><td>0.084</td><td>4.80</td><td>42</td><td>17.7</td></tr><tr><td>GR12</td><td>Site II</td><td>0.176</td><td>7.04</td><td>100.8</td><td>53</td></tr><tr><td>LP-01</td><td>Site III</td><td>0.06</td><td>2.12</td><td>139 ± 3</td><td>1.3 ± 0.1</td></tr><tr><td>LP-1B</td><td>Site III</td><td>0.07</td><td>3.15</td><td>118 ± 4</td><td>8.8 ± 1.2</td></tr><tr><td>LP-02</td><td>Site III</td><td>0.16</td><td>2.92</td><td>24 ± 0.1</td><td>10.5 ± 2.6</td></tr><tr><td>LP-2g</td><td>Site III</td><td>0.27</td><td>4.01</td><td>18 ± 1</td><td>13.2 ± 5.0</td></tr><tr><td>LP-3w</td><td>Site III</td><td>0.65</td><td>14.0</td><td>2.2 ± 0.5</td><td>92 ± 10</td></tr><tr><td>LP-4j</td><td>Site III</td><td>0.08</td><td>2.97</td><td>126 ± 1</td><td>6.5 ± 0.9</td></tr><tr><td>LP-5d</td><td>Site III</td><td>0.08</td><td>1.60</td><td>61 ± 2</td><td>5.4 ± 3.7</td></tr><tr><td>LP-A</td><td>Site III</td><td>0.05</td><td>2.13</td><td>255 ± 3</td><td>2.2 ± 1.0</td></tr><tr><td>ABM-ABK</td><td>Kaolinite (1)</td><td>0.87</td><td>10.1</td><td>1.4 ± 0.2</td><td>160 ± 7</td></tr><tr><td>COS</td><td>Alps (2)</td><td>0.028</td><td>0.80</td><td>499 ± 7</td><td>0.32 ± 0.03</td></tr></tbody></table>

(1) The in-phase conductivity for kaolinite is given at a pore water conductivity close to 0.1 S m⁻¹ (25 °C, NaCl) and at 100 Hz. ABM-ABK is a pure kaolinite in suspension.

(2) COS (Cosmic Ridge, Mont-Blanc, French Alps) is a granite sample from the study of Coperey et al. (2019) (pore water conductivity 0.026 S m⁻¹ at 25 °C).

The in-phase conductivity can be written as $\sigma' \approx \sigma_w/F + \sigma_S$ where $F$ (dimensionless) denotes the (intrinsic) formation factor and $\sigma_S$ (S m⁻¹) denotes the surface conductivity. In order to determine the values of $F$ and $\sigma_S$, we invert the laboratory data ($\sigma'$, $\sigma_w$) in the log-log domain (logo', $\log\sigma_w$) (i.e. to be sure the low-salinity data are correctly accounted for in the determination of the surface conductivity, see Zhang et al. 2023, for a complete description of the approach and Fig. 8 for an example). The surface conductivity values and formation factor values obtained using this procedure are reported in Table 1 for each core sample.

# 3 MODEL AND INTERPRETATION OF LABORATORY DATA

## 3.1. Complex conductivity model

We first assume that the granite can be considered as an isotropic, homogeneous and linear material and we assume that the electromagnetic fields are harmonic with a pulsation frequency $\omega$. The

total current density $J$ (A m⁻²) is defined through Ampere's law $\nabla \times H = J$ where $H$ is auxiliary magnetic field (A m⁻¹) and $J$ includes the contributions of electromigration, electrodiffusion plus a displacement current density. Usually the contribution of electromigration and electrodiffusion can be written through a complex conductivity term $\sigma^*$ leading to

$$J = \sigma^* E + \frac{\partial D}{\partial t}. \qquad (2)$$

The electric displacement field $D$ (electric induction, in C m⁻²) is related to the electrical field $E$ by $D = \varepsilon * E$ where $\varepsilon$ (F m⁻¹) denotes the complex-valued permittivity. The expression of the effective conductivity $\hat{\sigma}^*$ and effective permittivity $\hat{\varepsilon}^*$ are given as,

$$\nabla \times H = \hat{\sigma}^* E (= i\omega\hat{\varepsilon}^* E), \qquad (3)$$

where $i$ denotes the pure imaginary number ($i^2 = -1$). Furthermore, if we assume $K$-dielectric polarization mechanisms and $N$-induced polarization mechanisms, both described by Cole–Cole parametric
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Data set. (a) Pictures of the 13 core samples of data set 1 from Brittany (France). (b) Pictures of the 10 core samples of data set 2 Site II) from the Vosges (France). The core samples belong to the so-called ‘coral red’ facie, which corresponds to a massive two-mica monzogranite with an isotropic texture. (c) Pictures of the eight core samples of data set 3 (Site III) from the Montagne Noire (Lampy, France). Some of the core samples are highly altered.</figcaption></figure>

models, we have:

$$ \hat{\varepsilon}^* = \varepsilon_{\infty} + \sum_{k=1}^{K} \frac{\varepsilon_k^0 - \varepsilon_k^{\infty}}{1 + (i\omega\tau_k)^{c_k}} - \frac{i}{\omega} \left( \sigma_{\infty} - \sum_{n=1}^{N} \frac{\sigma_n^{\infty} - \sigma_n^0}{1 + (i\omega\tau_n)^{c_n}} \right), \quad (4) $$

where the high-frequency conductivity and permittivity are given by

$$ \sigma_{\infty} = \sum_{n=1}^{N} \sigma_{n}^{\infty}, \qquad (5) $$

$$ \varepsilon_{\infty} = \sum_{k=1}^{K} \varepsilon_{k}^{\infty}. \qquad (6) $$

In these equations, $\tau_k$ (in s) denotes a relaxation time for the dielectric polarization process labelled $k$, $\varepsilon_k^0$ and $\varepsilon_k^{\infty}$ denote the low-frequency and high-frequency dielectric constants for the polarization process $k$ ($\Delta\varepsilon_k = \varepsilon_k^0 - \varepsilon_k^{\infty}$ is called the dielectric increment), and $0 \le c_k \le 1$ denotes the Cole Cole exponent for the dielectric polarization process $k$, $\sigma_n^0$ and $\sigma_n^{\infty}$ denotes the low-frequency and high-frequency conductivities associated with the induced polarization contribution $n$ ($M_n^j = \sigma_n^0 - \sigma_n^{\infty}$ is called the normalized chargeability of contribution $n$), $\tau_n$ denotes the relaxation time for the $\alpha$-contribution $n$ and $0 \le c_n \le 1$ denotes the Cole Cole exponent $n$. In Appendix A, we discuss further the nomenclature used for the induced polarization processes in presence or absence of metallic particles.

In this paper, we consider simply one polarization process ($n=1$) and one dielectric polarization process ($k=1$). The induced polarization model is associated with the polarization of the electrical double layer coating the clay particles present in the granite while the dielectric polarization model is associated with the Maxwell-Wagner-Sillars polarization mechanism associated with the discontinuity of the displacement current at the interface between the different phases of the porous composite (e.g. Sillars 1937).

We first focus on the induced polarization contribution. Granites are characterized by a broad-scale variations in their crack distribution (Velde et al. 1991). For porous media characterized by multiscale heterogeneities, the CPA (Constant Phase Angle) model has been used by various researchers in the literature to characterize their complex conductivity (e.g. Van Voorhis et al. 1973; Weller et al. 1996; Revil et al. 2017c). The complex conductivity associated with induced polarization is then written as,

$$ \sigma^*(\omega) = \left( \frac{i\omega}{\omega_n} \right)^b \sigma_n, \qquad (7) $$

where $\sigma_n$ denotes the amplitude of the conductivity at the angular frequency $\omega_n$ ($\omega = 2\pi f$ where $f$ is the frequency). With the sign convention that $\sigma^*(\omega) = \sigma'(\omega) - i\sigma''(\omega)$, the in-phase and quadrature conductivities are given by,

$$ \sigma'(\omega) = \left( \frac{\omega}{\omega_n} \right)^b \sigma_n \cos \left( \frac{\pi}{2} b \right). \qquad (8) $$
<table><caption>**Table 2.** Petrophysical properties of the 33 core samples used in this study. In-phase conductivity, quadrature conductivity and normalized chargeability values at 25°C. The normalized chargeability $M_n$ is computed from the difference of the in-phase conductivity between 10 mHz and 10 kHz.</caption><thead><tr><th>Full name</th><th>Origin</th><th>σ' 10 mHz 10<sup>-4</sup> S m<sup>-1</sup></th><th>σ' 10 kHz 10<sup>-4</sup> S m<sup>-1</sup></th><th>M<sub>n</sub> 10<sup>-4</sup> S m<sup>-1</sup></th><th>σ' 32 Hz 10<sup>-4</sup> S m<sup>-1</sup></th><th>σ' 32 Hz 10<sup>-4</sup> S m<sup>-1</sup></th></tr></thead><tbody><tr><td>BARA203</td><td>Site I</td><td>34</td><td>48</td><td>14.45</td><td>45</td><td>0.78</td></tr><tr><td>CHA201</td><td>Site I</td><td>9</td><td>13</td><td>4.14</td><td>11</td><td>0.12</td></tr><tr><td>RHU227</td><td>Site I</td><td>46</td><td>61</td><td>15.62</td><td>55</td><td>1.39</td></tr><tr><td>RHU228</td><td>Site I</td><td>20</td><td>27</td><td>6.64</td><td>24</td><td>0.56</td></tr><tr><td>RHU221</td><td>Site I</td><td>29</td><td>39</td><td>9.99</td><td>36</td><td>0.55</td></tr><tr><td>RHU223A</td><td>Site I</td><td>23</td><td>36</td><td>12.28</td><td>30</td><td>0.62</td></tr><tr><td>RHU223B</td><td>Site I</td><td>11</td><td>16</td><td>4.69</td><td>14</td><td>0.29</td></tr><tr><td>RHU224</td><td>Site I</td><td>28</td><td>42</td><td>14.01</td><td>37</td><td>0.81</td></tr><tr><td>RHU225</td><td>Site I</td><td>184</td><td>221</td><td>36.52</td><td>207</td><td>3.01</td></tr><tr><td>STJU201</td><td>Site I</td><td>14</td><td>22</td><td>7.68</td><td>19</td><td>0.15</td></tr><tr><td>VI208A</td><td>Site I</td><td>15</td><td>21</td><td>6.61</td><td>19</td><td>0.16</td></tr><tr><td>VI208B</td><td>Site I</td><td>14</td><td>18</td><td>4.63</td><td>17</td><td>0.09</td></tr><tr><td>VI209</td><td>Site I</td><td>16</td><td>23</td><td>6.93</td><td>20</td><td>0.14</td></tr><tr><td>GR01</td><td>Site II</td><td>4.62</td><td>5.40</td><td>0.78</td><td>4.86</td><td>0.07</td></tr><tr><td>GR02</td><td>Site II</td><td>3.04</td><td>4.27</td><td>1.23</td><td>3.83</td><td>0.07</td></tr><tr><td>GR03</td><td>Site II</td><td>0.80</td><td>1.43</td><td>0.63</td><td>1.15</td><td>0.02</td></tr><tr><td>GR04</td><td>Site II</td><td>1.21</td><td>2.01</td><td>0.80</td><td>1.45</td><td>0.04</td></tr><tr><td>GR05</td><td>Site II</td><td>142</td><td>170</td><td>27.9</td><td>160.8</td><td>1.80</td></tr><tr><td>GR06</td><td>Site II</td><td>32.5</td><td>42.8</td><td>10.3</td><td>38.7</td><td>0.76</td></tr><tr><td>GR07</td><td>Site II</td><td>12.4</td><td>15.8</td><td>3.4</td><td>13.9</td><td>0.27</td></tr><tr><td>GR08</td><td>Site II</td><td>20.6</td><td>26.4</td><td>5.8</td><td>24.2</td><td>0.45</td></tr><tr><td>GR09</td><td>Site II</td><td>35.0</td><td>47.7</td><td>12.7</td><td>44.5</td><td>0.71</td></tr><tr><td>GR12</td><td>Site II</td><td>53.9</td><td>65.6</td><td>11.6</td><td>61.7</td><td>0.69</td></tr><tr><td>LP-01</td><td>Site III</td><td>1.1</td><td>2.0</td><td>0.9</td><td>1.6</td><td>0.03</td></tr><tr><td>LP-1B</td><td>Site III</td><td>2.7</td><td>4.3</td><td>1.6</td><td>3.5</td><td>0.07</td></tr><tr><td>LP-02</td><td>Site III</td><td>22.6</td><td>26.9</td><td>4.3</td><td>25.3</td><td>0.19</td></tr><tr><td>LP-2g</td><td>Site III</td><td>12.8</td><td>15.9</td><td>3.1</td><td>14.1</td><td>0.12</td></tr><tr><td>LP-3w</td><td>Site III</td><td>198.2</td><td>261.7</td><td>63.5</td><td>233.1</td><td>2.37</td></tr><tr><td>LP-4j</td><td>Site III</td><td>4.8</td><td>6.0</td><td>1.2</td><td>5.6</td><td>0.07</td></tr><tr><td>LP-5d</td><td>Site III</td><td>4.5</td><td>5.2</td><td>0.7</td><td>4.8</td><td>0.04</td></tr><tr><td>LP-A</td><td>Site III</td><td>0.5</td><td>1.7</td><td>0.8</td><td>1.3</td><td>0.03</td></tr><tr><td>ABM-ABK</td><td>Kaolinite</td><td>245.9</td><td>277.0</td><td>31.1</td><td>255</td><td>3.2</td></tr><tr><td>COS</td><td>Alps</td><td>0.93</td><td>1.03</td><td>0.10</td><td>0.94</td><td>0.01</td></tr></tbody></table>

(1) Measurements at a pore water conductivity of 0.013 S m⁻¹ at 25 °C.

$$ \sigma''(\omega) = -\left(\frac{\omega}{\omega_n}\right)^b \sigma_n \sin\left(\frac{\pi}{2}b\right). \qquad (9) $$

respectively and where $\varphi = \pi b/2$ (in rad) denotes the constant phase angle or shift between the current density and the electrical field. Since the phase is small (<50 mrad), we have the following approximations:

$$ \cos\left(\frac{\pi}{2}b\right) \approx 1, \qquad (10) $$

$$ \sin\left(\frac{\pi}{2}b\right) \approx \frac{\pi}{2}b = \varphi, \qquad (11) $$

$$ \tan\left(\frac{\pi}{2}b\right) \approx \varphi \approx \frac{\sigma''_n}{\sigma_n}, \qquad (12) $$

where $\sigma''_n$ denotes the amplitude of the quadrature conductivity at angular frequency $\omega_n$. Using the frequencies instead the angular frequencies, this yields,

$$ \sigma'(f) = (f)^b \sigma_n(1\text{Hertz}), \qquad (13) $$

$$ \sigma''(f) = (f)^b \sigma''_n(1\text{Hertz}). \qquad (14) $$

In the framework of the CPA model, the amplitude of the quadrature conductivity at the geometric mean of two frequencies $f_1$ and $f_2$ and the normalized chargeability defined as the difference between the in-phase conductivity at the frequency $f_2(> f_1)$ and the in-phase conductivity at the lower frequency $f_1$, we can connect the

quadrature conductivity and the normalized chargeability with (Van Voorhis et al. 1973; Revil et al. 2017c)

$$ \sigma''(\sqrt{f_1 f_2}) \approx -\frac{M_n(f_1, f_2)}{\alpha}, \qquad (15) $$

and $\alpha \approx (2/\pi) \ln A$ and $A$ denotes the number of decades between $f_1$ and $f_2$ (for three decades, we have $A = 10^3$ and $\alpha \approx 4.4$). If we further assume that all the relaxation times are contained inside 6 orders of magnitude (accounting for the largest and the smallest polarization length scales of the granite), the largest value of $\alpha$ is therefore given by $\alpha \approx 9$ (using $\alpha \approx (2/\pi) \ln A$, 6 orders of magnitude $A = 10^6$). We assume then that the quadrature conductivity can be written as

$$ \sigma''_n(\sqrt{f_1 f_2}) \approx -\frac{M_n(f_1, f_2)}{9}. \qquad (16) $$

A core sample of granite cannot be self-affine over all the scales. There is necessarily a large and small scales defining the limits for the self-affine character of the material. For frequencies below and above the frequencies required to polarize these largest and smallest polarization length scales, the conductivity is expected to reach the asymptotic limits $\sigma_0$ and $\sigma_\infty$, respectively. The total normalized chargeability and the chargeability are then defined as

$$ M_n \equiv \sigma_\infty - \sigma_0, \qquad (17) $$
<table><caption>**Table 3.** Petrophysical properties of the 33 core samples used in this study. In-phase conductivity, quadrature conductivity and normalized chargeability values at 25 °C. The normalized chargeability $M_n$ is computed from the difference of the in-phase conductivity between 10 mHz and 100 Hz.</caption><thead><tr><th>Full name</th><th>Origin</th><th>σ' 10 mHz 10<sup>-4</sup> S m<sup>-1</sup></th><th>σ' 100 Hz 10<sup>-4</sup> S m<sup>-1</sup></th><th>M<sub>n</sub> 10<sup>-4</sup> S m<sup>-1</sup></th><th>σ' 1 Hz 10<sup>-4</sup> S m<sup>-1</sup></th><th>σ' 1 Hz 10<sup>-4</sup> S m<sup>-1</sup></th></tr></thead><tbody><tr><td>BARA203</td><td>Site I</td><td>33.9</td><td>45</td><td>11.1</td><td>41.8</td><td>0.789</td></tr><tr><td>CHA201</td><td>Site I</td><td>8.57</td><td>10.8</td><td>2.23</td><td>9.94</td><td>0.0705</td></tr><tr><td>RHU227</td><td>Site I</td><td>45.7</td><td>56.1</td><td>10.4</td><td>51.9</td><td>1.24</td></tr><tr><td>RHU228</td><td>Site I</td><td>20.4</td><td>24.1</td><td>3.7</td><td>22.5</td><td>0.503</td></tr><tr><td>RHU221</td><td>Site I</td><td>29.2</td><td>36.3</td><td>7.1</td><td>34.2</td><td>0.59</td></tr><tr><td>RHU223A</td><td>Site I</td><td>23.5</td><td>30.4</td><td>6.9</td><td>27.8</td><td>0.519</td></tr><tr><td>RHU223B</td><td>Site I</td><td>11.2</td><td>14.1</td><td>2.9</td><td>13.2</td><td>0.267</td></tr><tr><td>RHU224</td><td>Site I</td><td>28.1</td><td>37.4</td><td>9.3</td><td>34.4</td><td>0.752</td></tr><tr><td>RHU225</td><td>Site I</td><td>184</td><td>210</td><td>26</td><td>200</td><td>3.04</td></tr><tr><td>STJU201</td><td>Site I</td><td>14</td><td>19.1</td><td>5.1</td><td>18.2</td><td>0.088</td></tr><tr><td>VI208A</td><td>Site I</td><td>14.9</td><td>18.7</td><td>3.8</td><td>17.9</td><td>0.089</td></tr><tr><td>VI208B</td><td>Site I</td><td>13.5</td><td>16.7</td><td>3.2</td><td>15.8</td><td>0.0514</td></tr><tr><td>VI209</td><td>Site I</td><td>16.2</td><td>19.7</td><td>3.5</td><td>18.8</td><td>0.0659</td></tr><tr><td>GR01</td><td>Site II</td><td>4.62</td><td>4.91</td><td>0.29</td><td>4.76</td><td>0.0613</td></tr><tr><td>GR02</td><td>Site II</td><td>3.04</td><td>3.89</td><td>0.85</td><td>3.61</td><td>0.0607</td></tr><tr><td>GR03</td><td>Site II</td><td>0.8</td><td>1.17</td><td>0.37</td><td>1.04</td><td>0.0196</td></tr><tr><td>GR04</td><td>Site II</td><td>1.21</td><td>1.48</td><td>0.27</td><td>1.37</td><td>0.0274</td></tr><tr><td>GR05</td><td>Site II</td><td>142</td><td>162</td><td>20</td><td>155</td><td>2.77</td></tr><tr><td>GR06</td><td>Site II</td><td>32.5</td><td>39.2</td><td>6.7</td><td>36.3</td><td>0.883</td></tr><tr><td>GR07</td><td>Site II</td><td>12.4</td><td>14.1</td><td>1.7</td><td>13.3</td><td>0.225</td></tr><tr><td>GR08</td><td>Site II</td><td>20.6</td><td>24.5</td><td>3.9</td><td>22.8</td><td>0.493</td></tr><tr><td>GR09</td><td>Site II</td><td>35</td><td>45.0</td><td>10</td><td>42.2</td><td>0.709</td></tr><tr><td>GR12</td><td>Site II</td><td>53.9</td><td>62.2</td><td>8.3</td><td>59.7</td><td>0.792</td></tr><tr><td>LP-01</td><td>Site III</td><td>1.31</td><td>1.84</td><td>0.53</td><td>1.67</td><td>0.0189</td></tr><tr><td>LP-1B</td><td>Site III</td><td>2.23</td><td>2.96</td><td>0.73</td><td>2.8</td><td>0.0339</td></tr><tr><td>LP-02</td><td>Site III</td><td>23.3</td><td>26.2</td><td>2.9</td><td>25.6</td><td>0.161</td></tr><tr><td>LP-2g</td><td>Site III</td><td>38.4</td><td>42.6</td><td>4.2</td><td>41.4</td><td>0.21</td></tr><tr><td>LP-3w</td><td>Site III</td><td>66.1</td><td>78.4</td><td>12.3</td><td>75.7</td><td>0.41</td></tr><tr><td>LP-4j</td><td>Site III</td><td>2.4</td><td>2.84</td><td>0.44</td><td>2.73</td><td>0.0212</td></tr><tr><td>LP-5d</td><td>Site III</td><td>7.43</td><td>7.94</td><td>0.51</td><td>7.81</td><td>0.0328</td></tr><tr><td>LP-A</td><td>Site III</td><td>0.52</td><td>0.73</td><td>0.209</td><td>0.682</td><td>0.00603</td></tr><tr><td>ABM-ABK</td><td>Kaolinite</td><td>250</td><td>263</td><td>12.4</td><td>255</td><td>0.0105</td></tr><tr><td>COS</td><td>Alps</td><td>0.93</td><td>—</td><td>—</td><td>—</td><td>—</td></tr></tbody></table>

<figure><img src="image_6.png" /><figcaption>**Figure 4.** Observed alteration path between CEC and porosity $\phi$. The plain line corresponds to the fit of the following relationship CEC(meq/100g) = $m_1 + m_2(1 - \exp(-m_3\phi))$corresponding therefore to the alteration path (essentially kaolinization) with $m_1 = 0.25 \pm 0.20$, $m_2 = 13 \pm 2$ and $m_3 = 2.5 \pm 0.9$ ($r^2 = 0.80$). Huitti *et al.* (1996) used three granite core samples characterized as fresh, weathered and altered and the CEC used here is reported for sodium.</figcaption></figure>

$$M \equiv \frac{M_n}{\sigma_0} = \frac{\sigma_{\infty} - \sigma_0}{\sigma_0}. \qquad (18)$$

While $M_n$ can be formally identified to the metal factor in the literature (see Appendix A), this terminology has been abandoned because it was a misnomer. In the context of the dynamic Stern layer model describing the polarization of the inner component of the electrical double layer coating the surface of the grains, we have (e.g. Revil *et al.* 2017c)

$$\sigma_{\infty} = \frac{1}{F} \sigma_w + \frac{\rho_g B \text{ CEC}}{F \phi}, \qquad (19)$$

$$M_n = \frac{\rho_g \lambda \text{CEC}}{F \phi}, \qquad (20)$$

where $F$ (dimensionless) denotes the formation factor related to porosity $\phi$ (dimensionless) by Archie's law $F = \phi^{-m}$, $m > 1$ being the so-called porosity (cementation) exponent (Archie, 1942), $\sigma_w$ (S m$^{-1}$) denotes the conductivity of the pore water, CEC denotes the Cation Exchange Capacity of the porous material (usually expressed in meq/100 g where meq means milliequivalent moles of surface electrical charges or expressed in or C kg$^{-1}$), $\rho_g$ (kg m$^{-3}$) denotes the grain density, and $B$ and $\lambda$ (both in m$^2$ s$^{-1}$ V$^{-1}$) denotes the two effective ionic mobilities for conduction and polarization (see Revil *et al.* 2017c). With the previous equations and assuming that the frequency dependence of the in-phase conductivity is rather small, we can use the following approximations to provide a complete description of the in-phase and quadrature conductivities:
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Complex conductivity (in phase and quadrature conductivities) spectra for the 13 core samples of data set 1 (Site I). The measurements are here shown at the smallest salinity (0.1 S m⁻¹, NaCl, 25 °C). The samples with the highest in-phase and quadrature conductivities are located in the fault core and are characterized by heavy alteration of the plagioclase. The high-frequency asymptotic behaviour is partly influenced by the Maxwell–Wagner–Sillars (MWS) polarization.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 6.** Complex conductivity spectra for the 10 core samples of data set 2 (Site II). The high-frequency asymptotic behaviour is partly controlled by the MWS polarization.</figcaption></figure>

$$ \sigma'(\omega) = (f)^b \left[ \frac{1}{F} \sigma_w + \frac{\rho_g B \text{ CEC}}{F \phi} \right], \quad (21) $$

$$ \sigma''(\omega) = (f)^b \left(\frac{1}{\alpha}\right) \frac{\rho_g \lambda \text{CEC}}{F \phi}. \quad (22) $$

Eqs (21) and (22) describe the induced polarization component. We need to add to this polarization component the MWS polarization. The effective conductivity can be written as $\hat{\sigma}^* = \sigma^* - i\omega\varepsilon^*$ where,

$$ \varepsilon^* = \varepsilon_{\text{MW}}^{\infty} + \frac{\varepsilon_{\text{MW}}^0 - \varepsilon_{\text{MW}}^{\infty}}{1 + i\omega\tau_{\text{MW}}}, \quad (23) $$

$$ \tau_{\text{MW}} = \frac{\varepsilon_{\text{S}} + 2\varepsilon_{\text{w}}}{\sigma_{\text{S}} + 2\sigma_{\text{w}}}. \quad (24) $$

When the pore water conductivity dominates the surface conductivity $\sigma_S$, the relaxation time is given by $\tau_{MW} = \varepsilon_w/\sigma_w$. For a pore water conductivity of 0.1 S m⁻¹, we have $\tau_{MW} = \varepsilon_w/\sigma_w = 80 \times 8.854 \times 10^{-12} \text{ C V}^{-1} \text{ m S m}^{-1} \approx 10^{-8} \text{ s} (\sim 100 \text{ MHz}). The high-frequency dielectric constant can be written as,

$$ \varepsilon_{\text{MW}}^{\infty} \approx \frac{\varepsilon_{\text{w}}}{F} + \varepsilon_{\text{s}}. \quad (25) $$

The quantities $\varepsilon_w$ ($\approx 81 \varepsilon_0$) and $\varepsilon_S$ denote the dielectric constant of the water phase and solid, respectively. For a mixture of minerals, there are many ways to compute the effective dielectric constant of the mixture. For each type of mineral, the permittivity is related to the grain density $\rho_S$ by $\varepsilon_S/\varepsilon_0 \approx 0.00191\rho_S$. For silica grains, alkali feldspars, and plagioclases, $\rho_S \approx 2700 \text{ kg m}^{-3}$ yields a relative permittivity of $\sim 5.0$ (4.5 for quartz according to Bottom 1972). A full description of the MWS contribution is out of the scope
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Complex conductivity (in phase and quadrature conductivities) spectra for the eight core samples of data set 3 (Site III). The measurements are here shown at the smallest salinity (0.1 S m⁻¹, NaCl, 25 °C). The high-frequency asymptotic behaviour is partly influenced by the MWS polarization.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Example of rock sample conductivity (1 Hz) versus pore water conductivity for four selected core samples from Data set 1. These rock samples are characterized by distinct porosities and cation exchange capacities (CEC). The plain lines correspond to the linear conductivity model for which the rock conductivity depends on the bulk conductivity (proportional to the pore water conductivity) and the surface conductivity, which depends on the CEC. The fit between such linear conductivity model and the data allows to determine the (intrinsic) formation factor *F* and the surface conductivity *σ*<sub>S</sub>.</figcaption></figure>

of this paper because it would require an expression for the ε<sub>MW</sub><sup>0</sup>
component as well as higher frequency measurements. However, it
should be pointed out that this contribution may be important in
the interpretation of induced polarization effects associated with
electromagnetic induction (Katzer & Macnae 2012; Macnae 2016).

**3.2. Comparison with the experimental data**

We plot the formation factors of the core samples versus the con-
nected porosities in Fig. 9. The data of Fig. 9 can be fitted with an
Archie's law with a cementation exponent given by m = 1.70 ± 0.02

<figure><img src="image_11.png" /><figcaption>**Figure 9.** Formation factor versus connected porosity for the 33 core sam-
ples described in Table 1. The plain line corresponds to the fit with Archie's
law. Note that the value of the cementation exponent (m = 1.70 ± 0.02) is
substantially smaller than for volcanic rocks for which m = 2.16 ± 0.02 (see
Revil *et al.* 2023).</figcaption></figure>

indicating that the porosity of the granite core samples is likely a
crack-type porosity. This value is much smaller than the cementation
exponent associated with extrusive volcanic rocks (m = 2.16 ± 0.02,
Zhang *et al.* 2023).

The plot of the normalized chargeability versus the quadrature conductivity of the 33 core samples is shown in Fig. 10. The plain lines correspond to the fit for a linear relationship between the normalized chargeability and the quadrature conductivity at the geometric mean of the two frequencies used to determine the normalized chargeability. Granites are multiscale materials and characterized by a hierarchical structure of cracks. The frequency range 10 mHz and 100 Hz corresponds to 4 decades, A = 10⁴ and this
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Normalized chargeability versus quadrature conductivity for the 33 core samples described in Table 1. The plain lines correspond to the fit for a linear relationship between the normalized chargeability and the quadrature conductivity at the geometric mean of the two frequencies used to determine the normalized chargeability. (a) Normalized chargeability between 10 mHz and 10 kHz, quadrature conductivity at 10 Hz. (b) Normalized chargeability between 10 mHz and 100 Hz, quadrature conductivity at 1 Hz.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Figure 11.** Normalized chargeability versus surface conductivity for the 33 core samples described in Table 1. The plain line corresponds to the fit for a linear relationship between the normalized chargeability and the surface conductivity.</figcaption></figure>

yields $\alpha \approx 6$. The observed value is slightly stronger with $\alpha = 9$ as shown in Fig. 10(b).

We analyse now the relationship between the surface conductivity $\sigma_S$ (associated with conduction in the electrical double layer) and the CEC (see last term of eq. 19, $\sigma_S = B\rho_g\text{CEC}/(F\phi)$). According to eq. (19), the surface conductivity is therefore proportional to the CEC divided by the bulk tortuosity obtained by the product of the formation factor by the connected porosity. For the pyrite-free samples, the relationship between the normalized chargeability and the CEC is given by $M_n = \lambda\rho_g\text{CEC}/(F\phi)$, where $\lambda$ (in m$^2$ s$^{-1}$ V$^{-1}$) is another effective mobility related to the double layer. It follows that

<figure><img src="image_14.png" /><figcaption>**Figure 12.** Normalized chargeability (S m$^{-1}$) versus normalized cation exchange capacity (CEC in meq/100 g divided by the bulk tortuosity defined as the product between the formation factor and the connected porosity) for the 33 core samples described in Table 1. The plain line corresponds to the fit for a linear relationship between the normalized chargeability and the normalized chargeability for the pyrite-free samples corresponding to Site III.</figcaption></figure>

the surface conductivity can be related to the normalized chargeability by (Revil 2013; Revil et al. 2017a) $M_n/\sigma_S = \lambda/B \equiv R$, where R is a non-dimensional parameter independent of temperature and saturation or textural properties. The proportionality between the surface conductivity and the normalized chargeability is demonstrated in Fig. 11. The trend shown in Fig. 11 between the normalized chargeability and the surface conductivity provides the value of the dimensionless parameter $R = 0.20$, higher than the
<figure><img src="image_15.png" /><figcaption>**Figure 13.** Time-domain decay curves in terms of partial chargeabilities versus the elapsed time after the shutdown of the primary current (the injection time is 1 s). (a) Type I. (b) Type II. (c) Type III. (d) Toarcian clay-rock used for comparison.</figcaption></figure>

value of *R* for smectite-dominated rocks (*R* = 0.10, see Revil *et al.* 2017a, b; Revil *et al.* 2019; Revil *et al.* 2021). Coming back to the linear trend fitted in Fig. 12 between the normalized chargeability and the ratio CEC/(*F*φ), equation (5) yields λ(Na⁺, 25 °C) = (1.7 ± 0.5) × 10⁻¹⁰ m² s⁻¹ V⁻¹. Using *R* = 0.20, this leads to B(Na⁺, 25 °C) = λ(Na⁺, 25 °C)/*R* = (0.85 ± 0.3) × 10⁻⁹ m² s⁻¹ V⁻¹.

## 3.3. Time-domain versus frequency-domain normalized chargeability

In field applications of the induced polarization method, we usually use time-domain induced polarization with a current injection period *T* = 1 s. The chargeability is determined by integrating an early window of the voltage decay curve after shutting down the electrical current. Then the normalized chargeability is obtained by the product of the conductivity with the chargeability. Our goal in this section is to compare the normalized chargeability obtained through such an approach with the normalized chargeability obtained in the frequency-domain through the difference of the in-phase conductivity between two frequencies.

We use the samples from Site I to compare the two normalized chargeabilities. First, we performed laboratory measurements at the

lowest salinity of time-domain induced polarization measurements with a current injection of *T* = 1 s. The partial chargeabilities decay curves are shown in Fig. 13. We observe a fast decay followed by a slower decay. The fast decay is possibly associated with the MWS polarization while the slower decay (after *t* = 0.1 s) is likely associated with the double layer polarization of interest. Therefore, we use the time window [0.20–0.22] s to determine the chargeability of each sample. As mentioned above the normalized chargeability is simply obtained by the product of the chargeability time the conductivity at the same salinity. These Time-Domain (TD) normalized chargeabilities are compared with the Frequency-Domain (FD) normalized chargeabilities in Fig. 14. We observe that the two are proportional to each other with a correction factor of *a* = 10 excluding the poorly altered core samples for which the MWS component remains important even below 100 Hz. This correction factor will be used in the next section to interpret field time-domain induced polarization data in terms of tomograms of the water content and CEC.

# 4 FIELD APPLICATIONS

In this section, we apply the petrophysical model discussed in the previous sections to a field induced polarization survey performed
<figure><img src="image_16.png" /><figcaption>**Figure 14.** Partial normalized chargeability from TD measurements versus the normalized chargeability determined from the FD measurements using the difference between the in-phase conductivity at 0.01 Hz and 100 Hz. The linear trend is marked by the plain line. It corresponds to an amplification factor *a* = 10.</figcaption></figure>

in the Vosges Massif (Northeastern France). Our goal is to show
that we can provide reasonable porosity (water content at satura-
tion) and CEC tomograms from the conductivity and normalized
chargeability tomograms.

**4.1. Geology**

The Vosges Massif represents a key segment of the Variscan orо-
genic belt in Western Europe (Fig. 3). The Variscan's orogeny
took place in the Devonian to the Lower Permian (380–280 My).
This continental collision has involved the convergence of several
microcontinents, including Avalonia and Armorica, between the
Gondwana landmass to the south and Laurussia to the north (e.g.
Franke 2000). As a result of this collision, two distinct lithotec-
tonic provinces have been recognized: (i) The Northern Vosges is
a part of the Saxothurungian microcontinent and (ii) the Southern
Vosges, which belongs to the Moldanubian microcontinent and rep-
resents the metamorphic root of the orogeny (Edel et al. 2018).
The late-orogenic extensional phase associated with the collapse of
the overthickened Variscan chain between 325 and 300 My facil-
itated the emplacement of granitic and granodioritic plutons. This
arrangement indicates a progressive magmatic evolution from early
mafic magmas to younger more differentiated intrusions, such as
the Senones granite (Tabaud 2012), which is investigated in this
paper.

The Senones granitic pluton was emplaced during the late-
orogenic extensional phase (Fig. 15). Using the K–Ar and Ar–Ar
methods on hornblende, its age is 325 ± 4 and 328 ± 4 Ma, re-
spectively (Altherr et al. 2000; Cocherie & Legendre 2007). The
intrusive Senones granite exhibits petrographic and geochemical
characteristics typical of I-type granites, which are derived from
the partial melting of meta-igneous infracrystal sources (Altherr
et al. 2000). This pluton shows a petrographic heterogeneity char-
acterized by two distinct facies. Notably, the ‘dead leaf’ and the

'coral red' varieties, which differ in their mineralogical composi-
tions and colorations, ranging from light grey pink to reddish-brown,
respectively.

The Vieux–Moulin quarry is the focus of the geophysical survey reported in this section. It presents only samples belonging to the ‘coral red’ facies. This facies is a massive two-mica monzogranite with an isotropic texture and no preferential mineral orientation. The texture is granular with a tendency toward porphyritic development. It consists of potassic feldspars’s phenocrysts that can reach several centimeters in size, imparting the characteristic pink colour of the facies. The minerals are interlocking and medium-grained, featuring subhedral white plagioclase feldspars, biotite arranged in clusters and quartz. Progressive alteration of these facies leads to the transformation of feldspars into clay minerals (likely kaolinite) and the micas become partially chloritized. The original texture of the granite can still be observed but with a significant increase in the pore size with alteration. The intense alteration of the granite in the upper section of the quarry suggests prolonged exposure to surface weathering processes. Within the quarry, the weathered granitic re-golith is characterized by a thickness of 25 m below the ground surface with a maximum depth of 100 m in the southwestern part of the pluton.

**4.2. Field survey**

The field survey at the Vieux–Moulins quarry was performed with a total of 25 V-receivers Fullwavers stations (see Fig. 16). These receivers record continuously the two-surface components of the electrical field at the ground surface. Those receivers have been deployed to cover the known granitic outcrops as well as the surrounding area (see Fig. 17). Each receiver is de-ployed in an L-shape forming triplets of electrodes MN (form-ing 2 dipoles). MN electrodes form two dipoles normal to each other with a common electrode (Fig. 16). The distance between the electrodes of a recording dipole is 50 ± 2 m (Figs 16 and 17).

The electrical current is injected/retrieved at pairs of electrodes AB (generally B fixed and A mobile across the investigated area and around). The source signal is produced by a VIP-5000 squared-signal injector, and recorded in series by an I-Fullwaver (Fig. 16). The total number of electrodes in the survey is 105, with 75 MN electrodes in triplets and 30 AB current electrodes in pairs (Fig. 17). The topography and coordinates of each electrode have been recorded with a Garmin Etrex 30 GPS (Global Positioning System) with a precision of 0.2 m. Usually, we may anticipate 3-D effects from the measurements obtained near abrupt surface topography changes. That said, in this study the changes in the topography (few tens of meters at most) are small with respect to the depth of investigation of this study (400 m) so the overall topographic effects are expected to be rather small and accounted through the use of a precise digital elevation map.

The injection array is performed along two lines plus two circu-
lar profiles increasing the resolution within the 3-D grid (Fig. 16).
All injection signals are a sum of 1 s on–off signals for a total
duration in the range of 5–10 min per measurement. This duration
is used in to allow for stacking and a sufficient signal-to-noise ra-
tio for the measured electrical field. We obtained a total of 1300
ABMN-quadrupoles with a good signal-to-noise ratio. Some typi-
cal decay curves are shown in Fig. 18 indicating a very good quality
of the IP data. From those measurements, a first filtering process
<figure><img src="image_17.png" /><figcaption>**Figure 15.** Mapping and geological context of the study area. (a) Synthetic map of the Vosges Massif showing the main geological units and tectonic structures (modified after Fluck *et al.* 1991; Tabaud 2012). (b) Simplified geological map of the Senones region, highlighting the main lithological formations and local geological structures. (c) Detailed geological map of the Vieux-Moulin quarry, showing the precise locations of the receiver boxes and injection points used for the deep induced polarization survey.</figcaption></figure>

occurs both on the apparent resistivity and apparent chargeability data. This filtering is done within the Prosys III software removing negative apparent resistance and apparent chargeability values and performing a filtering analysis on outliers and on the decay curves for the secondary voltage. This filtering process leads to 80 per cent of the values being kept, giving us a data set of 1040 filtered ABMN-quadrupoles. The maximum pseudo-depth is located at 816.14 m below the ground surface. In order to display the data density used for the inversion, a report point distribution with the display of the digital elevation model used for topography is provided in Fig. 19 (see also Glaser *et al.* 2021, for example).

### 4.3. Inversion and interpretation

The filtered data set is inverted within the Res3DInv software (Loke & Barker 1996) with a 5 × 8 m grid in the x and y horizontal directions and with 26 layers in the vertical z direction ranging from 10.68 to 816 m below the ground surface. The last error change for stopping iterations in the inversion is chosen at 1 per cent and the inversion is stopped at the seventh iteration. The final tomograms are shown in 3-D in Fig. 20 for the conductivity and normalized chargeability (the inverted data set shows an RMS error of 25 per cent for the apparent resistivity values and 1.3 per cent for the apparent chargeability). The high RMS for the resistivity is mostly due to the
<figure><img src="image_18.png" /><figcaption>**Figure 16.** Sketch of a typical fullwaver survey. The Fullwaver system is comprised of 2 types of units: (i) the I-Fullwaver unit used to record time-series of the injected current and (ii) the V-Fullwaver boxes used to measure the resulting electric field. Each of the V-Fullwaver stations has three electrodes $P_1$, $P_2$ and $P_3$. The electric field tangential component $E_1$ is measured on the first channel between $P_1$ and $P_2$, while the normal component of the electric field $E_2$ is measured on the second channel between $P_2$ and $P_3$. The quantity $E_1$ denotes the component of $E$ along the horizontal direction and $E_2$ is the component of $E$ along the vertical direction. The current is injected between the electrodes A and B, which are connected by a cable with electrode B fixed while the position of electrode A is switched between different locations (modified from Soueid Ahmed & Revil 2018).</figcaption></figure>

large variety of the electrofacies within the area of interest with unsaturated and fully saturated weathered granites, fresh granite and conductive sandstone. It is also due to temporal saturation variations in highly permeable altered areas, where deviation in voltage can sometimes increase daily due to the change in dryness of the shallow subsurface. These two effects lead to resistivity spanning over 4 orders of magnitude in the area. A 2-D cross-section is cut in the 3-D tomogram and shown in Fig. 21.

In Fig. 22, we plot the normalized chargeability versus the conductivity. To be consistent with the laboratory data, the normalized chargeability needs to be corrected by an amplification factor $a = 8$ consistent with the laboratory findings. Furthermore, the fact that the data are close to the line defined by the ratio $R = 0.20$ (as described in Section 3 above) implies that the conductivity data are dominated by the surface conductivity. The amplification factor is here to correct for the fact that in the field we use a period $T = 1$ s for the current injection while an integration over the entire frequency spectrum would require a much longer period.

From the petrophysical section, we have the following expression for the conductivity and normalized chargeability

$$ \sigma \approx \phi^m \sigma_w + \phi^{m-1} \rho_g B \text{ CEC}, \quad (26) $$

$$ M_n = \phi^{m-1} \lambda \rho_g \text{CEC}. \quad (27) $$

Using the correction factor $a$ for the CEC determination, we obtain the following relationships for the porosity and CEC (see Fig. 23 for a description of the algorithm used to interpret the field data),

$$ \phi = \left( \frac{\sigma}{\sigma_w} - \frac{M_n}{\sigma_w R} \right)^{1/m}, \quad (28) $$

$$ \text{CEC} = \frac{a M_n}{\phi^{m-1} \rho_g \lambda}. \quad (29) $$

Since $\sigma$, $\sigma_w$, $M_n$ and $\lambda$ have roughly the same temperature dependence (e.g. Revil et al. 2017c) and $R$, $\rho_g$ and $m$ are temperature independent, we can perform the computation by bringing all the values at a reference temperature of 25 °C. The pore water conductivity $\sigma_w$ is taken analogously from a borehole reaching same ages Variscan granites in the Rhine basin nearby with pore waters in chemical equilibrium at depth with the granitic basement (see Sanjuan et al. 2020). This yields $\sigma_w$ values in the narrow range 0.086–0.130 S m⁻¹ at 25 °C. In the following, we take a pore water conductivity of $\sigma_w = 0.10$ S m⁻¹ at 25 °C. Using this fluid conductivity and $m = 1.70$, we can compute CEC and porosity tomograms,
<figure><img src="image_19.png" /><figcaption>**Figure 17.** Position of the electrodes in the induced polarization survey. The stations with the L-shapes ($M_1$, N, $M_2$) formed the receivers (each receiver is characterized by three electrodes forming an L-shape). The other electrodes correspond to the current electrodes (B fixed and A mobile). CBA, CBB, CBC and CBD correspond to the position of the fixed current electrode B. The central area corresponds to the granite quarry. The figure exhibits the area covered by the 3-D tomogram and the AB cross-section.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Figure 18.** Five examples of decay curves for the integrated secondary voltage after the shutdown of the primary current. These fives curves are representative of the entire data set of the field survey. They were selected for corresponding apparent resistivities in the range 360 Ohm m (curve 1) to 6800 Ohm m (curve 4). Each curve corresponds to a total of 100 to 400 stacks corresponding in turn to a maximum record of 10 min per quadrupole ABMN.</figcaption></figure>

which are shown in Fig. 24. The results are consistent with the laboratory data in terms of range of values for the porosity and CEC.

Coming back to the interpretation of the data, we have mentioned above (from Fig. 22b) that the data implies the dominant role of surface conductivity in the interpretation of the conductivity data. This implies that we cannot use Archie's law as a conductivity equation to interpret the field data as erroneously done in many field studies. We can confirm this point by adding the field conductivity value of the pore water conductivity (0.086–0.130 S m⁻¹ at 25 °C) on the petrophysical data of Site II. This is shown in Fig. 25. This figure shows that for the range of *in situ* pore water conductivities, the conductivity is indeed dominated by the surface conductivity contribution.

Finally, we can try to interpret the resulting tomograms in terms of structural and lithological features. Four superficial formations are identified, being weathered granites with varying degrees of saturation, and sandstone lenses (see Fig. 26a). The rest of the cross-section is interpreted as fresher granite with varying low degrees of fracturation/alteration.

Furthermore, the changes in resistivity and porosity across faults has been well-documented in the literature (e.g. Rucker *et al.* 2009; Glaser *et al.* 2021 and references therein). The granite batholith is interpreted as cross-cut by faults labelled a, b, c, d and e. These faults are characterized by lower resistivities and higher porosities. Structurally, from geological mapping, three major faults have been noted in the investigated area (Fig. 15). Two of these faults
<figure><img src="image_21.png" /><figcaption>**Figure 19.** Report points for the 3-D survey located below the digital elevation model acquired by drone. (a) View from above. (b) Side view. (c) Side view. The report points allow a 3-D determination of the resistivity and chargeability.</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>**Figure 20.** 3D tomogram at the seventh iteration in the Gauss–Newton inversion process. (a) Electrical conductivity tomogram (RMS error 25 per cent). (b) Normalized chargeability tomogram (RMS error of 1.3 per cent for the apparent chargeability).</figcaption></figure>

have 120–130 N orientations and are likely associated with the corridors a and c observed in the 3-D induced polarization model (Fig. 15). Fault e could be likely the N150-160 fault of the geological map, which is intersecting the model at its north-western edge. The b and d faults may be associated with previously unnoticed faults, with 3-D respective structural coherence with the N10 Rhine rift extension and N120 regional structural directions (Fig. 15). A view of sensitivities per volume is also provided in Fig. 26(b), showing a high confidence in the results especially for structures b, c and d and a good confidence for a and e structures.

## 5 CONCLUSIONS

We have developed a database of spectral induced polarization measurements of 33 granitic core samples. The measurements were done in the frequency range 10 mHz to 45 kHz using a very sensitive impedance meter. The complex conductivity spectra shows two components, one associated with the polarization of the non-metallic grains (especially kaolinite) and a high frequency contribution associated with MWS polarization (above ~10–100 Hz) and characterized by an effective dielectric constant. The in-phase conductivity is the sum of two contributions. The first contribution is associated with conduction in the bulk pore space controlled by the
<figure><img src="image_23.png" /><figcaption>**Figure 21.** 2-D cross-sections along Profile AB along an SW-NE direction for the conductivity and normalized chargeability. (a) Electrical conductivity tomogram. When surface conductivity dominates, the conductivity of the fresh core samples is between $10^{-3}$ and $10^{-5}$ S m$^{-1}$ (see Fig. 7) in agreement with the tomogram. (b) Normalized chargeability tomogram. The normalized chargeability of the fresh core samples is between $10^{-4}$ and $10^{-5}$ S m$^{-1}$ (see Fig. 11), above one order of magnitude by the distribution exhibited by the tomogram, the normalized chargeability being here not corrected by the amplification factor $a = 8$.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>**Figure 22.** Normalized chargeability versus conductivity data. (a) Comparison between laboratory and field data without the use of the amplification factor. (b) Comparison between laboratory and field data with the use of the amplification factor ($a = 10$, see Fig. 14). The line corresponds to the case for which the surface conductivity dominates the conductivity response of the rock sample using a dimensionless parameter $R = 0.20$ (see Fig. 11). The fact that the field data are close to or just below this line means that the conductivity data are dominated by their surface conductivity component at the field scale.</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>**Figure 23.** Flow chart to obtain the porosity and CEC tomograms from the electrical conductivity and normalized chargeability tomograms. The amplification factor *a* = 8 is used only for the CEC tomogram but not to the porosity tomogram. The pore water conductivity *σ*<sub>w</sub> can be obtained through its measurement of the pore water conductivity in boreholes. The two mobilities in the equations are given by λ(Na<sup>+</sup>, 25 °C) = (1.7 ± 0.5) × 10<sup>-10</sup> m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup> and B(Na<sup>+</sup>, 25 °C) = (0.85 ± 0.3) × 10<sup>-9</sup> m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>. Their ratio is equal to *R* = 0.20.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>**Figure 24.** 2-D cross-sections of the porosity and CEC tomograms along Profile AB along an SW-NE direction. (a) Porosity tomogram. The porosity of the fresh core samples could be between 1 and 6 per cent in agreement with the tomogram. (b) Cation Exchange Capacity tomogram. For the fresh core samples, the CEC was shown to be between 1 to 2 meq/100 g (Fig. 4).</figcaption></figure>
<figure><img src="image_27.png" /><figcaption>**Figure 25.** Electrical conductivity of the rock versus the pore water conductivity for the samples of Site II sampled at the test site. The vertical (yellow) band corresponds to the range of *in situ* pore water conductivity (at 25 °C). We conclude that there is likely a strong influence of the surface conductivity in the field data but bulk conductivity (associated with conduction in the bulk pore space) cannot be neglected.</figcaption></figure>

conductivity of the pore water and the formation factor, which is related to the (connected) porosity by Archie's law with a porosity exponent $m = 1.70 \pm 0.02$. The second contribution (surface conductivity) is related to the CEC of the material and the bulk tortuosity (product of the formation factor by the porosity). The low-frequency component of the quadrature conductivity is related to the normalized chargeability, which is in turn proportional to the surface conductivity. The normalized chargeability and the surface conductivity are proportional to each other with a ratio $R = 0.20$. Our model is able to explain all the observations for the granites investigated in this paper.

A field application is provided to demonstrate how the petrophysical model developed for granitoids can be applied to time-domain induced polarization field data in order to image alteration (CEC) and water content (porosity) down to a depth of ~400 m. It will be useful in the future to connect the spectral induced polarization characteristics of granites to their hydraulic, mechanical and seismic properties. Such work could also be applied to the prospection of lithium along the Variscan belt of Europe, a point that we let for further applications.

<figure><img src="image_28.png" /><figcaption>**Figure 26.** Interpretation and sensitivity of the AB cross-section. (a) Interpretation of the cross-section AB. The cross-section displays five compartments labelled 1 to 5 separated by faults zones (labelled a, b, c and d). The porous bodies at the ground surface correspond to partially saturated sandstones and granite regolith. (b) Normalized sensitivity per volume for the resistivity.</figcaption></figure>
DATA AVAILABILITY

The data used in this manuscript are available upon request to the
corresponding author.

ACKNOWLEDGMENTS

We thank the CNRS for supporting this work, Egon Zimmerman for
the construction of the ZELSIP04-V02 impedance meter, and Lucas
Le Blanc for his help with the rock samples of Site I. We thank the
students (Aurélie Aillaud, Guillaume Beringuier, Elena Caussarieu,
Zahraa Hamieh, Charlie Magniez, Elise Masse) of the ‘Géologie-
Energies’ option of the class 106 of the Ecole Nationale Supérieure
de Géologie for their help in the field. We thank the mayor of the
Vieux-Moulin municipality, Loic Firtion, as well as the quarry staff
Director, Remi Artur, as well as the Lingenheld Group for granting
us access to the site. We thank Dan Glaser and James Macnae for
their careful and constructive reviews of our manuscript and the
Editor for the professional handling of our manuscript. We thank
Lithium de France and ADEME (Agence De l’Environnement et
de la Maîtrise de l’Energie) for funding the PhD thesis of Pierre
Cosme, and Région Grand-Est and CNRS (Centre National de la
recherche Scientifique) for funding the PhD thesis of Loris Piolat.

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## APPENDIX A. TERMINOLOGY: A GUIDE THROUGH THE MAZE

We discuss in this section the terminology used for certain terms used in our paper and how they compare to the old and recent literature. The definitions of the normalized chargeability and chargeability are,

$$M_n = \sigma_{\infty} - \sigma_0, \qquad (A1)$$

$$M = \frac{\rho_0 - \rho_{\infty}}{\rho_0} = \frac{\sigma_{\infty} - \sigma_0}{\sigma_{\infty}}, \qquad (A2)$$

where $\rho_0 = 1/\sigma_0$ and $\rho_{\infty} = 1/\sigma_{\infty}$ denote the DC resistivity and instantaneous resistivity, respectively. Eq. (A2) is discussed for instance in Wait (1959) and Shuey & Johnson (1973). Lesmes & Frye

(2001), coined the term ‘normalized chargeability’ 25 yr ago and used by many authors at the onset of the millennium (e.g. Slater & Glaser 2003). The normalized chargeability $M_n$ reflects the dispersion of the conductivity (i.e. the frequency-dependence of the conductivity itself). As shown in the main text, it is proportional to the quadrature conductivity, which is another measurement of the polarization process.

Using mixing theory and the self-consistent approximation, Revil et al. (2015a, b) obtained the following formula for the chargeability of a porous composite with two types of polarizable particles: clay mineral coated with their electrical double layer (index b) and metallic particles (especially semiconductors, index m):

$$M = 1 - (1 - M_m)(1 - M_b), \qquad (A3)$$

where the chargeabilities associated with the pyrite (for small content <20 per cent vol. fraction) and clay particles are given by:

$$M_m \approx \frac{9}{2}\varphi_m, \qquad (A4)$$

$$M_b \approx \frac{\rho_g \lambda \text{CEC}}{\phi \sigma_w + \rho_g \text{BCEC}}, \qquad (A5)$$

respectively and where $\varphi_m$ denotes the volumetric content of metallic particles. The parameters entering eq. (A5) are defined in the main text and the CEC is proportional to the clay content at a given clay mineralogy (Mao et al. 2016). If the presence of metallic particles dominates the polarization response of the composite, then the chargeability is simply proportional to the volumetric amount of metallic particles $\varphi_m$. If the presence of clay particles dominates, then the normalized chargeability (not the chargeability) is proportional to the CEC, since

$$M_n \approx \frac{\rho_g \lambda \text{CEC}}{F \phi}. \qquad (A6)$$

In this case, the normalized chargeability $M_n$ is proportional to the clay content (Mao et al. 2016). Since induced polarization has been historically developed for the exploration of ores, it is normal to see that this is the chargeability has been imaged and not the normalized chargeability. That said, in environmental studies, it is more legitimate to plot the normalized chargeability to image the clay content.

To make things very confusing, a metal factor was introduced by Marshall & Madden (1959) and later Hallof (1964). It is defined as

$$M_f \equiv \frac{M}{\rho} = M\sigma = M_n, \qquad (A7)$$

that is, the metal factor is defined as the chargeability divided by the resistivity which corresponds actually to the normalized chargeability. Marshall & Madden (1959) established the proportionality between the metal factor and the quadrature conductivity. Tomograms of the metal factor is supposed to emphasize where both low-resistivity and high-chargeability regions coexist. However, sulfides and even graphites (e.g. Mao et al. 2016; Abdulsamad et al. 2020) are often disseminated and the ores will appear to be resistive and correlated with high chargeability and its use has therefore quickly been dismissed in ore exploration. In the realms of hydrogeophysics and environmental geosciences, it is obvious, then, that the terminology ‘metal factor’ is very misleading to plot a parameter related to the clay content. Lesmes & Frye (2001) wrote ‘Keller (1959) proposed a similar normalization for the time domain chargeability $MN = M/\rho$. Keller called this normalized chargeability the ‘specific capacity.’ We also report our normalized chargeability measurements in units of $S \text{ m}^{-1}$.’
The FE (Frequency Effect) is usually defined as (Vinegar & Waxman 1984)

$$FE = \frac{\rho'(f_1) - \rho'(f_2)}{\rho'(f_1)}, \qquad (A8)$$

($\rho'$ is the real part of the complex resistivity) and where $f_1$ and $f_2$ are two frequencies with $f_2 > f_1$. Alternatively, we could write the FE as (Revil et al. 2017c)

$$FE \approx \frac{\sigma'(f_2) - \sigma'(f_1)}{\sigma'(f_2)}. \qquad (A9)$$

Eqs (A8) and (Aç) are close to each but strictly equivalent as discussed in Revil et al. (2017c). Typically the PFE (Percentage

Frequency Effect) is determined as the dispersion of the in-phase resistivity or the in-phase conductivity over one decade in frequency. If ($f_2, f_1$) are extended to the instantaneous and DC conditions (i.e. ($\infty_2, 0$)), FE is simply the chargeability $M$. Generally speaking FE $\le M$. Similarly, in the text, we use a partial normalized chargeability defined as,

$$M_n(f_1, f_2) = \sigma'(f_2) - \sigma'(f_1), \qquad (A10)$$

and we have $M_n(f_1, f_2) < M_n(0, \infty) = M_n$. Under the assumption that the spectra can be modelled with the CPA parametric model, it is straightforward to connect the FE or partial normalized chargeability to the chargeability or normalized chargeability.