# A mechanistic model for the complex conductivity of clay materials.

## II. Comparison to experimental data

Alexis Maineult¹, Philippe Leroy²*, Aida Mendieta³, Damien Jougnot⁴

¹Laboratoire de Géologie, Ecole Normale Supérieure / CNRS UMR8538, PSL Research University, 75005 Paris, France.

²BRGM, French Geological Survey, 45060 Orléans, France.

³Université de Paris, Institut de Physique du Globe de Paris, CNRS, 75005 Paris, France.

⁴Sorbonne Université, CNRS, EPHE, UMR 7619 METIS, 75005 Paris, France.

*Corresponding author, p.leroy@brgm.fr

*Intended for publication in Geophysical Journal International*

## SUMMARY

Clays are very abundant minerals in the Earth's crust and express a high conductivity response that can be observed at the scale of the geological formation by electrical and electromagnetic methods. However, these minerals have a complex microstructure that renders difficult the quantitative petrophysical interpretation of the electrical field measurements. In this study, we developed a new approach to interpret spectral induced polarisation (SIP) signals measured on clay materials in terms of microstructural and electrical double layer (EDL) properties, including surface conductivity, using the physical model recently developed by Leroy *et al.* (2024). With a restrained set of optimised physical-chemical parameters, i.e., the fraction of the counter-charge and cation effective ion mobility in the Stern layer, number of stacked sheets per montmorillonite particle, and clay aggregate
effective shape and size distribution, our model well reproduces the measured laboratory SIP
spectra in the mHz to kHz frequency range on kaolinite, illite and montmorillonite muds at
different NaCl concentrations. Our results suggest that most of the EDL counter-charge
controlling SIP spectra is located in the Stern layer on the basal surfaces. First Archie's law
explains the in-phase conductivity measurements provided that the contribution of the diffuse
layer to water conductivity is properly included. The ratio of imaginary surface conductivity
to real surface conductivity is highly dependent on frequency and slightly dependent on
salinity. Finally the measured quadrature conductivity is proportional but not necessarily
equal to the imaginary surface conductivity. Our study is a step forward to better understand
the complex electrical conductivity of clays.

**Key words:** Electrical properties, Induced polarisation, Microstructure,

# 1 INTRODUCTION

Clays are fined-grained materials (with particle size below 2 µm) having a very high specific
surface area (SSA) (Grim 1962). One gram of clay mineral can cover a surface typically
comprised between 10 and 1000 m² (see the review in Revil & Leroy 2004). Montmorillonite
clay, with a SSA of about 800 m² g⁻¹, has the highest known SSA among major natural
minerals (Tournassat et al. 2015). Clay minerals are very common in the Earth's crust,
especially in geologic deposits, terrestrial weathering environments like regolith, and marine
sediments (Thiry 2000). They are the main constituent of soils and extended sedimentary
stratigraphic layers (Schroeder 2018). For instance, illite and smectite alone constitute about
30% of all sedimentary rocks on Earth (Garrels & Mackenzie 1971) and extra-terrestrial
smectite was even observed by the Curiosity rover on Mars (Bristow et al. 2018).
Clay particles have a unique lamellar microstructure (e.g., Sposito 1989). They are hydrous layer aluminosilicates made by a stack of octahedral (O) alumina sheet sandwiched between two tetrahedral (T) silica sheets (e.g., montmorillonite and illite particles) or of tetrahedral silica and octahedral alumina sheets (e.g., kaolinite particles) (Grim 1962). Phyllosilicate clay minerals have a broad range of specific surface areas, from typically $10-20 \text{ m}^2 \text{ g}^{-1}$ (chlorite and kaolinite), $100-200 \text{ m}^2 \text{ g}^{-1}$ (illite) to $700-800 \text{ m}^2 \text{ g}^{-1}$ (montmorillonite) (e.g., Tournassat et al. 2003; Revil & Leroy 2004; Hassan et al. 2006; Tournassat et al. 2015). Clays are also characterized by a high negative surface charge when in contact with water, in particular when isomorphic substitutions (replacement of $Si^{4+}$ by $Al^{3+}$ or $Fe^{3+}$ ions in tetrahedral sheets and of $Al^{3+}$ by $Mg^{2+}$ or $Fe^{2+}$ ions in octahedral sheets) occur in, e.g., illite and montmorillonite crystals (García-Romero et al. 2021). For instance, the surface charge density of hydrated montmorillonite is typically between -0.15 and -0.1 $\text{C m}^{-2}$ (Tournassat et al. 2011).

High specific surface area and negative surface charge are responsible for the high measured cation exchange capacity (CEC) of clays, typically between 0.1 and $1.5 \text{ meq g}^{-1}$ (Bergaya & Vayer 1997), which confer them remarkable adsorption properties (Tournassat et al. 2011). High specific surface area and negative surface charge of clays are also responsible for the very low permeability of clay materials (clay geologic formations have permeability values typically between $10^{-23}$ and $10^{-17} \text{ m}^2$ (Neuzil 1994)). Compacted clay rocks such as argillites have a very low permeability at saturation, typically around $10^{-21} \text{ m}^2$ (Jougnot et al. 2010a), because their pores contain a significant content of nanopores filled by the electrical double layer (EDL) surrounding the clay mineral surface in contact with water and compensating the usual negative surface charge of the mineral (Leroy et al. 2007).

Materials containing clays are used in many geo-environmental and engineering applications (see for example the references inside Leroy et al. (2017) and Ural (2018) in geotechnics). For instance, clay membrane barriers (natural and synthetic clays) adsorb salt ions (Ali et al.
2021), metallic pollutants (Gehin et al. 2007) and organic pollutants (Kausar et al. 2018) and hence are used to improve water quality (Ewis et al. 2022). In absence of connected fractures and of the significant presence of more permeable minerals than clays, ion diffusion over geological time scale is the main transport process through clay materials (Rotenberg et al. 2007). Clay geological formations (e.g., marls, shale, mudstone, argillite and claystone), because of their confining properties, serve as cap rocks above green (or de-carbonated) energies (e.g., from stored H₂ as an energy vector, and geothermal energy (Guisseau et al. 2007)) and carbonated energies (e.g., from oil and gas (Li et al. 2006)). Clay rocks such as argillites are used for the geological storage of high-level nuclear waste (Yven et al. 2007) and CO₂ (Tao et al. 2022). Clay membrane barriers, e.g., swelling bentonites, develop plasticity when wet and act as containers of industrial and domestic wastes (Malusis et al. 2003), and are also used for the geological storage of high-level nuclear waste (Leroy et al. 2006). The presence of swelling clays in the soil and underground can also induce adverse effects. For instance, the swelling of smectites due to clay hydration results to quick clay landslides and the shrinkage of smectites due to clay dehydration can fracture buildings, an issue or paramount importance in civil engineering (Khaldoun et al. 2009). The swelling of smectites also complicates subsurface oil and gas extraction (Anderson et al. 2010).

Clays in contact with water express a high excess of electrical conductivity on their surface because of the large number of moving counter-ions in the EDL (Obrien & Rowlands 1993; Rasmusson et al. 1997; Revil & Leroy 2001; Leroy & Revil 2004). In geophysics, electrical methods (e.g., Revil et al. 2005) and electromagnetic methods (e.g., Porté et al. 2023), typically undertaken on the ground surface or in boreholes, are very sensitive to the high conductivity of clays. These methods, used conjointly with geochemical and hydrogeological methods (e.g., using physico-chemical analyses and reactive transport modelling), can map in real-time and in 2 or 3 dimensions the state (e.g., water content, clay content, salinity) and
transport properties (e.g., diffusive transport, permeability) and fluid and mineralogical distributions of geological formations (e.g., Kemna et al. 2012; Binley et al. 2015; Peshtani et al. 2024).

Among electrical and electromagnetic methods in geophysics, spectral induced polarisation (SIP) is based on the injection of a low-frequency (typically from mHz to kHz) sinusoidal electrical current through the investigated medium and to the measurement of the resulting voltage (Pelton et al. 1978). The measured voltage magnitude and phase-shift between injected current and measured voltage give access to the complex conductivity of the investigated medium, i.e. to the imaginary part of the complex conductivity of the investigated medium in addition to the real part traditionally inferred from electrical resistivity tomography (ERT) (e.g., Jougnot et al. 2010b; Binley et al. 2015). Under the influence of the applied low-frequency electrical field, ions in the EDL electromigrate, polarise the EDL surrounding the clay minerals and diffuse back to their equilibrium position, increasing both conductivity magnitude and phase-shift or real and imaginary conductivity measurements (de Lima & Sharma 1992). Clays have a complex microstructure (e.g., Massat et al. 2016; Chaaya et al. 2023) and their high conductivity response (notably the contributions from the clay particle surface and the pore water) is actually not completely understood in geophysics because most petrophysical SIP models (e.g., Vinegar & Waxman 1984; Börner et al. 1993; Weller et al. 2013) do not consider explicitly clay microstructure and EDL properties that are established at the nanometric scale (Okay et al. 2014; Qi & Wu 2022, 2024). Indeed, these petrophysical models are not mechanistic ones because they are not based on ionic transport equations (like Poisson-Nernst-Planck equations, see for example Bücker et al. 2019). They are established at the scale of the representative elementary volume (REV) that is much larger than the scale of clay particles (de Lima & Sharma 1992). They are based on phenomenological equivalent circuit models that consider water conduction
(resistor) in parallel with surface conduction (a resistor in series with a capacitor for taking into account polarisation). In addition, the diffuse layer of thin clay particles such as illite and montmorillonite particles can be larger than the particle thickness hence not contributing to particle surface conductivity as defined by O'Konski (1960) (Leroy & Revil 2009; Leroy et al. 2017). In addition, Na-montmorillonite particle contains hydrated and mobile Na⁺ ions in the interlayer space between TOT sheets (e.g., Bourg et al. 2003; Tournassat & Steefel 2019) that can contribute to the measured high electrical conductivity response (e.g., Rotenberg et al. 2005; Greathouse et al. 2016).

In this work, we use the mechanistic SIP model recently developed in a companion paper by Leroy et al. (2024), which explicitly considers clay microstructural and electrochemical properties, to understand the SIP measurements by Mendieta et al. (2021) on kaolinite, illite and montmorillonite mud samples at different NaCl concentrations. Mud porosity, electrical formation factor, water conductivity and clay surface conductivity as well as aggregate size distributions are extracted from the comparison of the model to the measurements. Section 2 of our manuscript presents the model and the optimisation procedure used to determine the best values of the parameters. Section 3 shows the comparison of the predictions to the experimental data and a discussion including a comparison to the petrophysical model by Weller et al. (2013) for clayey materials.
## 2 SIP MODEL AND OPTIMIZATION PROCEDURE

### 2.1 Methodology

<figure><img src="image_1.png" /><figcaption>**Figure 1.** Sketch of a clay mud containing montmorillonite aggregates with an effective diameter $d_{eff}$ showing the clay particles interacting with each other and the electrical conductivities of the different components of the aggregate (not to scale) (modified from Leroy *et al.* 2024).</figcaption></figure>

A wet clay mud contains clay particles that are interacting each other to form aggregates (Coussot & Piau 1994). The EDL of clays is typically made of a “compact” Stern layer with a solid-like viscosity and containing mostly counter-ions and of a diffuse layer with a liquid-like viscosity and containing an excess of counter-ions and a deficiency of co-ions (Tournassat *et al.* 2009). We assume that the Stern layer and the interlayer space (in the case of Na-montmorillonite) as well as the diffuse layer and the bulk water (not influenced by clay surface charge) contribute to the electrical conductivity of the mud (Fig. 1). In addition, in our model, displacement current is considered in addition to conduction current (for more related information read section 3 of Leroy *et al.* 2024), as indicated by the asterisks on the right side
of the sigma symbols in Fig. 1. The diffuse layer (like bulk water) is assumed to contribute to water (volume) conductivity and not to the surface conductivity of the particle because the diffuse layer can be thicker than the particle (the thickness of a TOT platelet of sodic montmorillonite (Na-Mt) is ~9.5 Å (Tournassat et al. 2015), see related discussion in section 2 of Leroy et al. 2024). The mobile counter-ions in the Stern layer around clay particles and in the interlayer space of Na-Mt particles are assumed to be responsible for the surface conductivity of the interacting clay particles within clay aggregates controlling low-frequency polarisation (Leroy et al. 2017; Leroy et al. 2024). In the case of Na-Mt, our model assumes that the interlayer space behaves like an EDL where water molecules have smaller permittivity than in the bulk water and the diffuse layer (for more related information read section 2.2.1 and section 3.2 of Leroy et al. 2024). The model considers the influence of minerals less conductive and polarisable than the main clay mineral like quartz, feldspars or carbonates on the modelled complex conductivity by considering them not conductive and polarisable compared to the main clay mineral. Finally, Maxwell-Wagner polarisation is simulated from the computed complex conductivities of the liquid and solid components of the mud using the self-consistent model of Bruggeman-Hanai-Sen (BHS) (Mendelson & Cohen 1982).
<figure><img src="image_2.png" /><figcaption>**Figure 2.** Sketch showing the input and output parameters of the SIP model (explained here including in Tables 2 and 3 and in Leroy *et al.* 2024) and of the optimisation procedure versus the SIP measurements reported in Mendieta *et al.* (2021).</figcaption></figure>

The SIP model is a mechanistic model based on the Nernst-Planck equations for ion migration and back diffusion coupled to the Stokes equations for the effect of water flow (electroosmosis) and to the Laplace and Poisson-Boltzmann equations for the ion distribution in the EDL (Leroy *et al.* 2024). It considers explicitly clay microstructure and electrochemical properties. It has been developed to understand the complex conductivity measurements on clay materials and to extract from them their petrophysical properties (porosity, aggregate size distribution) and geophysical properties (surface conductivity, water conductivity, electrical formation factor) that can be used to estimate material transport properties (ion diffusivity, permeability) (e.g., Jougnot *et al.* 2010b; Peshtani *et al.* 2024).

Some input parameters of the model are from the measured experimental conditions in the
laboratory that are: the initial salt -here NaCl concentration in bulk water (before evaporation, see Mendieta *et al.* 2021) ($C_{NaCl}^0$), pH of bulk water, cation exchange capacity (CEC), dry clay powder mass ($\delta_s$), sample volume ($V_t$), temperature in degree Celsius ($T_C$), sample masses before ($\delta_t^0$), and after evaporation ($\delta_t^1$), and volume fraction of the main clay mineral in the solid phase ($\xi_c$) (Fig. 2 and Table 1) (Mendieta *et al.* 2021).

<table><caption>**Table 1.** Experimental input parameters of the SIP model.</caption><thead><tr><th>Clays</th><th>Kaolinite</th><th>Illite</th><th>Green Mt</th><th>Red Mt</th></tr></thead><tbody><tr><td>$C_{NaCl}^0$ (M)</td><td>0.01</td><td>0.1</td><td>0.01</td><td>0.1</td></tr><tr><td>$\delta_s$ (g)</td><td>373</td><td>374</td><td>404</td><td>438</td></tr><tr><td>$V_t$ (cm³)</td><td>331</td><td>331</td><td>338</td><td>338</td></tr><tr><td>$T$ (°C)</td><td>21.8</td><td>16.9</td><td>21.9</td><td>16.4</td></tr><tr><td>$\delta_t^0$ (g)</td><td>409</td><td>604</td><td>406</td><td>505</td></tr><tr><td>$\delta_t^1$ (g)</td><td>266</td><td>317</td><td>225</td><td>278</td></tr></tbody></table>

Table 2 presents the input parameters of the SIP model. All computations were carried out using Matlab R2019. The value of $D_i^b$ is taken from measured ion self-diffusion coefficients in distilled water at a temperature of 298.15 K (e.g., $D_{Na^+}^b$ (298) = 1.33 m² s⁻¹ and $D_{Cl^-}^b$ (298) = 2.03 m² s⁻¹) and considers the effect of temperature increasing $D_i^b$ and of salinity decreasing $D_i^b$ for Na⁺ and Cl⁻ ions here (Leroy *et al.* 2015). More details on the input parameters are given in Leroy *et al.* 2024. The aggregate size distribution $f(d_k)$, $k=1...L$ (L being the number of different sized aggregates), was computed prior to the optimisation procedure using the method outlined in Leroy *et al.* (2017) based on the calculation of the partial chargeability $m_k$, i.e., $f(d_k) = m_k / \sum_{k=1}^{L} m_k$ (cf. Appendix A for the computation of the $m_k$).

For each complex conductivity spectrum, three parameters (case of kaolinite and illite) or four
parameters (case of montmorillonite) were optimised: the fraction of the EDL surface charge density contributing to polarisation ($f_Q$, $0 \le f_Q \le 1$), the effective ion mobility along the mineral surface of the counter-ions contributing to electrochemical polarisation ($\beta_s$), the number of stacked sheets per clay particle ($n_c$, for montmorillonite only), and the cementation exponent characterizing the shape of the clay aggregates ($m$) (in theory, $1.2 \le m \le 4$ (Mendelson & Cohen 1982)). We assume that the partition coefficient $f_Q$ corresponds to the fraction of the mineral surface charge density compensated by the Stern layer. The $\beta_s$ quantity was adjusted by matching simulated to measured SIP spectra using an ion mobility ratio between sodium ion mobility in the Stern layer and in bulk and distilled water at a temperature $T$ of 298 K, $mr_s = \beta_s(298) / \bar{\beta}_{Na^+}^b(298)$ with $0 \le mr_s \le 1$.

<table><caption>**Table 2.** Symbols, definitions and units of the input parameters of the SIP model.</caption><thead><tr><th>Symbol</th><th>Definition</th><th>Unit</th></tr></thead><tbody><tr><td>C<sup>0</sup><sub>NaCl</sub></td><td>initial NaCl concentration in bulk water</td><td>mol L<sup>-1</sup> (M)</td></tr><tr><td>pH</td><td>potential of hydrogen ion in bulk water</td><td>dimensionless</td></tr><tr><td>CEC</td><td>cation exchange capacity</td><td>meq g<sup>-1</sup></td></tr><tr><td>δ<sub>s</sub></td><td>clay powder mass</td><td>g</td></tr><tr><td>V<sub>t</sub></td><td>sample volume</td><td>cm<sup>3</sup></td></tr><tr><td>T<sub>c</sub></td><td>temperature</td><td>°C</td></tr><tr><td>δ<sup>0</sup><sub>t</sub></td><td>sample mass before evaporation</td><td>g</td></tr><tr><td>δ<sup>1</sup><sub>t</sub></td><td>sample mass after evaporation</td><td>g</td></tr><tr><td>ξ<sub>c</sub></td><td>solid volume fraction of the main clay mineral</td><td>dimensionless</td></tr><tr><td>τ<sub>k</sub><sup>(1)</sup></td><td>relaxation time of the k<sup>th</sup> sized aggregate</td><td>s</td></tr><tr><td>f(d<sub>k</sub>)<sup>(1)</sup></td><td>aggregate size distribution</td><td>dimensionless</td></tr><tr><td>f<sub>Q</sub><sup>(2)</sup></td><td>fraction of the EDL surface charge that polarises</td><td>dimensionless</td></tr><tr><td>β<sub>s</sub><sup>(2)</sup></td><td>mobility of the counter-ions contributing to polarisation</td><td>m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup></td></tr><tr><td>n<sub>c</sub><sup>(2)</sup></td><td>number of stacked sheets per particle</td><td>dimensionless</td></tr><tr><td>m<sup>(2)</sup></td><td>cementation exponent of the clay aggregate</td><td>dimensionless</td></tr><tr><td>SS<sub>e</sub></td><td>specific surface area of the edge surfaces</td><td>m<sup>2</sup> g<sup>-1</sup></td></tr><tr><td>SS<sub>b</sub></td><td>specific surface area of the basal surfaces</td><td>m<sup>2</sup> g<sup>-1</sup></td></tr></tbody></table>
<table><tr><td>ρ<sub>s</sub></td><td>solid volumetric mass density</td><td>kg m<sup>-3</sup></td></tr><tr><td>t<sub>in</sub></td><td>interlayer thickness</td><td>m</td></tr><tr><td>D<sup>b</sup><sub>i</sub></td><td>ion self-diffusion coefficient</td><td>m<sup>2</sup> s<sup>-1</sup></td></tr><tr><td>η<sub>w</sub></td><td>liquid water dynamic viscosity</td><td>Pa s</td></tr><tr><td>ε<sub>w</sub></td><td>liquid water dielectric permittivity</td><td>F m<sup>-1</sup></td></tr><tr><td>K<sub>w</sub></td><td>dissociation constant of water molecules</td><td>dimensionless</td></tr><tr><td>M</td><td>effect of the diffuse layer on Stern layer polarisation</td><td>dimensionless</td></tr></table>

The number of TOT (“T” means tetrahedral and “O” means octahedral) sheets per montmorillonite particle ($n_c$) was adjusted by matching computed to measured SIP spectra in the range 1 to 10 (Leroy et al. 2024) as long as clay coagulation does not occur. Otherwise $n_c$ can increase to values typically between 10 and 150 for electrolytes containing $Na^+$ as dominant cation (Michot et al. 2013; Leroy et al. 2015).

We did not consider pH changes due to $H^+$ or $OH^-$ ions or ions released from the interlayer space of montmorillonite or mineral dissolution. Indeed, the initial lowest NaCl concentration in bulk water was relatively high ($C_{NaCl}^0 \approx 10^{-2}$ M), making $Na^+$ and $Cl^-$ ions the dominant ion species in bulk water during the SIP measurements of Mendieta et al. (2021). As a consequence, the pH was fixed to 7 in our computations.

For the fitting procedure of the complex conductivity, we first applied a simulated annealing method (see for instance Maineult 2016), and then we refined the solution using the simplex method (function `fminsearch` in Matlab®). The following cost-function was minimized:

$$y = 1 - R^2 = \frac{\sum_{i=1}^{N_{\sigma}} (|\sigma_{meas}^i *| - |\sigma_{mod}^i *|)^2}{\sum_{i=1}^{N_{\sigma}} (|\sigma_{meas}^i *| - |\langle \boldsymbol{\sigma}_{meas} * \rangle|)^2} + \frac{\sum_{i=1}^{N_{\sigma}} (\theta_{meas}^i - \theta_{mod}^i)^2}{\sum_{i=1}^{N_{\sigma}} (\theta_{meas}^i - \langle \boldsymbol{\theta}_{meas} \rangle)^2}, \quad (1)$$

where $R^2$ is the coefficient of determination, $N_{\sigma}$ is the number of conductivity

<footnote>(1) Parameter value a priori optimised by inverting the relaxation time distribution $f(\tau_k)$.</footnote>

<footnote>(2) Parameter value optimised by comparing model predictions to measurements.</footnote>
measurements at different frequencies (for a given mud at a given pore water salinity), $|\sigma_{meas}^i|$ and $|\sigma_{mod}^i|$ are the $i^{th}$ measured and modelled complex conductivity magnitude, respectively, and $\langle|\sigma_{meas}^i|\rangle$ is the arithmetic mean of the measured complex conductivity magnitudes. In Eq. 1, $\theta_{meas}^i$ and $\theta_{mod}^i$ are the $i^{th}$ measured and modelled minus phase-shift, respectively, and $\langle\theta_{meas}^i\rangle$ is the arithmetic mean of the measured minus phase-shifts between injected current and measured voltage and defined for the complex conductivity as

$$ \sigma^* = \sigma' + i\sigma'' = |\sigma^*|\exp(i\theta). \quad (2) $$

<table><caption>**Table 3.** Symbols, definitions and units of the output (deduced from the model) parameters.</caption><thead><tr><th>Symbol</th><th>Definition</th><th>Unit</th></tr></thead><tbody><tr><td>d<sub>k</sub><sup>(1)</sup></td><td>Aggregate effective diameter</td><td>m</td></tr><tr><td>C<sub>NaCl</sub><sup>b</sup></td><td>NaCl concentration in bulk water</td><td>mol L<sup>-1</sup></td></tr><tr><td>φ<sub>a</sub></td><td>Aggregate porosity</td><td>dimensionless</td></tr><tr><td>φ</td><td>Mud porosity</td><td>dimensionless</td></tr><tr><td>Θ<sub>w</sub><sup>d</sup></td><td>Water volume fraction of the diffuse layer</td><td>dimensionless</td></tr><tr><td>Ξ<sub>in</sub><sup>(2)</sup></td><td>Volume of the interlayer space over the diffuse layer and interlayer space volumes</td><td>dimensionless</td></tr><tr><td>σ<sub>s</sub><sup>*</sup></td><td>Surface conductivity of the aggregates</td><td>S m<sup>-1</sup></td></tr><tr><td>σ<sub>w</sub></td><td>Water conductivity (<|σ<sub>w</sub><sup>*</sup>|>)</td><td>S m<sup>-1</sup></td></tr><tr><td>σ<sub>w</sub><sup>b</sup></td><td>Bulk water conductivity</td><td>S m<sup>-1</sup></td></tr><tr><td>σ<sub>w</sub><sup>d</sup><sup>(3)</sup></td><td>Diffuse layer water conductivity</td><td>S m<sup>-1</sup></td></tr><tr><td>φ<sub>d</sub></td><td>Electrical potential at the start of the diffuse layer</td><td>mV</td></tr><tr><td>B<sub>Na<sup>+</sup></sub><sup>d</sup></td><td>Average sodium ion mobility in the diffuse layer</td><td>m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup></td></tr><tr><td>Q<sub>v</sub></td><td>Aggregate excess of charge per unit pore volume</td><td>C m<sup>-3</sup></td></tr></tbody></table>

The output parameters of the SIP model are reported in Table 3. Four output parameters are of particular importance to compare the model predictions to the petrophysical measurements

<footnote>(1) Parameter value computed from adjusted $\beta_s$ value and inverted $\tau_k$ value.</footnote>

<footnote>(2) For montmorillonite only.</footnote>

<footnote>(3) Thick diffuse layer assumption (see related discussion in section 3 of Leroy *et al.* 2024).</footnote>
and to the parameters of petrophysical models developed at larger scale than the EDL scale (e.g., Weller et al. 2013). These parameters are mud clay porosity (φ) and cementation exponent of the clay aggregates (m) given access to the electrical formation factor (F) according to first Archie's law, water conductivity (σw) and clay surface conductivity (σs*).

In addition, the average effective ion mobility of Na⁺ cation in the diffuse layer (B<sub>Na⁺</sub><sup>d</sup>) is shown in Table 3 because Na⁺ is the major ionic component influencing diffuse layer conductivity here. The B<sub>Na⁺</sub><sup>d</sup> quantity (Leroy et al. 2024) was computed from

$$B_{Na^+}^d = \beta_{Na^+}^d + \frac{1}{\chi_d} \int_{x=0}^{x=\chi_d} \beta_{eo}^d(x) dx, \quad (3)$$

where β<sub>Na⁺</sub><sup>d</sup> is the Na⁺ ion mobility due to electromigration (considered to be equal to the related ion mobility in bulk water, i.e. β<sub>Na⁺</sub><sup>d</sup> = β<sub>Na⁺</sub><sup>b</sup>), χ<sub>d</sub> is the diffuse layer thickness, x is the distance from the beginning of the diffuse layer and normal to the mineral wall, and β<sub>eo</sub><sup>d</sup> is the electroosmotic contribution to surface conductivity. According to Leroy et al. (2015) χ<sub>d</sub> = 2χ<sub>D</sub> where χ<sub>D</sub> is the Debye length that is inversely proportional to bulk water ionic strength (χ<sub>D</sub> = κ⁻¹, see Eq. 25). The quantities β<sub>Na⁺</sub><sup>d</sup> and β<sub>eo</sub><sup>d</sup>(x) were computed considering their dependence on bulk water salinity, temperature and of the electrical potential in the diffuse layer for β<sub>eo</sub><sup>d</sup>(x) (see Section 2.2.1).

We consider that the interlayer space of our montmorillonite samples, i.e., Na-Mt containing hydrated Na⁺ cations, can occupy a significant part of the volume of the mud. Mud porosity and the relative part of the volume of the interlayer space of Na-Mt over the interlayer space and diffuse layer volume were computed using
$$ \phi = \frac{V_{in} + V_{w}^{d} + V_{w}^{b}}{V_{t}}, \qquad (4) $$

$$ \Xi_{in} = \frac{V_{in}}{V_{in} + V_{w}^{d}}, \qquad (5) $$

where, in the numerator of Eq. 4, the Stern layer volume, $V_s$, was considered equal to 0 m³ (because the Stern layer can be considered part of the solid and of the small Stern layer thickness, $d_s \approx 0.5$ nm (Leroy et al. 2024), and to decrease the number of model parameters), $V_{in}$ is the computed interlayer space volume (we took an interlayer space thickness $t_{in} = 1$ nm for Na-Mt) and $V_w^d$ is the computed diffuse layer volume. Concerning $t_{in}$, according to the review paper of Tournassat et al. (2015), the thickness of a TOT layer is around 9.5 Å and the basal spacing at $d_{001}$ is equal to about 20.5 Å for the four-layer hydrate state. Therefore, the corresponding thickness of the interlayer space, $t_{in}$, is equal to $20.5 - 9.5 = 11 \approx 10$ Å (1 nm).

## 2.2 Model

### 2.2.1 Electrochemical properties

<figure><img src="image_3.png" /><figcaption>**Figure 3.** Sketch showing the electrochemical properties of the basal surface of montmorillonite in the case of a surrounding fully dissociated monovalent electrolyte like</figcaption></figure>
NaCl. The quantities $Q_b$ and $Q_d$ are the surface charge densities of the Stern plane and of the diffuse layer, respectively.

The electrical potential at the beginning of the diffuse layer (from the mineral surface) ($\phi_d$) (Fig. 3, for more details, see Fig. 5 of Leroy et al. 2024) is calculated from the measured temperature ($T$) (in K), bulk water salinity ($C_b$), mineral surface charge density ($Q_0$) and partition coefficient $f_Q$ values. First the bulk water salinity ($C_b$) (in mol L$^{-1}$, M) was computed from the initial NaCl concentration ($C_{NaCl}^0$) and the measured sample masses before and after evaporation ($\delta_t^0$ and $\delta_t^1$, respectively). It was given by:

$$C_b = C_{NaCl}^0 \frac{\delta_t^0}{\delta_t^1}, \qquad (6)$$

where $C_b = C_{NaCl}^b$. The electrolyte was assumed fully dissociated implying

$$C_b = C_{Na^+}^b, \qquad (7)$$

and

$$C_b = C_{Cl^-}^b. \qquad (8)$$

The complex conductivity model depends on the specific surface area of the main clay mineral where cation exchange reactions occur ($SS$) (in meq g$^{-1}$) (Leroy et al. 2024). Among the $SS$ values reported in Mendieta et al. (2021) (see supplementary material), only the specific surface area of illite measured by the BET (Brunauer-Emmett-Teller) method was considered for the complex conductivity modelling because the BET method cannot measure the internal specific surface area of smectite (Metz et al. 2005) and only the illite sample does not contain a significant smectite content according to X-ray diffraction (XRD) measurements (see Table 1 of Mendieta et al. 2021). The BET method probes the outer specific surface area
of clays by measuring adsorption isotherm of non-polar gases such as nitrogen or argon at low pressure and boiling temperature (77 K) (Tournassat *et al.* 2003). The specific surface area of the mineralogically heterogeneous kaolinite sample (containing significant smectite and illite contents, 4 % and 3 %, respectively) was adjusted by matching simulated to measured SIP spectra and the specific surface area of the green and red montmorillonite samples, which can be approximated by the total specific surface area of the basal surfaces (SS<sub>b</sub>), was taken from crystallography (Table 4).

<table><caption>**Table 4.** Input microstructural, electrochemical, and mineralogical parameters</caption><thead><tr><th>Clay samples</th><th>Kaolinite</th><th>Illite</th><th>Green Mt</th><th>Red Mt</th></tr></thead><tbody><tr><td>SS (m<sup>2</sup> g<sup>-1</sup>)</td><td>29.78<sup>(1)</sup></td><td>101.6<sup>(2)</sup></td><td>760<sup>(3)</sup></td><td>760<sup>(3)</sup></td></tr><tr><td>CEC (meq g<sup>-1</sup>)</td><td>0.10<sup>(4)</sup></td><td>0.20<sup>(4)</sup></td><td>1.32<sup>(2)</sup></td><td>1.35<sup>(2)</sup></td></tr><tr><td>ξ<sub>c</sub> (%)</td><td>0.91<sup>(2)</sup></td><td>0.67<sup>(2)</sup></td><td>0.90<sup>(2)</sup></td><td>0.66<sup>(2)</sup></td></tr><tr><td>Q<sub>0</sub> (C m<sup>-2</sup>)<sup>(5)</sup></td><td>-0.324</td><td>-0.190</td><td>-0.168</td><td>-0.171</td></tr></tbody></table>

(1) Adjusted by matching simulated to measured SIP spectra.

(2) Measured (from Mendieta *et al.* (2021)); note that for kaolinite, we took into account smectite and illite contents.

(3) Leroy *et al.* (2015) (crystallography).

(4) Computed according to XRD measurements and theoretical CEC values.

(5) Computed according to the CEC and the specific surface area (Eq. 9).

The Q<sub>0</sub> quantity was calculated from the CEC and associated specific surface area (SS) (Zhou & Gunter 1992) using

$$Q_0 = \frac{-eN_A \text{CEC}}{10^3 \text{SS}}, \qquad (9)$$

where e is the electron (or elementary) charge (~1.602×10<sup>-19</sup> C) and N<sub>A</sub> is the Avogadro number (~6.022×10<sup>23</sup> mol<sup>-1</sup>). The CEC values reported in Mendieta *et al.* (2021) for kaolinite and illite samples (0.22 and 0.47 meq g<sup>-1</sup>, respectively) were too high to obtain realistic surface charge density measurements (see supplementary material). We thought that the CEC measurements reported in Mendieta *et al.* (2021) overestimate the CEC of kaolinite and in a lesser extent of illite samples – possibly because only the fraction of clay particles with size
below 1 micrometre was considered for the CEC measurements. Indeed, the CEC increases when the size of the investigated particles decreases (Shim et al. 2003). As a consequence, for kaolinite and illite samples, we ran our SIP model with theoretical CEC values using

$$CEC = \sum_{i=1}^{N_c} w_i CEC_i, \qquad (10)$$

where $N_c$ is the number of the different clay minerals, $w_i$ is the solid content of clay mineral i given by XRD measurements and $CEC_i$ is its CEC. In Eq. 10, we used a CEC value of 0.05 meq g⁻¹ for kaolinite, 0.25 meq g⁻¹ for illite, and 1 meq g⁻¹ for smectite (Revil & Leroy 2004, and Table 4). With the theoretical CEC values for kaolinite and illite samples and the measured CEC values of montmorillonites, we obtained $Q_0 = -0.324 \ C \ m^{-2}$ for kaolinite, $Q_0 = -0.190 \ C \ m^{-2}$ for illite, $Q_0 = -0.168 \ C \ m^{-2}$ for green montmorillonite, and $Q_0 = -0.171 \ C \ m^{-2}$ for red montmorillonite samples (Table 4 and Eq. 9). Using these values divided by the electron charge, we estimated that kaolinite, illite, green and red montmorillonite samples have ~2.014, 1.186, 1.046, and 1.067 elementary charges nm⁻² on their surface, respectively, which is in the range of the 1 to 3 elementary charges nm⁻² expected for clay minerals (Revil & Leroy, 2004; see their Figure 2).

The electrical potential at the beginning of the diffuse layer $φ_d$ is required to compute (i) the electrical conductivity of the diffuse layer, (ii) the conductivity of the part of the interlayer space of Na-Mt not contributing to polarisation (see Leroy et al. 2024), and (iii) the effect of the diffuse layer on Stern layer polarisation according to Lyklema et al. (1983). These latter authors assumed that the counter-ions in the diffuse layer decrease the relaxation time associated with the polarised Stern layer. In our model, this effect can be modelled through the computation of the M parameter (see Leroy et al. 2024). The $φ_d$ potential is calculated
from the measured temperature T (in K, T = T<sub>C</sub> + 273.15), C<sub>b</sub>, Q<sub>0</sub> and f<sub>Q</sub> values by solving

$$ \left| \sqrt{4\overline{\varepsilon_w k_B T 1000 N_A C_b} \left[ \cosh\left(\frac{e\varphi_d}{k_B T}\right) - 1 \right]} - Q_0(1-f_Q) \right| = 0, \quad (11) $$

where $\overline{\varepsilon_w}$ represents the average water permittivity in the diffuse layer (on clay particle surface) and interlayer space (in the clay particle) of Na-Mt (in F m<sup>-1</sup>), and $k_B$ is the Boltzmann's constant (~1.381×10<sup>-23</sup> J K<sup>-1</sup>).

It should be noted that the water permittivity term ($\overline{\varepsilon_w}$) intervening in Eq. 11 is assumed to be related to the bulk water permittivity ($\varepsilon_w$) and to the permittivity of the interlayer space of Na-Mt that is assumed to be half the bulk water permittivity (see discussion in Tournassat et al. 2009 and in section 3.2 of Leroy et al. 2024). We assume that the water permittivity of the diffuse layer around clay particles is equal to bulk water permittivity. The term $\overline{\varepsilon_w}$ is calculated from the water permittivities in the diffuse layer and interlayer space weighted by their respective relative volume fraction, i.e.

$$ \overline{\varepsilon_w} = \frac{\varepsilon_w}{V_w^d + V_{in}} \left( V_w^d + \frac{V_{in}}{2} \right), \quad (12) $$

where $\varepsilon_w$ depends on temperature (in the temperature range 2-87°C, for more related information read Leroy et al. 2024). In Eq. 12, the $V_w^d$ and $V_{in}$ quantities were determined from

$$ V_w^d = 2\delta_s \xi_c SS_0 \chi_D, \quad (13) $$

where $SS_0$ is the outer specific surface area of clay particles (in m<sup>2</sup> g<sup>-1</sup>) and $\chi_D = \kappa^{-1}$ is the
Debye length (in m), and

$$V_{in} = \frac{\delta_s \xi_c SS}{2n_c} (n_c - 1) t_{in}. \qquad (14)$$

We considered in Eq. 13 from Leroy et al. (2024) that the outer specific surface areas of the clay particles in kaolinite and illite samples are equal to SS (i.e. $SS_o = SS$) and that the outer specific surface areas of the clay particles in montmorillonite samples are equal to $SS/n_c$ (i.e. $SS_o = SS/n_c$). In the case of montmorillonite, we assumed in Eq. 14 from Leroy et al. (2024) that $SS = SS_b$ with $SS_b$ being the total specific surface area of the basal surfaces (see Appendix A of Leroy et al. 2024, for details).

### 2.2.2 Surface conductivity and effective size distribution of the aggregates

The SIP model considers that the clay mud is composed of the solid part of the clay aggregates (including Stern layers on particle surfaces) and of Na-Mt interlayer spaces, bulk and diffuse layer waters (case of the thick diffuse layer assumption in the model of Leroy et al. 2024), and weakly conductive and polarisable minerals (Fig. 1). Because Na-Mt particles contain hydrated and mobile Na⁺ cations in their interlayer space (Bourg et al. 2003), we assumed that the interlayer space of Na-Mt behaves like an EDL containing a Stern and a diffuse layer. In our model, the excess of charge per unit pore volume of the clay aggregate ($Q_v$) (in C m⁻³) is the main parameter controlling electrochemical polarisation, like in the model of Vinegar & Waxman (1984). The $Q_v$ quantity was computed according to the solid volumetric mass density ($\rho_s$) (in kg m⁻³), computed porosity of the clay aggregate ($\phi_a$), and measured CEC using
$$Q_v = \rho_s \left( \frac{1 - \phi_a}{\phi_a} \right) e N_A \text{ CEC}, \qquad (15)$$

where $\phi_a$ depends on the calculated volumes of the different components of the aggregate

$$\phi_a = \frac{V_w^d + V_w^b}{V_{sc} + V_w^d + V_w^b}, \qquad (16)$$

for kaolinite, and

$$\phi_a = \frac{V_w^d}{V_{sc} + V_w^d}, \qquad (17)$$

for illite, and

$$\phi_a = \frac{V_{in} + V_w^d}{V_{sc} + V_{in} + V_w^d}, \qquad (18)$$

for montmorillonite such as Na-Mt, where $V_{sc}$ and $V_w^b$ are the computed volumes of the solid clay layers (in m³) and bulk water, respectively (for more details, see Appendix A of Leroy et al. 2024)

$$V_s = \frac{10^{-3} \delta_s}{\rho_s}, \qquad (19)$$

$$V_{sc} = V_s \xi_c, \qquad (20)$$

$$V_w^b = 10^{-6} V_t - V_s - V_w^d - V_{in}, \qquad (21)$$

where we took $\rho_s = 2700$ kg m⁻³ for the solid volumetric mass density (Tournassat et al. 2016) and we neglected illite interlayer space and Stern layer volumes in the calculation of the aggregate porosity to decrease the number of model parameters. It should be noted that we
assumed in Eqs 17 and 18 that illite and montmorillonite aggregates contain only diffuse layer (and interlayer space for Na-Mt) in their pores. Therefore, in these conditions, computed aggregate porosity $\phi_a$ is not necessarily identical to computed mud porosity $\phi$ because mud can also contain bulk water (see Eq. 4). In Leroy et al. (2024), we considered that a pure kaolinite aggregate also contains bulk water, but, in the SIP experiment of Mendieta et al. (2021), kaolinite sample contains significant illite and smectite solid contents (3 % and 4 %, respectively) that may explain why we had to consider only diffuse layer water in the clay aggregates of the kaolinite sample ($V_w^b = 0 m^3$ in Eq. 16).

The surface conductivity of the clay aggregate due to Stern layer polarisation ($\sigma_s^*$; in S m$^{-1}$), was determined as a function of the angular frequency $\omega$ (in rad s$^{-1}$) using

$$ \sigma_s^*(\omega) = \frac{2}{3} \phi_a \beta_s f_Q Q_V \sum_{k=1}^{L} f(d_k) \frac{i\omega\tau_k}{1 + i\omega\tau_k} + i\omega\alpha\rho_s\epsilon_0, \quad (22) $$

where $f(d_k)$ is the aggregate size distribution, with $k=1...L$ ($L$ being the number of different aggregate effective diameters and $d_k$ being the effective diameter of the $k^{th}$ aggregate), $i$ is the imaginary number ($i^2 = -1$) and $\tau_k$ is the relaxation time of the polarised Stern layers around interacting clay particles within the $k^{th}$ aggregate (in s) (Leroy et al. 2024). In Eq. 22, $\alpha = 0.00191 m^3 kg^{-1}$ is an empirical value that was used by Olhoeft (1981) to adjust solid dielectric permittivity spectra in absence of conducting materials, such as magnetic or metallic minerals, and Fe Ti sulphides oxides such as ilmenite, $\epsilon_0 \approx 8.854 \times 10^{-12} F m^{-1}$ is the permittivity of vacuum, and $\kappa = \chi_D^{-1}$ is the inverse of the Debye length (in m$^{-1}$).

According to Leroy et al. 2024, aggregate effective size distribution was computed from
$$d_k = \sqrt{8 \frac{k_B T}{e} \beta_S M \tau_k}, \qquad (23)$$

$$M = 1 - \frac{f_Q Q_0}{k_B T \overline{\varepsilon_w \kappa \cosh\left(\frac{e \varphi_d}{2k_B T}\right)}}, \qquad (24)$$

$$\kappa = \sqrt{\frac{2e^2 10^3 N_A C_b}{\overline{\varepsilon_w k_B T}}}. \qquad (25)$$

### 2.2.3 Water conductivity

Our model calculates water conductivity ($\sigma_w^*$) from the computed bulk water conductivity ($\sigma_w^b$*) and diffuse layer water (and Na-Mt interlayer space) conductivity ($\sigma_w^d$*) using the lower bound of the self-consistent model developed by Hashin & Shtrikman (1962):

$$\sigma_w^* = \sigma_w^b * \left[ 1 + \frac{3\Theta_w^d (\sigma_w^d * - \sigma_w^b * )}{3\sigma_w^b * + (1 - \Theta_w^d) (\sigma_w^d * - \sigma_w^b * )} \right], \qquad (26)$$

where $\Theta_w^d$ is the volume fraction of diffuse layer water over diffuse layer and bulk waters. In Eq. 26, the condition $|\sigma_w^d *| > |\sigma_w^b *|$ that is usually considered on particle surface (explaining surface conductivity (Revil & Glover 1997; Lyklema & Minor 1998)) should be satisfied.

Bulk water and diffuse layer conductivities were calculated from

$$\sigma_w^b * = e10^3 N_A (\beta_{Na^+}^b + \beta_{Cl^-}^b) C_b + i\omega\varepsilon_w, \qquad (27)$$

$$\sigma_w^d * = \frac{e10^3 N_A C_b}{\chi_d} \int_{x=0}^{x=\chi_d} \left\{ B_{Na^+}^d(x) \exp\left[-\frac{e\varphi(x)}{k_B T}\right] + B_{Cl^-}^d(x) \exp\left[\frac{e\varphi(x)}{k_B T}\right] \right\} dx + i\omega\varepsilon_w, \qquad (28)$$
$$B_{Na^+}^d(x) = \beta_{Na^+}^b + \beta_{eo}^d(x), \qquad (29)$$

$$B_{Cl^-}^d(x) = \beta_{Cl^-}^b - \beta_{eo}^d(x), \qquad (30)$$

$$\beta_{eo}^d(x) = \frac{\varepsilon_w}{\eta_w} [\varphi(x) - \varphi_d], \qquad (31)$$

where $\beta_{Na^+}^b$ and $\beta_{Cl^-}^b$ are the Na$^+$ and Cl$^-$ ion mobilities (in m$^2$ s$^{-1}$ V$^{-1}$) due to ion electromigration under the applied electrical field that were corrected of temperature using water dynamic viscosity measurements, $\eta_w$ (in Pa s$^{-1}$), and salinity (see Appendix B of Leroy et al. 2024). In Eq. 28, $B_{Na^+}^d$ and $B_{Cl^-}^d$ are the effective ion mobilities of Na$^+$ and Cl$^-$ ions in the diffuse layer considering electroosmosis and $\varphi$ is the electrical potential in the diffuse layer (in V). The electrical potential $\varphi$ was computed by solving analytically the Poisson-Boltzmann equation (see Section 3.4 of Leroy et al. 2024):

$$\varphi(x) = \frac{4k_B T}{e} \tanh^{-1} \left[ \tanh \left( \frac{e\varphi_d}{4k_B T} \right) \exp(-\kappa x) \right]. \qquad (32)$$

Because we used the diffuse layer electrical conductivity model based on the $\varphi_d$ potential calculation, we did not use here the diffuse layer electrical conductivity model based on the calculation of the average electrical potential in the diffuse layer ($\varphi_m$), which is reported in Okay et al. (2014) and in Leroy et al. (2024). Indeed, we assume here that the Poisson-Boltzmann equation is more accurate to estimate the electrical potential distribution in the diffuse layer than using directly an average electrical potential. In addition, the calculation of the $\varphi_d$ potential allows directly computing the effect of the diffuse layer on Stern layer polarisation represented in the model by the M parameter (Eq. 24).
### 2.2.4 Mud clay complex conductivity

In our model, the background conductivity of clay mud (not considering surface conductivity effect) ($\sigma_0^*$) is corrected of the effect of minerals less conductive than water using the upper bound of the self-consistent model developed by Hashin & Shtrikman (1962):

$$ \sigma_0^* = \sigma_w^* \left[ 1 - \frac{3(1-c)}{(3-c)} \right], \qquad (33) $$

where c is the volume fraction of conductive water over the volume of considered not conductive minerals and conductive water. The effect of magnetite, which is a metallic mineral that may be more conductive than clays, was neglected in our model because of its small content (1 %) in red montmorillonite sample (Mendieta et al. 2021).

After computing $\sigma_0^*$, the self-consistent assumption of the differential effective medium theory (Bruggeman-Hanai-Sen (BHS) model; Mendelson & Cohen 1982; Sen 1984) was used to calculate clay mud complex conductivity ($\sigma^*$) considering inclusions of randomly oriented solid clay aggregates of cementation exponent m with volume fraction $d\Omega_s$ and surface conductivity $\sigma_s^*$ in the background medium up to reach the solid clay content in the sample ($\Omega_s = 10^6 V_{sc} / V_t$):

$$ L = \frac{3 + \sqrt{9 + 36m^2 - 60m}}{6m}, \qquad (34) $$

$$ d\sigma^* = \frac{\sigma^* (\sigma_w^* - \sigma_0^*) [(1+3L)\sigma_s^* + (5-3L)\sigma_0^*]}{3 [L\sigma_s^* + (1-L)\sigma_0^*] [(1-L)\sigma_s^* + (1+L)\sigma_0^*]} \frac{d\Omega_s}{1-\Omega_s}, \qquad (35) $$

$$ \sigma^* = \sigma^* + d\sigma^*, \qquad (36) $$
$$ \Omega_s = \Omega_s + d\Omega_s, \qquad (37) $$

where L is the depolarisation factor of inclusions ($0 \le L \le 1$), $\Omega_s = 0$ and $\sigma^*=\sigma_0^*$ initially and $d\Omega_s = 10^6 V_{sc} / (V_t N_s)$ with $N_s$ being the number of randomly oriented solid inclusions. The procedure outlined in Eqs 34-37 is sensitive to the value of the cementation exponent $m$ and allows also simulating Maxwell-Wagner polarisation (Leroy & Revil 2009).

Aggregate eccentricity ($e_c$) was also deduced from the depolarisation factor L calculation (Mendelson & Cohen 1982) based on optimised cementation exponent $m$ value (Eq. 34) by solving

$$ \left| L - \frac{1+e_c^2}{e_c^3} (e_c - \tan^{-1} e_c) \right| = 0, \qquad (38) $$

and then, according to Mendelson & Cohen (1982), the height of the $k^{th}$ aggregate (in m) with effective diameter $d_k$ was given by

$$ h_k = \sqrt{\frac{d_k^2}{1+e_c^2}}. \qquad (39) $$

## 2.3 Optimisation procedure

First of all, it should be noted that the measured conductivity $\sigma_{meas}^*(\omega)$ is not equal to $\sigma_{mod}^*(\omega)$ ($\sigma_{mod}^*(\omega)$ being the prediction of the model developed in Leroy et al. (2024)) but it is equal to $\sigma_{mod}^*(\omega)+i\omega C_c$. The $i\omega C_c$ term is introduced here to take into account the contact impedance of the potential electrodes and electromagnetic coupling effects between cables during the measurements (Ingeman-Nielsen & Baumgartner 2006; Florsch et al. 2014;
Huisman et al. 2016). It should be also noticed that the $C_c$ values determined during the optimization procedure (see Appendix B) are different from the $C$ values obtained during the Debye decomposition procedure (see Appendix A). Indeed, in the model developed in Leroy et al. (2024), the displacement current is taken into account. The $C_c$ values obtained here are therefore the effects of the contact impedance of the potential electrodes and electromagnetic coupling only. The corrected spectra are the spectra from which the $i\omega C_c$ curve has been subtracted.

For each clay type, we used a simulated annealing algorithm (see Maineult 2016) and then the simplex algorithm to determine the best values of the model parameters using the spectra at 0.01 M NaCl initially and 0.10 M NaCl initially together. For kaolinite, the “common” parameter was SS, the “individual” parameters were $f_Q$, $mr_S$, and $m$, and the coupling coefficient $C_c$ (here, “common” means that the value is not dependent on salinity, and “individual” means that there is a value of the parameter per salinity, see Tables 4 and 5). For illite, all the parameters, i.e., $f_Q$, $mr_S$, $m$, and $C_c$ were individual. For montmorillonite, the common parameter was $mr_S$, the individual ones were $f_Q$, $m$, and $C_c$, and the values of $n_c$ were determined by a systematic exploration. Note that red montmorillonite at 1 M NaCl initially was optimised separately. We assumed a constant value of $mr_S$ in the case of montmorillonite for the two lowest salinities because of the stronger influence of the isomorphic substitutions in the solid crystal compared to kaolinite and illite, inducing a constant negative attractive force slowing down cations with the same strength in the Stern layer.
# 3 COMPARISON TO EXPERIMENTAL DATA AND DISCUSSION

## 3.1 SIP data representation and model parameter optimisation

The complex conductivity magnitude and minus phase-shift can be expressed as a function of the real part of the complex conductivity (in-phase conductivity) and the imaginary part of the complex conductivity (quadrature conductivity) using:

$$|\sigma^*| = \sqrt{\sigma'^2 + \sigma''^2}, \qquad (40)$$

$$\theta = \tan^{-1}\left(\frac{\sigma''}{\sigma'}\right), \qquad (41)$$

It is easier to understand in-phase and quadrature conductivities rather than magnitude and minus phase-shift of the complex conductivity because $\sigma'$ is mostly sensitive to conduction current and $\sigma''$ is only sensitive to polarisation current (Okay et al., 2014), whereas $|\sigma^*|$ and $\theta$ are a combination of both $\sigma'$ and $\sigma''$. In addition, because $|\sigma^*| \gg |\sigma''|$, $|\sigma^*| \approx |\sigma'|$ (Eq. 40). Therefore, in the following, we only discussed complex conductivity magnitude or in-phase conductivity, and not complex conductivity magnitude and in-phase conductivity (see Fig. 4 for instance). Nevertheless, we present $|\sigma^*|$, $\theta$, and $\sigma'$, $\sigma''$ because complex conductivity measurements are usually represented in terms of $|\sigma^*|$ and $\theta$. The magnitude and minus phase-shift of the complex conductivity as well as the in-phase conductivity and the quadrature conductivity and the aggregate effective size distribution of each clay sample are reported in Figs 4 to 8. The results of the optimisation procedure are shown in Tables 5 and 6 (optimised parameter values are given with four digits after coma to enable people to reproduce the results).

The parameter values were well constrained (uncertainties below 10 % of the main value) (see
supplementary material). Varying infinitesimally these parameter values from their optimised values induces a more than 50 % increase of the cost function (Eq. 1) (except for the case of the number of platelets per particle ($n_c$) for red montmorillonite at 1 M NaCl initially).

The effective ion mobility of Na⁺ counter-ion in the Stern layer of the main clay mineral was computed from $mr_s$ and the temperature $T$ (K) (Revil et al. 1998), with:

$$ \beta_s = mr_s \bar{\beta}_{Na^+}^b (298) [1 + v_s (T - 298.15)], \quad (42) $$

where $\bar{\beta}_{Na^+}^b (298) \approx 5.19 \times 10^{-8} \text{ m}^2 \text{ s}^{-1} \text{ V}^{-1}$ and $v_s = 0.037 \text{ K}^{-1}$ (Leroy et al. 2024).

Mineral surface charge influences surface ion mobility of counter-ions in the Stern layer and interlayer space by slowing down counter-ion species and water molecules (Bourg & Sposito 2011; Greathouse et al. 2016; Bouhadja & Skelton 2018). In our model, the magnitude of the surface charge density of the main clay mineral ($Q_0$) is fixed and linearly proportional to the CEC (Eq. 9). In the case of pure kaolinite, the surface ion mobility of counter-ions in the Stern layer can decrease with increasing salinity because of the stronger surface charge density attracting and slowing down more strongly counter-ions in the Stern layer (Bouhadja & Skelton 2018). Indeed, when salinity increases, at constant pH, deprotonation of surface hydroxyl sites and the increasing concentration of counter-ions in the Stern layer increases the magnitude of the negative surface charge density of the mineral (Sahai & Sverjensky 1997). In the case of 0.01 and 0.1 M NaCl initially, the constant CEC of montmorillonite minerals when salinity changes (due to the permanent and negative structural surface charge) implies an electrostatic attraction force slowing down counter-ions with the same strength whatever salinity. However, at initial salinities of 0.1 M NaCl and 1 M NaCl, Na⁺ effective ion mobility on the surface of illite and red montmorillonite, respectively, can decrease compared to those at lower initial salinities because of particle coagulation and the higher Na⁺ concentration
combined with the more hydrophobic clay surface (Greathouse *et al.* 2016). In addition, our kaolinite sample contains a significant solid content of smectite (4 %) and illite (3 %), which may control the electrochemical properties of the sample due to the higher structural surface charges and considerably higher specific surface areas of montmorillonite (typically 700-800 m² g⁻¹) and illite (typically 100-200 m² g⁻¹) particles than kaolinite particle (typically 10-20 m² g⁻¹) (Tournassat *et al.* 2003; Hassan *et al.* 2006; Tournassat *et al.* 2015) (see related discussion in Leroy *et al.* 2024). Therefore, we assumed that Na⁺ effective ion mobility on clay particle surface in the kaolinite sample is not changing significantly with salinity (see Table 5).

## 3.2 Results and discussion

### 3.2.1 SIP and aggregate size distribution spectra

<figure><img src="image_4.png" /><figcaption>**Figure 4.** Complex conductivity of kaolinite muds at two NaCl concentrations (C<sub>b</sub> = 0.0154 M and C<sub>b</sub> = 0.191 M). Measurements (filled circles) and modelling results (curves). **a:** raw real part, **c:** raw imaginary part, **e:** raw phase, **b, d, f:** same as **a, c, e** but after removing the</figcaption></figure>
coupling term (in **b**, the in-phase conductivity at the lowest frequency is also removed).

<figure><img src="image_5.png" /></figure>

**Figure 5.** Complex conductivity of illite muds at two NaCl concentrations (C<sub>b</sub> = 0.0180 M and C<sub>b</sub> = 0.182 M). Measurements (filled circles) and modelling results (curves). **a:** raw real part, **c:** raw imaginary part, **e:** raw phase, **b, d, f:** same as **a, c, e** but after removing the coupling term (in **b**, the in-phase conductivity at the lowest frequency is also removed).

<figure><img src="image_6.png" /></figure>
<figure><img src="image_7.png" /><figcaption>**Figure 6.** Complex conductivity of green montmorillonite muds at two NaCl concentrations (C<sub>b</sub> = 0.0153 M and C<sub>b</sub> = 0.146 M). Measurements (filled circles) and modelling results (curves). **a:** raw real part, **c:** raw imaginary part, **e:** raw phase, **b, d, f:** same as **a, c, e** but after removing the coupling term (in **b**, the in-phase conductivity at the lowest frequency is also removed).</figcaption></figure>

<figure><img src="image_8.png" /></figure>
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Complex conductivity of red montmorillonite muds at three NaCl concentrations (C<sub>b</sub> = 0.0171 M, C<sub>w</sub> = 0.154 M, and C<sub>b</sub> = 1.51 M). Measurements (filled circles) and modelling results (curves). **a:** raw real part, **c:** raw imaginary part, **e:** raw phase, **b, d, f:** same as **a, c, e** but after removing the coupling term (in **b**, the in-phase conductivity at the lowest frequency is also removed).</figcaption></figure>

After running the optimisation procedure, our model reproduces very well the experimental data (see Figs 4 to 7 and the cost-function values reported in Table 5). The model is able to reproduce the observed increase of in-phase conductivity σ' when salinity increases by considering increasing water conductivity σ<sub>w</sub> (Table 6). We also showed in Table 6 the computed bulk conductivity values considering the presence of not conducting minerals σ<sub>0</sub> = <|σ<sub>0</sub> *|> (Eq. 33). Measured data corrected from coupling effects show more reliable
behaviour at high frequency. The measured increase of $\sigma'$ when frequency increases, which can be clearly seen for frequencies above 0.1 Hz, was also reproduced in our model by simulating Stern layer polarisation. As shown earlier by Okay et al. (2014) and Leroy et al. (2017), the observed quadrature conductivity $\sigma''$ is less sensitive to salinity than the observed in-phase conductivity because of the structural charge influence on polarisation. Indeed, the permanent and negative structural charge adsorbs a large quantity of mobile cations in the Stern layer on the surface of clay particles, which controls electrochemical polarisation and then quadrature conductivity ($\sigma''$). Observed quadrature conductivity has a tendency to increase when salinity increases. We explain such observation by the increasing excess of charge per unit pore volume in the Stern layers within the aggregate contributing to polarisation, i.e., the product $f_Q Q_v$ (Tables 5 and 6). However, effective ion mobility of Na⁺ counter-ion in the Stern layer on clay surface, $\beta_s$, also influences electrochemical polarisation. In our model, the decrease of $\beta_s$ when salinity increases explains the observed decrease of the quadrature conductivity of illite at 0.1 M NaCl initially and of red montmorillonite at 1 M NaCl initially. The $\beta_s$ values are also considerably smaller, i.e. about two orders of magnitude smaller, than the ion mobility value of Na⁺ ion in bulk and distilled water, in agreement with Revil (2012). We think that these $\beta_s$ values are effective Na⁺ ion mobility values and do not correspond directly to the local Na⁺ ion mobility value in the Stern layer. The latter can be inferred from molecular dynamic simulations. According to e.g. Tournassat et al. (2009) and Bourg & Sposito (2011), its value is considerably larger than our $\beta_s$ values, and is about half Na⁺ ion mobility value in bulk and distilled water. Leroy et al. (2017) were able to reproduce complex conductivity measurements on dispersed montmorillonite suspensions (less than one hundred grams of solid clay per litre of dispersion) using the same model as we, considering $\beta_s \cong 0.42 \bar{\beta}_{Na^+}^b$ (298). As we were
dealing here with considerably more concentrated clay muds (around one thousand grams of solid clay per litre of mud, Table 1), our opinion is that the interaction between clay particles decreases Na⁺ ion effective mobility in the Stern layer. Our computations also reproduce minus phase-shift measurements that depend on the ratio of σ" to σ'.

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Aggregate effective size distributions inferred from the computation of the complex conductivity of clay muds at three initial NaCl concentrations (blue: 0.01 M NaCl, red: 0.1 M NaCl, green: 1 M NaCl initially; top: *M* = 1 according to the original model by Schwarz (1962); bottom: *M* computed by Lyklema *et al.* (1983) model).</figcaption></figure>

Fig. 8 shows the aggregate size distributions inferred from the comparison of the model predictions to the corresponding experimental data. Aggregate size distributions are in the µm range expected for clay aggregates (Tombacz & Szekeres 2004, 2006; Jiang *et al.* 2022) and are relatively similar between each type of clay except for illite that shows two distinguishable peaks. It looks that the presence of a high smectite content in the kaolinite sample can explain similarity in observable aggregate size distributions between kaolinite and montmorillonite samples. In the illite sample, the presence of an additional peak for large sizes can be explained by the presence of large grains like calcite grains coated by illite particles (calcite solid content of 12%, Mendieta *et al.* 2021). On the contrary to what has been observed in Leroy *et al.* 2017 for Na-Mt clay dispersions, aggregate size distributions do
not exhibit here large changes when salinity increases. In particular, we did not see here a peak corresponding to larger aggregates due to clay particle coagulation at the highest salinities. The clay muds investigated here are considerably more concentrated than the dispersed suspensions used in Leroy *et al.* 2017 (~1000 g versus ~10 g clay powder per sample litre). Due to the considerably higher clay content with interacting clay particles, the SIP data showed here are flatter in frequency with larger magnitudes and are less-changing with salinity especially in the low-frequency range (typically below 1 Hz) than the SIP data reported in Leroy *et al.* (2017). It would be interesting to validate or not our modelling results by aggregate size distribution measurements using high resolution characterisation techniques such as X-ray microtomography (Massat *et al.* 2016) and wide angle X-ray scattering (Chaaya *et al.* 2023).

### 3.2.2 Output model parameter values (optimised)

<table><caption>**Table 5.** Output model parameter values (optimised in bold) inferred from the computation of the complex conductivity of clay muds at different NaCl concentrations and corresponding cost function (y) values.</caption><thead><tr><th>Clay</th><th>C<sub>b</sub><br>(M)</th><th>f<sub>Q</sub></th><th>Q<sub>β</sub><sup>(1)</sup><br>(C m<sup>-2</sup>)</th><th>10<sup>2</sup>mr<sub>s</sub></th><th>10<sup>10</sup>β<sub>s</sub><sup>(2)</sup><br>(m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</th><th>n<sub>c</sub></th><th>m</th><th>10<sup>3</sup>y</th></tr></thead><tbody><tr><td rowspan="2">Kaolinite</td><td>1.540×10<sup>-2</sup></td><td><strong>0.5135</strong></td><td>0.1664</td><td><strong>7.394</strong></td><td>33.63</td><td>1</td><td><strong>1.500</strong></td><td>0.43</td></tr><tr><td>1.908×10<sup>-1</sup></td><td><strong>0.5347</strong></td><td>0.1732</td><td><strong>6.979</strong></td><td>26.12</td><td>1</td><td><strong>1.500</strong></td><td>0.22</td></tr><tr><td rowspan="2">Illite</td><td>1.804×10<sup>-2</sup></td><td><strong>0.8265</strong></td><td>0.1570</td><td><strong>1.268</strong></td><td>5.794</td><td>1</td><td><strong>1.697</strong></td><td>0.67</td></tr><tr><td>1.816×10<sup>-1</sup></td><td><strong>0.8267</strong></td><td>0.1571</td><td><strong>0.722</strong></td><td>2.475</td><td>1</td><td><strong>1.697</strong></td><td>0.50</td></tr><tr><td rowspan="2">Green Mt</td><td>1.528×10<sup>-2</sup></td><td><strong>0.7900</strong></td><td>0.1327</td><td><strong>3.166</strong></td><td>13.61</td><td><strong>2</strong></td><td><strong>1.500</strong></td><td>0.60</td></tr><tr><td>1.457×10<sup>-1</sup></td><td><strong>0.7213</strong></td><td>0.1212</td><td><strong>3.166</strong></td><td>10.59</td><td><strong>2</strong></td><td><strong>2.031</strong></td><td>0.43</td></tr><tr><td rowspan="2">Red Mt</td><td>1.708×10<sup>-2</sup></td><td><strong>0.7592</strong></td><td>0.1298</td><td><strong>3.796</strong></td><td>17.89</td><td><strong>2</strong></td><td><strong>1.500</strong></td><td>0.53</td></tr><tr><td>1.541×10<sup>-1</sup></td><td><strong>0.7509</strong></td><td>0.1284</td><td><strong>3.796</strong></td><td>13.41</td><td><strong>2</strong></td><td><strong>1.922</strong></td><td>1.47</td></tr><tr><td></td><td>1.509</td><td><strong>0.8388</strong></td><td>0.1434</td><td><strong>0.595</strong></td><td>2.720</td><td><strong>25</strong></td><td><strong>2.367</strong></td><td>4.78</td></tr></tbody></table>

<footnote>(1) Using $Q_0$ values (Table 4) and optimised $f_Q$ values and Eq. 43.</footnote>

<footnote>(2) Using optimised $mr_s$ values and Eq. 42.</footnote>
In our model, the fraction of the counter-charge in the Stern layer was defined as

$$f_Q = \frac{Q_\beta}{Q_\beta + Q_d} = -\frac{Q_\beta}{Q_0}, \qquad (43)$$

where $Q_\beta$ is the surface charge density of the Stern layer and $Q_d$ is the surface charge density of the diffuse layer (Leroy et al. 2024). We consider here a constant CEC and a mineral surface charge density $Q_0$ for each type of clay sample (kaolinite, illite and montmorillonite). Therefore, $f_Q$ changes induce associated $Q_\beta$ changes and an increasing $f_Q$ means an increasing surface charge density of the Stern layer. Because the permanent structural charge of the basal surfaces controls the EDL properties of the clays, the value of $Q_\beta$ is high and relatively constant, as we can see in Table 5.

The Stern layer makes up more than 50 % of the counter-charge on the external clay particle surface, at least for the basal surfaces of illite and montmorillonite containing a high structural charge (Tournassat et al. 2009; Bourg & Sposito 2011; Tournassat et al. 2015; Tinnacher et al. 2016). We found that about 51 % to 84 % of the surface charge density of the EDL polarizes ($f_Q$ values in Table 5), confirming that the Stern layer polarisation process may control the electrochemical polarisation of our clay mud samples as suggested earlier by Obrien & Rowlands (1993) for kaolinite, Rasmusson et al. (1997) for montmorillonite and later by Revil and co-workers (e.g., Leroy & Revil, 2009; Revil 2012; Okay et al. 2014) for clayey materials. The $f_Q$ values globally increase when salinity increases, especially for kaolinite and red montmorillonite at 1 M NaCl initially. Kaolinite presents also smaller $f_Q$ values than illite and green and red montmorillonite. The obtained $f_Q$ values are in accordance with the fact that kaolinite has a smaller structural charge than illite and
montmorillonite inducing potentially weaker ion adsorption in the Stern layer and higher sensitivity to bulk pore water chemical composition. Indeed, the presence of a strong structural charge on the basal surfaces of illite and montmorillonite particles induces a strong electrostatic force attracting and condensing counter-ions and water molecules in the Stern layer (Bourg & Sposito 2011) that may be weaker in the case of kaolinite particle carrying a weak structural charge on its silica basal surface (Leroy et al. 2024). As a consequence, Stern layer adsorption should be weaker and $f_Q$ smaller in pure kaolinite sample than in pure illite and montmorillonite samples. In addition, a significant part of the surface of a kaolinite particle (edge and alumina basal surfaces) is occupied by silica and alumina hydroxyl surface sites (>Si-OH, >Al-OH) exchanging proton with $Na^+$ ion from bulk water in the Stern layer (Tombacz & Szekeres 2006). In that situation, the adsorption of $Na^+$ in the Stern layer increases the magnitude of the negative mineral surface charge density (Leroy & Revil 2004). Therefore, at fixed bulk pore water pH, increasing bulk water NaCl concentration increases $H^+$ to $Na^+$ exchange in the Stern layer and the magnitude of $Q_0$, leading to higher $Na^+$ adsorption in the Stern layer and potentially higher $f_Q$ values for pure kaolinite. However, our kaolinite sample is not pure and contains significant amounts of smectite (4%) and illite (3%) presenting a high structural charge according to XRD measurements, therefore increasing and limiting the expected increase of $f_Q$ with salinity. We also found that illite has higher $f_Q$ values than montmorillonite when NaCl concentrations in bulk water are similar in accordance with the stronger surface charge density of illite ($Q_0$ = -0.190 C m⁻², Table 4) than montmorillonite ($Q_0$ = -0.168 C m⁻² for green Mt and $Q_0$ = -0.171 C m⁻² for red Mt, Table 4) inducing stronger $Na^+$ adsorption in the Stern layer. The evolution of $f_Q$ values with salinity for montmorillonite is in agreement with the trend reported by Leroy *et al.* (2015). There is a slight decrease of $f_Q$ at intermediate salinity that was also reported by Leroy *et al.* (2015),
due to the use of Na⁺ activity coefficient of value below 1 in their Stern layer surface complexation model. At the highest salinity about 1 M NaCl, we, like Leroy *et al.* (2015), also observed the highest *f*<sub>Q</sub> value. Nevertheless, we observed higher *f*<sub>Q</sub> values than in Leroy *et al.* (2015) (they reported *f*<sub>Q</sub> between 0.625 and 0.7 or 0.75 considering or not Na⁺ activity coefficient). This can be explained by the higher amount of clay in our samples that favour higher Stern layer adsorption due to clay charged surfaces that are closer each to other.

Surface charge slows down counter-ions and water molecules on the clay surface (Greathouse *et al.* 2016). Kaolinite has a higher ion mobility ratio (*mr*<sub>s</sub>) and effective ion mobility in the Stern layer (*β*<sub>s</sub>) than montmorillonite, and montmorillonite has a higher *β*<sub>s</sub> than illite, at similar salinity (Table 5). This can be explained by the fact that kaolinite has in theory a weaker structural charge than montmorillonite (Grim 1962), and montmorillonite has a weaker structural charge than illite (Ghasemi & Sharifi 2021). Nevertheless, our kaolinite sample is not pure and we did not have such weak surface charge density for it in our model (Q<sub>0</sub> = -0.324 C m<sup>-2</sup>, Table 4). For illite, the *mr*<sub>s</sub> value significantly decreases when salinity increases from 1.804×10<sup>-2</sup> to 1.816×10<sup>-1</sup> M NaCl. For red montmorillonite, we observe the same behaviour when salinity increases from 1.541×10<sup>-1</sup> to 1.509 M NaCl (Table 5). We attribute this decrease of effective ion mobility in the Stern layer by the effect of increasing salinity and clay particle coagulation (Tombacz & Szekeres 2006). Clay particle coagulation should arrive earlier for illite than for montmorillonite because of the smaller surface area of illite where the structural charge inducing an EDL is expressed (Tournassat *et al.* 2015). Indeed, two similar clay particles with the same charged surface are repelled by each other due to the electrostatic diffuse layer force (Tombacz & Szekeres 2004). This electrostatic force weakens when salinity increases because of the compression of the diffuse layer leading to clay coagulation (Bouhaik *et al.* 2013). The larger the surface area is, the larger the
associated diffuse layer repulsive force is expressed between the two charged surfaces, and
the larger is the salinity necessary to suppress it.

The number of stacked TOT sheets per particle, $n_c$, is 2 for the two montmorillonites at the two lower salinities, in agreement with the experimental observations of Schramm & Kwak (1982) on Na-Mt suspensions. At the highest salinity, we found $n_c = 25$ in agreement with the experimental observations of Michot *et al.* (2013) where they reported that about 30 TOT sheets are stacked per particle when Na-Mt particles strongly coagulate in 1 M NaNO₃ electrolyte. In our calculations, we used $n_c = 1$ for kaolinite and illite (Table 5), but, in reality, the number of sheets per particle according to our model is around 200 for kaolinite particles in kaolinite sample and around 6 for illite particles in illite sample (see supplementary material). These values are in the range of $n_c$ expected values for kaolinite ($10 \le n_c \le 200$) and illite ($5 \le n_c \le 20$) particles (Leroy *et al.* 2024).

The cementation exponent *m*, which depends on the shape of the aggregates (Table S1 in supplementary material), globally increases when salinity increases. Compression of the diffuse layer and increasing particle coagulation when salinity increases can explain such results. The optimised *m* value is also higher for illite and montmorillonite than for kaolinite, for which our model predicts a spherical shape for the aggregates (*m* = 1.5; Table 5). Illite and montmorillonite have more elongated particles than kaolinite (Tournassat *et al.* 2015), which can explain such *m* values. These results showing in average increasing *m* values in the ascending order of kaolinite, illite and montmorillonite are in accordance with what was found previously by Leroy & Revil (2009) and Okay *et al.* (2014). However, these authors found higher *m* values with *m* = 2 for kaolinite, *m* ∈ {2,3} for illite (only reported in Leroy & Revil (2009)) and *m* ∈ {3,4} for smectite. The difference we obtained between Leroy & Revil (2009), Okay *et al.* (2014) and our model concerning the *m* values can be explained by the
fact that we consider here a polarising porous clay aggregate with an excess of charge per unit pore volume ($Q_v$) in agreement with Vinegar & Waxman (1984) and more recently with Leroy et al. (2017) – on the contrary to Leroy & Revil (2009) and Okay et al. (2014) who considered a polarising but not porous aggregate.

### 3.2.3 Output model parameter values (deduced from the model)

At low salinity (0.01 M NaCl initially), our model predicts that the diffuse layer occupies most of the connected pore space of illite and montmorillonite and controls diffuse layer conductivity $\sigma_w^d$ and water conductivity $\sigma_w$ (see $\Theta_w^d$ values in Table 6). At higher salinity (0.1 M NaCl initially), it predicts diffuse layer compression and that more connected pore space is occupied by bulk water, which controls bulk water conductivity $\sigma_w^b$ and then $\sigma_w$ (Tournassat & Appelo 2011). Even at 0.1 M NaCl, our model predicts that montmorillonite samples contain a significant pore water volume occupied by the diffuse layer. It should be noted that we consider that the montmorillonite interlayer space is filled by diffuse layer. Despite simulating diffuse layer compression, computed pore water volume fraction of the diffuse layer remains significant even at the highest salinity (1 M NaCl initially). Indeed the calculated interlayer space volume fraction over diffuse layer and interlayer space volume increases when initial NaCl concentration increases from 0.1 to 1 M NaCl ($\Xi_i$, Eq. 5, Table 6).
<table><caption>**Table 6.** Output model parameter values (not optimised) inferred from the computation of the complex conductivity of clay muds.</caption><thead><tr><th>Clay</th><th>C<sub>b</sub><br>M</th><th>σ<sub>0</sub><br>S m<sup>-1</sup></th><th>σ<sub>w</sub><br>S m<sup>-1</sup></th><th>σ<sub>w</sub><sup>b</sup><br>S m<sup>-1</sup></th><th>σ<sub>w</sub><sup>d</sup><br>S m<sup>-1</sup></th><th>φ<sub>d</sub><br>mV</th><th>Θ<sub>w</sub><sup>d</sup></th><th>Ξ<sub>i</sub></th><th>10<sup>8</sup> B<sup>d</sup><sub>Na+</sub><br>m<sup>2</sup> s<sup>-1</sup></th><th>φ<sub>a</sub></th><th>10<sup>-7</sup> Q<sub>V</sub><br>C m<sup>-3</sup></th></tr></thead><tbody><tr><td>Kaolinite</td><td>0.015<br>0.191</td><td>0.297<br>1.632</td><td>0.326<br>1.790</td><td>0.171<br>1.595</td><td>4.875<br>7.046</td><td>-155.6<br>-89.73</td><td>0.252<br>0.072</td><td>-</td><td>11.01<br>6.686</td><td>0.238<br>0.101</td><td>6.60<br>23.17</td></tr><tr><td>Illite</td><td>0.018<br>0.182</td><td>0.288<br>1.073</td><td>0.400<br>1.559</td><td>0.200<br>1.499</td><td>0.567<br>1.759</td><td>-75.31<br>-30.71</td><td>0.658<br>0.242</td><td>-</td><td>7.664<br>4.405</td><td>0.555<br>0.283</td><td>4.188<br>13.24</td></tr><tr><td>Gr. Mt</td><td>0.015<br>0.146</td><td>0.555<br>1.625</td><td>0.591<br>1.726</td><td>0.164<br>1.197</td><td>0.591<br>1.883</td><td>-84.26<br>-47.53</td><td>1.000<br>0.800</td><td>0.198<br>0.238</td><td>7.814<br>5.011</td><td>0.723<br>0.684</td><td>13.24<br>15.95</td></tr><tr><td>Red Mt</td><td>0.017<br>0.154<br>1.509</td><td>0.602<br>1.352<br>8.216</td><td>0.796<br>1.738<br>10.55</td><td>0.194<br>1.298<br>10.59</td><td>0.796<br>1.879<br>10.47</td><td>-90.06<br>-43.50<br>-13.34</td><td>1.000<br>0.783<br>0.378</td><td>0.214<br>0.243<br>0.960</td><td>8.425<br>4.972<br>3.048</td><td>0.706<br>0.679<br>0.507</td><td>14.70<br>16.67<br>34.29</td></tr></tbody></table>

Computed Na⁺ ion mobility in the diffuse layer, $B_{Na⁺}^d$, decreases when salinity increases. This results from (i) the mutual interaction of oppositely-charged ions that diminishes the mobility of Na⁺ and Cl⁻ ($\beta_{Na⁺}^b$ and $\beta_{Cl⁻}^b$ respectively) (see Table S2 and Leroy et al. 2015) and (ii) the compression of the diffuse layer that causes a smaller increase of the diffuse layer conductivity due to electroosmosis (Dufreche et al. 2001 and Eqs 28-32). Magnitude of simulated electrical potential at the beginning of the diffuse layer, $\phi_d$, decreases when salinity increases due to diffuse layer compression. The values we found for $\phi_d$ in the case of montmorillonite are in agreement with the values reported by Leroy et al. (2015) for the montmorillonite basal surface (Table 6). Illite has smaller calculated $\phi_d$ values than montmorillonite, in agreement with Horikawa et al. (1988) and Leroy & Revil (2009), due to stronger ion adsorption in the Stern layer (higher $f_Q$ value, Table 5). The $\phi_d$ potentials we obtained for kaolinite may be too large compared to the $\phi_d$ potentials for illite and montmorillonite (Leroy & Revil 2004; Tombacz & Szekeres 2006). We attribute this result to the heterogeneity of our kaolinite sample, which contains significant solid contents of
smectite (4%) and illite (3%). In our model, the surprisingly large mineral surface charge density ($Q_0 = -0.324 \text{ C m}^{-2}$) and low fractions of the counter-charge in the Stern layer ($f_Q = 0.52$ at 0.01 M NaCl initially and $f_Q = 0.53$ at 0.1 M NaCl initially) induce large $\phi_d$ potentials. Finally, we observe an increase of the excess of charge per unit pore volume in the aggregate ($Q_v$) when salinity increases, due to the decrease of the aggregate porosity $\phi_a$ (Table 6 and Eq. 15).

### 3.2.4 Comparison to the model of Weller et al. (2013)

The electrical formation factor $F$ ($F \ge 1$) is an important petrophysical parameter in geophysics (e.g., Revil & Leroy 2004). It is defined by the ratio of water conductivity to saturated sample conductivity in absence of surface conductivity (Archie 1942), i.e.

$$F = \lim_{\sigma_s \to 0} \frac{\sigma_w}{\sigma'} \qquad (44)$$

Hence, $F=1$ means that the current lines in water are not modified by the solid matrix and a high formation factor value like $F=10$ means that the current lines in water are deformed by the solid matrix, thus decreasing sample capability to let current passing through the sample, and therefore its electrical conductivity. The electrical formation factor provides therefore paramount information on pore structure and investigated porous medium transport properties like ion diffusive properties and permeability (e.g., Revil & Cathles 1999; Leroy et al. 2006; Jougnot et al. 2009; Revil & Florsch 2010; Al Khalifah et al. 2020; Peshtani et al., 2024). In addition, Archie (1942) in his first law showed that the electrical formation factor can be computed from sample porosity $\phi$ and cementation exponent $m$ when surface conductivity can be neglected, i.e.
$$F = \phi^{-m}. \qquad (45)$$

It is usually assumed that it is very difficult to determine the formation factor of clays such as montmorillonite because of the high conductivity on their surface compared to bulk water conductivity (Revil et al. 1998; Leroy & Revil 2009; Weller et al. 2013). For argillites and mudstones, it is also difficult to determine *F* given how low the permeability is and how difficult it is to saturate such tight pore spaces (Neuzil, 1994; Jougnot et al. 2010a; Peshtani et al. 2024). To reduce the effect of surface conductivity, it is common to increase the water conductivity (de Lima & Sharma 1992; Weller et al. 2013; Li et al. 2016). But in the case of clay muds, this changes their structure because of diffuse layer compression (Leroy et al. 2017; Okay et al. 2014). In addition, in the case of nanoporous clays such as montmorillonite, it is very difficult to estimate accurately water conductivity, because of the presence of the diffuse layer and interlayer space filling a significant part of the pores (Leroy & Revil 2009).

To quantify the surface conductivity contribution, we used the Dukhin number, defined by $Du = |\sigma_s^*/\sigma_w^*|$. In our samples, it is less than 0.1 (Fig. 9), meaning that in our model the surface conductivity contribution (due to the Stern layer) to the measured magnitude of sample conductivity can be neglected. Therefore Archie's first law should apply here.
<figure><img src="image_11.png" /><figcaption>**Figure 9.** Dukhin number as a function of frequency and salinity. Blue: 0.01 M NaCl, red: 0.1 M NaCl, and green: 1 M NaCl initially.</figcaption></figure>

Our model allows us to estimate the electrical formation factor of our clay muds in the two ways presented previously (Eqs 44 and 45). For that purpose we chose in-phase conductivity measurements on clay muds ($\sigma'$) at low frequency (here 1.46 Hz) where surface conductivity effect on in-phase conductivity measurements can be neglected. It should be noted again that in our model the diffuse layer only contributes to water conductivity and not to surface conductivity according to O'Konski (1960) because the diffuse layer can be thicker than the particle (see related discussion in section 3 of Leroy *et al.* 2024). Water conductivity ($\sigma_w$) and mud porosity ($\phi$) were computed using the method outlined in Section 2. The cementation exponent $m$ was obtained by optimisation (Section 2.1).
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Measured in-phase conductivity of the samples at a frequency of 1.46 Hz as a function of the computed water conductivity (a) and measured in-phase conductivity of the samples at a frequency of 1.46 Hz as a function of the computed bulk water conductivity (b).</figcaption></figure>

Fig. 10a shows that first Archie's law (represented here by Eq 44) can indeed be used to interpret in-phase conductivity measurements on clay muds at low frequency (typically ≤ 1 Hz) when clay water conductivity is computed considering the diffuse layer contribution. However, first Archie's law cannot be used to interpret in-phase conductivity measurements on clay muds at low frequency when only bulk water conductivity is known because $\sigma_w^b$ underestimates clay water conductivity especially at low salinity (0.01 M NaCl initially) (Fig10b and 11). It is therefore necessary to consider surface conductivity in parallel to bulk conductivity to explain in-phase conductivity measurements at low frequency when using bulk water conductivity only. For instance, Weller *et al.* (2013) considered $\sigma' = \sigma_w^b / F + \sigma_s'$ to explain in-phase conductivity measurements at low frequency on clayey materials.
<figure><img src="image_13.png" /><figcaption>**Figure 11.** Computed water conductivity versus computed bulk water conductivity.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Figure 12.** Electrical formation factor deduced from the model ($F = \phi^{-m}$) compared to the electrical formation factor deduced from the model and the measurement ($F = \sigma_w / \sigma'(1.46 \text{ Hz})$) (a). Calculated porosity versus measured porosity (b).</figcaption></figure>

<table><caption>**Table 7.** Computed clay mud porosity ($\phi$) and measured porosity of the sample ($\phi_{meas}$) (from Mendieta et al. [2021]).</caption><thead><tr><th>Clays</th><th>Kaol.</th><th>Kaol.</th><th>Illite</th><th>Illite</th><th>G. Mt</th><th>G. Mt</th><th>R. Mt</th><th>R. Mt</th><th>R. Mt</th></tr></thead><tbody><tr><td>C<sub>b</sub> (M)</td><td>0.0154</td><td>0.1908</td><td>0.0180</td><td>0.1816</td><td>0.0153</td><td>0.1457</td><td>0.0171</td><td>0.1541</td><td>1.5091</td></tr><tr><td>φ</td><td>0.584</td><td>0.583</td><td>0.560</td><td>0.522</td><td>0.701</td><td>0.709</td><td>0.613</td><td>0.641</td><td>0.643</td></tr><tr><td>φ<sub>meas</sub></td><td>0.57</td><td>0.56</td><td>0.54</td><td>0.56</td><td>0.68</td><td>0.71</td><td>0.61</td><td>0.62</td><td>0.51</td></tr></tbody></table>
The predictions of the electrical formation factor using $F = \phi^{-m}$ are in relative agreement with the experimental measurements using $F = \sigma_w / \sigma'(@1.46 Hz)$ (Fig. 11a). Most F discrepancies are observed for illite at 0.1 M NaCl initially and red montmorillonite at 1 M NaCl initially where particle coagulation occurs. It should be noted that our model using $F = \phi^{-m}$ reproduces very well the corresponding “observed” F measurements when using $m = 1.5$ in our computations (spherical aggregates) and that our F prediction discrepancy from the “observed” one increases when $m$ increases (Table 5). Computed clay mud porosities are also in agreement with the related measurements reported in Mendieta et al. (2021) except for the clay mud porosity of red montmorillonite at 1 M NaCl initially, which is surprisingly low ($\phi \approx 0.51$) compared to the other clay mud porosity measurements ($\phi \approx 0.61$ in average) and the model prediction ($\phi \approx 0.64$) (Figs 12b and Table 7). We can attribute this observed porosity difference by the fact that we considered that the interlayer space of red Mt contributes to sample porosity whereas this type of porosity was not measured during the experiment of Mendieta et al. (2021). It is not surprising that the clay mud porosities computed by our model are in agreement with the corresponding measurements because we know the mass of clay powder in our model ($\delta_s$), the solid volumetric mass density ($\rho_s$) ($\rho_s \approx 2700 \text{ kg m}^{-3}$), and the sample volume ($V_t$). Indeed:

$$ \phi = \frac{V_w}{V_t} = \frac{V_t - 10^6 V_s}{V_t} = \frac{V_t - 10^3 \delta_s / \rho_s}{V_t}. \qquad (46) $$
<figure><img src="image_15.png" /><figcaption>**Figure 13.** Real part (a-d) and imaginary part (e-h) of the surface conductivity of the clay samples at different salinities as a function of frequency. Ratio of the imaginary part to the real part of the surface conductivity of the clay samples at different salinities as a function of frequency (i-l). Blue: 0.01 M NaCl, red: 0.1 M NaCl, and green: 1 M NaCl initially.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>**Figure 14.** Measured sample quadrature conductivity as a function of the imaginary part of surface conductivity at different salinities (blue: 0.01 M NaCl, red: 0.1 M NaCl, and green: 1 M NaCl initially).</figcaption></figure>

Our model can also be used to determine the surface conductivity of clay aggregates $\sigma_s^* = \sigma_s' + i\sigma_s''$ (Eq 22). Fig. 13 shows modelled $\sigma_s'$ and $\sigma_s''$, with $\sigma_s'$ being stronger in magnitude than $\sigma_s''$ for frequency above 1 Hz. Weller *et al.* (2013) considered that the EDL on clay particle surface controls surface conductivity. They introduced the parameter $l = \sigma_s''/\sigma_s'$. Our modelled $l$ ratio strongly decreases when the frequency increases in particular at low frequencies due to the high increase of $\sigma_s'$ when frequency increases (Fig. 13). It is also sensitive to salinity, especially for frequencies below 1 Hz. Our modelling results also show that computed surface quadrature conductivity $\sigma_s''$ is not equal to the
measured sample quadrature conductivity $\sigma''$ (Fig. 14). Indeed, we found that $\sigma'' = a\sigma_s''$
with $a$ being a real constant, which is not necessarily equal to 1, contrary to what was
assumed by Weller *et al.* (2013). Furthermore, the slope $a$ increases when salinity increases,
especially for montmorillonite. Such results can be explained by the self-consistent model
used here (BHS model) to compute mud conductivity from surface and water conductivities
in the case of very low Dukhin number (Fig. 9) and large contribution of the diffuse layer and
interlayer space of Na-Mt to water conductivity (Table 6).

# 4 CONCLUSIONS AND PERSPECTIVES

We have developed a new methodology to explain laboratory spectral induced polarisation (SIP) measurements on kaolinite, illite, and montmorillonite concentrated muds at different NaCl concentrations, using a mechanistic model taking into account microstructural and electrochemical properties. The model considers explicitly the influence of the mobile ions in the electrical double layer (EDL) covering clay particle external surfaces and in the interlayer space of montmorillonite on complex conductivity. We assume that the diffuse layer contributes to water conductivity and not to surface conductivity because the diffuse layer can be thicker than the clay particle. In our model, the counter-ions in the Stern layers around interacting clay particles, which form aggregates, and the mobile counter-ions in the interlayer space of montmorillonite particles control the electrochemical polarisation and the surface conductivity. One limitation of our model is that it does not take into account full diffuse layer polarisation as well as the polarisation due diffuse layer interaction, which can lead to a membrane-like polarisation effect – a further work in that direction would be therefore necessary to complete our approach.

Distribution of relaxation times is first inverted from the measured SIP spectra using the
Debye decomposition approach. It serves to get the partial chargeability distribution and then the effective size distribution of the clay aggregates. Afterwards an optimization procedure is used to adjust the parameters of the model by comparing model predictions to measured SIP spectra for each type of clay at different salinities (NaCl). These parameters are the fraction of the counter-charge and the effective ion mobility (tangentially to the clay surface) of Na⁺ counter-ions in the Stern layer, the number of stacked sheets per montmorillonite particle, and the cementation exponent of the aggregates characterising their shape.

Our model is able to reproduce rather well the measured SIP spectra of concentrated clay muds using optimised parameter values, in agreement with the corresponding data reported in the physical-chemistry literature, except for two parameters: the effective size distribution of the clay aggregates and the effective ion mobility in the Stern layer. As mentioned, these parameters are effective ones. Due to the large concentration of interacting clay particles in the investigated samples (typically around 1 thousand grams of solid clay powder per litre of sample) compared to clay dispersions (particle concentration typically below 1 hundred grams of solid clay powder per litre of sample), it was not possible to estimate accurately the size distribution of the aggregates from measured SIP spectra and to use directly the ion mobility values deduced from molecular dynamic simulations, which are significantly too high.

Our approach suggests that most of the EDL counter-charge controlling SIP spectra on clayey materials is located in the Stern layer, especially on the basal surface of illite and montmorillonite. We also obtained a very low value for the effective ion mobility of Na⁺ counter-ions in the Stern layer, between one and two orders of magnitude below the corresponding ion mobility in bulk and distilled water (but in agreement with values reported in the geophysical literature). The numbers of stacked montmorillonite sheets per particle are in agreement with the values reported in the physical-chemistry literature including when particle coagulation occurs. In our model, clay aggregates have spherical and spheroidal
shapes and their elongation increases when salinity increases for montmorillonite.

We showed that first Archie's law can be used to explain in-phase conductivity measurements at low frequency (typically ≤ 1 Hz) on clayey materials when water conductivity can be modelled to include the contribution of the diffuse layer. We also extracted surface conductivity from measured SIP spectra and reported their dependence on salinity and frequency for each type of clay. We found that the ratio of the imaginary part of surface conductivity to its real part is highly dependent on frequency and can also depend on salinity at frequencies below 1 Hz. Finally, we observed from our model that the measured quadrature conductivity is proportional, but not identical, to the imaginary part of surface conductivity.

It would be interesting to use our approach to infer the state and transport properties of unconsolidated and consolidated clay materials from SIP spectra. Among them, clay content, bulk water salinity, ion diffusivity and permeability would be of particular importance. In addition, our results should be confirmed using purely numerical SIP models developed at the scale of the clay particle combined with high resolution imaging techniques.

## ACKNOWLEDGMENTS

This study was carried out within the framework of the ANR EXCITING and IMAGE projects (grants ANR-17-CE06-0012 and ANR-21-CE04-0013, respectively, funded by the French National Research Agency). Philippe Leroy warmly thanks Francis Claret for his support and Sylvain Grangeon for the fruitful discussions. The work of Philippe Leroy was also supported by the EURAD work package DONUT (EC grant agreement No. 847593). We thank the editor, Léa Levy, Klaudio Peshtani and an anonymous referee for reviewing our manuscript.
DATA AVAILABILITY STATEMENT

The data used in this study are available in the Zenodo repository
(http://doi.org/10.5281/zenodo.4050345).

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APPENDIX A. COMPUTATION OF THE DEBYE DECOMPOSITION

Following the Debye decomposition, the complex conductivity σ* can be written as:

$$ \sigma^*(\omega_k) = \sigma_\infty + (\sigma_0 - \sigma_\infty) \sum_{j=1}^{N_\tau} \frac{m_j}{1 + i\omega_k \tau_j} + i\omega_k C, \quad (A1) $$

where $\sigma_\infty$ is the limit value of $|\sigma^*|$ at “infinite” frequency, $\sigma_0$ is the limit value of $|\sigma^*|$ at zero frequency, $\omega_k = 2\pi f_k$ is the kth pulsation or angular frequency with $f_k$, $k=1...N_f$, being the frequencies at which the measurements were done, $\tau_j$, $j=1...N_\tau$, are the relaxation times, and $m_j$, $j=1...N_\tau$, is the partial chargeability distribution. The last term in Eq. A1 was introduced to take into account the high-frequency trend of the spectrum (see Florsch et al. 2014), due to Maxwell-Wagner polarisation, and/or contact impedance of the potential electrodes, and/or electromagnetic coupling effects between acquisition cables.

We chose the $\tau_j$, $j=1...N_\tau$, values in such a way that $N_\tau$ is less than $N_f/2$ and that the number of points per decade is an entire number. The $\tau_j$ values are logarithmically distributed between $\tau_{min}$ and $\tau_{max}$, with

$$ \left\{ \begin{array}{l} \tau_{\min} = 10^{\left\lfloor \log_{10} \left( \frac{1}{2\pi f_{N_f}} \right) \right\rfloor} \\ \\ \tau_{\max} = 10^{\left\lfloor \log_{10} \left( \frac{1}{2\pi f_j} \right) \right\rfloor} \end{array} \right. \quad (A2) $$

Using a simulated annealing procedure (see Maineult 2016) we searched for $\sigma_\infty$, $m_j$ (for $j=1...N_\tau$) and C. This approach supposes that $\sigma_0$ is known, since the term $(\sigma_0 - \sigma_\infty)$ cannot be separated from the m terms in practice. To determine the value of $\sigma_0$, we first fitted the measured spectra with a double Cole-Cole model:
$$ \sigma^*(\omega) = \sigma_0 \left[ 1 + \frac{m_{CC,1}}{1-m_{CC,1}} \left( 1 - \frac{1}{1+(i\omega\tau_{CC,1})^{c_{CC,1}}} \right) + \frac{m_{CC,2}}{1-m_{CC,2}} \left( 1 - \frac{1}{1+(i\omega\tau_{CC,2})^{c_{CC,2}}} \right) \right] + i\omega C_{CC} \quad (A3) $$

using the simplex algorithm to search for $\sigma_0$, $m_{CC,1}$, $m_{CC,2}$ (the chargeabilities associated with the first and second peak, respectively), $\tau_{CC,1}$, $\tau_{CC,2}$ (the associated relaxation times), $c_{CC,1}$, $c_{CC,2}$ (the associated Cole-Cole exponents) and $C_{CC}$. We then injected the value of $\sigma_0$ into the simulated annealing procedure.

## APPENDIX B. CORRECTION OF THE CONTACT IMPEDANCE AND ELECTROMAGNETIC COUPLING EFFECTS ON THE MEASURED SIP DATA

Tables B1 shows the values of the coupling coefficient $C_c$ computed to remove the contact impedance and electromagnetic coupling effects from the complex conductivity measurements.

<table><caption>**Table B1.** Contact impedance and electromagnetic coupling effects correction of the SIP measurements</caption><thead><tr><th>Clays</th><th>Kaol.</th><th>Kaol.</th><th>Illite</th><th>Illite</th><th>G. Mt</th><th>G. Mt</th><th>R. Mt</th><th>R. Mt</th><th>R. Mt</th></tr></thead><tbody><tr><td>C<sub>b</sub> (M)</td><td>0.0154</td><td>0.1908</td><td>0.0180</td><td>0.1816</td><td>0.0153</td><td>0.1457</td><td>0.0171</td><td>0.1541</td><td>1.5091</td></tr><tr><td>log C<sub>c</sub></td><td>-7.874</td><td>-7.372</td><td>-7.863</td><td>-7.642</td><td>-7.322</td><td>-6.949</td><td>-7.141</td><td>-7.000</td><td>-8.072</td></tr></tbody></table>