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

Philippe Leroy, Alexis Maineult, Aida Mendieta, Damien Jougnot

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Philippe Leroy, Alexis Maineult, Aida Mendieta, Damien Jougnot. A mechanistic model for the complex conductivity of clay materials. I. Theory. Geophysical Journal International, 2024, 10.1093/gji/ggae411/7905067. hal-04813665

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https://brgm.hal.science/hal-04813665v1

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A mechanistic model for the complex conductivity of clay materials.

I. Theory

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Philippe Leroy¹,*, Alexis Maineult², Aida Mendieta³, Damien Jougnot⁴

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¹ BRGM, French Geological Survey, 45060 Orléans, France.

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² Laboratoire de Géologie, Ecole Normale Supérieure / CNRS UMR 8538, PSL Research

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University, 75005 Paris, France.

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

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

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*Corresponding author, p.leroy@brgm.fr

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Intended for publication in Geophysical Journal International
27 Clay materials are ubiquitous in the Earth's continental and oceanic crusts. They are
28 characterized by a large specific surface area, and, in contact with water, they have
29 remarkable adsorption, catalytic and containment properties. Clay materials also exhibit a
30 high electrical conductivity response associated with their large surface. However,
31 sedimentary clay minerals have a complex microstructure and electrochemistry that are not
32 fully understood and make the quantitative petrophysical interpretation of geoelectrical
33 measurements difficult. In this study, we have developed a new mechanistic model to better
34 understand and predict the complex conductivity of clay materials from their microstructure
35 and electrical double layer properties. For the conductive component, our model considers ion
36 electromigration in bulk water, clay electrical diffuse layer, and interlayer space, if any.
37 Concerning polarisation, it takes into account ion back-diffusion in the Stern layer and in the
38 interlayer space, if any, and also Maxwell-Wagner effect. Water and clay surface
39 conductivities, formation factor, cementation exponent and sample electrically connected
40 porosity can be extracted from the comparison of the model to experimental data. This study
41 is a step forward to better understand and quantify the complex conductivity of clays observed
42 during electrical and electromagnetic measurements, from laboratory to geological formation
43 scales.

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45 **Keywords:** Electrical properties, Induced polarisation, Microstructure.

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<table><thead><tr><th>Symbol</th><th>Meaning</th><th>Unit</th></tr></thead><tbody><tr><td>a<sub>i</sub><sup>b</sup></td><td>ion i activity in bulk water</td><td>dimensionless</td></tr><tr><td>α</td><td>empirical value to compute solid permittivity</td><td>m<sup>3</sup> kg<sup>-1</sup></td></tr><tr><td>β<sub>i</sub><sup>b</sup></td><td>ion mobility in bulk water due to electro-migration</td><td>m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup></td></tr><tr><td>β<sub>eo</sub><sup>d</sup></td><td>mean ion electro-osmotic mobility in the diffuse layer</td><td>m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup></td></tr><tr><td>β<sub>s</sub></td><td>mobility of the counter-ions in the Stern layer</td><td>m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup></td></tr><tr><td>β<sub>i</sub><sup>b</sup></td><td>ion mobility in bulk and distilled water due to electro-migration</td><td>m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup></td></tr><tr><td>B<sub>i</sub><sup>d</sup></td><td>ion i mobility in the diffuse layer with electro-osmosis</td><td>m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup></td></tr><tr><td>B<sub>d</sub></td><td>mean ion mobility in the diffuse layer with electro-osmosis (thin diffuse layer assumption)</td><td>m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup></td></tr><tr><td>~B<sub>d</sub></td><td>mean ion mobility in the diffuse layer with electro-osmosis around the aggregate (thin diffuse layer assumption)</td><td>m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup></td></tr><tr><td>c</td><td>volume fraction of conductive 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>C<sub>d</sub></td><td>differential capacitance of the diffuse layer</td><td>F m<sup>-2</sup></td></tr><tr><td>C<sub>i</sub><sup>b</sup></td><td>ion i molar concentration</td><td>mol L<sup>-1</sup> (M)</td></tr><tr><td>C<sub>salt</sub><sup>b</sup></td><td>salt molar concentration</td><td>mol L<sup>-1</sup> (M)</td></tr><tr><td>C<sub>NaCl</sub><sup>0</sup></td><td>initial NaCl concentration in bulk water</td><td>mol L<sup>-1</sup> (M)</td></tr><tr><td>C<sub>salt</sub><sup>0</sup></td><td>initial salt concentration in bulk water</td><td>mol L<sup>-1</sup> (M)</td></tr><tr><td>C<sub>i</sub><sup>d</sup></td><td>average ion concentration in the diffuse layer</td><td>mol L<sup>-1</sup> (M)</td></tr><tr><td>C<sub>i</sub><sup>b</sup></td><td>cation concentration from dissolved minerals and montmorillonite interlayer space</td><td>mol L<sup>-1</sup> (M)</td></tr><tr><td>C<sub>zj</sub><sup>b</sup></td><td>anion concentration from dissolved minerals and montmorillonite interlayer space</td><td>mol L<sup>-1</sup> (M)</td></tr><tr><td>χ<sub>d</sub></td><td>diffuse layer thickness</td><td>m</td></tr><tr><td>χ<sub>D</sub></td><td>Debye length</td><td>m</td></tr><tr><td>d</td><td>diameter of the spherical particle</td><td>m</td></tr><tr><td>d<sub>k</sub></td><td>effective diameter of the k<sup>th</sup> aggregate</td><td>m</td></tr><tr><td>d<sub>s</sub></td><td>Stern layer thickness</td><td>m</td></tr><tr><td>δ<sub>s</sub></td><td>clay powder mass</td><td>g</td></tr><tr><td>δ<sub>sc</sub></td><td>main clay solid mass</td><td>g</td></tr><tr><td>δ<sub>t</sub><sup>0</sup></td><td>sample mass before evaporation</td><td>g</td></tr><tr><td>δ<sub>t</sub><sup>1</sup></td><td>sample mass after evaporation</td><td>g</td></tr><tr><td>D<sub>i</sub><sup>b</sup></td><td>ion i self-diffusion coefficient</td><td>m<sup>2</sup> s<sup>-1</sup></td></tr><tr><td>D<sub>i</sub><sup>b</sup></td><td>ion i self-diffusion coefficient in distilled (bulk) water</td><td>m<sup>2</sup> s<sup>-1</sup></td></tr><tr><td>e</td><td>elementary (or electron) charge</td><td>C</td></tr><tr><td>ε<sub>c</sub></td><td>aggregate eccentricity</td><td>dimensionless</td></tr><tr><td>ε</td><td>medium permittivity</td><td>F m<sup>-1</sup></td></tr><tr><td>ε<sub>0</sub></td><td>vacuum permittivity</td><td>F m<sup>-1</sup></td></tr><tr><td>ε<sub>in</sub></td><td>interlayer space permittivity</td><td>F m<sup>-1</sup></td></tr></tbody></table>
<table>
  <tr>
    <td>ε<sub>s</sub></td>
    <td>solid permittivity</td>
    <td>F m<sup>-1</sup></td>
  </tr>
  <tr>
    <td>ε<sub>w</sub></td>
    <td>liquid water permittivity</td>
    <td>F m<sup>-1</sup></td>
  </tr>
  <tr>
    <td>ε<sub>w</sub></td>
    <td>average permittivity of the diffuse layer and interlayer space</td>
    <td>F m<sup>-1</sup></td>
  </tr>
  <tr>
    <td>E</td>
    <td>applied electrical field</td>
    <td>V m<sup>-1</sup></td>
  </tr>
  <tr>
    <td>E<sub>0</sub></td>
    <td>initial applied electrical field</td>
    <td>V m<sup>-1</sup></td>
  </tr>
  <tr>
    <td>f(d<sub>k</sub>)</td>
    <td>aggregate size distribution of the k<sup>th</sup> aggregate</td>
    <td>dimensionless</td>
  </tr>
  <tr>
    <td>f<sub>Q</sub></td>
    <td>fraction of the counter-charge in the Stern layer</td>
    <td>dimensionless</td>
  </tr>
  <tr>
    <td>k<sub>B</sub></td>
    <td>Boltzmann's constant</td>
    <td>J K<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>φ</td>
    <td>mud porosity</td>
    <td>dimensionless</td>
  </tr>
  <tr>
    <td>φ<sub>a</sub></td>
    <td>aggregate porosity</td>
    <td>dimensionless</td>
  </tr>
  <tr>
    <td>η<sub>w</sub></td>
    <td>liquid water dynamic viscosity</td>
    <td>Pa s</td>
  </tr>
  <tr>
    <td>h<sub>k</sub></td>
    <td>height of the k<sup>th</sup> aggregate</td>
    <td>m</td>
  </tr>
  <tr>
    <td>H</td>
    <td>magnetic field strength</td>
    <td>A m<sup>-1</sup></td>
  </tr>
  <tr>
    <td>i</td>
    <td>imaginary number</td>
    <td>dimensionless</td>
  </tr>
  <tr>
    <td>I</td>
    <td>ionic strength</td>
    <td>mol L<sup>-1</sup> (M)</td>
  </tr>
  <tr>
    <td>κ</td>
    <td>inverse of the Debye length</td>
    <td>m<sup>-1</sup></td>
  </tr>
  <tr>
    <td>φ</td>
    <td>electrical potential in the diffuse layer</td>
    <td>V</td>
  </tr>
  <tr>
    <td>φ<sub>d</sub></td>
    <td>electrical potential at the d-plane</td>
    <td>V</td>
  </tr>
  <tr>
    <td>φ<sub>m</sub></td>
    <td>average electrical potential in the diffuse layer</td>
    <td>V</td>
  </tr>
  <tr>
    <td>J<sub>C</sub></td>
    <td>conduction current density</td>
    <td>A m<sup>-2</sup></td>
  </tr>
  <tr>
    <td>J<sub>D</sub></td>
    <td>displacement current density</td>
    <td>A m<sup>-2</sup></td>
  </tr>
  <tr>
    <td>pH</td>
    <td>potential of hydrogen ion in bulk water</td>
    <td>dimensionless</td>
  </tr>
  <tr>
    <td>q</td>
    <td>mean charge of the counter-ions in the Stern layer</td>
    <td>C</td>
  </tr>
  <tr>
    <td>q<sub>i</sub></td>
    <td>ion i charge</td>
    <td>C</td>
  </tr>
  <tr>
    <td>Q<sub>0</sub></td>
    <td>mineral surface charge density</td>
    <td>C m<sup>-2</sup></td>
  </tr>
  <tr>
    <td>Q<sub>0</sub><sup>e</sup></td>
    <td>edge surface mineral surface charge density</td>
    <td>C m<sup>-2</sup></td>
  </tr>
  <tr>
    <td>Q<sub>0</sub><sup>b</sup></td>
    <td>basal surface mineral surface charge density</td>
    <td>C m<sup>-2</sup></td>
  </tr>
  <tr>
    <td>Q<sub>β</sub></td>
    <td>Stern layer surface charge density</td>
    <td>C m<sup>-2</sup></td>
  </tr>
  <tr>
    <td>Q<sub>d</sub></td>
    <td>diffuse layer surface charge density</td>
    <td>C m<sup>-2</sup></td>
  </tr>
  <tr>
    <td>Q<sub>EDL</sub></td>
    <td>electrical double layer surface charge density</td>
    <td>C m<sup>-2</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>
  <tr>
    <td>Q<sub>V</sub><sup>0</sup></td>
    <td>excess of charge per unit Stern and diffuse layer volumes</td>
    <td>C m<sup>-3</sup></td>
  </tr>
  <tr>
    <td>Q<sub>V</sub><sup>d</sup></td>
    <td>excess of charge per unit diffuse layer volume</td>
    <td>C m<sup>-3</sup></td>
  </tr>
  <tr>
    <td>Q<sub>V</sub><sup>n</sup></td>
    <td>excess of charge per unit interlayer space volume</td>
    <td>C m<sup>-3</sup></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>L</td>
    <td>depolarisation coefficient</td>
    <td>dimensionless</td>
  </tr>
  <tr>
    <td>m</td>
    <td>cementation exponent of the clay aggregate</td>
    <td>dimensionless</td>
  </tr>
  <tr>
    <td>m<sub>i</sub><sup>d</sup></td>
    <td>electro-osmotic effect on diffuse layer conductivity</td>
    <td>dimensionless</td>
  </tr>
  <tr>
    <td>mr<sub>s</sub></td>
    <td>ion mobility ratio between in the Stern layer and distilled water</td>
    <td>dimensionless</td>
  </tr>
</table>
<table>
  <tr>
    <td>M</td>
    <td>effect of the diffuse layer on Stern layer polarization</td>
    <td>dimensionless</td>
  </tr>
  <tr>
    <td>n<sub>c</sub></td>
    <td>number of stacked sheets per particle</td>
    <td>dimensionless</td>
  </tr>
  <tr>
    <td>N</td>
    <td>number of ion species in bulk and diffuse layer waters</td>
    <td>dimensionless</td>
  </tr>
  <tr>
    <td>N<sub>A</sub></td>
    <td>Avogadro's number</td>
    <td>mol<sup>-1</sup></td>
  </tr>
  <tr>
    <td>N<sub>d<sub>k</sub></sub></td>
    <td>Number of different sized aggregates</td>
    <td>dimensionless</td>
  </tr>
  <tr>
    <td>N<sub>s</sub></td>
    <td>number of solid inclusions in the modelling</td>
    <td>dimensionless</td>
  </tr>
  <tr>
    <td>v</td>
    <td>empirical coefficient for temperature dependence of ion mobility</td>
    <td>K<sup>-1</sup></td>
  </tr>
  <tr>
    <td>pK<sub>w</sub></td>
    <td>minus log of dissociation constant of water molecules</td>
    <td>dimensionless</td>
  </tr>
  <tr>
    <td>ρ<sub>s</sub></td>
    <td>solid volumetric mass density</td>
    <td>kg m<sup>-3</sup></td>
  </tr>
  <tr>
    <td>σ<sup>(1)</sup></td>
    <td>medium electrical conductivity</td>
    <td>S m<sup>-1</sup></td>
  </tr>
  <tr>
    <td>σ<sub>0</sub><sup>(1)</sup></td>
    <td>medium bulk conductivity (excluding polarisation effect)</td>
    <td>S m<sup>-1</sup></td>
  </tr>
  <tr>
    <td>σ<sub>s</sub><sup>(1)</sup></td>
    <td>"surface" conductivity due to Stern layer polarisation</td>
    <td>S m<sup>-1</sup></td>
  </tr>
  <tr>
    <td>σ<sub>w</sub><sup>(1)</sup></td>
    <td>water conductivity</td>
    <td>S m<sup>-1</sup></td>
  </tr>
  <tr>
    <td>σ<sub>w</sub><sup>b (1)</sup></td>
    <td>bulk water conductivity</td>
    <td>S m<sup>-1</sup></td>
  </tr>
  <tr>
    <td>σ<sub>w</sub><sup>d (1)</sup></td>
    <td>diffuse layer water conductivity</td>
    <td>S m<sup>-1</sup></td>
  </tr>
  <tr>
    <td>S<sub>in</sub></td>
    <td>surface area of the interlayer space</td>
    <td>m<sup>2</sup></td>
  </tr>
  <tr>
    <td>S<sub>o</sub></td>
    <td>outer surface area of the clay particle</td>
    <td>m<sup>2</sup></td>
  </tr>
  <tr>
    <td>SS</td>
    <td>total specific surface area of the clay particle</td>
    <td>m<sup>2</sup> g<sup>-1</sup></td>
  </tr>
  <tr>
    <td>SS<sub>e</sub></td>
    <td>specific surface area of the edge surface</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 surface</td>
    <td>m<sup>2</sup> g<sup>-1</sup></td>
  </tr>
  <tr>
    <td>SS<sub>o</sub></td>
    <td>total outer specific surface area of the clay particle</td>
    <td>m<sup>2</sup> g<sup>-1</sup></td>
  </tr>
  <tr>
    <td>Σ<sub>s</sub><sup>d</sup></td>
    <td>Surface conductivity of the diffuse layer</td>
    <td>S</td>
  </tr>
  <tr>
    <td>Σ<sub>s</sub></td>
    <td>Surface conductivity of the Stern layer</td>
    <td>S</td>
  </tr>
  <tr>
    <td>t</td>
    <td>time</td>
    <td>s</td>
  </tr>
  <tr>
    <td>t<sub>in</sub></td>
    <td>interlayer thickness</td>
    <td>m</td>
  </tr>
  <tr>
    <td>τ</td>
    <td>relaxation time of the spherical particle</td>
    <td>s</td>
  </tr>
  <tr>
    <td>τ<sub>k</sub></td>
    <td>relaxation time of the k<sup>th</sup> sized aggregate</td>
    <td>s</td>
  </tr>
  <tr>
    <td>T</td>
    <td>absolute temperature</td>
    <td>K</td>
  </tr>
  <tr>
    <td>T<sub>C</sub></td>
    <td>temperature in degree Celsius</td>
    <td>°C</td>
  </tr>
  <tr>
    <td>g</td>
    <td>surface area weighting coefficient</td>
    <td>dimensionless</td>
  </tr>
  <tr>
    <td>V<sub>in</sub></td>
    <td>interlayer space volume</td>
    <td>m<sup>3</sup></td>
  </tr>
  <tr>
    <td>V<sub>s</sub></td>
    <td>solid volume</td>
    <td>m<sup>3</sup></td>
  </tr>
  <tr>
    <td>V<sub>sc</sub></td>
    <td>solid clay layer volume</td>
    <td>m<sup>3</sup></td>
  </tr>
  <tr>
    <td>V<sub>so</sub></td>
    <td>weakly conductive other solid volume</td>
    <td>m<sup>3</sup></td>
  </tr>
  <tr>
    <td>V<sub>s</sub></td>
    <td>Stern layer volume</td>
    <td>m<sup>3</sup></td>
  </tr>
  <tr>
    <td>V<sub>t</sub></td>
    <td>sample volume</td>
    <td>cm<sup>3</sup></td>
  </tr>
  <tr>
    <td>V<sub>w</sub><sup>d</sup></td>
    <td>volume of diffuse layer water</td>
    <td>m<sup>3</sup></td>
  </tr>
  <tr>
    <td>V<sub>w</sub><sup>b</sup></td>
    <td>volume of bulk water</td>
    <td>m<sup>3</sup></td>
  </tr>
  <tr>
    <td>ω</td>
    <td>angular frequency of the applied electrical current</td>
    <td>rad s<sup>-1</sup></td>
  </tr>
</table>
<table><tr><td>Ω<sub>s</sub></td><td>solid content in the sample</td><td>dimensionless</td></tr><tr><td>x</td><td>distance from the d-plane</td><td>m</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>z</td><td>mean valence of the counter-ions</td><td>dimensionless</td></tr><tr><td>z<sub>i</sub></td><td>ion i valence</td><td>dimensionless</td></tr></table>

(1) Followed by “*” means including displacement current.

48

49

ORIGINAL UNEDITED MANUSCRIPT
50 **1 INTRODUCTION**

51 Clay minerals or clays are fine-grained materials (particle size below 2 µm) that are very
52 common in geologic deposits, terrestrial weathering environments (e.g., regolith), and marine
53 sediments (Sposito 1989). They have a high specific surface area (SSA), the highest known
54 among major natural minerals (Leroy & Revil 2009). For instance, one gram of
55 montmorillonite (Mt), which belongs to the smectite group, can cover a surface of roughly
56 800 m² (Tournassat & Appelo 2011). Unlike other high-SSA natural minerals such as zeolites
57 and manganese and iron (hydr)oxides, clay minerals are frequently the main components of
58 soils and extended sedimentary stratigraphic layers, making around 40% of the minerals in
59 sedimentary rocks (Schroeder 2018).

60 Unique properties of clays are due to their microstructure – among them high adsorption
61 capability, swelling, catalysis, plasticity when wet, and very low permeability (Grim 1962).
62 These minerals are hydrous layer aluminosilicates, which form a diverse group constituting
63 the greater part of the phyllosilicate family of minerals. Clays are mostly made of aluminum,
64 silicon, and oxygen atoms interacting with water molecules on the mineral surface, sometimes
65 with variable amounts of iron, alkali metals (e.g., Na⁺ or K⁺), and alkaline earth metals (e.g.,
66 Ca²⁺ or Mg²⁺) (Tournassat et al. 2015). They have a layered structure containing an
67 octahedral (O) alumina sheet sandwiched between two tetrahedral (T) silica sheets (case of Mt
68 and illite) or a tetrahedral silica and octahedral alumina sheet (case of kaolinite) that are
69 stacked to form a particle (Leroy & Revil 2004).

70 Clay minerals have remarkable adsorption properties reflected by their high cation exchange
71 capacities (CEC), typically between 0.1 and 1.5 meq g⁻¹ for, in ascending order, chlorite,
72 kaolinite, illite and Mt (Revil & Leroy 2004). Isomorphic substitutions in the crystal are, e.g.,
73 the replacement of Al³⁺ by Mg²⁺ or Fe²⁺ ions in octahedral sheets or Si⁴⁺ by Al³⁺ or Fe³⁺ ions
in tetrahedral sheets with most of the substitutions occurring in the octahedral sheets for illite and Mt (Leroy *et al.* 2017b). Isomorphic substitutions are responsible for a high permanent negative surface charge mostly expressed on the basal clay surfaces (Bourg & Sposito 2011).

This surface charge, conjointly with the high specific surface area of illite (typically 100-200 m² g⁻¹ compared to a kaolinite SSA of typically 10-20 m² g⁻¹) and Mt (typically 700-800 m² g⁻¹) (Revil & Leroy 2004), are at the origin of the high measured CEC of illite (~0.2-0.3 meq g⁻¹) and Mt (~0.7-1.5 meq g⁻¹) (Revil & Leroy 2004; Tournassat *et al.* 2015). Clays such as Mt also have swelling properties due to the presence of hydrated cations such as Na⁺ ions adsorbed in the interlayer space between TOT layers (Massat *et al.* 2016) and catalytic properties because of their high reactive specific surface area (Ismadji *et al.* 2015). Clay minerals also have rheological properties and develop plasticity when wet because of a molecular film of water surrounding clay particles, but become hard, brittle and non-plastic upon drying or firing (Guggenheim & Martin 1995).

Clays play a major role in the terrestrial biogeochemical cycling of metal cations (e.g., K⁺ or Ca²⁺) and in the chemical homeostasis of the oceans (Sposito *et al.* 1999) because they are abundant in Earth's soil and subsurface and of their high chemical reactivity (regarding, e.g., adsorption and catalysis) (Sposito *et al.* 1999). These minerals also provide an efficient confinement environment for biomolecules and might have enhanced biochemical reactions to develop life during early life evolution on Earth (Yang *et al.* 2013) and possibly on Mars (Azua-Bustos *et al.* 2020). Due to clay rheological properties, shale beds (sedimentary rocks composed of very fine clay particles) have preserved a remarkable range of fossils, e.g., hard body parts, soft tissues, feathers, prints, and burrows (Petraglia *et al.* 2014). Therefore these clayey rocks provide records of Earth's evolutionary history and inspired Charles Darwin for his theory of evolution (Charlet *et al.* 2017). Clay minerals also provide indicators of the
98 evolution of aqueous conditions and possible habitats for life on ancient Mars (Bristow et al.
99 2018).

100 In addition, clays are used in a broad range of engineering processes, for instance as
101 lubricants in oil industry (Udonne 2011), industrial catalysts for the synthesis of organic
102 compounds (Nagendrappa 2011), and for pharmaceutical drug delivery (Viseras et al. 2010).
103 Since the Neolithic period, clay minerals have also been employed for pottery (Vandiver et al.
104 1989). Special attention is currently devoted to the pharmacological functions of clays and
105 their biomedical applications in pelotherapy, wound healing, regenerative medicine,
106 antimicrobial, and dermocosmetics (Viseras et al. 2019). Natural and synthetic clays are also
107 used as semi-permeable membrane barriers adsorbing metallic pollutants (Gehin et al. 2007),
108 salt ions (water desalinisation (Ali et al. 2021)) and organic pollutants (Kausar et al. 2018) to
109 improve the quality of water resources. They are also used as containers in landfills because
110 of their confining properties towards industrial and domestic wastes (Malusis et al. 2003).

111 Clay-rich (argillaceous) rocks, including marls, shale, mudstone, argillite and claystone
112 (herein referred to as clay rocks), are sedimentary formations with a very low permeability
113 (typically between 10⁻²³ and 10⁻¹⁷ m², see Neuzil 1994). The low permeability of clay rocks is
114 a consequence of consolidated fine-grained sediments containing a large fraction of nanopores
115 (Yven et al. 2007) with a loss of pore connectivity during diagenetic or metamorphic rock
116 reactions (Charlet et al. 2017). Because of their confining properties, clay geological
117 formations act as cap rocks above oil and gas (“carbonated” energies) (Li et al. 2006) and
118 geothermal energy (“decarbonated” energy) (Guisseau et al. 2007) reservoirs. They are also
119 frequently used for the geological storage of H₂ (energy “vector”) (Liu et al. 2022), CO₂
120 (Busch et al. 2016) and high-level nuclear waste (Sellin & Leupin 2013). Smectite minerals
121 can adsorb and store CO₂ in their interlayer space to mitigate greenhouse gas emissions (Bø
122 Hunvik et al. 2021). The swelling of smectite minerals in contact with an aqueous solution
can also produce strong adverse effects in the extraction of oil and gas (Karaborni et al. 1996) and of “green” geothermal energy (Xu et al. 2006), and is of major concern in civil engineering due to quick clay landslides (Khaldoun et al. 2009).

When clay mineral is in contact with water, an electrical double layer (EDL) lies on the mineral outer surface to compensate the usual negative surface charge not compensated by interlayer cations (Bourg et al. 2007). Some clays like kaolinite can have a positive surface charge in acidic conditions because of the adsorption of protons on surface oxygen (Tombacz & Szekeres 2006). The clay EDL is 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 (Leroy & Revil 2004). The EDL controls the distribution and displacement of electrical charges on the surface of clays because of the high surface charge magnitude and specific surface area of clays (Tournassat & Steefel 2019). Therefore, clay EDL has a strong influence on clay electrical conductivity (Waxman & Smits 1968).

In geophysics, electrical and electromagnetic methods are very sensitive to electrical charge transport in clay materials, especially in the EDL at the interface between solid clay and bulk pore water (e.g., Revil & Leroy 2001; Jougnot et al. 2010a; Qi & Wu 2022). Any perturbation of chemical, electrical, hydrostatic and thermal nature in a geological system containing a significant fraction of clays should be theoretically captured by these methods (Revil & Leroy 2004; Revil et al. 2005; Revil & Linde 2006). Notably, the electrical current injected into the ground containing clay materials moves the electrical charges in the pore water, especially those in the diffuse part of the clay EDL and in the stagnant part of the clay EDL (Stern layer), which in turn influence ground electrical conductivity (ability of the medium to carry electric current) (e.g., Revil 2012; Okay et al. 2014; Qi & Wu 2024). In the laboratory, a tension typically ± 5V and a resulting weak electrical current typically in the mA range is
148 injected when investigating pluricentimetric samples (Zimmermann *et al.* 2008). In the field, 149 considerably higher voltages and currents are typically injected into the ground depending on 150 the depth of the investigation (Kemna *et al.* 2012). The measured electrical and magnetic 151 fields and related conductivities and susceptibilities can thus be used to estimate the 152 petrophysical, state and transport properties of clay materials in geological formations 153 (Triantafilis & Lesch 2005; Glaser *et al.* 2023; Porté *et al.* 2023).

154 A main inherent drawback of the conventional direct current (DC) resistivity survey is its 155 ambiguity in distinguishing between the effects of groundwater salinity, clay content, and 156 pore space geometry (Binley & Kemna 2005; Binley & Slater 2020). Among the electrical 157 and electromagnetic methods, one named “induced polarisation” (IP) is of particular interest 158 for geophysicists due to its increasing sensitivity to subsurface processes over the last 20 159 years (Kemna *et al.* 2012; Binley *et al.* 2015). The induced polarisation method is related to 160 the reversible “storage” of electrical charges in porous media (Revil & Skold 2011). It was 161 discovered more than 100 years ago by Conrad Schlumberger (as far as we know). After 162 cutting the electrical current injected into the ground containing sulphide ores, Schlumberger 163 observed that the related voltage did not disappear immediately and instead decreased slowly 164 with time (Schlumberger 1920). Integrating the measured voltage in excess over the time after 165 switching off the current allows estimating the chargeability of the investigated porous 166 medium (Pelton *et al.* 1978). In the case of clay materials, the chargeability is sensitive to the 167 excess of electrical conductivity in the clay EDL compared to in bulk water, which is 168 responsible for clay “surface” conductivity controlled by clay microstructural and 169 electrochemical properties (Leroy & Revil 2009; Leroy *et al.* 2017b; Revil *et al.* 2023). The 170 IP method gives therefore access not only to the DC (“zero frequency”) conductivity or 171 resistivity but it also gives access to the chargeability of the investigated clayey medium 172 (Revil & Florsch 2010; Revil 2012; Qi & Wu 2024). These physical quantities, due to their
dependence on clay pore water and surface conductivity, can deliver important information on the petrophysical properties of clay materials. These properties are, for instance, mineralogy, pore structure, water and clay contents, and transport properties including ion diffusivity and rock permeability (Jougnot et al. 2009, 2010b; Revil & Florsch 2010; El Alam et al. 2023; Revil et al. 2023).

The spectral induced polarisation (SIP) method belongs to the broader family of IP methods. Rather than switching off and on the injected current, a sinusoidal, low-frequency (typically in the mHz to kHz frequency range) electrical current is injected into the ground and the resulting voltage is measured (Marshall & Madden 1959). There are different ways of doing TDIP and SIP measurements in the laboratory and in the field. In the case of the Wenner alpha configuration, two current electrodes are used to inject the current and two potential electrodes located between the current electrodes and at equal separation distance are used to measure the resulting voltage in the investigated porous medium (e.g., Martin et al. 2020). Other frequently employed configurations are dipole-dipole, pole-dipole or Wenner-Schlumberger ones (Aizebeokhai & Oyeyemi, 2014).

During SIP measurements on clay materials, the magnitude of the measured voltage depends on ion fluxes induced by the imposed electrical field in the pore water especially on those in the clay diffuse layer and in the clay Stern layer (ion electromigration processes) (e.g., Leroy & Revil 2009; Okay et al. 2014). The measured phase-shift between injected current and measured voltage depends on ion counteracting diffusion fluxes to reestablish their distribution at thermodynamic equilibrium (Fig. 1) (e.g., Lyklema et al. 1983; Grosse & Delgado 2010). Indeed, when the frequency of the injected sinusoidal current is sufficiently low (typically below 1 kHz), the EDL polarizes (i.e. counter-ions in the EDL have time to electromigrate tangentially to the particle surface) to accumulate on one side of the particle, and to diffuse back to their initial position (Fig. 1a, Schwarz 1962). These ion movements,
which can be characterized by a relaxation time, thus create a complex surface conductivity
increasing the electrical conductivity of the medium (e.g., Leroy et al. 2008; Leroy et al. 2017a). When the frequency of the injected sinusoidal current is higher (typically above 1 kHz), there is a charge build-up at the boundary between two phases presenting different conductivities and permittivities, like solid clay and water, responsible for another polarisation mechanism, Maxwell-Wagner polarisation, and increasing the electrical conductivity of the medium (de Lima & Sharma 1992). Nanoporous clay materials are characterized by a broad distribution of relaxation times due to Stern layer polarisation influencing in particular medium and high-frequency SIP measurements (Leroy & Revil 2009).

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Sketch of EDL polarisation at low frequencies of the injected current for a negatively charged particle. Model of Lyklema *et al.* (1983) (generalized Schwarz model) showing the capacitive effect of the polarized thin layer usually assumed to be the Stern layer (surface diffusion processes) (a., modified from Leroy *et al.* 2017b). Model of Grosse & Delgado (2010) showing the formation of ion clouds in excess and deficiency outside the EDL compared to in the surrounding electrolyte due to EDL (here diffuse layer) polarisation (concentration polarisation, volume diffusion processes) (b., modified from Grosse & Delgado 2010).</figcaption></figure>

Petrophysical models have been developed to understand quantitatively SIP laboratory and field IP measurements on clayey media. For instance, Vinegar & Waxman (1984) developed a
petrophysical SIP model for shaly sands based on an equivalent circuit model. Following Waxman & Smits (1968), their model assumes that the measured real part of the complex conductivity, the measured in-phase conductivity, can be understood quantitatively considering two conductances in parallel, one associated with electromigrating ions in the bulk pore water and one associated with electromigrating cation counter-ions on the clay surface. Following the membrane polarisation model of Marshall & Madden (1959), they considered in addition two conductances in series, one related to clay counter-ion displacement in clay-free zones and one related to electrolyte blockage by the clay-site membranes, to explain imaginary, quadrature conductivity measurements on shaly sands between 1 Hz and 1 kHz. In the model of Vinegar & Waxman 1984, the in-phase and quadrature (or out-of-phase) conductivities depend on $Q_v$, which is the excess of counter-charge (canceling clay surface charge) per unit pore volume and is calculated from the measured porosity and clay cation exchange capacity. The SIP model of Vinegar & Waxman (1984) is elegant and easy to use but its main drawback is, in our opinion, its establishment at the macroscale of the centimetric sample that does not allow it to explicitly consider clay microstructure and EDL properties. de Lima & Sharma (1992) developed mechanistic SIP models for clay materials considering explicitly Stern layer (Schwarz 1962) or diffuse layer (Fixman 1980) polarisation and Maxwell-Wagner polarisation (Sen et al. 1981). The SIP model of de Lima & Sharma 1992 is more physical and more difficult to handle than the SIP model of Vinegar & Waxman (1984). In our opinion its main drawback is its inability to capture clay microstructure with very elongated interacting clay particles on the contrary to the spherical clay particle considered in the model of de Lima & Sharma (1992). Most of the following SIP models for clay materials were inspired by the works of Vinegar & Waxman (1984) and de Lima & Sharma (1992) (e.g., Leroy & Revil 2009; Jougnot et al. 2010a; Revil 2012; Okay et al. 2014; Weller et al. 2015; Leroy et al. 2017b; Revil et al. 2023; Qi & Wu
244 2024). For instance, Leroy *et al.* (2017b) considered a Stern layer polarisation model for dispersed clay particles interacting with each other and forming aggregate. However, they did not consider Maxwell-Wagner polarisation. In addition, the role of the interlayer space of Na-montmorillonite (Na-Mt) containing hydrated and mobile Na⁺ cations (Bourg *et al.* 2003) on clay conduction and polarisation during SIP measurement was not investigated in the previously reported works. In our knowledge, most of the current SIP models are not considering explicitly clay microstructure and EDL properties. The microstructural and electrical properties of the different components of the clay material, e.g., bulk pore water, diffuse layer, Stern layer, and interlayer space, are not clearly described.

253 In this study, we propose a new mechanistic model to interpret SIP measurements on unconsolidated clay materials (e.g., muds) as a function of salinity and temperature from clay microstructural and EDL properties. Our model is an extension of the SIP model of Leroy *et al.* (2017b). It considers bulk pore water and diffuse layer conductivities as well as Stern layer and Maxwell-Wagner polarisations including the contribution of the interlayer space of Na-Mt, which both were neglected in Leroy *et al.* (2017b) due to the low concentration of clay particles in the suspension (less than 60 g of clay powder per liter of electrolyte). Porosity, cementation exponent, electrical formation factor, surface conductivity and water conductivity can be extracted from our model compared to experimental data in order to better understand the electrical properties of clay materials and to test the assumptions of SIP petrophysical models developed at larger scale (e.g., Weller *et al.* 2013). Section 2 of our manuscript explains the microstructural and electrochemical properties of the main clay minerals, i.e. kaolinite, illite and montmorillonite clays. Section 3 presents the theory of our mechanistic SIP model that is compared to the experimental data of Mendieta *et al.* (2021) in the companion paper by Maineult *et al.* (2024).
269 **2 CHARACTERISTICS OF KAOLINITE, ILLITE AND MT PARTICLES**

270 **2.1. Microstructural properties**

271 Clay minerals are characterized by a wide variety of mineralogies, sizes and shapes, and
272 heterogeneities in their surface properties (the edge surfaces behave differently than the basal
273 surfaces) (Tombacz & Szekeres 2006; Brigatti et al. 2013). Clay particles (or tactoids) are a
274 stack of platelets that are very elongated (Tournassat et al. 2015). These particles have
275 therefore a high aspect ratio with a thickness considerably smaller than length for illite and Mt
276 particles. Typical dimensions of clay platelets and particles are indicated in the following text.

277 The chemical formula of kaolinite is Al₂Si₂O₅(OH)₄ (Grim 1962). A kaolinite particle is a
278 stack of several silica tetrahedral (T) and alumina octahedral (O) sheets linked by hydrogen
279 bonds (1:1 clay). The thickness of a TO sheet is ~7 Å (Fig. 2a) and its length is typically
280 between 200 and more than 1000 nm (Fig. 3a) (Tournassat et al. 2015). The specific surface
281 areas (SSA) of the TO platelets (or layers) can be estimated from crystallography (Leroy &
282 Revil 2004). Indeed, a square TO platelet with a thickness of 7.2 Å, a length of 500 nm and a
283 solid volumetric density of 2.62 g cm⁻³ (Hassan et al. 2006) has a total specific surface area of
284 about 1063 m² g⁻¹ from which the edge surface share is about 3 m² g⁻¹ and the basal surface
285 share is about 1060 m² g⁻¹. According to the previous calculation and Hassan et al. 2006 and
286 Tournassat et al. 2015, the edge specific surface area of a TO platelet is about 2-7 m² g⁻¹ and
287 the basal specific surface area of a TO platelet is about 1060-1070 m² g⁻¹. The edge SSA of
288 individual TO layers is therefore roughly 150 to 500 times lower than their basal SSA. In
289 addition, a kaolinite particle is considerably thicker and larger in average than an illite and Mt
290 particle (Fig. 3). Kaolinite particle thickness is typically from less than 7 nm to more than 140
291 nm (Tournassat et al. 2015). The number of stacked TO layers of a kaolinite particle can
292 therefore range from 10 to more than 200.
<figure><img src="image_3.png" /><figcaption>**Figure 2.** Sketch of (a.) kaolinite, and (b.) illite and Mt platelets (modified from Leroy & Revil 2009).</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Sketch of (a.) kaolinite, (b.) illite and (c.) Mt particles (modified from Tournassat *et al.* 2015).</figcaption></figure>

The chemical formula of illite is $(K_2H_3O)(Al,Mg,Fe)_2(Si,Al)_4O_{10}[(OH)_2,(H_2O)]$ and the chemical formula of Mt is $(Na,Ca)_{0.33}(Al,Mg)_2Si_4O_{10}(OH)_2 \cdot n(H_2O)$ (Grim 1962). Illite and Mt particles are stacks of TOT sheets of thickness ~9.5 Å (Fig. 2b) and length between 50 and 100 nm for illite (Fig. 3b) and between 50 and 1000 nm for Mt (Fig. 3c) (Tournassat *et al.* 2015). TOT sheets exhibit a permanent negative charge on their basal surfaces and are bonded by non-solvated cations (e.g, $K^+$) for illite and fully solvated cations (e.g., $Na^+$ and/or $Ca^{2+}$) for swelling clays such as Mt (2:1 clays) (Tournassat *et al.* 2015). From crystallographic consideration, it can be estimated that the total specific surface area of a TOT layer is about 780 $m^2 g^{-1}$, from which the edge surface share is about 10-30 $m^2 g^{-1}$ and the basal surface
share is about 750-770 m² g⁻¹ (Leroy et al. 2015). The edge SSA of individual TOT layers is therefore roughly 20 to 80 times lower than their basal SSA. The SSA of clay layers are thus dominated by the contribution of the basal surfaces, as the edge SSA represents only a minor part of the total SSA. Furthermore, illite particle thicknesses are typically from 5 nm to 20 nm and Mt particle thicknesses are typically from 1 nm to above 10 nm (Tournassat et al. 2015). Therefore, Mt particles are more elongated than illite particles. Illite particles typically consist of 5 to 20 stacked TOT layers. The TOT layers of Mt particles can become completely delaminated. The number of TOT layers per Mt particle rarely exceeds 10 in water (Tournassat & Appelo 2011). The number of TOT layers per Mt particle increases with decreasing water chemical potential and also tends to increase with the valence of the counter-ion in the interlayer space (Tournassat et al. 2015). According to viscosity and light-transmission experiments, Na-Mt dispersions in pure water contain particles made up of 1-2 TOT layers, whereas Ca-Mt or Mg-Mt dispersions in pure water contain particles made up of 6-10 TOT layers (Schramm & Kwak 1982).

The total SSA of a kaolinite particle, typically 10-20 m² g⁻¹ (where the edge surface shares around 15% of it, around 1.5-3 m² g⁻¹ (Hassan et al. 2006)), is considerably smaller than the total SSA of an illite or Mt particle, which is typically between 100 and 200 m² g⁻¹ for illite and between 750 and 800 m² g⁻¹ for Na-Mt (Revil & Leroy 2004; Hassan et al. 2006; Tournassat & Appelo 2011; Tournassat et al. 2015). Due to the irregular morphology of clay mineral layers, the edge surfaces of different layers in a single particle may be misaligned. Moreover, translational and rotational disorder between adjacent layers makes the structure turbostratic at the scale of individual clay mineral particles. Clay minerals have different morphologies: kaolinite and well-crystallized illite have a tendency toward hexagonal and elongated hexagonal morphologies, respectively, whereas Mt and less well-crystallized illite have mostly irregular platy or lath-shaped morphologies (Fig. 4).
<figure><img src="image_5.png" /><figcaption>**Figure 4.** Scanning electron microscopy (SEM) images of dried kaolinite (a., modified from Christidis 2010), illite (b., modified from Mc Bride 2012), and Mt (c., modified from Christidis 2010) particles. All images are at the same scale.</figcaption></figure>

**2.2. Electrochemical properties**

The different faces of clay particles exhibit different electrochemical properties (Tombacz & Szekeres 2006; Bourg *et al.* 2007; Tournassat *et al.* 2016a). The electrochemical properties of kaolinite particles are mainly controlled by the edge surfaces and the alumina basal surfaces due to the weak negative structural charge on the silica basal surfaces (Tombacz & Szekeres 2006) (Fig. 3a). The electrochemical properties of Mt particles and in lesser extent of illite particles are mainly controlled by the basal surfaces due to their larger specific surface area than the edge surfaces and to the structural charge (Tombacz & Szekeres 2004; Leroy & Revil 2009; Tournassat *et al.* 2011; Tournassat *et al.* 2013) (Figs 3b and 3c). In the following, we define $Q_0$ as the surface charge density of the clay mineral (in C m⁻²), $Q_0^b$ as the basal surface charge density, $Q_0^e$ as the edge surface charge density, $Q_i$ as the negative and permanent structural surface charge density due to isomorphic substitutions, and $Q_{EDL}$ as the surface charge density of the EDL.
The quantity $Q_0$ is an accurate indicator of the EDL properties controlling the IP response because of the electroneutrality condition (Hunter 1981) implying

$Q_0 + Q_{EDL} = 0,$

and then

Therefore, according to Eq. 2, a high clay surface charge density magnitude implies a high EDL surface charge density magnitude and potentially a high IP real conductivity response (due to EDL conduction) and a high IP imaginary conductivity response (due to EDL polarisation) (Leroy & Revil 2009). The quantity $Q_0$ depends on the contributions of the edge and basal surfaces considering

$Q_{EDL} = -Q_0.$

(2)

$Q_0 = \frac{Q_0^e SS_e + Q_0^b SS_b}{SS}$

(3)

where $SS_e$ is the edge specific surface area (in m² g⁻¹), $SS_b$ is the basal specific surface area, and $SS$ is the total specific surface area, $SS = SS_e + SS_b$.

The surface charge density $Q_0^e$ is controlled by the amphoteric aluminol, >Al-OH, and silanol, >Si-OH, hydroxyl surface sites that are sensitive to pH and salinity of the surrounding electrolyte due to proton desorption and adsorption on mineral surface (e.g., >S-OH ↔ >S-O⁻ + H⁺ where >S = >Al or >Si) and salt counter-ion adsorption and desorption in the Stern layer (e.g., >S-O⁻ + M⁺ ↔ >S-O⁻-M⁺ where M⁺=Na⁺) (Leroy & Revil 2004; Tombacz & Szekeres 2004; Tombacz & Szekeres 2006; Tournassat et al. 2016a). The alumina basal surface of kaolinite particle is also sensitive to pH and salinity (Tombacz & Szekeres 2006). In absence of strong $Q_i$ influence and dissolution/precipitation reactions, $Q_0^e$ magnitude

$$Q_0 + Q_{EDL} = 0,$$

$$Q_{EDL} = -Q_0.$$

$$Q_0 = \frac{Q_0^e SS_e + Q_0^b SS_b}{SS},$$
should increase when pH moves from the point of zero charge (PZC, pH value where the surface charge involving only amphoteric sites is zero). Indeed, in these conditions, when pH decreases from PZC, mineral edge surface charge should become more positive because of the increasing adsorption of protons H⁺, and when pH increases from PZC, mineral edge surface charge should become more negative because of the increasing desorption of protons H⁺ (Leroy *et al.* 2011). In these conditions also, |Q₀| should increase when salinity increases because of the adsorption of salt counter-ions in the Stern layer (Leroy *et al.* 2013). The replacement of chemically adsorbed protons by salt cations mostly adsorbed further away from the mineral surface in the Stern layer increases the magnitude of the negative surface charge (the surface charge due to negative oxygen atoms is not compensated by H⁺ anymore, see Fig. 5).

<figure><img src="image_6.png" /></figure>

**Figure 5.** Sketch of the electrochemical properties of (a.) kaolinite, (b.) illite and Mt particles in contact with a 1:1 electrolyte like NaCl or KCl (modified from Leroy & Revil 2009). The electrical potential φ₉ is assumed to be located at the *d*-plane delimiting the beginning of the diffuse layer from the mineral surface. In Figure 5b, the white color in the interlayer space means that cations are non-solvated and the blue color in the interlayer space means that cations are fully-solvated.
Illite and Mt particles exhibit a permanent and negative structural charge on their basal surface ($Q_0^b$ value typically between -0.20 C m⁻² and -0.10 C m⁻²) due to isomorphic substitutions within the mineral crystal (Grim 1962; Bourg et al. 2007; Leroy et al. 2007; Leroy et al. 2017b). This structural surface charge is partly compensated by non-solvated cations (e.g., $K⁺$ or $NH_4⁺$ ions with low hydration energy) and hydrated cations (e.g., $Na⁺$ and/or $Ca^{2+}$ions) in the interlayer space of illite and Mt (dioctahedral smectite), respectively (Melkior et al. 2009; Tournassat et al. 2015). In most of natural Mts, $Na⁺$ and $Ca^{2+}$ ions commonly coexist in the interlayer space (Tournassat et al. 2011). Hydrated cations such as $Na⁺$ ions have a significant mobility in the interlayer space of Mt (for instance, around 20% of their mobility in bulk water according to the molecular dynamic simulations of Tournassat et al. 2009) whereas non-solvated cations such as $K⁺$ ions have a negligible mobility in the interlayer space of illite (Ruiz Pestana et al. 2016). Therefore, in the following, we consider that the interlayer space of Mts such as Na-Mt is conductive and polarizable in the mHz to kHz frequency range on the contrary to the illite interlayer space.

The behavior of the clay surface charge is therefore more complicated than for oxides such as $SiO_2$ (Leroy et al. 2013) or $TiO_2$ (Leroy et al. 2011) or carbonates (Li et al. 2016) because, as written previously, the different faces have not the same electrochemical properties (Tournassat et al. 2013). For instance, the PZC of the edge face of Na-Mt is around 6.5 whereas its basal faces do not exhibit a PZC value because of the strong influence of the negative structural surface charge (Tombacz & Szekeres 2004). The basal faces (or surfaces) of illite and Mt are less sensitive to pH and salinity than their edge face because the hydroxyl sites are less abundant and of the strong influence of the structural surface charge (Tournassat et al. 2015). In a system where protons and hydroxide ions are the only aqueous species, the point of zero net proton charge (PZNPC) of a mineral defines the pH at which the positively
and negatively charged functional groups on its surface are equal (Hao *et al.* 2018). For clay minerals presenting both pH-dependent surface charge and a permanent structural surface charge, the PZNPC is a better indicator of the state of the surface charge than the PZC (Tombacz & Szekeres 2004). The PZNPC of clays decreases to lower pH with increasing, e.g., NaCl concentration in bulk water (not influenced by the electrically charged surface) because of the structural charge influence, which is stronger for, e.g., Na-Mt or illite than for kaolinite (Avena & De Pauli 1998; Tombacz & Szekeres 2006). Tombacz & Szekeres (2006) reported that the PZNPC of Na-Mt ranges from around 8.75 at a salinity of 0.01 M NaCl to around 6.75 at a salinity of 1 M NaCl and that the PZNPC of kaolinite ranges from around 6.75 at a salinity of 0.01 M NaCl to around 5.25 at a salinity of 1 M NaCl (see their Fig. 6). This confirms that the structural charge influence on clay surface charge is stronger for Na-Mt than for kaolinite and hence that the surface charge of kaolinite is more sensitive to pH and salinity compared to the surface charge of Na-Mt. Using measured PZNPC values, Hao *et al.* (2018) showed that the surface charge of kaolinite is more sensitive to pH and salinity compared to the surface charge of Mt and also illite in a lesser extent (a significant part of the structural surface charge of illite is compensated by non-solvated K⁺ counter-ions).

The outer mineral surface charge, i.e., the mineral surface charge not compensated by the interlayer counter-ions, is compensated by counter-ions adsorbed in the “compact” Stern layer that has a solid-like water viscosity and is located between the mineral surface and the shear (or slipping) plane when water flow occurs on the mineral surface (Revil & Leroy 2001). The remaining outer mineral surface charge is compensated by counter-ions in excess and co-ions in deficiency compared to their concentration in bulk water that follow a Boltzmann type distribution in the diffuse part of the EDL (Leroy & Revil 2004) (Fig. 5). The high negative surface charge on the basal faces of Mt and illite favors counter-ion condensation within the Stern layer (Leroy & Revil 2009). This is the reason why the fraction of the surface charge
compensated by the Stern layer on the external basal surface of such clay minerals ($f_Q$) is typically above 50% (Leroy & Revil 2004; Tombacz & Szekeres 2004; Leroy et al. 2007; Leroy & Revil 2009; Leroy et al. 2015). The $f_Q$ quantity is of paramount importance for SIP models because they consider that the Stern and diffuse layers have different contributions to measured SIP data (e.g., de Lima & Sharma 1992; Leroy & Revil 2009; Revil 2012; Okay et al. 2014; Leroy et al. 2017b; Qi & Wu 2024). It is defined by:

$$f_Q = \frac{Q_\beta}{Q_\beta + Q_d} = -\frac{Q_\beta}{Q_0}, \qquad (4)$$

where $Q_\beta$ and $Q_d$ are the surface charge densities of the Stern and diffuse layers (in C m⁻²), respectively. The $f_Q$ quantity can be estimated using ion adsorption experiments and surface complexation models (e.g., basic Stern, triple layer or extended Stern layer models) (e.g., Yates et al. 1974; Hiemstra & VanRiemsdijk 1996; Sahai & Sverjensky 1997). In addition, in our knowledge, there is no study on the fraction of the counter-charge controlled by the Stern layer on the edge surfaces of illite and Mt. However, we can expect, in average, lower $f_Q$ values for the edge surfaces than for the external basal surfaces of illite and Mt and higher $f_Q$ values for the edge surfaces of these minerals than for the surfaces of kaolinite because of the structural charge influence on the edge surfaces of illite and Mt. Indeed, Secor & Radke (1985) followed by, e.g., Bourg et al. (2007), Tournassat et al. (2013), and Tournassat et al. (2016a) showed that there is a spill-over effect of the electrostatic potential on the external basal surface onto the electrostatic potential on the edge surface, which can increase the magnitude of the edge negative surface charge in neutral to basic pH conditions and favor Stern layer ion condensation. (Grosse & Delgado 2010)
In water, clay particles in contact with each other form aggregates in particular because of EDL interactions between particles (Fig. 6) (Mc Bride & Baveye 2002). Attractive edge-to-face interactions (when pH < PZC of the edge faces) and repulsive face-to-face interactions as well as edge-to-edge interactions regulate the size and shape of the clay aggregate and its internal porosity, which are strongly depend on pH and salinity (e.g., Leroy *et al.* 2015, Leroy *et al.* 2017b, Rasmusson *et al.* 1997, Sondi *et al.* 1996, Tombacz & Szekeres 2006). The critical coagulation concentration (CCC) is defined as the minimum concentration of salt counter-ions to induce coagulation of colloidal particles. Above the CCC, which is typically between 0.01 and 0.1 mol L⁻¹ (M) for dispersed clays in NaCl electrolyte, clay particles coagulate due to diffuse layer compression and form larger particles with more interlayer space for illite and Mt than at lower salinities (e.g., Sondi *et al.* 1996, Rasmusson *et al.* 1997, Tombacz & Szekeres 2004, Tombacz & Szekeres 2006, Leroy *et al.* 2015, Leroy *et al.* 2017b, Mendieta *et al.* 2021).

<figure><img src="image_7.png" /><figcaption>**Figure 6.** Sketch of a Na-Mt aggregate containing interacting clay particles (a., modified from Leroy *et al.* 2015, in reality particles are thinner than those represented here) and Cryo-SEM images of high-pressure frozen and freeze-fractured SWy-2 bentonite in distilled water with 5% (wt/wt) solid content (b., Na-ion exchanged gel, modified from Mouzon *et al.* 2016).</figcaption></figure>
The surface conductivity influencing IP measurements is defined as the excess of electrical conductivity in the EDL compared to in bulk water integrated over the thickness of the EDL (Revil & Glover 1997; Lyklema & Minor 1998). According to O'Konski 1960, surface conductivity contributes to particle conductivity when the thickness (or radius) of the particle is larger than the thickness of the EDL. However, the EDL can be thicker than clay such as illite or Na-Mt particle (section 2.1.), meaning that in that case it can contribute to water conductivity (Dufreche et al. 2001; Leroy & Revil 2009). In addition, the electrochemical properties and surface conductivity of kaolinite are more sensitive to pH and salinity than the electrochemical properties and surface conductivity of Mt and illite in a lesser extent (Leroy & Revil 2004). Furthermore, because a kaolinite TO sheet has a weaker structural surface charge and a kaolinite particle has a smaller specific surface area than an illite or a Mt particle, kaolinite has weaker electrochemical and adsorption properties, and hence surface conductivity than illite and Mt (Leroy & Revil 2004; Tombacz & Szekeres 2006; Okay et al. 2014). Mt surface conductivity should be higher than illite surface conductivity because Mt has a larger reactive specific surface area than illite (Leroy & Revil 2009). This should be particularly true for Na-Mt containing hydrated and mobile Na⁺ cations in its interlayer space. This conclusion also means that the laboratory CEC measurement, which depends on the product of the surface charge density (Q₀) with the reactive specific surface area (Leroy et al. 2007; Okay et al. 2014), cannot be used to characterize the electrochemical and IP properties of clays such as kaolinite because of the strong dependence of Q₀ on the bulk pore water composition (pH, salinity). Indeed, CEC laboratory measurements are done using a particular bulk water composition (Khaled & Stucki 1991) that does not correspond necessarily to the bulk pore water composition in the laboratory samples during SIP measurements and in-situ during field geophysical measurements. However, the laboratory CEC measurement could be safely employed to characterize the electrochemical and IP properties of Mt and in a lesser
extent of illite and soils, sediments and rocks containing a significant fraction of these
minerals because of the strong influence of the permanent and negative structural charge
(Vinegar & Waxman 1984; Revil 2012; Weller et al. 2013; Okay et al. 2014; Leroy et al.
2017b).

505

## 3 THE PETROPHYSICAL SIP MODEL

After presenting the microstructural and electrochemical properties of kaolinite, illite, and Mt clays, we explain here the petrophysical SIP model. During SIP measurement, a sinusoidal electrical current in the mHz to kHz frequency range is injected into the clay material and the resulting voltage is recorded in order to measure the complex conductivity of the clay material. In this section, we briefly explain the SIP model developed by Leroy et al. (2017b) for dilute Mt suspensions. We then explain the extension of this SIP model to clay materials containing significant content of kaolinite, illite or Mt considering interlayer space conduction and polarisation for Na-Mt and Maxwell-Wagner polarisation.

Our conductivity model is based on Ampere's law, one of the four Maxwell's equations. This law considers that the magnetic field strength is influenced by conduction current satisfying Ohm's law and displacement current due to the time-varying electrical field (Maxwell 1892), that is

$$ \nabla \times \mathbf{H} = \mathbf{J}_C + \mathbf{J}_D, \qquad (5) $$

where $\mathbf{H}$ is the magnetic field strength (in A m⁻¹), and $\mathbf{J}_C$ is the electrical current density due to conduction current (in A m⁻²) and $\mathbf{J}_D$ is the electrical current density due to displacement current, which are defined by
$$J_C = \sigma E, \qquad (6)$$

522 and

$$J_D = \frac{\partial D}{\partial t}, \qquad (7)$$

523 where $\sigma$ is the electrical conductivity of the investigated medium (in S m⁻¹), $E$ is the applied

524 electrical field (in V m⁻¹), $D$ is the dielectric displacement (in C m⁻²), and $t$ is the time (in s).

525 The applied sinusoidal electrical field and the dielectric displacement are respectively defined

526 by

$$E = E_0 \exp(i\omega t), \qquad (8)$$

527 and

$$D = \varepsilon E, \qquad (9)$$

528 where $i$ is the imaginary number ($i^2 = -1$), $\omega$ is the angular frequency of the applied electrical

529 current (in rad s⁻¹; $\omega = 2\pi f$ with $f$ being the frequency in Hz), and $\varepsilon$ is the permittivity of the

530 investigated medium (in F m⁻¹). By using Eqs 6 to 9, we finally obtain for the total current

531 density

$$J_t = (\sigma + i\omega\varepsilon)E, \qquad (10)$$

532 and then for the medium conductivity

$$\sigma^* = \sigma + i\omega\varepsilon. \qquad (11)$$

533

534
## 3.1. Model assumptions

A clay aggregate can be decomposed into different compartments: bulk (electroneutral) water, diffuse layer water, Stern layers and interlayer spaces of particles, which exhibit different electrical conductivities (Leroy & Revil 2009; Okay et al. 2014; Leroy et al. 2017b) (Fig. 6).

Following the works of Revil and co-workers (e.g., Leroy & Revil 2009; Jougnot et al. 2010a; Revil 2012; Okay et al. 2014), the SIP model of Leroy et al. (2017b) assumes that the bulk water and the diffuse layer water surrounding clay particles are continuous at the scale of the laboratory centimetric measurement and hence do not contribute to the polarisation of the particles. According to Leroy et al. (2017b), the bulk and diffuse layer waters only contribute to conductivity at zero frequency (direct current, DC conductivity) influencing clay material in-phase conductivity (real component of clay material complex conductivity). Similarly to the SIP models of Revil and co-workers, the model of Leroy et al. (2017b) considers that the Stern layers around clay particles are discontinuous at the scale of the centimetric laboratory measurement and polarize, influencing clay material in-phase conductivity and quadrature conductivity (imaginary component of clay material complex conductivity). Leroy et al. 2017b assumed that the clay material polarizes at the scale of the clay aggregate, meaning that they considered the polarisation of the Stern layers on the clay particles in contact with each other. Similarly to the works of Revil and co-workers, Leroy et al. (2017b) did not consider the adsorption/desorption kinetics of the ions in the Stern layer as well as their exchange with the ions in the diffuse layer and bulk water because they assumed that these processes are instantaneous compared to the movements of the ions in the EDL and bulk water.

Following the works of, e.g., Dufreche et al. (2001), Bourg et al. (2003), Rotenberg et al. (2005), and Cadene et al. (2006), we consider that some fully hydrated ions (such as Na⁺ ions
and/or Ca²⁺ ions in a lesser extent (Tournassat et al. 2009; Bourg & Sposito 2011; Greathouse et al. 2016)) are mobile in the interlayer space of Mt particle and hence contribute to low-frequency conduction and polarisation in clay particles in contact with each other.

Our model has been developed to extract porosity, electrical formation factor, clay aggregate size distribution, and water and surface conductivities from complex conductivity measurements. It was used to interpret SIP measurements reported in Mendieta et al. (2021) from the measured experimental conditions (Maineult et al. 2024). For instance, prior to the SIP measurements, Mendieta et al. (2021) measured the mass of the dry clay powder, i.e. the solid mass ($\delta_s$) (in g), before mixing it with a NaCl electrolyte of known salt concentration ($C_{NaCl}^0$) (in mol L⁻¹, M). It should be noted that the SIP model presented here can be generalized to any type of electrolyte but the case of a NaCl electrolyte is considered here in order to compare in Maineult et al. (2024) the model to the experimental data reported in Mendieta et al. 2021 on kaolinite, illite and Mt muds. In these experiments, the (total) volume of the sample ($V_t$) (in m³), the temperature ($T_c$) (in °C) ($T_c = T - 273.15$ where T is the absolute temperature in K), and the (total) masses of the sample before and after water evaporation ($\delta_t^0$ and $\delta_t^1$, respectively) (in g), were also recorded. In our model, measured $\delta_s$ and $V_t$ values as well as the measured volume fraction of the main clay mineral in the solid phase ($\xi_c$) (by X-ray diffraction, XRD) are used to compute the volumes of the different components of the clay material intervening in the SIP model (Appendix A). Measured $T_c$ values are used to compute the model parameter values as a function of temperature (Appendix B), and measured $C_{NaCl}^0$, $\delta_t^0$ and $\delta_t^1$ values are used to calculate NaCl concentration in bulk water after evaporation and bulk water chemical composition (Appendix C). Na⁺ and Cl⁻ ion mobilities in bulk water intervening in the SIP model are also computed as a function of NaCl concentration (Appendix B) because of the mutual influence
of cations and anions decreasing their mobility and hence the related electrical conductivity (Leroy *et al.* 2015).

586

## 3.2. Calculation of the EDL properties

We have developed here a new approach to directly compute the electrical potential at the *d*-plane (delimiting the beginning of the diffuse layer from the mineral surface) ($\varphi_d$) (in V, Fig. 4), from the fraction of the counter-charge in the Stern layer ($f_Q$) and the mineral surface charge density ($Q_0$) (in C m⁻²). The mineral surface charge density is computed from ratio of the measured cation exchange capacity (CEC) (in meq g⁻¹) to the specific surface area where cation exchange reactions occur on clay mineral surface (SS) (in m² g⁻¹) (Okay *et al.* 2014).

The SS quantity can be estimated using crystallography (Leroy & Revil 2004). In the case of Na-Mt, it can be approximated by the total specific surface area of the basal surfaces (SS$_b$) (Leroy *et al.* 2015; Leroy *et al.* 2017b). Clay particles exhibit different electrical potentials on their surfaces (Bourg *et al.* 2007; Tournassat *et al.* 2013). Our approach considers a dominating electrical potential at the *d*-plane on the clay surface ($\varphi_d$) and not a distribution of $\varphi_d$ potentials on the clay surface, which will make the modeling considerably more complex.

The advantage of our approach is also that it does not need an electrostatic surface complexation model, that can be very complicated, to estimate $\varphi_d$ from bulk water chemical composition (see for example Tournassat *et al.* 2013).
1910; Chapman 1913; Grahame 1947). This electrical potential is commonly assumed to be equal to the zeta potential inferred from electrokinetic measurements when a shear or slipping plane involving polar solvent flow occurs on the surfaces (von Smoluchowski 1905; von Smoluchowski 1921). Knowing the $\phi_d$ potential value is useful here to compute two processes considered in the SIP model: (i) the effect of the diffuse layer on Stern layer polarisation through the $M$ parameter value ($M \ge 1$) and (ii) the electrical conductivity of the diffuse layer (that will be fully explained in Sections 3.3 and 3.4). According to Lyklema *et al.* (1983), ions in the diffuse layer counterbalance locally ions in the polarized Stern layer to ensure local electroneutrality. This effect decreases the length of the pathway of the migrating ions in the Stern layer and hence it decreases the relaxation time of the polarized Stern layer (Fig. 1, Fig. 7b and Eq. 15).

Ions in the diffuse layer counteracting ions in the Stern layer contributing to polarisation could in principle not contribute also to conduction via electromigration. However, on the contrary to what is shown by Lyklema *et al.* (1983) about the EDL of a single particle, the diffuse layers around clay particles are probably connected and may fill a significant part of the pore space in clay materials when the clay concentration in the sample is large. For instance, clay muds can contain a large mass of dry clay powder per liter of sample. In the SIP experiments of Mendieta *et al.* (2021), clay muds contain about one thousand grams of dry clay powder per liter of sample. This means that the diffuse layers are not necessarily disconnected and hence solely counteracting Stern layer polarisation. For this reason, we consider in the following that the diffuse layer contributes to DC conductivity implying that $M = 1$ theoretically (no effect of diffuse layer on Stern layer polarisation). However, we also consider here a hybrid approach assuming that the diffuse layer can locally (at the particle scale) counter-balance Stern layer polarisation through the computation of $M \ge 1$ and contribute also to conduction via electromigration (at the sample scale).
The following function is minimized to estimate $\varphi_d$ from $f_Q$ and $Q_0$ (Appendix D) (knowing the bulk water chemical composition):

$$ \left| \sqrt{2\overline{\epsilon_w k_B T} 10^3 N_A \sum_{i=1}^{N} C_i^b} \left[ \exp\left(-\frac{q_i \varphi_d}{k_B T}\right) - 1 \right] - Q_0(1-f_Q) \right| = 0, \quad (12) $$

where $\overline{\epsilon_w}$ represents the mean water permittivity (in F m⁻¹) including in the interlayer space of Na-Mt (we consider that $\epsilon_w$ has the same value in the diffuse layer and bulk water, see Appendix B for $\epsilon_w$ values), and $k_B$ is the Boltzmann's constant (~1.381×10⁻²³ J K⁻¹). In Eq. 12, $N$ represents the number of different ion species $i$ in bulk water with ion concentration $C_i^b$ (in mol L⁻¹, M) and charge $q_i = \pm ez_i$ (in C) where “+” and “-” stand for cations and anions, respectively, $e$ is the elementary (or electron) charge (~1.602×10⁻¹⁹ C), $z_i$ is the ion valence (e.g., $z_i = 1$ for sodium or chloride ion and $z_i = 2$ for calcium, magnesium or sulfate ion), and $N_A$ is the Avogadro's number (~6.022×10²³ mol⁻¹).

In the case of Na-Mt, we assume in Eq. 12 the same electrical potential $\varphi_d$ in the interlayer space and diffuse layer on the external clay surface (Dufreche et al. 2001). However, for Na-Mt, the water permittivity value intervening in the calculation of $\varphi_d$ (Eqs 12 or 15) and of the differential capacitance of the diffuse layer $C_d$ (in F m⁻²) (Hunter 1981) to compute $M$ (Eq. 16) could be significantly smaller in the interlayer space than in the diffuse layer on the external clay surface due to the stronger electrical field (Hiemstra & Van Riemsdijk 2006) and the weaker level of hydration (Wander & Clark 2008) in the interlayer space. In our knowledge, the water permittivity value is not accurately known in the interlayer space of Na-Mt and should be between clay solid permittivity value ($\epsilon_s \approx 5\epsilon_0$ where $\epsilon_0$ is vacuum permittivity, see Section 3.3) and bulk water permittivity value ($\epsilon_w \approx 78.3\epsilon_0$ for distilled
water at a temperature of 298.15 K, see Appendix B). For that reason, we consider in the following that the water permittivity value in the interlayer space of Na-Mt is half the value in bulk water, i.e. $ε_{in} = ε_{w} / 2$. In Eqs 12 or 15 and 16, water permittivity is therefore computed using

$$ \overline{\epsilon_w} = \frac{\epsilon_w V_w^d + \epsilon_{in} V_{in}}{V_w^d + V_{in}} = \frac{\epsilon_w}{V_w^d + V_{in}} \left( V_w^d + \frac{V_{in}}{2} \right), \quad (13) $$

where $V_w^d$ is the calculated diffuse layer volume (in m³) and $V_{in}$ is the calculated interlayer space volume when it contains mobile ions such as Na⁺ ions (see Appendix A).

If $Q_0$ is considered homogeneous on the mineral surface, it can be computed as a function of the measured CEC and SS (Okay et al. 2014) using

$$ Q_0 = -\frac{eN_A \text{CEC}}{10^3 \text{SS}}, \quad (14) $$

then when using Eq. 14, Eq. 12 becomes

$$ \left| \sqrt{2\overline{\epsilon_w}k_B T 10^3 N_A \sum_{i=1}^N C_i^b \left[ \exp\left(-\frac{q_i\varphi_d}{k_B T}\right) - 1 \right]} - \frac{eN_A \text{CEC}}{10^3 \text{SS}}(1-f_Q) \right| = 0. \quad (15) $$

The M parameter value (Lyklema et al. 1983) is computed from $f_Q$ and $\varphi_d$ according to

$$ M = 1 + \frac{qQ_{\beta}}{k_B T C_d}, \quad (16) $$

where q is the mean charge of the counter-ions in the Stern layer (in C; q = ±ez with z is the mean valence of the counter-ions). The $Q_β$ quantity in Eq. 16 is determined as a function of $f_Q$ and the surface charge density on the mineral surface ($Q_0$) using (Appendix D)
$$Q_{\beta} = -f_{Q}Q_{0}, \qquad (17)$$

where $Q_0$ can be computed as a function of the measured CEC and SS using Eq. 7. The differential capacitance of the diffuse layer (Eq. 16) is determined from $\phi_d$ (Leroy et al. 2017a) according to

$$C_d = -\frac{\partial Q_d}{\partial \phi_d} = \sqrt{\frac{\overline{\epsilon_w} 10^3 N_A \sum_{i=1}^{N} q_i C_i^b \exp\left(-\frac{q_i \phi_d}{k_B T}\right)}{2k_B T \sqrt{10^3 N_A \sum_{i=1}^{N} C_i^b \left[\exp\left(-\frac{q_i \phi_d}{k_B T}\right) - 1\right]}}, \qquad (18)$$

and for a symmetric electrolyte of valence z, Eq. 18 reduces to

$$C_d = \overline{\epsilon_w} \kappa \cosh\left(\frac{ze\phi_d}{2k_B T}\right), \qquad (19)$$

where $\kappa = \chi_D^{-1}$ is the inverse of the Debye length (in m⁻¹) (see Eq. A5 (Lyklema et al. 1983)), and then the M parameter value is computed from $f_Q$ and $Q_0$ or CEC and SS according to Eqs 12 to 18 or 19. The case M = 1 corresponds to the Schwarz’s model (Schwarz 1962) with no effect of the diffuse layer on Stern layer polarisation. Leroy et al. (2017b) found that M ≈ 4 for the basal surface of Na-Mt, meaning that, in this case, ions in the diffuse layer significantly decrease the length of the pathways and then the relaxation time of ions in the polarized Stern layer according to the generalized Schwarz model of Lyklema et al. (1983).

### 3.3. Aggregate conductivity due to surface conductivity

A mineral surface in contact with a polar solvent like water and submitted to a low-frequency electrical field expresses a surface conductivity response due to migrating ions in the EDL (O’Konski 1960). Surface conductivity (or surface conductance) is defined as a surface
quantity in Siemens and not as a volume quantity in Siemens per meter. However, we will define it in the following as a volume quantity because of its widespread use as a volume quantity like water conductivity in the geophysical community. Like our water conductivity model, our “surface” conductivity model is based on Ampere’s law (Eq. 5) (e.g., Schwarz 1962; de Lima & Sharma 1992; Chelidze & Gueguen 1999; Leroy et al. 2008). The resulting surface conductivity is written as (Eq. 11)

$$ \sigma_s^*(\omega) = \sigma_s(\omega) + i\omega\varepsilon_s, \qquad (20) $$

where the first term (a complex quantity when considering EDL polarisation) arises from conduction current and the second term arises from displacement current in the presence of an applied sinusoidal electrical field with $\varepsilon_s$ being the solid dielectric constant (in F m⁻¹). In our model, we assume that clay aggregates control the surface conductivity and induced polarisation response of the material. Therefore, the symbol $\sigma_s^*$ reported in Eq. 20 for the clay aggregate conductivity corresponds to the symbol used by most IP practitioners for the surface conductivity of the whole material. In addition, in Eq. 20, we do not consider anisotropy in the calculation of the surface conductivity. Therefore, our model is particularly adapted to unconsolidated clay materials like uncompacted clay muds.

Clay particles assemble themselves to make clay aggregate, which can be considered a spherical object with a surface area made of the surface area of the particles constituting the aggregate at first approximation. Leroy et al. (2017b) developed petrophysical relationships to compute $\sigma_s$ in Eq. 20 from the Mt aggregate microstructural and electrochemical properties. They considered that the surface conductivity of a Mt aggregate is similar to the surface conductivity of an equivalent spherical particle with an amplified surface area $S$ (in m²) compared to its surface area if it is considered perfectly smooth and non-porous. In the model of Leroy et al. (2017b), $S$ is related to the specific surface area of all the polarising clay
704 platelets inside the aggregate (Fig. 6a). In addition, the SIP model of Leroy *et al.* (2017b) uses
705 the generalized Schwarz model developed by Lyklema *et al.* (1983) to simulate the effect of
706 the diffuse layer on the polarisation of the Stern layer surrounding clay particle (Fig. 1a).

707 It is actually very difficult and time-consuming to numerically compute the induced
708 polarisation response of interacting clay particles because of (i) their high aspect ratio, (ii)
709 heterogeneities in their surface electrical properties, and (iii) the physical-chemical
710 mechanisms involved in the interaction (interacting diffuse and Stern layers) (Tournassat et
711 al. 2013; Tournassat et al. 2015; Bücker et al. 2019). For this reason, we have to make the
712 assumption that the polarisation of the Stern layers surrounding interacting clay particles is
713 equivalent to the polarisation of a porous clay aggregate with an effective diameter
714 surrounded by a Stern layer (for more information read Leroy et al. 2017b). In other words, in
715 our model, the Stern layers of the interacting clay particles are assumed continuous at the
716 scale of the aggregate (see also discussion in Jougnot et al. 2010a).

717 It should be noted that, according to some molecular dynamic (MD) simulations, most Na⁺
718 and Ca²⁺ counter-ions in the Stern layer on the Mt basal surface may be adsorbed beyond the
719 first hydration layer plane as outer-sphere surface complexes, keeping their hydration shell
720 (e.g., Tournassat et al. 2009; Bourg & Sposito 2011). These MD simulations show that these
721 counter-ions may behave in terms of ion diffusivity like ions in the diffuse layer. Therefore,
722 according to these MD simulations, we can expect that, for a single clay particle, Stern layer
723 polarisation is not necessarily different from diffuse layer polarisation on the basal surface of
724 Mt.

725 We can attribute diffuse layer polarisation to a volume diffusion mechanism involving
726 migrating ions in the diffuse layer and ions clouds in bulk water (Fig. 1b) and Stern layer
727 polarisation to a surface diffusion mechanism associated with migrating ions on particle
surface (Fig. 1a) (Grosse & Delgado 2010). Delgado et al. (1998) showed that the
contribution of the volume diffusion mechanism to the measured dielectric dispersion of
polymer latex particles in the kHz to MHz frequency range decreases compared to the
contribution of the surface diffusion mechanism because the space available for volume
diffusion in bulk water decreases faster than the space available for surface diffusion when the
particle concentration in solution increases (here particle volume fraction ranges from 3 to
16%). Concentrated colloidal suspensions like clay muds have solid volume fractions
typically above 10% (Richter & Sevick-Muraca 2000). Our model is adapted to saturated clay
materials presenting typically a high concentration of clay particles with a high fraction of
counter-charge in the Stern layer ($f_Q$). For this reason, we believe that the polarisation
models of Schwarz (1962) and Lyklema et al. (1983) considering surface diffusion is suitable
to simulate EDL polarisation of clay materials.

According to Leroy *et al.* (2017b), the low-frequency surface conductivity of clay aggregates
is written as

$$ \sigma_s^* = \frac{2}{3} \phi_a \beta_s f_Q Q_v \sum_{k=1}^{L} f(d_k) \frac{i \omega \tau_k}{1 + i \omega \tau_k} + i \omega \alpha \rho_s \epsilon_0, \quad (21) $$

where $\phi_a$ is the porosity of the aggregate (dimensionless, between 0 and 1), $\beta_s$ is the mean
ion mobility of the counter-ions moving along the particle surface contributing to low-
frequency polarisation (in $m^2 s^{-1} V^{-1}$), and $Q_v$ is the excess of counter-charge of the EDL per
unit pore volume of the aggregate (in $C m^{-3}$). In Eq. 21, $f(d_k), k=1...L$, is the discretized
distribution of the effective sizes of the aggregates with $d_k$ being the effective diameter of the
$k^{th}$ aggregate (in m) and $L$ being the number of different effective diameters of the aggregates
in the model. In Eq. 21 also, $\tau_k$ is the relaxation time of the polarized Stern layers within the
$k^{th}$ aggregate (in s), $\alpha = 0.00191 m^3 kg^{-1}$ is an empirical value that was used by Olhoeft
(1981) to adjust solid permittivity spectra, $\rho_s$ is the solid volumetric mass density (in kg m⁻³), and $\varepsilon_0 \cong 8.854 \times 10^{-12}$ F m⁻¹ is the vacuum permittivity. According to the crystallographic unit cell, $\rho_s \cong 2700$ kg m⁻³ for clays (~2600, 2700, and 2800 kg m⁻³ for kaolinite, smectite and illite, respectively (Leroy & Revil 2009); ~2710 and 2840 kg m⁻³ for Kunipia-G Na-Mt (Massat et al. 2016) and MX80 bentonite – mostly made of Na-Mt (Tournassat & Appelo 2011)). Using $\varepsilon_s \cong \alpha \rho_s \varepsilon_0$ (Eqs 20 and 21), the $\varepsilon_s \cong 4.57\varepsilon_0$ value obtained accordingly is close to the $\varepsilon_s \cong 5.5\varepsilon_0$ value reported by Robinson (2004) for Mt using time domain reflectometry.

Determining $\rho_s$ by crystallography, i.e. dividing the mass of the solid by the estimated related volume, is a challenging task for clays because of the difficulty to accurately measure the effective thickness of a clay layer $t$. Tournassat et al. (2016b) used molecular dynamic simulations to estimate $t$. They found $\rho_s = 2930 \pm 100$ kg m⁻³ for swelling clays such as Mt, which is significantly higher than the Mt $\rho_s$ value estimated by pycnometry, ~2700 kg m⁻³ that can be representative of the volumetric mass density value of the solid part of a non-swelling clay such as illite (value down to 2600 kg m⁻³).

In our model $\phi_a$ (Eq. 21) is not necessarily the sample porosity $\phi$ because we assume that some types of clay aggregates like illite and Mt aggregates contain only diffuse layer and interlayer space waters (case of Na-Mt) and the sample can also contain bulk water. The word “sample” is referred here to the investigated porous medium at the scale of the representative elementary volume (REV) where the measured electrical properties are considered homogeneous (Revil & Leroy 2004).

Aggregate porosity $\phi_a$ is calculated from the computed volumes of the different components of the aggregate:
$$ \phi_a = \frac{V_S + V_w^d + V_w^b}{V_{sc} + V_S + V_w^d + V_w^b}, \qquad (22) $$

772 for kaolinite, and

$$ \phi_a = \frac{V_S + V_w^d}{V_{sc} + V_{in} + V_S + V_w^d}, \qquad (23) $$

773 for illite, and

$$ \phi_a = \frac{V_{in} + V_S + V_w^d}{V_{sc} + V_{in} + V_S + V_w^d}, \qquad (24) $$

774 for Mt such as Na-Mt, where $V_w^b$, $V_S$, and $V_{sc}$ are the computed volumes of the bulk water (in m³), Stern layer, and solid clay layers, respectively (Appendix A). Kaolinite aggregates are assumed to contain bulk and diffuse layer waters (Eq. 22). Illite and Mt aggregates are assumed to contain only diffuse layer and interlayer space waters (case of Na-Mt) because of their higher specific surface areas and surface charges and smaller pores than kaolinite aggregates (Leroy & Revil 2009; Okay et al. 2014; Leroy et al. 2015) (Eqs 23 and 24). In Eq. 23, we assume that $K^+$ ions migrate and diffuse very slowly along the surface of the interlayer space of illite compared to in bulk water (Ruiz Pestana et al. 2016) and hence that they do not contribute to low-frequency polarisation. We also assume that they do not contribute to low-frequency conduction. In Eq. 24, we assume that $Na^+$ ions migrate and diffuse significantly along the surface of the interlayer space of Na-Mt compared to in bulk water (Bourg et al. 2003; Zhang et al. 2014) and hence that they contribute to low-frequency polarisation (Rotenberg et al. 2005; Cadene et al. 2006). We also assume that the part of $Na^+$ ions in the interlayer space not contributing to low-frequency polarisation contributes to low-frequency conduction. Mt particles containing immobile or very slow ions in the interlayer space can be considered similar to illite particles.
790 Sample porosity can be computed using

$$ \phi = \frac{V_{in} + V_{w}^{d} + V_{w}^{b}}{V_{t}}, \qquad (25) $$

791 where $V_t$ is the measured total volume of the sample during laboratory SIP measurements.

792 In Eq. 21, the $Q_v$ quantity is given by

$$ Q_v = \rho_s \left( \frac{1 - \phi_a}{\phi_a} \right) e N_A \text{ CEC.} \qquad (26) $$

793 Eq. 21 uses a parallel mixing formula for the modelling of the surface conductivity ($\sigma_s$) of the clay aggregates with different sizes, i.e., to compute $\sigma_s$, the surface conductivity response of each aggregate of diameter $d_k$ is weighted by its volume relative to the total volume of the aggregates (Lesmes & Morgan 2001). As stated by, e.g., Lesmes & Morgan (2001), the parallel mixing formula gives an upper bound for the computed complex surface conductivity of different sized particles. The geometric mean mixing formula, which arises from random networks consisting of both parallel and serial connections (Madden 1976), would have been more realistic to compute $\sigma_s$ in our model because it gives an average value for the computed complex surface conductivity of different sized particles. However, the parameter optimization procedure to obtain the relaxation time distribution intervening in the SIP model from the SIP measurement is the Debye decomposition approach here (for more details on the optimization procedure, see Maineult et al. 2024). The parallel mixing formula gives for $\sigma_s$ an Eq. equivalent to the Eq. inferred from the Debye decomposition approach (Lesmes & Morgan 2001). This is the reason why our model uses the parallel mixing formula for $\sigma_s$ similarly to Lesmes & Morgan (2001).
The relaxation time $\tau_k$ (Eq. 21) is the time necessary for the counter-ions in the Stern layers within the $k^{th}$ aggregate to diffuse back to their equilibrium position once the Stern layers are polarized due to ion electromigration (Lyklema et al. 1983; de Lima & Sharma 1992; Chelidze & Gueguen 1999) (see Fig. 1a). It is related to the effective diameter and surface ion mobility using

$$ \tau_k = \frac{d_k^2 |q|}{8k_B T \beta_s M}, \qquad (27) $$

Eq. 27 shows that, for each aggregate of effective size $d_k$, the generalized Schwarz model of Lyklema et al. (1983) considers one relaxation time $\tau_k$ (see Fig. 7). Therefore, our model simulates a Debye type response for the increase of the real and imaginary conductivity response due to the low-frequency polarisation of the aggregate. Our modelling approach is based on (i) fitting the complex conductivity spectra to obtain the relaxation time distribution $f(\tau_k)$, $k=1...L$ and the aggregate size distribution $f(d_k)$, $k=1...L$ (for more details on the optimization procedure see Leroy et al. 2017b and Maineult et al. 2024) and (ii) computing the aggregate effective size $d_k$ from the obtained relaxation time $\tau_k$ and the optimized $\beta_s$ value obtained by successfully comparing measured to predicted complex conductivity spectra using

$$ d_k = \sqrt{8 \frac{k_B T}{|q|} \beta_s M \tau_k}, \qquad (28) $$

meaning that for given values of $\beta_s$ and $\tau_k$, increasing $M$ increases the calculated effective diameter of the $k^{th}$ aggregate.
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>**Fig. 7.** Sketch of a Mt aggregate with an effective diameter $d_{eff}$ showing the clay particles interacting with each other and the electrical conductivities of the different components (not in scale) (a). Sketch of the equivalent Stern layer polarisation model of Leroy *et al.* (2017b) around a negatively charged spherical particle in contact with a monovalent electrolyte (not in scale) (b) (modified from Leroy *et al.* 2017b).</figcaption></figure>

In Section 3.4, we will explain that the diffuse layer of clay particles, especially illite and Na-Mt particles, contributes to the volume conductivity of the clay material and not to its surface conductivity because the diffuse layer thickness can be larger than the clay particle thickness (see Section 2.1) (O'Konski 1960; Dufreche *et al.* 2001). However, at high ionic strength *I* (typically ≥ 1 M), the diffuse layer is highly compressed. For instance, the Debye length characterizing the diffuse layer thickness is ~3 Å at *T* = 298.15 K and ionic strength *I* = 1 M (see Eqs A5 and B16). In that case, the diffuse layer thickness can be smaller than the thickness of the clay layer and the assumption of the diffuse layer contributing to the volume conductivity of the clay material does not necessarily hold because the diffuse layer can contribute to the surface conductivity of the particle (O'Konski 1960). For larger clay particles such as kaolinite particles, the diffuse layer can be thinner than the particle thickness for smaller ionic strengths compared to illite and Mt particles and hence contribute to surface conductivity according to O'Konski (1960). For this reason, in this condition, Eq. 21 becomes (Leroy *et al.* 2008; Leroy *et al.* 2017b) (Appendix D):
$$ \sigma_s^* = \frac{2}{3} \phi_a \sum_{k=1}^{N_{dk}} f(d_k) \left[ \beta_s f_Q \frac{i\omega\tau_k}{1+i\omega\tau_k} + \tilde{B}_d (1-f_Q) \right] Q_v + i\omega\alpha\rho_s\epsilon_0, \qquad (29) $$

where $\tilde{B}_d$ is the ion mobility in the diffuse layer including electro-osmosis (in m² s⁻¹ V⁻¹)
(Revil & Glover 1997), which can be calculated according to

$$ \tilde{B}_d = \frac{1}{\mathcal{G}} \left( \beta_i^b + \frac{2\overline{\varepsilon_w k_B T}}{\eta_w |q_i|} \right), \qquad (30) $$

where $\mathcal{G}$ ($\mathcal{G} \ge 1$) is a weighting coefficient that can be considered equal to the ratio of the surface area of the clay particles within the clay aggregate to the surface area of the equivalent non-porous spherical particle (Appendix D), $\beta_i^b$ is the ion mobility in the diffuse layer due to electro-migration (in m² s⁻¹ V⁻¹) (we assume $\beta_i^d = \beta_i^b$) and $q_i$ is the charge of the dominating ion in the diffuse layer (in C). For a 1:1 electrolyte such as NaCl electrolyte at a temperature of 298.15 K, the second term of Eq. 30 describing electro-osmotic effect yields $m_i^d \approx 3.78 \times 10^{-8}$ m² s⁻¹ V⁻¹ using $\overline{\varepsilon_w} = 78.3\varepsilon_0$, which is in the same order of magnitude than the ion mobility values in bulk and distilled water due to electro-migration ($\beta_{Na^+}^b = 5.17 \times 10^{-8}$ m² s⁻¹ V⁻¹ and $\beta_{Cl^-}^b = 7.89 \times 10^{-8}$ m² s⁻¹ V⁻¹), meaning that electro-osmosis cannot be neglected for the computation of the surface conductivity of the diffuse layer. It should be noted that it may be complicated to distinguish between Stern and diffuse layer contributions to surface conductivity especially at high ionic strengths due to diffuse layer compression and to the fact that most counter-ions adsorbed in the Stern layer on the basal clay surface may be adsorbed as outer-sphere surface complexes.

862

863
## 3.4. Water conductivity

We consider that the bulk water and the diffuse layer water are continuous at the scale of the SIP laboratory measurement and hence that they only contribute to the in-phase conductivity of the clay material (see Section 3.1). The self-consistent model developed by Hashin & Shtrikman (1962) is used here to compute clay water conductivity including the diffuse layer contribution in the case of the thick diffuse layer assumption. It is written as

$$ \sigma_w^* = \sigma_w^b * \left[ 1 + \frac{3\Theta_w^d (\sigma_w^d * - \sigma_w^b * )}{3\sigma_w^b * + (1 - \Theta_w^d) (\sigma_w^d * - \sigma_w^b * )} \right], \quad (31) $$

where $\sigma_w^b *$ is the bulk water conductivity (in S m⁻¹) (including displacement current), $\sigma_w^d *$ is the diffuse layer water conductivity (including displacement current), and $\Theta_w^d$ is the volume fraction of diffuse layer water over diffuse layer and bulk waters. Hashin & Shtrikman (1962) presented two models as the upper and lower bounds to calculate the effective property that depends on which one of the two components is seen as the matrix or as the inclusion. In Eq. 31 the bulk water is the matrix and the diffuse layer water is the inclusion during the calculation. This equation is the lower bound obtained by Hashin & Shtrikman (1962). Eq. 31 has been developed under the condition $|\sigma_w^d *| > |\sigma_w^b *|$ that is usually satisfied because there is an excess of mobile counter-ions in the diffuse layer compared to in bulk water to compensate the surface charge not compensated by counter-ions in the Stern layer (Bourg & Sposito 2011). In our model, the $\Theta_w^d$ quantity in Eq. 31 is determined from the computed volumes of the different components of the aggregate (Appendix A). It is given by

$$ \Theta_w^d = \frac{V_w^d}{V_w^b + V_w^d}, \quad (32) $$

for kaolinite and illite, and
$$ \Theta_w^d = \frac{V_w^d + V_{in}}{V_w^b + V_w^d + V_{in}}, \qquad (33) $$

for Mt such as Na-Mt, where the interlayer space of Na-Mt is considered contributing of the volume conductivity of the clay material similarly to the diffuse layer. Despite the fact that the thickness of the interlayer space of Na-Mt is on the same order of magnitude than the thickness of a TOT layer (~ 1 nm), we do not consider here that the part of the interlayer space of Na-Mt that is not polarizing contributes to the surface conductivity of Na-Mt. Indeed, it would be too complicated to consider at this stage of our model that the interlayer space would contribute to the surface conductivity of the clay particles while the diffuse layer would contribute to the water conductivity. In addition, our diffuse layer surface conductivity model is more accurate for low $\phi_d$ potentials (see Appendix D) whereas our diffuse layer volume conductivity model is adapted to any $\phi_d$ potentials when the electrolyte is symmetric (Leroy et al. 2015) (see discussion related to Eq. 38).

In Eq. 31, the conductivity of the bulk water is calculated from ion concentration in bulk water ($C_i^b$) considering conduction current from ion electro-migration and displacement current (Leroy et al. 2008)

$$ \sigma_w^b* = e10^3 N_A \sum_{i=1}^{N} z_i \beta_i^b C_i^b + i\omega\epsilon_w. \qquad (34) $$

In our model, simulated bulk water conductivity depends on temperature and salinity. Indeed, in Eq. 34, computed ion mobilities in bulk water ($\beta_i^b$) consider temperature and salinity (here NaCl concentration) effects, and bulk water permittivity ($\epsilon_w$) considers temperature effect (Appendix B). Computed ion concentration in bulk water ($C_i^b$) also considers temperature effect via the simulation of the temperature dependence of the dissociation constant of water
molecules K<sub>w</sub> (Eq. B2). For other types of ions than Na<sup>+</sup> and Cl<sup>-</sup> ions (here H<sup>+</sup>, OH<sup>-</sup>, and Ca<sup>2+</sup> and SO<sub>4</sub><sup>2-</sup> ions), β<sub>i</sub><sup>b</sup> (298) is also computed as a function of the measured ion self-diffusion coefficient in distilled water at a temperature of 298.15 K D&#773;<sub>i</sub><sup>b</sup> (298) (in m<sup>2</sup> s<sup>-1</sup>) reported in the phreeqc.dat database of the geochemical software PhreeqC (Parkhurst & Appelo 2013).

Similarly to bulk water conductivity, the volume conductivity of the diffuse layer water is determined from

$$ \sigma_w^d * = \frac{e l 10^3 N_A}{\chi_d} \int_{x=0}^{x=\chi_d} \sum_{i=1}^{N} z_i B_i^d(x) C_i^b \exp\left[-\frac{q_i \varphi(x)}{k_B T}\right] dx + i \omega \overline{\epsilon_w}, \quad (35) $$

$$ B_i^d(x) = \beta_i^b \pm \beta_{eo}^d(x), \quad (36) $$

$$ \beta_{eo}^d(x) = \frac{\epsilon_w}{\eta_w} [\varphi(x) - \varphi_d], \quad (37) $$

$$ \varphi(x) = \frac{4k_B T}{ez} \tanh^{-1} \left[ \tanh \left( \frac{ez\varphi_d}{4k_B T} \right) \exp(-\kappa x) \right], \quad (38) $$

where x is the distance from the beginning of the diffuse layer on the mineral side (in m), χ<sub>d</sub> = 2χ<sub>D</sub> (Leroy et al. 2015) is the diffuse layer thickness (with χ<sub>D</sub> being the Debye length, Eq. A5) (in m), B<sub>i</sub><sup>d</sup> is the ion mobility in the diffuse layer considering electro-osmosis (in m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>), and κ = χ<sub>D</sub><sup>-1</sup> (Leroy et al. 2015; Leroy et al. 2017b). Liquid water dynamic viscosity η<sub>w</sub> (in Pa s<sup>-1</sup>) intervening in the calculation of the effect of electro-osmosis on surface conductivity (Eqs 36 and 37) is calculated as a function of temperature (Appendix B). In Eq. 36, “+” and “-” stand for cations and anions, respectively. Eqs 35-38 show interesting features for the conductivity calculations compared to the surface conductivity model based
on Bikerman's equation (Eqs D12-D14). Eq. 36 shows that electro-osmosis associated with cations increases conductivity whereas it shows that electro-osmosis associated with anions decreases conductivity because the solvent flow due to the excess of cationic charge is in the direction of the electrical field and the solvent flow due to the excess of anionic charge is in the direction opposed to the electrical field (Leroy et al. 2015). Eqs 37-38 also show that electro-osmotic effect on conductivity is larger when the $\varphi_d$ potential is larger hence when ionic strength decreases at fixed pH or when pH is further away from IEP (pH value where the zeta potential is zero) at fixed ionic strength. Bikerman's Eq. considers that both cations and anions increase conductivity due to electro-osmosis with a value independent on the $\varphi_d$ potential (Eqs D12 to D14).

When the electrolyte is not symmetric, the following equation, based on the Debye-Hückel approximation (Appendix D), is used to calculate the electrical potential distribution in the diffuse layer instead of Eq. 38:

$$ \varphi(x) = \varphi_d \exp(-\kappa x). \qquad (39) $$

Leroy et al. (2017b) computed the diffuse layer volume conductivity by calculating an average (or mean) electrical potential in the diffuse layer $\varphi_m$ (see Appendix E). In that case, Eqs 35-38 become

$$ \sigma_w^d = e10^3 N_A \sum_{i=1}^{N} z_i B_i^d C_i^b \exp\left(-\frac{q_i \varphi_m}{k_B T}\right) + i\omega\overline{\epsilon_w}, \qquad (40) $$

$$ B_i^d = \beta_i^b \pm \beta_{eo}^d, \qquad (41) $$

$$ \beta_{eo}^d = \frac{\epsilon_w}{\eta_w}(\varphi_m - \varphi_d), \qquad (42) $$
where $\varphi_m$ is computed according to Eq. E8 for kaolinite and illite and Eq. E7 for Mt with mobile ions in the interlayer space such as Na-Mt. Compared to Eqs 35-38, Eqs 40-42 and E8 or E7 have the advantage of not needing to know the electrical potential distribution in the diffuse layer. But, these Eqs can be more approximate because they do not consider the dependence on the distance from the beginning of the diffuse layer, which can lead to less accurate results for the electrical potential distribution, especially at high ionic strengths when the diffuse layer is highly compressed.

940

### 3.5. Clay material conductivity

In our model, minerals other than the main clay mineral are not considered conductive compared to water and the main clay mineral. The effect of weakly conductive minerals on the modeled background conductivity ($\sigma_0$*), which does not consider surface conductivity of the particles, is considered here using the self-consistent model developed by Hashin & Shtrikman (1962)

where

$$ \sigma_0^* = \sigma_w^* \left[ 1 - \frac{3(1-c)}{(3-c)} \right], \qquad (43) $$

$$ c = \frac{V_w^d + V_w^b}{V_{so} + V_S + V_w^d + V_w^b} \qquad (44) $$

for kaolinite, and

$$ c = \frac{V_w^d + V_w^b}{V_{so} + V_{in} + V_S + V_w^d + V_w^b} \qquad (45) $$

for illite, and
$$c = \frac{V_{in} + V_{w}^{d} + V_{w}^{b}}{V_{so} + V_{in} + V_{s} + V_{w}^{d} + V_{w}^{b}}, \qquad (46)$$

for Mt such as Na-Mt, where c is the volume fraction of conductive water and $V_{so}$ (in m³) is the volume of weakly conductive minerals (compared to water and the main clay mineral; the subscript “so” means “solid other”) that are not considered conductive. The Stern layer is considered not conductive too here.

The differential effective medium (DEM) theory (Bruggeman 1935; Hanai 1968; Sen 1984) is used to compute the complex conductivity of the clay material after calculating the complex conductivity of the clay particles and of the background medium. In the DEM theory, clay particles are embedded in a self-similar way in the background medium (Sen et al. 1981). First, a particle is added to the background medium and the influence of the particle upon the conductivity of the medium is calculated. Then, the particle/background medium mixture is added to another particle in the background medium and the conductivity of the new particle/background medium mixture is calculated, and so on following an iterative procedure until the desired porosity is reached (see Fig. 1 of Sen et al. 1981). The DEM theory assumes that the current crosses all the pores in the investigated sample, i.e. that they are all interconnected electrically speaking. It works well to describe the electrical properties (conductivity and permittivity) of porous media with small and isolated contact regions between the particles. It also simulates Maxwell-Wagner polarisation effect (Leroy & Revil 2009).

The DEM model uses the computed background conductivity ($\sigma_0$*), aggregate surface conductivity ($\sigma_s$*) and volume fraction of the solid part of the main clay mineral in the investigated sample ($\Omega_s = V_{sc}/V_t$) ($V_{sc}$ in m³ is the volume of the solid part of the main clay mineral and $V_t$ in m³ is the sample volume, see Appendix A) to compute clay material
complex conductivity. The cementation exponent *m* characterizing the shape of the aggregates should also be considered in the DEM model. For a sphere, *m* = 1.5 (Sen *et al.* 1981), and the *m* value increases when the aspect ratio (width to height ratio) of the particles or grains of the material with a *σ*<sub>s</sub> * conductivity increases. For unconsolidated sands packed in the laboratory, *m* = 1.3, consolidated sandstones, 1.8 ≤ *m* ≤ 2, and 1.2 ≤ *m* ≤ 4 for most minerals with goethite and clay particles exhibiting the highest *m* values (Mendelson & Cohen 1982).

In our model, DEM computations are presented as follows. The initial electrical conductivity of the clay material is the background conductivity, i.e. σ* = σ₀* initially. Inclusions are clay aggregates with a given shape that have a surface conductivity σs* and volume fraction dΩs and that are randomly oriented in the medium. The depolarisation factor of inclusions, L (0 ≤ L ≤ 1), which is necessary for the calculation of σ* according to the DEM theory (Mendelson & Cohen 1982), is calculated from

$$L = \frac{3 + \sqrt{9 + 36m^2 - 60m}}{6m}, \qquad (47)$$

and, finally, clay material conductivity is calculated iteratively using

$$d\sigma^* = \frac{\sigma^* (\sigma_s^* - \sigma_0^*) [(1+3L)\sigma_s^* + (5-3L)\sigma_0^*]}{3 [L\sigma_s^* + (1-L)\sigma_0^*] [(1-L)\sigma_s^* + (1+L)\sigma_0^*]} \frac{d\Omega_s}{1-\Omega_s}, \qquad (48)$$

$$\sigma^* = \sigma^* + d\sigma^*, \qquad (49)$$

$$\Omega_s = \Omega_s + d\Omega_s, \qquad (50)$$

where Ωs = 0 (and σ* = σ₀*) initially and dΩs = Vsc / (VtNs) with Ns being the number of solid inclusions to reach the volume fraction of the solid part of the main mineral in the sample. Compared to the integrated form of the DEM model, the iterative form shown here
gives more accurate complex conductivity predictions when the inclusion eccentricity increases compared to spherical inclusions (Sen et al. 1981; Mendelson & Cohen 1982; Sen 1984).

Our model allows determining the effective diameter of the kth clay aggregate, $d_k$ (Eq. 28).

The following equation (Mendelson & Cohen 1982) can be solved numerically to determine aggregate eccentricity ($e_c$) from the depolarisation factor ($L$) calculation

$$ \left| L - \frac{1+e_c^2}{e_c^3} (e_c - \tan^{-1} e_c) \right| = 0, \qquad (51) $$

and the height of the kth aggregate is then given by

$$ h_k = \sqrt{\frac{d_k^2}{1+e_c^2}}. \qquad (52) $$

## 4 CONCLUSIONS

We have developed a new mechanistic model to understand and predict laboratory and field spectral induced polarisation (SIP) measurements on clay materials including kaolinite, illite and montmorillonite minerals from their microstructural and electrochemical properties.

Water and surface conductivities, electrical formation factor, cementation exponent, sample porosity, and clay aggregate effective size distribution can be extracted from the comparison of the model to experimental data.

Our model simulates the complex conductivity (for frequencies typically from 1 mHz to tens of kHz) of the different components of the clay material. It considers the polarisation of the Stern layers around clay particles forming aggregates and contributing to surface conductivity as well as montmorillonite interlayer space conduction and polarisation and Maxwell-Wagner
1007 polarisation. The diffuse layer is assumed to contribute to the water conductivity or to the
1008 surface conductivity of the particle depending notably if the diffuse layer is thicker or not to
1009 the clay particle. Our model assumes one main clay mineral contributing to conduction and
1010 polarisation in the low-frequency range. It also uses parameter values that can be measured in
1011 the laboratory independently to the SIP measurements like temperature (for temperatures
1012 between 2 and 87°C where water is liquid), cation exchange capacity (CEC), clay powder
1013 mass, initial salinity, and sample volume. The parameters to be adjusted are the fraction of the
1014 counter-charge and the ion effective mobility in the Stern layer, the number of stacked solid
1015 layers per particle for montmorillonite and the cementation exponent characterizing the shape
1016 of the aggregates.

1017 The comparison of the model predictions to the experimental data including the optimization
1018 procedure and the related discussion will be presented in a companion paper. Our model can
1019 test the validity of some petrophysical models developed at larger scale that are commonly
1020 used to interpret laboratory and field SIP measurements. In addition, its predictions can be
1021 checked by purely numerical SIP models developed at the scale of the clay particle and
1022 constrained by high-resolution characterization measurements.

1023

1024 ACKNOWLEDGMENTS

1025 This study was carried out within the framework of the ANR EXCITING and IMAGE
1026 projects (grants ANR-17-CE06-0012 and ANR-21-CE04-0013, respectively, funded by the
1027 French National Research Agency). Philippe Leroy warmly thanks Francis Claret for his
1028 support and Christophe Tournassat for the fruitful discussions. The work of Philippe Leroy
1029 was also supported by the EURAD work package DONUT (EC grant agreement No. 847593).
1030 We thank the editor, Léa Levy, and the two reviewers, Viacheslav Emelianov and Youzheng

1031 Qi, for providing thorough and constructive reviews of our manuscript.

1032 DATA AVAILABILITY STATEMENT

1033 No data were used for this theoretical paper.

1034 REFERENCES

1035 Aizebeokhai, A. P. & Oyeyemi, K. D., 2014. The use of the multiple-gradient array for geoelectrical resistivity and induced polarization imaging, *Journal of Applied Geophysics*, **111**, 364-376, doi:10.1016/j.jappgeo.2014.10.023.

1038 Ali, E. S., Askalany, A. A., Harby, K., Diab, M. R., Hussein, B. R. M. & Alsaman, A. S., 2021. Experimental adsorption water desalination system utilizing activated clay for low grade heat source applications, *Journal of Energy Storage*, **43**, 103219, doi:10.1016/j.est.2021.103219.

1042 Avena, M. J. & De Pauli, C. P., 1998. Proton adsorption and electrokinetics of an Argentinean montmorillonite, *Journal of Colloid and Interface Science*, **202**, 195-204, doi:10.1006/jcis.1998.5402.

1045 Azua-Bustos, A., Fairén, A. G., Silva, C. G., Carrizo, D., Fernández-Martínez, M. Á., Arenas-Fajardo, C., Fernández-Sampedro, M., Gil-Lozano, C., Sánchez-García, L., Ascaso, C., Wierzchos, J. & Rampe, E. B., 2020. Inhabited subsurface wet smectites in the hyperarid core of the Atacama Desert as an analog for the search for life on Mars, *Sci Rep*, **10**, doi:10.1038/s41598-020-76302-z.

1050 Bandura, A. V. & Lvov, S. N., 2006. The Ionization Constant of Water over Wide Ranges of Temperature and Density, *Journal of Physical and Chemical Reference Data*, **35**, 15-30, doi:10.1063/1.1928231.

1053 Bernard, O., Kunz, W., Turq, P. & Blum, L., 1992. Self-Diffusion in Electrolyte-Solutions Using the Mean Spherical Approximation, *Journal of Physical Chemistry*, **96**, 398-403, doi:10.1021/J100180a074.

1056 Bikerman, J. J., 1940. Electrokinetic equations and surface conductance. A survey of the diffuse double layer theory of colloidal solutions *Transactions of the Faraday Society*, **35**, 154-160, doi:10.1039/TF9403500154.

1059 Binley, A., Hubbard, S. S., Huisman, J. A., Revil, A., Robinson, D. A., Singha, K. & Slater, L. D., 2015. The emergence of hydrogeophysics for improved understanding of subsurface processes over multiple scales, *Water Resources Research*, **51**, 3837-3866, doi: 10.1002/2015WR017016.

1063 Binley, A. & Kemna, A., 2005. DC resistivity and induced polarization methods. in *Hydrogeophysics*, pp. 129-156, ed. Hubbard, Y. R. a. S. S. Springer, New York,

1065 Binley, A. & Slater, L., 2020. *Resistivity and Induced Polarization: Theory and Applications to the Near-Surface Earth*, edn, Vol., Cambridge University Press, doi:10.1017/9781108685955.

1068 Bø Hunvik, K. W., Loch, P., Wallacher, D., Kirch, A., Cavalcanti, L. P., Rieß, M., Daab, M., Josvanger, V., Grätz, S., Yokaichiya, F., Knudsen, K. D., Rodrigues Miranda, C., Breu, J. & Fossum, J. O., 2021. CO2 Adsorption Enhanced by Tuning the Layer
1071 Charge in a Clay Mineral, Langmuir, 37, 14491-14499,
1072 doi:10.1021/acs.langmuir.1c02467.

1073 Bourg, I. C., Bourg, A. C. M. & Sposito, G., 2003. Modeling diffusion and adsorption in
1074 compacted bentonite: a critical review, *Journal of Contaminant Hydrology*, **61**, 293-
1075 302, doi:10.1016/s0169-7722(02)00128-6.

1076 Bourg, I. C. & Sposito, G., 2011. Molecular dynamics simulations of the electrical double
1077 layer on smectite surfaces contacting concentrated mixed electrolyte (NaCl-CaCl2)
1078 solutions, *Journal of Colloid and Interface Science*, **360**, 701-715,
1079 doi:10.1016/j.jcis.2011.04.063.

1080 Bourg, I. C., Sposito, G. & Bourg, A. C. M., 2007. Modeling the acid-base surface chemistry
1081 of montmorillonite, *Journal of Colloid and Interface Science*, **312**, 297-310,
1082 doi:10.1016/j.jcis.2007.03.062.

1083 Brigatti, M., Galán, E. & Theng, B., 2013. 2 Chapter 2 - Structure and Mineralogy of Clay
1084 Minerals in *Handbook of Clay Science*, pp. 21-81, ed. Bergaya, F. L., G. Elsevier,
1085 Oxford,

1086 Bristow, T. F., Rampe, E. B., Achilles, C. N., Blake, D. F., Chipera, S. J., Craig, P., Crisp, J.
1087 A., Des Marais, D. J., Downs, R. T., Gellert, R., Grotzinger, J. P., Gupta, S., Hazen, R.
1088 M., Horgan, B., Hogancamp, J. V., Mangold, N., Mahaffy, P. R., McAdam, A. C.,
1089 Ming, D. W., Morookian, J. M., Morris, R. V., Morrison, S. M., Treiman, A. H.,
1090 Vaniman, D. T., Vasavada, A. R. & Yen, A. S., 2018. Clay mineral diversity and
1091 abundance in sedimentary rocks of Gale crater, Mars, *Science Advances*, **4**, eaar3330,
1092 doi:10.1126/sciadv.aar3330.

1093 Bruggeman, D. A. G., 1935. Berechnung verschiedener physikalischer Konstanten von
1094 heterogenen Substanzen, *Annalen der Physik*, **416**, 636-664,
1095 doi:10.1002/andp.19354160705.

1096 Bücker, M., Flores Orozco, A., Undorf, S. & Kemna, A., 2019. On the Role of Stern- and
1097 Diffuse-Layer Polarization Mechanisms in Porous Media, *Journal of Geophysical
1098 Research: Solid Earth*, **124**, 5656-5677, doi:10.1029/2019jb017679.

1099 Busch, A., Bertier, P., Gensterblum, Y., Rother, G., Spiers, C. J., Zhang, M. & Wentinck, H.
1100 M., 2016. On sorption and swelling of CO2 in clays, *Geomechanics and Geophysics
1101 for Geo-Energy and Geo-Resources*, **2**, 111-130, doi:10.1007/s40948-016-0024-4.

1102 Cadene, A., Rotenberg, B., Durand-Vidal, S., Badot, J. C. & Turq, P., 2006. Dielectric
1103 spectroscopy as a probe for dynamic properties of compacted smectites, *Physics and
1104 Chemistry of the Earth*, **31**, 505-510, doi:10.1016/j.pce.2006.04.002.

1105 Chapman, D. L., 1913. A contribution to the theory of electrocapillarity *Philosophical Magazine Serie 6*, **25**, 475-481, doi: 10.1080/14786440408634187.

1107 Charlet, L., Alt-Epping, P., Wersin, P. & Gilbert, B., 2017. Diffusive transport and reaction in
1108 clay rocks: A storage (nuclear waste, CO2, H2), energy (shale gas) and water quality
1109 issue, *Advances in Water Resources*, **106**, 39-59,
1110 doi:10.1016/j.advwatres.2017.03.019.

1111 Chelidze, T. L. & Gueguen, Y., 1999. Electrical spectroscopy of porous rocks: a review - I.
1112 Theoretical models, *Geophysical Journal International*, **137**, 1-15,
1113 doi:10.1046/j.1365-246x.1999.00799.x.

1114 Christidis, G. E., 2010. Industrial Clays, 341-414, doi:10.1180/EMU-notes.9.9.

1115 de Lima, O. A. L. & Sharma, M. M., 1992. A generalized Maxwell-Wagner theory for
1116 membrane polarization in shaly sands, *Geophysics*, **57**, 431-440,
1117 doi:10.1190/1.1443257.

1118 Delgado, A. V., Arroyo, F. J., Gonzalez-Caballero, F., Shilov, V. N. & Borkovskaya, Y. B.,
1119 1998. The effect of the concentration of dispersed particles on the mechanisms of low-
1120 frequency dielectric dispersion (LFDD) in colloidal suspensions, *Colloids and*
1121 *Surfaces a-Physicochemical and Engineering Aspects*, **140**, 139-149,
1122 doi:10.1016/S0927-7757(97)00272-0.

1123 Dufreche, J. F., Marry, V., Bernard, O. & Turq, P., 2001. Models for electrokinetic
1124 phenomena in montmorillonite, *Colloids and Surfaces a-Physicochemical and
1125 Engineering Aspects*, **195**, 171-180, doi:10.1016/S0927-7757(01)00840-8.

1126 El Alam, J., Revil, A. & Dick, P., 2023. Influence of the water content on the complex
1127 conductivity of bentonite, *Engineering Geology*, **322**,
1128 doi:10.1016/j.enggeo.2023.107183.

1129 Fixman, M., 1980. Charged Macromolecules in External Fields. I. The Sphere, *Journal of Chemical Physics*, **72**, 5177-5186, doi:10.1063/1.439753.

1131 Gehin, A., Greneche, J. M., Tournassat, C., Brendle, J., Rancourt, D. G. & Charlet, L., 2007.
1132 Reversible surface-sorption-induced electron-transfer oxidation of Fe(II) at reactive
1133 sites on a synthetic clay mineral, *Geochimica Et Cosmochimica Acta*, **71**, 863-876,
1134 doi:10.1016/j.gca.2006.10.019.

1135 Glaser, D. R., Barrowes, B. E., Shubitidze, F. & Slater, L. D., 2023. Laboratory investigation
1136 of high-frequency electromagnetic induction measurements for macro-scale relaxation
1137 signatures, *Geophysical Journal International*, **235**, 1274-1291,
1138 doi:10.1093/gji/ggad298.

1139 Gouy, G., 1910. Sur la constitution de la charge électrique à la surface d'un électrolyte,
1140 *Annales de Physique (Paris) Série 4*, **9**, 457-468, doi:
1141 10.1051/jphystap:019100090045700.

1142 Grahame, D. C., 1947. The Electrical Double Layer and the Theory of Electrocapillarity,
1143 *Chemical Reviews*, **41**, 441-501, doi:10.1021/cr60130a002.

1144 Greathouse, J. A., Cygan, R. T., Fredrich, J. T. & Jerauld, G. R., 2016. Molecular Dynamics
1145 Simulation of Diffusion and Electrical Conductivity in Montmorillonite Interlayers,
1146 *The Journal of Physical Chemistry C*, **120**, 1640-1649, doi:10.1021/acs.jpcc.5b10851.

1147 Grim, R. E., 1962. *Applied Clay Mineralogy*, edn, Vol., McGraw-Hill, New York,

1148 Grosse, C. & Delgado, A. V., 2010. Dielectric dispersion in aqueous colloidal systems,
1149 *Current Opinion in Colloid & Interface Science*, **15**, 145-159,
1150 doi:10.1016/j.cocis.2009.11.004.

1151 Guggenheim, S. & Martin, R. T., 1995. Definition of Clay and Clay Mineral: Joint Report of
1152 the AIPEA Nomenclature and CMS Nomenclature Committees, *Clays and Clay
1153 Minerals*, **43**, 255-256, doi:10.1346/ccmn.1995.0430213.

1154 Guisseau, D., Mas, P. P., Beaufort, D., Girard, J. P., Inoue, A., Sanjuan, B., Petit, S., Lens, A.
1155 & Genter, A., 2007. Significance of the depth-related transition montmorillonite-
1156 beidellite in the Bouillante geothermal field (Guadeloupe, Lesser Antilles), *American
1157 Mineralogist*, **92**, 1800-1813, doi:10.2138/am.2007.2398.

1158 Hanai, T., 1968. 5 Electrical properties of emulsions. in *Emulsions Science*, pp. 354-477, ed.
1159 Sherman, P. Academic Press, New York,

1160 Hao, W., Flynn, S. L., Alessi, D. S. & Konhauser, K. O., 2018. Change of the point of zero
1161 net proton charge (pHPZNPC) of clay minerals with ionic strength, *Chemical
1162 Geology*, **493**, 458-467, doi:10.1016/j.chemgeo.2018.06.023.

1163 Hashin, Z. & Shtrikman, S., 1962. A Variational Approach to the Theory of the Effective
1164 Magnetic Permeability of Multiphase Materials, *Journal of Applied Physics*, **33**, 3125-
1165 3131, doi:10.1063/1.1728579.

1166 Hassan, M. S., Villieras, F., Gaboriaud, F. & Razafitianamaharavo, A., 2006. AFM and low-
1167 pressure argon adsorption analysis of geometrical properties of phyllosilicates,
1168 *Journal of Colloid and Interface Science*, **296**, 614-623,
1169 doi:10.1016/j.jcis.2005.09.028.
1170 Hiemstra, T. & Van Riemsdijk, W. H., 2006. On the relationship between charge distribution, surface hydration, and the structure of the interface of metal hydroxides, *Journal of Colloid and Interface Science*, **301**, 1-18, doi:10.1016/j.jcis.2006.05.008.

1171 Hiemstra, T. & VanRiemsdijk, W. H., 1996. A surface structural approach to ion adsorption: The charge distribution (CD) model, *Journal of Colloid and Interface Science*, **179**, 488-508, doi:10.1006/jcis.1996.0242.

1172 Hiemstra, T. & VanRiemsdijk, W. H., 1996. A surface structural approach to ion adsorption: The charge distribution (CD) model, *Journal of Colloid and Interface Science*, **179**, 488-508, doi:10.1006/jcis.1996.0242.

1173 Hiemstra, T. & VanRiemsdijk, W. H., 1996. A surface structural approach to ion adsorption: The charge distribution (CD) model, *Journal of Colloid and Interface Science*, **179**, 488-508, doi:10.1006/jcis.1996.0242.

1174 Hiemstra, T. & VanRiemsdijk, W. H., 1996. A surface structural approach to ion adsorption: The charge distribution (CD) model, *Journal of Colloid and Interface Science*, **179**, 488-508, doi:10.1006/jcis.1996.0242.

1175 Hiemstra, T. & VanRiemsdijk, W. H., 1996. A surface structural approach to ion adsorption: The charge distribution (CD) model, *Journal of Colloid and Interface Science*, **179**, 488-508, doi:10.1006/jcis.1996.0242.

1176 Hiemstra, T. & VanRiemsdijk, W. H., 1996. A surface structural approach to ion adsorption: The charge distribution (CD) model, *Journal of Colloid and Interface Science*, **179**, 488-508, doi:10.1006/jcis.1996.0242.

1177 Hiemstra, T. & VanRiemsdijk, W. H., 1996. A surface structural approach to ion adsorption: The charge distribution (CD) model, *Journal of Colloid and Interface Science*, **179**, 488-508, doi:10.1006/jcis.1996.0242.

1178 Ismadji, S., Soetaredjo, F. E. & Ayucitra, A., 2015. *Clay Materials for Environmental Remediation*, edn, Vol. 25, Springer, Surabaya, doi:10.1007/978-3-319-16712-1.

1179 Ismadji, S., Soetaredjo, F. E. & Ayucitra, A., 2015. *Clay Materials for Environmental Remediation*, edn, Vol. 25, Springer, Surabaya, doi:10.1007/978-3-319-16712-1.

1180 Jougnot, D., Ghorbani, A., Revil, A., Leroy, P. & Cosenza, P., 2010a. Spectral induced polarization of partially saturated clay-rocks: a mechanistic approach, *Geophysical Journal International*, **180**, 210-224, doi:10.1111/j.1365-246X.2009.04426.x.

1181 Jougnot, D., Ghorbani, A., Revil, A., Leroy, P. & Cosenza, P., 2010a. Spectral induced polarization of partially saturated clay-rocks: a mechanistic approach, *Geophysical Journal International*, **180**, 210-224, doi:10.1111/j.1365-246X.2009.04426.x.

1182 Jougnot, D., Ghorbani, A., Revil, A., Leroy, P. & Cosenza, P., 2010a. Spectral induced polarization of partially saturated clay-rocks: a mechanistic approach, *Geophysical Journal International*, **180**, 210-224, doi:10.1111/j.1365-246X.2009.04426.x.

1183 Jougnot, D., Revil, A. & Leroy, P., 2009. Diffusion of ionic tracers in the Callovo-Oxfordian clay-rock using the Donnan equilibrium model and the formation factor, *Geochimica Et Cosmochimica Acta*, **73**, 2712-2726, doi:10.1016/j.gca.2009.01.035.

1184 Jougnot, D., Revil, A. & Leroy, P., 2009. Diffusion of ionic tracers in the Callovo-Oxfordian clay-rock using the Donnan equilibrium model and the formation factor, *Geochimica Et Cosmochimica Acta*, **73**, 2712-2726, doi:10.1016/j.gca.2009.01.035.

1185 Jougnot, D., Revil, A. & Leroy, P., 2009. Diffusion of ionic tracers in the Callovo-Oxfordian clay-rock using the Donnan equilibrium model and the formation factor, *Geochimica Et Cosmochimica Acta*, **73**, 2712-2726, doi:10.1016/j.gca.2009.01.035.
1218 Leroy, P. & Revil, A., 2009. A mechanistic model for the spectral induced polarization of clay materials, *Journal of Geophysical Research-Solid Earth*, **114**, 1-21, doi:10.1029/2008jb006114.

1219 Leroy, P., Revil, A., Altmann, S. & Tournassat, C., 2007. Modeling the composition of the pore water in a clay-rock geological formation (Callovo-Oxfordian, France), *Geochimica Et Cosmochimica Acta*, **71**, 1087-1097, doi:10.1016/j.gca.2006.11.009.

1220 Leroy, P., Revil, A., Kemna, A., Cosenza, P. & Ghorbani, A., 2008. Complex conductivity of water-saturated packs of glass beads, *Journal of Colloid and Interface Science*, **321**, 103-117, doi:10.1016/j.jcis.2007.12.031.

1221 Leroy, P., Tournassat, C., Bernard, O., Devau, N. & Azaroual, M., 2015. The electrophoretic mobility of montmorillonite. Zeta potential and surface conductivity effects, *Journal of Colloid and Interface Science*, **451**, 21-39, doi:10.1016/j.jcis.2015.03.047.

1222 Leroy, P., Tournassat, C. & Bizi, M., 2011. Influence of surface conductivity on the apparent zeta potential of TiO2 nanoparticles, *Journal of Colloid and Interface Science*, **356**, 442-453, doi:10.1016/j.jcis.2011.01.016.

1223 Leroy, P., Weigand, M., Mériguet, G., Zimmermann, E., Tournassat, C., Fagerlund, F., Kemna, A. & Huisman, J. A., 2017b. Spectral induced polarization of Na-montmorillonite dispersions, *Journal of Colloid and Interface Science*, **505**, 1093-1110, doi:10.1016/j.jcis.2017.06.071.

1224 Lesmes, D. P. & Morgan, F. D., 2001. Dielectric spectroscopy of sedimentary rocks, *Journal of Geophysical Research-Solid Earth*, **106**, 13329-13346, doi:10.1029/2000jb900402.

1225 Li, S., Leroy, P., Heberling, F., Devau, N., Jougnot, D. & Chiaberge, C., 2016. Influence of surface conductivity on the apparent zeta potential of calcite, *Journal of Colloid and Interface Science*, **468**, 262-275, doi:10.1016/j.jcis.2016.01.075.

1226 Li, Z., Dong, M., Li, S. & Huang, S., 2006. CO2 sequestration in depleted oil and gas reservoirs—caprock characterization and storage capacity, *Energy Conversion and Management*, **47**, 1372-1382, doi:10.1016/j.enconman.2005.08.023.

1227 Lide, D. R., 1990. *CRC Handbook of Chemistry and Physics* edn, Vol., CRC Press, Boca Raton,

1228 Liu, J., Wang, S., Javadpour, F., Feng, Q. & Cha, L., 2022. Hydrogen Diffusion in Clay Slit: Implications for the Geological Storage, *Energy & Fuels*, **36**, 7651-7660, doi:10.1021/acs.energyfuels.2c01189.

1229 Lyklema, J., Dukhin, S. S. & Shilov, V. N., 1983. The relaxation of the double-layer around colloidal particles and the low-frequency dielectric-dispersion. Part I. Theoretical Considerations, *Journal of Electroanalytical Chemistry*, **143**, 1-21, doi:10.1016/S0022-0728(83)80251-4.

1230 Lyklema, J. & Minor, M., 1998. On surface conduction and its role in electrokinetics, *Colloids and Surfaces* a-Physicochemical and Engineering Aspects, **140**, 33-41, doi:10.1016/S0927-7757(97)00266-5.

1231 Madden, T. R., 1976. Random Networks and Mixing Laws, *Geophysics*, **41**, 1104-1125, doi:10.1190/1.2035907.

1232 Maineult, A., Leroy, P., Mendieta, A. & Jougnot, D., 2024. A mechanistic model for the complex conductivity of clay materials. II. Comparison to experimental data, submitted to *Geophysical Journal International*.

1233 Malusis, M. A., Shackelford, C. D. & Olsen, H. W., 2003. Flow and transport through clay membrane barriers, *Engineering Geology*, **70**, 235-248, doi:10.1016/s0013-7952(03)00092-9.

1234 Marshall, D. J. & Madden, T. R., 1959. Induced Polarization, a Study of Its Causes, *Geophysics*, **24**, 790-816, doi:10.1190/1.1438659.
1267 Martin, T., Günther, T., Orozco, A. F. & Dahlin, T., 2020. Evaluation of spectral induced polarization field measurements in time and frequency domain, *Journal of Applied Geophysics*, **180**, doi:10.1016/j.jappgeo.2020.104141.

1268
1269 Massat, L., Cuisinier, O., Bihannic, I., Claret, F., Pelletier, M., Masrouri, F. & Gaboreau, S., 2016. Swelling pressure development and inter-aggregate porosity evolution upon hydration of a compacted swelling clay, *Applied Clay Science*, **124-125**, 197-210, doi:10.1016/j.clay.2016.01.002.

1274 Maxwell, J. C., 1892. *A Treatise on Electricity and Magnetism*, third edition edn, Vol., Oxford University Press, London,

1276 Mc Bride, E. F., 2012. Petrology of the Eureka Quartzite (Middle and Late Ordovician), Utah and Nevada, U.S.A., *Rocky Mountain Geology*, **47**, 81-111, doi:10.2113/gsrocky.47.2.81.

1279 Mc Bride, M. B. & Baveye, P., 2002. Diffuse double-layer models, long-range forces, and ordering in clay colloids, *Soil Science Society of America Journal*, **66**, 1207-1217, doi:10.2136/sssaj2002.1207.

1282 Melkior, T., Gaucher, E. C., Brouard, C., Yahiaoui, S., Thoby, D., Clinard, C., Ferrage, E., 1999. Na+ and HTO diffusion in compacted bentonite: Effect of surface chemistry and related texture, *Journal of Hydrology*, **370**, 9-20, doi:10.1016/j.jhydrol.2009.02.035.

1286 Mendelson, K. S. & Cohen, M. H., 1982. The effect of grain anisotropy on the electrical-
1287 properties of sedimentary-rocks, *Geophysics*, **47**, 257-263, doi:10.1190/1.1441332.

1288 Mendieta, A., Jougnot, D., Leroy, P. & Maineult, A., 2021. Spectral Induced Polarization Characterization of Non-Consolidated Clays for Varying Salinities—An Experimental Study, *Journal of Geophysical Research: Solid Earth*, **126**, doi:10.1029/2020jb021125.

1292 Mouzon, J., Bhuiyan, I. U. & Hedlund, J., 2016. The structure of montmorillonite gels
1293 revealed by sequential cryo-XHR-SEM imaging, *Journal of Colloid and Interface
1294 Science*, **465**, 58-66, doi:10.1016/j.jcis.2015.11.031.

1295 Nagendrappa, G., 2011. Organic synthesis using clay and clay-supported catalysts, *Applied Clay Science*, **53**, 106-138, doi:10.1016/j.clay.2010.09.016.

1297 Neuzil, C. E., 1994. How permeable are clays and shales?, *Water Resources Research*, **30**, 145-150, doi:10.1029/93wr02930.

1299 O'Konski, C. T., 1960. Electric properties of macromolecules. V. Theory of ionic polarization in polyelectrolytes, *The Journal of Physical Chemistry*, **64**, 605-619, doi:10.1021/j100834a023.

1302 Okay, G., Leroy, P., Ghorbani, A., Cosenza, P., Camerlynck, C., Cabrera, J., Florsch, N. & Revil, A., 2014. Spectral induced polarization of clay-sand mixtures: Experiments and modeling, *Geophysics*, **79**, E353-E375, doi:10.1190/Geo2013-0347.1.

1305 Olhoeft, G. R., 1981. Electrical properties of rocks. in *Physical properties of rocks and minerals*, pp. 257-339, ed. Y. S. Touloukian, W. R. J., and R. F. Roy. McGraw-Hill, New York,

1308 Parkhurst, D. L. & Appelo, C. A. J., 2013. 43 Description of Input and Examples for PHREEQC Version 3--a Computer Program for Speciation, Batch-reaction, One-dimensional Transport, and Inverse Geochemical Calculations. in *U.S. Geological Survey Techniques and Methods, Book 6, Modeling Techniques*, pp. 497, ed. Reston, U. S. G. S., Denver,

1313 Pelton, W. H., Ward, S. H., Hallof, P. G., Sill, W. R. & Nelson, P. H., 1978. Mineral Discrimination and Removal of Inductive Coupling with Multifrequency-Ip, *Geophysics*, **43**, 588-609, doi:10.1190/1.1440839.
1316 Petraglia, M. D., Ashton, N., Lewis, S. G., De Groote, I., Duffy, S. M., Bates, M., Bates, R.,
1317 Hoare, P., Lewis, M., Parfitt, S. A., Peglar, S., Williams, C. & Stringer, C., 2014.
1318 Hominin Footprints from Early Pleistocene Deposits at Happisburgh, UK, *Plos One*,
1319 **9**, e88329, doi:10.1371/journal.pone.0088329.

1320 Porté, J., Bretaudieu, F. & Girard, J F., 2023. 3-D complex resistivity imaging using
1321 controlled source electromagnetic data: a multistage procedure using a second order
1322 polynomial parametrization, *Geophysical Journal International*, **233**, 839-860,
1323 doi:10.1093/gji/ggac486.

1324 Pride, S., 1994. Governing Equations for the Coupled Electromagnetics and Acoustics of
1325 Porous-Media, *Physical Review B*, **50**, 15678-15696,
1326 doi:10.1103/PhysRevB.50.15678.

1327 Qi, Y. & Wu, Y., 2022. Electrical Conductivity of Clayey Rocks and Soils: A Non-Linear
1328 Model, *Geophysical Research Letters*, **49**, doi:10.1029/2021gl097408.

1329 Qi, Y. & Wu, Y., 2024. Induced Polarization of Clayey Rocks and Soils: Non-Linear
1330 Complex Conductivity Models, *Journal of Geophysical Research: Solid Earth*, **129**,
1331 doi:10.1029/2023jb028405.

1332 Rasmusson, M., Rowlands, W., OBrien, R. W. & Hunter, R. J., 1997. The dynamic mobility
1333 and dielectric response of sodium bentonite, *Journal of Colloid and Interface Science*,
1334 **189**, 92-100, doi:10.1006/jcis.1997.4793.

1335 Revil, A., 2012. Spectral induced polarization of shaly sands: Influence of the electrical
1336 double layer, *Water Resources Research*, **48**, 1-23, doi:10.1029/2011WR011260.

1337 Revil, A., Cathles, L. M., Losh, S. & Nunn, J. A., 1998. Electrical conductivity in shaly sands
1338 with geophysical applications, *Journal of Geophysical Research-Solid Earth*, **103**,
1339 23925-23936, doi:10.1029/98JB02125.

1340 Revil, A. & Florsch, N., 2010. Determination of permeability from spectral induced
1341 polarization in granular media, *Geophysical Journal International*, **181**, 1480-1498,
1342 doi:10.1111/j.1365-246X.2010.04573.x.

1343 Revil, A., Ghorbani, A., Jougnot, D. & Yven, B., 2023. Induced polarization of clay-rich
1344 materials — Part 1: The effect of desiccation, *Geophysics*, **88**, MR195-MR210,
1345 doi:10.1190/geo2022-0510.1.

1346 Revil, A. & Glover, P. W. J., 1997. Theory of ionic-surface electrical conduction in porous
1347 media, *Physical Review B*, **55**, 1757-1773, doi:10.1103/PhysRevB.55.1757.

1348 Revil, A. & Leroy, P., 2001. Hydroelectric coupling in a Clayey Material, *Geophysical
1349 Research Letters*, **28**, 1643-1646, doi:10.1029/2000GL012268.

1350 Revil, A. & Leroy, P., 2004. Constitutive equations for ionic transport in porous shales,
1351 *Journal of Geophysical Research-Solid Earth*, **109**, 1-19, doi:10.1029/2003jb002755.

1352 Revil, A., Leroy, P. & Titov, K., 2005. Characterization of transport properties of argillaceous
1353 sediments: Application to the Callovo-Oxfordian argillite, *Journal of Geophysical
1354 Research-Solid Earth*, **110**, B06202, doi:10.1029/2004jb003442.

1355 Revil, A. & Linde, N., 2006. Chemico-electromechanical coupling in microporous media, *J
1356 Colloid Interface Sci*, **302**, 682-694, doi:10.1016/j.jcis.2006.06.051.

1357 Revil, A. & Skold, M., 2011. Salinity dependence of spectral induced polarization in sands
1358 and sandstones, *Geophysical Journal International*, **187**, 813-824, doi:10.1111/j.1365-246X.2011.05181.x.

1360 Richter, S. M. & Sevick-Muraca, E. M., 2000. Characterization of concentrated colloidal
1361 suspensions using time-dependent photon migration measurements, *Colloids and
1362 Surfaces A: Physicochemical and Engineering Aspects*, **172**, 163-173,
1363 doi:10.1016/s0927-7757(00)00581-1.
1364 Robinson, D. A., 2004. Measurement of the solid dielectric permittivity of clay minerals and granular samples using a time domain reflectometry immersion method, *Vadose Zone Journal*, **3**, 705-713, doi:10.2136/vzj2004.0705.

1365 Rotenberg, B., Cadéne, A., Dufrêche, J. F., Durand-Vidal, S., Badot, J. C. & Turq, P., 2005. An Analytical Model for Probing Ion Dynamics in Clays with Broadband Dielectric Spectroscopy, *The Journal of Physical Chemistry B*, **109**, 15548-15557, doi:10.1021/jp051586k.

1367 Ruiz Pestana, L., Kolluri, K., Head-Gordon, T. & Lammers, L. N., 2016. Direct Exchange Mechanism for Interlayer Ions in Non-Swelling Clays, *Environmental Science & Technology*, **51**, 393-400, doi:10.1021/acs.est.6b04747.

1371 Sahai, N. & Sverjensky, D. A., 1997. Evaluation of internally consistent parameters for the triple-layer model by the systematic analysis of oxide surface titration data, *Geochimica Et Cosmochimica Acta*, **61**, 2801-2826, doi:10.1016/S0016-7037(97)00128-2.

1378 Schlumberger, C., 1920. *Study of underground electrical prospecting*, edn, Vol., Paris,

1379 Schramm, L. L. & Kwak, J. C. T., 1982. Influence of Exchangeable Cation Composition on the Size and Shape of Montmorillonite Particles in Dilute Suspension, *Clays and Clay Minerals*, **30**, 40-48, doi:10.1346/Ccmn.1982.0300105.

1382 Schroeder, P. A., 2018. *Clays in the Critical Zone*, edn, Vol., Cambridge University Press, Cambridge, UK, doi:10.1017/9781316480083.

1384 Schurr, J. M., 1964. On the Theory of the Dielectric Dispersion of Spherical Colloidal Particles in Electrolyte Solution, *The Journal of Physical Chemistry*, **68**, 2407-2413, doi:10.1021/j100791a004.

1387 Schwarz, G., 1962. A theory of the low-frequency dielectric dispersion of colloidal particles in electrolyte solution, *The Journal of Physical Chemistry*, **66**, 2636-2642, doi:10.1021/j100818a067.

1390 Secor, R. B. & Radke, C. J., 1985. Spillover of the diffuse double layer on montmorillonite particles, *Journal of Colloid and Interface Science*, **103**, 237-244, doi:10.1016/0021-9797(85)90096-7.

1393 Sellin, P. & Leupin, O. X., 2013. The Use of Clay as an Engineered Barrier in Radioactive-Waste Management – A Review, *Clays and Clay Minerals*, **61**, 477-498, doi:10.1346/ccmn.2013.0610601.

1396 Sen, P. N., 1984. Grain Shape Effects on Dielectric and Electrical-Properties of Rocks, *Geophysics*, **49**, 586-587, doi:10.1190/1.1441695.

1398 Sen, P. N., Scala, C. & Cohen, M. H., 1981. A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beads, *Geophysics*, **46**, 781-795, doi:10.1190/1.1441215.

1401 Sondi, I., Biscan, J. & Pravdik, V., 1996. Electrokinetics of pure clay minerals revisited, *Journal of Colloid and Interface Science*, **178**, 514-522, doi:10.1006/jcis.1996.0146.

1403 Sposito, G., 1989. *The Chemistry of Soils* edn, Vol., Oxford University Press, New York,

1404 Sposito, G., Skipper, N. T., Sutton, R., Park, S. h., Soper, A. K. & Greathouse, J. A., 1999. Surface geochemistry of the clay minerals, *Proceedings of the National Academy of Sciences*, **96**, 3358-3364, doi:10.1073/pnas.96.7.3358.

1407 Tombacz, E. & Szekeres, M., 2004. Colloidal behavior of aqueous montmorillonite suspensions: the specific role of pH in the presence of indifferent electrolytes, *Applied Clay Science*, **27**, 75-94, doi:10.1016/j.clay.2004.01.001.

1410 Tombacz, E. & Szekeres, M., 2006. Surface charge heterogeneity of kaolinite in aqueous suspension in comparison with montmorillonite, *Applied Clay Science*, **34**, 105-124, doi:10.1016/j.clay.2006.05.009.
1413 Tournassat, C. & Appelo, C. A. J., 2011. Modelling approaches for anion-exclusion in
1414 compacted Na-bentonite, *Geochimica Et Cosmochimica Acta*, **75**, 3698-3710,
1415 doi:10.1016/j.gca.2011.04.001.

1416 Tournassat, C., Bizi, M., Braibant, G. & Crouzet, C., 2011. Influence of montmorillonite
tactoid size on Na-Ca cation exchange reactions, *Journal of Colloid and Interface
Science*, **364**, 443-454, doi:10.1016/j.jcis.2011.07.039.

1419 Tournassat, C., Bourg, I. C., Steefel, C. I. & Bergaya, F., 2015. Surface Properties of Clay
Minerals, pp. 5-31, doi:10.1016/b978-0-08-100027-4.00001-2.

1421 Tournassat, C., Chapron, Y., Leroy, P., Bizi, M. & Boulahya, F., 2009. Comparison of
1422 molecular dynamics simulations with triple layer and modified Gouy-Chapman
1423 models in a 0.1 M NaCl-montmorillonite system, *Journal of Colloid and Interface
1424 Science*, **339**, 533-541, doi:10.1016/j.jcis.2009.06.051.

1425 Tournassat, C., Davis, J. A., Chiaberge, C., Grangeon, S. & Bourg, I. C., 2016a. Modeling the
1426 Acid-Base Properties of Montmorillonite Edge Surfaces, *Environmental Science &
1427 Technology*, **50**, 13436-13445, doi:10.1021/acs.est.6b04677.

1428 Tournassat, C., Gaboreau, S., Robinet, J. C., Bourg, I. & Steefel, C. I., 2016b. Impact of
1429 microstructure on anion exclusion in compacted clay media, **22**, 137-149,
1430 doi:10.1346/cms-wls-21.11.

1431 Tournassat, C., Grangeon, S., Leroy, P. & Giffaut, E., 2013. Modeling specific pH dependent
1432 sorption of divalent metals on montmorillonite surfaces. A review of pitfalls, recent
1433 achievements and current challenges, *American Journal of Science*, **313**, 395-451,
1434 doi:10.2475/05.2013.01.

1435 Tournassat, C. & Steefel, C. I., 2019. Reactive Transport Modeling of Coupled Processes in
1436 Nanoporous Media, *Reviews in Mineralogy and Geochemistry*, **85**, 75-109,
1437 doi:10.2138/rmg.2019.85.4.

1438 Triantafilis, J. & Lesch, S. M., 2005. Mapping clay content variation using electromagnetic
1439 induction techniques, *Computers and Electronics in Agriculture*, **46**, 203-237,
1440 doi:10.1016/j.compag.2004.11.006.

1441 Udonne, J. D., 2011. A comparative study of recycling of used lubrication oils using
1442 distillation, acid and activated charcoal with clay methods, *Journal of Petroleum and
1443 Gas Engineering*, **2**, 12-19, doi:10.5897/JPGE.9000001.

1444 Vandiver, P. B., Soffer, O., Klima, B. & Svoboda, J., 1989. The Origins of Ceramic
1445 Technology at Dolni Věstonice, Czechoslovakia, *Science*, **246**, 1002-1008,
1446 doi:10.1126/science.246.4933.1002.

1447 Vinegar, H. J. & Waxman, M. H., 1984. Induced Polarization of Shaly Sands, *Geophysics*, **49**, 1267-1287, doi:10.1190/1.1441755.

1449 Viseras, C., Carazo, E., Borrego-Sánchez, A., García-Villén, F., Sánchez-Espejo, R., Cerezo, P. & Aguzzi, C., 2019. Clay Minerals in Skin Drug Delivery, *Clays and Clay Minerals*, **67**, 59-71, doi:10.1007/s42860-018-0003-7.

1452 Viseras, C., Cerezo, P., Sanchez, R., Salcedo, I. & Aguzzi, C., 2010. Current challenges in
1453 clay minerals for drug delivery, *Applied Clay Science*, **48**, 291-295,
1454 doi:10.1016/j.clay.2010.01.007.

1455 Von Smoluchowski, M., 1905. Zur theorie der elektrischen kataphorese und der
1456 oberflachenleitung, *Physikalische Zeitschrift*, **6**, 529-531.

1457 Von Smoluchowski, M., 1921. edn, Vol. II, pp. 366-428, L. Graetz, Leipzig,

1458 Wander, M. C. F. & Clark, A. E., 2008. Structural and Dielectric Properties of Quartz-Water
1459 Interfaces, *Journal of Physical Chemistry C*, **112**, 19986-19994,
1460 doi:10.1021/Jp803642c.

1461 Waxman, M. H. & Smits, L. J. M., 1968. Electrical conductivities in oil bearing shaly sands,
*Society of Petroleum Engineers Journal*, **8**, 107-122, doi:10.2118/1863-A.
1463 Weller, A., Slater, L., Huisman, J. A., Esser, O. & Haegel, F. H., 2015. On the specific polarizability of sands and sand-clay mixtures, *Geophysics*, **80**, A57-A61, doi:10.1190/Geo2014-0509.1.

1466 Weller, A., Slater, L. & Nordsiek, S., 2013. On the relationship between induced polarization and surface conductivity: Implications for petrophysical interpretation of electrical measurements, *Geophysics*, **78**, D315-D325, doi:10.1190/Geo2013-0076.1.

1469 Xu, T., Sonnenthal, E., Spycher, N. & Pruess, K., 2006. TOUGHREACT—A simulation program for non-isothermal multiphase reactive geochemical transport in variably saturated geologic media: Applications to geothermal injectivity and CO2 geological sequestration, *Computers & Geosciences*, **32**, 145-165, doi:10.1016/j.cageo.2005.06.014.

1474 Yang, D., Peng, S., Hartman, M. R., Gupton-Campolongo, T., Rice, E. J., Chang, A. K., Gu, Z., Lu, G. Q. & Luo, D., 2013. Enhanced transcription and translation in clay hydrogel and implications for early life evolution, *Sci Rep*, **3**, doi:10.1038/srep03165.

1477 Yates, D. E., Levine, S. & Healy, T. W., 1974. Site-binding Model of the Electrical Double Layer at the Oxide/Water interface, *Journal of the Chemical Society, Faraday Transactions*, **70**, 1807-1818, doi:10.1039/F19747001807.

1480 Yven, B., Sammartino, S., Geraud, Y., Homand, F. & Villieras, F., 2007. Mineralogy, texture and porosity of Callovo-Oxfordian argillites of the Meuse/Haute-Marne region (eastern Paris Basin), *Mémoires de la Société géologique de France*, **178**, 73-90.

1483 Zhang, L., Lu, X., Liu, X., Zhou, J. & Zhou, H., 2014. Hydration and Mobility of Interlayer Ions of (Nax, Cay)-Montmorillonite: A Molecular Dynamics Study, *The Journal of Physical Chemistry C*, **118**, 29811-29821, doi:10.1021/jp508427c.

1486 Zimmermann, E., Kemna, A., Berwix, J., Glaas, W., Munch, H. M. & Huisman, J. A., 2008. A high-accuracy impedance spectrometer for measuring sediments with low polarizability, *Measurement Science & Technology*, **19**, 105603, doi:10.1088/0957-0233/19/10/105603.

1490

1491
1492 APPENDIX A. COMPUTED VOLUMES OF THE COMPONENTS OF THE CLAY

1493 MATERIAL

1494 The volume of the solid part of the clay sample (not considering the interlayer space) (in m³)

1495 is given by

$$V_s = \frac{10^{-3} \delta_s}{\rho_s}. \qquad (A1)$$

1496 The volumes of the solid part of the main clay mineral and of the remaining minerals in the

1497 sample are determined, respectively, from

$$V_{sc} = V_s \xi_c, \qquad (A2)$$

$$V_{so} = V_s (1 - \xi_c), \qquad (A3)$$

1498 assuming that the minerals have roughly the same volumetric mass density.

1499 The volume of the diffuse layer water is calculated using

$$V_w^d = S_o \chi_d, \qquad (A4)$$

1500 where $S_o$ is the outer surface area of the main clay mineral in the mud (in m²) and $\chi_d = 2\chi_D$.

1501 The Debye length is calculated as a function of the bulk water chemical composition using

$$\chi_D = \sqrt{\frac{\epsilon_w k_B T}{e^2 10^3 N_A \sum_{i=1}^{N} z_i^2 C_i^b}}. \qquad (A5)$$

1502 The quantity $S_o$ in Eq. A4 is given by

$$S_o = \delta_{sc} S S_o, \qquad (A6)$$
where $\delta_{sc} = \delta_s \xi_c$ is the solid mass of the main clay mineral in the mud (in g) and $SS_o$ is the outer specific surface area of the main clay mineral in the mud (in m² g⁻¹), which is computed using

$$SS_o = SS_e + \frac{SS_b}{n_c}, \qquad (A7)$$

where $n_c$ is the number of solid layers per clay particle (Tournassat et al. 2011). In Eq. A7, the $SS_e$ quantity can be neglected for Mt and illite (see Section 2.1.).

The volume of the Stern layers is calculated using

$$V_s = S_o d_s, \qquad (A8)$$

where $d_s$ is the Stern layer thickness (in m) that can be approximated to 0.5 nm for the basal surface of illite and Mt (Bourg & Sposito 2011) and possibly to the same value for the edge surface (more work is necessary for the Stern layer modelling on the edge surface). Most of the time, the Stern layer volume can be neglected compared to the diffuse layer volume, but, in the case of compacted clays (dry density typically above 1600 kg m⁻³ (Massat et al. 2016)) and saline bulk aqueous solutions (ionic strengths typically ≥ 0.1 M (Leroy et al. 2015)), the diffuse layer can be highly compressed, and the Stern layer volume can be higher or equal to the diffuse layer volume and hence it cannot be neglected.

The volume of the interlayer space (for illite and Mt) is determined by

$$V_{in} = \frac{\delta_{sc} SS_b}{2n_c (n_c - 1) t_{in}}, \qquad (A9)$$

where $t_{in}$ is the thickness of the interlayer space (in m). The quantity $t_{in} \approx 10$ Å for Na-Mt and $t_{in} \approx 3$ Å for illite (crystal diameter of K⁺ (Leroy et al. 2010)). But because K⁺ ions are non-
1520 solvated and for the mud we can take the mass of the clay powder containing already K⁺ ions
1521 for Vₛ, we can assume tᵢₘ ≅ 0 Å for illite (Tournassat et al. 2015) (tᵢₘ ≅ 0 Å also for
1522 kaolinite).

1523 Finally, the volume of bulk water is calculated according to the volume of the sample minus
1524 the estimated volumes of the other components of the mud:

$$
V_w^b = V_t - V_s - V_w^d - V_s - V_{in}. \tag{A10}
$$

1525

1526
1527 APPENDIX B. COMPUTATION OF MODEL PARAMETER VALUES FROM
TEMPERATURE AND SALINITY

1528 In the model, the liquid water permittivity, for temperatures ranging from 2°C to 87°C (Lide
1529 1990) and minus the decimal logarithm of the dissociation constant of water molecules (
1530 pK<sub>w</sub> = −log K<sub>w</sub>), for temperatures ranging from 0°C to 100°C (Bandura & Lvov 2006), are
1531 computed as a function of the temperature in degree Celsius using

$$
\epsilon_w = \epsilon_0 (a_0 + a_1 T_C + a_2 T_C^2 + a_3 T_C^3), \quad (B1)
$$

$$
pK_w = b_0 + b_1 T_C + b_2 T_C^2 + b_3 T_C^3, \tag{B2}
$$

1532 where $a_0 = 87.895$, $a_1 = -4.0029 \times 10^{-1}$, $a_2 = 8.6132 \times 10^{-4}$, $a_3 = -9.565 \times 10^{-7}$, $b_0 = 14.95$,
1533 $b_1 = -4.38 \times 10^{-2}$, $b_2 = 2.32 \times 10^{-4}$, and $b_3 = -6.4 \times 10^{-7}$ in order to reproduce the related
1534 measurements.

1535 The ion i mobility in bulk water at a temperature T (in K) is calculated from
$$T = T_c + 273.15, \qquad (B3)$$

$$\beta_i^b(T) = \beta_i^b(298) \frac{T}{298} \frac{\eta_w(298)}{\eta_w}, \qquad (B4)$$

where $\beta_i^b(298)$ is the ion $i$ mobility in bulk water at a temperature of 298.15 K, $\eta_w(298)$ is the liquid water dynamic viscosity at a temperature of 298.15 K (~0.8905×10⁻³ Pa s), and $\eta_w$ is the liquid water dynamic viscosity at a temperature $T$.

Ions of opposite charge decrease ion mobility in bulk water because of their mutual interaction (e.g., Bernard et al. 1992). Na⁺ and Cl⁻ ion mobilities in bulk water at a temperature of 298.15 K are determined as a function of NaCl concentration in bulk water using

$$\beta_{Na^+}^b(298) = (a_0 + a_1 C_{NaCl}^b + a_2 C_{NaCl}^{b^2}) 10^{-8}, \qquad (B5)$$

for $C_{NaCl}^b \le 0.1 \text{ M}$,

$$\beta_{Na^+}^b(298) = (a_0' + a_1' C_{NaCl}^b) 10^{-8}, \qquad (B6)$$

for $C_{NaCl}^b > 0.1 \text{ M}$,

$$\beta_{Cl^-}^b(298) = (b_0 + b_1 C_{NaCl}^b + b_2 C_{NaCl}^{b^2}) 10^{-8}, \qquad (B7)$$

for $C_{NaCl}^b \le 0.1 \text{ M}$, and

$$\beta_{Cl^-}^b(298) = (b_0' + b_1' C_{NaCl}^b) 10^{-8}, \qquad (B8)$$

for $C_{NaCl}^b > 0.1 \text{ M}$,
where $a_0 = 5.07490$, $a_1 = -9.55617$, $a_2 = 7.73145$, $a'_0 = 4.32333$, $a'_1 = -1.07333$, $b_0 = 7.77572$, $b_1 = 10.77742$, $b'_0 = 6.92256$, and $b'_1 = 1.14556$ in order to reproduce the corresponding values simulated by the mean spherical approximation (MSA) method (Bernard *et al.* 1992; Leroy *et al.* 2015). For other types of ions (here $H^+$, $OH^-$, and $Ca^{2+}$ and $SO_4^{2-}$ ions from the interlayer space and dissolved clay minerals, according to Leroy *et al.* 2017b), we use for ion *i* mobility in bulk water

$$ \beta_i^b (298) = \frac{\overline{D_i} (298) |q_i|}{k_B 298}, \qquad (B9) $$

where $\overline{D_i} (298)$ is the measured ion *i* self-diffusion coefficient in distilled (bulk) water at a temperature of 298.15 K (in $m^2 s^{-1}$) (Table B1 (Parkhurst & Appelo 2013)) (Eq. B9 is the Nernst-Einstein equation).

Table B1

Ion self-diffusion coefficient values in distilled water from the phreeqc.dat database

<table><thead><tr><th></th><th>Na<sup>+</sup></th><th>Cl<sup>-</sup></th><th>H<sup>+</sup></th><th>OH<sup>-</sup></th><th>Ca<sup>2+</sup></th><th>SO<sub>4</sub><sup>2-</sup></th></tr></thead><tbody><tr><td>DN&#773;<sub>i</sub> (298) (10<sup>-9</sup> m s<sup>-1</sup>)</td><td>1.33</td><td>2.03</td><td>9.31</td><td>5.27</td><td>0.793</td><td>1.07</td></tr></tbody></table>

In Eq. B3, the liquid water dynamic viscosity at a temperature $T$ (in K) is computed from

$$ \eta_w = \frac{\eta_w (293)}{10^{A/B}}, \qquad (B10) $$

$$ A = 1.37023(T - 293.15) + 8.36 \times 10^{-4} (T - 293.15)^2, \qquad (B11) $$

$$ B = T - 164.15, \qquad (B12) $$
where $\eta_w$ (293) is the liquid water dynamic viscosity at a temperature of 293 K (~1.002×10⁻³ Pa s).

The ion mobility of the counter-ions in the Stern layer at a temperature T (K) is calculated from

$$ \beta_s = \beta_s(298)[1 + v_s(T - 298.15)], \qquad (B13) $$

$$ \beta_s = \bar{\beta}_i^b(298)mr_s[1 + v_s(T - 298.15)], \qquad (B14) $$

where $v_s = 0.037$ K⁻¹ for sodium ion (Revil et al. 1998) (there are other $v_s$ values for monovalent and divalent cations reported in Revil et al. 1998) and $mr_s$ is the mobility ratio between a ion in the Stern layer and the same ion in bulk and distilled water at a temperature of 298.15 K (I is the ionic strength in M),

$$ mr_s = \frac{\beta_s(298)}{\bar{\beta}_i^b(298)}, \qquad (B15) $$

$$ \bar{\beta}_i^b(298) = \lim_{I \to 0} \beta_i^b(298), \qquad (B16) $$

$$ I = 0.5 \sum_{i=1}^{N} z_i^2 C_i^b. \qquad (B17) $$

The Stern layer of clays is mostly occupied by ions with the same sign of the charge. This is the reason why we use in Eqs B14 and B15 $\bar{\beta}_i^b(298)$ instead of $\beta_i^b(298)$. Indeed, because of the mutual interaction of anions and cations in bulk water, $\beta_i^b(298)$ decreases when salt concentration increases, which is not expected in the Stern layer of clays containing usually
1572 mostly cations counter-ions. In our model, $\overline{\beta}_i^b$ (298) is a constant that is not decreasing with

1573 salinity on the contrary to $\beta_i^b$ (298).

1574

## 1575 APPENDIX C. COMPUTED BULK WATER CHEMICAL COMPOSITION

1576 In our model, NaCl concentration in bulk water after evaporation, $C_{\text{NaCl}}^b$, is determined by:

$$C_{\text{NaCl}}^{b} = C_{\text{NaCl}}^{0} \frac{\delta_{t}^{0}}{\delta_{t}^{1}}. \qquad (\text{C1})$$

1577 HCl and NaOH electrolytes can also be added to NaCl electrolyte to change the pH. Hence,

1578 their concentrations after evaporation become

$$C_{\text{HCl}}^{b} = C_{\text{HCl}}^{0} \frac{\delta_{t}^{0}}{\delta_{t}^{1}}, \qquad (\text{C2})$$

$$C_{\text{NaOH}}^{b} = C_{\text{NaOH}}^{0} \frac{\delta_{t}^{0}}{\delta_{t}^{1}}. \qquad (\text{C3})$$

1579 Ion concentrations in bulk water, $C_i^b$, are calculated according to:

$$a_{H^+}^{b} = 10^{-pH}, \qquad (\text{C4})$$

$$a_{OH}^{b} = K_{w} / a_{H^{+}}^{b}, \qquad (\text{C5})$$

$$C_{H^{+}}^{b} \approx a_{H^{+}}^{b}, \qquad (\text{C6})$$

$$C_{OH}^{b} \approx a_{OH}^{b}, \qquad (\text{C7})$$
$$C_{\text{Cl}^-}^b = C_{\text{NaCl}}^b + C_{\text{HCl}}^b + \overline{C_{\text{Cl}^-}^b}, \qquad (C8)$$

$$C_{\text{Na}^+}^b = C_{\text{NaCl}}^b + C_{\text{NaOH}}^b + \sum_i -z_i \overline{C_{z_i^+}^b} + \sum_{i \neq j} z_j \overline{C_{z_j^-}^b} - C_{\text{H}^+}^b + C_{\text{OH}^-}^b + C_{\text{Cl}^-}^b, \qquad (C9)$$

where $\overline{C_{z_i^+}^b}$ and $\overline{C_{z_j^-}^b}$ are ion (cation and anion) concentrations from dissolved minerals and Mt interlayer space. It should be noted that more complex electrolytes can be considered in our model, but here we focus on an electrolyte containing mostly Na⁺, Cl⁻ and leached ions because we will show in Maineult et al. 2024) the comparison of our SIP model predictions to the corresponding measurements reported by Mendieta et al. (2021).

## APPENDIX D. COMPUTED ELECTRICAL POTENTIAL AT THE D-PLANE AND SURFACE CONDUCTIVITY OF THE DIFFUSE LAYER

The $\varphi_d$ potential is calculated by minimizing the following function:

$$|Q_{d1} - Q_{d2}| = 0, \qquad (D1)$$

$$Q_{d1} = \sqrt{2\overline{\varepsilon_w} k_B T 10^3 N_A \sum_{i=1}^{N} C_i^b \left[ \exp\left(-\frac{q_i \varphi_d}{k_B T}\right) - 1 \right]}, \qquad (D2)$$

$$Q_{d2} = -Q_0 (1 - f_Q), \qquad (D3)$$

where $Q_{d1}$ is the surface charge density of the diffuse layer calculated from the Gouy-Chapman theory (Hunter 1981) based on the assumption that the electrical potential in the diffuse layer does not change with the distance from the mineral surface at the end of the
1592 diffuse layer, i.e. $(d\varphi/dx)_{x=\chi_d} = 0$ V m⁻¹. The expression of the surface charge density of the

1593 diffuse layer $Q_{d2}$ (Eq. D3) was determined from

$$f_Q = \frac{Q_\beta}{Q_\beta + Q_{d2}}, \quad \text{and} \quad \text{(D4)}$$

$$Q_{d2} = -(Q_0 + Q_\beta), \quad \text{(D5)}$$

1594 which make

$$f_Q = -\frac{Q_\beta}{Q_0}, \quad \text{(D6)}$$

1595 and finally

$$Q_{d2} = -Q_0(1 - f_Q). \quad \text{(D7)}$$

1596 Eq. D5 is the electroneutrality condition.

1597 According to Schurr 1964, the low-frequency conductivity of a spherical particle of diameter

1598 *d* due to EDL polarisation and conduction is given by

$$\sigma_s = \frac{4}{d} \left( \sum_s \frac{i\omega\tau}{1 + i\omega\tau} + \sum_s^d \right), \quad \text{(D8)}$$

1599 where the first term in brackets represents conductivity from the polarisation of the bounded

1600 ion layer with surface conductivity $\Sigma_s$ (in S) (Schwarz 1962) and the second term in brackets

1601 represents conductivity from ion electro-migration in the diffuse layer with surface

1602 conductivity $\Sigma_s^d$ (Leroy et al. 2008).

1603 Leroy et al. 2017b) showed that the first term in brackets of Eq. D8 significantly

1604 underestimates the complex conductivity magnitude of clay aggregates because it does not
consider the surface area of the clay particles inside the aggregate. To reproduce the measured
Na-Mt complex conductivity magnitude, they therefore proposed to multiply this term by the
ratio of the surface area of the clay particles inside the aggregate ($S_o$) to the surface area of
the non-porous spherical particle with the same effective diameter $d_k$ ($S_k$). We used the same
method than Leroy et al. 2017b) to calculate aggregate conductivity considering EDL
polarisation and conduction in the case here of a thin diffuse layer. After some algebraic
developments, we obtained the following equation for aggregate conductivity

$$ \sigma_s = \frac{2}{3} \phi_a \beta_s f_Q Q_v \frac{i\omega\tau}{1+i\omega\tau} - \frac{2}{3g} \phi_a \frac{\sum_s^d}{Q_0} Q_v. \qquad (D9) $$

In Eq. D9, the surface conductivity of the diffuse layer can be calculated from

$$ \Sigma_s^d = 10^3 N_A \int_0^{\chi_d} \left\{ \sum_{i=1}^{N} |q_i| \left[ \beta_i^d C_i^d(x) - \beta_i^b C_i^b \right] + q_i C_i^d(x) \beta_{eo}^d(x) \right\} dx, \qquad (D10) $$

where the first term of Eq. D10 represents diffuse layer surface conductivity due to ion
electro-migration and the second term of Eq. D10 is diffuse layer surface conductivity due to
electro-osmosis (Leroy et al. 2011). By assuming that ion concentration in the diffuse layer
follows a Boltzmann's distribution, i.e. $C_i^d(x) = C_i^b \exp[-q_i \varphi(x) / k_B T]$ and that ion mobility
in the diffuse layer is equal to ion mobility in bulk water, i.e. $\beta_i^d = \beta_i^b$, and using Eq. 29, Eq.
D10 reduces to

$$ \Sigma_s^d = 10^3 N_A \int_0^{\chi_d} \left\{ \sum_{i=1}^{N} |q_i| \beta_i^b C_i^b \left\{ \exp\left[-\frac{q_i \varphi(x)}{k_B T}\right] - 1 \right\} + q_i C_i^b \exp\left[-\frac{q_i \varphi(x)}{k_B T}\right] \frac{\overline{\varepsilon_{w}}}{\eta_w} [\varphi(x) - \varphi_d] \right\} dx, \qquad (D11) $$
1619 where the electrical potential distribution in the diffuse layer $\varphi(x)$ is given by Eq. 37 or Eq.

1620 38.

1621 According to Revil & Glover 1997), Eq. D11 can be simplified to

$$ \Sigma_s^d = \chi_d 10^3 N_A \sum_{i=1}^{N} |q_i| B_i^d C_i^b \left[ \exp\left(-\frac{q_i \varphi_d}{2k_B T}\right) - 1 \right], \quad (D12) $$

$$ B_i^d = \beta_i^b + m_i^d, \quad (D13) $$

1622 where $m_i^d$ (in m$^2$ s$^{-1}$ V$^{-1}$) in Eq. D13 is related to the increase of electrical conductivity of the

1623 diffuse layer due to electro-osmosis (Bikerman 1940; Revil & Glover 1997; Leroy et al.

1624 2011). The term $m_i^d$ is written as

$$ m_i^d = \frac{2\overline{\varepsilon_w k_B T}}{\eta_w |q_i|}. \quad (D14) $$

1625 Eqs D12-D14 have been developed considering a symmetric electrolyte where cation valence

1626 $z_+$ is equal to anion valence $z_-$ (Pride 1994; Lyklema & Minor 1998). In the case of an

1627 asymmetric electrolyte, Eqs D12-D14 can be used to compute the diffuse layer surface

1628 conductivity despite being more approximate than in the case of a symmetric electrolyte

1629 (Revil & Glover 1997). Eqs D12-D14 are strictly valid for $|\varphi_d| < k_B T / (ze)$ where $z$ is the

1630 valence of the symmetric electrolyte ($k_B T / e \approx 25.7$ mV at a temperature of 298.15 K).

1631 Indeed, they are based on the Debye-Hückel (D-H) approximation, i.e. $|\varphi| \ll k_B T / (ze)$, to

1632 compute the electrical potential distribution in the diffuse layer from the $\varphi_d$ potential and the

1633 Poisson-Boltzmann (P-B) equation, i.e. $\varphi(x) = \varphi_d \exp(-x / \chi_d)$ (Hunter 1981; Pride 1994). In

1634 our model, we consider that the diffuse layer can contribute to the surface conductivity of the

1635 aggregate when the diffuse layer is thinner than the particle. For illite and Mt, this occurs
typically when the diffuse layer is highly compressed (see end of section 3.3) at high ionic strengths typically ≥ 0.1 M where the $φ_d$ potential magnitude should be low satisfying $|φ_d| < k_B T / (ze)$ (Sondi et al. 1996; Leroy et al. 2015). Therefore, in this condition, Eqs D12-D14 are valid for our calculation of the surface conductivity of the diffuse layer within the aggregate. It should be noted that Eq. D11 with Eq. 37 or 38 or Eqs D12-D14 can be used to compute the diffuse layer surface conductivity of kaolinite and illite particles at typically lower ionic strengths than Mt particles because kaolinite and in a lesser extent illite particles are thicker than Na-Mt particles and they do not have interlayer space containing hydrated and mobile cations.

If the ion mobilities are similar in the diffuse layer, Eq. D12 can be reduced to

$$ \Sigma_s^d = B_d Q_d, \qquad (D15) $$

and Eq. D9 finally becomes

$$ \sigma_s = \frac{2}{3} \phi_a \left[ \beta_s f_Q \frac{i\omega\tau}{1+i\omega\tau} + \frac{B_d}{\vartheta} (1-f_Q) \right] Q_v, \qquad (D16) $$

$$ \sigma_s = \frac{2}{3} \phi_a \left[ \beta_s f_Q \frac{i\omega\tau}{1+i\omega\tau} + \tilde{B}_d (1-f_Q) \right] Q_v. \qquad (D17) $$

The $\vartheta$ ratio can be computed using Eq. A6

$$ \vartheta = \frac{S_o}{S_k} = \frac{\delta_{sc} SS_o}{\pi d_k^2}. \qquad (D18) $$
1650

1651 APPENDIX E. COMPUTED AVERAGE ELECTRICAL POTENTIAL IN THE
1652 DIFFUSE LAYER AND INTERLAYER SPACE OF NA-MT

1653 We use the volume averaging method to compute $\phi_m$ as a function of CEC and bulk water
1654 chemical composition (Revil & Leroy 2004; Leroy et al. 2007; Leroy et al. 2017b). We
1655 assume that the (usually negative) surface charge of the clay mineral is compensated by the
1656 excess of counter-charge per unit pore volume in the Stern and diffuse layers and interlayer
1657 space of Na-Mt, i.e.

1658 where $S_{in}$ is the surface area of the interlayer space (in m²), $Q_V^{in}$ is the excess of charge per
1659 unit interlayer space volume (in C m⁻³) and $Q_V^o$ is the excess of charge per unit Stern and
1660 diffuse layer volumes. In the case of kaolinite and illite, the first two terms in Eq. E1 can be
1661 assumed to be zero (no interlayer space and interlayer space containing non-solvated K⁺ ions,
1662 respectively). In the case of Na-Mt, we assume that $Q_V^{in} = Q_V^o$. Eq. E1 then becomes

1663 If $Q_0$ and $Q_V^o$ are assumed constant throughout the pore surface and volume, respectively,
1664 and, according to Eq. A6, Eq. E2 simplifies to:

1665 for Na-Mt, and

$$ \frac{1}{V_t} \left( \int_{0}^{S_{in}} Q_0 dS + \int_{0}^{V_{in}} Q_V^{in} dV + \int_{0}^{S_o} Q_0 dS + \int_{0}^{V_S + V_w^d} Q_V^o dV \right) = 0, \qquad (E1) $$

$$ \frac{1}{V_t} \left( \int_{0}^{S_{in}+S_o} Q_0 dS + \int_{0}^{V_{in}+V_S+V_w^d} Q_V^o dV \right) = 0. \qquad (E2) $$

$$ Q_V^o = - \frac{\delta_{sc} SS}{V_{in} + V_S + V_w^d} Q_0, \qquad (E3) $$
$$Q_V^o = - \frac{\delta_{sc}SS}{V_S + V_w^d} Q_0, \qquad (E4)$$

1666 for kaolinite and illite. The excess of charge per unit pore volume in the diffuse layer and 1667 interlayer space of Na-Mt (contributing solely to conduction due to ion electro-migration) is 1668 written as:

$$Q_V^d = 10^3 N_A \sum_{i=1}^{N} q_i \overline{C_i^d}, \qquad (E5)$$

$$Q_V^d = (1 - f_Q) Q_V^o, \qquad (E6)$$

1669 and, finally, using Eqs E3, E5, and E6 for Na-Mt and Eqs E4, E5, and E6 for kaolinite and 1670 illite, the $\varphi_m$ potential was computed from

$$10^3 N_A \sum_{i=1}^{N} q_i C_i^b \exp\left(-\frac{q_i \varphi_m}{k_B T}\right) + (1-f_Q) \frac{\delta_{sc}SS}{V_S + V_S^d + V_w^d} Q_0 = 0, \qquad (E7)$$

1671 for Na-Mt, and

$$10^3 N_A \sum_{i=1}^{N} q_i C_i^b \exp\left(-\frac{q_i \varphi_m}{k_B T}\right) + (1-f_Q) \frac{\delta_{sc}SS}{V_S + V_w^d} Q_0 = 0, \qquad (E8)$$

1672 for kaolinite and illite, where $\overline{C_i^d} = C_i^b \exp(-q_i \varphi_m / k_B T)$ is the average ion concentration in 1673 the diffuse layer and interlayer space of Na-Mt or in the diffuse layer of kaolinite and illite, 1674 which follows a Boltzmann's distribution. The $Q_0$ quantity is expressed as a function of CEC 1675 using Eq. 14. We assume in Eq. E7 that the interlayer space of Na-Mt behaves like the EDL 1676 on the clay external surface. This assumption could be considered rather crude but we do not 1677 want to complicate more our model and hence add additional parameter values to be fitted in 1678 our model.

1679