# The influence of the fractal dimension on the complex conductivity of porous materials

S Qiang, André Revil, X Shi, A Maineult, J Richard, H Wang

► To cite this version:

S Qiang, André Revil, X Shi, A Maineult, J Richard, et al.. The influence of the fractal dimension on the complex conductivity of porous materials. Geophysical Journal International, 2025, 243, 10.1093/gji/ggaf329. hal-05298114

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# The influence of the fractal dimension on the complex conductivity of porous materials

S. Qiang,<fnref n="1" /> A. Revil,<fnref n="2" /> X. Shi,<fnref n="1" /> A. Maineult,<fnref n="3" /> J. Richard<fnref n="2" />,<fnref n="4" /> and H. Wang<fnref n="5" />

Accepted 2025 August 19. Received 2025 August 18; in original form 2025 January 3

## SUMMARY

The bulk component of the electrical conductivity of a porous material is related to the (connected) porosity and saturation by power-law functions defining the first and second Archie's laws. Recently, it was shown that for porous materials with fractal characteristics, the power-law exponent of Archie's law could be related to the fractal dimension of such materials. Similarly, the real and imaginary parts of the complex-valued surface conductivity are not just proportional to the specific surface area and saturation of the material but to power-law functions of these properties defining two additional 'interfacial' Archie's laws, which are called the third (saturated case) and fourth (unsaturated case) Archie's laws in this paper. These new laws have been poorly recognized and studied so far. A number of porous materials and especially clay-rich media are multiscale materials characterized by broad distributions of particle and pore sizes. We extend Archie's laws concept to describe the complex conductivity of such materials. We use both numerical simulations in fractal porous materials as well as published experimental data sets to propose a unified physical interpretation of the exponents entering the four Archie's type power-law relationships, which offer an updated complex conductivity model for natural porous media.

**Key words:** Hydrogeophysics; Electrical properties; Fractals and multifractals.

## 1. INTRODUCTION

In the realm of hydrogeophysics and petroleum engineering, induced polarization is increasingly used to non-intrusively characterize and monitor various properties and processes in the subsurface of the Earth (e.g. Vinegar & Waxman 1984; Tarasov & Titov 2007; Binley et al. 2015; Piolat et al. 2023; Revil et al. 2023a; Peshtani et al. 2024). In spectral induced polarization, the conductivity of an isotropic time-invariant porous material is defined as a frequency-dependent complex number $\sigma^* = \sigma' - i\sigma''$ ($i^2 = -1$, $i$ denotes the pure imaginary number) with an in-phase component $\sigma'$ (in $S m^{-1}$) characterizing conduction and a quadrature (out-of-phase, positive) component $\sigma''$ (in $S m^{-1}$) characterizing the ability of rocks to store reversibly electrical charges under the influence of the primary electrical field or current (Vinegar & Waxman 1984; Titov et al. 2002; Revil et al. 2017b).

When the effects of the Electrical Double Layer (EDL) around the grains can be neglected, Archie (1942) proposed the well-known and accepted first Archie's law between the so-called (resistivity)

formation factor $F$ and the (connected) porosity $\phi$ (both dimensionless). According to this relationship, the bulk conductivity is linearly related to the pore water conductivity through a property called the formation factor $F$. The formation factor depends on the (connected) porosity $\phi$ according to a power-law relationship $1/F = \phi^m$ called the first Archie's law in the realm of petrophysics. The porosity exponent $m$ (dimensionless, usually loosely referred to the cementation exponent) is usually close to 2 but can be comprised between 1 and 4 (Friedman 2005).

Neglecting the effects of the EDL and for partially saturated materials, the ratio between the bulk conductivity at a given saturation $S_w$ and the conductivity at full saturation (the inverse of the so-called resistivity index) is equal to a power-law function of saturation, that is, $S_w^n$ where $n$ (dimensionless) denotes the saturation exponent. The saturation exponent $n$ is also typically comprised between 1 and 4 and very often $n \approx m$ (Revil 2013). These power laws, considered by many research scientists as empirical in nature, are actually well-explained by theories such as the differential effective medium theory (Sen et al. 1981; Mendelson & Cohen 1982; Cosenza et al.

<footnote n="1">¹Key Laboratory of Surficial Geochemistry of Ministry of Education, School of Earth Sciences and Engineering, Nanjing University, Nanjing 210023, China.
E-mail: shixq@nju.edu.cn</footnote>

<footnote n="2">²Université Savoie Mont-Blanc, CNRS, UMR CNRS 5204, EDYTEM, 73370 Le Bourget du Lac, France. E-mail: andre.revil@univ-smb.fr</footnote>

<footnote n="3">³Laboratoire de Géologie, Ecole Normale Supérieure, CNRS UMR 8538, PSL Research University, 75005 Paris, France</footnote>

<footnote n="4">⁴Nāga Geophysics, 229 Rue Joseph Fontanet, 73000 Chambéry, France</footnote>

<footnote n="5">⁵College of Geophysics, Chengdu University of Technology, Chengdu 610051, China</footnote>
2021), the continuum percolation theory (Hunt 2004) and the fractal theory (Katz & Thompson 1985; Wei et al. 2015; Cai et al. 2017; Wang & Liu 2017). Researchers have observed that m and n vary with different pore textures (e.g. Friedman 2005; Niu & Zhang 2018; Sun & Torres-Verdín 2022), wettabilities and pore fluid distributions (Anderson 1986; Han et al. 2009; Montaron 2009; Maineult et al. 2018; Qiang et al. 2022), and fractal dimensions (Wei et al. 2015; Chen et al. 2023; Zhu et al. 2023).

In clay-rich media, clay minerals are surrounded (like all minerals) by an EDL. The excess charges contained in the diffuse and Stern layers are responsible for an excess conductivity called the surface (interfacial) conductivity. Working on the results of Winsauer and McCardell, Waxman & Smits (1968) developed a linear conductivity model in which the so-called surface conductivity acts in parallel to the bulk conductivity described by Archie (1942). In modern petrophysics, this assumption works only in the so-called high-salinity asymptotic limit above the so-called isoconductivity point (e.g. Bernabé & Revil 1995).

Furthermore, in presence of the EDL coating the surface of the minerals, the porous material is also able to store reversibly electrical charges at the grain or pore scales in presence of an applied (external) electrical field. This phenomenon is called induced polarization in geophysics (Schlumberger 1920; Vinegar & Waxman 1984). Induced polarization is an active galvanometric method that was initially developed in France by Conrad Schlumberger (1920).

Waxman & Smits (1968) demonstrated that the two Archie's laws remain valid in presence of an EDL because they are not defined as conductivity laws as erroneously written in a growing number of papers in the literature, especially in hydrogeophysics. They are actually defined as intrinsic relationships between the dynamic totuosity and the porosity and saturation (e.g. Johnson et al. 1986; Revil et al. 2019). Surface conductivity can be connected to the surface per pore volume ratio or to the cation exchange capacity (Waxman & Smits 1968; Tenchov 1991; Titov et al. 2002). In partially saturated media (water is considered to be the wetting phase with respect to the solid phase), and according to Waxman & Smits (1968) (see also Waxman & Thomas 1974), surface conductivity depends on saturation according to a power-law relationship $S_w^p$ with $p = n - 1$.

An induced polarization model of clay-rich media (called the Dynamic Stern Layer model) has been developed to account for saturation (Revil et al. 2023a), anisotropy (Revil et al. 2023b), the presence of semiconductors like pyrite and magnetite (Revil et al. 2024). This model was recently used by El Alam et al. (2024) to discuss the electrical properties of sand-bentonite mixture characterized by very high Cation Exchange Capacities (CEC, in C kg⁻¹). Furthermore, El Alam et al. (2025) demonstrated that induced polarization can be used to monitor the process of illitization of smectite in bentonite with hyperalkaline pore waters.

Clayey materials have a multiscale nature (e.g. Krohn 1988; Sun et al. 2020). The grain or pore sizes of these materials can range over several orders of magnitude (Ougier-Simonin et al. 2016; Philipp et al. 2017) forming various assemblages such as tactoids and aggregates (Dor et al. 2020). This multiscale nature is expected to impact the complex conductivity of these materials. Meanwhile, recently derived conductivity models point out the dependence of the exponents m and n on the pore textures or pore water phase morphologies of the saturated or unsaturated porous media. Vinegar & Waxman (1984) showed that the quadrature conductivity depends on saturation according to the same power-law relationship as surface conductivity, that is, according to $S_w^p$ (with $p = n - 1$) pointing out the key-role of the EDL on these two properties (Vinegar &

Waxman 1984; Revil 2012). The $p = n - 1$ relationship between the saturation exponents fits well for most low-CEC and water-wet porous media (Revil 2013; Deng et al. 2018). We restrict ourselves in this paper to water-wet porous media.

Slater & Lesmes (2002), Revil & Florsch (2010) and Weller et al. (2010) have pointed out that the surface conductivity, the quadrature conductivity, and the normalized chargeability depend on the specific surface area of porous media (see also Revil 2012, 2013; Weller & Slater 2015; Cosenza et al. 2021, for further discussions on this topic). Linear and power-law relationships have been observed without a consensus on this dependence. Recently, it has been demonstrated through fractal theory that the surface conductivity of fractal-type porous media depends on the specific surface area according to a power-law relationship (Wang & Revil 2020). The power-law exponent of this relationship is theoretically related to the porosity exponent entering Archie's law and to the pore volume fractal dimension (Wang & Revil 2020). This relationship is called an interfacial Archie's law in this work to emphasize its significance, as it exhibits a similar power-law correlation to classical Archie's law for the bulk conductivity, but extends it to the petrophysical relationships between surface conductivity and interfacial properties of porous media. However, such type of relationship has not been demonstrated for the quadrature conductivity or the normalized chargeability so far. Furthermore, the exponent p that describes the dependence of quadrature/surface conductivity and normalized chargeability on saturation still lacks an accepted physical interpretation associated with interfacial properties (Qiang et al. 2024).

In this work, we first extend the interfacial Archie's law to the relationships between quadrature conductivity (or normalized chargeability) and specific surface area. The resulting relationships are referred to as the third and fourth Archie's laws, corresponding to saturated and unsaturated conditions, respectively. Numerical simulations with ideal and granular fractal geometries, as well as multiple core sample data sets, are used to test the model predictions. Meanwhile, core samples with mercury porosimetry data are used to test the dependence of the 'specific surface area exponent' q in the interfacial Archie's law on the pore volume fractal dimension. We then propose a unified physical interpretation of the four power-law exponents associated with complex conductivity.

## 2. THEORY

### 2.1. An interfacial Archie's law for quadrature conductivity

We first consider a water-saturated isotropic and time-invariant porous material. The complex conductivity of such medium is defined as $\sigma^* = \sigma' - i\sigma'$ where $\sigma'$ and $\sigma'$ denote the in-phase and quadrature conductivities (both defined as positive). The conductivity at the scale of a representative elementary volume of a porous rock or soil can be obtained by volume-averaging the local Ohm's law valid in the pore space of the porous medium. This upscaling procedure yields the following macroscopic conductivity equation,

$$ \mathbf{J} = \sigma \mathbf{E}, \qquad (1) $$

$$ \sigma' = \frac{\sigma_w}{F} + \frac{\Sigma_S}{F_S}, \qquad (2) $$

$$ \Sigma_S = \int_0^\infty (\sigma(\chi) - \sigma_w) d\chi, \qquad (3) $$
<figure><img src="image_2.png" /><figcaption>**Figure 1.** 2-D sketch of the simulation domain and the local dissipation of energy $\mathbf{j}e_b$ for an example porous medium with insulating grains (in white), j denotes the local current density, $\hat{e}_b$ denotes the normalized local electrical field. The left and right boundaries are potential boundaries with potential difference. Top and bottom boundaries are insulating boundaries. The grey lines with arrows are current lines. The dark area in both pore space and grain surface indicates higher local dissipation of energy. The uneven distribution of dissipation of energy indicates that pore space and surface in different positions would contribute differently to the formation factor F and the surface formation factor $F_S$. Thus, the complex conductivity is not dependent solely on the porosity or specific surface area, but also on the pore and interface geometries of the domain.</figcaption></figure>

where J is the conduction current density (A m⁻²), E (V m⁻¹) is the harmonic electrical field, ΣS (S) is called the specific surface conductance or conductivity and is a characteristic of the surface of the grains, χ (m) is the radial distance normal to the surface of the grain (χ =0), σ(χ) (S m⁻¹) is the local conductivity in the EDL coating the surface of the grains, and σw (S m⁻¹) is the pore water conductivity (i.e. in the pore space outside the double layer). The inverse of bulk and surface formation factors entering eq. (2) are defined by volume averaging the local Joule dissipation of energy (Wang & Revil 2020),

$$ \frac{1}{F} = \frac{1}{V} \int_{V_p} |\hat{e}_b|^2 dV_p, \qquad (4) $$

$$ \frac{1}{F_S} = \frac{1}{V} \int_S |\hat{e}_b|^2 dS, \qquad (5) $$

where $\hat{e}_b$ (dimensionless) denotes the local electrical field normalized by the applied electrical field and $dV_p$ denotes a pore volume element of the connected pore volume $V_p$ (m³), and $dS$ denotes a surface element of the total surface area S (m²) between the solid and water phases. The quantity V (m³) denotes the volume of interest through which the macroscopic electrical field is applied. Eqs (4) and (5) are simply obtained by upscaling the dissipation of energy (see Johnson et al. 1986; Revil & Glover 1998) and assuming that the specific surface conductivity ΣS implies a small perturbation of the local electrical field close to the mineral surface. In this case, $\hat{e}_b$ is simply obtained by solving the Laplace equation for the electrical potential in the pore space with microscopic and macroscopic boundary conditions (see Fig. 1).

It has been recognized in the last three decades that the EDL properties influence in the same way both the surface conductivity and

the quadrature conductivity (or the normalized chargeability, see for instance Vinegar & Waxman 1984). Staying in the high-salinity asymptotic limit (where pore space is the dominant path for the electro-migration of the cations, see Johnson et al. 1986; Bernabé & Revil 1995), we generalize eq. (2) to the complex conductivity σ* by writing:

$$ \mathbf{J} = \sigma^* \mathbf{E}, \qquad (6) $$

$$ \sigma^* = \frac{\sigma_w}{F} + \frac{\Sigma_S^*}{F_S}, \qquad (7) $$

The complex conductivity σ* is therefore frequency-dependent with a kernel of relaxation times that depends on the distribution of the polarization length scales of the material. For porous media characterized by multiscale pore radius distributions (e.g. power-law or loguniform distribution), we expect that the corresponding spectra obey a Constant Phase Angle (CPA) model. What is elegant in the form of eq. (7) is that σw characterizes the properties of the pore water while ΣS* characterizes the properties of the surface of the clay particles and both are independent of the pore-space geometry (Niu et al. 2016).

To show how the two formation factors F and F_S can be numerically obtained by upscaling the local Joule dissipation of energy, we show in Fig. 1 the local dissipation of energy $\wp$ in a granular porous material computed as:

$$ \wp = \mathbf{j} \cdot e_b, \qquad (8) $$

where j denotes the local current density (Revil et al. 2018). The dissipation of energy distributes along both the pore space and the interface of the grains. This observation combined with eqs (4) and (5) implies that the bulk and surface conductivities are not just dependent on the porosity and the specific surface area, which corresponds to geometrical quantities defined as:

$$ \phi = \frac{1}{V} \int_{V_p} dV_p, \qquad (9) $$

$$ \frac{S}{V_p} = \frac{1}{\phi} \frac{1}{V} \int_S dS, \qquad (10) $$

respectively. The bulk and surface conductivities also depend on the geometry and strength of the local electrical field. We can therefore write the associated relationships in terms of the first and third Archie's laws as,

$$ \frac{1}{F} = a\phi^m, \qquad (11) $$

$$ \frac{1}{F_S} = a_S \left( \frac{\phi S}{V_p} \right)^q, \qquad (12) $$

respectively, and where m denotes the Archie's first exponent (porosity exponent) and q is called the specific surface area exponent in Wang & Revil (2020) and hereinafter. These two exponents define the decoupling effects between the 'dynamic' properties F and F_S and the geometrical properties φ and S/V, respectively. The term dynamic is here used to say that the pore space and the surface area of the solid/water interface are dynamically connected by the current lines (see Johnson et al. 1986). Note that while F is scale-invariant, this is not the case for F_S. The quantity a (dimensionless, Winsauer et al. 1952) is a constant fixed to 1 to ensure 1/F = 1 when φ → 1, while a_S (dimensionless) is also a constant (see definition in Wang & Revil 2020). We use a_S to replace the parameter 1/a_S in the original work of Wang & Revil (2020) for simplification, and we also use q to replace the exponent m_S in Wang & Revil (2020) to highlight
the significance of this exponent as an independent parameter with different physical mechanisms from the cementation factor m.

Wang & Revil (2020) developed a surface conductivity model for fractal-based porous media neglecting the effect of polarization. The (intrinsic) pore volume fractal dimension Df used in their work describes the self-similarity of the porous media (see Barnsley et al. 1988, for details). It is defined as Df = ln(N)/ln(L) where N is the number of fractal objects with a downscaling factor L. Then, Wang & Revil (2020) obtained the following expressions of m and q:

$$m \ge 1, \qquad (13)$$

$$q = m + \frac{m-1}{D - D_f - 1} \le 1, \qquad (14)$$

where D = 3 (dimensionless) denotes the Euclidean dimension of the embedding volume in 3-D. Eq. (14) is valid under the assumption that Df > D - m⁻¹, which holds for most natural porous media (see for instance through the data sets of Zhang & Weller 2014). Note that if m and q are independently determined from experimental complex conductivity data (which is indeed the case), Df can be independently determined. This cannot be done only with (in-phase) conductivity measurements, but with complex conductivity measurements, this is straightforward thanks to the relationship between surface conductivity and normalized chargeability as discussed further below. This point is the first result of importance in the present paper that will be used to interpret experimental data and determine their pore volume fractal dimension Df. The derivation of eq. (14) is based on systems with fractal pore volumes and smooth grain surfaces (Pape et al. 1982). Accordingly, this study does not consider the intrinsic surface roughness of grains. Nevertheless, incorporating a fractal surface would modify the specific surface area exponent q, without altering the power-law form of the interfacial Archie's law.

## 2.2. Dependence on the cation exchange capacity

A grain immersed in water is characterized by an EDL coating its surface. The charge density of the EDL counters the (usually negative) charge density on its surface. The double layer comprises the Stern layer of sorbed counterions and the diffuse layer for which the concentrations (activities) are controlled by Boltzmann distributions in the Coulombic field created by the mineral surface charge density (see Fig. 2 for a sketch of EDL). The Outer Helmholtz Plane (OHP) is defined as the boundary between the Stern layer and the diffuse layer (e.g. Leroy et al. 2013). For non-colloidal clay-assemblages, we consider that the diffuse layer of clay particles is overlapping and cannot be easily polarized. We consider that the Stern layer is discontinuous and can be regarded as a leaking capacitance. For a given clay particle, we write the complex specific surface conductance as,

$$\Sigma_S^*(\omega) = \Sigma_S^\infty - \frac{\Sigma_S^\infty - \Sigma_S^0}{1 + (i\omega\tau)^{\frac{1}{2}}}, \qquad (15)$$

where ω = 2πf (in rad s⁻¹) denotes the angular frequency, f (Hz) is the measured frequency and τ (s) denotes the relaxation time related to the particle size (see Tarasov & Titov 2013 and Weller & Slater 2022 for an extensive discussion around the Cole–Cole functions). The exponent ½ (Warburg exponent) in eq. (15) is justified by the leaking character of the Stern layer because of sorption/desorption of the counterions in and from the Stern layer (Revil et al. 2014). The relaxation time is given by τ = r²/(2D) with r (m) is the grain radius when taking grain as the characteristic polarization length scale and

D (m² s⁻¹) is the electro-diffusion coefficient for the counterions of the Stern layer. This diffusion coefficient can be related to the mobility of the counterions of the Stern layer according to the Einstein equation. The instantaneous and DC surface conductance are given by

$$\Sigma_S^\infty = \Sigma_S^d + \Sigma_S^s. \qquad (16)$$

$$\Sigma_S^0 = \Sigma_S^d, \qquad (17)$$

respectively. The diffuse and Stern layer contributions are written as (see Fig. 2c),

$$\Sigma_S^d = \beta_{(+)} (1 - f_S) Q_S. \qquad (18)$$

$$\Sigma_S^s = \beta_{(+)}^s f_S Q_S. \qquad (19)$$

where β_{(+)} and β_{(+)} (in m² s⁻¹ V⁻¹) denote the mobility of the counterions in the diffuse and Stern layers, respectively (in the diffuse layer the mobilities are the same as in the free water outside the double layer), f_S (dimensionless) denotes the fraction of counterions in the Stern and therefore 1 - f_S denotes the fraction of counterions in the diffuse layer, and Q_S (in C m⁻²) denotes the total surface charge density of the double layer. With these definitions and after some straightforward algebraic manipulations, the apparent mobilities of the counterions for surface conduction B (in m² s⁻¹ V⁻¹) and for Stern layer associated with the quadrature conductivity λ (in m² s⁻¹ V⁻¹) are related to the mobilities of the Stern and diffuse layer by (see Revil et al. 2017d),

$$B = \beta_{(+)} (1 - f_S) + \beta_{(+)}^s f_S. \qquad (20)$$

$$\lambda = \beta_{(+)}^s f_S. \qquad (21)$$

Now that we have defined the two mobilities, we can come back to the expressions of the instantaneous conductivity and normalized chargeability of the porous material given by

$$\sigma_{\infty} = \frac{\sigma_w}{F} + \frac{\Sigma_S^{\infty}}{F_S}, \qquad (22)$$

$$M_n = \frac{\Sigma_S^\infty - \Sigma_S^0}{F_S}, \qquad (23)$$

respectively. We can write these equations as,

$$\sigma_{\infty} = \frac{\sigma_w}{F} + \frac{1}{F_S} (B Q_S), \qquad (24)$$

$$M_n = \frac{\lambda Q_S}{F_S}, \qquad (25)$$

Next, we consider the relationship between the specific surface area S_v and counterion content per volume Q_vφ (expressed in C m⁻³) where Q_v denote the volumetric excess charge density (see definition in Waxman & Smits 1968). If we replace S_v in the interfacial Archie's law with the more easily obtainable ratio between Q_vφ and surface charge density Q_S, the interfacial Archie's law is then written as:

$$\frac{1}{F_S} = a_S \left( \frac{Q_v \phi}{Q_S} \right)^q, \qquad (26)$$

and therefore we have,

$$\sigma_{\infty} = \frac{\sigma_w}{F} + a_S B Q_S^{1-q} (Q_v \phi)^q, \qquad (27)$$

$$M_n = a_S \lambda Q_S^{1-q} (Q_v \phi)^q. \qquad (28)$$

this implies M_n ∝ (Q_vφ)q when the remaining part of eq. (28) is constant for a specific type of material.
<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 2.** Structure of the electrical double layer. (a) Mineral grain with a charge density $Q_S$ and surrounded by an electrical double layer. (b) Structure of the double layer with the Stern and diffuse layers, OHP stands for the Outer Helmholtz Plane, corresponding to the interface between the Stern and diffuse layers at distance $x_S$ from the mineral surface. (c) Specific surface conductance associated with the Stern and diffuse layers.</figcaption></figure>

## 2.3. Constant phase angle (CPA) model

Now what happens if the pore radius follows a wide multiscale distribution? For instance, we can consider that the grain size $r$ follows a power-law, lognormal or loguniform distribution. In this case, the relaxation times also follow the same distribution. This assumption is true for clayey materials because of their multi-scale nature as discussed above (see Fig. 3). First, we recall that the CPA model has been used by various researchers in the literature in induced polarization studies (e.g. Van Voorhis et al. 1973; Weller et al. 1996). The amplitude of the conductivity can be

written as,

$$ \sigma(\omega) = \left( i \frac{\omega}{\omega_n} \right)^b \sigma_n, \qquad (29) $$

where $\sigma_n$ is the amplitude of the conductivity at the angular frequency $\omega_n$ ($\omega = 2\pi f$ and $i^2 = -1$) and $b$ (fractional) is the frequency exponent (Bates et al. 1986). With the sign convention that $\sigma^*(\omega) = \sigma'(\omega) - i\sigma''(\omega)$, the in-phase and quadrature
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Example of clayey porous material with a variety of scales for the heterogeneities. (a, b) Crystals with nanoporosity (interfoliar porosity for 2:1 clay minerals). (c, d) Microporosity associated with tactoids and aggregates. (e) Additional heterogeneity scales with the presence of silt and sand and macroporosity. The scales are just provided as a guide.</figcaption></figure>

conductivities are given by,

$$ \sigma'(\omega) = \left(\frac{\omega}{\omega_n}\right)^b \sigma_n \cos\left(\frac{\pi}{2}b\right), \quad (30) $$

$$ \sigma''(\omega) = \left(\frac{\omega}{\omega_n}\right)^b \sigma_n \sin\left(\frac{\pi}{2}b\right), \quad (31) $$

respectively and where $\varphi = \pi b/2$ (rad) denotes the constant phase angle or shift between the current density and the electrical field. Since the phase is small (<50 mrad, $b \ll 1$), we have the following approximations

$$ \cos\left(\frac{\pi}{2}b\right) \approx 1, \quad (32) $$

$$ \sin\left(\frac{\pi}{2}b\right) \approx \frac{\pi}{2}b = \varphi, \quad (33) $$

$$ \tan\left(\frac{\pi}{2}b\right) \approx \varphi \approx \frac{\sigma''_n}{\sigma_n}, \quad (34) $$

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

$$ \sigma'(\omega) = \left(\frac{f}{f_n}\right)^b \sigma_n(f_n), \quad (35) $$

$$ \sigma''(\omega) = \left(\frac{f}{f_n}\right)^b \sigma''_n(f_n). \quad (36) $$

where $f_n$ is the reference frequency (usually $f_n = 1$ Hz).

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

Voorhis et al. 1973; Revil et al. 2017d):

$$ \sigma''(\sqrt{f_1 f_2}) \approx \frac{M_n(f_1, f_2)}{\alpha}, \quad (37) $$

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

$$ \sigma''_n \approx \frac{M_n}{9}. \quad (38) $$

If we consider now the dynamic Stern later model developed by Revil et al. (2017a, b, c), we have in absence of metallic grains,

$$ \sigma'(f) = (f)^b \left[ \frac{1}{F} \sigma_w + a_S B Q_S^{1-q} (Q_v \phi)^q \right], \quad (39) $$

$$ \sigma''(f) = (f)^b \frac{a_S \lambda Q_S^{1-q} (Q_v \phi)^q}{9}. \quad (40) $$

These equations describe the in-phase and quadrature conductivity dependence with the frequency $f$.

# 3. NUMERICAL SIMULATIONS

## 3.1. Pore size distribution and CPA model

To study the impact of the type of pore radius distribution on the phase behaviour of the complex conductivity, we ran some 3-D tube network simulations on a cubic mesh. To do so, we extended the methodology for 2-D square networks which was developed by Maineult et al. (2017, 2018), which was also successfully applied
<figure><img src="image_6.png" /><figcaption>**Figure 4.** An example of results for a 50 × 50 2-D square network in the DC case. (a) Lognormal distribution of the radius (standard-deviation 0.5), (b) random network, (c) computed electrical potential field and (d) computed electrical current (expressed in percentage of the maximum of the absolute value of the local electrical current).</figcaption></figure>

in Maineult et al. (2018) and Mendieta et al. (2023). To summarize, we consider here a 20 × 20 × 20 cubic network of tubes having the same length but different radii r.

For each frequency f, we imposed a potential difference ΔV between two opposed faces, and no fluxes for the four remaining lateral faces. Applying Kirchhoff's law (Kirchhoff 1845) at each node of the network, that is, the sum of the electrical fluxes at a node is equal to zero, we obtain a linear system of equations whose solution provides the electrical potential at each node. Then it is possible to compute the electrical fluxes in each tube linking two nodes using Ohm's law. To illustrate the procedure, Fig. 4 shows the case of a 2-D square network in the DC case, with a random network that obeys a lognormal distribution of pore radius, and the associated electrical potential field and electrical currents. From the ΔV value and the total electrical current outflowing from the network (i.e. the sum of the longitudinal fluxes arriving at the downstream face), I(f), we can finally calculate the amplitude and phase spectra (see Maineult et al. 2017, 2018, for details). As in Maineult et al. (2018),

we used a Pelton model in conductivity for each tube:

$$ \sigma^* = \sigma_0 \left( 1 + \frac{M}{1-M} \left( 1 - \frac{1}{1 + (i2\pi f \tau)^{1/2}} \right) \right), \quad (41) $$

where $\sigma_0 = 0.01 \text{ S m}^{-1}$ denotes the DC conductivity, $M = 0.1 \text{ V V}^{-1}$ is the chargeability and $\tau$ the relaxation time, related to the radius and a diffusion coefficient $D = 10^{-9} \text{ m}^2 \text{ s}^{-1}$ (see Revil et al. 2021) by $\tau = r^2/(2D)$, and c the Cole–Cole exponent that we took equal to 0.5 (Warburg model, see explanations in Revil et al. 2014). Other values of the Cole–Cole exponents are investigated in the Supplementary Information.

The radii were randomly distributed, using two types of distributions. The first one is the lognormal distribution (as used in Maineult et al. 2017), whose cumulative function is given by:

$$ P(\text{Log}_{10}(r) \le X) = \frac{1}{2} + \frac{1}{2}\text{erf}\left(\frac{X - \text{Log}_{10}(r_{\text{peak}})}{\text{SD}\sqrt{2}}\right), \quad X \in \mathbb{R}, \quad (42) $$
<figure><img src="image_7.png" /><figcaption>**Figure 5.** Amplitude and phase spectra for the 3-D lognormal case used to describe the pore size distribution (using a standard-deviation increasing from 0.01 to 1.5). We observe that the resulting spectra do not obey a constant phase model.</figcaption></figure>

where SD stands for the Standard-Deviation. For the $r_{peak}$ value we took 10 µm.

The second one is a loguniform distribution, for which the logarithm of the radius is uniformly distributed between two values, with a cumulative function provided by:

$$P(\text{Log}_{10}(r) \le X) = \frac{X - \text{Log}_{10}(r_{\min})}{\text{Log}_{10}(r_{\max}) - \text{Log}_{10}(r_{\min})}, \quad X \in [\text{Log}_{10}(r_{\min}), \text{Log}_{10}(r_{\max})]. \quad (43)$$

The cumulative probability as a function of $\text{Log}_{10}(r)$ for these two distributions is shown in Fig. S1 in the Supporting Information. The computed spectra are shown in Figs 5 and 6 for the lognormal and loguniform cases, respectively. In the first case, the phase spectra are bell-shaped, with a decreasing maximum value and an increasing width with a wider pore radii distribution, as also evidenced by the decrease of the Cole–Cole exponent with increasing SD (Fig. 7a), similar to what was reported in Maineult et al. (2017). In the second case, when the pore radius distribution width increases, the maximum value of the phase also decreases, but phase tends to have a constant value in the classical frequency range investigated in the laboratory (i.e. 1 mHz to 1 kHz). That is, the complex conductivity follows a CPA model resulting from the superposition of the Warburg model associated with each polarization length. We estimated the b value as the maximum value of the phase divided by $\pi/2$: it significantly decreases with a wider pore radii distribution (Fig. 7) (see the Supporting Information file for the cases c = 0.25, 0.75 and 1.0 in Figs S2–S7).

Note that in this section, we only proved the correlations between the loguniform pore radius distribution (rather than the power-law or fractal distribution) and CPA response. However, the simulated complex conductivity spectra are indeed controlled by the constric- tivity function of the electrical conductivity assigned for each tube (e.g. eq. 41), and it is still not clear which form of constrictiv- ity function is more in line with the natural rock or soil samples. Thus, we can not yet conclude here whether it is natural materials

<figure><img src="image_8.png" /><figcaption>**Figure 6.** Amplitude and phase spectra for the 3-D loguniform case for the pore sizes (using $\text{Log}_{10}(r_{\min})$ decreasing from −6 to −7.5 and $\text{Log}_{10}(r_{\max})$ increasing from −4 to −2.5). We observe that the resulting complex con- ductivity spectra obey more and more to a constant phase model when the loguniform distribution corresponds to a broad pore size distribution.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 7.** Evolution of the Cole–Cole exponent as a function of the standard deviation for the spectra obtained using the lognormal distribution (a), and evolution of the b-value as a function of $\text{Log}_{10}(r_{\max}/r_{\min})$ for the spectra obtained using the loguniform distribution (b).</figcaption></figure>
with loguniform or power-law pore size distributions that would yield a CPA response in practice. Future work is needed to further explore the complex conductivity spectra shape under power-law distributions using different constrictivity functions. Nevertheless, the result has proved that the polarization of different materials is closely related to their pore textures, which provides a foundation for the following research on Interfacial Archie's law.

## 3.2. Numerical approach for interfacial Archie's law verification

Despite the upscaling of the dissipation of energy, the surface formation factor $F_S$ could also be determined by the ratio between the quadrature conductance $\Sigma_S''$ and quadrature conductivity of porous media, which could be solved using numerical approach. Following eq. (7) and $\Sigma_S^* = \Sigma_S' - i\Sigma_S'$, where $\Sigma_S^*$ is the complex surface conductance. The quadrature conductivity could be written as:

$$ \sigma'' = \frac{\Sigma_S''}{F_S}. \qquad (44) $$

Then the surface formation factor $F_S$ is obtained as:

$$ F_S = \frac{\Sigma_S''}{\sigma''}. \qquad (45) $$

where $\Sigma_S''$ denotes the quadrature surface conductance. Note $\Sigma_S''$ could be affected by factors such as pore water salinity, temperature and frequency. It is therefore enough to fix the value of $\Sigma_S''$ and simulate numerically the electrical problem to compute the value of the surface formation factor with the routine in eq. (45), and providing another verification of the interfacial Archie's law.

Here, we utilized the numerical simulations method, which calculates the complex conductivity of porous media by upscaling the complex surface conductance of the EDL on the grain surface to the pore or solid phase (Niu & Zhang 2017; Niu et al. 2020). We use the following workflow for the numerical solution of $F_S$:

(1) The concerned domain is first assigned with water and solid phases according to the desired porous media texture. The side length of the domain and boundary conditions used for the 2-D and 3-D models are identical to Fig. 1 (boundary insulating conditions on all faces except where we assign the A and B current electrodes). We solve the electrical potential obeying Laplace equation.

(2) The dynamic pore size length $\Lambda$ (in m, see Johnson et al. 1986) and the effective particle size $\Pi$ (in m) of each domain with a specific pore structure are calculated thanks to the finite element code COMSOL Multiphysics® (version 5.4) as:

$$ \frac{2}{\Lambda} = \frac{\int_S |\hat{\varepsilon}_b|^2 dS}{\int_{V_p} |\hat{\varepsilon}_b|^2 dV_p}, \qquad (46) $$

$$ \frac{2}{\Pi} = \frac{\int_S |\hat{\varepsilon}_b|^2 dS}{\int_{V_s} |\hat{\varepsilon}_b|^2 dV_s}. \qquad (47) $$

(3) The extra complex conductivity that will be added to the water (increment $\Delta\sigma_w^*$) or solid phase (increment $\Delta\sigma_s^*$) in addition to their intrinsic values are calculated as:

$$ \Delta\sigma_w^* = \frac{2\Sigma_S^*}{\Lambda}, \qquad (48) $$

$$ \Delta\sigma_s^* = \frac{2\Sigma_S^*}{\Pi}. \qquad (49) $$

(4) After adding the complex conductivity increments to the corresponding phases, $\sigma^*$ is obtained by calculating the relative ratio of

the average applied electrical field to resulting current density, that is, $\sigma^* = J/E$ using the finite element code COMSOL Multiphysics. (5) Finally, the surface formation factor is determined by the ratio of the assigned quadrature surface conductance $\Sigma_S''$ to the simulated quadrature conductivity $\sigma''$.

## 3.3. Interfacial Archie's law in ideal fractal models

In this section, we test the prediction of the surface formation factor with both interfacial Archie's law and the scaling numerical simulations for ideal fractal models. The term 'ideal' indicates that the fractal models have a consistent scaling unit (generator) and deterministic scale invariance. The models evaluated here include the Sierpinski Carpet (SC, Allouche & Shallit 2003) with stages 1-4, Menger sponge (MS) with stages 1-3, and the fractal model B in the work by Roy & Tarafdar (1997), which has nine small cubes out of the total 27 are solid (RTB, Roy and Tarafdar fractal model, see Fig. 1c in their works) with stages 1-3. All fractals are incorporated into a 1 mm domain as described in Fig. 1 (2-D for SC and 3-D for MS and RTB). The sketches of the second stage of SC and the generators of MS and RTB are shown in the subplot of Fig. 8. Fig. 8 plots the surface formation factor predicted by four different approaches:

(1) The pore or grain upscaled complex conductivity simulation (red circle and grey triangle, eq. 45) in Section 3.2. The surface conductance $\Sigma_S^*$ for all model is set to $5.2 \times 10^{-10} + 2.7 \times 10^{-10}i$ S using eq. (15) with $B = 5.2 \times 10^{-8} \text{ m}^2 \text{ s}^{-1} \text{ V}^{-1}$, $Q_S = 0.025 \text{ C m}^{-2}$, and $D_S = 3.8 \times 10^{-12} \text{ m}^2 \text{ s}^{-1}$ while the water conductivity $\sigma_w$ is set to $10 \text{ S m}^{-1}$ to meet the high-salinity asymptotic limit for the interfacial Archie's law.

(2) The interfacial Archie's law based on the fractal theory (black cross), we first calculate the value of $q$ by using eq. (14) with the fitted values of both $m$ and $D_f$ for each type of fractal model, then, since $a_S$ is a parameter (not adjustable for specific pore structure) that cannot be determined by numerical modeling, the value of $a_S$ for each model series is acquired by fitting the simulated $F_S$ using the Joule dissipation of energy and $S_v$ with a power-law distribution. The values of $a_S$ are close to each other for different types of fractal models (see values in Table S1 in the Supporting Information file) since $a_S$ is obtained from the value of $F_S$ when $S_v=1$, which is a small fraction of the surface. Additionally, we can get the surface formation factor by eq. (12) with known specific surface area using the calculated value of $q$, and the fitted value of $a_S$ as described above.

(3) The Waxman-Smits model from Waxman & Smits (1968) (see the W-S, blue rectangles).

(4) The differential effective medium model from Bussian (1983, DEM, cyan star). More details regarding these two models can be explicitly found in the Supporting Information and Wang & Revil (2020).

The $F_S$ predicted by the interfacial Archie's law and pore upscaling quadrature conductivity simulations are consistent with the $F_S$ simulated using the Joule dissipation of energy (the black solid line) with $R^2 = 0.993$ and 0.987, respectively. While the $F_S$ predicted by grain upscaling complex conductivity simulations (grey triangles) is slightly higher than the simulated value with $R^2 = 0.970$. The significant overestimation of $F_S$ by the Waxman-Smits model implying that the surface of the fractal models is less tortuous than the pore space, which is different from the capillary model with two resistive elements in parallel as assumed by the Waxman-Smits
<figure><img src="image_10.png" /><figcaption>**Figure 8.** Comparison of the surface formation factor $F_S$ obtained by different methods for ideal fractal materials. The simulated $F_S$ on x-axis is obtained by integrating the dissipation of energy along the solid water interface. The predicted $F_S$ is obtained by (1) the pore or grain upscaling complex conductivity simulation (filled circles and filled inverted triangles, eq. 45), (2) interfacial Archie's law based on the fractal theory (crosses, eq. 14), (3) the Waxman-Smits (W-S) model (filled rectangles) and (4) the Differential Effective Medium (DEM) model (stars, see Wang & Revil 2020 for details). The subplots show the sketches of the ideal fractal models used to generate the data points. SC denotes the Siepinski Carpet (Allouche & Shallit 2003), MS denotes the Menger sponge and RTB denotes the fractal of Roy & Tarafdar (1997). The alignment of the data points with the simulated $F_S$ confirms the validity of interfacial Archie's law for ideal fractal materials.</figcaption></figure>

model (for a complete discussion, see Bernabé & Revil 1995). By contrast, the DEM model underestimates $F_S$ because it assumes the media to be spheres (or circles in 2-D) inclusions embedded in the pore space, in which the pore surface is more tortuous compared

with the pore space. These different hypothetical pore textures may explain why the two models have opposite behaviours.

Meanwhile, the decrease in the pore water conductivity below the isoconductivity point can lead to a variation in the apparent value of $F_S$. For example, the surface formation factor calculated by the grain upscaling method varies with changing water/grain conductivity ratio due to the macroscale salinity effect, as shown in Fig. S8 in the Supporting Information file (we use the fourth stage of the SC fractal model as an example). However, both the derivation of interfacial Archie's law and all pore water salinity used in our simulations are in the high salinity asymptotic limit, for which the bulk conductivity in the pore space corresponds to the dominant conduction mechanism for the conduction process (Bernabé & Revil 1995). Thus, we will not discuss further the salinity dependence of $F_S$ for different numerical modelling method and their underlying mechanism (more details can be found in Revil et al. 2019).

### 3.4. Interfacial Archie's law in granular materials with fractal characteristics

To test the interfacial Archie's law in non-ideal fractal materials, which show certain self-similarities at some length scales but do not have consistent generators and deterministic scale invariance, we construct a series of 2-D granular materials with fractal characteristics. Like ideal fractal models, the granular materials with fractal characteristics are also set to have different stages. For each stage, different amounts of granular grains are added to the domain (see Fig. 9). The radius $r_s$ (m) and amount $N_s$ (dimensionless) of grains added at each stage follow:

$$N_s = \frac{C_s}{r_s D_s}, \qquad (50)$$

where $C_s$ (dimensionless) is a constant of proportionality, $D_s$ (dimensionless) is the fractal dimension of the grain radius distribution. Note that $D_s$ is conceptually different from the previous pore volume fractal dimension $D_v$. For the construction of each granular material series, we set the value of $N_s$ as [4,8,16,32,64] for the five stages, and calculated the corresponding $r_s$ of added grains using

<figure><img src="image_11.png" /><figcaption>**Figure 9.** Sketch of 2-D granular materials with fractal characteristics at different stages for the verification of interfacial Archie's law. The parameter $D_s$ and $C_s$ control the size of grains (shown in black) added to the pore space at each stage. The box counting pore volume fractal dimensions $D_b$, cementation factor $m$ and specific surface area exponent $q$ vary with different geometry stages and series, hence expecting different specific surface area exponent $q$.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Relationship between the inverse of surface formation factor $1/F_s$ and specific surface area $S_v$ for granular materials with fractal characteristics (see two examples in Fig. 9). The plotted series have $D_s$ in eq. (50) ranging from 1 to 1.3 with 0.05 step and $C_s$ equals to $(4 \times 10^{-4})^{D_s}$. Each data series denotes an average result of 10 parallel realizations due to the randomness in the pore geometry constructions. The solid lines indicate interfacial Archie's law (power law) fit with $R^2 > 0.99$. The deviation in the curves shows that granular materials with different fractal features have different specific surface area exponents $q$.</figcaption></figure>

eq. (50). Fig. 9 shows two cases of the evaluated schemes with different $D_s$ and $C_s$. During the construction of the porous media, the grain positions at each stage are randomly selected but the interval between each grain follows a minimum of $1 \times 10^{-4}$ mm to each other to smooth grid meshing of the finite element solution. The number of added grains $N_s$ in a few cases may change due to space limitations, which would distort the grain size distribution and finally the corresponding fractal behavior. Together with the random positions of added grains at each stage, the final models will not exhibit consistent fractal behavior across all length scales or stages. However, the pore geometry still shows fractal behavior at certain length scales, thus could yield granular geometries with different apparent (average) pore volume fractal dimensions, which is also the case for some natural porous materials.

Fig. 10 plots the resulting inverse of surface formation factor acquired using eq. (5) versus specific surface area for different granular materials, with the set $D_s$ in eq. (50) ranging from 1 to 1.3 with a 0.05 step and $C_s$ equals to $(4 \times 10^{-4})^{D_s}$. Each data series in Fig. 10 denotes an average result of 10 parallel realizations due to the randomness in the pore geometry construction. While all the plotted data series exhibit power-law behaviour, the deviation of the curves indicates different specific surface area exponents $q$, which are controlled by the pore geometries and the local dissipation of energy.

Despite the relationship between surface formation factor and specific surface area, another key point of the interfacial Archie's law is the dependence of the specific surface area exponent $q$ on the pore volume fractal dimension $D_f$, as demonstrated by eq. (14). Unfortunately, the pore volume fractal dimension $D_f$ could not be determined for porous media that do not exhibit consistent generators and deterministic scale invariance (Li et al. 2022). There is also

<figure><img src="image_13.png" /><figcaption>**Figure 11.** Comparison between the estimated box-counting dimensions $D_b$ and the predicted pore volume fractal dimensions $D_f$ of the 2-D granular materials with fractal characteristics. $D_f$ is predicted from the fitted specific surface exponent $q$ and $m$ through eq. (14). The box-counting pore volume fractal dimension $D_b$ is estimated from the corresponding pore geometry at their final stage. The results contain granular materials with fractal characteristics series with setting $D_s$ range from 1 to 2, and $C_s$ from $(2E - 4)^{D_s}$ to $(6E - 4)^{D_s}$ in eq. (50). The solid line indicates a linear fit with a regression coefficient $R^2 = 0.85$. As we cannot determine the true pore volume fractal dimensions $D_f$ of the granular materials with fractal characteristics, the linear correlation between the two fractal dimensions indicates the potential of using specific surface area exponent $q$ to predict the fractal properties based on eq. (14).</figcaption></figure>

no generally accepted method to prove whether the calculated $D_f$ value is precise (Wei et al. 2015). Alternatively, we introduce here another definition of fractal dimension: the box-counting pore volume dimension $D_b$ (also called Minkowski-Bouligand dimension, see Li et al. 2009). The box-counting method determines the fractal dimension by covering the pore space with a series of rectangular boxes. For a fractal target with self-similarity, the number $N(\varepsilon)$ (dimensionless) of boxes of size $\varepsilon$ (dimensionless, equal to the number of pixels along the box sides) needed to cover a fractal set should follow a power law with the exponent equal to the box-counting fractal dimension:

$$D_b = \lim_{\varepsilon \to 0} \frac{\log N(\varepsilon)}{\log \frac{1}{\varepsilon}}, \qquad (51)$$

where $D_b \le D$. In practice, $D_b$ is obtained from the slope of the linear fitting of $\log N(\varepsilon)$ versus $\log(1/\varepsilon)$ for set values of $\varepsilon$, see Fig. S9 in Supporting Information for the corresponding box-counting curves of the pore geometry series in the first row of Fig. 9. The estimated box-counting pore volume dimension $D_b$ could restore the 'apparent' self-similarity of the pore volume, thus, the predicted pore volume fractal dimension $D_f$ and $D_b$ should be somehow correlated (Falconer 1988). In fact, different types of fractal dimensions of a porous medium could be inherently correlated, even though they have different target phases and calculation methods (Dathe & Thullner 2005). Fig. 11 shows a comparison of the predicted $D_f$ and $D_b$, despite the data from the curves shown in Fig. 10, the result from granular materials with fractal characteristics with set $D_s$ ranging from 1 to 2, and $C_s$ from $(2 \times 10^{-4})^{D_s}$ to $(6 \times 10^{-4})^{D_s}$ in
eq. (50) are also plotted. The pore volume fractal dimension $D_f$ for each pore geometry series is estimated by eq. (14) using the fitted $q$ and $m$ from $1/F_S \sim (S_v)^q$ and $1/F \sim \phi^m$ curves, respectively. The box-counting pore volume fractal dimensions $D_b$ are estimated from the corresponding pore volume at the final stage of each series as apparent values (see Fig. 9). The relationship between the two fractal dimensions is confirmed by their linear correlation. The scattered nature of the data points in Fig. 11 is caused by the randomness of the pore geometry generation process, which leads to uncertainties in the fitted $q$ and $m$, as well as in the estimated $D_b$ at the final stage. Note that we are unable to prove the validity of eq. (14) using Fig. 11 because we cannot have the true pore volume fractal dimensions $D_f$. However, we can validate eq. (14) from a qualitative point of view by demonstrating the feasibility of using specific surface area exponent $q$ to predict pore fractal properties, which is confirmed by the linear correlation between $D_b$ and predicted $D_f$ in Fig. 11.

# 4. COMPARISON WITH LABORATORY DATA

## 4.1. Interfacial Archie's law in natural clayey materials

This section aims to assess the applicability of interfacial Archie's law for natural clayey materials. If we assume that the apparent mobilities of the counterions for Stern layer quadrature conductivity $\lambda$ and surface charge density $Q_s$ are fixed for clayey materials (see Revil et al. 2017d), eq. (25) can be directly equivalent to

$$M_n = a'_S S_v^q, \qquad (52)$$

where $a'_S$ is a new coefficient incorporating the surface properties of clay particles. Fig. 12 plots the normalized chargeability versus the specific surface area $S_v$ of clayey material data sets from multiple literatures (Börner & Schön 1991; Lesmes & Frye 2001; Binley et al. 2005; Weller et al. 2011; Revil et al. 2013; Robinson et al. 2018; Mendieta et al. 2021; Peshtani et al. 2022, 2024). The specific surface area $S_v$ for the data sets except for Revil et al. (2013) and Mendieta et al. (2021) is obtained from $S_{por,\phi}$, while for the remaining two data set $S_v$ is estimated from the specific surface area per mass $S_{sp}$ by $\rho_g(1 - \phi)S_{sp}$, where $\rho_g$ is grain density and set to 2650 kg m$^{-3}$. Some of the datasets plotted in Fig. 12 are obtained thanks to the collection and correction by Weller & Slater (2015) of the relevant information. Following the linear correlation between $\sigma''$ and $M_n$, the normalized chargeabilities for the data reported by Börner & Schön (1991), Lesmes & Frye (2001), Binley et al. (2005) and Revil et al. (2013) are obtained using the quadrature conductivity data with $M_n = 8\sigma''$ (Revil et al. 2017d), the quadrature conductivity measurements for those data sets are conducted at around 1 Hz with an equivalent $\sigma_w = 0.1$ S m$^{-1}$. While for Mendieta et al. (2021), $M_n$ is obtained by $\sigma_{\infty} - \sigma'_0$ with the low and high frequencies to be $1.4 \times 10^{-3}$ Hz and $1.5 \times 10^3$ Hz, respectively. For Peshtani et al. (2022, 2024) and Robinson et al. (2018), $M_n$ is obtained directly from the original papers. The interfacial Archie's law fits for these data sets yield a power-law curve with $q = 0.66 \pm 0.12$ (95 per cent confidence bounds) and $R^2 = 0.52$ for data sets excluding the mudstones. While we got a $q = 0.79 \pm 0.20$ (less than 1) and $R^2 = 0.76$ for the mudstones data alone.

Meanwhile, considering the proportional relationship between the counterion content per volume $Q_v\phi$ and specific surface area, we have:

$$M_n = a'_S (Q_v \phi)^q, \qquad (53)$$

<figure><img src="image_14.png" /><figcaption>**Figure 12.** Relationship between normalized chargeability $M_n$ and specific surface area $S_v$ for clayey materials. Data compiled from multiple literature (Börner & Schön 1991; Lesmes & Frye 2001; Binley et al. 2005; Weller et al. 2011; Revil et al. 2013; Robinson et al. 2018; Mendieta et al. 2021; Peshtani et al. 2022, 2024). The plotted data fits well with the interfacial Archie's law (solid and grey line) with a specific surface area exponent $q = 0.66$ for clayey materials data excluding mudstones and $q = 0.79$ for mudstones data alone.</figcaption></figure>

where $a'_S$ also depends on the interface properties of porous media. When $q$ is close to 1, these relationships are consistent with the proposed linear models between $M_n$ and the specific surface area of clayey materials (see Revil 2013; Revil et al. 2017d). Fig. 13(a) plots the normalized chargeability versus $Q_v\phi$ for a broad range of clayey material data sets. Among the data sets, the clayey materials are collected by Revil et al. (2015), as well as four data points from Mendieta et al. (2021). The rammed earth and soil data are from Abdulsamad et al. (2020) and the soil data are from Netherlands samples (unpublished). The normalized chargeability in all data sets except the unpublished soils is also obtained using the quadrature conductivity data with $\alpha = 8$. The interfacial Archie's law fit for all data sets in Fig. 13(a) yields a $q = 0.77 \pm 0.12$ and $R^2 = 0.55$. Together with the fitted $q = 0.66$ in Fig. 12, the nature of the specific surface area exponents $q$ to be close to 1 indeed leads to the coexistence of the observed linear or power-law relationships between the normalized chargeability (or quadrature conductivity) and the specific surface area.

## 4.2. Pore volume fractal dimensions for clayey materials and volcanic rocks

In this section, we assess the possibility of using complex conductivity to predict the pore volume fractal dimension information of the real core samples based on the interfacial Archie's law. First, Fig. 13(b) shows the formation factors versus the corresponding porosities of the data sets used in Fig. 13(a) in log-log plots. The fit of Archie's law yields a cementation factor $m = 1.76$ for a broad
<figure><img src="image_15.png" /><figcaption>**Figure 13. (a).** Relationship between normalized chargeability $M_n$ and the counterion content per volume $Q_v\phi$ for various materials. The data set for clayey materials is collected by Revil et al. (2015). The rammed earth and soils are from Abdulsamad et al. (2020) and the soils are unpublished data from Netherlands samples. The plotted data fit well with interfacial Archie's law (solid line) with a specific surface area exponent $q = 0.77$. (b). Relationship between the formation factor and the porosity for the core samples and the fit with the empirical Archie's law. The data sets yield a cementation factor $m = 1.76$. Together with the fitted $q = 0.77$ in panel (a), we obtain a mean value of the pore volume fractal dimension $D_f$ equal to ~2.76.</figcaption></figure>

range of clayey materials. Meanwhile, from eq. (14), we have:

$$D_f = \frac{1-q}{q-m} + D, \quad (54)$$

and $q = 1$ follows from $D_f = D$. Combining the value of $q = 0.77$, we obtain a pore volume fractal dimension $D_f = 2.76$ for clayey materials in Fig. 13(a) as an average value. However, although the clayey material data in Figs 12 and 13(a) basically follow power-law relationships, the data points still show some scattered nature at the same $S_v$ or $Q_v\phi$, and the fitted specific surface area exponents $q$ (the slope of the fitting curve) and $a'_S$ vary with different independent datasets (Weller et al. 2010). The deviation is more evident when considering the coloured clayey materials data points and the grey mudstone data in Fig. 12. This is reasonable because $q$ as defined in eq. (14) varies with different materials which exhibit different textures (e.g. $D_f$ in different length scales, $m$, and intrinsic $a'_S$). Thus, we note here that the $q$ determined in Figs 12 or 13(a) are only mean values (and not unique) for the presented wide variety of materials. Meanwhile, the exponent $q$ for each dataset alone may also contain uncertainties due to the scattered nature of the limited measured data. For example, the Shahejie sandstones data set in Fig. 12 (Blue rectangle, Peshtani et al. 2022) alone yields $q = 0.25 \pm 0.17$, which yields a $D_f$ between 2.48 and 2.60 using mean $m = 1.87$. Thus, it is essential to consider these uncertainties in $q$ when evaluating the prediction accuracy and practical application of eq. (54).

To further test the practicality of eq. (54), we present the normalized chargeability versus the counterion content per volume for a data set of altered volcanic rocks in Fig. 14(a). These altered volcanic rocks exhibit different mineralogy (Revil et al. 2021) and hence are expected to have significantly different fractal features from the clayey materials. The plot data include the samples from Humu'ula saddle region, Kilauea volcano, Hawai'i (Revil et al. 2021), la Soufrière and Papandayan volcanoes (Revil et al. 2022), Whakaari, Yellowstone, and Furnas volcanoes (Ghorbani et al. 2018). The normalized chargeability of all samples except for Kilacuea, la Soufrière, and Papandayan volcanoes is obtained

using the quadrature conductivity data with $M_n = \alpha\sigma''$ with $\alpha = 4.42$ (Revil et al. 2022). As demonstrated by Fig. 14(a), the fitted specific surface area exponent $q$ of the volcanic rocks is close to 1 ($1.08 \pm 0.14$). The $q > 1$ here can be attributed to the presence of minerals (e.g. magnetite) with higher magnetic susceptibility and polarization strength in the samples, such as those from la Soufrière and Yellowstone area (Ghorbani et al. 2018; Peshtani et al. 2022). While some sandstones and sand-clay mixtures have also been observed to exhibit $q > 1$ (Weller et al. 2010), this may also be due to the non-fractal behaviour of the materials at certain length scales or to the fitting uncertainties arising from limited data points; however, the exact reasons remain speculative and require further laboratory tests. In Fig. 14(b), the plotted formation factors versus the corresponding porosities yield a cementation factor $m$ around 2.40, indicating a more tortuous pore network for these samples than the clayey materials with $m = 1.76$. Together with the fitted specific surface area exponent $q$ around 1 in Fig. 14(a), we have a value of $D_f$ very close to 3 for these altered volcanic rocks, which is higher than that for the clayey materials. This is reasonable because $D_f = 3$ corresponds to a uniformly distributed pore radius that fulfills the porous medium, which is consistent with the extremely tortuous pore networks of the volcanic rocks. This comparison of the predicted $D_f$ shows that the proposed Interfacial Archie's law, while can not provide a uniquely determined fractal dimension for data with uncertainties, can help to characterize their fractal properties qualitatively.

### 4.3. Use of the capillary pressure curve

The ability of complex conductivity measurements to predict the fractal properties could be further assessed and verified using single core samples. As discussed before, the pore volume fractal dimension $D_f$ could not be determined for those natural core samples without a deterministic scale invariance, so it is impossible to make a quantitative comparison between the real and predicted $D_f$ of a given core sample. Alternatively, we introduce here another
<figure><img src="image_16.png" /><figcaption>**Figure 14.** (a). Relationship between normalized chargeability $M_n$ and the counterion content per volume $Q_v\phi$ for altered volcanic rocks. The plot data includes the samples from Humu'ula saddle region and Kilauea volcano, Hawai'i (Revil et al. 2021), la Soufrière and Papandayan volcanoes (Revil et al. 2022), Whakaari, Yellowstone, and Furnas volcanoes (Ghorbani et al. 2018). The plotted data fit the interfacial Archie's law (solid line) with a specific surface area exponent $q = 1.08$. (b). Relationship between the formation factor and the porosity for the altered volcanic rocks and fits with the empirical Archie's law. The data set yields a cementation factor $m = 2.40$. Together with $q$ in panel (a) we acquire a pore volume fractal dimension $D_f$ very close to 3.</figcaption></figure>

parameter called ‘capillary pressure fractal dimension’ $D_{cp}$. This parameter is a pore volume fractal dimension derived from the capillary pressure curves during drainage experiments (e.g. Zhang & Weller 2014). There are plenty of routines to determine $D_{cp}$, which may yield correlated but different results for the same core sample. Here we use the routine proposed by Friesen & Mikula (1987) because it organizes the capillary pressure fractal dimension $D_{cp}$ calculations based on the ideal Menger Sponge fractal structure, which is also one of the models classically used in the derivation of the interfacial Archie's law (Wang & Revil 2020). The proposed equation connecting $D_{cp}$ and the capillary pressure curves is:

$$ \log\left(-\frac{dV_p}{dr}\right) \sim (2 - D_{cp}) \log(r), \qquad (55) $$

where $V_p$ (m³) is the pore volume with a fixed pore radius $r$ (m). For the mercury porosimetry data, the corresponding pore volume at a given pressure can be replaced by the mercury saturation $S_{Hg}$ (dimensionless). Meanwhile, following the modified Young-Laplace equation (Jurin 1717):

$$ P = \frac{2\sigma \cos(\theta)}{r}, \qquad (56) $$

where $P$ (in Pa) denotes the capillary pressure in a cylindrical pore, $\sigma$ (N m⁻¹) denotes the surface tension, $\theta$ (rad) denotes the contact angle between the injected and replaced fluids. Eq. (56) indicates that $r \propto 1/P$. Then by inserting eq. (56) into eq. (55), we have:

$$ \frac{dS_{Hg}}{dP} \propto P^{(D_{cp}-2)} \left(-\frac{1}{P^2}\right) = P^{-(4-D_{cp})}, \qquad (57) $$

and:

$$ \log(dS_{Hg}/dP) = (D_{cp} - 4) \log(P) + \log(C), \qquad (58) $$

<figure><img src="image_17.png" /><figcaption>**Figure 15.** The log(dS<sub>Hg</sub>/dP) versus log(P) plot for the sandstone sample S490 from Revil et al. (2014) and volcanic rock HG583 from Revil et al. (2017c). The capillary pressure fractal dimension $D_{cp}$ is estimated by fitting the incorporated equation. Only the data that exhibit linear trends ($R^2 > 0.95$) are used.</figcaption></figure>

where C is a constant, and the capillary pressure fractal dimension $D_{cp}$ can be derived from the slope of log(dS<sub>Hg</sub>/dP)-log P curve. Note that for natural porous materials, there may not be a constant linear relationship between log(dS<sub>Hg</sub>/dP) and log(P) in the log-log plots along the whole capillary range (Wang et al. 2019). See Fig. 15, the data points become scattered and irregular in high and
<figure><img src="image_18.png" /><figcaption>**Figure 16.** Relationships between the capillary pressure pore fractal dimension $D_{cp}$ and predicted pore volume fractal dimension $D_f$ of single samples in Revil *et al.* (2014) for sandstones and mudstone (light and dark crosses) and Revil *et al.* (2017c) for volcanic rocks (black circles). The grey solid line corresponds to the 1:1 reference line. The linear correlations for each data set indicate that we could predict the fractal properties of porous materials with complex conductivity measurements, $Q_v\phi$ or specific surface area and porosity data.</figcaption></figure>

low $\log(P)$ ranges. This means that their pore volume does not exhibit a consistent capillary pressure fractal dimension $D_{cp}$ across all pore radii. Thus, the data are screened first to remove those with capillary curves that do not show linear trends (linear fitting $R^2$ higher than 0.95) in the log-log plots over a wide pressure range (>20 data points).

To test the fractal dimension predictions with interfacial Archie's law, we access two data sets from Revil *et al.* (2014, 2017c) with simultaneous core complex conductivity measurements and mercury injection pressure curves. The first dataset contains four sandstones and mudstone (S498, S490, S493 and S439) samples and the second concerns a series of fresh and altered volcanic rocks. Fig. 16 plots the comparison between the predicted pore volume fractal dimension $D_f$ and capillary pressure fractal dimension $D_{cp}$ for the screened data sets. The specific surface area exponent $q$ used for $D_f$ determination is calculated by eq. (53), where the value of $a'_S$ for the sandstones and mudstones is taken from the clayey materials in Fig. 13(a). For the volcanic rocks, $a'_S$ is taken from the volcanic dataset itself. $D_{cp}$ is determined by fitting the $\log(dS_{tig}/dP)$ – $\log(P)$ curves of each sample. Notably, the $D_{cp}$ in Fig. 16 for sandstone and mudstones slightly overestimate the predicted $D_f$, and the $D_{cp}$ for the volcanic rocks is much smaller than the predicted $D_f$. This deviation between $D_{cp}$ and predicted $D_f$ are also observed in core sample datasets. For instance, the predicted $D_f$ value between 2.48 and 2.60 for Shahejie formations (blue rectangles) in Fig. 12 is much smaller than the measured $D_{cp}$ range from 2.769 to 2.986 (Zhang & Weller 2014). There are many possible reasons for these deviations in addition to the data uncertainties. For example, the value of $D_{cp}$ would change with the pressure range chosen (Friesen & Mikula 1987; Ding *et al.* 2017), while as shown in Fig. 15, the linear part data chosen for the $D_{cp}$ estimation for the volcanic samples HG583 indeed could not reveal the overall slope (related to fractal properties) of the data points, despite the effective

capillary pressure range for HG583 being neither as wide as the sandstone samples S490. $D_{cp}$ may also deviate from $D_f$ because of the bottleneck effect that causes a distortion of the true pore volume distribution, which shows remarkable restriction in the access of larger pores for the volcanic rocks. On the other hand, the presence of certain minerals (e.g. magnetite, see Revil *et al.* 2017c) in the volcanic samples can increase the predicted $D_f$ by influencing the fitted exponent $q$, thereby enlarging the difference between $D_{cp}$ and predicted $D_f$.

The difference between $D_{cp}$ and predicted $D_f$ again emphasizes that, due to the large uncertainties within the measured data, the proposed approach is intended here for the relative estimation of the fractal properties of natural materials, rather than the accurate determination of a specific type of fractal dimensions. Accordingly, the aim of this section is solely to test whether the pore volume fractal dimension $D_f$ predicted by the interfacial Archie's law are related to the estimated capillary pressure fractal dimension $D_{cp}$, rather than to directly compare unique values. For each dataset in Fig. 16, the two fractal dimensions are linearly correlated (see the dashed lines indicating linear fits for sandstones/mudstones and volcanic rocks with $R^2 = 0.88$ and 0.73, respectively). According to Revil *et al.* (2017c), it has been observed that for shallow fresh volcanic rocks (the depth of the sample is shown by the sample number in Fig. 16), their quadrature conductivity at constant $Q_v\phi$ varies with the tortuosity of rock samples. This phenomenon is reflected in the fractal dimensions in Fig. 16, where the fractal dimensions (and corresponding specific surface area exponents) increase with the sample depth of the volcanic rock, indicating the dependence of the pore volume fractal dimensions $D_f$ on the tortuosity of porous media (Coleman & Vassilicos 2008; Wei *et al.* 2015). These results for single core samples again highlight the potential of complex conductivity measurements to predict fractal properties: We could first have the value of specific surface area exponent $q$ through the plot of quadrature conductivity or normalized chargeability versus specific surface area, together with the cementation factor $m$ acquired through in-phase conductivity measurements, we get the predicted pore volume fractal dimensions by eq. (54). This fractal dimension information can further help us to determine the permeability and other hydrogeological parameters, which are critical for the characterization and identification of subsurface porous media (Katz & Thompson 1985; Yu & Liu 2004; Cai *et al.* 2020).

# 5. DISCUSSION

## 5.1. Interfacial Archie's law for unsaturated case

The interfacial (third) Archie's law for the saturated case denotes a power-law relationship between the surface or quadrature conductivity (or normalized chargeability) and the specific surface area of a porous material characterized by a fractal dimension. The extension of this power-law relationship to the unsaturated case is rather attractive because it could offer the geometrical information of pore water distributions, which can be of great importance in fields such as oil recovery (Wang *et al.* 2018) or organic pollutant removal (Wang *et al.* 2023). However, due to the inability of simple fractal models to simulate the complex water-air-solid phase interactions in unsaturated conditions, the fourth interfacial Archie's law is difficult to verify numerically. The situation could be more challenging in laboratory data sets because the fractal dimensions of the pore volume occupied by water in unsaturated cases are rarely
measured with induced polarization data. Thus, we propose aninterfacial Archie's law for the unsaturated case (or the so-called fourth Archie's law) only for comparison with the first and the second Archie's laws. Taking the quadrature conductivity as an example, we write:

$$ \frac{\sigma_{(S_w)''}}{\sigma_{(S_w)=1}''} = \frac{F_{S(S_w=1)}}{F_{S(S_w)}} = \left( \frac{S_v(S_w)}{S_v(S_w=1)} \right)^t \quad (59) $$

where $S_v(S_w)$ denotes the effective interfacial surface area between the water and solid phases. The exponent $t$ is the so-called specific surface area—saturation exponent, which is empirical so far due to the lack of mathematical explanation. The equation is valid when the specific surface area of the smooth air/water interface is much smaller than that of the rough solid/water interface. As water-air and water-solid contacts usually exhibit different fractal and electrochemical behaviour, the extension of eq. (59) to both air/water and solid/water interface may involve adding an extra power law to the model with different exponents. It is noted that the values of $t$ and the exponent $q$ in the third Archie's law are expected to be close to each other as they both describe the dependence of quadrature conductivity on the specific surface area (or interface area between water and grain phase in the unsaturated case). This hypothesis is deduced from the broadly acknowledged similarity between the Archie's exponents for the bulk components of in-phase conductivity in saturated (m) and unsaturated (n) cases (e.g. Gkortsas et al. 2018). For instance, the observed $m$ is found to be equal to $n$ for many materials (Revil 2013), and the theoretical interpretations of both $m$ and $n$ are focused on the change rate of the conductive phase (Glover 2009, 2017; Yue 2019; Qiang et al. 2022). However, the values of $t$ and $q$ can be different for the same porous materials depending on the pore space structures and pore water distributions. Further mathematical and statistical verification of such speculative relationships requires more realistic pore network models with fractal characteristics and laboratory evidence. We keep this task for future investigations.

## 5.2. A new interpretation of the Archie's law exponents

The interfacial Archie's law indicates that the quadrature conductivity, surface conductivity and normalized chargeability of porous media are directly controlled by their specific surface area $S_v$, rather than by porosity $\phi$ or saturation $S_w$. However, we can still write them in the form of porosity- or saturation-dependence by introducing an assumption that the specific surface area of porous media is correlated with their porosity. For example, for a two-component composite when the pore space acts as the host of the incrementally non-overlapping solid grains (see for instance the numerical porous media in Fig. 9), the specific surface area of the material could be directly related to the fraction of the solid phase $(1 - \phi)$ (see Du Plessis 2007; Hussaini & Dvorkin 2021a), considering the case that the added grain size is not fixed, we can write the relationship in the form of a power-law:

$$ S_v(\phi) \propto (1 - \phi)^{\lambda_q}, \quad (60) $$

where $\lambda_q$ (dimensionless) is defined as an exponent describing the rate of change of the relative specific surface area with the fraction of the solid phase. The value of $\lambda_q$ varies among different materials (Hussaini & Dvorkin 2021a, b), which is controlled by their particle or pore size distribution (Du Plessis 2007; Tinni et al. 2014). It is easy to know that $\lambda_q = 1$ for grain inclusions all with the same size and patterns. According to eq. (12), the interfacial Archie's law (the

third Archie's law) can be written as:

$$ \frac{1}{F_S} = a_S(S_v)^q, \quad (61) $$

and based on the assumption of relationship between $S_v$ and $1 - \phi$ (eq. 60) for media in which pore space acts as the host of the solid grains:

$$ \frac{1}{F_S} \propto ((1 - \phi)^{\lambda_q})^q. \quad (62) $$

Next, considering the case where porous media are modelled as solids with cavity inclusions as used in the effective medium model (Eshelby 1957), which is also the case for some porous materials (Hussaini & Dvorkin 2021a), the specific surface area could be directly related to $\phi$ rather than $(1 - \phi)$ (Mavko et al. 2020), so we have:

$$ \sigma'_s, \sigma'', M_n \propto \phi^{\lambda_q q}, \quad (63) $$

where $\sigma'_s$ denotes the (in-phase) surface conductivity. Compared with the existing direct empirical relationship between surface/quadrature conductivity, normalized chargeability and porosity: $\sigma'_s, \sigma'', M_n \propto \phi^q$, we have:

$$ v = \lambda_q q. \quad (64) $$

Similar derivations could be made to the unsaturated case, where the saturation exponent $p$ for $\sigma'_s, \sigma'',$ and $M_n$ can be written as the product of two exponents:

$$ p = \lambda_p t, \quad (65) $$

where $\lambda_p$ denotes the rate of change of interface surface area between water and solid phases with saturation.

The idea that the exponents are the product of two parameters indicates that these exponents are controlled by two mechanisms. The first part $\lambda_q$ or $\lambda_p$ is the change rate of (effective) specific surface area with porosity or water content, and the latter part $q$ or $t$ is how these additional interface areas contribute to the porous medium 'ability to restore reversible charges'. Eq. (64) can be verified using the numerical simulation results of both the saturated ideal and granular materials with fractal characteristics. We estimate the $\lambda_q, q$ and the equivalent exponent $v$ for each pore geometry series by fitting the $(1 - \phi) \sim S_v, S_v \sim 1/(F_S)$ and $(1 - \phi) \sim 1/(F_S)$ curves with power laws, respectively. Note that these fitted exponents are not the true values for specific pore geometry at a specific stage, but an effective value of the whole construction process. Fig. 17 shows that the fitted $v$ is close to the product of the fitted $\lambda_q$ and $q$. Since existing fractal models cannot simulate the complex interaction between saturation and specific surface area in unsaturated systems, we did not verify eq. (65) for unsaturated conditions numerically. However, the further verifications and mathematical derivation of the exponent $t$ might be done by more complex and realistic pore models with fractal characteristics.

As follows, we consider the similar power-law relationship between the bulk components of the in-phase conductivity and porosity or water content, following eqs (64) and (65), the first and second (Archie's) exponents that related the bulk components of the in-phase conductivity can also be written as:

$$ m = \lambda_m m_b, \quad (67) $$

$$ n = \lambda_n n_b, \quad (68) $$

where $\lambda_n$ and $\lambda_m$ denote the rate of change of the 'connected' porosity or water content, respectively with their intrinsic value.
<figure><img src="image_19.png" /><figcaption>**Figure 17.** Verification of the two-part interpretation of the exponent in the power-law relationship between quadrature conductivity and porosity. *v* denotes the exponent taken from the 1/*F*<sub>S</sub> ~ (1 − *φ*)<sup>*v*</sup> curves, which is supposed to be controlled by two physical mechanisms as *v* = *λ*<sub>q</sub>*q*, where *λ*<sub>q</sub> denotes the change rate of specific surface area with the volume fraction of solid phase, taken from the *S*<sub>v</sub> ~ (1 − *φ*)<sup>*v*·q</sup> curves, *q* denotes the specific surface area exponents, taken from the 1/*F*<sub>S</sub> ~ *S*<sub>v</sub><sup>*q*</sup> curves. Both the results from the ideal and granular materials with fractal characteristics are plotted.</figcaption></figure>

The concept of ‘connected’ here denotes the fraction of porosity or water content that effectively contributes to the conduction of charge carriers, which could also refer to the ‘effective’ porosity or water content as defined in Koponen *et al.* (1997). *λ*<sub>n</sub> and *λ*<sub>m</sub> are also related to the percolation effect of the conductivity, where the change rate of the ‘connected’ porosity or water content varies significantly near the percolation threshold (Navarre-Sitchler *et al.* 2009). The latter parts *n*<sub>b</sub> and *m*<sub>b</sub> in eqs (67) and (68) denote how the connected porosity or water content contributes to the ‘ability to conduct charges’ of the porous media, that is, the same additional ‘connected’ porosity or water content may contribute differently to

conductivity according to the different local distribution of the dis-
sipation of energy. Similarly, Glover (2009, 2017) interpreted the
exponents *m* and *n* as the rate of change of the fractional connect-
edness with (1) porosity or saturation and (2) connectivity of pore
or pore water phase. The former part of this interpretation is con-
ceptually consistent with λ_n or λ_m and the latter could be somehow
revealed by the definition of m_b or n_b.

Considering eqs (64), (65), (67) and (68), we can summarize
that there is a universal theoretical interpretation of the power-law
exponents related to the different components of the complex con-
ductivity. The first component of these exponents is the change rate
of the controlling physical parameter (such as connected porosity
or water content and specific surface area) with the true porosity or
saturation, and the second is how significant the controlling physical
parameter would contribute to the ability of the porous medium to
conduct or restore electrical charges. The two components are con-
trolled by the local potential distribution of the porous material in
the macroscopic electrical field. We acknowledged that it is difficult
to test whether eqs (64), (65), (67) and (68) can really contribute
to providing a robust prediction of saturation exponents because
verifying the unified expressions for each component is difficult,
despite there being plenty of works focused on their analytic deter-
minations (Sen 1984; Wei et al. 2015; Zhang et al. 2021). However,
through these simple equations, we provide a two-part physical in-
terpretation of the power-law exponents, which helps us to better
understand the physics of exponents and find the possible parame-
ters such as pore volume fractal dimensions that can have an impact.
These findings are important for a broad range of applications in
goelectrical data interpretation.

**5.3. The empirical relationship p = n - 1**

We come back now to the empirical relationship between the power-
law exponents for the bulk and quadrature (surface) components of
the complex conductivity in unsaturated conditions, that is the re-
lationship *p* = *n* − 1. This relationship has been observed in many
natural porous media (Vinegar & Waxman 1984; Revil 2013, but
not applicable for some scenarii such as when pore water salin-
ity changed with saturation, e.g. desiccations as in Zhang *et al.*
2019). This relationship could be equivalent to *v* = *m* − 1 for the

<figure><img src="image_20.png" /><figcaption>**Figure 18.** The fitted exponent (a) *v* and (b) *q* versus the fitted cementation exponent *m* for both saturated ideal fractals and granular materials with fractal characteristics. The light circles indicate granular materials with uniform grains where *λ*<sub>q</sub> = 1. The data follows the relationship *q* = *m* − 1 well, rather than *v* = *m* − 1 (the latter corresponds to the well-established relationship *p* = *n* − 1 for the unsaturated case, Vinegar & Waxman 1984). Considering *v* = *λ*<sub>q</sub>*q*, the results indicate that the empirical *v* = *m* − 1 relationship is built on a true assumption that *λ*<sub>q</sub> for most porous materials is very close to 1.</figcaption></figure>
saturated case. Following the interfacial Archie's law, the relationships between $v$ and $m$ are controlled by both the change rate of specific surface area with porosity and the pore volume fractal dimensions. Fig. 18 plots the fitted $v$ and $q$ versus $m$ of the ideal fractal models and granular materials with fractal characteristics (the three red circles indicate the models with uniform grains added at each stage with $\lambda_p = 1$). Unexpectedly, the data follow the relationship $q = m - 1$ well, rather than $v = m - 1$. Thus, it is possible that the empirical relationship $v = m - 1$ arises from the natural materials characterized by $\lambda_p \approx 1$ and when $S/V_p$ do not change significantly with porosity. This is indeed the case for some natural porous materials where the total interface area is expected to change linearly with the porosity (Mavko et al. 2020; Hussaini & Dvorkin 2021a).

Fig. 18 only shows the case when $m$ is smaller than 2. Considering that the interfacial Archie's law suggests that $q$ should be smaller than one, the $v = m - 1$ relationships at higher $m$ necessitate $\lambda_q$ to be larger than 1, which is reasonable because porous media with higher $m$ usually corresponds to large change rate of pore connectivity with porosity (or saturation, see Yue 2019; Qiang et al. 2022), and thus larger change rate of specific surface area $\lambda_q$. The two-part interpretation of the exponents mentioned in the Discussion Section 5.2 could help to explain the deviated observations of the $p = n - 1$ relationship for some unsaturated natural material. For example, Schmutz et al. (2012) observe a negative $p$ with $p \approx n - 2.5$ for the presence of strong water-repellent oil in pore spaces. As the solid phase of a strong oil-wet system is usually preferentially wetted by the oil phase, few water-solid interfaces could be generated with increasing water content. Thus the $\lambda_p$ for this case would be close to 0. Meanwhile, the water phase co-exists with strong water-repellent oil would be highly irregular, corresponding to large pore volume fractal dimensions $D_f$ (Lin et al. 2022) that may increase the value of $t$. Deng et al. (2018) found that some porous media with very high Cation Exchange Capacities (> 100 meq/100 g) would have a high value of the exponent $p$. This is related to the fact that materials with high Cation Exchange Capacities usually correspond to large specific surface area (Revil 2012), and hence complex pore interface structures within limited sample volume. These complex pore paths may lead to a large values of $\lambda_p$ for saturation changes and hence higher $p$-values.

## 6. CONCLUSIONS

We verified the power-law relationship between the surface formation factor and specific surface area (also known as the interfacial Archie's law) of porous materials. Numerical simulations of multiple fractal geometries and natural core sample datasets including clayey materials and volcanic rocks were evaluated. The results highlight the potential to distinguish and predict the fractal properties of various types of porous media by the values of Archie's exponents acquired from their complex conductivity, porosity and specific surface area. We also proposed a universal interpretation of the four power-law (Archie's law) exponents related to the bulk, surface, and quadrature components of complex conductivity, which can be divided into the dynamics of effective physical parameters (such as connected porosity or water content and specific surface area) and their contribution to the porous media's ability to conduct or restore electrical charges. Meanwhile, a pore network simulation proved that the complex conductivity of materials with a wide pore radius distribution would follow a CPA model resulting from the superposition of the Warburg model (this works for other values of the Cole–Cole exponent used to describe the local impedance at

the pore scale). These findings enhance our understanding of the links between different components of complex conductivity and the microscopic and multiscale properties of porous materials, which in turn would promote the interpretation of induced polarization data in field conditions. Moving forwards, incorporating the grain surface fractal dimensions into the interfacial Archie's law could be of great interest and expand its application scope. We could also expand the present framework to build additional numerical tests and teach a neural network to predict the values of the four exponents.

## ACKNOWLEDGMENTS

The work was supported by the National Natural Science Foundation of China: Basic Research Project for Young Scholars (Doctoral candidate, 424B2021), National Key Research and Development Program of China (2023YFC3706505) and the National Natural Science Foundation of China (42472318, 42272276). XS was partly supported by the Fundamental Research Funds for the Central Universities (020614380217 and 020614380181). AR was supported by the PALLAS project between ASNR, the CNRS and the BDI. We thank the Editor Tina Martin for the professional handling of our manuscript. We also thank the two referees for their fruitful and constructive comments.

## SUPPORTING INFORMATION

Supplementary data are available at *GJIRAS* online.

**Table S1.** The calculated and predicted values of $F_S$ used in Fig. 8 of the manuscript. Sierpinski carpet (SC), Menger sponge (MS) and model B from Roy & Tarafdar (1997, RTB). $F_{S1}$ (mm): simulated; $F_{S2}$: calculated by pore upscaling method; $F_{S3}$: calculated by grain upscaling method; $F_{S4}$: calculated by interfacial Archie's law; $F_{S5}$: calculated by Waxman–Smits model; $F_{S6}$: calculated by DEM model. $a_s$ and $q$ is the fitted parameter and specific surface area exponent used for $F_{S4}$ determination.

**Figure S1.** (a) Cumulative probability as a function of $\log10(r)$ for the log-normal distribution (eq. 42). (b) Cumulative probability as a function of $\log10(r)$ for the loguniform distribution (eq. 43). The legend indicates the slope, equal to $1/(\log10(r_{max}) - \log10(r_{min}))$.

**Figure S2.** Amplitude spectra for the lognormal case used to describe the pore size distribution (using a standard-deviation increasing from 0.01 to 1.5), for various values of the Cole–Cole exponent $c$. Note that here we use $30 \times 30 \times 30$ networks.

**Figure S3.** Phase spectra for the lognormal case used to describe the pore size distribution (using a standard-deviation increasing from 0.01 to 1.5), for various values of the Cole–Cole exponent $c$. Note that here we use $30 \times 30 \times 30$ networks.

**Figure S4.** Evolution of the Cole–Cole exponent as a function of the standard deviation for the spectra obtained using the lognormal distribution, for various values of the Cole–Cole exponent $c$. Note that here we use $30 \times 30 \times 30$ networks.

**Figure S5.** Amplitude spectra for the loguniform case for the pore sizes (using $\log_{10}(r_{min})$ decreasing from -6 to -7.5 and $\log_{10}(r_{max})$ increasing from -4 to -2.5), for various values of the Cole–Cole exponent $c$. Note that here we use $30 \times 30 \times 30$ networks.

**Figure S6.** Phase spectra for the loguniform case for the pore sizes (using $\log_{10}(r_{min})$ decreasing from -6 to -7.5 and $\log_{10}(r_{max})$ increasing from -4 to -2.5), for various values of the Cole–Cole exponent $c$. Note that here we use $30 \times 30 \times 30$ networks.
**Figure S7.** Evolution of the b-value as a function of $log_{10}(r_{max}/r_{min})$ for the spectra obtained using the loguniform distribution, for various values of the Cole–Cole exponent c. Note that here we use 30 × 30 × 30 networks.

**Figure S8:** Simulated surface formation factor by grain upscaling method versus pore water conductivity $\sigma_w$ out of the high salinity asymptotic limit $\sigma_w > 1$ S/m( for this case), which will not be discussed in this work. The dashed line shows the value of $F_s$ near high salinity asymptotic limit.

**Figure S9:** Relationship between the number of boxes $N(\epsilon)$ and box size $\epsilon$ (equal to the number of pixels along the side) when estimating the box-counting fractal dimensions of the five stages shown in the first column in Fig. 9 of the manuscripts.

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## DATA AVAILABILITY

The data used in this paper could be obtained through a request to the corresponding author.

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