# Electrical conductivity model for reactive porous media under partially saturated conditions with hysteresis effects

Mariangeles Soldi, Flore Rembert, Luis Guarracino, Damien Jougnot

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Mariangeles Soldi, Flore Rembert, Luis Guarracino, Damien Jougnot. Electrical conductivity model for reactive porous media under partially saturated conditions with hysteresis effects. Advances in Water Resources, 2024, pp.104815. 10.1016/j.advwatres.2024.104815. insu-04701070v2

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Electrical conductivity model for reactive porous media under
partially saturated conditions with hysteresis effects

M. Soldi¹,³*, F. Rembert², L. Guarracino¹ and D. Jougnot³

¹Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Consejo Nacional de Investigaciones Científicas y Técnicas, La Plata, Argentina

²Univ. Orléans, CNRS, BRGM, ISTO, UMR 7327, F-45071 Orléans, France

³Sorbonne Université, CNRS, EPHE, UMR 7619 METIS, 75005, Paris, France

*E-mail: msoldi@fcaglp.unlp.edu.ar

This paper has been published in Advances in Water Resources: Soldi, M., Rembert, F., Guarracino, L., Jougnot, D. (2024) Electrical conductivity model for reactive porous media under partially saturated conditions with hysteresis effects. Advances in Water Resources. Volume 193, 2024, 104815, ISSN 0309-1708, https://doi.org/10.1016/j.advwatres.2024.104815.

Abstract

The electrical conductivity of a porous medium is strongly controlled by the
structure of the medium at the microscale as the pore configuration governs the
distribution of the conductive fluid. The pore structure thus plays a key role since
different geometries translate in variations of the fluid distribution, causing different
behaviours measurable at the macroscale. In this study, we present a new physically-
based analytical model derived under the assumption that the pore structure can
be represented by a bundle of tortuous capillary tubes with periodic variations of
their radius and a fractal distribution of pore sizes. By upscaling the microscale
properties of the porous medium, we obtain expressions to estimate the total and
relative electrical conductivity. The proposed pore geometry allows us to include
the hysteresis phenomenon in the electrical conductivity estimates. The variations
on these estimates caused by pore structure changes due to reactive processes are
accounted by assuming a uniform dissolution of the pores. Under this hypothesis,
we describe the evolution of the electrical conductivity during reactive processes.
The expressions of the proposed model have been tested with published data from
different soil textures, showing a satisfactory agreement with the experimental data.
Hysteretic behavior and mineral dissolution are also successfully addressed. By in-
cluding hysteresis and mineral dissolution/precipitation in the estimates of the elec-
trical conductivity, this new analytical model presents an improvement as it relates
those macroscopic physical phenomena to its origins at the microscale. This opens
up exciting possibilities for studies involving electrical conductivity measurements
to monitor water movement, and hysteretic and reactive processes in porous media.

Highlights

* New electrical conductivity model for partially saturated reactive porous media.
* Pore structure composed of capillaries with periodic reductions of their radii.

* Variations in pore structure significantly influence the electrical conductivity.

* Hysteretic behavior is included from geometrical effects, pore radii reductions.

* Geometrical impact on the pores due to mineral dissolution is successfully described.

**Keywords:** Electrical conductivity, Unsaturated flow, Vadose zone, Fractal distribution, Reactive porous media

# 1 Introduction

The water content and dynamics in geological media are largely controlled by their petrophysical properties (such as permeability and porosity) which often exhibit a high degree of spatial variability due to the irregularities of the pore space structure. Knowledge of these properties is important to understand flow and transport processes as they control the movement and storage of fluids. This fact is highlighted for porous media containing carbonates, since dissolution and precipitation processes can drastically change the size and shape of the pore spaces and the degree of their interconnection (e.g., Noiriel et al., 2004; Leger and Luquot, 2021). Therefore, understanding better the effect of pore geometry at the microscale is a key to describe and predict petrophysical and transport properties at the macroscale.

During the last two decades, the use of geophysical methods to non-invasively characterize hydrological processes has been extensively developed (e.g., Binley et al., 2015; Hermans et al., 2023; Loiseau et al., 2023). Among the geophysical methods, the electrical and electromagnetic methods are very useful since they can provide quantitative information on the structure, water content, or fluid composition of the porous media through the measurement of electrical conductivity. Electrical resistivity tomography (ERT), induced polarization (IP), and electromagnetic induction (EMI) are some of the hydrogeophysical methods that have increasingly been used to monitor flow and transport quantitatively (e.g., Revil et al., 2012; Binley and Slater, 2020). Among many other applications of these methods related to groundwater in porous media, it can be mentioned the application of ERT to monitor critical saturated conditions for landslide occurrence (e.g., Olabode et al., 2020; Wicki and Hauck, 2022), the use of IP to describe contaminant plumes and to monitor their remediation (e.g., Deceuster and Kaufmann, 2012; Schwartz and Furman, 2012), or the capability of EMI to detect and monitor groundwater flow (e.g., Wilson et al., 2002; Zhu et al., 2010; Doolittle and Brevik, 2014).

The electrical conductivity $\sigma$ (S m$^{-1}$) of the porous medium is one of the petrophysical properties of interest measured with the electrical methods and electromagnetic methods. Many models exist to express the electrical conductivity of a water-saturated porous media (for a review, see Glover (2015) and Schön (2015)). Among the most simple ones, the pore space can be seen as an electrical circuit in parallel, where $\sigma$ is linked to the electrical currents that arise when charge carriers (i.e., ions) flow through the media and
66 can be expressed as:

$$ \sigma = \frac{\sigma_w}{F} + \sigma_s \qquad (1) $$

where $\sigma_w$ (S m⁻¹) is the electrical conductivity of the fluid, F (no units) the formation factor and $\sigma_s$ (S m⁻¹) the surface conductivity. This model was first introduced by Patnode and Wyllie (1950) who discovered that surface conductivity $\sigma_s$ cannot always be neglected in reservoir rocks. The first term in Eq. (1) corresponds to the contribution from the electrical conduction in the liquid phase (i.e., electrolyte) whereas the second term is the contribution due to the electrical conduction in the liquid-solid interface, the so-called electrical double layer (EDL, e.g., Hunter, 1981; Leroy and Revil, 2004). The electrical conductivity of the electrolyte $\sigma_w$ has a magnitude that depends on its chemical composition and the ionic concentrations (e.g., Glover, 2015). Experimental and theoretical works, e.g., the three-resistor model (Wyllie and Southwick, 1954; Lévy et al., 2018), the BHS (Bruggeman-Hanai-Sen) model (Sen et al., 1981), and the recent models of (Qi and Wu, 2022, 2024), have shown that the surface conductivity $\sigma_s$ depends on the water and the EDL electrical conductivities. However, when the surface conductivity $\sigma_s$ can be neglected, that is when the electrolytic conduction is much larger than the surface one, the electrical conductivity $\sigma$ is controlled mainly by the pore space geometry in which the charge carriers (i.e., usually ions) can move it. That is the porous medium connectivity and the size and shape of the pores (e.g. Olsen, 2011; Revil, 2013b; Jougnot et al., 2018). The dependency of the porous medium electrical conductivity to its microstructure and the electrolyte conductivity is taken into account through the formation factor, a quantitative proxy for the medium geometry. Petrophysical models aim to establish an expression for the formation factor based on pore structure parameters. Among the existing approaches, models capturing the true pore space topology play a major role when considering the partially saturated conditions. To identify physically-based structural parameters defining the formation factor, several studies have analyzed the effect of pore structure in electrical properties directly related to the electrical conductivity. Müller-Huber et al. (2015) examined the effect of changes in the pore radius on the electrical conductivity. They studied a straight pore channel whose radius increases exponentially between its two extremes showing that pore geometry has a strong effect on the electrical conductivity. For carbonate rocks, Regnet et al. (2019) analyzed the influence of microstructural patterns on petrophysical properties (electrical conductivity, porosity, and permeability). They emphasized the importance of pore space to estimate these properties through many parameters such as pore size and shape, grain contacts or cracks, pore network connectivity, and mineralogy. Rembert et al. (2020) derived a model that estimates the electrical conductivity and the formation factor as a function of both tortuosity and constrictivity of the pore structure. They observed that the constrictivity follows a monotonous variation either during dissolution or precipitation processes.

Among the models to predict the electrical conductivity of porous media for both saturated and partially saturated conditions, the most widely used is the one proposed by Archie (1942). Assuming negligible surface conductivity, this empirical model provides a
106 relationship between the medium and the electrolyte electrical conductivities given by

$$ \sigma = \sigma_w S_w^n \phi^m \qquad (2) $$

107 where $S_w$ (no units) is the water saturation and $\phi$ (no units) the porosity of the medium; 108 and exponents $n$ and $m$ (no units) are the so-called saturation and cementation factors, 109 respectively. Several authors have found that these empirical factors depend on the geometry of the pore system or its connectivity (e.g., Knight and Endres, 2005; Glover, 110 111 112 113 114 115 116 117 118 119

$$ \sigma = \phi^m [\sigma_w S_w^n + (\phi^{-m} - 1)\sigma_s]. \qquad (3) $$

120 Several phenomena such as hysteresis effects and reactive processes affect the electrical conductivity (e.g. Binley and Kemna, 2005). Hysteresis refers to the non-unique relationship between electrical conductivity and pressure head or water saturation, which 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143
hysteresis loop. They remarked that neglecting the hysteresis leads to dramatic errors between the measured and the simulated parameters. Therefore, it is necessary to fully account for the hysteresis behavior for an accurate flow description and electrical conductivity prediction when performing a numerical simulation. Furthermore, a considerably large amount of research studies has been conducted to analyze the different aspects of reactive behaviors of carbonate rocks at total saturation conditions, such as wormholes and their complex relationship with transport properties or changes in the dissolution rate constant (e.g., Alkattan et al., 1998; Liu et al., 2005; Kaufmann and Dreybrodt, 2007; Noiriel et al., 2004; Garing et al., 2015). Among experimental works on the electrical monitoring of dissolution/precipitation processes, Wu et al. (2010) monitored calcite precipitation on a glass beads column and observed the evolution of this process from electrical conductivity data. These results have been confirmed by other studies (e.g., Saneiyan et al., 2019; Izumoto et al., 2020, 2022). Garing et al. (2014) characterized natural carbonate samples presenting high heterogeneity resulting from many dissolution and precipitation processes by measuring the electrical conductivity. However, they did not monitor electrical conductivity changes caused by these reactive processes. Cherubini et al. (2019) analyzed the increase in the electrical conductivity values due to the dissolution of calcite in limestones samples. In the numerical field, Niu and Zhang (2019) simulated the dissolution and precipitation on digital representations of microstructural images to study the electrical conductivity. Even though they observed changes in this electrical property values, no significant variations were estimated in other petrophysical parameters, such as porosity. More recently, Rembert et al. (2023a) studied the impact of conduits formation in limestone samples on the electrical conductivity measurements due to calcite dissolution and the relationship between electrical properties and the evolution of structural parameters over time.

The framework of capillary tube models has been useful to provide valuable insights into different transport phenomena occurring in porous media from the microscopic standpoint (e.g., Celia et al., 2004; Jackson, 2010; Jougnot et al., 2012; Luo et al., 2023; Thanh et al., 2023). In addition, petrophysical properties can be directly calculated in these type of models by simply analyzing the capillary tubes geometry (e.g., Guarracino et al., 2014; Rembert et al., 2020; Soldi et al., 2017, 2022). Because of these advantages, capillary tube models have been extensively used considering different pore size distributions to upscale the petrophysical properties from the contribution of one capillary (e.g., multi-modal, quasi-fractal, fractal or gaussian, see Jougnot et al. (2019)). Among them, the fractal distributions have proven to be efficient to represent porous media due to their simplicity and capacity to describe a wide range of natural geological media (e.g. Guarracino, 2006; Yu, 2008; Guarracino and Quintana, 2009; Ghanbarian-Alavijeh et al., 2011). Recently, Soldi et al. (2022) derived an analytical model to estimate the hydraulic properties of porous media under partially saturated conditions. They assumed piecewise sinusoidal variations of the capillary radii along its length which allowed them a more realistic description of the porous media. Indeed, the periodic reductions assumed on the pore structure are defined by two geometrical parameters whose range of values allows the representation of the hydraulic properties for different soil textures.
Electrical properties have been successfully studied within the capillary tube models and fractal distributions. In the last years, Thanh et al. (2019, 2020) derived electrical conductivity models for total and partially saturated porous media accounting for surface conductivity. For both models, they assumed that the bundle of capillaries follows a fractal pore size distribution and that each capillary is represented by a tortuous cylindrical tube of constant radius. Chen et al. (2023) derived a theoretical correlation between the formation factor and tortuous pore structure with a wide distribution of pore sizes. The authors also consider cylindrical tubes of constant radii, however, they propose that tortuosity follows a fractal law. Saafan et al. (2023) developed a model to describe the electrical conductivity for a reservoir porous medium where the fluid phases are water and oil. They represented the partially saturated pore system using tortuous fractal capillaries with constant square or triangular cross-sectional areas. Nevertheless, these models are lacking an important microstructure parameter which significantly controls the transport and petrophysical properties of porous media, the constrictivity. This parameter characterizes the so-called bottleneck effect that means that the wide spaces of the pores or pore bodies are connected through pore throats (i.e. non-constrictive and constrictive lengths). The previously described models do not account for this type of variations in the capillary radii along their length which should be considered to provide a more accurate description of porous media in the framework of capillary bundles. In this work, we derive an analytical model that considers a realistic pore geometry to estimate the electrical conductivity under partially saturated conditions. Based on the framework of capillary tube models and the geometry proposed by Soldi et al. (2022), the pore space is represented by a bundle of cylindrical capillaries with varying aperture along their pore length to take into account the structure composed of pore throats and pore bodies. This geometry has the advantage to represent a wide range of pore geometries due to the variable lengths of the pore throats and pore bodies. The proposed model is an extension of the model developed by Rembert et al. (2020) that estimates the electrical conductivity for total saturated porous media. Their model considers the pores as tortuous cylindrical tubes with sinusoidal variations of their aperture. Assuming a fractal pore size distribution that can be either filled by water or by air, we obtain mathematical closed-form expressions to estimate the saturated and relative electrical conductivity from the upscaling procedure. The macroscopic expression relies on geometrical parameters defining the pore structure (e.g., the constrictivity), tortuosity, pore-size distribution, and radii of the pores. The pore throats and pore bodies present in the proposed geometry allow us to include the hysteresis phenomenon in the relative electrical conductivity. Considering the structural variations that reactive processes can cause, we describe the temporal evolution of the saturated electrical conductivity by assuming a constant dissolution of the pores. The performance of the model to reproduce experimental data is tested against different sets of laboratory data of soils with varied textures.
## 2 Electrical conductivity model

The present model is derived based on the pore geometry proposed by Soldi et al. (2022) and follows the work developed by Rembert et al. (2020). Within the framework of capillary tube models, we consider a porous medium represented by an ensemble of tortuous constrictive capillaries in a cylindrical representative elementary volume (REV) of radius $R_{REV}$ (m) and length L (m). Since we focus on the porous microstructure, we consider that the surface conductivity is negligible. However, similar developments could be done including the surface conductivity (e.g. Thanh et al., 2020; Luo et al., 2023).

### 2.1 Pore space description

At the microscopic scale, we consider that each pore of the capillary bundle is represented by a tortuous cylindrical tube with varying aperture (see Fig. 1). All the pores are assumed to have the same length l (m) and tortuosity $\tau$ ($\tau = l/L$, no units). The pore radius varies along all the length of each capillary between its radius value R (m) and its pore throat radius value aR (m) where a (no units) is the radial factor that quantifies the reduction in the pore radius due to the presence of the constrictivity. A sinusoidal piecewise geometry is assumed with a wavelength $\lambda$ (m) which replicates along the pore length (i.e. $l \gg \lambda$). Thus, the varying radius $r(x)$ (m) of a capillary can be expressed as (Soldi et al., 2022):

$$r(x) = \begin{cases} (1+a)\frac{R}{2} + (1-a)\frac{R}{2}\sin\left(\frac{\pi x}{\lambda(1-c)}\right) & \text{if } x \in [0, \lambda(1-c)) \\ (1+a)\frac{R}{2} + (1-a)\frac{R}{2}\sin\left(\frac{\pi[x-\lambda(1-2c)]}{\lambda c}\right) & \text{if } x \in [\lambda(1-c), \lambda), \end{cases} \quad (4)$$

where c (no units) is the length factor which represents the constrictive fraction of $\lambda$ (i.e. the segment of $\lambda$ where the radius is reduced). The geometrical factors a and c vary between 0 and 1. Note that the limit values a = 0 and a = 1 correspond to the cases of periodically closed pores and cylindrical pores of constant radius R, respectively. Also note that if c = 0.5, the proposed geometry is consistent with the geometry proposed by Guarracino et al. (2014) and Rembert et al. (2020) that considers equal lengths for the constrictive and non-constrictive fractions of the wavelength.

The porous structure of the medium is assumed to be a bundle of capillary tubes with different sizes of radii varying from a minimum pore radius $R_{min}$ to a maximum pore radius $R_{max}$. Based on the fractal theory for porous media (e.g. Tyler and Wheatcraft, 1990; Yu et al., 2003; Liang et al., 2014), the pore size distribution can be described by the following fractal law:

$$N(R) = \left(\frac{R_{REV}}{R}\right)^D, \quad (5)$$

where $N(R)$ (no units) is the number of pores whose radii are greater than or equal to R and D (no units) is the pore size fractal dimension. Considering the classical fractal object called Sierpinski carpet, Tyler and Wheatcraft (1990) show that this parameter D
<figure><img src="image_2.png" /><figcaption>Figure 1: (a) Sketch of the representative elementary volume represented by a bundle of tortuous sinusoidal capillary tubes, and (b) the pore geometry considered for each piecewise sinusoidal capillary.</figcaption></figure>

260

261

varies between 1 and 2 for different textures of porous media where its highest values are associated with the finest textured soils.

262

263

Then, the number of pores whose radii are within the infinitesimal range $R$ and $R+dR$ is obtained from Eq. (5) by differentiating with respect to $R$:

$$ -dN(R) = DR_{REV}^{D}R^{-D-1}dR, \qquad (6) $$

264

265

where the minus sign implies that the pore number increases when the pore radius decreases (e.g. Yu et al., 2003; Yu, 2008).

## 2.2 Electrical conductivity model

266

### 2.2.1 Electrical model at saturation

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270

To derive the expressions of the new electrical conductivity model, we first describe the electrical properties of a single capillary. For a water saturated capillary of radius $R$, its electrical conductance $\Sigma_p(R)$ (S) can be calculated as:

$$ \Sigma_p(R) = \int_0^l \frac{1}{\sigma_w \pi r^2(x)} dx. \qquad (7) $$

271

By replacing Eq. (4) in Eq. (7) yields:

$$ \Sigma_p(R) = \frac{\sigma_w \pi R^2 f(a, c)}{\tau L}, \qquad (8) $$

272

$$ f(a, c) = \frac{2a^{3/2}}{1+a} \left\{ 1 + (2c-1) \left[ \frac{4\sqrt{a}(1-a)}{\pi(1+a)^2} + \frac{2}{\pi} \tan^{-1} \left( \frac{1-a}{2\sqrt{a}} \right) \right] \right\}^{-1}. \qquad (9) $$

273

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275

The factor $f(a, c)$ (no units) varies between 0 and 1, and quantifies the reduction in the conductance of each pore due to the presence of the pore throats (i.e. the so-called constrictivity).
The contribution of each single capillary conductivity $\sigma_p(R)$ (S m$^{-1}$) to the porous medium conductivity is then obtained by multiplying the pore conductance (Eq. (8)) with a geometric factor $f_g = \pi R_{REV}^2/L$ (m) (Hem and Minear, 2012; Rembert et al., 2020):

$$ \sigma_p(R) = \frac{\sigma_w R^2 f(a, c)}{\tau R_{REV}^2}. \qquad (10) $$

According to Ohm's law, the electric current $i_p(R)$ (A) flowing between the edges of a capillary of radius $R$ is directly proportional to the electrical voltage difference $\Delta V$ (V) across them:

$$ i_p(R) = \Sigma_p(R)\Delta V = \frac{\sigma_w \pi R^2 f(a, c)}{\tau L} \Delta V. \qquad (11) $$

For a fully saturated porous medium, the total electric current $I^{REV}$ (A) flowing through the REV is obtained from Kirchhoff's current law, by summing the electric currents contributions of all the capillaries over the entire range of pore sizes:

$$ I^{REV} = \int_{R_{min}}^{R_{max}} i_p(R) dN(R). \qquad (12) $$

Substituting Eqs. (6) and (11) in Eq. (12) and integrating it yields to

$$ I^{REV} = - \frac{\sigma_w \pi R_{REV}^D D f(a, c)}{\tau L(2-D)} (R_{max}^{2-D} - R_{min}^{2-D}) \Delta V. \qquad (13) $$

On the other hand, based on Ohm's law at the REV scale, the total electric current $I^{REV}$ flowing through the porous medium can be expressed as:

$$ I^{REV} = -\sigma^{REV} \pi R_{REV}^2 \frac{\Delta V}{L}, \qquad (14) $$

where $\sigma^{REV}$ (S m$^{-1}$) is the electrical conductivity of the porous medium which under total saturated conditions is redefined as the saturated electrical conductivity $\sigma_{sat}^{REV}$. Combining Eqs. (13) and (14), an expression for $\sigma_{sat}^{REV}$ is obtained:

$$ \sigma_{sat}^{REV} = \frac{\sigma_w D f(a, c)}{\tau R_{REV}^{2-D}(2-D)} (R_{max}^{2-D} - R_{min}^{2-D}). \qquad (15) $$

The capability to estimate the electrical conductivity $\sigma_{sat}^{REV}$ of a porous medium as a function of the medium's porosity $\phi$ is significantly valuable in petrophysics. For the proposed model, a relationship between these two macroscopic properties can be obtained by considering that for this porous medium geometry its porosity $\phi$ is given by (Soldi et al., 2022):

$$ \phi = \frac{D \tau f_v(a, c)}{R_{REV}^{2-D}(2-D)} (R_{max}^{2-D} - R_{min}^{2-D}) \qquad (16) $$
where $f_v(a, c) = (1+a)^2/4 + (1-a)^2/8 + (1-a^2)(1-2c)/\pi$ is a factor that quantifies the reduction in the porosity of the REV due to the presence of the constrictivities (the subscript v stands for volume). Then, combining Eqs. (15) and (16) yields:

$$ \sigma_{sat}^{REV} = \frac{\sigma_w f_{\sigma}(a, c)\phi}{\tau^2} \qquad (17) $$

where $f_\sigma$ is a dimensionless factor that varies between 0 and 1, and is defined as the ratio between the factor $f$ and $f_v$. The exact expression of $f$ given in Eq. (9) can be reduced similar to Soldi et al. (2022) and $f_\sigma$ can be expressed as:

$$ f_{\sigma}(a, c) = \frac{16\pi^2 a^{3/2}(1+a)}{[\pi(1+a)^2 + 2(2c-1)(1-a)(1+\sqrt{a})^2][2\pi(1+a)^2 + \pi(1-a)^2 + 8(1-a^2)(1-2c)]} \qquad (18) $$

The variation of the factor $f_\sigma$ as a function of the radial factor $a$ is shown in Fig. 2, for different values of the length factor $c$. Note that the presence of the pore throats affects significantly the electrical conductivity of the REV $\sigma_{sat}^{REV}$ through the $f_\sigma$ values which depends on $a$. Also note that if $a=0$, $f_\sigma=0$ and $\sigma_{sat}^{REV} = 0$, no electrical conduction takes place in the REV as the pores are periodically closed. In that case, $f_v \neq 0$ which means that the REV has a porosity value but the pores are not connected. In order to represent this type of porous media using Archie (1942), $\sigma = 0$ only in the case that $\phi = 0$. Therefore, Archie (1942) cannot represent this media, while the proposed model can describe media with a porosity that corresponds to poorly connected pores, and low electrical conductivity. The expression of the factor $f_\sigma$ given by Eq. (18) can be reduced while taking into account the main features of the factor's behavior and it yields

$$ g_{\sigma}(a, c) = \frac{8a^{3/2}}{(1+a)[(1+a)^2 - (1-a)^2(1-6c+6c^2)]} \qquad (19) $$

This simplified expression is easier to evaluate and, therefore, its implementation in numerical codes is straightforward.

Eqs. (15) and (17) are closed-form expressions that describe the saturated electrical conductivity $\sigma_{sat}^{REV}$ of the medium which depend on the water electrical conductivity, porosity and model parameters with physical or geometrical meaning. It is interesting to observe that the estimates of $\sigma_{sat}^{REV}$ are strongly influenced by the parameters $a$ and $c$ of the pore geometry through the factor $f_\sigma$. In the limit case of $a=1$ and $\tau=1$, Eq. (17) becomes $\sigma_{sat}^{REV} = \sigma_w\phi$ which is consistent with Archie's law expression for $m=1$ where the porous medium is composed by a bundle of non-tortuous capillaries of constant radii (e.g., Clennell, 1997). For tortuous capillary bundles, the relationship between tortuosity and porosity can be expressed by the Bruggeman relation, i.e., $\tau^2 = \phi^{-1/2}$ (Holzer et al., 2023). Substituting this relation in Eq. (17), we obtain a porosity with an exponent equal to 1.5 which is consistent with the $m$ value observed by Sen et al. (1981) for ideally packed spheres. Note also that, if $c=0.5$, the resulting expressions of Eqs. (15) and (17) are similar to the ones proposed by the model of Rembert et al. (2020).
<figure><img src="image_3.png" /><figcaption>Figure 2: Dimensionless constrictivity factor $f_σ$ (Eq. (18)) as a function of the radial factor $a$ for different constant values of the length factor $c$.</figcaption></figure>

### 2.2.2 Effect of partial saturation and hysteresis

Under the hypothesis of neglecting surface conduction, the electrical conductivity of the REV under partially saturated conditions can be expressed as

$$ \sigma^{REV} = \sigma_{sat}^{REV} \sigma_{rel} \quad (20) $$

being $\sigma_{rel}$ (no units) the relative electrical conductivity of the REV. To obtain the expression of $\sigma_{rel}$, we consider a similar derivation procedure as for the total saturated porous medium but with capillaries that can be either fully saturated by water or by air (e.g., Soldi et al., 2017; Thanh et al., 2020). Following the procedure proposed by Guarracino et al. (2014), we assume that the REV is fully saturated at its initial state and then is drained when subjected to a pressure head $h$ (m). To determine which pores will be desaturated by the applied pressure head, $h$, under a Darcy's flow regime (i.e., low Reynolds number), we consider the link between the radius of the water-filled pore $R_h$ (m) to the pressure head $h$ given by (e.g. Jurin, 1717)

$$ h = \frac{2T_s \cos(\gamma)}{\rho_w g R_h}, \quad (21) $$

where $T_s$ (N m⁻¹) is the surface tension of the water, $\gamma$ (°) the contact angle, $\rho_w$ (kg m⁻³) the water density and $g$ (m s⁻²) the gravitational acceleration. A capillary tube becomes fully desaturated if the radius of its pore throat $aR$ is greater than the radius $R_h$ defined by Eq. (21). Thus, for the drainage process, it is reasonable to assume that pores with radii $R$ between $R_{min}$ and $R_h/a$ will remain fully saturated and contribute to the electric
current flowing through the REV. If we now consider an imbibition process where the REV is dry at the beginning and flooded by a pressure head $h$, the capillary tubes that will be fully saturated in this case are those whose radii $R$ are smaller than $R_h$. Then, considering the ranges of saturated pores for each process, the electrical current streaming through the REV (Eq. (12)) can be expressed as:

$$I^{REV} = \int_{R_{min}}^{R^{*}} i_{p}(R)dN(R) \qquad (22)$$

where $R^* = R_h/a$ for the drainage and $R^* = R_h$ for the imbibition. Combining Ohm's law at macroscale (Eq. (14)) with Eq. (20), and comparing it with the result of integrating Eq. (22), we obtain the expression for the relative electrical conductivity $\sigma_{rel}$:

$$\sigma_{rel}(R^*) = \frac{R^{*2-D} - R_{min}^{2-D}}{R_{max}^{2-D} - R_{min}^{2-D}} \qquad (23)$$

Substituting Eq. (21) in (23), the relative electrical conductivity can be expressed as function of the pressure head for a drainage process as:

$$\sigma_{rel}^{d}(h) = \frac{(ah)^{D-2} - h_{max}^{D-2}}{h_{min}^{D-2} - h_{max}^{D-2}} \qquad (24)$$

where $h$ varies between $\frac{h_{min}}{a}$ and $\frac{h_{max}}{a}$, and for an imbibition process, it yields:

$$\sigma_{rel}^{w}(h) = \frac{h^{D-2} - h_{max}^{D-2}}{h_{min}^{D-2} - h_{max}^{D-2}} \qquad (25)$$

where $h$ varies between $h_{min}$ and $h_{max}$. By inspection of Eq. (24), note that if $h < \frac{h_{min}}{a}$, $\sigma_{rel}^d = 1$, since the REV is fully saturated as the pressure head applied is lower than the one necessary to start draining the larger pores, and in the case of $h > \frac{h_{max}}{a}$, $\sigma_{rel}^d = 0$ as the pressure head applied is large enough to drain the REV. Similar results can be obtained by analyzing Eq. (25) for the imbibition process, $\sigma_{rel}^d = 1$ and $\sigma_{rel}^d = 0$, when $h < h_{min}$ and $h > h_{max}$, respectively. Note that the $\sigma_{rel}$ expression differs between the drainage and imbibition processes due to the presence of the pore radius reduction defined by the radial factor $a$. Therefore, these irregularities present in the pore geometry allow to introduce the hysteresis phenomenon in the electrical conductivity model. This fact has also been remarked by Singha et al. (2007), who explain solute transport hysteresis effects in the electrical conductivity though a bicontinuum model composed by mobile (i.e., well-connected) and immobile (i.e., poorly-connected) pores. Fig. 3 shows the hysteresis cycle on this electrical property for different values of the radial factor $a$. For the remaining parameters of Eqs. (24) and (25), we considered the following constant values: $D = 1.5$, $h_{min} = 0.01$ m and $h_{max} = 10$ m. It can be seen that the radial factor $a$ significantly affects the main drying curves of the relative electrical conductivity while no variations are produced on the main wetting curves of this electrical property since they are independent of $a$. Indeed, low values of $a$ produce a shift of the main drying $\sigma_{rel}$ curves to higher values
<figure><img src="image_4.png" /><figcaption>Figure 3: Hysteresis cycle of the relative electrical conductivity as function of pressure head for different values of the radial factor a. The solid and dashed lines correspond to the main relative electrical conductivity curves for drainage and imbibition experiments (Eqs. (24) and (25)), respectively. Note that all the imbibition curves are superimposed.</figcaption></figure>

of the pressure head as higher pressure head values are needed to drain more constrictive pores. Also note that, in the limit case of a approaching 1, the two $\sigma_{rel}$ curves tend to reduce their distance, as it can be expected since this situation represents capillary tubes of constant radii and thus no hysteretic phenomenon will be observed.

For partially saturated conditions, the behaviour of the relative electrical conductivity is often described by its dependence with the water saturation $S_w$ (no units). Within the framework of capillary tube models, the effective saturation $S_e$ (no units) is the variable most used since the tubes are fully saturated or empty, and is related to $S_w$ through

$$S_w = (1 - S_r)S_e + S_r \qquad (26)$$

where $S_r$ (no units) is the residual water saturation. Recently, Soldi et al. (2022) developed an analytical model for the hydraulic properties of a porous medium with the geometry considered for the proposed electrical model. Note that saturation is obtained from an equation which is identical to Eq. (23). Therefore, for the proposed model the relationship between $\sigma_{rel}$ and $S_e$ yields:

$$\sigma_{rel}(S_e) = S_e. \qquad (27)$$

From a theoretical point of view, it can be observed that Eq. (27) results in a non-hysteretic function, while Eqs. (24) and (25) account for the hysteresis phenomenon.

### 2.2.3 Evolution of the model in reactive media

The electrical conductivity can be affected by dissolution and precipitation processes particularly in porous media composed by carbonate minerals. When a fluid flows through a
reactive porous medium produces changes in the structure of the pore space causing variations of its volume and surface. In order to estimate the resulting geometrical changes, we assume that each capillary tube is dissolved uniformly so that its circular cross-sectional shape is retained while only increasing its radius. Then, to evaluate how dissolution affects the pore structure, we assume that the volume change of a pore with time t is proportional to the pore surface $S_p$ that is in contact with the reactive fluid and to the dissolution rate $\alpha$ (Freedman et al., 2004; Guarracino et al., 2014):

$$ \frac{dV_p(R)}{dt} = S_p(R)\alpha(R). \qquad (28) $$

Whereas dissolution will only occur in water-saturated pores, for further developments and similar to the film depositional model proposed by Freedman et al. (2004), we assume that this reactive process affects all pore radii. Note that in Eq. (28), the dissolution rate depends on the pore radius value ($\alpha(R)$) since large pores enlarge faster than small pores during dissolution (e.g. Schechter and Gidley, 1969). We assume a linear dependence between these variables (i.e. $\alpha(R) = \tilde{\alpha}R$) since a constant rate would reflect an extremely slow dissolution independent of the actual occuring dissolution along the flow path. In the other extreme, a relationship with a higher exponent would reflect a very aggressive dissolution which could lead to wormholes (e.g., Noiriel et al., 2004). Under this linear rate hypothesis, the fractal dimension and the radial and length factors can be assumed constant during the reactive process. Integrating Eq. (28), we obtain the evolution with time of a pore radius (Guarracino et al., 2014):

$$ R(t) = R(t_0)e^{\beta(t-t_0)} \qquad (29) $$

being $t_0$ (h) the initial time of the dissolution process and factor $\beta$ (h$^{-1}$) a function of the model parameters $a$, $c$ and $\tilde{\alpha}$ (h$^{-1}$) given by (Soldi et al., 2024):

$$ \beta = \frac{\tilde{\alpha}[4\pi(1+a)c + (1-a)(1-2c)]}{2\pi(1+a)^2 + \pi(1-a)^2 + 8(1-a^2)(1-2c)}. \qquad (30) $$

Note that for a dissolution process both $\tilde{\alpha}$ and $\beta$ are positive parameters. Under the same hypotheses and considering that parameter $\tilde{\alpha} < 0$, we can derive a similar expression to Eq. (29) where $\beta < 0$ (i.e. a decreasing radius with time) which allows us to describe the reduction of the pore volume with time that occurs during a precipitation process.

The evolution of the electrical conductivity during reactive processes can be obtained by considering the changes of the pore radii $R_{min}$ and $R_{max}$ with time. Substituting Eq. (29) in Eq. (15), the temporal variation of this macroscopic property for saturated conditions can be expressed as:

$$ \sigma_{sat}^{REV}(t) = \sigma_{sat}^{REV}(t_0)e^{\beta(2-D)(t-t_0)} \qquad (31) $$

where $\sigma_{sat}^{REV}(t_0)$ is the electrical conductivity of the REV at the initial time $t_0$ of the dissolution process. For partially saturated conditions, at each time of a dissolution
process, the pressure head needed to drain or flood the REV is related to the size of the pore throat and pore body radii for the drainage and imbibition processes, respectively. Therefore, the relative electrical conductivity (Eqs. (24) and (25)) depends on time and for a drainage process it can be expressed as

$$ \sigma_{rel}^{d}(h,t) = \frac{(ah)^{D-2}e^{\beta(D-2)(t-t_0)} - h_{max}^{D-2}(t_0)}{h_{min}^{D-2}(t_0) - h_{max}^{D-2}(t_0)} \quad (32) $$

where h varies between $h_{min}(t_0)e^{-\beta(t-t_0)}/a$ and $h_{max}(t_0)e^{-\beta(t-t_0)}/a$. While for an imbibition process, the relative electrical conductivity yields

$$ \sigma_{rel}^{w}(h,t) = \frac{h^{D-2}e^{\beta(D-2)(t-t_0)} - h_{max}^{D-2}(t_0)}{h_{min}^{D-2}(t_0) - h_{max}^{D-2}(t_0)} \quad (33) $$

where h varies between $h_{min}(t_0)e^{-\beta(t-t_0)}$ and $h_{max}(t_0)e^{-\beta(t-t_0)}$. The terms $h_{min}(t_0)$ and $h_{max}(t_0)$ are the minimum and maximum pressure heads at the beginning of the dissolution process. Given the exponential nature of Eqs. (29), (31), (32), and (33), we perform a parametric analysis of the exponent parameter $\beta$ (Eq. (30)). Fig. 4 shows the role of the model's geometric parameters (a and c) in the estimates of parameter $\beta$ for a constant dissolution rate of $\tilde{\alpha} = 0.0001 h^{-1}$. Note that for a high value of c and low values of a (i.e., a long pore-throat length and important pore radius reduction), the values of parameter $\beta$ increase significantly as dissolution will produce a significant opening of the pore. Therefore, as the flow increases through the capillaries, so does the electrical conductivity. It can also be observed that for high values of parameter a and low values of c (i.e., capillaries with a slight pore throat), parameter $\beta$ has the lowest value. This means that the radius of the pores will increase with the dissolution without significant changes occurring in the constrictive geometry as the capillary is almost a cylindrical tube. From Eq. (30) it can be seen that the value of $\beta$ remains constrained in the range $0 \le \beta \le \frac{4\pi-1}{3\pi-8}\tilde{\alpha} \approx 8.12\tilde{\alpha}$. Indeed, as the magnitude of the dissolution rates $\tilde{\alpha}$ are very low, parameter $\beta$ cannot take a significant value. Note also that when the radii of the pores increase, the pressure head needed to drain or flood the REV decreases, and the hysteresis cycle of the final stage of dissolution is shifted to the left of the cycle from the beginning of the process.

The model proposed in this section is developed within the framework of capillary tube models. One of the limitations that affect the models based on this framework is that lateral connection between the capillaries is not considered. There are no intersection points among the pores and they all have parallel directions. However, many models of electrical properties have been developed in this framework and successfully proven to provide good estimates of electrical and hydraulic properties (e.g., see comparison between 2D pore networks and capillary bundle models in Jougnot et al., 2019). A second point of concern in the model regards the relatively high number of model parameters compared to previous models such as the empirical model of Archie (1942) which has only two parameters (cementation and saturation exponents). Nevertheless, the proposed theoretical model can be used for any porous media while Archie (1942) cannot represent
<figure><img src="image_5.png" /><figcaption>Figure 4: Parametric analysis of the factor $\beta$ as a function of the radial factor $a$ for different constant values of the length factor $c$ and a constant value of the dissolution rate $\tilde{\alpha} = 0.0001 h^{-1}$.</figcaption></figure>

media with high porosity and low electrical conductivity. Beyond that, a step forward of the proposed model over the classical is that it accounts for phenomena such as the hysteresis effects or dissolution/precipitation. One last point of the proposed petrophysical model is that these reactive processes are considered through their effect on the pore geometry changes, while chemical variations, such as changes on salinity or pH, are neglected. These variations were taken into account in the geochemical model proposed by Rembert et al. (2022), and in the works of Leroy et al. (2017) and Heberling et al. (2021) who developed geochemical models that consider surface alterations. Note that, while other electrical models can be used to estimate the electrical conductivity, the proposed model can explicitly relate parameters linked to the microstructure to the estimates of the property at the macroscale, and with only one set of them is also capable to account for hysteresis effects and reactive processes.

## 3 Model comparison with experimental data

The proposed model ability to estimate the electrical conductivity is tested against different sets of experimental data from the research literature. These data sets consist of measured electrical conductivity or relative electrical conductivity values for different soil textures.
## 3.1 Electrical conductivity data for different soil textures

In order to test the estimates of the proposed model, we selected experimental data sets from different soil textures: a sandy loam from Amente et al. (2000), a sand from Inoue et al. (2000), a loam from Doussan and Ruy (2009) and a packing of mica particles from Friedman and Robinson (2002). These data series consist of electrical conductivity or relative electrical conductivity values as a function of saturation. As mentioned is Section 2, the proposed model assumes that the surface electrical conduction is negligible and the electrical conductivity is due to the conduction in the porous water. However, the values of surface conductivity for soils can impact the experimentally measured values of soil conductivity. We account for the contribution of the surface conductivity $\sigma_s$ as a parallel conductivity with an adjustable value, therefore, considering Eqs. (15), (27) and (26), the model expression as function of water saturation becomes:

$$ \sigma^{REV}(S_w) = \frac{\sigma_w f_{\sigma}(a, c) \phi}{\tau^2(1 - S_r)} (S_w - S_r) + \sigma_s. \quad (34) $$

Equation (34) is also compared with the Waxman and Smits (1968) model:

$$ \sigma_{WS}(S_w) = \frac{S_w^n}{F} \left( \sigma_w + \frac{\sigma_s^{WS}}{S_w} \right) \quad (35) $$

where the formation factor $F$ is related to porosity $\phi$ through $F = \phi^{-m}$. The relative electrical conductivity values $\sigma_{rel}^{REV}(S_w)$ can be estimated, for both models, as the quotient between the electrical conductivity value as function of saturation $\sigma^{REV}(S_w)$ and its value at total saturation $\sigma^{REV}(S_w = 1)$. The proposed model is fitted by an exhaustive search method by minimizing the normalized mean square error (NMSE) between the calculated and the experimentally measured values. The comparison between the proposed model and the different experimental data is shown in Figure 5 and the model of Waxman and Smits is also shown. Tables 1 and 2 list the best fitted parameters of the proposed model ($a, c, \tau, S_r$ and $\sigma_s$) and Waxman and Smits model ($n, m$ and $\sigma_s^{WS}$), respectively. The values of $\phi$ and $\sigma_w$ required by the models are listed in Table 1. By comparing the values of parameter $\sigma_s$ and $\sigma_s^{WS}$, it can be observed that the differences between them is lower for the most conductive electrolytes (i.e., high $\sigma_w$) since in those cases the electrical conduction is more significant in the liquid phase than the electrical conduction occurring in the liquid-solid interface. Waxman and Smits (1968) accounted the surface electrical conduction by assuming that this contribution depends inversely with water saturation (Eq. (35)). Therefore, the electrical conduction takes place near the mineral surface for low saturations, while for high saturations the electrical conduction through the electrolyte is the most important contribution. However, the surface conductivity is also dependent on the salinity, the mobility of the ions, and the surface charge density of the EDL (Revil, 2013a). In the proposed model (Eq. (34)), $\sigma_s$ is treated as a constant for simplicity, not varying with water saturation as assumed by previous works (e.g., Mualem and Friedman, 1991; Friedman, 2005; Laloy et al., 2011; Breede et al., 2011). Even though the differences in the developments of the models, note that the proposed model is able
<figure><img src="image_6.png" /><figcaption>Figure 5: Electrical conductivity as function of water saturation for different data sets: a) a sandy loam from Amente et al. (2000), b) a packing of mica particles from Friedman and Robinson (2002), c) a sand from Inoue et al. (2000) and d) a loam from Doussan and Ruy (2009).</figcaption></figure>

to fairly reproduce the behaviour of the data for the different soil textures and is in good agreement with the previous model of Waxman and Smits.

The experimental samples from Weerts et al. (1999) correspond to four undisturbed soil cores of a loamy sand (named PS9, PS10, PS11 and PS13) where the electrical conductivity was measured as function of water saturation. Figure 6 illustrates the fit of the proposed model (Eqs. (17) and (27)) for the four samples and the best-fitted parameters are listed in Table 3 which were found using the same method as for the previous experimental data sets. It can be observed that the proposed model produces a fairly good agreement with the data sets for all the samples. From the comparison of the fitted model parameter values, note that even though their values differ between the different samples, they vary between close values which can be expected as the samples are from a same soil type. The pore geometry described by these parameters presents smooth changes between the throat and wide pore body. Indeed, the high values of *c* and *a* represent that the pore radius varies moderately. We can compare our model performance to the previously published model of Thanh et al. (2020) to fit these experimental data. They derived a model based on a capillary bundle of constant radii pores and proposed a similar number of fitting parameters. Note that our new model performs better than the
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The data from Tartrat et al. (2019) consist on electrical conductivity-water saturation values measured in five core samples. These samples were made of mixes of a background illitic clay material and pyrite grains. The electrical measurements were performed on each sample during their desiccation. This process causes a variation on the water electrical conductivity that cannot be neglected as salinity increases during the decrease of the saturation due to evaporation. To account this phenomenon on the estimates of the electrical conductivity model (Eq. (34)), we assume that the water electrical conductivity increases inversely with saturation during the desiccation process, $\sigma_w(S_w) = \sigma_w(S_w = 1)/S_w$ where the water electrical conductivity at saturation is $\sigma_w(S_w = 1) = 0.115 \text{ S m}^{-1}$ (Tartrat et al., 2019). The performance of the proposed model for the different samples is shown in Figure 7 and Table 4 lists the best fitted parameter values. The porosity value needed to estimate $\sigma_{sat}^{REV}$ was taken from Tartrat et al. (2019) being its value $\phi = 0.65$. The proposed model is able to fairly well reproduce the behaviour of all the experimental data sets. The values obtained for the geometrical parameters $a$ and $c$ that define the constrictivities of the pores vary between the different samples. However, they remain in a narrow range of values, from 0.40 to 0.63 for $a$, and from 0.36 to 0.49 for $c$. These ranges of $a$ and $c$ values represent pores with a fairly significant constrictivity (i.e. a large fraction of the pore with reduced radius) which can be expected since clays present these reductions in the pore structure due to the packing of its fined grains.

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Saafan et al. (2023) measured electrical conductivity variations with water saturation in low-permeability sandstone samples, called S6, S9, S13, S14, S16 and S18. The test of the proposed model relies on fitting the electrical conductivity data sets with Eq. (34). As sandstones usually present very low surface conductivities, we considered negligible its contribution to the electrical conductivity. Using an exhaustive search method as previously, we found the best-fitted parameters which are listed in Table 5. To fit these parameters we considered the value of the water electrical conductivity $\sigma_w = 5.38 \text{ S m}^{-1}$ which was taken from Saafan et al. (2023). Figure 8 shows predicted and measured data where it can be observed that the proposed model produces a fairly good agreement for all the samples. The low values of parameter $a$ and high values of $c$ fitted for the proposed model represent pore geometries with a strong reduction of the pore radii along a significant length of the capillary. This pore geometry can be expected since the enclosure of the pore radii hinders the water flow which translates in low permeability samples.

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## 3.2 Electrical conductivity during dissolution

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The performance of the model to describe the dissolution phenomenon on the electrical conductivity is tested with experimental data obtained by Rembert et al. (2023a). They measured the evolution of the electrical conductivity $\sigma_{sat}^{REV}$ with time on two limestone samples, named E04 and E05, during dissolution. Moreover, they also measured the variations of porosity $\phi$ and permeability $k$ of the sample due to this process. Based on the expressions of $\phi$ and $k$ developed by Soldi et al. (2022) and considering the changes
<figure><img src="image_7.png" /><figcaption>Figure 6: Comparison between the electrical conductivity proposed model (solid lines) and experimental data sets (points) from a loamy sand (Weerts et al., 1999).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Figure 7: Comparison between the electrical conductivity proposed model (solid lines) and experimental data sets (points) measured in a illitic clay material with pyrite grains from Tartrat et al. (2019).</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 8: Comparison between the electrical conductivity proposed model (solid lines) and experimental data sets (points) from a low-permeability sandstone from Saafan et al. (2023).</figcaption></figure>

<table><caption>Table 1: Values of the parameters used to estimate the electrical conductivity for experimental data from Amente et al. (2000), Inoue et al. (2000), Friedman and Robinson (2002) and Doussan and Ruy (2009). The porosity $\phi$ and water electrical conductivity $\sigma_w$ values were taken from their works.</caption><thead><tr><th rowspan="2">Sample</th><th rowspan="2">&phi; (-)</th><th rowspan="2">&sigma;<sub>w</sub> (S m<sup>-1</sup>)</th><th colspan="5">Proposed model parameters</th><th rowspan="2">NMSE</th></tr><tr><th>a (-)</th><th>c (-)</th><th>&tau; (-)</th><th>S<sub>r</sub> (-)</th><th>&sigma;<sub>s</sub> (S m<sup>-1</sup>)</th></tr></thead><tbody><tr><td>Sandy loam</td><td>0.40</td><td>0.565</td><td>0.59</td><td>0.84</td><td>1.40</td><td>0.100</td><td>10.0&times;10<sup>-4</sup></td><td>0.0167</td></tr><tr><td>Sand</td><td>0.45</td><td>0.058</td><td>0.66</td><td>0.71</td><td>1.36</td><td>0.056</td><td>2.00&times;10<sup>-4</sup></td><td>0.0280</td></tr><tr><td>Mica particles</td><td>0.77</td><td>0.510</td><td>0.78</td><td>0.87</td><td>1.12</td><td>0.387</td><td>1.80&times;10<sup>-4</sup></td><td>0.3430</td></tr><tr><td>Loam</td><td>0.44</td><td>0.072</td><td>0.55</td><td>0.65</td><td>1.54</td><td>0.028</td><td>0.45&times;10<sup>-4</sup></td><td>0.3127</td></tr></tbody></table>

<table><caption>Table 2: Fitted Waxman and Smits parameter values to estimate the electrical conductivity for experimental data. The porosity $\phi$ and water electrical conductivity $\sigma_w$ values are the values listed in Table 1.</caption><thead><tr><th>Sample</th><th>n (-)</th><th>m (-)</th><th>&sigma;<sub>s</sub><sup>WS</sup> (S m<sup>-1</sup>)</th><th>NMSE</th><th>Source</th></tr></thead><tbody><tr><td>Sandy loam</td><td>1.70</td><td>1.66</td><td>32.0&times;10<sup>-4</sup></td><td>0.0147</td><td>Amente et al. (2000)</td></tr><tr><td>Sand</td><td>2.00</td><td>1.53</td><td>16.5&times;10<sup>-3</sup></td><td>0.0034</td><td>Inoue et al. (2000)</td></tr><tr><td>Mica particles</td><td>2.60</td><td>2.73</td><td>97.0&times;10<sup>-4</sup></td><td>0.3050</td><td>Friedman and Robinson (2002)</td></tr><tr><td>Loam</td><td>1.60</td><td>2.62</td><td>77.4&times;10<sup>-3</sup></td><td>0.3144</td><td>Doussan and Ruy (2009)</td></tr></tbody></table>
<table><caption>Table 3: Values of the parameters used to estimate the electrical conductivity for the experimental data from Weerts et al. (1999). The porosity $\phi$ and water electrical conductivity $\sigma_w$ values were taken from their work.</caption><thead><tr><th>Sample</th><th>φ<br>(-)</th><th>σ<sub>w</sub><br>(S m<sup>-1</sup>)</th><th colspan="4">Proposed model parameters</th><th>NMSE</th></tr><tr><th></th><th></th><th></th><th>a (-)</th><th>c (-)</th><th>τ (-)</th><th>S<sub>r</sub> (-)</th><th></th></tr></thead><tbody><tr><td>PS9</td><td>0.41</td><td>0.48</td><td>0.72</td><td>0.62</td><td>1.23</td><td>0.17</td><td>1.16×10<sup>-2</sup></td></tr><tr><td>PS10</td><td>0.37</td><td>0.90</td><td>0.73</td><td>0.67</td><td>1.21</td><td>0.26</td><td>1.07×10<sup>-2</sup></td></tr><tr><td>PS11</td><td>0.41</td><td>1.34</td><td>0.69</td><td>0.65</td><td>1.27</td><td>0.28</td><td>2.90×10<sup>-2</sup></td></tr><tr><td>PS13</td><td>0.40</td><td>2.25</td><td>0.74</td><td>0.58</td><td>1.28</td><td>0.27</td><td>2.50×10<sup>-2</sup></td></tr></tbody></table>

<table><caption>Table 4: Values of the parameters used to estimate the electrical conductivity for the experimental data from Tartrat et al. (2019).</caption><thead><tr><th>Sample</th><th colspan="5">Proposed model parameters</th><th>NMSE</th></tr><tr><th></th><th>a (-)</th><th>c (-)</th><th>τ (-)</th><th>S<sub>r</sub> (-)</th><th>σ<sub>s</sub> (S m<sup>-1</sup>)</th><th></th></tr></thead><tbody><tr><td>1</td><td>0.53</td><td>0.46</td><td>1.55</td><td>0.34</td><td>0.026</td><td>4.40×10<sup>-2</sup></td></tr><tr><td>2</td><td>0.47</td><td>0.49</td><td>1.51</td><td>0.44</td><td>0.028</td><td>4.00×10<sup>-2</sup></td></tr><tr><td>3</td><td>0.62</td><td>0.36</td><td>1.67</td><td>0.50</td><td>0.036</td><td>3.30×10<sup>-2</sup></td></tr><tr><td>4</td><td>0.58</td><td>0.40</td><td>1.50</td><td>0.47</td><td>0.039</td><td>5.50×10<sup>-2</sup></td></tr><tr><td>5</td><td>0.40</td><td>0.59</td><td>1.37</td><td>0.42</td><td>0.042</td><td>5.10×10<sup>-2</sup></td></tr><tr><td>6</td><td>0.63</td><td>0.42</td><td>1.28</td><td>0.39</td><td>0.050</td><td>5.30×10<sup>-2</sup></td></tr></tbody></table>

<table><caption>Table 5: Values of the parameters used to estimate the electrical conductivity for the experimental data from Saafan et al. (2023).</caption><thead><tr><th>Sample</th><th>φ<br>(-)</th><th colspan="4">Proposed model parameters</th><th>NMSE</th></tr><tr><th></th><th></th><th>a (-)</th><th>c (-)</th><th>τ (-)</th><th>S<sub>r</sub> (-)</th><th></th></tr></thead><tbody><tr><td>S6</td><td>0.084</td><td>0.14</td><td>0.79</td><td>1.67</td><td>0.35</td><td>3.12×10<sup>-2</sup></td></tr><tr><td>S9</td><td>0.076</td><td>0.13</td><td>0.71</td><td>1.42</td><td>0.31</td><td>2.06×10<sup>-2</sup></td></tr><tr><td>S13</td><td>0.102</td><td>0.20</td><td>0.54</td><td>1.47</td><td>0.21</td><td>3.55×10<sup>-2</sup></td></tr><tr><td>S14</td><td>0.074</td><td>0.17</td><td>0.65</td><td>1.62</td><td>0.26</td><td>4.05×10<sup>-2</sup></td></tr><tr><td>S16</td><td>0.068</td><td>0.16</td><td>0.75</td><td>1.22</td><td>0.04</td><td>1.60×10<sup>-2</sup></td></tr><tr><td>S18</td><td>0.078</td><td>0.15</td><td>0.60</td><td>1.28</td><td>0.27</td><td>4.23×10<sup>-2</sup></td></tr></tbody></table>
of the pore radii with time (Eq. (29)), these two properties can be estimated during dissolution by (Soldi et al., 2024):

$$ \phi(t) = \phi(t_0)e^{\beta(2-D)(t-t_0)} \quad (36) $$

$$ k(t) = k(t_0)e^{\beta(4-D)(t-t_0)} \quad (37) $$

where $\phi(t_0)$ and $k(t_0)$ are the porosity and permeability of the REV at the initial time $t_0$ of the dissolution process.

The time series that we selected to fit the model correspond to a stage of the experiment when the dissolution has been already occurring since the data at the initial and final stages are out of the hypotheses of the model. The data from the beginning of the process is disregarded due to the strong dissolution, and the data from the final stages were measured after the complete percolation and the sample which then cannot be considered as a REV. Eqs. (31), (36) and (37) are fitted using an exhaustive search method by minimizing the weighted normalized error between the calculated and measured values of $\sigma_{sat}^{REV}$, $\phi$ and $k$ simultaneously. Figure 9 shows the fit of the proposed model to these petrophysical properties and the values of the fitted parameters are listed in Table 6. To estimate $\phi(t_0)$, $k(t_0)$ and $\sigma_{sat}^{REV}(t_0)$ in Eqs. (36), (37), and (31), the model also requires a $R_{REV}$ and $\sigma_w$ values for which we consider the values from Rembert et al. (2023a), $R_{REV} = 9$ mm, $\sigma_w = 1200$ µS cm$^{-1}$ for E04 and $\sigma_w = 1000$ µS cm$^{-1}$ for E05. Note that the evolution over time of $\phi$, $k$, and $\sigma_{sat}^{REV}$ of the experimental data can be satisfactorily reproduced by the proposed model, as shown in Fig. 9(a-b), (d-c) and (e-f) respectively. It can be observed from the values of the best fitted parameters (Table 6) that they do not change significantly between the two samples. Moreover, the fitted minimum and maximum radii are consistent with the values obtained from statistical computations on tomographic images of the samples (Rembert et al., 2023a) while the fitted dissolution rate may be slightly higher than the one calculated in Léger et al. (2022a). It is important to notice that we analyse the evolution of these macroscopic properties due to the changes in the pore geometry caused by the dissolution. Therefore, variations of other parameters during the process, such as ionic concentration of the pore water, are neglected. The experimental data used in this section from Rembert et al. (2023a) account for water electrical conductivity measured at the outlet. They observed that the electrical conductivity of carbonate samples presents a significant increase at the beginning of the dissolution followed by a gentle slope that decreases its magnitude. This fact can be related to an initial strong dissolution regime that diminishes with time since water electrical conductivity is stable over this period.

To highlight the link between the evolution of hydraulic and electrical properties during the dissolution, the permeability and electrical conductivity are plotted as functions of the porosity. Fig. 10 shows the relationship between these properties for the data sets and the proposed model. We use the fitted parameters of the proposed model from Table 6. In addition, we compare them with the most classical equation to represent permeability-porosity and electrical conductivity-porosity data, the Kozeny-Carman equation (Kozeny, 1927; Carman, 1937) and Archie (1942). Following the Kozeny-Carman equation, perme-
ability can be expressed in terms of porosity as:

$$k = p \frac{\phi^3}{(1 - \phi)^2}, \qquad (38)$$

where $p$ is a fitting parameter that depends on the specific internal surface area, an empirical geometrical parameter, and the tortuosity. The best agreement between the data and the Kozeny-Carman equation is obtained for $p = 1.732 \times 10^{-12}$ ($NMSE = 6.514 \times 10^{-3}$) and $p = 2.7949 \times 10^{-12}$ ($NMSE = 7.161 \times 10^{-2}$), for E04 and E05, respectively. It can be observed that the proposed model provides better results than the Kozeny-Carman equation for both samples (see Figs. 10a and b). In fact, the $NMSE$ values for the proposed model are $NMSE = 1.595 \times 10^{-5}$ for E04 and $NMSE = 3.401 \times 10^{-3}$ for E05 which are significantly smaller than for the Kozeny-Carman equation.

The relationship between the electrical conductivity and porosity given by Archie (1942) in Eq. (2), which under fully water-saturated conditions yields:

$$\sigma = \sigma_w \phi^m, \qquad (39)$$

where $m$ is the cementation exponent. For the $\sigma_w$ value, we considered the values from Rembert et al. (2023a), previously mentioned for the fitting of the proposed model. The value of this fitted parameter is $m = 2.679$ ($NMSE = 1.490 \times 10^{-2}$) and $m = 2.712$ ($NMSE = 2.390 \times 10^{-2}$), for E04 and E05, respectively. Figs. 10c and d show the comparison between the data, the proposed model, and Archie (1942). It can be seen that the model from Archie (1942) has a poor performance in reproducing the data behavior while the proposed model predicts closer values. The $NMSE$ values for the proposed model are $NMSE = 7.378 \times 10^{-5}$ for E04 and $NMSE = 1.276 \times 10^{-4}$ for E05. Note that the $NMSE$ values between the Archie (1942) and the proposed model differs between 2 and 3 orders of magnitude.

From a geometrical approach, the proposed model can represent the changes of the pore structure which results in a larger porosity, permeability, and electrical conductivity of the porous medium. The comparison with classical models shows that they cannot reproduce the variations that affect these properties of the porous media under dissolution. Therefore, the proposed analytical model represents a step forward to estimate the petrophysical properties during reactive processes.

# 4 Discussion and conclusions

Based on the framework of capillary tubes approach, we present an analytical model to determine the electrical conductivity of a porous medium represented by tortuous capillaries with varying aperture. The periodic fluctuations of the capillary radii follow a sinusoidal piecewise behaviour along the pore length. This pore geometry determines periodical fractions of the capillaries with pore throats and pore bodies defined by two geometrical parameters, the radial $a$ and length $c$ factors, which allow us to represent a wide range of pore geometries. The model has analytical closed-form expressions that
<table><caption>Table 6: Values of the parameters used to estimate the electrical conductivity for the experimental data from Rembert et al. (2023a).</caption><thead><tr><th rowspan="2">Sample</th><th colspan="6">Proposed model parameters</th><th colspan="2">NMSE</th></tr><tr><th>a<br>(-)</th><th>c<br>(-)</th><th>D<br>(-)</th><th>τ<br>(-)</th><th>R<sub>max</sub><br>(mm)</th><th>R<sub>min</sub><br>(mm)</th><th>α̃<br>(h<sup>-1</sup>)</th><th></th></tr></thead><tbody><tr><td>E04</td><td>0.20</td><td>0.87</td><td>1.31</td><td>1.36</td><td>0.45</td><td>1.26×10<sup>-4</sup></td><td>0.0046</td><td>1.814×10<sup>-3</sup></td></tr><tr><td>E05</td><td>0.28</td><td>0.82</td><td>1.29</td><td>1.25</td><td>0.86</td><td>5.50×10<sup>-4</sup></td><td>0.0055</td><td>3.521×10<sup>-3</sup></td></tr></tbody></table>

<figure><img src="image_10.png" /><figcaption>Figure 9: Model fit to the time evolution of: a-b) porosity, c-d) permeability and e-f) electrical conductivity data for samples E04 and E05 from Rembert et al. (2023a).</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 10: Comparison among the estimates of: a) and b) permeability-porosity for the proposed model and the Kozeny-Carman Equation, c) and d) electrical conductivity-porosity for the proposed model and Archie's law or samples E04 and E05 from Rembert et al. (2023a).</figcaption></figure>
depend on parameters which all have a physical or geometrical meaning. In the particular case of $c = 0.5$, the proposed model is consistent with the model developed by Rembert et al. (2020) for a porous medium under water saturated conditions. In addition, the proposed model addresses partially saturated conditions and more complex processes.

The periodic throats of the pore structure allow us to introduce the hysteresis phenomenon in the relative electrical conductivity when expressed as a function of the pressure head. The resulting expressions for drainage and imbibition experiments also depend on only four independent model parameters (the fractal dimension $D$, the radial factor $a$, the minimum $R_{min}$ and maximum $R_{max}$ radii). Hysteretic behaviour of the electrical conductivity has been observed in experimental and numerical studies as function of saturation (e.g. Knight, 1991; Zhang et al., 2017; Maineult et al., 2018). However, from our theoretical results, hysteresis phenomena are not visible when expressed as a function of the saturation. Phenomena observed by other studies may be related to different effects, such as contact angle hysteresis, disconnection of the fluid phase (i.e., snap off), or film flow (e.g. Jury et al., 1991; Blunt et al., 2002; Spiteri et al., 2008).

In our model, the reactivity of the porous medium is introduced by considering the variations in the structure of the capillaries. Therefore, it should be noted that the model focuses only on the geometrical changes that these processes produce on the porous medium. The proposed model is compared against experimental data of electrical conductivity, porosity, and permeability during dissolution. This comparison shows that the model is satisfactorily able to reproduce the evolution of these three petrophysical properties with time simultaneously.

The performance of our model to estimate electrical conductivity is tested against different sets of experimental data. The proposed relationship between the relative electrical conductivity and effective saturation can fairly reproduce the data from different soil textures and a packing of mica particles. Moreover, when estimating the electrical conductivity for several samples of a same type of soil, the fitting parameter values vary in a narrow range for each soil. For the loamy sand case, the high values of the radial factor $a$ and length factor $c$ represent pores whose radii vary slightly between their constrictive and non-constrictive fractions which results in less tortuous paths due a higher connection between the fractions (e.g. Xing et al., 2021). For the illitic clay samples, the values of $a$ are low while the tortuosity values are higher than the ones for the loamy sand. This result is consistent with this type of soil that may present more constrictive pores and complex flow paths (i.e. more tortuous) due to the packaging of its finer grains. While in the case of the sandstone samples, the significant low values of $a$ and high values of $c$ represent strong constrictivities of the pores (i.e. high pore radii reductions over a large length of the pores) causing that the water flow hinders and a high decrease of the permeability.

The model developed in this study represents an improvement over available models within the framework of capillary tube models by its capability to describe a wide variety of porous space geometries. In fact, it could be used to represent micropores which may have long and narrow necks (i.e. high $c$ value and low $a$ value) connected to larger and short pore bodies (e.g. Sun and Torres-Verdín, 2022). When compared to most
widely used models, the proposed model is consistent with the empirical model of Archie (1942) for the cases of non-constrictive capillaries, and is in good agreement with the model of Waxman and Smits (1968) for simple rock characterization. Even though the expression of these classical models to describe the electrical conductivity might be less complex, they cannot account for hysteresis effects which are of significant importance for unsaturated flow, nor for reactive phenomena that can predominate in carbonate media. On the contrary, the proposed model can represent both of these phenomena and with one set of the model parameters. This new analytical model to estimate the electrical conductivity is developed under the same framework and assumptions used previously to derive hydraulic properties (permeability, porosity). From a geometrical approach, we are capable of describing petrophysical and flow properties while accounting for their characteristic physical phenomena such as hysteresis and reactive processes. Thus, this work represents a significant step toward a unified model for many important petrophysical properties, and therefore, a way to characterize hydraulic properties of porous media from electrical measurements.

## 5 Notation List

<table><thead><tr><th>Symbol</th><th>Description</th><th>Units</th></tr></thead><tbody><tr><td>REV</td><td>Representative elementary volume</td><td>-</td></tr><tr><td>σ</td><td>Electrical conductivity</td><td>S m<sup>-1</sup></td></tr><tr><td>σ<sub>w</sub></td><td>Electrical conductivity of the electrolite</td><td>S m<sup>-1</sup></td></tr><tr><td>F</td><td>Formation factor</td><td>-</td></tr><tr><td>σ<sub>s</sub></td><td>Surface electrical conductivity</td><td>S m<sup>-1</sup></td></tr><tr><td>φ</td><td>Porosity</td><td>-</td></tr><tr><td>S<sub>w</sub></td><td>Water saturation</td><td>-</td></tr><tr><td>n, m</td><td>Archie's saturation and cementation factors, respectively</td><td>-</td></tr><tr><td>R</td><td>Radius of a circular tube</td><td>m</td></tr><tr><td>l</td><td>Pore length</td><td>m</td></tr><tr><td>τ</td><td>Tortuosity</td><td>-</td></tr><tr><td>a</td><td>Radial factor of the constrictivity</td><td>-</td></tr><tr><td>c</td><td>Length factor of the constrictivity</td><td>-</td></tr><tr><td>λ</td><td>Wavelength</td><td>m</td></tr><tr><td>r(x)</td><td>Pore radius variation along the longitudinal variable x</td><td>m</td></tr><tr><td>R<sub>REV</sub></td><td>REV radius</td><td>m</td></tr><tr><td>R<sub>min</sub>, R<sub>max</sub></td><td>Minimum and maximum pore radii, respectively</td><td>m</td></tr><tr><td>N(R)</td><td>Number of pores of radius equal or larger than R</td><td>-</td></tr><tr><td>D</td><td>Fractal dimension</td><td>-</td></tr><tr><td>L</td><td>REV length</td><td>m</td></tr><tr><td>Σ<sub>p</sub></td><td>Electrical conductance of a pore</td><td>S</td></tr></tbody></table>

(It continues in the following page).
<table><thead><tr><th>Symbol</th><th>Description</th><th>Units</th></tr></thead><tbody><tr><td><i>f</i></td><td>Reduction factor in a pore conductance due to the presence of constrictivities</td><td>-</td></tr><tr><td><i>σ</i><sub>p</sub></td><td>Electrical conductivity of a pore</td><td>S m<sup>-1</sup></td></tr><tr><td><i>i</i><sub>p</sub></td><td>Electric current in a pore</td><td>A</td></tr><tr><td>Δ<i>V</i></td><td>Electrical voltage difference</td><td>V</td></tr><tr><td><i>I</i><sup>REV</sup></td><td>Electric current in the REV</td><td>A</td></tr><tr><td><i>σ</i><sup>REV</sup></td><td>Electrical conductivity of the REV</td><td>S m<sup>-1</sup></td></tr><tr><td><i>σ</i><sup>REV</sup><sub>sat</sub></td><td>Electrical conductivity of the total saturated REV</td><td>S m<sup>-1</sup></td></tr><tr><td><i>f</i><sub>v</sub></td><td>Reduction factor in the porosity due to the presence of constrictivities</td><td>-</td></tr><tr><td><i>f</i><sub>σ</sub></td><td>Reduction factor in <i>σ</i><sup>REV</sup> due to the presence of constrictivities</td><td>-</td></tr><tr><td><i>h</i></td><td>Pressure head</td><td>m</td></tr><tr><td><i>T</i><sub>s</sub></td><td>Surface tension</td><td>N m<sup>-1</sup></td></tr><tr><td>γ</td><td>Contact angle</td><td>degrees</td></tr><tr><td>ρ<sub>w</sub></td><td>Water density</td><td>kg m<sup>-3</sup></td></tr><tr><td><i>g</i></td><td>Gravitational acceleration</td><td>m s<sup>-2</sup></td></tr><tr><td><i>R</i><sub>h</sub></td><td>Pore radius related to pressure head <i>h</i></td><td>m</td></tr><tr><td><i>R</i><sup>*</sup></td><td>Integration variable</td><td>m</td></tr><tr><td><i>σ</i><sub>rel</sub></td><td>Relative electrical conductivity of the REV</td><td>-</td></tr><tr><td><i>σ</i><sup>d</sup><sub>rel</sub>,<i>σ</i><sup>w</sup><sub>rel</sub></td><td>Main drying and wetting relative electrical conductivity, respectively</td><td>-</td></tr><tr><td><i>h</i><sub>min</sub>, <i>h</i><sub>max</sub></td><td>Minimum and maximum pressure heads, respectively</td><td>m</td></tr><tr><td><i>S</i><sub>e</sub>, <i>S</i><sub>r</sub></td><td>Effective and residual water saturations, respectively</td><td>-</td></tr><tr><td><i>V</i><sub>p</sub></td><td>Pore volume</td><td>m<sup>3</sup></td></tr><tr><td><i>t</i></td><td>Time</td><td>h</td></tr><tr><td><i>S</i><sub>p</sub></td><td>Pore surface</td><td>m<sup>2</sup></td></tr><tr><td>α</td><td>Dissolution rate</td><td>m<sup>3</sup> h<sup>-1</sup></td></tr><tr><td>α̃</td><td>Dissolution rate</td><td>h<sup>-1</sup></td></tr><tr><td>β</td><td>Dissolution factor</td><td>h<sup>-1</sup></td></tr><tr><td><i>σ</i><sub>WS</sub></td><td>Electrical conductivity from Waxman and Smits (1968)</td><td>S m<sup>-1</sup></td></tr><tr><td><i>k</i></td><td>Permeability</td><td>m<sup>2</sup></td></tr></tbody></table>

704 Acknowledgements

This research was carried out as part of M. Soldi's postdoctoral fellowship funded by the Consejo Nacional de Investigaciones Científicas y Técnicas (Argentine). The authors strongly thank the Editor Graham Sander, and the reviews of Youzheng Qi and two anonymous reviewers for their careful assessment of our work and their valuable comments and suggestions which helped to improve the manuscript.
710 CRediT authorship contribution statement

711 Mariangeles Soldi: Conceptualization, Methodology, Investigation, Formal analysis, Writing - original draft, Writing – review & editing. Flore Rembert: Investigation, Formal analysis, Methodology, Writing – review & editing. Luis Guarracino: Conceptualization, Methodology, Writing – review & editing, Investigation, Supervision. Damien Jougnot: Conceptualization, Methodology, Writing – review & editing, Investigation, Supervision.
712
713
714
715

716 Data availability

717 No original data, everything has been published before.

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