3. The freezing temperature of red clay or clayey silt is lower than that of quartz sand for identical initial conditions. As the ionic concentration increases or the initial water saturation decreases, the freezing temperature decreases. Additionally, red clay or clayey silt exhibits significantly higher electrical conductivity compared to quartz sand, and the increase in initial water saturation or initial salinity results in higher electrical conductivity over the entire temperature range. Furthermore, we observe that three SFCC modeling parameters are not only related to the pore volumetric contents of the pore, but also related to the initial conditions of the porous media (i.e., water saturation and salinity). However, four geometrical parameters related to pore size ($r_{max}$, $\alpha$, and $D_f$) as well as pore tortuosity ($D_e$) remains stable over the entire water saturation and salinity ranges, which means that these parameters are mainly affected by the size distribution of pores and pore irregularities.

4. The true and apparent formation factors can be determined by the proposed electrical conductivity model. In this study, we compare the evolution of the apparent formation factor as a function of porosity and as a function of water saturation. It was observed that the proposed model correctly reproduces the tendencies of the experimental data. It is worth noting that the true formation factor is unrelated to water saturation, whereas the apparent formation factor decreases as water saturation increases. Besides, we also compare the dependence of the apparent formation factor in the unsaturated frozen porous medium on temperature. The predicted results from the proposed model are largely in agreement with the experimental data.

The proposed capillary bundle model involves four fitting parameters ($r_{max}$, $\alpha$, $D_f$, and $D_e$), each with its own physical meaning. The maximum radius and the ratio of the minimum radius to the maximum radius can be determined from analyzing the PSD of the samples using the NMR method or mercury intrusion porosimetry. In this study, to reduce the number of parameters in the surface complexation model, we adopted the same equilibrium constant values and the total surface site density for different types of minerals (i.e., quartz sand, red clay, and clayey salt). However, to account for the significant differences in the specific surface conductivity among different mineral surfaces (e.g., montmorillonite, kaolinite, and illite), we will need to extend the proposed model in future research. Furthermore, in actual media, particularly in complex porous media like soils, sediments, and rocks, the pore structure can be significantly intricate, involving interconnections between pores. As a result, we will need to consider the effects of these interconnected pores, such as crossed cylindrical pores. Moreover, because the linear additive model of pore water conductivity is primarily efficient at low salinities, its application may result in an overestimation of pore water conductivity across various salinity levels. Given the absence of an appropriate physically-based model for negative temperatures at high concentrations, this opens up a new avenue for exploring enhancements in our future research.

Electrical conductivity measurements are highly valuable in permafrost research, as they can help estimate physical properties of frozen porous media such as temperature, liquid water content, and salinity. The proposed model and corresponding fitting parameters provide strong support for these measurements, particularly when investigating interactions among different physical properties. Electrical conductivity measurements offer crucial information, contributing to a more comprehensive understanding of subsurface permafrost characteristics. However, in real-field data, it is often necessary to distinguish their effects by combining other alternative monitoring methods (e.g., induced polarization and dielectric measurements) or physical parameters (e.g., complex conductivity and dielectric permittivity), since these electrophysical parameters change significantly when the water freezes (Duvillard et al., 2018; Léger et al., 2023).

## Appendix A: Electrical Double-Layer Model of Mineral-Water Interface

We consider that mineral grains are in contact with a simple binary symmetric 1:1 electrolyte (e.g., NaCl) and that the pH is in the range 6–8. We obtain the distribution of the counterions and coions of EDL at the mineral-water interface (e.g., Leroy et al., 2008; Revil & Glover, 1997). In this work, as our experimental measurements of the electrical conductivity are made in the case of neutral pH (6–8), we neglect the effect of the relative equilibrium constants for the second protonation at the mineral-water interface, but the second protonation reaction should be considered in an acid environment. Additional assumption is that the zeta potential of mineral-water interface is equal to the electrical potential at the shear plane (d-plane in Figure 1e).

The electrochemical double-layer in the mineral-water interface consists of a Stern layer with strongly bounded ions (including inner and outer sphere surface complexes) and a diffuse layer with mobile ions (Figure 1e). The counterions of the outer sphere surface complexes in the Stern layer keep a certain mobility as in the diffuse layer. With this “dynamic Stern layer assumption”, the specific surface conductance at the mineral surface is composed
of two contributions: from the excess counterions contained in the outer sphere surface complexes of the Stern layer and from the diffuse layer. Therefore, the surface charge density $Q_{s0}$ (in C/m²) at the mineral surface, the surface charge density $Q_{sβ}$ (in C/m²) at the Stern layer, and the equivalent surface charge density $Q_{ss}$ (in C/m²) related to the excess of charge at the diffuse layer can be computed according to respectively:

$$Q_{s0} = -\frac{e_0 \Gamma_s^0}{\alpha_s} \left[ K_{s2} C_{Na^+}^f \exp\left(-\frac{e_0 \varphi_{s\beta}}{k_b T}\right) + 1 \right], \quad (A1)$$

$$Q_{s\beta} = \frac{e_0 \Gamma_s^0}{\alpha_s} K_{s2} C_{Na^+}^f \exp\left(-\frac{e_0 \varphi_{s\beta}}{k_b T}\right), \quad (A2)$$

$$Q_{ss} = -\sqrt{8\epsilon_f k_b T C_f} \sinh\left(\frac{e_0 \varphi_{sd}}{2k_b T}\right), \quad (A3)$$

with

$$\alpha_s = 1 + K_{s1} C_{H^+}^f \exp\left(-\frac{e_0 \varphi_{s0}}{k_b T}\right) + K_{s2} C_{Na^+}^f \exp\left(-\frac{e_0 \varphi_{s\beta}}{k_b T}\right), \quad (A4)$$

where $\Gamma_s^0$ is the total surface site density of the mineral grain (in sites nm⁻²), $K_{s1}$, $K_{s2}$ are the intrinsic equilibrium constants of different types of amphoteric reactions in the mineral-water interface (values are shown in Table S1 in Supporting Information S1). In this study, we do not take into account the influence of different mineral surfaces on the specific surface conductance, thus we take the same equilibrium constant at the mineral-water interface for any type of materials (red clay, quartz sand and clayey silt) to compute the electrochemical properties (surface charge density, zeta potential and specific surface conductance) of the mineral surface in order to decrease the number of parameters when modeling the electrokinetic properties. The symbols $C_{H^+}^f$, $C_{Na^+}^f$ and $C_f$ (in mol/L) represent the concentrations of the protons, the counterions ($Na^+$ in the free electrolyte) and the salinity in the free electrolyte, $C_{Na^+}^f = C_f$ and $C_{H^+}^f = 10^{-pH}$, $\epsilon_f$ is the pore water permittivity ($\epsilon_f = 81\epsilon_0$, $\epsilon_0 \sim 8.85 \times 10^{-12} \text{ F m}^{-1}$), $\epsilon_0$ represents the elementary charge (taken positive, $\epsilon_0 = 1.6 \times 10^{-19} \text{ C}$), $k_b$ is the Boltzmann constant ($1.381 \times 10^{-23} \text{ J K}^{-1}$), $\varphi_{s0}$, $\varphi_{s\beta}$, and $\varphi_{sd}$ (in V) are the electrical potential at the mineral surface (0-plane), at the $\beta$- and $d$-planes, respectively. The electrical potentials are related by the series capacitor at the 0-, $\beta$- and $d$-planes. This yields

$$\varphi_{s0} - \varphi_{s\beta} = \frac{Q_{s0}}{C_{s1}}, \quad (A5)$$

and

$$\varphi_{s\beta} - \varphi_{sd} = -\frac{Q_{ss}}{C_{s2}}, \quad (A6)$$

where $C_{s1}$ and $C_{s2}$ are the (constant) integral capacities of the inner and outer parts of the Stern layer in the mineral-water interface, respectively, and the values of these capacities are listed in Table S1 in Supporting Information S1 (Leroy et al., 2008).

The global electroneutrality equation for EDL in the mineral grain and water interface is

$$Q_{s0} + Q_{s\beta} + Q_{ss} = 0. \quad (A7)$$

On the basis of the total surface site density ($\Gamma_s^0$) of the mineral surface, and of the associated equilibrium constants for surface reactions of mineral surface given by Leroy et al. (2008), we can determine the surface charge density and electrical potential at mineral surface, Stern layer and diffuse layer for different ionic concentrations based on Equations A1 to A7.
of two contributions: from the excess counterions contained in the outer sphere surface complexes of the Stern layer and from the diffuse layer. Therefore, the surface charge density $Q_{s0}$ (in C/m²) at the mineral surface, the surface charge density $Q_{sβ}$ (in C/m²) at the Stern layer, and the equivalent surface charge density $Q_{ss}$ (in C/m²) related to the excess of charge at the diffuse layer can be computed according to respectively:

$$Q_{s0} = -\frac{e_0 \Gamma_s^0}{\alpha_s} \left[ K_{s2} C_{Na^+}^f \exp\left(-\frac{e_0 \varphi_{s\beta}}{k_b T}\right) + 1 \right], \quad (A1)$$

$$Q_{s\beta} = \frac{e_0 \Gamma_s^0}{\alpha_s} K_{s2} C_{Na^+}^f \exp\left(-\frac{e_0 \varphi_{s\beta}}{k_b T}\right), \quad (A2)$$

$$Q_{ss} = -\sqrt{8\epsilon_f k_b T C_f} \sinh\left(\frac{e_0 \varphi_{sd}}{2k_b T}\right), \quad (A3)$$

with

$$\alpha_s = 1 + K_{s1} C_{H^+}^f \exp\left(-\frac{e_0 \varphi_{s0}}{k_b T}\right) + K_{s2} C_{Na^+}^f \exp\left(-\frac{e_0 \varphi_{s\beta}}{k_b T}\right), \quad (A4)$$

where $\Gamma_s^0$ is the total surface site density of the mineral grain (in sites nm⁻²), $K_{s1}$, $K_{s2}$ are the intrinsic equilibrium constants of different types of amphoteric reactions in the mineral-water interface (values are shown in Table S1 in Supporting Information S1). In this study, we do not take into account the influence of different mineral surfaces on the specific surface conductance, thus we take the same equilibrium constant at the mineral-water interface for any type of materials (red clay, quartz sand and clayey silt) to compute the electrochemical properties (surface charge density, zeta potential and specific surface conductance) of the mineral surface in order to decrease the number of parameters when modeling the electrokinetic properties. The symbols $C_{H^+}^f$, $C_{Na^+}^f$ and $C_f$ (in mol/L) represent the concentrations of the protons, the counterions ($Na^+$ in the free electrolyte) and the salinity in the free electrolyte, $C_{Na^+}^f = C_f$ and $C_{H^+}^f = 10^{-pH}$, $\epsilon_f$ is the pore water permittivity ($\epsilon_f = 81\epsilon_0$, $\epsilon_0 \sim 8.85 \times 10^{-12}$ F m⁻¹), $\epsilon_0$ represents the elementary charge (taken positive, $\epsilon_0 = 1.6 \times 10^{-19}$ C), $k_b$ is the Boltzmann constant ($1.381 \times 10^{-23}$ J K⁻¹), $\varphi_{s0}$, $\varphi_{s\beta}$, and $\varphi_{sd}$ (in V) are the electrical potential at the mineral surface (0-plane), at the $\beta$- and $d$-planes, respectively. The electrical potentials are related by the series capacitor at the 0-, $\beta$- and $d$-planes. This yields

$$\varphi_{s0} - \varphi_{s\beta} = \frac{Q_{s0}}{C_{s1}}, \quad (A5)$$

and

$$\varphi_{s\beta} - \varphi_{sd} = -\frac{Q_{ss}}{C_{s2}}, \quad (A6)$$

where $C_{s1}$ and $C_{s2}$ are the (constant) integral capacities of the inner and outer parts of the Stern layer in the mineral-water interface, respectively, and the values of these capacities are listed in Table S1 in Supporting Information S1 (Leroy et al., 2008).

The global electroneutrality equation for EDL in the mineral grain and water interface is

$$Q_{s0} + Q_{s\beta} + Q_{ss} = 0. \quad (A7)$$

On the basis of the total surface site density ($\Gamma_s^0$) of the mineral surface, and of the associated equilibrium constants for surface reactions of mineral surface given by Leroy et al. (2008), we can determine the surface charge density and electrical potential at mineral surface, Stern layer and diffuse layer for different ionic concentrations based on Equations A1 to A7.
Since the major contribution of the specific surface conductance at the mineral-water interface is in the diffuse layer of mobile ions and in the Stern layer of bound counterions, the specific surface conductance $\Sigma_{sw}$ (in S) is then computed using the following equation, relating to the surface charge density and to the temperature (Revil, 2012):

$$ \Sigma_{sw} \approx (\beta_{0\_Na}^{\text{ref}} Q_{s\beta} + \beta_{s\_Na}^{\text{ref}} Q_{ss}) [1 + \alpha_T (T - T_{\text{ref}})], \quad (A8) $$

where $\beta_{0\_Na}^{\text{ref}}$ and $\beta_{s\_Na}^{\text{ref}}$ ($= \beta_{Na}^{\text{ref}}$) are the effective reference mobilities of the counterions in the Stern layer and in the diffuse layer at the reference temperature ($T_{\text{ref}} = 298$ K), respectively (in $m^2 s^{-1} V^{-1}$, values are shown in Table S1 in Supporting Information S1) (S. Li et al., 2016).

## Appendix B: Electrical Double-Layer Model of Bulk Ice-Water Interface

Water molecules are organized in a tetrahedral lattice where hydrogen bonds provide the majority of the lattice energy in hexagonal ice Ih (hexagonal ice crystal). The arrangement of atoms of the ice lattice takes two different forms: one where the two hydrogen atoms bond covalently every oxygen atom and the other where one hydrogen atom is contained in every O-O vertex. Daigle (2021) and Kallay et al. (2003) considered the development of the surface chemical reactions for two types of surface sites (1 pK or 2 pK), and the distribution of charge within the electrical double-layer in the bulk ice-water interface in a manner analogous to existing modes for metal oxides in aqueous solutions.

In this study, we consider these two types of surface sites that exhibit vastly different behavior (very different pK) and compute their electrochemical properties in EDL of the interface between the bulk ice and liquid water. We consider that the electrolyte surrounding the ice is NaCl, and we ignore the adsorption and transport of chloride ions at bulk ice-water interface during the surface reaction, as well as that pH is in the range of 6–8. Due to the counterions in the Stern layer are prevented from penetrating into the bulk ice surface at the bulk ice-water interface, the Stern layer is assumed to be immediately adjacent to the bulk ice surface. We also ignore the influence of the second protonation and the corresponding equilibrium constant for the bulk ice-water interface. We consider four types of reactions: deprotonation of surface hydroxyls, ion association reaction on deprotonated sites (e.g., the adsorption of sodium cations), sorption of the protons, and adsorption of sodium cations on dangling O atoms. Therefore, the surface charge density $Q_{i0}$ at the surface of the bulk ice (0-plane) as well as the equivalent surface charge density $Q_{is}$ at the diffuse layer can be determined according to respectively

$$ Q_{i0} = - \frac{e_0 \Gamma_i^0}{\alpha_i} \left[ \left( K_{i1} + \frac{1}{K_{i3}} \right) \frac{1}{C_{H^+}^f} \exp \left( \frac{e_0 \varphi_{i0}}{k_b T} \right) \right], \quad (B1) $$

$$ Q_{is} = -\sqrt{8\epsilon_f k_b T C_f} \sinh\left(\frac{e_0 \varphi_{id}}{2k_b T}\right), \quad (B2) $$

with

$$ \alpha_i = 1 + \left( K_{i1} + \frac{1}{K_{i3}} \right) \frac{1}{C_{H^+}^f} \exp\left( \frac{e_0 \varphi_{i0}}{k_b T} \right) + \frac{K_{i2}}{K_{i3}} \frac{C_{Na^+}^f}{C_{H^+}^f} \exp\left[ \frac{e_0(\varphi_{i0} - \varphi_{id})}{k_b T} \right], \quad (B3) $$

where $\Gamma_i^0$ (in sites $nm^{-2}$) represents the total surface site density of the bulk ice surface, $K_{i1}$, $K_{i2}$ and $K_{i3}$ are the intrinsic equilibrium constants of different types of amphoteric reactions in the bulk ice-water interface are listed in Table S1 in Supporting Information S1; $\varphi_{i0}$ and $\varphi_{id}$ (in V) are the electrical potential at the 0-plane (bulk ice surface) and at the d-plane (zeta potential), respectively. The electrical potentials are associated by the series capacitor at the 0-plane and d-plane. This yields

$$ \varphi_{i0} - \varphi_{id} = \frac{Q_{i0}}{C_i}, \quad (B4) $$
Since the major contribution of the specific surface conductance at the mineral-water interface is in the diffuse layer of mobile ions and in the Stern layer of bound counterions, the specific surface conductance $\Sigma_{sw}$ (in S) is then computed using the following equation, relating to the surface charge density and to the temperature (Revil, 2012):

$$ \Sigma_{sw} \approx (\beta_{0\_Na}^{\text{ref}} Q_{s\beta} + \beta_{s\_Na}^{\text{ref}} Q_{ss}) [1 + \alpha_T (T - T_{\text{ref}})], \quad (A8) $$

where $\beta_{0\_Na}^{\text{ref}}$ and $\beta_{s\_Na}^{\text{ref}}$ ($= \beta_{Na}^{\text{ref}}$) are the effective reference mobilities of the counterions in the Stern layer and in the diffuse layer at the reference temperature ($T_{\text{ref}} = 298$ K), respectively (in $m^2 s^{-1} V^{-1}$, values are shown in Table S1 in Supporting Information S1) (S. Li et al., 2016).

## Appendix B: Electrical Double-Layer Model of Bulk Ice-Water Interface

Water molecules are organized in a tetrahedral lattice where hydrogen bonds provide the majority of the lattice energy in hexagonal ice Ih (hexagonal ice crystal). The arrangement of atoms of the ice lattice takes two different forms: one where the two hydrogen atoms bond covalently every oxygen atom and the other where one hydrogen atom is contained in every O-O vertex. Daigle (2021) and Kallay et al. (2003) considered the development of the surface chemical reactions for two types of surface sites (1 pK or 2 pK), and the distribution of charge within the electrical double-layer in the bulk ice-water interface in a manner analogous to existing modes for metal oxides in aqueous solutions.

In this study, we consider these two types of surface sites that exhibit vastly different behavior (very different pK) and compute their electrochemical properties in EDL of the interface between the bulk ice and liquid water. We consider that the electrolyte surrounding the ice is NaCl, and we ignore the adsorption and transport of chloride ions at bulk ice-water interface during the surface reaction, as well as that pH is in the range of 6–8. Due to the counterions in the Stern layer are prevented from penetrating into the bulk ice surface at the bulk ice-water interface, the Stern layer is assumed to be immediately adjacent to the bulk ice surface. We also ignore the influence of the second protonation and the corresponding equilibrium constant for the bulk ice-water interface. We consider four types of reactions: deprotonation of surface hydroxyls, ion association reaction on deprotonated sites (e.g., the adsorption of sodium cations), sorption of the protons, and adsorption of sodium cations on dangling O atoms. Therefore, the surface charge density $Q_{i0}$ at the surface of the bulk ice (0-plane) as well as the equivalent surface charge density $Q_{is}$ at the diffuse layer can be determined according to respectively

$$ Q_{i0} = - \frac{e_0 \Gamma_i^0}{\alpha_i} \left[ \left( K_{i1} + \frac{1}{K_{i3}} \right) \frac{1}{C_{H^+}^f} \exp \left( \frac{e_0 \varphi_{i0}}{k_b T} \right) \right], \quad (B1) $$

$$ Q_{is} = -\sqrt{8\epsilon_f k_b T C_f} \sinh\left(\frac{e_0 \varphi_{id}}{2k_b T}\right), \quad (B2) $$

with

$$ \alpha_i = 1 + \left( K_{i1} + \frac{1}{K_{i3}} \right) \frac{1}{C_{H^+}^f} \exp \left( \frac{e_0 \varphi_{i0}}{k_b T} \right) + \frac{K_{i2}}{K_{i3}} \frac{C_{Na^+}^f}{C_{H^+}^f} \exp \left[ \frac{e_0 (\varphi_{i0} - \varphi_{id})}{k_b T} \right], \quad (B3) $$

where $\Gamma_i^0$ (in sites $nm^{-2}$) represents the total surface site density of the bulk ice surface, $K_{i1}$, $K_{i2}$ and $K_{i3}$ are the intrinsic equilibrium constants of different types of amphoteric reactions in the bulk ice-water interface are listed in Table S1 in Supporting Information S1; $\varphi_{i0}$ and $\varphi_{id}$ (in V) are the electrical potential at the 0-plane (bulk ice surface) and at the d-plane (zeta potential), respectively. The electrical potentials are associated by the series capacitor at the 0-plane and d-plane. This yields

$$ \varphi_{i0} - \varphi_{id} = \frac{Q_{i0}}{C_i}, \quad (B4) $$
where $C_i$ is the (constant) integral capacity of the Stern layer in the interface between bulk ice and liquid water film, and $C_i = 1 \text{ F m}^{-2}$ from Daigle (2021).

The electroneutrality condition for EDL in the bulk ice-water interface is given by

$$Q_{i0} + Q_{is} = 0. \qquad (B5)$$

With the total surface site density ($\Gamma_i^0$) of bulk ice surface, and the associated equilibrium constants for surface reactions of bulk ice-water interface (Daigle, 2021; Kallay et al., 2003), we can determine the surface charge density and the electrical potential at the bulk ice surface and the diffuse layer for different salinities based on Equations B1 to B5.

We consider that the mobile ions in the diffuse layer affect the specific surface conductance in the electrical double-layer at the bulk ice-water interface. Therefore, we can determine the specific surface conductance $\Sigma_{iw}$ (in S) as a function of surface charge density (Revil, 2012):

$$\Sigma_{iw} \approx \beta_{s\_Na}^{\text{ref}} Q_{is} [1 + \alpha_{T} (T - T_{\text{ref}})]. \qquad (B6)$$

## Appendix C: Electrical Double-Layer Model of Air-Water Interface

Leroy et al. (2012) developed a unified surface complexation model that describes the electrochemical properties of the gas/water interface at a low ionic concentration, which is similar to the traditional “one site/two pK” model (Davis et al., 1978). We hence consider the air-water film interface in contact with a 1:1 aqueous electrolyte such as NaCl and the pH of fluid is assumed to equal 6–8. Since the available experimental measurements of the electrical conductivity are of neutral pH (6–8), we only consider the chemical reaction linked with the surface adsorption of protons and chloride (because the chloride anion is closer to the surface than the sodium cation at high ionic strengths). According to the surface complexation model of the air-water film interface derived by Leroy et al. (2012), the surface charge density $Q_{a0}$ at the surface of air (the sum of the surface site densities of charged surface groups), and the equivalent surface charge density $Q_{as}$ at the diffuse layer can be determined as follows respectively

$$Q_{a0} = -\frac{e_0 \Gamma_a^0}{\alpha_a} [K_{a1} K_{a2} C_{H^+}^f C_{Cl^-}^f + 1], \qquad (C1)$$

$$Q_{as} = -\sqrt{8\varepsilon_f k_b T C_f} \sinh\left(\frac{e_0 \varphi_{a0}}{2k_b T}\right), \qquad (C2)$$

with

$$\alpha_a = 1 + K_{a1} C_{H^+}^f \exp\left(-\frac{e_0 \varphi_{a0}}{k_b T}\right) + K_{a1} K_{a2} C_{H^+}^f C_{Cl^-}^f, \qquad (C3)$$

where $\Gamma_a^0$ (in sites $\text{nm}^{-2}$) is the total surface site density of the air-water interface, $K_{a1}$, and $K_{a2}$ are the associated equilibrium constants for the different reactions of the air-water interface, their values are shown in Table S1 in Supporting Information S1; the symbol $C_{Cl^-}^f$ (in mol/L) represents the salinity of the chloride electrolyte in the free electrolyte, $C_{Cl^-}^f = C_f \varphi_{a0}$ and $\varphi_{ad}$ are, respectively, the electrical potential at the 0-plane and at the d-plane (in V). The electrical potentials are given by

$$\varphi_{a0} - \varphi_{ad} = \frac{Q_{a0}}{C_a}, \qquad (C4)$$

where $C_a$ is the (constant) integral capacity of the Stern layer in the air-water interface, and $C_a = 0.0258 \text{ F m}^{-2}$ obtained by Leroy et al. (2012).
where $C_i$ is the (constant) integral capacity of the Stern layer in the interface between bulk ice and liquid water film, and $C_i = 1 \text{ F m}^{-2}$ from Daigle (2021).

The electroneutrality condition for EDL in the bulk ice-water interface is given by

$$Q_{i0} + Q_{is} = 0. \qquad (B5)$$

With the total surface site density ($\Gamma_i^0$) of bulk ice surface, and the associated equilibrium constants for surface reactions of bulk ice-water interface (Daigle, 2021; Kallay et al., 2003), we can determine the surface charge density and the electrical potential at the bulk ice surface and the diffuse layer for different salinities based on Equations B1 to B5.

We consider that the mobile ions in the diffuse layer affect the specific surface conductance in the electrical double-layer at the bulk ice-water interface. Therefore, we can determine the specific surface conductance $\Sigma_{iw}$ (in S) as a function of surface charge density (Revil, 2012):

$$\Sigma_{iw} \approx \beta_{s\_Na}^{\text{ref}} Q_{is} [1 + \alpha_{T} (T - T_{\text{ref}})]. \qquad (B6)$$

## Appendix C: Electrical Double-Layer Model of Air-Water Interface

Leroy et al. (2012) developed a unified surface complexation model that describes the electrochemical properties of the gas/water interface at a low ionic concentration, which is similar to the traditional “one site/two pK” model (Davis et al., 1978). We hence consider the air-water film interface in contact with a 1:1 aqueous electrolyte such as NaCl and the pH of fluid is assumed to equal 6–8. Since the available experimental measurements of the electrical conductivity are of neutral pH (6–8), we only consider the chemical reaction linked with the surface adsorption of protons and chloride (because the chloride anion is closer to the surface than the sodium cation at high ionic strengths). According to the surface complexation model of the air-water film interface derived by Leroy et al. (2012), the surface charge density $Q_{a0}$ at the surface of air (the sum of the surface site densities of charged surface groups), and the equivalent surface charge density $Q_{as}$ at the diffuse layer can be determined as follows respectively

$$Q_{a0} = -\frac{e_0 \Gamma_a^0}{\alpha_a} [K_{a1} K_{a2} C_{H^+}^f C_{Cl^-}^f + 1], \qquad (C1)$$

$$Q_{as} = -\sqrt{8\epsilon_f k_b T C_f} \sinh\left(\frac{e_0 \varphi_{ad}}{2k_b T}\right), \qquad (C2)$$

with

$$\alpha_a = 1 + K_{a1} C_{H^+}^f \exp\left(-\frac{e_0 \varphi_{a0}}{k_b T}\right) + K_{a1} K_{a2} C_{H^+}^f C_{Cl^-}^f, \qquad (C3)$$

where $\Gamma_a^0$ (in sites $\text{nm}^{-2}$) is the total surface site density of the air-water interface, $K_{a1}$, and $K_{a2}$ are the associated equilibrium constants for the different reactions of the air-water interface, their values are shown in Table S1 in Supporting Information S1; the symbol $C_{Cl^-}^f$ (in mol/L) represents the salinity of the chloride electrolyte in the free electrolyte, $C_{Cl^-}^f = C_f \varphi_{a0}$ and $\varphi_{ad}$ are, respectively, the electrical potential at the 0-plane and at the d-plane (in V). The electrical potentials are given by

$$\varphi_{a0} - \varphi_{ad} = \frac{Q_{a0}}{C_a}, \qquad (C4)$$

where $C_a$ is the (constant) integral capacity of the Stern layer in the air-water interface, and $C_a = 0.0258 \text{ F m}^{-2}$ obtained by Leroy et al. (2012).
The global electroneutrality condition for the electrical double-layer at the air-water interface implies the following:

$$Q_{a0} + Q_{as} = 0. \qquad (C5)$$

On the basis of the total surface site density ($\Gamma_a^0$) of air surface, the associated equilibrium constants for surface reactions of air-water interface are given by Leroy et al. (2012). Then we can determine the surface charge density and electrical potential at the bulk ice surface and the diffuse layer for different concentrations based on Equations C1 to C5.

Leroy et al. (2012) developed a specific surface conductance model including the contribution of proton electro-migration at the 0-plane in the Stern layer (the effect of the sorption of the protons) and the contribution of the counterions electro-migration in the diffuse layer. Considering a relatively neutral pH environment, thus we only consider the influence of the counterions electro-migration in the diffuse layer on the surface conductivity at the air-water interface. Therefore, we can determine the relationship between specific surface conductance $\Sigma_{aw}$ (in S) and surface charge density in the diffuse layer:

$$\Sigma_{aw} \approx \beta_{s\_Na}^{\text{ref}} Q_{as} [1 + \alpha_{T}(T - T_{\text{ref}})]. \qquad (C6)$$

## Appendix D: Apparent Formation Factor

The formation factor ($F$) is a strictly geometrical parameter that can be used to describe the geometry of the pore space (e.g., Archie, 1942; Coperey, Revil, Abdulsamad, et al., 2019; Revil, Coperey, et al., 2018; Revil et al., 2014; Revil, Soueid Ahmed, & Matthai, 2018). In absence of surface conductivity and when the pore space is occupied only by liquid water (i.e., no ice, no air), the formation factor can be defined as the ratio of the actual pore water conductivity to the electrical conductivity of the porous medium. From Equation 20, we can obtain the formation factor as a function of the porosity:

$$F = \frac{\left(\tau_g^{\text{eff}}\right)^2}{\phi} \frac{(D_e - D_f + 1)}{(3 - D_e - D_f)} \frac{[1 - \alpha^{3-D_e-D_f}]}{\{1 - \alpha^{D_e-D_f+1}\}}. \qquad (D1)$$

In the following, we also define an apparent formation factor ($F_a$) that depends on the water saturation and the unfrozen water content. The use of the apparent formation factor is frequent in the literature to highlight the effect of the water saturation (e.g., Maineult et al., 2018) or unfrozen water content (e.g., Coperey, Revil, Abdulsamad, et al., 2019). Following our model description, the apparent formation factor under partial water saturation above the freezing temperature is obtained as

$$F_a(S_{w,0}) = \frac{\left(\tau_g^{\text{eff}}\right)^2}{\phi} \frac{(D_e - D_f + 1)}{(3 - D_e - D_f)} \frac{[1 - \alpha^{3-D_e-D_f}]}{\left\{[\alpha^{3-D_e-D_f} + S_{w,0}(1 - \alpha^{3-D_e-D_f})]^{\frac{D_e-D_f+1}{3-D_e-D_f}} - \alpha^{D_e-D_f+1}\right\}}. \qquad (D2)$$

And the apparent formation factor as a function of the unfrozen water saturation at freezing temperature is given by

$$F_a(S_u) = \frac{\left(\tau_g^{\text{eff}}\right)^2}{\phi} \frac{(D_e - D_f + 1)}{(3 - D_e - D_f)} \frac{[1 - \alpha^{3-D_e-D_f}]}{\left\{[\alpha^{3-D_e-D_f} + S_u(1 - \alpha^{3-D_e-D_f})]^{\frac{D_e-D_f+1}{3-D_e-D_f}} - \alpha^{D_e-D_f+1}\right\}}. \qquad (D3)$$

Equation D1 illustrates that the formation factor above the freezing temperature is linked with the geometrical tortuosity ($\tau_g^{\text{eff}}$) and the microstructural parameters of the REV ($D_e$, $D_f$, $\phi$, $\alpha$), but is independent of the water saturation, and it is inversely proportional to the porosity in a similar form as reported in Thanh et al. (2019, 2020).
The global electroneutrality condition for the electrical double-layer at the air-water interface implies the following:

$$Q_{a0} + Q_{as} = 0. \qquad (C5)$$

On the basis of the total surface site density ($\Gamma_a^0$) of air surface, the associated equilibrium constants for surface reactions of air-water interface are given by Leroy et al. (2012). Then we can determine the surface charge density and electrical potential at the bulk ice surface and the diffuse layer for different concentrations based on Equations C1 to C5.

Leroy et al. (2012) developed a specific surface conductance model including the contribution of proton electro-migration at the 0-plane in the Stern layer (the effect of the sorption of the protons) and the contribution of the counterions electro-migration in the diffuse layer. Considering a relatively neutral pH environment, thus we only consider the influence of the counterions electro-migration in the diffuse layer on the surface conductivity at the air-water interface. Therefore, we can determine the relationship between specific surface conductance $\Sigma_{aw}$ (in S) and surface charge density in the diffuse layer:

$$\Sigma_{aw} \approx \beta_{s\_Na}^{\text{ref}} Q_{as} [1 + \alpha_T (T - T_{\text{ref}})]. \qquad (C6)$$

## Appendix D: Apparent Formation Factor

The formation factor ($F$) is a strictly geometrical parameter that can be used to describe the geometry of the pore space (e.g., Archie, 1942; Coperey, Revil, Abdulsamad, et al., 2019; Revil, Coperey, et al., 2018; Revil et al., 2014; Revil, Soueid Ahmed, & Matthai, 2018). In absence of surface conductivity and when the pore space is occupied only by liquid water (i.e., no ice, no air), the formation factor can be defined as the ratio of the actual pore water conductivity to the electrical conductivity of the porous medium. From Equation 20, we can obtain the formation factor as a function of the porosity:

$$F = \frac{\left(\tau_g^{\text{eff}}\right)^2}{\phi} \frac{(D_e - D_f + 1) \left[1 - \alpha^{3-D_e-D_f}\right]}{(3 - D_e - D_f) \left\{1 - \alpha^{D_e-D_f+1}\right\}} \qquad (D1)$$

In the following, we also define an apparent formation factor ($F_a$) that depends on the water saturation and the unfrozen water content. The use of the apparent formation factor is frequent in the literature to highlight the effect of the water saturation (e.g., Maineult et al., 2018) or unfrozen water content (e.g., Coperey, Revil, Abdulsamad, et al., 2019). Following our model description, the apparent formation factor under partial water saturation above the freezing temperature is obtained as

$$F_a(S_{w,0}) = \frac{\left(\tau_g^{\text{eff}}\right)^2}{\phi} \frac{(D_e - D_f + 1)}{(3 - D_e - D_f)} \frac{\left[1 - \alpha^{3-D_e-D_f}\right]}{\left\{\left[\alpha^{3-D_e-D_f} + S_{w,0}(1 - \alpha^{3-D_e-D_f})\right]^{\frac{D_e-D_f+1}{3-D_e-D_f}} - \alpha^{D_e-D_f+1}\right\}} \qquad (D2)$$

And the apparent formation factor as a function of the unfrozen water saturation at freezing temperature is given by

$$F_a(S_u) = \frac{\left(\tau_g^{\text{eff}}\right)^2}{\phi} \frac{(D_e - D_f + 1)}{(3 - D_e - D_f)} \frac{\left[1 - \alpha^{3-D_e-D_f}\right]}{\left\{\left[\alpha^{3-D_e-D_f} + S_u(1 - \alpha^{3-D_e-D_f})\right]^{\frac{D_e-D_f+1}{3-D_e-D_f}} - \alpha^{D_e-D_f+1}\right\}} \qquad (D3)$$

Equation D1 illustrates that the formation factor above the freezing temperature is linked with the geometrical tortuosity ($\tau_g^{\text{eff}}$) and the microstructural parameters of the REV ($D_e$, $D_f$, $\phi$, $\alpha$), but is independent of the water saturation, and it is inversely proportional to the porosity in a similar form as reported in Thanh et al. (2019, 2020).
However, the apparent formation factor is related not only to the microstructural parameters of the porous medium, but also to the water saturation ($S_{w,0}$) or to the unfrozen water saturation ($S_u$) from Equations D2 and D3.

Notations

<table><tr><td>T</td><td>Temperature of the REV (K)</td></tr><tr><td>T<sub>0</sub></td><td>Freezing temperature of bulk water (K)</td></tr><tr><td>T<sub>m</sub></td><td>Freezing temperature of the porous medium (K)</td></tr><tr><td>T<sub>ref</sub></td><td>Reference temperature (K)</td></tr><tr><td>T<sub>s</sub></td><td>Liquid water surface tension (N m<sup>-1</sup>)</td></tr><tr><td>r</td><td>Radius of the capillary tube (m)</td></tr><tr><td>r<sub>1</sub></td><td>Radius of the capillary tube filled with liquid water (m)</td></tr><tr><td>r<sub>2</sub></td><td>Radius of the capillary tube occupied by bulk ice (m)</td></tr><tr><td>r<sub>3</sub></td><td>Radius of the capillary tube occupied by air (m)</td></tr><tr><td>r<sub>i</sub></td><td>Critical freezing radius (either filled with liquid water or bulk ice) of the REV (m)</td></tr><tr><td>r<sub>h</sub></td><td>Critical freezing radius (either occupied by bulk ice or air) of the REV (m)</td></tr><tr><td>r<sub>max</sub></td><td>Maximum radius of the capillary tube (m)</td></tr><tr><td>r<sub>min</sub></td><td>Minimum radius of the capillary tube (m)</td></tr><tr><td>g</td><td>Acceleration of gravity (m s<sup>-2</sup>)</td></tr><tr><td>α</td><td>Ratio of the minimum pore radius to the maximum pore radius (–)</td></tr><tr><td>β</td><td>Air-water-solid contact angle (–)</td></tr><tr><td>σ<sub>sl</sub></td><td>Ice-water interfacial free energy (J m<sup>-2</sup>)</td></tr><tr><td>σ<sub>w</sub></td><td>Electrical conductivity of pore water (S/m)</td></tr><tr><td>σ</td><td>Electrical conductivity of the porous medium (S/m)</td></tr><tr><td>L</td><td>Length of the sample (m)</td></tr><tr><td>L<sub>f</sub></td><td>Latent heat of phase transformation (J/kg)</td></tr><tr><td>L<sub>e</sub></td><td>Length of the capillary (tortuous length) (m)</td></tr><tr><td>L<sub>0</sub></td><td>REV length (m)</td></tr><tr><td>ρ<sub>i</sub></td><td>Density of ice (kg m<sup>-3</sup>)</td></tr><tr><td>ρ<sub>w</sub></td><td>Liquid water volume density (kg m<sup>-3</sup>)</td></tr><tr><td>ρ<sub>g</sub></td><td>Grain density (kg m<sup>-3</sup>)</td></tr><tr><td>R</td><td>Resistance value of experiments (Ω)</td></tr><tr><td>R(r)</td><td>Total resistance of the three types of capillary (Ω)</td></tr><tr><td>R<sub>1</sub></td><td>Resistance of the capillary filled with liquid water (r<sub>min</sub> &lt; r &lt; r<sub>i</sub>) (Ω)</td></tr><tr><td>R<sub>2</sub></td><td>Resistance of the capillary occupied by bulk ice (r<sub>i</sub> &lt; r &lt; r<sub>h</sub>) (Ω)</td></tr><tr><td>R<sub>3</sub></td><td>Resistance of the capillary occupied by air (r<sub>h</sub> &lt; r &lt; r<sub>max</sub>) (Ω)</td></tr><tr><td>R<sub>0</sub></td><td>Resistance of the porous media at REV scale (Ω)</td></tr><tr><td>Σ<sub>aw</sub></td><td>Specific surface conductance at the air-water surface (S)</td></tr></table>
However, the apparent formation factor is related not only to the microstructural parameters of the porous medium, but also to the water saturation ($S_{w,0}$) or to the unfrozen water saturation ($S_u$) from Equations D2 and D3.

Notations

<table><tr><td>T</td><td>Temperature of the REV (K)</td></tr><tr><td>T<sub>0</sub></td><td>Freezing temperature of bulk water (K)</td></tr><tr><td>T<sub>m</sub></td><td>Freezing temperature of the porous medium (K)</td></tr><tr><td>T<sub>ref</sub></td><td>Reference temperature (K)</td></tr><tr><td>T<sub>s</sub></td><td>Liquid water surface tension (N m<sup>-1</sup>)</td></tr><tr><td>r</td><td>Radius of the capillary tube (m)</td></tr><tr><td>r<sub>1</sub></td><td>Radius of the capillary tube filled with liquid water (m)</td></tr><tr><td>r<sub>2</sub></td><td>Radius of the capillary tube occupied by bulk ice (m)</td></tr><tr><td>r<sub>3</sub></td><td>Radius of the capillary tube occupied by air (m)</td></tr><tr><td>r<sub>i</sub></td><td>Critical freezing radius (either filled with liquid water or bulk ice) of the REV (m)</td></tr><tr><td>r<sub>h</sub></td><td>Critical freezing radius (either occupied by bulk ice or air) of the REV (m)</td></tr><tr><td>r<sub>max</sub></td><td>Maximum radius of the capillary tube (m)</td></tr><tr><td>r<sub>min</sub></td><td>Minimum radius of the capillary tube (m)</td></tr><tr><td>g</td><td>Acceleration of gravity (m s<sup>-2</sup>)</td></tr><tr><td>α</td><td>Ratio of the minimum pore radius to the maximum pore radius (–)</td></tr><tr><td>β</td><td>Air-water-solid contact angle (–)</td></tr><tr><td>σ<sub>sl</sub></td><td>Ice-water interfacial free energy (J m<sup>-2</sup>)</td></tr><tr><td>σ<sub>w</sub></td><td>Electrical conductivity of pore water (S/m)</td></tr><tr><td>σ</td><td>Electrical conductivity of the porous medium (S/m)</td></tr><tr><td>L</td><td>Length of the sample (m)</td></tr><tr><td>L<sub>f</sub></td><td>Latent heat of phase transformation (J/kg)</td></tr><tr><td>L<sub>e</sub></td><td>Length of the capillary (tortuous length) (m)</td></tr><tr><td>L<sub>0</sub></td><td>REV length (m)</td></tr><tr><td>ρ<sub>i</sub></td><td>Density of ice (kg m<sup>-3</sup>)</td></tr><tr><td>ρ<sub>w</sub></td><td>Liquid water volume density (kg m<sup>-3</sup>)</td></tr><tr><td>ρ<sub>g</sub></td><td>Grain density (kg m<sup>-3</sup>)</td></tr><tr><td>R</td><td>Resistance value of experiments (Ω)</td></tr><tr><td>R(r)</td><td>Total resistance of the three types of capillary (Ω)</td></tr><tr><td>R<sub>1</sub></td><td>Resistance of the capillary filled with liquid water (r<sub>min</sub> &lt; r &lt; r<sub>i</sub>) (Ω)</td></tr><tr><td>R<sub>2</sub></td><td>Resistance of the capillary occupied by bulk ice (r<sub>i</sub> &lt; r &lt; r<sub>h</sub>) (Ω)</td></tr><tr><td>R<sub>3</sub></td><td>Resistance of the capillary occupied by air (r<sub>h</sub> &lt; r &lt; r<sub>max</sub>) (Ω)</td></tr><tr><td>R<sub>0</sub></td><td>Resistance of the porous media at REV scale (Ω)</td></tr><tr><td>Σ<sub>aw</sub></td><td>Specific surface conductance at the air-water surface (S)</td></tr></table>
<table>
  <tr>
    <td>Σ<sub>sw</sub></td>
    <td>Specific surface conductance at the mineral-water surface (S)</td>
  </tr>
  <tr>
    <td>Σ<sub>iw</sub></td>
    <td>Specific surface conductance at the bulk ice-water surface (S)</td>
  </tr>
  <tr>
    <td>Σ (r<sub>1</sub>, r<sub>2</sub>, r<sub>3</sub>)</td>
    <td>Total conductance of the three capillaries (they are occupied by water, ice and air) (S)</td>
  </tr>
  <tr>
    <td>A<sub>REV</sub></td>
    <td>Cross-section area of REV (m<sup>2</sup>)</td>
  </tr>
  <tr>
    <td>f(r)</td>
    <td>Capillary tube distribution function of the REV (–)</td>
  </tr>
  <tr>
    <td>F</td>
    <td>Formation factor of the porous medium (–)</td>
  </tr>
  <tr>
    <td>F<sub>a</sub> (S<sub>w,0</sub>)</td>
    <td>Apparent formation factor above the freezing temperature (–)</td>
  </tr>
  <tr>
    <td>F<sub>a</sub> (S<sub>u</sub>)</td>
    <td>Apparent formation factor below the freezing temperature (–)</td>
  </tr>
  <tr>
    <td>S</td>
    <td>Area of the sample (m<sup>2</sup>)</td>
  </tr>
  <tr>
    <td>n</td>
    <td>Unfrozen exponent (–)</td>
  </tr>
  <tr>
    <td>N(≥r)</td>
    <td>Number of the capillaries (with radius larger than r) (–)</td>
  </tr>
  <tr>
    <td>N<sub>A</sub></td>
    <td>Avogadro's number (mol<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>φ</td>
    <td>Porosity of the REV (–)</td>
  </tr>
  <tr>
    <td>τ<sup>eff</sup><sub>g</sub></td>
    <td>Effective geometrical tortuosity of the porous medium (–)</td>
  </tr>
  <tr>
    <td>S<sub>w,0</sub></td>
    <td>Initial water saturation of the REV (–)</td>
  </tr>
  <tr>
    <td>S<sub>u</sub></td>
    <td>Unfrozen water saturation of the REV (–)</td>
  </tr>
  <tr>
    <td>h<sub>m</sub></td>
    <td>Initial pressure head (m)</td>
  </tr>
  <tr>
    <td>D<sub>f</sub></td>
    <td>Fractal dimension for pore space (–)</td>
  </tr>
  <tr>
    <td>D<sub>e</sub></td>
    <td>Fractal dimension of capillary tortuosity (–)</td>
  </tr>
  <tr>
    <td>a</td>
    <td>Physical parameters of the SFCC (–)</td>
  </tr>
  <tr>
    <td>b</td>
    <td>Physical parameters of the SFCC (–)</td>
  </tr>
  <tr>
    <td>c</td>
    <td>Physical parameters of the SFCC (–)</td>
  </tr>
  <tr>
    <td>CEC</td>
    <td>Cation exchange capacity (C/kg)</td>
  </tr>
  <tr>
    <td>e<sub>0</sub></td>
    <td>Elementary charge (C)</td>
  </tr>
  <tr>
    <td>k<sub>b</sub></td>
    <td>Boltzmann's constant (J K<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>ε<sub>f</sub></td>
    <td>Permittivity of the pore water (F m<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>C<sub>0</sub></td>
    <td>Initial ionic concentration of the REV (mol/L)</td>
  </tr>
  <tr>
    <td>C<sub>u</sub></td>
    <td>Ionic concentration of the REV in frost state (mol/L)</td>
  </tr>
  <tr>
    <td>C<sub>f</sub></td>
    <td>Salinity in the free electrolyte (mol/L)</td>
  </tr>
  <tr>
    <td>C<sup>f</sup><sub>H+</sub></td>
    <td>Ionic concentration of the protons in the free electrolyte (mol/L)</td>
  </tr>
  <tr>
    <td>C<sup>f</sup><sub>Na+</sub></td>
    <td>Ionic concentration of sodium cations in the free electrolyte (mol/L)</td>
  </tr>
  <tr>
    <td>C<sup>f</sup><sub>i</sub></td>
    <td>Concentration of ionic species i in the free electrolyte (mol/L)</td>
  </tr>
  <tr>
    <td>C<sup>f</sup><sub>Cl<sup>-</sup></sub></td>
    <td>Concentration of chloride in the free electrolyte (mol/L)</td>
  </tr>
  <tr>
    <td>q<sub>i</sub></td>
    <td>Charge of species i in the free electrolyte (C)</td>
  </tr>
  <tr>
    <td>B(T)</td>
    <td>Apparent mobility of the counterions for surface conduction (m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>λ(T)</td>
    <td>Apparent mobility of the counterions for the polarization (m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</td>
  </tr>
</table>
<table>
  <tr>
    <td>Σ<sub>sw</sub></td>
    <td>Specific surface conductance at the mineral-water surface (S)</td>
  </tr>
  <tr>
    <td>Σ<sub>iw</sub></td>
    <td>Specific surface conductance at the bulk ice-water surface (S)</td>
  </tr>
  <tr>
    <td>Σ (r<sub>1</sub>, r<sub>2</sub>, r<sub>3</sub>)</td>
    <td>Total conductance of the three capillaries (they are occupied by water, ice and air) (S)</td>
  </tr>
  <tr>
    <td>A<sub>REV</sub></td>
    <td>Cross-section area of REV (m<sup>2</sup>)</td>
  </tr>
  <tr>
    <td>f(r)</td>
    <td>Capillary tube distribution function of the REV (–)</td>
  </tr>
  <tr>
    <td>F</td>
    <td>Formation factor of the porous medium (–)</td>
  </tr>
  <tr>
    <td>F<sub>a</sub> (S<sub>w,0</sub>)</td>
    <td>Apparent formation factor above the freezing temperature (–)</td>
  </tr>
  <tr>
    <td>F<sub>a</sub> (S<sub>u</sub>)</td>
    <td>Apparent formation factor below the freezing temperature (–)</td>
  </tr>
  <tr>
    <td>S</td>
    <td>Area of the sample (m<sup>2</sup>)</td>
  </tr>
  <tr>
    <td>n</td>
    <td>Unfrozen exponent (–)</td>
  </tr>
  <tr>
    <td>N(≥r)</td>
    <td>Number of the capillaries (with radius larger than r) (–)</td>
  </tr>
  <tr>
    <td>N<sub>A</sub></td>
    <td>Avogadro's number (mol<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>φ</td>
    <td>Porosity of the REV (–)</td>
  </tr>
  <tr>
    <td>τ<sup>eff</sup><sub>g</sub></td>
    <td>Effective geometrical tortuosity of the porous medium (–)</td>
  </tr>
  <tr>
    <td>S<sub>w,0</sub></td>
    <td>Initial water saturation of the REV (–)</td>
  </tr>
  <tr>
    <td>S<sub>u</sub></td>
    <td>Unfrozen water saturation of the REV (–)</td>
  </tr>
  <tr>
    <td>h<sub>m</sub></td>
    <td>Initial pressure head (m)</td>
  </tr>
  <tr>
    <td>D<sub>f</sub></td>
    <td>Fractal dimension for pore space (–)</td>
  </tr>
  <tr>
    <td>D<sub>e</sub></td>
    <td>Fractal dimension of capillary tortuosity (–)</td>
  </tr>
  <tr>
    <td>a</td>
    <td>Physical parameters of the SFCC (–)</td>
  </tr>
  <tr>
    <td>b</td>
    <td>Physical parameters of the SFCC (–)</td>
  </tr>
  <tr>
    <td>c</td>
    <td>Physical parameters of the SFCC (–)</td>
  </tr>
  <tr>
    <td>CEC</td>
    <td>Cation exchange capacity (C/kg)</td>
  </tr>
  <tr>
    <td>e<sub>0</sub></td>
    <td>Elementary charge (C)</td>
  </tr>
  <tr>
    <td>k<sub>b</sub></td>
    <td>Boltzmann's constant (J K<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>ε<sub>f</sub></td>
    <td>Permittivity of the pore water (F m<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>C<sub>0</sub></td>
    <td>Initial ionic concentration of the REV (mol/L)</td>
  </tr>
  <tr>
    <td>C<sub>u</sub></td>
    <td>Ionic concentration of the REV in frost state (mol/L)</td>
  </tr>
  <tr>
    <td>C<sub>f</sub></td>
    <td>Salinity in the free electrolyte (mol/L)</td>
  </tr>
  <tr>
    <td>C<sup>f</sup><sub>H+</sub></td>
    <td>Ionic concentration of the protons in the free electrolyte (mol/L)</td>
  </tr>
  <tr>
    <td>C<sup>f</sup><sub>Na+</sub></td>
    <td>Ionic concentration of sodium cations in the free electrolyte (mol/L)</td>
  </tr>
  <tr>
    <td>C<sup>f</sup><sub>i</sub></td>
    <td>Concentration of ionic species i in the free electrolyte (mol/L)</td>
  </tr>
  <tr>
    <td>C<sup>f</sup><sub>Cl<sup>-</sup></sub></td>
    <td>Concentration of chloride in the free electrolyte (mol/L)</td>
  </tr>
  <tr>
    <td>q<sub>i</sub></td>
    <td>Charge of species i in the free electrolyte (C)</td>
  </tr>
  <tr>
    <td>B(T)</td>
    <td>Apparent mobility of the counterions for surface conduction (m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>λ(T)</td>
    <td>Apparent mobility of the counterions for the polarization (m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</td>
  </tr>
</table>
<table>
  <tr>
    <td>β<sub>i</sub></td>
    <td>Mobility of ionic species i in the free electrolyte (m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>β<sup>ref</sup><sub>i</sub></td>
    <td>Reference mobility of ionic species i in the free electrolyte (m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>β<sup>ref</sup><sub>Na</sub></td>
    <td>Reference mobility of ionic species Na<sup>+</sup> in the free electrolyte (m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>β<sup>ref</sup><sub>0_Na</sub></td>
    <td>Effective reference mobility of counterions in the Stern layer (m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>β<sup>ref</sup><sub>s_Na</sub></td>
    <td>Effective reference mobility of counterions in the diffuse layer (m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>β<sup>ref</sup><sub>Cl</sub></td>
    <td>Reference mobility of ionic species Cl<sup>-</sup> in the free electrolyte (m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>β<sup>ref</sup><sub>H</sub></td>
    <td>Reference mobility of ionic species H<sup>+</sup> in the free electrolyte (m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>β<sup>ref</sup><sub>OH</sub></td>
    <td>Reference mobility of ionic species OH<sup>-</sup> in the free electrolyte (m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>α<sub>T</sub></td>
    <td>Temperature factor (K<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>Γ<sup>0</sup><sub>s</sub></td>
    <td>Total surface site density of the mineral surface (Sites nm<sup>-2</sup>)</td>
  </tr>
  <tr>
    <td>Γ<sup>0</sup><sub>i</sub></td>
    <td>Total surface site density of the ice surface (Sites nm<sup>-2</sup>)</td>
  </tr>
  <tr>
    <td>Γ<sup>0</sup><sub>a</sub></td>
    <td>Total surface site density of the air surface (Sites nm<sup>-2</sup>)</td>
  </tr>
  <tr>
    <td>K<sub>s1</sub></td>
    <td>Equilibrium constant of deprotonation at the mineral-water interface (-)</td>
  </tr>
  <tr>
    <td>K<sub>s2</sub></td>
    <td>Equilibrium constant of sodium adsorption at the mineral-water interface (-)</td>
  </tr>
  <tr>
    <td>K<sub>i1</sub></td>
    <td>Equilibrium constant for deprotonation at the ice-water interface (-)</td>
  </tr>
  <tr>
    <td>K<sub>i2</sub></td>
    <td>Equilibrium constant for sodium sorption at the ice-water interface (-)</td>
  </tr>
  <tr>
    <td>K<sub>i3</sub></td>
    <td>Equilibrium constant for H<sup>+</sup> sorption on dangling O at the ice-water interface (-)</td>
  </tr>
  <tr>
    <td>K<sub>a1</sub></td>
    <td>Equilibrium constant associated with surface adsorption of protons at (&gt;X<sup>-</sup>) sites at the air-water interface (-)</td>
  </tr>
  <tr>
    <td>K<sub>a2</sub></td>
    <td>Equilibrium constant associated with surface adsorption of chloride at (&gt;XH<sup>+</sup>) sites at the air-water interface (-)</td>
  </tr>
  <tr>
    <td>K<sub>g</sub></td>
    <td>Geometrical factor of the sample (m)</td>
  </tr>
  <tr>
    <td>C<sub>s1</sub></td>
    <td>Capacities of the inner parts of the Stern layer in the mineral-water interface (F m<sup>-2</sup>)</td>
  </tr>
  <tr>
    <td>C<sub>s2</sub></td>
    <td>Capacities of the outer parts of the Stern layer in the mineral-water interface (F m<sup>-2</sup>)</td>
  </tr>
  <tr>
    <td>C<sub>i</sub></td>
    <td>Capacity of the Stern layer at the bulk ice-water interface (F m<sup>-2</sup>)</td>
  </tr>
  <tr>
    <td>C<sub>a</sub></td>
    <td>Capacities of the Stern layer in the air-water interface (F m<sup>-2</sup>)</td>
  </tr>
  <tr>
    <td>Q<sub>s0</sub></td>
    <td>Surface charge density at the surface of the mineral (mineral-water interface) (C)</td>
  </tr>
  <tr>
    <td>Q<sub>sβ</sub></td>
    <td>Surface charge density at the Stern layer (mineral-water interface) (C)</td>
  </tr>
  <tr>
    <td>Q<sub>ss</sub></td>
    <td>Surface charge density at the diffuse layer (mineral-water interface) (C)</td>
  </tr>
  <tr>
    <td>φ<sub>s0</sub></td>
    <td>Electrical potential at the mineral surface (mineral-water interface- 0-plane) (V)</td>
  </tr>
  <tr>
    <td>φ<sub>sβ</sub></td>
    <td>Electrical potential at the β-plane (mineral-water interface) (V)</td>
  </tr>
  <tr>
    <td>φ<sub>sd</sub></td>
    <td>Electrical potential at the d-plane (mineral-water interface) (V)</td>
  </tr>
  <tr>
    <td>Q<sub>i0</sub></td>
    <td>Surface charge density at the surface of the bulk ice (bulk ice-water interface) (C)</td>
  </tr>
  <tr>
    <td>Q<sub>is</sub></td>
    <td>Surface charge density at the diffuse layer (bulk ice-water interface) (C)</td>
  </tr>
  <tr>
    <td>φ<sub>i0</sub></td>
    <td>Electrical potential at the mineral surface (bulk ice-water interface- 0-plane) (V)</td>
  </tr>
  <tr>
    <td>φ<sub>id</sub></td>
    <td>Electrical potential at the d-plane (bulk ice-water interface) (V)</td>
  </tr>
</table>
<table>
  <tr>
    <td>β<sub>i</sub></td>
    <td>Mobility of ionic species i in the free electrolyte (m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>β<sup>ref</sup><sub>i</sub></td>
    <td>Reference mobility of ionic species i in the free electrolyte (m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>β<sup>ref</sup><sub>Na</sub></td>
    <td>Reference mobility of ionic species Na<sup>+</sup> in the free electrolyte (m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>β<sup>ref</sup><sub>0_Na</sub></td>
    <td>Effective reference mobility of counterions in the Stern layer (m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>β<sup>ref</sup><sub>s_Na</sub></td>
    <td>Effective reference mobility of counterions in the diffuse layer (m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>β<sup>ref</sup><sub>Cl</sub></td>
    <td>Reference mobility of ionic species Cl<sup>-</sup> in the free electrolyte (m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>β<sup>ref</sup><sub>H</sub></td>
    <td>Reference mobility of ionic species H<sup>+</sup> in the free electrolyte (m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>β<sup>ref</sup><sub>OH</sub></td>
    <td>Reference mobility of ionic species OH<sup>-</sup> in the free electrolyte (m<sup>2</sup> s<sup>-1</sup> V<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>α<sub>T</sub></td>
    <td>Temperature factor (K<sup>-1</sup>)</td>
  </tr>
  <tr>
    <td>Γ<sup>0</sup><sub>s</sub></td>
    <td>Total surface site density of the mineral surface (Sites nm<sup>-2</sup>)</td>
  </tr>
  <tr>
    <td>Γ<sup>0</sup><sub>i</sub></td>
    <td>Total surface site density of the ice surface (Sites nm<sup>-2</sup>)</td>
  </tr>
  <tr>
    <td>Γ<sup>0</sup><sub>a</sub></td>
    <td>Total surface site density of the air surface (Sites nm<sup>-2</sup>)</td>
  </tr>
  <tr>
    <td>K<sub>s1</sub></td>
    <td>Equilibrium constant of deprotonation at the mineral-water interface (–)</td>
  </tr>
  <tr>
    <td>K<sub>s2</sub></td>
    <td>Equilibrium constant of sodium adsorption at the mineral-water interface (–)</td>
  </tr>
  <tr>
    <td>K<sub>i1</sub></td>
    <td>Equilibrium constant for deprotonation at the ice-water interface (–)</td>
  </tr>
  <tr>
    <td>K<sub>i2</sub></td>
    <td>Equilibrium constant for sodium sorption at the ice-water interface (–)</td>
  </tr>
  <tr>
    <td>K<sub>i3</sub></td>
    <td>Equilibrium constant for H<sup>+</sup> sorption on dangling O at the ice-water interface (–)</td>
  </tr>
  <tr>
    <td>K<sub>a1</sub></td>
    <td>Equilibrium constant associated with surface adsorption of protons at (&gt;X<sup>-</sup>) sites at the air-water interface (–)</td>
  </tr>
  <tr>
    <td>K<sub>a2</sub></td>
    <td>Equilibrium constant associated with surface adsorption of chloride at (&gt;XH<sup>+</sup>) sites at the air-water interface (–)</td>
  </tr>
  <tr>
    <td>K<sub>g</sub></td>
    <td>Geometrical factor of the sample (m)</td>
  </tr>
  <tr>
    <td>C<sub>s1</sub></td>
    <td>Capacities of the inner parts of the Stern layer in the mineral-water interface (F m<sup>-2</sup>)</td>
  </tr>
  <tr>
    <td>C<sub>s2</sub></td>
    <td>Capacities of the outer parts of the Stern layer in the mineral-water interface (F m<sup>-2</sup>)</td>
  </tr>
  <tr>
    <td>C<sub>i</sub></td>
    <td>Capacity of the Stern layer at the bulk ice-water interface (F m<sup>-2</sup>)</td>
  </tr>
  <tr>
    <td>C<sub>a</sub></td>
    <td>Capacities of the Stern layer in the air-water interface (F m<sup>-2</sup>)</td>
  </tr>
  <tr>
    <td>Q<sub>s0</sub></td>
    <td>Surface charge density at the surface of the mineral (mineral-water interface) (C)</td>
  </tr>
  <tr>
    <td>Q<sub>sβ</sub></td>
    <td>Surface charge density at the Stern layer (mineral-water interface) (C)</td>
  </tr>
  <tr>
    <td>Q<sub>ss</sub></td>
    <td>Surface charge density at the diffuse layer (mineral-water interface) (C)</td>
  </tr>
  <tr>
    <td>φ<sub>s0</sub></td>
    <td>Electrical potential at the mineral surface (mineral-water interface- 0-plane) (V)</td>
  </tr>
  <tr>
    <td>φ<sub>sβ</sub></td>
    <td>Electrical potential at the β-plane (mineral-water interface) (V)</td>
  </tr>
  <tr>
    <td>φ<sub>sd</sub></td>
    <td>Electrical potential at the d-plane (mineral-water interface) (V)</td>
  </tr>
  <tr>
    <td>Q<sub>i0</sub></td>
    <td>Surface charge density at the surface of the bulk ice (bulk ice-water interface) (C)</td>
  </tr>
  <tr>
    <td>Q<sub>is</sub></td>
    <td>Surface charge density at the diffuse layer (bulk ice-water interface) (C)</td>
  </tr>
  <tr>
    <td>φ<sub>i0</sub></td>
    <td>Electrical potential at the mineral surface (bulk ice-water interface- 0-plane) (V)</td>
  </tr>
  <tr>
    <td>φ<sub>id</sub></td>
    <td>Electrical potential at the d-plane (bulk ice-water interface) (V)</td>
  </tr>
</table>
$Q_{a0}$ Surface charge density at the surface of the air (air-water interface) (C)

$Q_{as}$ Surface charge density at the diffuse layer (air-water interface) (C)

$\varphi_{a0}$ Electrical potential at the 0-plane (air-water interface) (V)

$\varphi_{ad}$ Electrical potential at the d-plane (air-water interface) (V)

## Data Availability Statement

The present study is supplemented by Supporting Information. The experimental data of frozen porous media for evaluating the evolution of electrical conductivity as a function of temperature are openly accessible in the Hydrogeophysics community via Zenodo under different initial water saturations and salinities: https://doi.org/ 10.5281/zenodo.7688602. Additionally, Matlab codes for calculating the predicted results of electrical conduc- tivity are also available through this link.

## Acknowledgments

The authors strongly thank the Editor, the Associate Editor, and the two reviewers (Philippe Leroy and an anonymous one) for kind and constructive comments. Support for this work was provided by China Scholarship Council (CSC) project (Grant 202106370053). This research was also supported by the National Natural Science Foundation of China (No. 52178376), and the Natural Science Foundation of Hunan Province (2022JJ10076). Furthermore, we would like to express our gratitude to the School of civil engineering (CSU) for granting us access to the laboratory experimental apparatus for electrical resistance experiments (LCR setup) and for SFCC experiments (NMR method).

## References

Amankwah, S. K., Ireson, A. M., Maulé, C., Brannen, R., & Mathias, A. (2021). A model for the soil freezing characteristic curve that represents the dominant role of salt exclusion. *Water Resources Research*, 57(8), e2021WR030070. https://doi.org/10.1029/2021WR030070

Andersland, O. B., & Ladanyi, B. (2004). *Frozen ground engineering* (2nd). Wiley.

Anderson, D. M. (1967). The interface between ice and silicate surfaces. *Journal of Colloid and Interface Science*, 25(2), 174–191. https://doi.org/10.1016/0021-9797(67)90021-5

Archie, G. E. (1942). The electrical resistivity log as an aid in determining some reservoir characteristics. *Transactions of the AIME*, 146(01), 54–62. https://doi.org/10.2118/942054-G

Bai, R., Lai, Y., Zhang, M., & Gao, J. (2018). Water-vapor-heat behavior in a freezing unsaturated coarse-grained soil with a closed top. *Cold Regions Science and Technology*, 155, 120–126. https://doi.org/10.1016/j.coldregions.2018.08.007

Banin, A., & Anderson, D. M. (1974). Effects of salt concentration changes during freezing on the unfrozen water content of porous materials. *Water Resources Research*, 10(1), 124–128. https://doi.org/10.1029/WR010i001p00124

Binley, A., Cassiani, G., Middleton, R., & Winship, P. (2002). Vadose zone flow model parameterisation using cross-borehole radar and resistivity imaging. *Journal of Hydrology*, 267(3–4), 147–159. https://doi.org/10.1016/S0022-1694(02)00146-4

Binley, A., Hubbard, S. S., Huisman, J. A., Revil, A., Robinson, D. A., Singha, K., & Slater, L. D. (2015). The emergence of hydrogeophysics for improved understanding of subsurface processes over multiple scales. *Water Resources Research*, 51(6), 3837–3866. https://doi.org/10.1002/2015WR017016

Breede, K., Kemna, A., Esser, O., Zimmermann, E., Vereecken, H., & Huisman, J. (2011). Joint measurement setup for determining spectral induced polarization and soil hydraulic properties. *Vadose Zone Journal*, 10(2), 716–726. https://doi.org/10.2136/vzj2010.0110

Bussière, L., Schmutz, M., Fortier, R., Lemieux, J. M., & Dupuy, A. (2022). Near-surface geophysical imaging of a thermokarst pond in the discontinuous permafrost zone in Nunavik (Québec), Canada. *Permafrost and Periglacial Processes*, 33(4), 353–369. https://doi.org/10.1002/ppp.2166

Chen, H., Chen, K., Yang, M. H., & Xu, P. (2020). A fractal capillary model for multiphase flow in porous media with hysteresis effect. *International Journal of Multiphase Flow*, 125, 103208. https://doi.org/10.1016/j.ijmultiphaseflow.2020.103208

Chen, K., Chen, W., & Liang, F. Y. (2023). Fractal-based hydraulic model of unsaturated flow in deformable soils considering the evolution of pore size distribution. *Journal of Hydrology*, 620, 129501. https://doi.org/10.1016/j.jhydrol.2023.129501

Christensen, A. F., He, H., Dyck, M. F., Lenore Turner, E., Chanasyk, D. S., Naeth, M. A., & Nichol, C. (2013). In situ measurement of snowmelt infiltration under various topsoil cap thicknesses on a reclaimed site. *Canadian Journal of Soil Science*, 93(4), 497–510. https://doi.org/10.4141/cjss2012-048

Clayton, W. S. (2017). In situ measurement of meltwater percolation flux in seasonal alpine snowpack using self-potential and capillary pressure sensors. *The Cryosphere Discussions*, 2017, 1–21. https://doi.org/10.5194/tc-2017-187

Clayton, W. S. (2021). Measurement of unsaturated meltwater percolation flux in seasonal snowpack using self-potential. *Journal of Glaciology*, 68(267), 25–40. https://doi.org/10.1017/jog.2021.67

Coperey, A., Revil, A., Abdulsamad, F., Stutz, B., Duvillard, P. A., & Ravanel, L. (2019). Low-frequency induced polarization of porous media undergoing freezing: Preliminary observations and modeling. *Journal of Geophysical Research: Solid Earth*, 124(5), 4523–4544. https://doi.org/10.1029/2018JB017015

Coperey, A., Revil, A., & Stutz, B. (2019). Electrical conductivity versus temperature in freezing conditions: A field experiment using a basket geothermal heat exchanger. *Geophysical Research Letters*, 46(24), 14531–14538. https://doi.org/10.1029/2019GL084962

Dafflon, B., Hubbard, S., Ulrich, C., Peterson, J., Wu, Y. X., Wainwright, H., & Kneafsey, T. J. (2016). Geophysical estimation of shallow permafrost distribution and properties in an ice-wedge polygon-dominated Arctic tundra region. *Geophysics*, 81(1), WA247–WA263. https://doi.org/10.1190/geo2015-0175.1

Daigle, H. (2021). Structure of the electrical double layer at the ice-water interface. *Journal of Chemical Physics*, 154(21), 214703. https://doi.org/10.1063/5.0048817

Davis, J. A., James, R. O., & Leckie, J. O. (1978). Surface ionization and complexation at the oxide/water interface. *Journal of Colloid and Interface Science*, 63(3), 480–499. https://doi.org/10.1016/S0021-9797(78)80009-5

Doussan, C., & Ruy, S. (2009). Prediction of unsaturated soil hydraulic conductivity with electrical conductivity. *Water Resources Research*, 45(10), W10408. https://doi.org/10.1029/2008WR007309

Duvillard, P. A., Magnin, F., Revil, A., Legay, A., Ravanel, L., Abdulsamad, F., & Coperey, A. (2021). Temperature distribution in a permafrost-affected rock ridge from conductivity and induced polarization tomography. *Geophysical Journal International*, 225(2), 1207–1221. https://doi.org/10.1093/gji/ggaa597
$Q_{a0}$ Surface charge density at the surface of the air (air-water interface) (C)

$Q_{as}$ Surface charge density at the diffuse layer (air-water interface) (C)

$\varphi_{a0}$ Electrical potential at the 0-plane (air-water interface) (V)

$\varphi_{ad}$ Electrical potential at the d-plane (air-water interface) (V)

## Data Availability Statement

The present study is supplemented by Supporting Information. The experimental data of frozen porous media for evaluating the evolution of electrical conductivity as a function of temperature are openly accessible in the Hydrogeophysics community via Zenodo under different initial water saturations and salinities: https://doi.org/ 10.5281/zenodo.7688602. Additionally, Matlab codes for calculating the predicted results of electrical conduc- tivity are also available through this link.

## Acknowledgments

The authors strongly thank the Editor, the Associate Editor, and the two reviewers (Philippe Leroy and an anonymous one) for kind and constructive comments. Support for this work was provided by China Scholarship Council (CSC) project (Grant 202106370053). This research was also supported by the National Natural Science Foundation of China (No. 52178376), and the Natural Science Foundation of Hunan Province (2022JJ10076). Furthermore, we would like to express our gratitude to the School of civil engineering (CSU) for granting us access to the laboratory experimental apparatus for electrical resistance experiments (LCR setup) and for SFCC experiments (NMR method).

## References

Amankwah, S. K., Ireson, A. M., Maulé, C., Brannen, R., & Mathias, A. (2021). A model for the soil freezing characteristic curve that represents the dominant role of salt exclusion. *Water Resources Research*, 57(8), e2021WR030070. https://doi.org/10.1029/2021WR030070

Andersland, O. B., & Ladanyi, B. (2004). *Frozen ground engineering* (2nd). Wiley.

Anderson, D. M. (1967). The interface between ice and silicate surfaces. *Journal of Colloid and Interface Science*, 25(2), 174–191. https://doi.org/10.1016/0021-9797(67)90021-5

Archie, G. E. (1942). The electrical resistivity log as an aid in determining some reservoir characteristics. *Transactions of the AIME*, 146(01), 54–62. https://doi.org/10.2118/942054-G

Bai, R., Lai, Y., Zhang, M., & Gao, J. (2018). Water-vapor-heat behavior in a freezing unsaturated coarse-grained soil with a closed top. *Cold Regions Science and Technology*, 155, 120–126. https://doi.org/10.1016/j.coldregions.2018.08.007

Banin, A., & Anderson, D. M. (1974). Effects of salt concentration changes during freezing on the unfrozen water content of porous materials. *Water Resources Research*, 10(1), 124–128. https://doi.org/10.1029/WR010i001p00124

Binley, A., Cassiani, G., Middleton, R., & Winship, P. (2002). Vadose zone flow model parameterisation using cross-borehole radar and resistivity imaging. *Journal of Hydrology*, 267(3–4), 147–159. https://doi.org/10.1016/S0022-1694(02)00146-4

Binley, A., Hubbard, S. S., Huisman, J. A., Revil, A., Robinson, D. A., Singha, K., & Slater, L. D. (2015). The emergence of hydrogeophysics for improved understanding of subsurface processes over multiple scales. *Water Resources Research*, 51(6), 3837–3866. https://doi.org/10.1002/2015WR017016

Breede, K., Kemna, A., Esser, O., Zimmermann, E., Vereecken, H., & Huisman, J. (2011). Joint measurement setup for determining spectral induced polarization and soil hydraulic properties. *Vadose Zone Journal*, 10(2), 716–726. https://doi.org/10.2136/vzj2010.0110

Bussière, L., Schmutz, M., Fortier, R., Lemieux, J. M., & Dupuy, A. (2022). Near-surface geophysical imaging of a thermokarst pond in the discontinuous permafrost zone in Nunavik (Québec), Canada. *Permafrost and Periglacial Processes*, 33(4), 353–369. https://doi.org/10.1002/ppp.2166

Chen, H., Chen, K., Yang, M. H., & Xu, P. (2020). A fractal capillary model for multiphase flow in porous media with hysteresis effect. *International Journal of Multiphase Flow*, 125, 103208. https://doi.org/10.1016/j.ijmultiphaseflow.2020.103208

Chen, K., Chen, W., & Liang, F. Y. (2023). Fractal-based hydraulic model of unsaturated flow in deformable soils considering the evolution of pore size distribution. *Journal of Hydrology*, 620, 129501. https://doi.org/10.1016/j.jhydrol.2023.129501

Christensen, A. F., He, H., Dyck, M. F., Lenore Turner, E., Chanasyk, D. S., Naeth, M. A., & Nichol, C. (2013). In situ measurement of snowmelt infiltration under various topsoil cap thicknesses on a reclaimed site. *Canadian Journal of Soil Science*, 93(4), 497–510. https://doi.org/10.4141/cjss2012-048

Clayton, W. S. (2017). In situ measurement of meltwater percolation flux in seasonal alpine snowpack using self-potential and capillary pressure sensors. *The Cryosphere Discussions*, 2017, 1–21. https://doi.org/10.5194/tc-2017-187

Clayton, W. S. (2021). Measurement of unsaturated meltwater percolation flux in seasonal snowpack using self-potential. *Journal of Glaciology*, 68(267), 25–40. https://doi.org/10.1017/jog.2021.67

Coperey, A., Revil, A., Abdulsamad, F., Stutz, B., Duvillard, P. A., & Ravanel, L. (2019). Low-frequency induced polarization of porous media undergoing freezing: Preliminary observations and modeling. *Journal of Geophysical Research: Solid Earth*, 124(5), 4523–4544. https://doi.org/10.1029/2018JB017015

Coperey, A., Revil, A., & Stutz, B. (2019). Electrical conductivity versus temperature in freezing conditions: A field experiment using a basket geothermal heat exchanger. *Geophysical Research Letters*, 46(24), 14531–14538. https://doi.org/10.1029/2019GL084962

Dafflon, B., Hubbard, S., Ulrich, C., Peterson, J., Wu, Y. X., Wainwright, H., & Kneafsey, T. J. (2016). Geophysical estimation of shallow permafrost distribution and properties in an ice-wedge polygon-dominated Arctic tundra region. *Geophysics*, 81(1), WA247–WA263. https://doi.org/10.1190/geo2015-0175.1

Daigle, H. (2021). Structure of the electrical double layer at the ice-water interface. *Journal of Chemical Physics*, 154(21), 214703. https://doi.org/10.1063/5.0048817

Davis, J. A., James, R. O., & Leckie, J. O. (1978). Surface ionization and complexation at the oxide/water interface. *Journal of Colloid and Interface Science*, 63(3), 480–499. https://doi.org/10.1016/S0021-9797(78)80009-5

Doussan, C., & Ruy, S. (2009). Prediction of unsaturated soil hydraulic conductivity with electrical conductivity. *Water Resources Research*, 45(10), W10408. https://doi.org/10.1029/2008WR007309

Duvillard, P. A., Magnin, F., Revil, A., Legay, A., Ravanel, L., Abdulsamad, F., & Coperey, A. (2021). Temperature distribution in a permafrost-affected rock ridge from conductivity and induced polarization tomography. *Geophysical Journal International*, 225(2), 1207–1221. https://doi.org/10.1093/gji/ggaa597
Duvillard, P. A., Revil, A., Qi, Y., Soueid Ahmed, A., Coperey, A., & Ravanel, L. (2018). Three-dimensional electrical conductivity and induced polarization tomography of a rock glacier. *Journal of Geophysical Research: Solid Earth, 123*(11), 9528–9554. https://doi.org/10.1029/2018JB015965

Giménez, G., Perfect, E., Rawls, W. J., & Pachepsky, Y. (1997). Fractal models for predicting soil hydraulic properties: A review. *Engineering Geology*, **48**(3–4), 161–183. https://doi.org/10.1016/S0013-7952(97)00038-0

Gleeson, T., Wang-Erlandsson, L., Porkka, M., Zipper, S. C., Jaramillo, F., Gerten, D., et al. (2020). Illuminating water cycle modifications and Earth system resilience in the Anthropocene. *Water Resources Research*, **56**(4), e2019WR024957. https://doi.org/10.1029/2019WR024957

Guarracino, L., & Jougnot, D. (2018). A physically based analytical model to describe effective excess charge for streaming potential generation in water saturated porous media. *Journal of Geophysical Research: Solid Earth*, **123**(1), 52–65. https://doi.org/10.1002/2017JB014873

Guarracino, L., Rötting, T., & Carrera, J. (2014). A fractal model to describe the evolution of multiphase flow properties during mineral dissolution. *Advances in Water Resources*, **67**, 78–86. https://doi.org/10.1016/j.adwatres.2014.02.011

Haarder, E. B., Jensen, K. H., Binley, A., Nielsen, L., Uglebjerg, T. B., & Looms, M. C. (2015). Estimation of recharge from long-term monitoring of saline tracer transport using electrical resistivity tomography. *Vadose Zone Journal*, **14**(7), 1–13. https://doi.org/10.2136/vzj2014.08.0110

Hauck, C., Bottcher, M., & Maurer, H. (2011). A new model for estimating subsurface ice content based on combined electrical and seismic data sets. *The Cryosphere*, **5**(2), 453–468. https://doi.org/10.5194/tc-5-453-2011

Hayashi, M., van der Kamp, G., & Schmidt, R. (2003). Focused infiltration of snowmelt water in partially frozen soil under small depressions. *Journal of Hydrology*, **270**(3–4), 214–229. https://doi.org/10.1016/S0022-1694(02)00287-1

He, H. L., Dyck, M., Zhao, Y., Si, B. C., Jin, H. J., Zhang, T. J., et al. (2016). Evaluation of five composite dielectric mixing models for understanding relationships between effective permittivity and unfrozen water content. *Cold Regions Science and Technology*, **130**, 33–42. https://doi.org/10.1016/j.coldregions.2016.07.006

Hermans, T., Goderniaus, P., Damien, J., Fleckenstein, J. H., Brunner, P., Nguyen, F., et al. (2023). Advancing measurements and representations of subsurface heterogeneity and dynamic processes: Towards 4D hydrogeology. *Hydrology and Earth System Sciences*, **27**(1), 255–287. https://doi.org/10.5194/hess-27-255-2023

Herring, T., Cey, E., & Pidlisecky, A. (2019). Electrical resistivity of a partially saturated porous medium at subzero temperatures. *Vadose Zone Journal*, **18**, 1–11. https://doi.org/10.2136/vzj2019.02.0019

Holloway, J., & Lewkowicz, A. (2019). Field and laboratory investigation of electrical resistivity-temperature relationships, Southern Northwest Territories. In *8th International conference on cold regions engineering and 8th Canadian permafrost conference* (pp. 64–72). https://doi.org/10.1061/9780784482599.008

Hu, S. J., Zhang, C., & Lu, N. (2023). Quantifying coupling effects between soil matric potential and osmotic potential. *Water Resources Research*, **59**(2), e2022WR033779. https://doi.org/10.1029/2022WR033779

IPCC. (2019). The ocean and cryosphere in a changing climate. IPCC special report on the ocean and cryosphere in a changing climate, 1155.

Jin, X., Yang, W., Gao, X. Q., Zhao, J. Q., Li, Z. C., & Jiang, J. X. (2020). Modeling the unfrozen water content of frozen soil based on the absorption effects of clay surfaces. *Water Resources Research*, **56**(12), e2020WR027482. https://doi.org/10.1029/2020WR027482

Jougnot, D., Ghorbani, A., Revil, A., Leroy, P., & Cosenza, P. (2010). Spectral induced polarization of partially saturated clay-rocks: A mechanistic approach. *Geophysical Journal International*, **180**(1), 210–224. https://doi.org/10.1111/j.1365-246X.2009.04426.x

Jougnot, D., Linde, N., Haarder, E. B., & Looms, M. C. (2015). Monitoring of saline tracer movement with vertically distributed self-potential measurements at the HOBE agricultural test site, Voulund, Denmark. *Journal of Hydrology*, **521**, 314–327. https://doi.org/10.1016/j.jhydrol.2014.11.041

Jurin, M. (1718). An account of some experiments shown before the Royal Society; with an enquiry into the cause of the ascent and suspension of water in capillary tubes. *Philosophical Transactions Royal Society*, **30**(355), 739–747. https://doi.org/10.1098/rstl.1717.0026

Kallay, N., Čop, A., Chibowski, E., & Holysz, L. (2003). Reversible charging of the ice-water interface: II. Estimation of equilibrium parameters. *Journal of Colloid and Interface Science*, **259**(1), 89–96. https://doi.org/10.1016/S0021-9797(02)00179-0

Kemna, A., Vanderborght, J., Kulessa, B., & Vereecken, H. (2002). Imaging and characterisation of subsurface solute transport using electrical resistivity tomography (ERT) and equivalent transport models. *Journal of Hydrology*, **267**(3–4), 125–146. https://doi.org/10.1016/S0022-1694(02)00145-2

Kou, J. Y. (2020). Study on the influence of soil microstructure on the hydraulic properties of soil (PhD Thesis). Central South University. (In Chinese with English abstract).

Kulessa, B., Chandler, D., Revil, A., & Essery, R. (2012). Theory and numerical modeling of electrical self-potential signatures of unsaturated flow in melting snow. *Water Resources Research*, **48**(9), W09511. https://doi.org/10.1029/2012WR012048

Kulessa, B., Hubbard, B., & Brown, G. H. (2003). Cross-coupled flow modeling of coincident streaming and electrochemical potentials and application to subglacial self-potential data. *Journal of Geophysical Research*, **108**(B8), 2381. https://doi.org/10.1029/2001JB001167

Lai, Z., Zhao, X., Tang, R., & Yang, J. (2021). Electrical conductivity-based estimation of unfrozen water content in saturated saline frozen sand. *Advances in Civil Engineering*, **21**, e8881304. https://doi.org/10.1155/2021/8881304

Lebeau, M., & Konrad, J. M. (2010). A new capillary and thin film flow model for predicting the hydraulic conductivity of unsaturated porous media. *Water Resources Research*, **46**(1), 1–15. https://doi.org/10.1029/2010WR009092

Lebeau, M., & Konrad, J. M. (2012). An extension of the capillary and thin film flow model for predicting the hydraulic conductivity of air-free frozen porous media. *Water Resources Research*, **48**(7), W07523. https://doi.org/10.1029/2012WR011916

Léger, E., Saintenoy, A., Serhir, M., Costard, F., & Grenier, C. (2023). Brief communication: Monitoring active layer dynamics using a lightweight nimble ground-penetrating radar system—a laboratory analogue test case. *The Cryosphere*, **17**(3), 1271–1277. https://doi.org/10.5194/tc-17-1271-2023

Leroy, P., Jougnot, D., Revil, A., Lassin, A., & Azaroual, M. (2012). A double layer model of the gas bubble/water interface. *Journal of Colloid and Interface Science*, **388**(1), 243–256. https://doi.org/10.1016/j.jcis.2012.07.029

Leroy, P., Revil, A., Kemna, A., Cosenza, P., & Ghorban, A. (2008). Complex conductivity of water-saturated packs of glass beads. *Journal of Colloid and Interface Science*, **321**(1), 103–117. https://doi.org/10.1016/j.jcis.2007.12.031

Li, M., Tang, Y. B., Bernabé, Y., Zhao, J. Z., Li, X. F., Bai, X. Y., & Zhang, L. H. (2015). Pore connectivity, electrical conductivity, and partial water saturation: Network simulations. *Journal of Geophysical Research: Solid Earth*, **120**(6), 4055–4068. https://doi.org/10.1002/2014JB011799

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

Loisel, J., Gallego-Sala, A. V., Amesbury, M. J., Magnan, G., Anshari, G., Beilman, D. W., et al. (2021). Expert assessment of future vulnerability of the global peatland carbon sink. *Nature Climate Change*, **11**(1), 70–77. https://doi.org/10.1038/s41558-020-00944-0
Duvillard, P. A., Revil, A., Qi, Y., Soueid Ahmed, A., Coperey, A., & Ravanel, L. (2018). Three-dimensional electrical conductivity and induced polarization tomography of a rock glacier. *Journal of Geophysical Research: Solid Earth, 123*(11), 9528-9554. https://doi.org/10.1029/2018JB015965

Giménez, G., Perfect, E., Rawls, W. J., & Pachepsky, Y. (1997). Fractal models for predicting soil hydraulic properties: A review. *Engineering Geology*, **48**(3-4), 161-183. https://doi.org/10.1016/S0013-7952(97)00038-0

Gleeson, T., Wang-Erlandsson, L., Porkka, M., Zipper, S. C., Jaramillo, F., Gerten, D., et al. (2020). Illuminating water cycle modifications and Earth system resilience in the Anthropocene. *Water Resources Research*, **56**(4), e2019WR024957. https://doi.org/10.1029/2019WR024957

Guarracino, L., & Jougnot, D. (2018). A physically based analytical model to describe effective excess charge for streaming potential generation in water saturated porous media. *Journal of Geophysical Research: Solid Earth*, **123**(1), 52-65. https://doi.org/10.1002/2017JB014873

Guarracino, L., Rötting, T., & Carrera, J. (2014). A fractal model to describe the evolution of multiphase flow properties during mineral dissolution. *Advances in Water Resources*, **67**, 78-86. https://doi.org/10.1016/j.adwatres.2014.02.011

Haarder, E. B., Jensen, K. H., Binley, A., Nielsen, L., Uglebjerg, T. B., & Looms, M. C. (2015). Estimation of recharge from long-term monitoring of saline tracer transport using electrical resistivity tomography. *Vadose Zone Journal*, **14**(7), 1-13. https://doi.org/10.2136/vzj2014.08.0110

Hauck, C., Bottcher, M., & Maurer, H. (2011). A new model for estimating subsurface ice content based on combined electrical and seismic data sets. *The Cryosphere*, **5**(2), 453-468. https://doi.org/10.5194/tc-5-453-2011

Hayashi, M., van der Kamp, G., & Schmidt, R. (2003). Focused infiltration of snowmelt water in partially frozen soil under small depressions. *Journal of Hydrology*, **270**(3-4), 214-229. https://doi.org/10.1016/S0022-1694(02)00287-1

He, H. L., Dyck, M., Zhao, Y., Si, B. C., Jin, H. J., Zhang, T. J., et al. (2016). Evaluation of five composite dielectric mixing models for understanding relationships between effective permittivity and unfrozen water content. *Cold Regions Science and Technology*, **130**, 33-42. https://doi.org/10.1016/j.coldregions.2016.07.006

Hermans, T., Goderniaus, P., Damien, J., Fleckenstein, J. H., Brunner, P., Nguyen, F., et al. (2023). Advancing measurements and representations of subsurface heterogeneity and dynamic processes: Towards 4D hydrogeology. *Hydrology and Earth System Sciences*, **27**(1), 255-287. https://doi.org/10.5194/hess-27-255-2023

Herring, T., Cey, E., & Pidlisecky, A. (2019). Electrical resistivity of a partially saturated porous medium at subzero temperatures. *Vadose Zone Journal*, **18**, 1-11. https://doi.org/10.2136/vzj2019.02.0019

Holloway, J., & Lewkowicz, A. (2019). Field and laboratory investigation of electrical resistivity-temperature relationships, Southern Northwest Territories. In *8th International conference on cold regions engineering and 8th Canadian permafrost conference* (pp. 64-72). https://doi.org/10.1061/9780784482599.008

Hu, S. J., Zhang, C., & Lu, N. (2023). Quantifying coupling effects between soil matric potential and osmotic potential. *Water Resources Research*, **59**(2), e2022WR033779. https://doi.org/10.1029/2022WR033779

IPCC. (2019). The ocean and cryosphere in a changing climate. IPCC special report on the ocean and cryosphere in a changing climate, 1155.

Jin, X., Yang, W., Gao, X. Q., Zhao, J. Q., Li, Z. C., & Jiang, J. X. (2020). Modeling the unfrozen water content of frozen soil based on the absorption effects of clay surfaces. *Water Resources Research*, **56**(12), e2020WR027482. https://doi.org/10.1029/2020WR027482

Jougnot, D., Ghorbani, A., Revil, A., Leroy, P., & Cosenza, P. (2010). Spectral induced polarization of partially saturated clay-rocks: A mechanistic approach. *Geophysical Journal International*, **180**(1), 210-224. https://doi.org/10.1111/j.1365-246X.2009.04426.x

Jougnot, D., Linde, N., Haarder, E. B., & Looms, M. C. (2015). Monitoring of saline tracer movement with vertically distributed self-potential measurements at the HOBE agricultural test site, Voulund, Denmark. *Journal of Hydrology*, **521**, 314-327. https://doi.org/10.1016/j.jhydrol.2014.11.041

Jurin, M. (1718). An account of some experiments shown before the Royal Society; with an enquiry into the cause of the ascent and suspension of water in capillary tubes. *Philosophical Transactions Royal Society*, **30**(355), 739-747. https://doi.org/10.1098/rstl.1717.0026

Kallay, N., Čop, A., Chibowski, E., & Holysz, L. (2003). Reversible charging of the ice-water interface: II. Estimation of equilibrium parameters. *Journal of Colloid and Interface Science*, **259**(1), 89-96. https://doi.org/10.1016/S0021-9797(02)00179-0

Kemna, A., Vanderborght, J., Kulessa, B., & Vereecken, H. (2002). Imaging and characterisation of subsurface solute transport using electrical resistivity tomography (ERT) and equivalent transport models. *Journal of Hydrology*, **267**(3-4), 125-146. https://doi.org/10.1016/S0022-1694(02)00145-2

Kou, J. Y. (2020). Study on the influence of soil microstructure on the hydraulic properties of soil (PhD Thesis). Central South University. (In Chinese with English abstract).

Kulessa, B., Chandler, D., Revil, A., & Essery, R. (2012). Theory and numerical modeling of electrical self-potential signatures of unsaturated flow in melting snow. *Water Resources Research*, **48**(9), W09511. https://doi.org/10.1029/2012WR012048

Kulessa, B., Hubbard, B., & Brown, G. H. (2003). Cross-coupled flow modeling of coincident streaming and electrochemical potentials and application to subglacial self-potential data. *Journal of Geophysical Research*, **108**(B8), 2381. https://doi.org/10.1029/2001JB001167

Lai, Z., Zhao, X., Tang, R., & Yang, J. (2021). Electrical conductivity-based estimation of unfrozen water content in saturated saline frozen sand. *Advances in Civil Engineering*, **21**, e8881304. https://doi.org/10.1155/2021/8881304

Lebeau, M., & Konrad, J. M. (2010). A new capillary and thin film flow model for predicting the hydraulic conductivity of unsaturated porous media. *Water Resources Research*, **46**(1), 1-15. https://doi.org/10.1029/2010WR009092

Lebeau, M., & Konrad, J. M. (2012). An extension of the capillary and thin film flow model for predicting the hydraulic conductivity of air-free frozen porous media. *Water Resources Research*, **48**(7), W07523. https://doi.org/10.1029/2012WR011916

Léger, E., Saintenoy, A., Serhir, M., Costard, F., & Grenier, C. (2023). Brief communication: Monitoring active layer dynamics using a lightweight nimble ground-penetrating radar system-a laboratory analogue test case. *The Cryosphere*, **17**(3), 1271-1277. https://doi.org/10.5194/tc-17-1271-2023

Leroy, P., Jougnot, D., Revil, A., Lassin, A., & Azaroual, M. (2012). A double layer model of the gas bubble/water interface. *Journal of Colloid and Interface Science*, **388**(1), 243-256. https://doi.org/10.1016/j.jcis.2012.07.029

Leroy, P., Revil, A., Kemna, A., Cosenza, P., & Ghorban, A. (2008). Complex conductivity of water-saturated packs of glass beads. *Journal of Colloid and Interface Science*, **321**(1), 103-117. https://doi.org/10.1016/j.jcis.2007.12.031

Li, M., Tang, Y. B., Bernabé, Y., Zhao, J. Z., Li, X. F., Bai, X. Y., & Zhang, L. H. (2015). Pore connectivity, electrical conductivity, and partial water saturation: Network simulations. *Journal of Geophysical Research: Solid Earth*, **120**(6), 4055-4068. https://doi.org/10.1002/2014JB011799

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

Loisel, J., Gallego-Sala, A. V., Amesbury, M. J., Magnan, G., Anshari, G., Beilman, D. W., et al. (2021). Expert assessment of future vulnerability of the global peatland carbon sink. *Nature Climate Change*, **11**(1), 70-77. https://doi.org/10.1038/s41558-020-00944-0
Luo, H. L., Jougnot, D., Jost, A., Teng, J. D., & Thanh, L. D. (2023). A capillary bundle model for the electrical conductivity of saturated frozen porous media. *Journal of Geophysical Research: Solid Earth*, 128(3), e2022JB025254. https://doi.org/10.1029/2022JB025254

Luo, H. L., Teng, J. D., Zhang, S., & Sheng, D. C. (2021). Study on the relationship between unfrozen water content and electrical conductivity in frozen soils. *Chinese Journal of Rock Mechanical and Engineering*, 40(5), 1068-1079. (in Chinese). Retrieved from http://rockmech.whrsm.ac.cn/CN/Y2021/V40/I5/1068

Maineult, A., Jougnot, D., & Revil, A. (2018). Variations of petrophysical properties and spectral induced polarization in response to drainage and imbibition: A study on a correlated random tube network. *Geophysical Journal International*, 212(2), 1398-1411. https://doi.org/10.1093/gji/ggx474

Minagawa, H., Nishikawa, Y., Ikeda, I., Miyazaki, K., Takahara, N., Sakamoto, Y., et al. (2008). Characterization of sand sediment by pore size distribution and permeability using proton nuclear magnetic resonance measurement. *Journal of Geophysical Research*, 113(B7), B07210. https://doi.org/10.1029/2007JB005403

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Niu, F. J., Li, A. Y., Luo, J., Lin, Z. J., Yin, G. A., Liu, M. H., et al. (2017). Soil moisture, ground temperatures, and deformation of a high-speed railway embankment in Northeast China. *Cold Regions Science and Technology*, 133, 7-14. https://doi.org/10.1016/j.coldregions.2016.10.007

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Oldenborger, G. A., & LeBlanc, A. M. (2018). Monitoring changes in unfrozen water content with electrical resistivity surveys in cold continuous permafrost. *Geophysical Journal International*, 72(4), 177-187. https://doi.org/10.1093/gji/ggy321

Oquist, M. G., Sparrman, T., Klemedtsson, L., Drotz, S. H., Grip, H., Schleucher, J., & Nilsson, M. (2009). Water availability controls microbial temperature responses in frozen soil CO₂ production. *Global Change Biology*, 15(11), 2715-2722. https://doi.org/10.1111/j.1365-2486.2009.01898.x

Pedrazas, M. N., Cardenas, M. B., Demir, C., Watson, J. A., Connolly, C. T., & McClelland, J. W. (2020). Absence of ice-bonded permafrost beneath an Arctic lagoon revealed by electrical geophysics. *Science Advances*, 6(43), eabb5083. https://doi.org/10.1126/sciadv.abb5083

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Rembert, F., Jougnot, D., & Guarracino, L. (2020). A fractal model for the electrical conductivity of water-saturated porous media during mineral precipitation-dissolution processes. *Advances in Water Resources*, 145, 103742. https://doi.org/10.1016/j.adwatres.2020.103742

Revil, A. (2012). Spectral induced polarization of shaly sands: Influence of the electrical double layer. *Water Resources Research*, 48(2), W02517. https://doi.org/10.1029/2011WR011260

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Revil, A., Linde, N., Cerepi, A., Jougnot, D., Matthäi, S., & Finsterle, S. (2007). Electrom kinetic coupling in unsaturated porous media. *Journal of Colloid and Interface Science*, 313(1), 315-327. https://doi.org/10.1016/j.jcis.2007.03.037

Revil, A., Soueid Ahmed, A., & Matthai, S. (2018). Transport of water and ions in partially water-saturated porous media: Part 3. Electrical conductivity. *Advances in Water Resources*, 121, 97-111. https://doi.org/10.1016/j.adwatres.2018.08.007

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Tai, B. W., Liu, J. K., & Chang, D. (2020). Experimental and numerical investigation on the sunny-shady slopes effect of three cooling embankments along an expressway in warm permafrost region, China. *Engineering Geology*, 269, 105545. https://doi.org/10.1016/j.enggeo.2020.105545

Tai, B. W., Liu, J. K., Yue, Z. R., Liu, J. Y., Tian, Y. H., & Wang, T. F. (2018). Effect of sunny-shady slopes and strike on thermal regime of subgrade along a high-speed railway in cold regions, China. *Engineering Geology*, 232, 182-191. https://doi.org/10.1016/j.enggeo.2017.09.002

Tang, L. Y., Wang, K., Jin, L., Yang, G. S., Jia, H. L., & Taoum, A. (2018). A resistivity model for testing unfrozen water content of frozen soil. *Cold Regions Science and Technology*, 153, 55-63. https://doi.org/10.1016/j.coldregions.2018.05.003

Teng, J. D., Kou, J. Y., Yan, X., Zhang, S., & Sheng, D. C. (2020). Parameterization of soil freezing characteristic curve for unsaturated soils. *Cold Regions Science and Technology*, 170(68), 102928. https://doi.org/10.1016/j.coldregions.2019.102928

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Teng, J. D., Liu, J. L., Zhang, S., & Sheng, D. C. (2022). Frost heave in coarse-grained soils: Experimental evidence and numerical modelling. *Géotechnique*, 73, 1-12. https://doi.org/10.1680/jgeot.21.00182

Thanh, L. D., Jougnot, D., Van Do, P., & Van Nghia, A. N. (2019). A physically based model for the electrical conductivity of water-saturated porous media. *Geophysical Journal International*, 219(2), 866-876. https://doi.org/10.1093/gji/ggz328

Thanh, L. D., Jougnot, D., Van Do, P., Tuyen, V. P., Ca, N. X., & Hien, N. T. (2020). A physically based model for the electrical conductivity of partially saturated porous media. *Geophysical Journal International*, 223(2), 993-1006. https://doi.org/10.1093/gji/ggaa307

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Luo, H. L., Jougnot, D., Jost, A., Teng, J. D., & Thanh, L. D. (2023). A capillary bundle model for the electrical conductivity of saturated frozen porous media. *Journal of Geophysical Research: Solid Earth*, 128(3), e2022JB025254. https://doi.org/10.1029/2022JB025254

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Wang, L. Y., Liu, J. K., Yu, X. B., Wang, T. F., & Feng, R. L. (2021). A simplified model for the phase composition curve of saline soils considering the second phase transition. *Water Resources Research*, 57(1), e2020WR028556. https://doi.org/10.1029/2020WR028556

Watanabe, K., & Flury, M. (2008). Capillary bundle model of hydraulic conductivity for frozen soil. *Water Resources Research*, 44(12), W12402. https://doi.org/10.1029/2008WR007012

Watanabe, K., & Mizoguchi, M. (2002). Amount of unfrozen water in frozen porous media saturated with solution. *Cold Regions Science and Technology*, 34(2), 103-110. https://doi.org/10.1016/S0165-232X(01)00063-5

Watanabe, K., & Osada, Y. (2017). Simultaneous measurement of unfrozen water content and hydraulic conductivity of partially frozen soil near 0°C. *Cold Regions Science and Technology*, 142, 79-84. https://doi.org/10.1016/j.coldregions.2017.08.002

Weerts, A. H., Bouten, W., & Verstraten, J. M. (1999). Simultaneous measurement of water retention and electrical conductivity in soils: Testing the Mualem-Friedman Tortuosity Model. *Water Resources Research*, 35(6), 1781-1787. https://doi.org/10.1029/1999WR900022

Wen, Z., Ma, W., Feng, W., Deng, Y., Wang, D., Fan, Z., & Zhou, C. (2012). Experimental study on unfrozen water content and soil matric potential of Qinghai-Tibetan silty clay. *Environmental Earth Sciences*, 66(5), 1467-1476. https://doi.org/10.1007/s12665-011-1386-0

Xiao, Z., Lai, Y. M., & Zhang, M. Y. (2018). Study on the freezing temperature of saline soil. *Acta Geotechnica*, 13(1), 195-205. https://doi.org/10.1007/s11440-017-0537-1

Yu, B., & Cheng, P. (2002). A fractal permeability model for bi-dispersed porous media. *International Journal of Heat and Mass Transfer*, 45(14), 2983-2993. https://doi.org/10.1016/S0017-9310(02)00014-5

Yu, B., Li, J., Li, Z., & Zou, M. (2003). Permeabilities of unsaturated fractal porous media. *International Journal of Multiphase Flow*, 29(10), 1625-1642. https://doi.org/10.1016/S0301-9322(03)00140-X

Zhang, C., & Liu, Z. (2018). Freezing of water confined in porous materials: Role of adsorption and unfreezable threshold. *Acta Geotechnica*, 13(5), 1203-1213. https://doi.org/10.1007/s11440-018-0637-6

Zhang, C., & Lu, N. (2018). What is the range of soil water density? Critical reviews with a unified model. *Reviews of Geophysics*, 56(3), 532-562. https://doi.org/10.1029/2018RG000597

Zhang, Y. Z., Ma, W., Wang, T. L., Cheng, B. Y., & Wen, A. (2019). Characteristics of the liquid and vapor migration of coarse-grained soil in an open-system under constant-temperature freezing. *Cold Regions Science and Technology*, 165, 102793. https://doi.org/10.1016/j.coldregions.2019.102793

Zhou, J., Meng, X., Wei, C., & Pei, W. (2020). Unified soil freezing characteristic for variably-saturated saline soils. *Water Resources Research*, 56(7), e2019WR026648. https://doi.org/10.1029/2019WR026648

## References From the Supporting Information

Heberling, F., Bosbach, D., Eckhardt, J. D., Fischer, U., Glowacky, J., Haist, M., et al. (2014). Reactivity of the calcite-water-interface, from molecular scale processes to geochemical engineering. *Applied Geochemistry*, 45, 158-190. https://doi.org/10.1016/j.apgeochem.2014.03.006

Heberling, F., Trainor, T. P., Lützenkirchen, J., Eng, P., Denecke, M. A., & Bosbach, D. (2011). Structure and reactivity of the calcite-water interface. *Journal of Colloid and Interface Science*, 354(2), 843-857. https://doi.org/10.1016/j.jcis.2010.10.047

Leroy, P., & Revil, A. (2004). A triple-layer model of the surface electrochemical properties of clay minerals. *Journal of Colloid and Interface Science*, 270(2), 371-380. https://doi.org/10.1016/j.jcis.2003.08.007

Tang, R., Zhou, G. Q., Jiao, W., & Ji, Y. K. (2019). Theoretical model of hydraulic conductivity for frozen saline/non-saline soil based on freezing characteristic curve. *Cold Regions Science and Technology*, 165, 102794. https://doi.org/10.1016/j.coldregions.2019.102794

Zimmermann, R., Dukhin, S., & Werner, C. (2001). Electrokinetic measurements reveal interfacial charge at polymer films caused by simple electrolyte ions. *The Journal of Physical Chemistry B*, 105(36), 8544-8549. https://doi.org/10.1021/jp004051u
Wang, C., Lai, Y., & Zhang, M. (2017). Estimating soil freezing characteristic curve based on pore-size distribution. *Applied Thermal Engineering, 124*, 1049-1060. https://doi.org/10.1016/j.applthermaleng.2017.06.006

Wang, L. Y., Liu, J. K., Yu, X. B., Wang, T. F., & Feng, R. L. (2021). A simplified model for the phase composition curve of saline soils considering the second phase transition. *Water Resources Research*, 57(1), e2020WR028556. https://doi.org/10.1029/2020WR028556

Watanabe, K., & Flury, M. (2008). Capillary bundle model of hydraulic conductivity for frozen soil. *Water Resources Research*, 44(12), W12402. https://doi.org/10.1029/2008WR007012

Watanabe, K., & Mizoguchi, M. (2002). Amount of unfrozen water in frozen porous media saturated with solution. *Cold Regions Science and Technology*, 34(2), 103-110. https://doi.org/10.1016/S0165-232X(01)00063-5

Watanabe, K., & Osada, Y. (2017). Simultaneous measurement of unfrozen water content and hydraulic conductivity of partially frozen soil near 0°C. *Cold Regions Science and Technology*, 142, 79-84. https://doi.org/10.1016/j.coldregions.2017.08.002

Weerts, A. H., Bouten, W., & Verstraten, J. M. (1999). Simultaneous measurement of water retention and electrical conductivity in soils: Testing the Mualem-Friedman Tortuosity Model. *Water Resources Research*, 35(6), 1781-1787. https://doi.org/10.1029/1999WR900022

Wen, Z., Ma, W., Feng, W., Deng, Y., Wang, D., Fan, Z., & Zhou, C. (2012). Experimental study on unfrozen water content and soil matric potential of Qinghai-Tibetan silty clay. *Environmental Earth Sciences*, 66(5), 1467-1476. https://doi.org/10.1007/s12665-011-1386-0

Xiao, Z., Lai, Y. M., & Zhang, M. Y. (2018). Study on the freezing temperature of saline soil. *Acta Geotechnica*, 13(1), 195-205. https://doi.org/10.1007/s11440-017-0537-1

Yu, B., & Cheng, P. (2002). A fractal permeability model for bi-dispersed porous media. *International Journal of Heat and Mass Transfer*, 45(14), 2983-2993. https://doi.org/10.1016/S0017-9310(02)00014-5

Yu, B., Li, J., Li, Z., & Zou, M. (2003). Permeabilities of unsaturated fractal porous media. *International Journal of Multiphase Flow*, 29(10), 1625-1642. https://doi.org/10.1016/S0301-9322(03)00140-X

Zhang, C., & Liu, Z. (2018). Freezing of water confined in porous materials: Role of adsorption and unfreezable threshold. *Acta Geotechnica*, 13(5), 1203-1213. https://doi.org/10.1007/s11440-018-0637-6

Zhang, C., & Lu, N. (2018). What is the range of soil water density? Critical reviews with a unified model. *Reviews of Geophysics*, 56(3), 532-562. https://doi.org/10.1029/2018RG000597

Zhang, Y. Z., Ma, W., Wang, T. L., Cheng, B. Y., & Wen, A. (2019). Characteristics of the liquid and vapor migration of coarse-grained soil in an open-system under constant-temperature freezing. *Cold Regions Science and Technology*, 165, 102793. https://doi.org/10.1016/j.coldregions.2019.102793

Zhou, J., Meng, X., Wei, C., & Pei, W. (2020). Unified soil freezing characteristic for variably-saturated saline soils. *Water Resources Research*, 56(7), e2019WR026648. https://doi.org/10.1029/2019WR026648

## References From the Supporting Information

Heberling, F., Bosbach, D., Eckhardt, J. D., Fischer, U., Glowacky, J., Haist, M., et al. (2014). Reactivity of the calcite-water-interface, from molecular scale processes to geochemical engineering. *Applied Geochemistry*, 45, 158-190. https://doi.org/10.1016/j.apgeochem.2014.03.006

Heberling, F., Trainor, T. P., Lützenkirchen, J., Eng, P., Denecke, M. A., & Bosbach, D. (2011). Structure and reactivity of the calcite-water interface. *Journal of Colloid and Interface Science*, 354(2), 843-857. https://doi.org/10.1016/j.jcis.2010.10.047

Leroy, P., & Revil, A. (2004). A triple-layer model of the surface electrochemical properties of clay minerals. *Journal of Colloid and Interface Science*, 270(2), 371-380. https://doi.org/10.1016/j.jcis.2003.08.007

Tang, R., Zhou, G. Q., Jiao, W., & Ji, Y. K. (2019). Theoretical model of hydraulic conductivity for frozen saline/non-saline soil based on freezing characteristic curve. *Cold Regions Science and Technology*, 165, 102794. https://doi.org/10.1016/j.coldregions.2019.102794

Zimmermann, R., Dukhin, S., & Werner, C. (2001). Electrokinetic measurements reveal interfacial charge at polymer films caused by simple electrolyte ions. *The Journal of Physical Chemistry B*, 105(36), 8544-8549. https://doi.org/10.1021/jp004051u