Numerical Simulations of Coupled Vapor, Water, and Heat Flow in Unsaturated Deformable Soils During Freezing and Thawing

Abstract

Freezing and thawing cycles significantly affect the mechanical and hydraulic behavior of soils, posing detrimental challenges for infrastructures in cold climates. This study develops and validates a coupled Thermal–Hydraulic–Mechanical (THM) model using COMSOL Multiphysics (Version 6.3) to demonstrate the complexities of vapor and water flux, heat transport, frost heave, and vertical stress build-up in unsaturated soils. The analysis focuses on fine sand, sandy clay, and silty clay by examining their varying susceptibilities to frost action. Silty clay generated the highest amount of frost heave and steepest vertical stress gradients due to its high-water retention and strong capillary forces. Fine sand, on the other hand, produced a minimal amount of frost heave and a polarized vertical stress distribution. The study also revealed that vapor flux is more noticeable in freezing fine sand, while silty clay produces the greatest water flux between the frozen and unfrozen zones. The study also assesses the impact of soil properties including the saturated hydraulic conductivity, the particle thermal conductivity, and particle heat capacity on the frost-induced phenomena. Findings show that reducing the saturated hydraulic conductivity has a greater impact on mitigating frost heave than other variations in thermal properties. Silty clay is most affected by these changes, particularly near the soil surface, while fine sand shows less noticeable responses.

1. Introduction

Freezing and thawing cycles significantly impact the mechanical and hydraulic properties of soils in cold climates. These processes influence various engineering challenges, such as the analysis and design of foundations that withstand frost heave, thaw settlement, and the generation of vertical stress build-up. These are important considerations for the maintenance of infrastructure such as roads, pipelines, and foundations [1,2,3,4]. By enlarge, frost heave occurs due to the ice lenses formation in freezing soils which are driven by water migration generated by cryosuction forces. This phenomenon causes upward displacement of the soil leading to surface deformation and potential structural damage [5]. Fine-grained soils, including silty clay and sandy clay, exhibit higher susceptibility to frost action compared to coarse-grained soils owing to their ability to retain water and support ice lens formation [6]. Vertical stress build-up generated during freeze–thaw cycles is dependent on soil type, confining pressures, and ice content [7].

There are few comparative studies that consider the impact of the soil freeze–thaw processes in various soil types. Specifically, these soils are conditioned by the vapor and water flux, the heat transport and resulting frost heave. One study, for example, considers a Thermal–Hydraulic–Mechanical–Chemical coupling model [8] to analyze the flow of water and vapor, the transfer of heat, frost heave, thaw settlement, and the transport of solutes during freeze–thaw cycles in sand, clay, and silt soils. That study assessed the influence on soil behavior of various parameters, such as the initial groundwater table depth, the hydraulic conductivity, the air temperature, and solute transport. Investigations on the influence of soil properties in various soil types, and on freezing and thawing processes via coupling of physical fields are generally limited. Lei et al. [9] presented a Thermo–Hydro–Mechanical coupled model that simulates the freezing process in saturated silty clay. That study explored the impacts of the initial void ratio, the hydraulic conductivity, and the thermal conductivity on a freezing saturated soil [10] developed a coupled Thermal–Hydro–Mechanical (THM) numerical model using the COMSOL Multiphysics software to investigate the effect of particle thermal conductivity on frost heave of unsaturated sands. Results showed that a higher particle thermal conductivity leads to greater frost heave and increased vertical strain, particularly near the soil surface. Moreover, ref. [11] developed a coupled Thermal–Hydro–Mechanical (THM) numerical model to investigate the effect of particle thermal conductivity and saturated hydraulic conductivity on the processes of freezing and thawing in unsaturated fine sands. That study investigated the influence of soil properties of saturated silty clay and unsaturated sand on the processes of freezing and thawing.

Despite these advancements, there is still a lack of comparative studies that examine the influence of soil properties across various soil types, in particular fine sands, by coupling physical fields and comparing their freezing–thawing behaviors. Fine sands, although recognized as frost-susceptible, have received less attention in numerical simulations. Armstrong and Cathy [12] analyzed data provided by the Canadian federal departments on frost damage to pavements. Their study demonstrated that soils with grain sizes characteristic similar to silt and fine sand are particularly prone to frost susceptibility. Previous researchers mainly examined fine sands experimentally [13,14]. Therefore, theoretical frameworks are necessary for numerically modeling fine sands, as demonstrated in the study by Soltanpour and Foriero [15]. Moreover, existing studies often focus on a single soil type without comparing different soil-type behaviors under freeze–thaw cycles. This presents a gap in the literature regarding how fundamental soil properties influence freezing–thawing behavior across various frost-susceptible soils.

To address this gap, this paper presents a coupled Thermo–Hydro–Mechanical (THM) model for explaining the interaction of water–vapor movement, heat transport and deformation in unsaturated soils during freezing and thawing cycles. This combined process is numerically solved using the FEM COMSOL Multiphysics software. The THM model proposed by this study is endorsed using experimental data from freeze–thaw cycles [14]. The validated model is then employed to examine the behavior of fine sand in cold regions via simulations with two types of frost-susceptible soils such as sandy and silty clay. To analyze this, water and vapor flow, heat transport, and frost heave during freezing and thawing cycles are assessed. The final focus of this study is to explore the effects of soil properties on frost heave and vertical stress build-up in three different soil types: fine sand, sandy clay, and silty clay. The findings of this research provide new insights into the role soil properties play in the process of freezing–thawing. This contributes to the understanding of soil–structure interactions in cold environments and promotes the design of more resilient foundations in frost-prone regions.

In modeling soil deformation under freeze–thaw cycles, a poroelastic constitutive relationship was assumed to model the soil deformation under freeze–thaw processes. The focus is on small-strain, thermally induced volume changes (e.g., frost heave and thaw settlement) rather than shear failure. This approach allows for an efficient simulation of stress development and deformation due to coupled thermal and hydraulic processes. More advanced plasticity-based constitutive models (e.g., Modified Cam Clay or Barcelona Basic Model) can be considered in future work to evaluate failure mechanisms under complex loading scenarios.

The hydraulic governing equations in this study are based on the van Genuchten model, which assumes instantaneous equilibrium between soil suction and water content. However, this assumption is not valid under highly transient hydraulic conditions, particularly during rapid freezing or thawing. In such cases, non-equilibrium effects occur due to a delayed response of water content to changes in suction, especially in coarse-textured soils where hydraulic redistribution is more dynamic. Recent experimental work by Yan et al. [16] has demonstrated that dynamic non-equilibrium behavior in soil water retention causes significant deviations from classical equilibrium models. However, the local equilibrium assumption remains a reasonable and computationally efficient simplification, particularly considering the emphasis of this study on gradual seasonal freeze–thaw cycles and macro-scale THM behavior. Future research may benefit from incorporating dynamic retention models to more accurately capture rapid freezing front movement and moisture redistribution in coarse or heterogeneous soils.

Moreover, the saturated hydraulic conductivity used in this study is closely linked to soil packing conditions such as porosity, dry density, and void ratio. According to the Kozeny–Carman equation [17,18], saturated hydraulic conductivity increases with higher porosity and larger particle sizes due to enhanced pore connectivity. Conversely, denser packing (lower porosity) reduces permeability and increases the soil’s air entry value (AEV), thereby influencing both the saturated and unsaturated hydraulic response. Since porosity also defines the saturated water content in the soil–water retention curve, it plays a crucial role in governing moisture migration and frost susceptibility. Although these simulations used typical literature values for each soil type [19], future work could explore the direct coupling between microstructural parameters and hydraulic properties to better capture freeze–thaw behavior under varying compaction states.

2. Methodology

Mathematical THM Model

The partial differential equation (phase-averaged) governing water and vapor transfer in unsaturated freezing soils is given as follows [20,21]:

$$\begin{gathered} ({\rho }_w{\epsilon }_{sp}{X}_f){\displaystyle \frac{\partial P_w}{\partial t}}+{\epsilon }_{sp}{\displaystyle \frac{\partial (S_i({\rho }_i-{\rho }_w))}{\partial T}}{\displaystyle \frac{\partial T}{\partial t}}+(S_i({\rho }_i-{\rho }_w)+(1-S_w){\rho }_v){\displaystyle \frac{\partial ({\epsilon }_{sp})}{\partial t}} \\ +{\epsilon }_{sp} {\rho }_v{\displaystyle \frac{\partial (1-S_w)}{\partial T}}{\displaystyle \frac{\partial T}{\partial t}}=-\nabla .({\rho }_w{u}_w-K_{vh}{\displaystyle \frac{d h}{d z}}-K_{vT}{\displaystyle \frac{dT}{d z}}) \end{gathered}$$

(1)

where subscripts $i$, $w$, and $v$, respectively refer to ice, water, and vapor; $\rho$ is the density (Kg/m³); ${\epsilon }_{s p}$ is the porosity of the soil; $X_f$ is the compressibility of water (=$4\times {10}^{-10}$ 1/Pa); $P_w$ is the pore water pressure; $t$ is time; $T$ is temperature (Kelvin); $S_i$ is the percentage of ice in the pores for a freezing soil, which is defined by the following formula: $S_i=1-{[1-(T-T_0)]}^{-3}$, where $T_0$ is the freezing temperature (=273.15 Kelvin) [22,23]; and $S_w$ is the degree of saturation, which is calculated by the following equation: $S_w={\theta }_r+({\theta }_s-{\theta }_r){\theta }_2(T)$, where ${\theta }_s$ is the volumetric saturated water content; ${\theta }_r$ is the volumetric residual water content; and ${\theta }_2(T)$ is a step function that is equal to 1 for $T>T_0$ and 0 for $T\le T_0$. The density of water vapor, ${\rho }_v$, (Kg/m³) is described by the following equation [20,24]:

$${\rho }_v=H_r\hspace{0.33em}{\rho }_{vs}=\exp ({\displaystyle \frac{hM g}{R T}})\hspace{0.33em}{\rho }_{vs}$$

(2)

where

$${\rho }_{vs}=\exp (31.37-6014.79 T^{ -1}-0.00792 T\hspace{0.17em}) \times 0.001 T^{-1}$$

(3)

is the density of saturated water vapor (Kg/m³); $H_r$ presents the dimensionless relative humidity;$$ represents the pore water pressure head (m); $M$ is the molecular weight of water, which is equal to 0.018 (Kg/mol); and R is the universal gas constant, which is equal to 8.341 (J/mol.K).

The water vapor concentration in air-filled pores is a key parameter in modeling vapor transport in unsaturated soils, especially when applying Fick’s law. In such conditions, the vapor flux $J_v$ is described by Fick’s law as follows [20]:

$$J_v=-D_{eff}\nabla C_{vr}$$

(4)

where $J_v$ is the vapor diffusion flux vector (Kg/m².s); $D_{eff}$ is the effective vapor diffusion coefficient (m²/s); and $C_{vr}$ is the vapor concentration in the air-filled pores (Kg/m³).

The vapor hydraulic conductivities resulting from the water head $K_{vh}$  (m/s) and temperature $K_{vT}$ (m/s) are expressed as follows [21,25]:

$$K_{vh}={\displaystyle \frac{D_v}{{\rho }_w}}\hspace{0.17em}{\rho }_{vs}({\displaystyle \frac{M g}{RT}}\hspace{0.17em}\hspace{0.17em}{H}_r)$$

(5)

and

$$K_{vT}={\displaystyle \frac{D_v}{{\rho }_w}}\hspace{0.17em}\eta \hspace{0.17em}{H}_r({\displaystyle \frac{d{\rho }_{vs}}{dT}})$$

(6)

where $g$ is gravitational acceleration (=9.81 m/s²) and $\eta \hspace{0.17em}$ is an enhancement factor, which is given by the following formula [21]:

$$\eta \hspace{0.17em}=9.5+3{\theta }_w{\theta }_s^{-1}-8.5\hspace{0.17em}(1/\exp \hspace{0.17em}{(1+2.6\hspace{0.17em}{(f_c)}^{-0.5}{\theta }_w{\theta }_s^{-1})}^4)$$

(7)

where $f_c$ is the clay mass fraction in the soil (dimensionless). The vapor diffusivity in the soil, $D_v$, (m²/s) is derived from the water vapor diffusivity of the air, $D_a$, (m²/s) as follows:

$$D_v=\hspace{0.17em}{\theta }_s^{-1}(D_a{\theta }_a^{2.5})$$

(8)

where

$$D_a\hspace{0.17em}=2.12\times {10}^{-5}\times {(T/273.15)}^2$$

(9)

In Equation (8), ${\theta }_a$ is the air-filled porosity [26]. Hence, the vapor flow in the unsaturated soil is presented as follows:

$$q_v=q_{vh}+q_{vT}=-K_{vh}{\displaystyle \frac{dh}{dz}}-K_{vT}{\displaystyle \frac{dT}{dz}}$$

(10)

The average water flow (Darcy flux), through the frozen unsaturated soil, as a consequence of the cryogenic suction and pressure gradient [22,27] is given as follows:

$$u_w=-{\displaystyle \frac{K_h}{{\rho }_{w}g}}({\displaystyle \frac{{\rho }_w{L}_f(T-T_0)}{T_0}}+(\nabla P_w+{\rho }_{w}gZ))$$

(11)

where $L_f$ is the latent heat of water freezing (=333.6 KJ/K) and $K_h$ is the unsaturated hydraulic conductivity (m/s), which is defined as follows [28]:

$$K_h=K_s{(S_e)}^{{\displaystyle \frac{1}{2}}}{\left[1-{(1-S_e^{{\displaystyle \frac{1}{m}}})}^m\right]}^2$$

(12)

$$S_e={\displaystyle \frac{\theta -{\theta }_r}{{\theta }_s-{\theta }_r}}=(1+{\left|\alpha h\right|}^n)-m$$

(13)

where $K_s$ is saturated hydraulic conductivity (m/s); $S_e$ is the degree of effective water saturation and, $\alpha$, $n$ and $m$ are van Genuchten empirical shape parameters ($m$ = 1 − $1/n$).

The governing equation for the thermal field, derived from the principles of energy conservation and the phase-averaged Fourier heat transport laws, is mathematically modeled by considering soil [8,29] as follows:

$$\begin{gathered} C_s{\displaystyle \frac{\partial T}{\partial t}}-L_f({\epsilon }_{sp}{\displaystyle \frac{\partial (S_i{\rho }_i)}{\partial T}}{\displaystyle \frac{\partial T}{\partial t}}+S_i{\rho }_i{\displaystyle \frac{\partial ({\epsilon }_{sp})}{\partial t}})+L_v({\epsilon }_{sp}{\displaystyle \frac{\partial ((1-S_w){\rho }_v)}{\partial T}}{\displaystyle \frac{\partial T}{\partial t}} \\ + \hspace{0.17em}(1-S_w){\rho }_v{\displaystyle \frac{\partial ({\epsilon }_{sp})}{\partial t}})=\nabla \cdot (K_{es}\nabla T)-{\rho }_w{C}_w{u}_w\hspace{0.33em}\nabla T-D_v\nabla {\rho }_v.(C_{v}T+L_v), \end{gathered}$$

(14)

$$C_s=(1-{\epsilon }_{sp}){\rho }_{sp}{C}_{sp}+n\hspace{0.17em}(S_w-S_i){\rho }_w{C}_w+{\epsilon }_{sp}(1-S_w){\rho }_v{C}_v+({\epsilon }_{sp}{S}_i){\rho }_i{C}_i,$$

(15)

and

$$K_{es}=K_{sp}^{ (1-{\epsilon }_{sp})}\hspace{0.33em}+K_w^{ {\epsilon }_{sp}(S_w-S_i)}\hspace{0.33em}+K_v^{ {\epsilon }_{sp}(1-S_w)}\hspace{0.33em}+K_i^{ {\epsilon }_{sp}{S}_i}$$

(16)

where $L_v$ is the latent heat of water vaporization, which is given as a temperature function of $L_{ v}=2.501\times {10}^6- 2369.2\hspace{0.33em}(\mathrm{T}-273.15)$ (J/Kg) [30,31]. $C_s$ (J/Kg. K) is expressed as a volumetric weighted mean of the heat capacities of soil particles, $C_{sp}$, water, $C_w$, vapor, $C_v$, and ice, $C_i$, (J/Kg. K).

According to Equation (16), the soil thermal conductivity, $K_{ es}$ (W/m. K), is computed as a weighted geometric mean by considering the thermal conductivities of soil particles, $K_{sp}$, water, $K_{ w}$, vapor, $K_{ v}$, and ice, $K_i$ (W/m. K).

The stress field is mathematically modeled by assuming that the inertial terms are not significant. Thus, the governing equation for static equilibrium assuming small-strains and a phase-averaged elastic soil medium is given as follows:

$$\nabla .\sigma -[{\rho }_{sp} g (1-{\epsilon }_{sP})+{\rho }_w g {\epsilon }_{sP}(S_w-S_i)+{\rho }_v g {\epsilon }_{sP}(1-S_w)+{\rho }_i g {\epsilon }_{sP}(S_i)]\hspace{0.33em}=0$$

(17)

$$\epsilon ={\displaystyle \frac{1}{2}}(\nabla D+\nabla D^T)$$

(18)

where $\sigma$ is the total stress tensor (N/m²); $D$ is the displacement vector (m); and $\epsilon$ is the strain tensor. The effect of temperature on the strain is captured with the following equation:

$${\epsilon }_{th}=\alpha (T)\hspace{0.33em}\Delta T$$

(19)

where ${\epsilon }_{th}$ is the thermal strain field as a function of the change in temperatures. In Equation (19), $\alpha (T)$ is the thermal expansion coefficient, which is $-1.95\times {10}^{-4}$ (1/K) for fine sand [32] and $1\times {10}^{-8}$ (1/K) for silty clay and sandy clay [8] and $\Delta T$ is the variation in temperature within the model.

The effective stress field is approximated by the classic linear poroelastic relationship described as follows [33,34,35]:

$${\sigma }^{\prime }=C:{\epsilon }_{el}+{\alpha }_{\beta }\hspace{0.33em}[\chi P_w+(1-\chi )P_a]$$

(20)

where ${\alpha }_{\beta }$ is the Biot–Willis coefficient; $C$ is the stiffness matrix of the soil matrix, which is a function of Young’s modules, $E$, and Poisson ratio, $\nu$; ${\epsilon }_{el}$ is the elastic strain; and $P_a$ is atmospheric pressure (=101 KPa). The coefficient of reduced suction $\chi$ is estimated [36] as follows:

$$\chi ={\displaystyle \frac{S_w}{(0.4S_w+0.6)}}$$

(21)

where $S_w$ is the saturation degree. In this study, the relationship between Young’s modules (MPa) and temperature (Kelvin) are given as follows:

$$E=\left(\begin{array}{l}4\times {10}^2{\left|T\right|}^{\hspace{0.17em}0.636}\hspace{1em}\hspace{0.17em}\hspace{0.17em}T\le T_0\\ E_{un}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}T>T_0\end{array}\right)$$

(22)

Here, the study by Zhu and Carbee [37] is adopted. In the above equation, $\hspace{0.17em}{E}_{un}$ is the Young’s modules of the unfrozen soil (MPa).

3. Validation of the THM Theory

Before using the mathematical THM model for prediction purposes, an accurate assessment of its performance in simulating laboratory results is conducted. This model assessment evaluates whether the proposed mathematical THM model provides the required precision when solving the governing system of coupled partial differential equations of frost heave. Generally, one achieves this by comparing the simulated results with experimental measurements obtained in the laboratory or field. The validation used in this study is based on a physical model of unsaturated fine sand subject to freeze–thaw cycles, conducted by Caicedo [14]. This physical model is composed of fine sand subgrade overlain by a 140 mm-thick asphalt layer. Figure 1 shows an FEM model representation of Caicedo’s test set-up. The FEM mesh measures 1 m in length, 0.85 m in height, and 0.1 m in width. Figure 1 also shows a symbolic graphic depiction of the imposed roller mechanical boundary conditions on the bottom and sides of the FEM mesh. The initial pore water pressure was established at −15.7 kPa.

The initial temperature was established at 293.15 K, corresponding to the recorded room temperature in the laboratory. Figure 1 and Figure 2 show the thermal boundary conditions used in the simulations, which were applied using a thermostatic bath and are based on laboratory measurements reported by Caicedo [14]. The hydraulic boundary conditions for the simulations were established based on a laboratory Mariotte bottle [14], which kept the water level near the bottom at 0.15 m. The water retention curve (WRC) for the fine sand utilized in the numerical FEM model is shown in Figure 3 and comes from measured data in the Caicedo tests [14].

Table 1. Parameters utilized in the THM numerical model.

Parameter	Value	Parameter	Value
${\rho }_{sp}$ (Kg/m3)	2646	$\alpha$ (1/m)	0.193
${\rho }_w$ (Kg/m3)	1000	$n$	5.015
${\rho }_i$ (Kg/m3)	917	$K_{s\hspace{0.17em}p}$ (W/m. K)	2.3
${\epsilon }_{sp}$	0.33	$K_w$ (W/m. K)	0.58
${\alpha }_{\beta }$	0.33	$K_v$ (W/m. K)	0.0676
$\hspace{0.17em}{E}_{un}$ (MPa)	59.42	$K_i$ (W/m. K)	2.2
$\nu$	0.45	$C_{ s\hspace{0.17em}p }$ (J/Kg.K)	1192.44
$K_s$ (m/s)	$32\times {10}^{-6}$	$C_w$ (J/Kg.K)	4181.3
${\theta }_s$ (m3/m3)	0.255	$C_i$ (J/Kg.K)	1960
${\theta }_r$ (m3/m3)	0.034	$C_v$ (J/Kg.K)	700

Note. Mechanical, thermal, and hydraulic properties complied from [14,38,39,40,41,42,43].

also presents the hydraulic parameters of the soil from the WRC curve utilized in the THM numerical models, including residual and saturated water contents as well as van Genuchten parameters [25,28]. Other parameters used in the THM numerical model are listed in Table 1.

As shown in Figure 4, the FEM simulated numerical profiles of heave, temperature, and pore water pressure are approximately consistent with the experimental results measured by Caicedo [14]. It follows from this that the previously presented theoretical framework (THM) and resulting numerical analysis are quite indicatory and realistic when evaluating the coupling between heat, water, and vapor in a deformable freezing soil. In Figure 4a, the maximum calculated heave value at the soil surface (the model upper boundary) is slightly over 11 mm, while the measured value is approximately 12 mm during the first freezing cycles. Figure 4b illustrates the temperature curves versus time at depths of 0, 150, 340, and 600 mm along a vertical profile, measured at a horizontal distance of 0.9 m [14]. Figure 4c demonstrates the pore water pressure changes versus time at depths of 140, 340, and 550 mm along a vertical profile at the horizontal distance of 0.93 m [14].

4. Thermal, Hydraulic, and Mechanical Coupling in Unsaturated Soils

Hereafter, the validated THM model is used to analyze water and vapor flux, heat transport, and frost heave in various soil types during the freezing and thawing cycles. To evaluate its performance, three frost-susceptible soils were selected: fine sand, sandy clay, and silty clay, as shown in

Table 2. The hydraulic, thermal, and mechanical properties compiled from [19,44,45].

Parameter		Sandy Clay	Silty Clay
Hydraulic properties	$K_s$ (m/s)	$24\times {10}^{-8}$	$33\times {10}^{-9}$
	${\theta }_s$ (m3/m3)	0.44	0.35
	${\theta }_r$ (m3/m3)	0.01	0.02
	$\alpha$ (1/m)	3.28	2.6
	$n$	1.54	2.3
	$f_c$	0.15	0.2
Thermal properties	$K_{s\hspace{0.17em}p}$ (W/m. K)	2.50	3.1
	$C_{ s\hspace{0.17em}p }$ (J/Kg.K)	840	1020
Mechanical properties	$\hspace{0.17em}{E}_{un}$ (MPa)	5.00	4.25
	$\nu$	0.27	0.27
	${\alpha }_{\beta }$	1	1
	${\epsilon }_{sp}$	0.33	0.33

[19,44,45].

The water retention curves (WRCs) for silty clay and sandy clay, used in the numerical simulations, are shown in Figure 5. These were respectively obtained by measuring the soil matric potentials and liquid water content at the laboratory. The initial and boundary conditions, in these cases, are identical to those utilized in the physical model of the above-mentioned fine sand [14].

Figure 6 displays the simulated heave outcomes for the three types of soil. As exhibited, the amount of heave of the silty clay soil is greater in value over time than that of the others. The largest heave of the silty clay occurs approximately at 13.5 mm, while the maximum value in fine sand is slightly over 11.0 mm. The plausible explanation for this result is that the fine particles in silty clay contribute to high-water retention and strong capillary forces. Consequently, the ensuing suction draws water from deeper layers toward the frozen region (see Figure 8a). This flow movement leads to ice lens formation and substantial frost heave.

Figure 7 shows the simulated heat flux results, for each soil type, at a depth of 140 mm beneath the surface. Silty clay is the most effective at conducting and storing heat, while fine sand has the lowest heat flux due to its large air-filled pores and lower water retention. Sandy clay manifests a similar heat−flux behavior to that of silty clay. The heat flux in the silty clay is mainly governed by heat conduction rather than convection, due to the low vapor flux in silty clay (Figure 8b).

Figure 8 displays the numerical results of water and vapor flux at the elevation of −10 mm (below soil surface) for each soil type. During the first freeze cycle, the largest water flux between the unfrozen and the frozen zones happens in the silty clay, again due to the strong capillary forces. However, the water flow in the find sand and sandy clay is not significant. As shown in Figure 8b, a considerable amount of vapor flow is observed in the fine sand, while a limited amount is discernable in the sandy clay and a negligible amount is present in the silty clay. This occurs because finer soil particles and a higher water content produce a high-water retention, especially in silty clay, and this obstructs vapor flow. This observation demonstrates that the vapor flux is significant in fine sand, thus corroborating the findings of Huang and Rudolph [8] and Zhang et al. [46].

5. Impact of Soil Properties on the Processes of Soil Freezing and Thawing

To evaluate the effect of soil properties on the unsaturated soil behavior during both freezing and thawing cycles, a series of numerical analyses were conducted on the previously studied soil types.

Table 3. Essential information and differences in simulation cases.

Cases	Saturated Hydraulic Conductivity	Soil Particles Thermal Conductivity	Soil Particles Heat Capacity
Fine Sand
FSRC0	$K_s$ = $32\times {10}^{-6}$	$K_{sp}$ = 2.3	$C_{sp}$ = 1192.4
FSC1	$0.1K_s$ = $32\times {10}^{-7}$	$K_{sp}$ = 2.3	$C_{sp}$ = 1192.4
FSC2	$K_s$ = $32\times {10}^{-6}$	$0.1K_{sp}$ = 0.23	$C_{sp}$ = 1192.4
FSC3	$K_s$ = $32\times {10}^{-6}$	$K_{sp}$ = 2.3	$1.5C_{sp}$ = 1788.6
Sandy Clay
SCRC0	$K_s$ = $24\times {10}^{-8}$	$K_{sp}$ = 2.50	$C_{sp}$ = 840
SCC1	$0.1K_s$ = $24\times {10}^{-9}$	$K_{sp}$ = 2.50	$C_{sp}$ = 840
SCC2	$K_s$ = $24\times {10}^{-8}$	$0.1K_{sp}$ = 0.25	$C_{sp}$ = 840
SCC3	$K_s$ = $24\times {10}^{-8}$	$K_{sp}$ = 2.50	$1.5C_{sp}$ = 1260
Silty Clay
SICRC0	$K_s$ = $33\times {10}^{-9}$	$K_{sp}$ = 3.1	$C_{sp}$ = 1020
SICC1	$0.1K_s$ = $33\times {10}^{-10}$	$K_{sp}$ = 3.1	$C_{sp}$ = 1020
SICC2	$K_s$ = $33\times {10}^{-9}$	$0.1K_{sp}$ = 0.31	$C_{sp}$ = 1020
SICC3	$K_s$ = $33\times {10}^{-9}$	$K_{sp}$ = 3.1	$1.5C_{sp}$ = 1530

provides the essential information and differences among the investigated cases. The reference cases (FSRC0, SCRC0, and SICRC0) correspond respectively to the analyses of fine sand, sandy clay, and silty clay of Section 4.

Figure 9 illustrates the simulated vertical stress build-up in fine sand (case FSRC0), sandy clay (case SCRC0), and silty clay (case SICRC0) on the last day of the first freezing cycle. The vertical stress values for the FSRC0 case range from approximately −43.50 kPa to 50 kPa, with the attainment of both tensile (positive) and compressive (negative) stresses. This polarized stress distribution is due to the fine sand’s high permeability, which allows for the rapid water movement and temperature changes. In the SCRC0 case, a moderate vertical stress concentration with a maximum value of around 124 kPa is present in the upper right-hand zone below the asphalt mixture layer. On the other hand, the bottom layer experiences relative low stress concentrations, which indicates a limited frost penetration. A plausible reason is due to the sandy clay’s lower permeability when compared to fine sand. For the SICRC0 case, two distinguishable zones exist. In the upper zone, which lies at depths between 0.25 m and 0.55 m from the surface, the values of vertical stresses range from 140 kPa to 67 kPa. During the freezing phase, this zone is more affected by frost heave and ice lens formation leading to concentrating stresses. In contrast, other zones then experience lower stresses with limited frost penetration.

6. Effect of Saturated Hydraulic Conductivity

Changes in saturated hydraulic conductivity considerably influence fluid movement in frozen soils [8,47]. Therefore, variations in saturated hydraulic conductivity are among the most significant variables affecting unsaturated soil behavior during freezing and thawing cycles. The saturated hydraulic conductivity in the cases of fine sand (FSC1), sandy clay (SCC1), and silty clay (SICC1) is presumed to be 10 times smaller than that of the reference cases (FSRC0, SCRC0, and SICRC0) given in Table 3. Typical FEM simulations are shown in Figure 10. The cases FSC1 and SCC1 show a reduction of 1.5 and 2 mm, respectively in maximum frost heave compared to the FSRC0 and SCRC0 scenarios, while the SICC1 case shows a reduction of 2.5 mm in maximum frost heave compared to the SICRC0 one. Therefore, the effect of the saturated hydraulic conductivity is quite evident in soils with higher water retention (e.g., silty clay).

Figure 11 illustrates the influence of the reduced saturated hydraulic conductivity on the simulated vertical stress build-up in fine sand (FSC1), sandy clay (SCC1), and silty clay (SICC1) for the last day of the first freezing cycle. These numerical results reflect the vertical stress build-up in fine sand (FSC1), sandy clay (SCC1), and silty clay (SICC1) on the last day of freezing of the first freeze cycle. According to Figure 9a and Figure 11a, the average vertical stress value near the soil surface, for the FSRC0 case, is approximately 30 kPa. This value decreases to zero at a depth of 0.6 m. In contrast, for the FSRC1 scenario, the average vertical stress is approximately 28 kPa and tends to be zero at a depth of 0.50 m. As shown in Figure 9b and Figure 11b, the average vertical stress near the surface for the SCRC0 case is approximately 115 kPa in the upper right-hand zone below the asphalt mixture layer. This value is greater than the vertical stress value of 94 kPa for the SCC1 case. The figure shows that this value remains constant up to an average depth of 0.50 m. The average vertical stress for the SICRC0 case is about 125 kPa for the range of depth from 0.25 m to 0.50 m, as illustrated in Figure 9c. As shown in Figure 11c, the vertical stress value for the SICC1 case reduces to 95 kPa and then continuously decreases to 75 kPa at an approximate depth of 0.45. As a result, one deduces that silty clay is the most sensitive to changes in saturated hydraulic conductivity both in terms of frost heave reduction and vertical stress build-up. These effects are less noticeable in fine sand, where simulated results yielded a minimal frost heave and vertical stress variation.

7. Effect of Soil Particles’ Thermal Conductivity

A decrease in the particle thermal conductivity of fine sand, sandy clay, and silty clay influences frost heave and vertical stress build-up. This decrease is triggered by a declining rate of heat transfer through the soil during the freeze−thaw cycle. The soil particle thermal conductivity in the cases of fine sand (FSC2), sandy clay (SCC2), and silty clay (SICC2) is presumed to be 10 times smaller than that of the reference cases (FSRC0, SCRC0, and SICRC0) given in Table 3. Typical FEM simulations are shown in Figure 12. As shown in Figure 12, the FSC2 and SCC2 cases show a 1 mm reduction in the maximum frost heave when compared to the SCRC0 and FSRC0 cases, while the SICC2 case shows a 2 mm reduction in maximum frost heave when compared to the SICRC0 case. Therefore, the effect of reducing the particle thermal conductivity on frost heave is less noticeable in soils with a lower water retention capability (e.g., fine sand). In contrast, soils with a higher water retention potential (e.g., silty clay) remain susceptible to frost heave even with lower particle thermal conductivity. However, one must emphasize that the values of frost heave reduction due to a decrease in particle thermal conductivity are smaller than those calculated for reductions in saturated hydraulic conductivity.

Figure 13 illustrates the influence of the soil particle thermal conductivity on the simulated vertical stress build-up in fine sand (FSC2), sandy clay (SCC2), and silty clay (SICC2) for the last day of the first freezing cycle. As shown in Figure 12a and Figure 13a, reducing the particle thermal conductivity has a slight influence on the vertical stress build-up in fine sands because they typically do not experience substantial frost heave. Therefore, the decrease in the vertical stress build-up is minimal in fine sands across the full depth. According to Figure 9a, the vertical stress close to the soil surface for the FSRC0 case is about 30 kPa, while it is about 23 kPa for the FSC2 case (see Figure 13a).

As shown in Figure 9b and Figure 13b, the average vertical stress build-up near the soil surface for the SCRC0 case is about 115 kPa in the upper right-hand zone below the asphalt mixture layer. Conversely, this value for the SCC2 case decreases to 94 kPa and remains steady up to a depth of 0.60 m. This indicates that an increased thermal conductivity leads to a deeper distribution of vertical stress in sandy clay. As illustrated in Figure 9c and Figure 13c, the average vertical stress for the SICRC0 case is 125 kPa within a depth ranging from 0.25 m to 0.50 m. In comparison, the average vertical stress for the SICC2 case is 95 kPa, which then reduces sharply to about 75 kPa at a depth of 0.35 m. These observations suggest that particle thermal conductivity plays a minor role in influencing unsaturated frozen soil behavior, particularly in soils with low water retention like fine sand. Moreover, sandy clay exhibits the same vertical stress value as silty clay but with a well-defined zone of higher vertical stress.

8. Effect of Soil Particles’ Heat Capacity

Changes in particle heat capacity influence frost heave and vertical stress build-up by affecting how much heat is required to change the temperature of the soil during the freeze−thaw process. An increase in particle heat capacity indirectly influences thermal expansion and frost heave by slowing down temperature changes. To investigate the effect of particle heat capacity, the cases for fine sand (FSC3), sandy clay (SCC3), and silty clay (SICC3) were investigated. These cases are assumed to have a particle heat capacity 1.5 times larger than that of the reference cases (FSRC0, SCRC0, and SICRC0) given in Table 3. Typical FEM simulations are shown in Figure 14. As shown in Figure 14, the FSC3 and SCC3 cases show an approximately 0.5 mm reduction in maximum frost heave when compared to the FSRC0 and SCRC0 cases, respectively. For the SICC3 case, this reduction amounts to 1.5 mm when compared to the SICRC0 case. Consequently, the effect of an increased particle heat capacity on frost heave is negligeable in soils having a low water retention potential (e.g., fine sand). This can be explained by the fact that an increase in the particle heat capacity indicates that the soil requires more thermal energy to produce a temperature change, thus reducing the rate of heat loss during freezing. As a result, a slower rate of freezing results in less frost heave.

Figure 15 shows the influence of an increased particle heat capacity on the simulated vertical stress build-up in fine sand (FSC3 case), sandy clay (SCC3 case), and silty clay (SICC3 case) on the last day of the first freezing cycle. The vertical stress build-up near the soil surface for the FSRC0 case is approximately 30 kPa as shown in Figure 9a. The vertical stress build-up for the FSC3 case reaches an average value of 35 kPa, throughout the soil depth beneath asphalt mixture layer as shown in Figure 15a. This shows that a rise in particle heat capacity results in slightly higher vertical stress in fine sand. In the other soil types, variations in the soil properties result in a reduction. This happens because of the high permeability of fine sand when compared with these other soils. When heat capacity increases, the rate of temperature changes slows down, delaying the freezing process. This delay allows more time for water to move towards the freezing front. Since fine sand has higher permeability than other soils, water can move efficiently through the pore spaces and accumulate in the freezing zone. According to Figure 9b, the average vertical stress build-up close to the soil surface for the SCRC0 case is 115 kPa in the upper right-hand zone below the asphalt mixture layer. In contrast, Figure 15b shows that this value for the SCC3 case reduces to about 94 kPa in this zone, which is similar to the SCC1 and SCC2 cases. Also, for the SCC1 and SCC2 cases, the vertical stress of 94 kPa extends to deeper depths as shown in Figure 11b and Figure 13b. The exception is the SCC3 case, which decreases to 85 kPa at a depth of 0.35 m (see Figure 15b). For silty clay (Figure 9c and Figure 15c), the average vertical stress value for the SICRC0 case is 125 kPa over a depth ranging from 0.25 m to 0.50 m. For the SCC3 case, this value is approximately 95 kPa, with an abrupt change to an approximate value of 75 kPa occurring at a depth of 0.40 m. Therefore, silty clay exhibits a well-defined zone of higher vertical stress compared to the other cases. This stress zone highlights the development of ice lenses in the silty clay, which is attributed to its low permeability and high frost susceptibility. Thus, an increase in the particle heat capacity limits vertical stress build-up for all soil types, except for fine sand below the asphalt mixture layer, by slowing the rate of temperature change during freezing.

9. Conclusions

This study employs a Thermal–Hydraulic–Mechanical (THM) coupled model, using COMSOL Multiphysics Version 6.1. FEM simulations were conducted to analyze water and vapor flux, heat transport, and frost heave during freeze−thaw cycles in unsaturated soils. The model validation using experimental data from [14] confirms its reliability in simulating processes such as frost heave, temperature changes, and pore water pressure under the processes of freezing and thawing cycles. The proposed THM numerical model was employed to simulate frost heave and vertical stress build-up during freezing and thawing cycles in fine sand, sandy clay, and silty clay under unsaturated conditions

The numerical simulations disclosed that silty clay experiences the most significant frost heave and heat flux over time. This heat flux is mostly governed by conduction rather than convection because of silty clay’s low vapor flux. Moreover, silty clay experiences the highest vertical stress build-up than the other soil types due to the more significant increase in its water flux. This is explained by its high-water retention and strong capillary forces that catalyze the flux between frozen and unfrozen zones.

Numerical simulations involving fine sand rendered a low frost heave and minimal vertical stress build-up variation with depth. This is partly due to the slower variation of the heat flux. On the other hand, the vapor flux in the find sand is much more evident than that in the silt clay or sandy clay.

The analysis in this study evaluated the influence of soil properties such as the saturated hydraulic conductivity, the particle thermal conductivity, and the particle heat capacity. Emphasis was placed on their effect in the previously mentioned three soils. This led to the following main conclusions:

• Reductions in frost heave, due to variations in particle thermal conductivity and particle heat capacity, are less significant than variations in saturated hydraulic conductivity.
• Silty clay experiences the greatest reduction in frost heave, particularly due to its low saturated hydraulic conductivity resulting from its high-water retention potential.
• The vertical stress values developed in fine sand include both tensile (positive) and compressive (negative) stresses. This polarized stress distribution results from the high permeability of fine sand, which allows for rapid water movement and temperature fluctuations.
• The highest continuous vertical stress build-up is observed in silty clay and sandy clay due to soil property variations. In contrast, fine sand shows a discontinuous vertical stress build-up.
• An increase in particle heat capacity reduces vertical stress build-up in all soil types, except for fine sand beneath asphalt mixture layer, by slowing the rate of temperature change during freezing.

10. Future Scope and Recommendations

This study examines the frost susceptibility of fine sand, sandy clay, and silty clay under unsaturated conditions. In particular, the effect of the saturated hydraulic conductivity, the particle thermal conductivity, and the particle heat capacity on the process. Consequently, not to deter from the goal of this study, the mechanical model was basic by assuming the soil as isotropic and linearly elastic. These assumptions may not adequately represent the complex, non-linear, and anisotropic behaviors typically observed in real-world geotechnical scenarios. As a result, the ability of the model to completely take in the complex interactions between soil structure, moisture migration, and thermal variations during freeze−thaw cycles may be limited, potentially affecting the simulation outcomes. Future studies should investigate more advanced material models that incorporate non-elastic behaviors and more complex mechanical interactions to more accurately represent real-world freezing and thawing processes. For instance, using viscoelastic or plastic material models could better capture stress−strain relationships during freeze−thaw cycles. Additionally, further validation with experimental data from other soil types would enhance the model’s applicability and accuracy.

Based on the findings, soil types such as silty clay and sandy clay should be used with caution due to their higher frost susceptibility and vertical stress build-up for road and pavement construction in cold regions. Engineering measures such as soil stabilization (e.g., with lime or cement), installation of proper drainage systems, and use of geosynthetics can significantly improve their performance. Fine sand, with its lower frost heave and more favorable stress distribution, may be more suitable in subgrade layers, provided it is compacted properly and protected against erosion. These insights can guide the selection and treatment of subgrade soils in frost-prone regions to enhance the durability and service life of pavement structures.