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Article

Water, Heat, Vapor Migration, and Frost Heaving Mechanism of Unsaturated Silty Clay During a Unidirectional Freezing Process

1
Gansu Hengtong Road & Bridge Engineering Co., Ltd., Lanzhou 730070, China
2
School of Chemistry and Chemical Engineering, Xi’an University of Science and Technology, Xi’an 710054, China
*
Author to whom correspondence should be addressed.
Symmetry 2025, 17(8), 1357; https://doi.org/10.3390/sym17081357
Submission received: 19 May 2025 / Revised: 25 July 2025 / Accepted: 4 August 2025 / Published: 19 August 2025

Abstract

Infrastructure development in permafrost regions continues to face growing challenges from frost heaves and thaw settlement. The traditional frost heave theory considers that soil freezing is caused by the migration of liquid water in the soil; however, existing engineering practice shows that the migration of water vapor during the freezing process cannot be neglected. Based on the hydrothermal–air migration theory of unsaturated soils and their frost heave mechanism, this study established a coupled hydrothermal–air frost heave model for unsaturated silty clay under unidirectional freezing conditions. The computational model was verified through indoor modelling tests. The entire process of water vapor migration, moisture accumulation, and condensation-induced ice formation in unsaturated silty clay was comprehensively reproduced by numerical simulation. The results showed that the moisture field is redistributed during the freezing process of unsaturated soil. The increase in volumetric ice content in the frozen zone is due mainly to the migration of water vapor. Liquid water and water vapor in the unfrozen zone migrate towards the freezing edge driven by the temperature gradient, where they accumulate, leading to a decrease in the unsaturated pore space and a decrease in the equivalent vapor content. This study’s results can provide theoretical support for frost damage prevention in unsaturated silty clay in permafrost regions.

1. Introduction

Frozen soil is a special type of soil (rock) with temperatures at or below 0 °C and containing ice [1,2]. Due to the existence of porous ice in frozen soil, the physical and mechanical properties of the soil are very sensitive to the influence of the external environment and climate. A series of complex physical and mechanical processes occur between frozen soil and structure foundations when the soil is subjected to the loads of the upper structures, leading to frost heaves, thawing, and other disasters [3,4,5].
With the large-scale construction of infrastructure projects in frozen soil regions, stricter control standards have been imposed on the deformation of foundations, airport runways, and highway, road, and railway subgrades. However, some new distress problems have emerged in the operation of related projects. For example, airport runways, railroads, and highway subgrades in cold regions still exhibit contracted road surface cracks, uneven settlement, increased water content of subgrades, and some freeze–thaw disasters after coarse-grained filler replacement [6,7]. Yao et al. [8,9,10] proposed the theory of a “pot cover effect” based on engineering distresses observed in airport runways. They believed that the frost heave of structures with upper cover layers, such as airport runways and high-speed railway subgrades, is caused by the migration of water vapor to the condensation phase below the cover layer, which turns into ice. The Harbin high-speed railway in China is the world’s first high-speed railroad constructed in middle- and high-latitude regions that contain frozen soil. Since the railway’s operation, the monitoring data have shown that there is a general frost heave in the roadbed of approximately 5 mm; local maximum frost heaves even reach 20 mm. Frost heave occurs mainly in Group A and Group B fillers and graded crushed stones [11]. Both those fillers and graded crushed stones belong to coarse-grained soil, with a mass moisture content generally below 5%, and the groundwater level is generally 3 to 5 m lower than the roadbed surface. As to what produces those frost heaves, Sheng et al. [12], Zhang et al. [13,14,15], and Niu et al. [16,17] believe that the migration of water vapor is the main factor. He et al. [11] showed that the amount of frost heave produced is approximately zero when water vapor is not considered, whereas 18.4 mm of frost heave is produced when there is water vapor migration and phase change. Saito et al. [18,19] studied the migration process of water and vapor in the unsaturated zone and proposed a new water–heat–vapor coupling model. The traditional theory of frost heaves posits that they are caused by the migration of liquid water in the soil; however, the Saito study demonstrated that the migration of water vapor during the freezing process of coarse-grained soils cannot be ignored. Foundation/subgrade soils in actual engineering are mainly unsaturated, and the mechanisms and calculation theories of water vapor migration, moisture accumulation, and condensation-induced ice formation in unsaturated soils require further study [20,21,22].
In this study, based on the hydrothermal–air migration theory of unsaturated soils combined with the frost heave mechanism of unsaturated soils, a coupled computational model of hydrothermal–air–frost heave (HTAF) for unsaturated soils under unidirectional freezing was established. That model was then verified through indoor modelling tests. The entire process of water vapor migration, moisture accumulation, and condensation-induced ice formation in unsaturated silty clay was comprehensively reproduced by numerical simulation.

2. Theoretical Model

2.1. Computational Modelling of Water–Heat–Vapor Migration in Unsaturated Soils Under Unidirectional Freezing Action

To construct a calculation model for the water–heat and frost heave of unsaturated soil under unidirectional freezing, the following basic assumptions were made:
  • Soil is a continuous, isotropic, and porous medium.
  • Water migration in unsaturated soil encompasses only the migration of liquid water and water vapor, disregarding ice migration.
  • Unsaturated frozen soil pores have connectivity, and the pressure borne by each point is atmospheric.
  • In the calculation of frost heave deformation, the soil is in a long-term static equilibrium state, and all stress and strain is caused by frost heaving.

2.1.1. Water and Vapor Migration Control Equations

During soil freezing, water, heat, and vapor move and redistribute themselves within the soil. This movement is driven by two main forces: the water potential gradient (or matric suction gradient) and the temperature gradient. These forces cause liquid water and vapor to travel away from the warmer parts of the soil toward the cold freezing front. The migration of liquid water adopts the Richards equation, taking into account both the temperature gradient and water head gradient, while the migration of water vapor is represented by Fick’s law, considering both isothermal and nonisothermal parts [21]. The equation controlling water vapor migration in unsaturated frozen soil driven by the water potential gradient and temperature gradient is [21,22]
θ t = θ l t + θ v t + ρ i ρ l θ i t = K l h ( h + z ) + K l T T + K v h h + K v T T ,
where θ is the total volumetric water content, θl is the liquid water volumetric content, θv is the equivalent water vapor volumetric content, θi is the ice volumetric content, θ = θl + ρiw θi, ρi is the liquid water density, ρw is the ice density, Klh is the isothermal hydraulic conductivity due to the head gradient (m∙s−1), KlT is the nonisothermal hydraulic conductivity due to the temperature gradient (m2∙s−1∙K−1), Kvh is the isothermal vapor conductivity (m∙s−1), and KvT is the nonisothermal vapor conductivity (m2∙s−1∙K−1).
The equivalent water vapor content θv in unsaturated soils can be expressed by [21,22]
θ v = ρ v ρ l n 0 θ l θ i ,
where n0 is the soil porosity.
The soil–water characteristic curve of unsaturated soil was selected using the van Genuchten model, and the relative saturation was used instead of the moisture content. The expression is [23]
S = 1 1 + ( α h ) n m ,
where h is the water head or matric suction head, S is the effective saturation of the soil, S = ( θ l θ r ) / ( θ s θ r ) , θl is the volume content of unfrozen water, θs and θr are the saturated moisture content and residual moisture content of the soil, respectively, α, m, and n are experimental fitting parameters, and the relation between m and n can be simplified as m = 1 − 1/n.
During the freezing process of soil, some liquid water will transform into ice. According to the Clapeyron equation, the relationship between pore water pressure and temperature is established and further introduced into Equation (3). The relationship between unfrozen water content and temperature in unsaturated frozen soil can be expressed as follows [24]:
θ l = θ θ r e ω ( T T 0 ) + θ r ,
where ω is the shape factor (°C−1) and T0 is the freezing temperature of the soil.
The isothermal hydraulic conductivity Klh can be expressed using the Mualem model as [25]
K l h = K s S l 1 1 S 1 / m m 2   ,
where Ks is the saturated permeability coefficient (m∙s−1), S is the relative saturation, and l is the adjustment coefficient, usually taken as 0.5.
During soil freezing, the pore ice in the soil blocks the migration channel of liquid water, so the isothermal hydraulic conductivity Klhʹ in frozen soil can be expressed by [26]
K l h = 10 10 θ i K l h .
The nonisothermal hydraulic conductivity KlT caused by the temperature gradient can be expressed by [25,26,27]
K l T = K l h h G w T 1 γ 0 d γ d T
γ = 75.6 0.1425 T 2.38 × 10 4 T 2 ,
where T is the temperature (°C), GwT is the parameter used to evaluate the influence of temperature on the soil–water characteristic curve, γ is the surface tension of soil moisture (J∙m−2), and γ0 is the surface tension of water, taken as 71.89 g/s2.
The isothermal vapor conductivity Kvh and nonisothermal vapor conductivity KvT can be expressed as [25,26,27]
K v h = D ρ v s M g H r ρ l R T
K v T = D η H r ρ l ρ v s T
D = θ v 7 / 3 θ s 2 θ v D a
ρ v s = exp ( 31.37 6014.79 / T 7.92 × 10 3 T ) 10 3 T
ρ v = ρ v s H r = ρ v s exp ( h M g / ( R T ) )
η = 9.5 + 3 θ l θ s 8.5 exp ( 1 + 2.6 f c ) θ l θ s 3
where ρvs is the saturated vapor density (kg∙m−3), ρv is the water vapor density (kg∙m−3), Hr is the relative humidity, H r = exp ( h M g / ( R T ) ) , M is the molar mass of water (0.018 kg∙mol−1), g is the gravitational acceleration (m∙s−2), R is the vapor constant (8.341 J∙mol−1∙K−1), D is the diffusion degree of water vapor in the soil, Da is the diffusion degree of water vapor in air, Da = 2.12 × 10−5 (T/273.15)2, η is the enhancement factor added to the steam diffusion term, and fc is the mass fraction of clay particles in the soil.

2.1.2. Temperature Control Equations

Considering the migration, evaporation, condensation, and phase change processes of water in unsaturated frozen soil, the governing equation for heat transfer in unsaturated frozen soil can be expressed as [28,29,30]
C T t + L l ρ l θ v t L i ρ i θ i t = ( λ T ) C l ( q l T ) C v ( q v T ) ρ l ( L l q v ) ,
where T is the temperature of the soil (°C), C is the total specific heat capacity of the soil (J∙m−3∙°C−1), λ is the thermal conductivity of the soil (W∙m−1∙°C−1), t is the time (s), ρi is the density of the ice (kg∙m−3), Ll is the latent heat of the evaporation of the water (J∙m−3), Lii is the latent heat of the freezing of the water (334.5 kJ∙kg−1), and Cl and Cv are the specific heat capacities of the liquid and vapor phases respectively (J∙m−3∙°C−1).
Concerning the theory related to porous media, the total heat capacity of permafrost can be expressed as
C = C s θ s + C l θ l + C i θ i + C v θ v ,
where Cs, Cl, Ci, and Cv are the specific heat capacities of soil particles, unfrozen water, ice, and water vapor (J∙m−3∙°C−1), respectively, which are 1.92, 4.18, 2.10, and 6.30 × 10−3 MJ∙m−3∙°C−1; and θs, θl, θi, and θv are the proportions of total volume of the soil, unfrozen water, ice, and water vapor, respectively. Water is a proportion of the total volume of the soil.
The thermal conductivity of frozen soil is
λ = λ s θ s λ w θ w λ v θ v λ i θ i ,
where λs, λw, λv, and λi are the thermal conductivities of soil particles, unfrozen water, water vapor, and ice, respectively (W∙m−1∙°C−1), and θs, θw, θv, and θi are the proportions of the total volume of the soil body accounted for by soil particles, unfrozen water, water vapor, and ice, respectively.
The latent heat of the evaporation of water as a function of temperature can be expressed as
L l = 2.501 × 10 6 2369.2 T .
To facilitate the theoretical modelling, the concept of a solid-phase rate was introduced, defining the solid-phase rate Bi(T) as the ratio of the pore ice content to the unfrozen water content in the soil [30]:
B i ( T ) = θ i θ l = ρ l ρ i T T f b 1 T < T f 0     T T f .
In summary, the water migration control Equation (1) and the temperature field control Equation (9) of unsaturated soil were combined with the ice−water phase transition equilibrium relation Equation (13); that is, the coupled computational model of the water–heat–vapor migration of unsaturated soil was constructed.

2.2. Calculation of Frost Heave Volume Change of Unsaturated Silty Clay

It was assumed that during the freezing process of unsaturated soils, only the effective pores underwent volume change due to freezing. Considering the effect of the soil pore ratio and saturation on freezing deformation, the freezing volume change εv can be calculated by [31]
ε v = η ν ν 1 1 S r e + 1 9 θ i S r e e e + 1
where η is the effective coefficient n = η n =   η e e + 1 , n is the porosity, n is the effective porosity, e is the pore ratio, η ( S r ) = 0                                                                 S r < S r 0           1 ( S r 1 ) q ( S r 0 1 ) q                   0 < S r < S r 0   , Sr0 is the saturation, Sr0 is the initial saturation, q is the test parameter, Sre is the effective saturation (the ratio of the volume of water in the effective pores of the soil body to the volume of the effective pores), S r e = ω u G s e S r η + 1 ω u G s e S r η S r , Gs is the specific gravity of the soil body, ωu is the unfrozen water content of the soil samples at temperature T, ω 0 ω u = T T f b       T < T f , θi is the ice content, and v and v′ are the molar volumes of the vapor before and after cooling, which are approximated using the Redlich–Kwong equation [27]
p = R T v k 2 k 1 T v ( v + k 2 ) ,
where p is the vapor pressure, Pa; R is the vapor constant, usually taken as 8.314 J/(mol∙K); T is the temperature (K); v is the vapor molar volume (m3/mol); and k1 and k2 are the correction factors.
The modulus of elasticity, Poisson’s ratio, cohesion, and angle of internal friction of the soil can be expressed as a function of temperature [28]:
E = a 1 + b 1 ( T 0 T ) 0.6                                           T T 0     a 1                                                                                                   T > T 0
λ = a 2 + b 2 ( T 0 T )                                                     T T 0     a 2                                                                                                       T > T 0
c = a 3 + b 3 ( T 0 T )                                                         T T 0     a 3                                                                                                           T > T 0
φ = a 4 + b 4 ( T 0 T )                                                       T T 0     a 4                                                                                                         T > T 0
where E is the modulus of elasticity, λ is the Poisson’s ratio, C is the cohesion, φ is the angle of internal friction, T is the temperature, and ai, bi are the test parameters.
Based on the coupled computational model of the water–heat–vapor migration of unsaturated soil, combined with Equation (14) for the frost heave change εv under the freeze–thaw cycle, a coupled HTAF model for unsaturated silty clay under unidirectional freezing conditions was constructed. COMSOL Multiphysics 5.6 has a very powerful multi-physics field coupling capability, overcoming the simplification limitations of DEM/FDEM on the continuum scale, and can achieve unified modelling from pore-scale physical mechanisms to engineering-scale behaviours [22]. Generalized partial differential equations from the mathematical module in COMSOL Multiphysics were used to numerically solve the above model. The expression of the coefficient-type partial differential equation in the software is
e a 2 u t 2 + d a u t + c 1 u α 1 u + γ 1 + β 1 u + a 1 u = f ,
where u is the basic variable of the control equation, ea is the mass coefficient, da is the damping coefficient, c1 is the diffusion coefficient, α1 is the conserved flux convection, γ1 is the conserved flux source, β1 is the convection coefficient, a1 is the absorption coefficient, and f is the source term.

3. Model Test and Validation

3.1. Model Test

The TMS9018 (Zhejiang Tomos Technology Co., Ltd., Hangzhou, China) low-temperature environment simulation system was used to conduct unidirectional freezing tests of the unsaturated silty clay. The test apparatus is shown in Figure 1, and a schematic diagram of the equipment structure is shown in Figure 2. Five temperature sensors and three 5TE moisture sensors were arranged along the height direction of the soil column, and one displacement sensor was placed at the top of the soil column. Real-time monitoring of the dynamic changes in temperature, liquid water content, and frost heave displacement within the silty clay during the unidirectional freezing process was conducted using temperature sensors, 5TE moisture sensors, and a displacement sensor. The testing accuracy of the temperature sensors was ±0.01 °C, that of the 5TE moisture sensors was ±1%, and that of the displacement sensors was ±0.01 mm in the model test. After the experiment, the spatial distribution of the total water content (sum of liquid water and ice content) of the silty clay was tested using a drying method.
The initial moisture content of the silty clay was 20%. The test sample was a cylindrical soil column with a diameter of 10 cm and a height of 10 cm, and its dry density was 1.57 g/cm3. The initial temperature of the silty clay was T0 = 1 °C, and the bottom temperature of the soil column was maintained at 1.0 °C. At the beginning of the experiment, the top temperature of the soil column was controlled at −3.0 °C.

3.2. Model Validation

The computational model for unidirectional freezing of the soil column was simplified as a 2D axisymmetric numerical model with a 1/2 section along the centreline of the top and bottom surfaces of the soil column. The model dimensions were 10 cm in height and 5 cm in width, as shown in Figure 3. The initial temperature was T0 = 1 °C, the temperature at the top of the model was controlled at −3.0 °C, the temperature at the bottom was controlled at 1.0 °C, and the left and right ends were zero flux boundaries [15,31]. The initial volumetric water content of the test soil was 0.30. The computational duration was 100 h, and the computational steps were 0.1 h. The hydraulic, thermodynamic, and mechanical parameters used in the computational model are shown in Table 1 [15,24,32,33].
Figure 4 shows the temperature variation curves of various measuring points in the silty clay over time during the freezing process. The figure shows that the temperature curves for each measuring point over time were generally consistent. During the early freezing period, the temperature declined rapidly over the first 10 h. From 10 to 40 h, there was a slow decline in temperature, and from 40 h, the temperature at each measuring point remained basically unchanged into the stabilization stage. As shown in the figure, there is a deviation between the simulated values and the test values at 20–40 h. The main reason is that a large number of parameters are required in the calculation model, and some parameters are directly quoted from the existing literature. However, overall, the degree of agreement between the simulated values and measured values was relatively high, and the variation pattern was basically consistent.
Figure 5 compares the measured and simulated values of the volumetric moisture content of the silty clay after freezing for 100 h. It can be seen that, except for some differences between the simulated and measured values near the upper boundary due to multiple reasons such as the high nonlinearity of the computational model, the simulation does not consider the freezing rate, and there exists ice segregation; the overall numerical simulation more accurately reflected the water migration phenomenon during the unidirectional freezing process of the silty clay. The measured and simulated values of the average volumetric moisture content in the frozen zone were 0.35 and 0.33, respectively, and the measured and simulated values of the average volumetric moisture content in the unfrozen zone were 0.28 and 0.25, respectively. The simulated and measured values in the frozen and unfrozen zones were closely aligned.
Figure 6 shows the variation curve of the frost heave displacement of the soil column over time. The figure shows that the simulated and measured values of the displacement were basically consistent, and the displacement also increased with the increase in freezing duration.
The frost heave growth rate during the entire experimental process varied significantly, with the highest growth rate occurring in the first 20 h, followed by slow growth from 20 to 60 h and gradual stabilization thereafter. The maximum measured frost heave displacement after 100 h of freezing was 1.1 mm, and the relative error between the simulated value and the measured value was 9%. Therefore, numerical simulation can well predict the dynamic development and final frost heave deformation of a silty clay sample under unidirectional freezing.
From the above analysis, it can be seen that the simulated temperature, moisture, and frost heave deformation of the silty clay column under unidirectional freezing were consistent with those of the corresponding experimental results, which verified the rationality of the coupled calculation model established in this paper.

4. Analysis of Simulation Results

The entire process of water–vapor migration, water accumulation, and condensation into ice during the freezing of unsaturated silty clay was calculated and analysed, with the calculation model and parameters being the same as those above.

4.1. Analysis of the Results of the Hydrothermal Field of the Freezing Process

Figure 7 shows the distribution of the temperature field of the silty clay at various times during the freezing, from which it can be seen that the freezing front of the soil gradually pushed downward with time, and the freezing depth of the specimen reached approximately 7.5 cm at 100 h; that meant the soil body with a height greater than 2.5 cm becomes frozen soil. Because the left and right boundaries of the model were adiabatic, there was no horizontal temperature gradient in the soil body, and the temperature remained constant horizontally.
Figure 8 shows the distribution of the volumetric water content (the sum of unfrozen water and ice content) of the silty clay during unidirectional freezing at various times. The figure shows clearly that the moisture field was redistributed during the unidirectional freezing of the soil column. From the distribution cloud diagrams of the volumetric water content of the specimens at 1 h, 10 h, and 100 h, it can be seen that as the freezing process developed, the water content in the frozen zone of the specimens gradually increased, whereas that in the unfrozen zone gradually decreased. That was due to the migration of liquid water and water vapor in the unfrozen soil to the freezing front driven by the temperature gradient, increasing the water content of the frozen zone. In contrast, the bottom of the test soil column was closed without water supply, resulting in a decrease in the water content of the unfrozen zone.
Figure 9 shows the distribution curves of volumetric water content along the height of the soil column for the 1st, 10th, 20th, 50th, and 100th h during the freezing process of the pulverized clay soil. The volumetric water content of the soil in the freezing zone gradually increased as the duration increased from 1 h to 100 h, and the volumetric water content of the soil in the unfrozen zone decreased, so the migration of the water to the freezing zone in the freezing process can be clearly seen.
Figure 10 shows the distribution of ice content in the silty clay at various stages of the freezing process. The figure shows clearly that there was a certain volume of ice content in the frozen zone, and the volume of ice content in the unfrozen zone was zero. During the freezing process, the volume of ice content in the frozen zone increased. The volumes of ice content at the 1st, 10th, and 100th h were 0.244, 0.268, and 0.273, respectively. The main reasons for the increase in the volume of ice content in the frozen zone were as follows: the unfrozen water continued to freeze during the continuous cooling process, and although the pore ice had a blocking effect on the migration of liquid water, the migration of water vapor still existed.
Figure 11 shows the distribution of the equivalent vapor content of silty clay at various times during the freezing process. The figure shows that at the beginning of the test, the temperature at the bottom of the soil column was higher, so its equivalent vapor content was larger. The equivalent vapor content near the freezing front was lower during the freezing process, which was due mainly to the migration of liquid water and vapor from the unfrozen zone to the freezing front, where they accumulated, resulting in a reduction in the unsaturated pore space at that height, and thus the equivalent vapor content at that height was lower. Comparing the equivalent vapor content at various times during the freezing process, it can be seen that the equivalent vapor content at the same height in the unfrozen zone gradually increased, and the equivalent vapor contents in the unfrozen zone at the 1st, 10th, and 100th h were approximately 1.368 × 10−6, 1.392 × 10−6, and 1.625 × 10−6, respectively. The equivalent vapor content in the freezing zone decreased with increasing freezing duration. That phenomenon was due mainly to the migration of liquid water and vapor to the freezing zone under the action of the temperature gradient, resulting in an increase in the space occupied by the vapor in the unfrozen zone. The equivalent vapor content of the unfrozen zone increased, the space occupied by the vapor in the freezing zone decreased, and the equivalent vapor content of the frozen zone decreased. That is, the equivalent vapor content in the unsaturated soil body was affected by the temperature and moisture of the soil body.
Figure 12 shows the distribution curves of the equivalent vapor content along the height of the soil column at various times. It can be clearly seen that at the initial freezing moment of the soil column, the equivalent vapor was more uniformly distributed along the soil column, but as the soil froze, the equivalent vapor content in the freezing zone decreased rapidly, and the equivalent vapor content in the unfrozen zone increased.

4.2. Analysis of Freezing Deformation Results of the Freezing Process

Figure 13 shows the distribution of frost heave displacement during the unidirectional freezing process of the silty clay. The figure shows that the frost heave of the soil column gradually increased with the increase in freezing duration in the initial stage of the experiment. The frost heave displacement in the first hour of freezing was small; the frost heave displacement on the top surface of the soil column was 0.3 mm. After freezing for 10 h, the ice content increased and gradually filled the unsaturated pores, and the frost heave displacement on the top surface of the soil column reached 0.7 mm. After freezing for 100 h, the frost heave displacement on the top surface of the soil column reached 1.2 mm, and the maximum measured frost heave displacement was 1.1 mm. The relative error between the simulated value and the measured value was 9%.

5. Conclusions and Discussion

1. This study established a coupled HTAF model for unsaturated silty clay under unidirectional freezing conditions. The entire process of water vapor migration, moisture accumulation, and condensation-induced ice formation in unsaturated silty clay was comprehensively reproduced by numerical simulation methods. The model can be applied to simulate and analyse the water, vapor, and frost heave deformation of unsaturated soil.
2. During the freezing process of the silty clay, the moisture field underwent significant redistribution, with the moisture content in the frozen zone gradually increasing to an average of 0.35 and the moisture content in the unfrozen zone gradually decreasing to an average of 0.28. As the freezing process progressed, the volume of the ice content within the frozen zone increased due primarily to the migration of water vapor.
3. The equivalent vapor content of the unsaturated silty clay was influenced by both soil temperature and moisture. During the freezing process, liquid water and water vapor in the unfrozen zone migrated towards the freezing front driven by the temperature gradient, where they accumulated. That resulted in a reduction in unsaturated pores, a decrease in equivalent vapor content, an increase in equivalent vapor content in the unfrozen zone, and a decrease in the frozen zone.
4. The growth rate of frost heave was the highest in the first 20 h of the freezing process, and the frost heave slowly increased from 20 to 60 h before gradually stabilizing. The maximum frost depth was 7.5 cm, and the maximum frost heave displacement was 1.1 mm.
5. The migration of water and vapor in soil generally occurs from the underground to the surface or from the surface to the underground, with relatively little lateral migration. Therefore, the one-dimensional model established in this paper can meet the numerical interpretation needs of water and vapor migration and heat transfer processes in unsaturated soil. However, this study has not conducted in-depth research on aspects such as the driving force of water migration and the applicability of soil constitutive models. In future studies, this model still needs to be further expanded to enable its application in the analysis of water–thermal–vapor–force coupling processes in practical engineering.

Author Contributions

Conceptualization, D.L.; original draft preparation, D.L. and H.W.; review and editing, H.W.; funding acquisition, H.W. All authors have read and agreed to the published version of the manuscript.

Funding

This work was supported by the research project of Gansu Hengtong Road & Bridge Engineering Co., Ltd., No. 2024-THTJ3-QT68.

Data Availability Statement

The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.

Conflicts of Interest

Author Dengzhou Li and Hanghang Wang were employed by Gansu Hengtong Road & Bridge Engineering Co., Ltd. The authors declare that this study received funding from the research project of Gansu Hengtong Road & Bridge Engineering Co., Ltd. The funder was not involved in the study design, collection, analysis, interpretation of data, the writing of this article, or the decision to submit it for publication.

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Figure 1. Test equipment.
Figure 1. Test equipment.
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Figure 2. Schematic diagram of the equipment structure.
Figure 2. Schematic diagram of the equipment structure.
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Figure 3. Computation model.
Figure 3. Computation model.
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Figure 4. Temperature variation curves over time (scattered circles in the figure represent experimental values, curves represent simulated values).
Figure 4. Temperature variation curves over time (scattered circles in the figure represent experimental values, curves represent simulated values).
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Figure 5. Variation in volumetric water content along the height of the soil column after freezing for 100 h.
Figure 5. Variation in volumetric water content along the height of the soil column after freezing for 100 h.
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Figure 6. Time variation curve of frost heave displacement of the silty clay soil column.
Figure 6. Time variation curve of frost heave displacement of the silty clay soil column.
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Figure 7. Temperature field distribution of silty clay during the unidirectional freezing process (the dashed arrow represents the freezing front’s progression).
Figure 7. Temperature field distribution of silty clay during the unidirectional freezing process (the dashed arrow represents the freezing front’s progression).
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Figure 8. Distribution of volumetric moisture content during the unidirectional freezing process of silty clay.
Figure 8. Distribution of volumetric moisture content during the unidirectional freezing process of silty clay.
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Figure 9. Variation curves of volumetric water content along the height of the soil column during freezing of silty clay.
Figure 9. Variation curves of volumetric water content along the height of the soil column during freezing of silty clay.
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Figure 10. Distribution of the volume of ice content of silty clay at various times during the freezing process.
Figure 10. Distribution of the volume of ice content of silty clay at various times during the freezing process.
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Figure 11. Distribution of equivalent vapor content during the freezing process of silty clay.
Figure 11. Distribution of equivalent vapor content during the freezing process of silty clay.
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Figure 12. Distribution curves of equivalent vapor content along the height of the soil column at various times.
Figure 12. Distribution curves of equivalent vapor content along the height of the soil column at various times.
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Figure 13. Distribution of frost heave displacement during the unidirectional freezing process of silty clay.
Figure 13. Distribution of frost heave displacement during the unidirectional freezing process of silty clay.
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Table 1. Thermal, hydraulic, and mechanical calculation parameters of soil in the calculation model.
Table 1. Thermal, hydraulic, and mechanical calculation parameters of soil in the calculation model.
ParametersUnitValueParametersUnitValue
Volumetric heat capacity of soil particles CnkJ∙m−3∙K−12160Residual moisture content θrm3∙m−30.05
Ice volume heat capacity CikJ∙m−3∙K−11874Saturated water content θsm3∙m−30.55
Volumetric heat capacity of water ClkJ∙m−3∙K−14180VG model parameters αm−10.149
Volumetric heat capacity of air CvkJ∙m−3∙K−11.21VG model parameters m10.267
Thermal conductivity of soil particles λnW∙m−1∙K−11.2Shape factor ω °C−10.522
Ice thermal conductivity λiW∙m−1∙K−12.22Coefficient of ice–water ratio b10.47
Thermal conductivity of water λlW∙m−1∙K−10.58Clay mass fraction fc10.7
Air thermal conductivity λvW∙m−1∙K−10.025Saturated permeability coefficient Ksm∙s−110−7
Soil particle density ρskg∙m−32700Modulus calculation factor a1MPa28
Ice density ρikg∙m−3917Modulus calculation factor b1126
Density of water ρlkg∙m−31000Poisson’s ratio calculation factor a210.4
Freezing temperature TfK272.85Poisson’s ratio calculation factor b21−0.0080
Latent heat of phase transition of ice and water LfkJ∙kg−1334.5Cohesion calculation factor a3MPa0.150
Gravity acceleration gm∙s−29.8Cohesion calculation factor b310.090
Molar mass of water Mkg∙mol−10.018Calculation factor for angle of internal friction a4122.0
Vapor constant RJ∙mol−1∙K−18.341Calculation factor for angle of internal friction b418.00
25 °C surface tension γ0g∙K−271.89
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Li, D.; Wang, H. Water, Heat, Vapor Migration, and Frost Heaving Mechanism of Unsaturated Silty Clay During a Unidirectional Freezing Process. Symmetry 2025, 17, 1357. https://doi.org/10.3390/sym17081357

AMA Style

Li D, Wang H. Water, Heat, Vapor Migration, and Frost Heaving Mechanism of Unsaturated Silty Clay During a Unidirectional Freezing Process. Symmetry. 2025; 17(8):1357. https://doi.org/10.3390/sym17081357

Chicago/Turabian Style

Li, Dengzhou, and Hanghang Wang. 2025. "Water, Heat, Vapor Migration, and Frost Heaving Mechanism of Unsaturated Silty Clay During a Unidirectional Freezing Process" Symmetry 17, no. 8: 1357. https://doi.org/10.3390/sym17081357

APA Style

Li, D., & Wang, H. (2025). Water, Heat, Vapor Migration, and Frost Heaving Mechanism of Unsaturated Silty Clay During a Unidirectional Freezing Process. Symmetry, 17(8), 1357. https://doi.org/10.3390/sym17081357

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