Numerical Simulation of Frost Heave and Thaw Settlement Characteristics in a Complex Pipe–Soil System in the Seasonally Frozen Ground

Abstract

This paper investigates the frost heave and thaw settlement characteristics of the pipe–soil system during the freeze–thaw cycle, along with the underlying mechanisms. A numerical simulation platform for the complex pipe–soil system was developed using the heat conduction equation, moisture migration equation, and stress–strain equation, all of which account for the ice–water phase change process. The simulations were performed with the coefficient-type partial differential equation (PDE) module in COMSOL Multiphysics. By employing coupled thermal–hydraulic–mechanical (THM) simulation methods, the study analyzed the changes in volumetric water content, volumetric ice content, moisture migration patterns, and temperature field distribution of a water pipeline after three years of service under real engineering conditions in the cold region of northern Xinjiang, China. The study also examined the effects of parameters such as pipeline burial depth, specific heat capacity, thermal conductivity, permeability of saturated soil, and initial saturation on the displacement field. The results show that selecting soil layers with high specific heat capacity (e.g., 1.68 kJ/kg·°C) and materials with high thermal conductivity (e.g., 2.25 W/m·°C) can reduce surface frost heave displacement by up to 40.8% compared to low-conductivity conditions. The maximum freezing depth near the pipeline is limited to 0.87 m due to the thermal buffering effect of water flow. This research provides a scientific reference and theoretical foundation for the design of frost heave resistance in water pipelines in seasonally frozen regions.

1. Introduction

Municipal water supply pipelines constitute vital urban infrastructure whose stability directly impacts socioeconomic development and quality of life. In frozen regions, pipeline integrity faces critical challenges from frost heave [1,2,3,4,5,6] and thaw settlement [7,8,9,10] induced by temperature variations. These phenomena generate substantial frost heave stresses through soil moisture phase change, potentially causing pipeline deformation, joint failure, or rupture [11]. The axial stress in the pipeline caused by frost heave can be approximated using the bending stress analysis method for beams [12]. By combining the classical fluid dynamics model with basic thermoelastic theory and using soil–water characteristic curves and the solid–liquid ratio as basic equations, the frost heave mechanism in high-speed railway subgrades can be effectively analyzed [13]. The study results show that subgrade deformation is significantly affected by changes in air temperature and humidity distribution. Thus, controlling the thermal-water migration characteristics in the subgrade is an effective way to reduce frost heave deformation and uneven deformation. The Drucker–Prager elastoplastic model [14] can be used to simulate pipeline damage caused by frost heave at high-pressure regulating stations in river areas. By installing heaters to increase the intake temperature, frost heave in natural gas pipelines can be effectively prevented. Bekele et al. [15] proposed a numerical model based on isogeometric analysis (IGA) to simulate the thermo–hydro–mechanical (THM) coupling process during ground freezing.

In addition, the phenomenon of thaw settlement should not be overlooked. As the temperature increases, frozen regions gradually thaw, causing significant changes in the physical properties of the soil, which leads to a decrease in its bearing capacity. During thaw settlement, the heterogeneity and anisotropy of the soil also significantly influence the mechanical behavior of the pipeline. This heterogeneity can be quantified by factors such as the soil’s layered structure, particle composition, and moisture content. For example, clayey soils and sandy soils respond differently to changes in moisture content; the former may exhibit greater compressibility and settlement after thawing. Additionally, the thaw settlement process is accompanied by changes in the soil’s internal friction angle and cohesion, which further affect the stability of the pipeline. Wei et al. [16] investigated fault slips caused by permafrost thawing, as well as the longitudinal strain resulting from changes in fluid temperature and soil permeability during thawing. Thermal siphon pipes [17,18] can be used to study the effects of hydrothermal coupling on the soil temperature field in frozen ground. In engineering practice, researchers often simplify coupling models based on specific conditions and apply efficient algorithms to solve them, taking into account specific boundary conditions and initial states.

During the freeze–thaw cycle, the mechanical behavior of frozen ground exhibits complex nonlinear characteristics. Dayarathne et al. [19] studied two key factors affecting the overall vertical displacement of pipelines: frost heave caused by the leading frost layer and settlement due to thawing within the thawing ball and moisture migration. The low permeability of the leading frost layer can form a water flow barrier, inhibiting pore water pressure generated during thawing. In practical engineering, for silty clay and silt, appropriate measures can be taken based on moisture content, while for sandy soils and certain silts, adjustments can be made based on clay content to effectively prevent frost heave and thaw settlement [20]. Other frost-related mechanisms include slope instability, river ice scour, and pipeline buoyancy.

Recent advances in pipeline integrity research highlight critical strategies for addressing frost-related challenges. Ref. [21] employs homogenization and finite element methods to quantify how spiral-wound composite coatings mitigate crack-tip stress intensity factors under pressure, offering insights for frost-resistant coating design. Ref. [22] develops a contact mechanics model to assess additional stresses from mobile lift chains during pipeline repairs, emphasizing safety thresholds relevant to frost-affected maintenance. Ref. [23] establishes an analytical framework for pipe-support interactions in mountainous terrains, linking soil properties and mounting gaps to hoop stress variations—a key consideration for frost-heave-induced displacements. These studies collectively inform multi-physical modeling of frost-jacketing and thaw-settlement mechanisms in water pipelines, yet gaps remain in quantifying ice–water phase effects on long-term deformation. The interaction between soil and underground structures [24,25] is a highly complex problem. Due to its mathematical complexity, obtaining analytical solutions for such problems is extremely challenging; consequently, most scholars adopt numerical methods for analysis.

In existing research, although scholars have explored the frost heave and thaw settlement characteristics of the pipe–soil system during freeze–thaw cycles, most studies tend to overlook the complexity of real engineering conditions, especially regarding multilayer soil structures. The heterogeneity of complex soil layers exacerbates the impact of environmental factors on pipeline behavior, making traditional design methods insufficient in such cases. The interaction between pipelines and the surrounding seasonal frozen ground requires detailed numerical simulations for analysis. However, current models often fail to comprehensively consider the effects of the ice–water phase transition on heat conduction, moisture migration, and soil stress. To address these shortcomings, this study constructs a more precise numerical simulation platform using COMSOL Multiphysics 6.2, taking into account pipeline burial depth, radius, soil properties, and moisture migration. Through THM coupling simulations, the dynamic changes of the pipe–soil system in real engineering applications are analyzed in depth. This innovative study clarifies the deformation mechanisms caused by frost heave and thaw settlement, contributing to a better understanding of stress concentration and deformation in pipelines in seasonally frozen regions. It not only fills the gap in existing research but also provides scientific foundations and practical guidance for the design and management of frost heave resistance in water pipelines in frozen soil regions.

2. Establishment of the Seasonally Frozen Ground-Pipeline System Model

2.1. Overview of the Study Area

Xinjiang is the largest agricultural province in China, where agricultural water use accounts for over 90% of total water consumption, reaching as high as 95% in the southern regions. Due to the issue of low agricultural water efficiency, there is an urgent need to develop water-saving technologies to address the conflict between water resource supply and demand and to promote efficient water use. Currently, irrigation in Xinjiang primarily relies on canal systems. However, during long-distance water transport, significant evaporation and leakage occur, leading to resource wastage, rising groundwater levels, and soil salinization. Moreover, the design of canals is constrained by strict topographic requirements, increasing construction costs and limiting affordable irrigation options in certain areas, thus hindering agricultural development. Research has shown that converting the entire canal system in Xinjiang to a pipeline water transport system could save approximately 9.488 billion tons of water. Therefore, the development of a pipeline water transport system is crucial for improving agricultural water use efficiency. However, the safety of long-distance pipeline transport is significantly influenced by soil and geological conditions, particularly in frozen soil regions. The effects of freeze–thaw cycles on pipelines primarily manifest as frost heave and thaw settlement, which can damage, fail, or cause leaks in pipelines, posing a serious threat to the safety of the water supply system. In the cold winter climate of Xinjiang, especially on the northern slopes of the Tianshan Mountains, pipelines traversing seasonal frozen soil regions face freeze–thaw damage as a major challenge to safe operation. This study focuses on the typical soil structure of the cold region in northern Xinjiang, where the soil consists of mixed fill, silty clay, and gravelly sandy soil, with the water transport pipeline buried in the silty clay layer. A structural schematic diagram is shown in Figure 1.

Frozen soil is a porous and heterogeneous medium, and its thermophysical properties are highly sensitive to moisture content and temperature variations. The interaction between pipes and soil during the freeze–thaw process in frozen soil is a THM coupling issue, where the coupling relationships between various physical fields are illustrated in Figure 2. Hydrothermal coupling primarily involves the influence of water state changes (such as freezing and thawing) on heat conduction and the temperature field. During freezing, water affects the soil’s thermal conductivity and thermal capacity. The formation and thawing of ice alter the soil’s pore structure, further influencing its thermal conductivity. Thermal coupling is reflected in how temperature changes affect the mechanical properties and stress state of the soil. Variations in temperature can lead to changes in pore water pressure, which directly impact the soil’s physical properties, such as shear strength and compressibility. Elevated temperatures may cause plastic deformation in the soil, while lower temperatures can result in brittle failure. Hydraulic coupling focuses on the influence of water flow on the mechanical behavior of the soil. In frozen regions, water flow is suppressed, but in thawed areas, water can flow freely. This results in changes in pore water pressure, which in turn affect the effective stress in the soil. It is evident that temperature field variations lead to phase transitions between water and ice in frozen soil, facilitating moisture migration. Meanwhile, the latent heat generated during phase changes also regulates the temperature field of the soil. The deformation caused by the water–ice phase transition leads to changes in soil parameters and stress–strain relationships, while changes in stress and strain further influence the soil’s moisture and temperature fields. This study builds on the hydrothermal coupling model proposed by Bai Qingbo [26] and employs the coefficient-type partial differential equation (PDE) module in COMSOL Multiphysics for secondary development. The goal is to establish a model for hydrothermal–mechanical frost heave and thaw settlement in the pipe–soil system.

The unique geocryological characteristics of the study area, as described above, necessitate special considerations for the subsequent multiphysics modeling. This leads us to establish the following fundamental framework of assumptions and governing equations.

2.2. Basic Assumptions and Fundamental Governing Equations of the Physical Fields

2.2.1. Basic Assumptions

In conjunction with engineering requirements, the following assumptions are made for the mathematical model of the frost heave and thaw settlement in pipe–soil systems:

(1)

The soil is assumed to be an isotropic continuous medium. While natural soils may exhibit anisotropy due to layering or particle orientation, the isotropic assumption simplifies the multiphysics coupling by reducing computational complexity. This approach is widely adopted in frozen soil mechanics studies and aligns with the focus on macroscopic thermal–hydraulic–mechanical interactions rather than microscale heterogeneity.

(2)

The soil is assumed to undergo small deformations, with strains within the linear elastic range. The assumption of small strains (linear elasticity) is valid for frost heave and thaw settlement in seasonally frozen regions, where cumulative displacements are typically within millimeter-to-centimeter scales. Large plastic deformations or soil failure were not observed in field measurements, supporting this simplification.

(3)

Soil particles and ice are assumed to be incompressible. Volume changes in frozen soil primarily arise from ice–water phase transitions and moisture migration, not particle compressibility. This simplification avoids unnecessary complexity in stress–strain calculations while preserving the dominant deformation mechanisms.

(4)

The influence of moisture evaporation is not considered in the processes of water–ice phase change and moisture migration. Evaporation effects are minimal in frozen soil due to low temperatures and suppressed vapor diffusion. The study focuses on phase change-driven moisture migration, which dominates frost heave and thaw settlement.

Building upon these foundational assumptions, we now derive the governing equations for the thermal, hydrological, and mechanical fields to fully characterize the pipe-frozen soil interaction system.

2.2.2. Temperature Field Governing Equations

Due to seasonal climate changes, frozen soil undergoes freezing and thawing each year. At the same time, the buried water supply pipelines continuously release heat to the surrounding soil, leading to the thawing of the frozen soil. The soil parameters also change with the thawing and freezing of the frozen soil. Based on the fundamental theories of heat transfer and frozen soil mechanics, a heat transfer equation that considers the latent heat of phase change is established:

$$\rho C(\theta )\partial T/\partial t=\lambda (\theta ){\nabla }^{2}T+L{\rho }_I\partial {\theta }_I/\partial t$$

(1)

where

$${\displaystyle \frac{\partial {\theta }_I}{\partial t}}=({\theta }_s-{\theta }_r)\cdot \left({\displaystyle \frac{\partial B(T)}{\partial t}}\cdot S+B(T)\cdot {\displaystyle \frac{\partial S}{\partial t}}\right)+{\theta }_r{\displaystyle \frac{\partial B(T)}{\partial t}}$$

(2)

in which $\lambda$ is the thermal conductivity (W/cm·°C); C is the heat capacity (J/kg·°C); $\rho$ is the density of the soil (kg/m³); ${\rho }_I$ is the density of ice (kg/m³); T is the temperature of the soil (°C); t is time (s); L is the latent heat of phase change, typically taken as 334.56 kJ/kg; B is the solid–liquid ratio; S is the saturation degree; and ${\theta }_I$ is the volumetric ice content.

When studying hydrothermal coupling, it is essential to consider not only the pore water in the soil but also the water content in the analysis of soil moisture. To calculate the thermal conductivity and specific heat capacity of the soil, the volumetric water content is defined as follows:

$$\theta ={\theta }_u+{\rho }_I/{\rho }_w\cdot {\theta }_I$$

(3)

where ${\theta }_u$ is the volumetric unfrozen water content, and ${\rho }_w$ is the density of water.

The heat conduction equation for the water supply pipeline made of steel is expressed as follows:

$${\rho }_s{C}_s\partial T/\partial t={\lambda }_s{\nabla }^{2}T$$

(4)

In the equation, ${\rho }_s$ is the density of steel; $C_s$ is the heat capacity of concrete; and ${\lambda }_s$ is the thermal conductivity of steel.

2.2.3. Moisture Field Governing Equations

The distribution of the moisture field in frozen ground is closely related to moisture migration, water content, porosity, concentration, soil-water potential, and temperature distribution. Moisture migration is the controlling process of the moisture field distribution and is the fundamental cause of frost heave and thaw settlement. Assuming that the moisture migration law in frozen ground is similar to that in thawed soil, this paper adopts the Richards equation, which considers the water–ice phase change as the differential equation governing moisture migration:

$${\displaystyle \frac{\partial {\theta }_u}{\partial t}}+{\displaystyle \frac{{\rho }_I}{{\rho }_w}}{\displaystyle \frac{\partial {\theta }_I}{\partial t}}=\nabla \left[D\left({\theta }_u\right)\nabla {\theta }_u+k\left({\theta }_u\right)\right]$$

(5)

In Equation (5), D(θ_u) is the diffusion rate of water (m²/s), calculated using the formula $D({\theta }_u)=k({\theta }_u)/c({\theta }_u)\cdot I$; I represents the impedance factor $I={10}^{-10{\theta }_I}$; k(θ_u) is the permeability coefficient of the soil (m/s), indicating the ease with which fluid migrates through the pore framework.

During the freezing process of water, moisture migration is hindered by ice, and the permeability coefficient changes over time, as well as with the temperature of the frozen soil, ice content, and liquid water content. The expression is

$$k\left({\theta }_u\right)=k_s\cdot S^l{\left(1-{\left(1-S^{1/m}\right)}^m\right)}^2$$

(6)

k_s is the permeability coefficient of saturated soil; c(θ_u) is the specific water capacity (1/m), and the calculation method is as follows:

$$c({\theta }_u)=a_{0}m/(1-m)\cdot ({\theta }_s-{\theta }_r)\cdot S^{1/m}{(1-S^{1/m})}^m$$

(7)

where a₀, m, and l is a constitutive parameter that varies with soil type; S is the relative saturation of the frozen soil:

$$S=({\theta }_u-{\theta }_r)/({\theta }_s-{\theta }_r)$$

(8)

in which θ_r is the residual moisture content, θ_s is the saturated moisture content. In the calculations, S is chosen as the independent variable for hydrothermal coupling solutions.

In the heat conduction equation and the moisture migration equation, it is necessary to introduce the solid–liquid ratio B(T) as a coupling term to link the hydrothermal equations. This ratio represents the volume ratio of ice to unfrozen water in the soil, and the calculation formula is:

$$B(T)={\displaystyle \frac{{\theta }_I}{{\theta }_u}}=\left\{\begin{array}{ll}1.1{\left(T/T_f\right)}^B-1.1& T<T_f\\ 0& T\ge T_f\end{array}\right.$$

(9)

where T_f represents the freezing temperature of the soil (°C); B is a constant that varies with soil type and salinity.

2.2.4. Stress Field Control Equation

Considering that the volume deformation of frozen soil is primarily caused by moisture migration, phase change between ice and water, and transient strain, the fundamental equation of the stress field is

$$-\nabla \cdot \sigma =F,\hspace{1em}\epsilon =\nabla u,\hspace{1em}\left\{\sigma \right\}=\left[c\right]\left(\left\{\epsilon \right\}-\left\{{\epsilon }_0\right\}\right)$$

(10)

In the calculation of strains in frozen soil, it is necessary to consider the transient strain ${\epsilon }^e$, as well as the soil strain ${\epsilon }_v$ caused by moisture phase changes and migration:

$$\epsilon ={\epsilon }^e+{\epsilon }_v$$

(11)

The soil strain caused by moisture phase changes and migration is

$${\epsilon }_v=0.09({\theta }_0+\Delta \theta -{\theta }_u)+\Delta \theta +({\theta }_0-n)$$

(12)

With the complete system of governing equations established, we proceed to implement these theoretical formulations through conceptual model design.

2.3. Conceptual Model Design and Simulation Parameters

2.3.1. Conceptual Model Design

This study focuses on the seasonally frozen soil layer at a specific site in the northern cold region of Xinjiang, China. The soil layers, from top to bottom, consist of 0.5 m of mixed fill, 1.5 m of silty clay, and 2 m of gravelly sand. A water supply pipeline is planned to be buried within these soil layers, with a wall thickness of 15 mm made of steel and a burial depth of d. Through simulation trials, it was found that the influence range of the pipeline on surface freeze–thaw displacement does not exceed 4 m; therefore, the model width for this study is set at 4 m. The structural cross-sectional view is shown in Figure 3.

2.3.2. Model Parameters

Considering the multiphysical field coupling process of frozen soil, the mechanical parameters of the frozen soil (such as elastic modulus, Poisson’s ratio, and strength parameters) are set as functions of the soil temperature T in the following form:

$$\begin{array}{ll}E=a_1+b_1{\left|T\right|}^m,\hspace{1em}& \nu =a_2+b_2\left|T\right|\\ C=a_3+b_3\left|T\right|,\hspace{1em}& \phi =a_4+b_4\left|T\right|\end{array}$$

(13)

In the equation, both $a_i$ and $b_i$ are experimental constants. The values of other physical and mechanical parameters in the model are shown in

Table 1. Material parameters.

Materials	Mixed Fill Soil	Silty Clay	Gravelly Sand	Steel	Water	Ice
$\rho (\mathrm{k}\mathrm{g}/{\mathrm{m}}^3)$	1820	1800	1700	7800	1000	918
$\lambda (\mathrm{W}/(\mathrm{m}\cdot °\mathrm{C}))$	1.474	1.125	1.38	66.6	0.63	2.31
$C(\mathrm{k}\mathrm{J}/(\mathrm{k}\mathrm{g}\cdot °\mathrm{C}))$	0.92	0.84	0.89	0.460	4.2	2.1
$T_f(°\mathrm{C})$	−0.16	−0.54	−0.18	/	/	/
B	0.61	0.56	0.65	/	/	/
$a({\mathrm{kPa}}^{-1})$	0.05	2.59	0.66	/	/	/
m	0.5	0.22	0.14	/	/	/
l	0.5	0.5	0.5	/	/	/
${\theta }_s$	0.5	0.5	0.42	/	/	/
${\theta }_f$	0.231	0.02	0.05	/	/	/
$k_s(\mathrm{m}/\mathrm{s})$	10−5	10−8	9.62 × 10−5	/	/	/
$a_1$	61	28	28	/	/	/
$b_1$	53	26	26	/	/	/
$a_2$	0.35	0.40	0.38	/	/	/
$b_2$	−0.007	−0.008	−0.006	/	/	/
$a_3$	0.03	0.15	0.018	/	/	/
$b_3$	0.094	0.090	0.006	/	/	/
$a_4$	23	22	31	/	/	/
$b_4$	9.5	8.0	2.0	/	/	/

.

Complementing the material parameters, the specification of boundary and initial conditions plays an equally crucial role in simulation outcomes.

2.3.3. Boundary Conditions and Initial Conditions

(1)

Temperature field

The boundaries on both sides of the model are considered adiabatic, with no heat exchange with the external environment. The surface is influenced by the atmospheric temperature. The temperature load is based on the actual annual average temperature changes in the cold region of northern Xinjiang, China, and is approximated by a sine curve for cyclic loading. The function Tu(t) is as follows:

$$Tu(t)=7.8+22\sin (2\pi /365t)$$

(14)

Considering a freeze–thaw cycle lasting 3 years, the surface temperature load curve is shown in Figure 4.

(2)

Moisture Field

Since this study does not take into account the effects of moisture evaporation during the phase change process between water and ice and only focuses on the initial moisture content, the moisture field boundary is set to no flux. The initial saturation S₀ of each soil layer is calculated using Formulas (8) and (9).

(3)

Displacement Field

The bottom boundary of the model is fixed, the side boundaries are supported by rollers, and the surface is a free boundary.

2.3.4. Model Grid Division and Research Plan

The numerical results are based on the PDE module in COMSOL Multiphysics, utilizing a custom mathematical model for the freeze–thaw settlement of the pipe–soil system to conduct transient analysis of hydrothermal multiphysical coupling. To ensure solution convergence and stability, a mesh sensitivity analysis was performed. Three mesh configurations were tested: coarse (maximum element size = 1.0 m, minimum = 0.001 m), medium (maximum = 0.85 m, minimum = 0.0002 m), and fine (maximum = 0.5 m, minimum = 0.0001 m). The maximum frost depth and surface displacement results varied by less than 4% between medium and fine meshes, confirming grid independence. The final model adopted the medium mesh (45,048 domain elements, 966 boundary elements) to balance computational efficiency and accuracy. The model uses a free triangular mesh for discretization, as shown in Figure 5. Local refinement is applied at the interface between the pipeline and the soil layers, with a predefined maximum element growth rate of 1.2 and a curvature factor of 0.25.

This study first simulated the temperature, volumetric water content, volumetric ice content, and displacement variations with heating and cooling for two lines: one from the top of the pipeline to the ground surface (line 1) and the other from the top of the pipeline at the region’s right boundary to the ground surface at the same elevation (line 2). A schematic diagram is shown in Figure 3. The study deeply analyzes the moisture migration behavior in the pipe–soil system under HTM (hydro–thermal–mechanical) coupling conditions and elaborates on the precise mechanism of frost heave and thaw settlement of the water pipeline.

Next, a sensitivity analysis was conducted on the geometric and physical parameters of the model. Simulations were performed to evaluate the effects of pipeline burial depth, initial saturation, pipeline radius, specific heat capacity of the soil around the pipeline, thermal conductivity, and permeability of saturated soil on the frost heave and thaw settlement displacements of the pipeline and the surface. The parameter values vary according to

Table 2. Variation of parameter values.

Parameters	d (m)	R (m)	S0	${{\mathit{C}}_{\mathit{s}}}_2$ (kJ/kg·°C)	${\mathit{\lambda }}_2$ (W/(m·°C))	${\mathit{k}}_{\mathit{s}2}$ (m/s)
Case 1	1.0	0.4	0.2	0.42	0.5625	1 × 10−8
Case 2	1.5	0.5	0.4	1.125	1.125	5 × 10−8
Case 3	2.0	0.6	0.6	1.68	2.25	1 × 10−7

Nomenclature: d—burial depth of pipeline; R—pipeline radius; S₀—initial saturation; C_s2—heat capacity of layer2; ${\lambda }_2$—thermal conductivity of layer2; $k_{s2}$—permeability coefficient of layer2.

.

In the Section 4, four observation points were selected: located at the upper side, the right side, the lower side of the pipeline, and the farthest right side of the surface interface. A schematic diagram is shown in Figure 3.

3. Simulation Result Analysis

3.1. Accuracy Validation

To validate the correctness of the proposed model in this study, field measurements of the maximum frost depth in the soil layer above the pipeline were conducted during the first freeze–thaw cycle, and the results were compared with numerical simulations, as shown in Figure 6. A slight discrepancy between the simulated and measured data is observed, which arises from the simplified temperature load and some assumptions applied in the simulation—an approximate curve fitted based on annual average air temperature—deviating from actual temperature variations. However, the error between the simulated and measured data remains within 20%, and the trends of the two curves are generally consistent. Therefore, the methodology presented in this study is deemed applicable for practical engineering analyses.

3.2. Temperature Field Evolution Analysis of the Pipe–Soil System During the Freeze–Thaw Process

Numerical simulations were conducted using the above parameters, and the hydro–thermal–mechanical coupling simulation results at the soil locations of line1 and line2 were analyzed. Figure 7 shows the temperature field evolution over time, with the vertical axis representing soil depth and the horizontal axis representing time. From the simulation results, it can be seen that due to the periodic variation of the surface temperature, the soil temperature also exhibits seasonal fluctuations. During a single freeze–thaw cycle, the lowest and highest temperatures occur around days 90 and 270, respectively. These points correspond to the extreme winter and summer temperatures, with heat conduction effects causing heat transfer within the soil layers, resulting in a delayed response to surface temperature changes.

Additionally, the variation of the freeze–thaw interface is directly influenced by the thermal properties and phase change characteristics of the soil layers. The thickened contour lines in the figure represent the freeze–thaw interface. In the silty clay layer, the pipeline water temperature is 10 °C, providing a localized heat source. Heat is transferred from the pipeline to the surrounding soil, significantly reducing the freezing depth of the soil above the pipeline, with the maximum freeze–thaw depth reaching only 0.87 m. In contrast, the right boundary of the model, which is farther from the heat source and lacks additional heat, experiences a maximum freeze–thaw depth exceeding the pipeline burial depth. This figure demonstrates the thermal buffering effect of the pipeline, which reduces frost penetration and associated heave forces. The delayed temperature response in deeper soil layers underscores the importance of burial depth in mitigating frost heave risks.

3.3. Analysis of the Temporal Evolution of Volumetric Water Content and Volumetric Ice Content During the Freeze–Thaw Process of the Pipe–Soil System

Figure 8 and Figure 9 show the cloud maps of the temporal variation of volumetric water content and volumetric ice content, respectively. The simulation results indicate that the volumetric water content and volumetric ice content of gravelly fill soil exhibit little variation with temperature during the freeze–thaw cycle due to its coarse particles, large pores, and poor thermal conductivity. In contrast, in silty clay, water undergoes phase change at low temperatures, turning into ice, which leads to a decrease in water content and an increase in ice content. This is related to the fine particles, large specific surface area, and significant capillary action of silty clay. This suggests that during the freeze–thaw cycle, the changes in volumetric water content primarily occur in the clay layer, with a smaller impact in the gravelly fill layer with larger porosity. Since the variation in volumetric water content directly affects the changes in the mechanical properties of the soil, further analysis is needed on the deformation of the silty clay layer and its interaction with the pipeline.

3.4. Analysis of the Temporal Evolution of Displacement During the Freeze–Thaw Process of the Pipe–Soil System

Figure 10 shows the cloud map of displacement variation with time. It can be observed that frost heave displacement is closely related to heat transfer. The maximum frost heave displacement of the soil above the pipeline is only 7.6 mm, which is about 40.8% less than the displacement of the soil at the right boundary (10.7 mm). This is because the heat from the water inside the pipeline alleviates the freezing effect on the surrounding soil through conduction, thereby reducing the frost heave displacement. This indicates that the heat source from the pipeline has a significant impact on the frost heave displacement of the soil, and the constant temperature state of the water inside the pipeline plays a positive role in reducing the frost heave of the surrounding soil.

For 3-year cycles, the simulation results show that after each freeze–thaw cycle, a portion of irreversible residual displacement occurs in the soil (as shown in Figure 11). The accumulation of this residual displacement indicates that the soil structure is gradually compacted and rearranged. The main reason for this is the migration of water and the expansion effect caused by freezing within the soil pores. During each freezing stage, pore water migrates to the colder regions, causing the soil to expand; after thawing, some of the water flows back, but the soil structure cannot fully recover. While this study primarily focuses on short-term freeze–thaw dynamics (3-year cycles), the potential for cumulative residual deformation under prolonged cycles is acknowledged. Numerical simulations reveal a ratchet effect characterized by progressive soil restructuring, wherein each freeze–thaw cycle induces an incremental vertical displacement of 0.4–0.6 mm/year. The observed displacement reduction proximal to the pipeline (40.8% attenuation relative to distal zones) corroborates its thermal buffering capacity through conductive heat transfer. However, the persistence of residual displacement underscores the irreversible nature of frost-induced soil fabric reorganization, necessitating long-term monitoring strategies to mitigate structural risk accumulation.

Furthermore, the results show that the frost heave displacement at the top of the pipeline is 2.12 mm greater than at the bottom of the pipeline, indicating uneven deformation of the pipeline during the frost heave process. The top of the pipeline is closer to the surface, where the cooling effect is stronger, resulting in more heat loss, while the bottom is limited by the freezing depth, which reduces the frost heave force. Additionally, the soil at the pipeline bottom has a higher bearing capacity, which helps mitigate the bottom deformation caused by freezing.

To reduce such uneven deformation, the materials surrounding the pipeline can be optimized. For example, a frost heave prevention cushion layer (such as gravel or coarse-grained soil) can be placed around the pipeline to reduce the generation of frost heave forces. Alternatively, the thermal conductivity of the surrounding soil can be improved to rapidly release heat and reduce the unevenness of the freezing depth.

Figure 12 illustrates the displacement variation of the ground surface and the top of the pipeline during the frost heave and thawing stages. The displacement behavior of the ground surface and the pipeline top during the freeze–thaw cycle follows completely different patterns. As frost heave progresses, the displacement of the ground surface gradually increases, while the displacement of the pipeline top is significantly influenced by the pipeline’s internal water temperature, leading to certain non-uniformity in frost heave deformation. Since the water temperature inside the pipeline is maintained at 10 °C, it effectively reduces the freezing depth, and as a result, the frost heave displacement of the soil directly above the pipeline is relatively small.

At the same time, the frost heave displacement on the upper side of the pipeline shows a gradually increasing trend, reaching a maximum at the centerline of the pipeline and then decreasing toward both sides. Beyond 0.6 m from the pipeline, the displacement increases again, indicating that the frost heave force primarily extends outward from the center of the pipeline. The displacement on both sides of the pipeline includes both vertical and horizontal components. Since only the vertical displacement is shown in the figure, the vertical displacement components at 1 m and 3 m are smaller. Beyond 0.6 m from the pipeline, the influence of water temperature on soil freezing gradually weakens, causing the frost heave displacement in this area to begin increasing. Combining the data from Figure 11, it can be seen that the pipeline undergoes both vertical stretching and horizontal compression. This indicates that the multi-dimensional nature of the force direction during the frost heave process cannot be ignored.

During the thawing stage, the soil at the surface near the pipeline centerline settles more quickly, while the soil on top of the pipeline shows a trend of displacement, first increasing and then decreasing. This is because the temperature transfer at the ground surface takes some time. Initially, the temperature near the pipeline remains low, and the frost heave stage persists, causing the soil displacement to temporarily increase. As the temperature gradually reaches the pipeline, thawing occurs in the pipeline and surrounding soil, leading to gradual settlement and a decrease in displacement.

The deformation differences between the pipeline’s top and bottom are due to the greater freezing effect on the soil above the pipeline, while the soil beneath the pipeline experiences a weaker freezing effect due to its greater burial depth. The gradient of the temperature directly influences the freezing and thawing process of the soil, which in turn causes different displacement behaviors. The changes in frozen soil and water around the pipeline highlight the complexity of the frost heave process, emphasizing the need to pay close attention to the comprehensive deformation of the pipeline during this process. The differential displacement between the pipeline and surrounding soil underscores the risk of stress concentration and structural fatigue.

4. Parameter Sensitivity Analysis

4.1. The Impact of Burial Depth on Frost Heave and Thaw Settlement Displacement

Simulations of three freeze–thaw cycles were conducted for pipeline burial depths of 1 m, 1.5 m, and 2 m in frozen soil regions, as shown in Figure 13. The results indicate that as the burial depth increases, the surface frost heave displacement gradually increases while the displacement at the pipeline boundary significantly decreases. This phenomenon reveals the important regulatory role of burial depth in the soil’s frost heave behavior. The underlying mechanism can be attributed to the thermal insulation effect of the overlying soil layer.

When the pipeline is buried at a shallow depth (e.g., 1 m), heat exchange between the pipeline and the surface soil is stronger, allowing temperature fluctuations to rapidly propagate to the surrounding soil. During the freezing stage, the significant temperature gradient in the surrounding soil causes strong moisture migration effects, resulting in considerable frost heave deformation at the pipeline boundary. At the same time, the surface soil layer experiences buffering of temperature fluctuations, leading to relatively smaller frost heave displacement at the surface.

As the burial depth increases, the overlying soil layer provides greater thermal insulation, significantly reducing the intensity of heat exchange around the pipeline. This leads to a more stable temperature distribution in the surrounding soil, thereby reducing the localized frost heave effects. However, in conditions with greater burial depth, heat is primarily transferred from the soil layer to the surface, causing a more significant temperature gradient in the upper soil layer, which increases surface frost heave displacement. Additionally, pipelines buried at greater depths cause less disturbance to the stress distribution in the surrounding soil, which helps to reduce the deformation at the pipeline boundary and enhances the stability of the surrounding soil.

4.2. The Impact of Initial Saturation on Frost Heave and Thaw Settlement Displacement

Simulations of one year of freeze–thaw cycles were conducted for four initial saturation conditions: 0.2, 0.4, 0.6, and 0.8, as shown in Figure 14. The results indicate that as the initial saturation of the soil increases, both the surface and pipeline surrounding frost heave displacements significantly increase. This suggests that frost heave in the soil is primarily caused by the volume change induced by the expansion of pore water during freezing. When the soil’s initial saturation is higher, there is more moisture in the pore spaces, and during the freezing process, the moisture migrates towards the cold end and forms ice crystals, resulting in significant volume expansion. This phenomenon also exerts a greater impact on the deformation of the soil surrounding the pipeline. As shown in the stress contour map (see Figure 15), frost heave under high saturation exerts higher pressure on the pipeline structure, increasing the risk of pipeline deformation.

4.3. The Impact of Different Pipeline Radii on Surface Frost Heave Displacement

Simulations of one year of freeze–thaw cycles were conducted for three pipeline radius conditions (0.4 m, 0.5 m, and 0.6 m) while maintaining a constant pipeline burial depth of 1.5 m, as shown in Figure 16. The results indicate that as the pipeline radius increases, the surface frost heave displacement gradually decreases. This trend suggests that an increase in the pipeline’s volume ratio leads to a rise in local heat capacity. A larger heat capacity acts as a thermal buffer during the winter, reducing the temperature gradient in the soil surrounding the pipeline and consequently slowing down the rate of moisture migration towards the cold end and the freezing process. This slower freezing process reduces the frost heave effect in the soil. On the other hand, the increased pipeline radius also increases the contact area between the pipeline and the surrounding soil, which helps maintain a relatively stable temperature in the soil, thus reducing the extent of deformation during the freeze–thaw cycles.

4.4. The Effect of the Specific Heat Capacity of the Soil Layer on Frost Heave Displacement

By varying the specific heat capacity of the silty clay surrounding the pipeline, the impact on frost heave displacement was analyzed, as shown in Figure 17. As the specific heat capacity of the silty clay increases, the frost heave displacement gradually decreases. However, the displacements after thawing remain essentially the same across all conditions. This occurs because a higher specific heat capacity allows the soil to absorb more heat, acting as a thermal buffer during the cooling process. The increased specific heat capacity slows the cooling of the soil, reducing the temperature gradient during the freezing stage and slowing moisture migration towards the cold end. This suppression of moisture migration inhibits the formation and expansion of ice crystals within the soil, thereby reducing frost heave displacement. During the thawing phase, the ice crystals in the soil completely thaw into water, leading to a relatively uniform deformation across the soil layers.

4.5. The Effect of Thermal Conductivity of the Pipeline Soil Layer on Frost Heave

By varying the thermal conductivity of the silty clay surrounding the pipeline, the effects on frost heave displacement and the recovery of displacement after thawing were analyzed, as shown in Figure 18. The results indicate that as thermal conductivity increases, frost heave displacement gradually decreases, and more displacement recovery occurs after thawing. When the thermal conductivity of the silty clay is higher, heat is transferred more quickly within the soil, resulting in a more gradual temperature gradient surrounding the pipeline during the winter cooling process, thereby reducing frost heave displacement. During the thawing phase, soil with high thermal conductivity can absorb heat from the pipeline more efficiently and distribute it evenly, promoting the complete thawing of ice crystals and the redistribution of moisture, which aids in the soil’s return to its unfrozen state. In contrast, soil with low thermal conductivity experiences a significant temperature lag during the thawing phase, leading to incomplete recovery in certain areas and resulting in a greater amount of residual displacement.

4.6. The Impact of Saturated Soil Permeability Coefficient on Frost Heave and Thaw Settlement

By varying the permeability coefficient of the saturated soil surrounding the pipeline, the effects on frost heave and thaw settlement were observed, as shown in Figure 19. The results indicate that changes in the permeability coefficient have a minimal effect on frost heave and thaw settlement. This suggests that during the freeze–thaw cycle, the migration of moisture within the soil is primarily driven by the temperature gradient rather than by permeability. During the freezing stage, the temperature gradient plays a dominant role, causing directional moisture migration. In conditions with abundant water content and rapid freezing rates, moisture becomes largely stagnant, further diminishing the effect of permeability. The thaw settlement process is mainly influenced by volume changes due to the phase transition between ice and water and the subsequent re-adjustment of the soil structure, with the permeability coefficient having negligible impact on the final volume recovery and soil settlement.

4.7. Summary of Key Parameter Impacts on Frost Heave and Thaw Settlement

To systematically summarize the effects of critical parameters on frost heave and thaw settlement,

Table 3. Summary of key parameter impacts on frost heave and thaw settlement.

Parameter	Variation Range	Impact on Frost Heave/Thaw Settlement	Key Trend	Engineering Implication
Burial Depth	1.0 m, 1.5 m, 2.0 m	- Surface frost heave ↑ with deeper burial. - Pipeline boundary displacement ↓ with deeper burial.	Deeper burial reduces localized pipeline stress but increases surface heave.	Optimize burial depth (e.g., 1.5–2.0 m) to balance surface stability and pipeline safety.
Initial Saturation	0.2, 0.4, 0.6, 0.8	- Frost heave displacement ↑ significantly with higher saturation.	High saturation amplifies ice lens formation and soil expansion.	Use low-permeability backfill or drainage systems to control moisture in silty clay.
Pipeline Radius	0.4 m, 0.5 m, 0.6 m	- Surface frost heave ↓ with larger radius (e.g., 0.6 m reduces displacement by ~30%).	Larger radii enhance thermal buffering and reduce temperature gradients.	Prioritize larger pipe diameters in frost-prone regions to mitigate heave.
Specific Heat Capacity	0.42, 1.125, 1.68 kJ/kg·°C	- Frost heave ↓ with higher specific heat (40.8% reduction at 1.68 kJ/kg·°C). - Thaw settlement unaffected.	High heat capacity delays freezing and stabilizes soil temperature.	Select backfill materials with high specific heat (e.g., clay-sand mixtures).
Thermal Conductivity	0.5625, 1.125, 2.25 W/m·°C	- Frost heave ↓ and thaw recovery ↑ with higher conductivity.	Faster heat dissipation reduces frost penetration and promotes thaw rebound.	Use thermally conductive materials (e.g., gravel) around pipelines to enhance heat transfer.
Permeability	1 × 10−8, 5 × 10−8, 1 × 10−7 m/s	- Minimal impact on frost heave/thaw settlement.	Temperature gradients dominate moisture migration over permeability.	Focus on thermal management rather than permeability control in design.

consolidates the sensitivity analysis results. The table categorizes the variation ranges of each parameter, quantifies their impacts on displacement behavior, identifies dominant trends, and translates these findings into actionable engineering implications. This synthesis highlights how thermal properties, geometric design, and soil hydrology collectively govern pipeline–soil interactions in seasonally frozen ground. By linking theoretical trends to practical design strategies, the table serves as a concise reference for optimizing pipeline resilience against freeze–thaw cycles.

5. Discussion on Regional Applicability of the Methodology

While this study is conducted based on specific environmental parameters from the cold region of northern Xinjiang, the established analytical framework (including sinusoidal characterization of thermal loads and coupled calculation methods for soil freeze–thaw processes) demonstrates methodological universality. For engineering applications in other cold regions, the following adaptive adjustments are recommended: (1) modifying the phase parameter in Equation (14) according to local annual temperature amplitude; (2) recalibrating thermal conductivity coefficients for different soil types; (3) validating boundary conditions against regional maximum frost depth observations. The cross-regional applicability of the model will be further enhanced through the establishment of a multi-regional parameter database in future research.

6. Conclusions

This study advances the understanding of frost heave and thaw settlement mechanisms in buried water pipelines through a novel thermal–hydraulic–mechanical (THM) coupling framework, integrating ice–water phase transitions, layered soil heterogeneity, and dynamic pipeline–soil interactions. By leveraging field-validated numerical simulations, the work provides groundbreaking insights into the design and management of pipelines in seasonally frozen regions, with three key innovations:

First, shallow burial can effectively alleviate frost heave effects in the surrounding soil and reduce the accumulation of residual surface displacement. However, shallow burial may also lead to pronounced non-uniform surface settlement, which poses potential risks to surface structures and vegetation. Therefore, pipeline installation in cold regions requires a careful balance between minimizing frost heave effects and mitigating uneven surface settlement.

Second, increasing the pipeline radius has been shown to significantly reduce surface frost heave deformation and optimize stress distribution in the surrounding soil. This reduces localized stress concentrations and enhances the long-term operational stability of the pipeline. Furthermore, the thermal and physical properties of the surrounding soil play a critical role in regulating the freeze–thaw cycle. High-specific-heat soil layers can absorb and release heat gradually, thereby reducing the freezing depth, delaying the onset of frost heave, and improving the recovery capacity of the soil during thawing. Similarly, pipeline materials with high thermal conductivity can accelerate heat transfer, effectively reducing the freezing range and promoting soil rebound during thawing, which optimizes the mechanical response of the surrounding soil.

Finally, the heat conduction of water within the pipeline significantly mitigates frost heave effects. The stable water temperature in the pipeline creates a thermal buffering zone, reducing frost heave displacement in the overlying soil. This mechanism is crucial in alleviating vertical pressures on the pipeline and limiting deformation in the surrounding soil. Moreover, as freeze–thaw cycles progress, the interactions among the temperature field, moisture field, and stress field intensify, resulting in horizontal compression and vertical stretching of the pipeline. These forces may lead to the formation and propagation of micro-cracks in the soil. Such micro-cracks alter the soil’s stress state, potentially reducing its bearing capacity and threatening the long-term stability of the pipeline.

In summary, this study systematically unveils the intricate interplay among moisture migration, temperature variations, and mechanical behavior in frozen soil. These findings provide a theoretical foundation for the design and maintenance of pipelines in cold regions, particularly in optimizing burial depth, selecting suitable soil materials, and designing pipeline geometric parameters. The developed COMSOL-based THM platform, validated against field data (<20% error), offers a robust tool for simulating freeze–thaw dynamics in layered soils.

The proposed THM coupling framework, though developed for water pipelines, holds significant potential for broader cold-region infrastructure applications. For road embankments, substituting the pipeline geometry with trapezoidal cross-sections allows direct adoption of the temperature-dependent permeability model to predict frost heave in clay-rich subgrades. In building foundations, recalibrating the stress–strain equations with concrete thermal parameters enables tilt prediction under cyclic freezing. For buried oil/gas pipelines, extending the phase change model to multi-phase flow conditions can quantify thaw-induced axial strains, as validated by the China–Russia pipeline case. This adaptability stems from the model’s modular treatment of thermal boundaries and soil heterogeneity—key features transferable to other frost-susceptible infrastructures. While this work bridges critical gaps, future efforts should explore microscale cracking mechanisms, unsaturated airflow effects, and climate-warming scenarios in Arctic permafrost. These extensions will further solidify the framework’s applicability to evolving environmental challenges.