Research on Frost Heaving Distribution of Seepage Stratum in Tunnel Construction Using Horizontal Freezing Technique

Abstract

During the horizontal freezing construction of a subway tunnel, the delay of the closure of the frozen wall occurs frequently due to the existence of groundwater seepage, which can be directly reflected by a freezing temperature field. Accordingly, the distribution of ground surface frost heaving displacement under seepage conditions will be different from that under hydrostatic conditions. In view of this, this paper uses COMSOL to realize the hydro–thermal coupling in frozen stratum under seepage conditions, then, the frost heaving distribution of seepage stratum in tunnel construction using horizontal freezing technique is researched considering the ice–water phase transition and orthotropic deformation characteristics of frozen–thawed soil by ABAQUS. The results show that the expansion speed of upstream frozen wall is obviously slower than that of the downstream frozen wall, and the freezing temperature field is symmetrical along the seepage direction. In addition, the ground frost heaving displacement field is asymmetrically distributed along the tunnel center line, which is manifested in that the vertical frost heaving displacement of the upstream stratum is less than that of the downstream stratum. The vertical frost heaving displacement of the ground surface decreases with the increase in tunnel buried depth, but the position of the maximum value remains unchanged as the tunnel buried depth increases. The numerical simulation method established in this paper can provide a theoretical basis and design reference for the construction of a subway tunnel in a water-rich stratum under different seepage using the artificial freezing technique.

1. Introduction

With the acceleration of urbanization, there is more and more urgency to develop underground resources to expand the space for human activity [1,2]. Nevertheless, various complex geological conditions, such as water-rich sand, loose gravelly soil and silty clay with frozen–thawed sensitive characteristics, are encountered in underground engineering, and traditional geotechnical engineering methods face difficulties in solving such problems. Artificial ground freezing (AGF) is a method of artificial cooling that absorbs heat from the soil around the underground space to be excavated, gradually turning the natural soil into frozen soil and thus forming a dense frozen soil body. So, with its unique waterproof effect, adaptability to any complex geological conditions, flexible use and other characteristics, the geotechnical engineering community have paid more and more attention to AGF, which is now widely used in complex geotechnical engineering construction [3,4,5].

During the tunnel freezing period, ice intrusions such as ice crystals and interlayers formed in stratum frozen area will increase the volume of soil which is the frost heaving phenomenon of stratum. This phenomenon will be amplified when the stratum exits groundwater seepage, which brings difficulty to tunnel construction and damages the environment [6,7,8]. In view of this, domestic and foreign scholars have carried out much research by means of numerical simulation, model tests and field measurement.

Hou [9] represented the freezing characteristic function by combining the Young–Laplace law with the van Genuchten model. Then, a thermal-mechanical model of saturated media considering dynamic water–ice phase equilibrium was established and validated thorough model tests under various seepage velocities. Zhou [10] studied the development law of frozen wall in the process of artificial ground freezing under the construction of combined formation seepage by model test. Ji [11] studied the evolution process of frost heaving force under thermo–mechanical coupling through a series of freezing tests under different constraint stiffness and temperature gradient, and proposed a method to measure formation frost heaving force under constraint conditions. Tang et al. [12] proposed a prediction method with porosity as a variable using the finite difference method, and analyzed the physical changes of silty soil during freezing. Cai [13] established an optimized prediction method for ground surface frost heaving during the freezing construction of a subway tunnel using AGF based on stochastic medium theory. Lai et al. [14,15] proposed a hydro–thermal coupling model considering ice–water phase change based on heat transfer theory and seepage theory. Wang et al. [16] studied the evolution law of artificial frozen wall under seepage conditions in saturated gravel formations. Ahmed [17] studied the diachronic evolution law of temperature field using COMSOL finite element software considering different seepage velocities (0.5, 1.0, 1.5 m/d). Li et al. [18] proposed a moisture–heat coupling model considering ice–water phase transition to quantify the temperature distribution in soil during the freezing process. Huang et al. [19] adopted the same model to optimize the positions of freezing pipes around circular tunnel in seepage stratum with the velocity of 2.0 m/d. Yang et al. [20] further revealed the development law of the frozen wall upstream, and downstream of the tunnel under seepage conditions, using a large-scale model test.

The above researchers have made great contributions to the distribution laws of artificial freezing temperature and displacement fields; however, the research on the frost heaving laws of seepage strata during the freezing construction period is still insufficient. Therefore, in this paper, the finite element software COMSOL is selected to realize the hydro–thermal coupling in frozen stratum under seepage conditions. Then, the numerical simulation software ABAQUS is selected to realize the thermo-mechanical coupling, considering the ice–water phase transition and orthotropic deformation characteristics of frozen–thawed soil. The HyperMesh tool is used to divide the grid of models in COMSOL and ABAQUS, which ensures the consistency of the grid, and the calculation results of the former are directly input into the latter program. Finally, the formation and distribution laws of the displacement field and displacement field of the stratum under the groundwater seepage conditions are researched by taking tunnel horizontal freezing engineering as the research object.

2. Hydro–Thermal Coupling Mathematical Model

2.1. Basic Parameters

The volume content of each component in the soil is shown in Figure 1.

Where φ is the soil skeleton gap, χ is the volume content of water in the gap.

The unfrozen water content in frozen soil is always in dynamic equilibrium with the temperature of frozen soil [21,22], and content can be determined by Equation (1). The volume content of pore water can be expressed as

$$w_u=w_0{\left(\frac{T_f}{T}\right)}^b$$

(1)

$$\chi =\left\{\begin{array}{l}1,\begin{array}{cc}& T>T_f\end{array}\\ {\left(\frac{T_f}{T}\right)}^b,T\le T_f\end{array}\right.$$

(2)

where w₀ denotes the initial water content of unfrozen soil; w_u denotes the free water content of frozen soil; T is soil temperature, °C; T_f is the initial freezing temperature of soil mass, °C; and b is the constant determined by soil properties.

2.2. Coupling of Temperature Field to Seepage Field

The temperature field during soil freezing is a transient heat transfer problem.it is assumed that ice, soil particles and water exist simultaneously in the freezing soil during the freezing process, and have different heat transfer characteristics.

According to the heat transfer and seepage principles [23], the control equations for the temperature and seepage fields considering the phase change are

$${\theta }_i{C}_i{\rho }_i\frac{\partial T}{\partial t}+\phi L{\rho }_l\frac{\partial \chi }{\partial T}\frac{\partial T}{\partial t}+{\theta }_l{C}_l{\rho }_l\frac{\partial T}{\partial t}+{\rho }_l{C}_l\left(\mathit{u}\mathbf{·}\nabla T\right)+{\theta }_s{C}_s{\rho }_s\frac{\partial T}{\partial t}=\nabla \left[\left({\theta }_i{k}_i+{\theta }_l{k}_l+{\theta }_s{k}_s\right)\nabla T\right]+Q$$

(3)

$${\rho }_l\left[{\alpha }_l\phi +{\alpha }_s\left(1-\phi \right)\right]\frac{\partial p}{\partial t}+\nabla \cdot \left({\rho }_l\mathit{u}\right)={\rho }_{l}q$$

(4)

$$\mathit{u}=-\frac{K\prime }{{\rho }_{l}g}\nabla p$$

(5)

$$K\prime =K_f+\left(K_0-K_f\right)\chi$$

(6)

where t is time; C_i, C_l and C_s denote the specific heat capacity of water, ice and soil particles; K_i, K_l and K_s are their thermal conductivity; ρ_i, ρ_l and ρ_s are their density; L denotes ice–water phase change latent heat; u denotes the relative velocity of water; Q denotes the total amount of equivalent heat source; α_s is the basic equivalent compression ratio; φ is porosity; q denotes the volume of flow; p is the osmotic pressure; K′ is hydraulic conductivity; and K_f and K₀ are the hydraulic conductivity of completely frozen area and unfrozen area respectively.

The change of absolute porosity during freezing can be expressed by the change of volume content of unfrozen water, therefore, Equation (4) can be expressed as

$${\rho }_l\left[{\alpha }_l{\theta }_l+{\alpha }_s\left(1-{\theta }_l\right)\right]\frac{\partial p}{\partial t}+\nabla \cdot \left({\rho }_l\mathit{u}\right)={\rho }_{l}q$$

(7)

The temperature of the outer wall of the freezing tube is the freezing temperature, where the temperature boundary at the inlet of the freezing tube is the inlet temperature, and the convection condition at the outlet of the freezing tube is set, as shown

$${\left.{\mathit{n}}_d\cdot \left(-k\nabla T\right)\right|}_{x=a_2}=0$$

(8)

where a₂ denotes the horizontal coordinate of the export of the freezing pipe, n_d denotes the axial direction vector at the export.

Setting the other boundaries to be adiabatic and not considering the heat transfer effect between the boundaries, the boundary conditions are expressed as

$${\left.-\mathit{n}\cdot \left(-k\nabla T\right)\right|}_{y=b_1,b_2}=0$$

(9)

where b₁ and b₂ are the ordinates of the boundary, and n n denotes the direction vector of the normal direction within the boundary.

The temperature boundary considering other boundary heat transfer effects can be expressed as

$$T=T_0$$

(10)

where T₀ is the temperature over time.

The initial conditions are

$${\left.p\right|}_{t=0}=p_0$$

(11)

where p₀ denotes the soil pressure distribution in the initial state.

Therefore, the mathematical model of hydro–thermal coupling during freezing process can be composed of Equation (3), Equation (7) and the above boundary conditions.

3. Thermo–Mechanical Coupling Mathematical Model

3.1. Freezing Temperature Field

In the process of artificial ground freezing, the change of the temperature of the soil will cause the change of water migration and stress state, and the stress field will cause the redistribution of the temperature field. For this reason, the coupling of temperature field and stress field should be considered in frost heaving deformation. The controlling equation for the temperature field can be expressed as [24,25]

$$C^{*}\frac{\partial T}{\partial t}=\frac{\partial }{\partial x}\left(k^{*}\frac{\partial T}{\partial x}\right)+\frac{\partial }{\partial y}\left(k^{*}\frac{\partial T}{\partial y}\right)$$

(12)

where $C^{*}$ is the equivalent volume specific heat; $k^{*}$ is the equivalent thermal conductivity; and T is the soil temperature; and t is the time. $C^{*}$ and $k^{*}$ are given by

$$C^{*}=\left\{\begin{array}{ll}{C}_f& T<T_d\\ \frac{C_f+C_u}{2}+\frac{L}{T_r-T_d}& T_d\le T\le T_r\\ C_u& T>T_r\end{array}\right.$$

(13)

$$k^{*}=\left\{\begin{array}{ll}{k}_f& T<T_d\\ k_f+\frac{k_u-k_f}{T_r-T_d}\left(T-T_d\right)& T_d\le T\le T_r\\ k_u& T>T_r\end{array}\right.$$

(14)

where C_f and C_u denote the volumetric heat capacity of frozen soil and unfrozen soil (i.e., C_f = ρ_f c_f, C_u = ρ_u c_u), ρ_f and ρ_u denote the density of frozen soil and unfrozen soil, and c_f and c_u denote the mass specific heat of frozen soil and unfrozen soil. k_f and k_u denote the thermal conductivity of frozen soil and unfrozen soil, respectively. T_d denotes the freezing temperature of soil; T_r denotes the thawing temperature of soil.

The initial condition of temperature field is

$${\left.T\right|}_{t=0}=T_0$$

(15)

where T₀ denotes the initial temperature of soil.

When the freezing pipe is transformed into one point, and the boundary condition of frozen pipe is expressed as

$${\left.T\right|}_{\left(x_p,y_p\right)}=T_c\left(t\right)$$

(16)

where (x_p, y_p) represents the central coordinate of the freezing pipe and T_c represents the brine temperature.

At the infinite distance of the frozen zone, the boundary condition is

$${\left.T\right|}_{\left(x=\infty , y=\infty \right)}=T_0$$

(17)

3.2. Coupling of Stress Field to Temperature Field

For the plane strain problem, the equilibrium equation of the stress field can be expressed as

$$\left\{\begin{array}{l}\frac{\partial {\sigma }_x}{\partial x}+\frac{\partial {\tau }_{xy}}{\partial y}=0\\ \frac{\partial {\tau }_{xy}}{\partial x}+\frac{\partial {\sigma }_x}{\partial y}=-\gamma \end{array}\right.$$

(18)

where γ is the gravity of soil.

The geometric equation can be expressed as

$$\left\{\begin{array}{l}{\epsilon }_x=\frac{\partial u}{\partial x}\\ {\epsilon }_x=\frac{\partial u}{\partial x}\\ {\epsilon }_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\end{array}\right.$$

(19)

Due to the frozen soil being defined as a linear elastomer, the mechanical properties of frozen soil change with temperature, considering its orthotropic characteristics. So, the thermo–mechanical coupling mathematical model can be expressed as

$$\left\{\begin{array}{l}{\epsilon }_x=\frac{1-\mu {\left(T\right)}^2}{E\left(T\right)}\left[{\sigma }_x-\frac{\mu \left(T\right)}{1-\mu \left(T\right)}{\sigma }_y\right]+{\sigma }_x^T\\ {\epsilon }_y=\frac{1-\mu {\left(T\right)}^2}{E\left(T\right)}\left[{\sigma }_y-\frac{\mu \left(T\right)}{1-\mu \left(T\right)}{\sigma }_x\right]+{\sigma }_y^T\\ {\epsilon }_{xy}=\frac{2\left[1-\mu \left(T\right)\right]}{E\left(T\right)}{\tau }_{xy}+{\epsilon }_{xy}^T\end{array}\right.$$

(20)

where ${\epsilon }_x^T$, ${\epsilon }_y^T$ and ${\epsilon }_{xy}^T$ are the instantaneous strain component caused by temperature.

To further effectively discuss the orthotropic deformation characteristics of frozen soil, generally, the frost heaving rate ${\epsilon }_{f/r}^T$ is used to measure the strength of frozen soil and the parameter is defined as the instantaneous volumetric strain related to temperature. As shown in Figure 2, the local coordinate system is established in the direction of temperature gradient in three directions (1, 2 and 3).

According to the linear elastic theory of frozen soil, it can be known that the instantaneous volumetric strain ${\epsilon }_f^T$ and elastic strain ${\epsilon }^{\theta }$ constitute the total strain ε of frozen soil when the soil begins to freeze and the instantaneous thermal strain component generated by temperature in the local coordinate system is expressed as

$$\left\{\begin{array}{ccc}{\epsilon }_{11}^T& {\epsilon }_{12}^T& {\epsilon }_{13}^T\\ {\epsilon }_{21}^T& {\epsilon }_{22}^T& {\epsilon }_{23}^T\\ {\epsilon }_{31}^T& {\epsilon }_{32}^T& {\epsilon }_{33}^T\end{array}\right\}=\left[\begin{array}{ccc}\eta & 0& 0\\ 0& \left(1-\eta \right)/2& 0\\ 0& 0& \left(1-\eta \right)/2\end{array}\right]{\epsilon }_f^T$$

(21)

where η is an orthotropic parameter. When η = 1, the frozen soil exhibits unidirectional deformation characteristics (i.e., in the direction of the temperature gradient).; when η = 1/3, the frozen soil exhibits isotropic deformation characterized. Therefore, whether unidirectional deformation or isotropic deformation, frozen soil has characteristics of orthotropic deformation.

Employing Hooke’s law, the bulk strain component of frozen soil can be expressed as

$$\left\{\begin{array}{l}{\epsilon }_{11}=\frac{1}{E}\left[{\sigma }_{11}-\mu \left({\sigma }_{22}+{\sigma }_{33}\right)\right]+\eta {\epsilon }_f^T\\ {\epsilon }_{22}=\frac{1}{E}\left[{\sigma }_{22}-\mu \left({\sigma }_{11}+{\sigma }_{33}\right)\right]+\frac{1}{2}\left(1-\eta \right){\epsilon }_f^T\\ {\epsilon }_{33}=\frac{1}{E}\left[{\sigma }_{33}-\mu \left({\sigma }_{11}+{\sigma }_{22}\right)\right]+\frac{1}{2}\left(1-\eta \right){\epsilon }_f^T\\ {\epsilon }_{12}=\frac{2\left(1+\mu \right)}{E}{\tau }_{12}\\ {\epsilon }_{23}=\frac{2\left(1+\mu \right)}{E}{\tau }_{23}\\ {\epsilon }_{13}=\frac{2\left(1+\mu \right)}{E}{\tau }_{13}\end{array}\right.$$

(22)

where E is the elastic modulus of frozen soil and μ is Poisson’s ratio of frozen soil.

For the plane strain problem, ε₃₃ = ε₁₃ = ε₂₃ = 0,

$${\sigma }_{33}=\mu \left({\sigma }_{11}+{\sigma }_{22}\right)-\frac{E}{2}\left(1-\eta \right){\epsilon }_f^T$$

(23)

Combined with Equations (22) and (23), bulk strain component can be expressed as

$$\left\{\begin{array}{l}{\epsilon }_{11}=\frac{1-{\mu }^2}{E}\left[{\sigma }_{11}-\frac{\mu }{1-\mu }{\sigma }_{22}\right]+\left[\eta +\frac{\mu }{2}\left(1-\eta \right)\right]{\epsilon }_f^T\\ {\epsilon }_{22}=\frac{1-{\mu }^2}{E}\left[{\sigma }_{22}-\frac{\mu }{1-\mu }{\sigma }_{11}\right]+\frac{1}{2}\left(1-\eta \right)\left(1+\eta \right){\epsilon }_f^T\\ {\epsilon }_{12}=\frac{2\left(1+\mu \right)}{E}{\tau }_{12}\end{array}\right.$$

(24)

From the above equation, the transiently thermal strain component is expressed as

$$\left\{\begin{array}{l}{\epsilon }_{11}^T=\left[\eta +\frac{\mu }{2}\left(1-\eta \right)\right]{\epsilon }_f^T\\ {\epsilon }_{22}^T=\frac{1}{2}\left(1-\eta \right)\left(1+\eta \right){\epsilon }_f^T\\ {\epsilon }_{12}^T=0\end{array}\right.$$

(25)

As shown in Figure 2, where θ represents the included angle between the coordinate axes (1 and x), and the transient thermal strain component is

$$\left\{\begin{array}{c}{\epsilon }_x^T\\ {\epsilon }_y^T\\ {\epsilon }_{xy}^T\end{array}\right\}=\left[\begin{array}{ccc}{\cos }^2\theta & {\sin }^2\theta & -2\sin \theta \cos \theta \\ {\sin }^2\theta & {\cos }^2\theta & 2\sin \theta \cos \theta \\ \sin \theta \cos \theta & -\sin \theta \cos \theta & {\cos }^2\theta -{\sin }^2\theta \end{array}\right]\left\{\begin{array}{c}{\epsilon }_{11}^T\\ {\epsilon }_{22}^T\\ {\epsilon }_{12}^T\end{array}\right\}$$

(26)

Substituting Equation (25) into (26), we can get

$$\left\{\begin{array}{l}{\epsilon }_x^T=\left\{{\cos }^2\theta \left[\eta +\frac{\mu }{2}\left(1-\eta \right)\right]+\frac{{\sin }^2\theta }{2}\left(1-\mu \right)\left(1+\mu \right)\right\}{\epsilon }_f^T\\ {\epsilon }_y^T=\left\{{\sin }^2\theta \left[\eta +\frac{\mu }{2}\left(1-\eta \right)\right]+\frac{{\cos }^2\theta }{2}\left(1-\mu \right)\left(1+\mu \right)\right\}{\epsilon }_f^T\\ {\epsilon }_{xy}^T=\frac{\sin \theta \cos \theta }{2}\left(3\eta -1\right){\epsilon }_f^T\end{array}\right.$$

(27)

From this, the calculation formula of instantaneous temperature strain component considering the orthotropic deformation characteristics of freeze-thaw soil is obtained.

4. Finite Element Calculation Model

4.1. Calculation Model

The tunnel constructed by horizontal freezing method is selected as the calculation model. The cross-section of the tunnel is circular, the buried depth is 15.0 m, the initial temperature of the soil is set as 20 °C, the upper displacement boundary of the model is a free boundary, the horizontal displacement is limited on the left and right sides, and the bottom is a fixed boundary.

To import the temperature field here into ABAQUS as a predefined field, first, the temperature field in COMSOL is derived in the form of numerical nodes, then a predefined field of temperature is established in ABAQUS, and finally the exported node data are imported into the initial predefined field in the form of a mapping field to obtain the cloud diagram of temperature field. The specific numerical simulation analysis process is shown in the Figure 3 below.

As shown in Figure 4 and Figure 5, the HyperMesh tool is used to divide the grid of models in COMSOL and ABAQUS which ensures the consistency of the grid, and the calculation results of the former are directly input into the latter program.

In order to combine the two finite element software more effectively, the temperature field obtained by hydro–thermal coupling in COMSOL is imported into ABAQUS as a predefined field (i.e., the temperature field results are transformed into nodal form). The cloud diagram of temperature field calculated by hydro–thermal coupling in COMSOL is shown in Figure 6, and that calculated by thermal–mechanical coupling in ABAQUS is shown in Figure 7. It can be seen from the above figure that the distribution of the temperature clouds calculated by both software are the same, and the freezing fronts show a non-uniform distribution along the flow direction due to the presence of the seepage field. The accuracy of the previously proposed thermal–hydraulic–mechanical coupled numerical simulation method is verified, thus realizing a more accurate and comprehensive study of the freezing and swelling law of seepage strata during the freezing construction period.

According to the orthogonal anisotropy of frozen soil, the freezing front will be formed only in the direction perpendicular to the temperature gradient during freezing process. The freezing front moves with the decrease in temperature. Thus, in this section, finite element software COMSOL and ABAQUS are selected to realize the hydro–thermal–mechanical coupling in frozen stratum under different seepage conditions. In addition, the ABAQUS frost heaving subroutine interface is introduced in the process of numerical analysis [26]. The process of calculating frost heaving deformation by subroutine interface is shown in Figure 8.

As shown in

Table 1. Model design.

Buried Depth	Seepage Velocities
	0 m/d	0.5 m/d	1.0 m/d	1.5 m/d	2.0 m/d
11 m	Model 11-1	11-2	11-3	11-4	11-5
13 m	Model 13-1	13-2	13-3	13-4	13-5
15 m	Model 15-1	15-2	15-3	15-4	15-5

, in order to analyze the influence of different seepage velocities and tunnel depths on surface deformation, five groundwater seepage velocities (0 m/d, 0.5 m/d, 1.0 m/d, 1.5 m/d and 2.0 m/d) and three tunnel depths (11 m, 13 m and 15 m) are selected.

4.2. Model Parameters

In all calculation models, the radius of the horizontal freezing pipe arrangement circle (R_d) is 4.0 m, a total of 22 freezing pipes with a wall temperature of −25 °C are arranged, and the effective design freezing thickness is 2.0 m. On the upper surface of the model, the convective heat transfer value between the surface and the air is set as 175 kcal/(m²·d·°C).

The thermal physical parameters and mechanical parameters of soil are shown in

Table 2. Physical parameters of soil.

Materials	Density/(kg·m−3)	Thermal Conductivity/(kcal·m−1·h−1·°C−1)	Latent Heat/(kcal·kg−1)	Specific Heat Capacity/(kcal·kg−1·°C−1)	Permeability Coefficient /(m·d−1)
Soil	2110	1.55	-	0.301	-
Frozen soil	-	1.792	12.736	-	5.3 × 10−4
Unfrozen soil	-	1.252	12.736	-	4.6 × 10−2
Water	1000	0.56	-	0.495	-
Ice	918	2.24	-	0.998	-

In this numerical simulation, T_trans= −1.0 °C, dT = 0.5 °C.

and

Table 3. Mechanical parameters of soil.

Names	Elastic Modulus (MPa)	Poisson’s Ratio	Cohesion (kPa)	Friction Angle (°)
Frozen soil	44.27 + 7.73|T|	0.25	-	-
Unfrozen soil	24	0.35	15	20

T is soil temperature.

. Considering the orthogonal anisotropy of soil, the deformation characteristic coefficient η is set as 0.9, and the frost heaving ratio is set as 1% [27].

4.3. Initial Boundary Conditions

4.3.1. Boundary Conditions and Initial Conditions of Seepage Field

(1) Dirichlet boundary condition

The inlet and outlet are set as constant pressure boundaries, and the corresponding value is taken according to the initial pressure distribution.

$$p=p_0$$

(28)

(2) Neumann boundary condition

$${\left.-n\left(\overrightarrow{u}p\right)\right|}_{y=b_1,b_2}=0$$

(29)

where b₁ and b₂ are the ordinates of the boundary, and n represents the internal normal direction.

(3) Initial condition

It is assumed that the seepage velocity in initial state is only from right to left perpendicular to the inlet boundary. To form a stable seepage velocity in initial state, initial seepage velocity control method is as follows

$$p\left(x,y,t\right)=p_0\left(x,y,0\right)$$

(30)

4.3.2. Boundary Conditions and Initial Conditions of Temperature Field

(1) Dirichlet boundary condition

$$T=T_{a_1}$$

(31)

where a₁ is the abscissa of the inlet boundary and the upper and lower boundaries are set as adiabatic boundaries, which can be expressed as

$${\left.n_d\left(-k\nabla T\right)\right|}_{y=b_1,b_2}=0$$

(32)

where b₁ and b₂ are the ordinates of boundary, and n is the inner normal direction vector of the boundary.

(2) Neumann boundary condition

$${\left.n_d\left(-k\nabla T\right)\right|}_{x=a_2}=0$$

(33)

where a₂ represents the abscissa of outflow boundary condition.

(3) Initial condition

$$T\left(x,y,t\right)=T_0\left(x,y,0\right)$$

(34)

where T₀ is the initial temperature of soil layer.

5. Discussion

5.1. Effect of Groundwater Seepage on Freezing Temperature Field

The temperature range of cloud diagram is set as −26~−1 °C (i.e., the freezing temperature of soil is −1 °C) to characterize the thickness of frozen wall in numerical simulation. Then, it can be found that the development of frozen front is uneven due to the influence of seepage. Therefore, a unified measure is established in numerical simulation by defining the thickness of frozen wall at different locations AB, BC, DE and EF which correspond to L_AB, L_BC, L_DE and L_EF, respectively, when the construction requirements are met. As shown in Figure 9, L_AB is the thickness from the center of freezing pipe to the upstream outer front, L_BC is the thickness from the center of the freezing pipe to the upstream inner front, L_DE is the thickness from the center of the freezing pipe to the downstream inner front and L_EF is the thickness from the center of the freezing pipe to the upstream outer front.

The frozen wall formed at three groundwater seepage velocities for the given boundary conditions are shown in Figure 10. The closure time of frozen wall at 0 m/d is only 9 d, while it is 24 d at 1.0 m/d and 61 d at 2.0 m/d. The faster the seepage is, the more cooling capacity generated by the upstream freezing pipes is brought to the downstream, and the later the upstream frozen wall closes, that is, the longer the closure time of frozen wall is.

Similarly, the time required to reach design frozen wall thickness for construction at different seepage velocities is also different, as shown in Figure 10. The time required to reach design frozen wall thickness at 0 m/d, 1.0 m/d and 2.0 m/d is 41d, 83d and 121d, respectively. Therefore, it will take more time to complete the freezing engineering in the seepage stratum with a large flow velocity. The temperature field at different seepage velocities is shown in Figure 11 and Figure 12.

Table 4. Relevant parameters of frozen wall at different flow velocity.

Seepage Velocity (m/d)	T1 (d)	T2 (d)	LAB (m)	LBC (m)	LDE (m)	LEF (m)
0	9	41	1.139	1.258	1.281	1.135
1.0	24	83	0.752	2.258	2.112	1.726
2.0	61	121	0.247	2.348	2.347	1.759

shows that when the minimum thickness of the frozen wall is satisfied (i.e., the construction requirements are met), the thickness of the upstream frozen wall L_AB decreases correspondingly with the increase in seepage velocity, and the thickness of the downstream frozen wall L_DE and L_EF increases correspondingly. However, the thickness variation of the upstream frozen wall L_BC does not vary linearly. This is because with the greater seepage velocity, the heat at the inner front of the upstream frozen wall is carried away, resulting in a change in the thickness of the frozen wall L_BC. Therefore, in the actual project, when the seepage velocity is large enough, sufficient attention should be paid to the upstream freezing wall and corresponding measures should be taken.

As shown in Figure 13, the development trend of frozen wall thickness at each position is similar at the same seepage velocity, the thickness L_AB is much smaller than L_BC, L_DE and L_EF when groundwater seepage occurs, and this phenomenon change linearly with the seepage. The thickness L_BC, L_DE and L_EF reached 1.600 m, 1.754 m and 1.364 m, respectively, when freezing for 83 d at 2.0 m/d of seepage velocity, while L_AB, only 0.410 m, is less than one-third of the other places’ thickness.

There are also great differences in the development speed of frozen wall thickness between the inner front and outer front. The development rates of frozen wall thickness L_AB and L_EF in the outer ring gradually decrease with freezing time. the thickness of frozen wall in upstream finally tends to be flat, and the phenomenon is related to the seepage velocity. When the thickness of frozen wall reached the construction standard, the development speed of the frozen wall thickness L_AB is 33.3 mm/d at 1.0 m/d of seepage, while the development rate of L_BC is only 0.5 mm/d. The frozen wall thickness L_BC and L_DE on the inner front increase linearly with freezing time, which is due to the closed area formed inside the freezing circle after the frozen wall intersects, which is basically no longer affected by the underground water.

5.2. Effect of Groundwater Seepage on Frost Heaving Displacement of Ground

In this section, taking the tunnel buried depth of 15 m as an example, five groundwater seepage velocities (0 m/d, 0.5 m/d, 1.0 m/d, 1.5 m/d and 2.0 m/d) are considered to explore the distribution law of stratum frost heaving displacement under the condition of groundwater seepage.

The development law of vertical displacement of the strata under different seepage velocities is shown in Figure 14. The vertical freezing expansion of the stratum is distributed asymmetrically along the center of the tunnel, which is due to the transfer of cold from the upstream freezing pipe to the downstream stratum under the action of seepage flow, resulting in the uneven thickness of the upstream and downstream freezing walls and the offset of the peak position of the stratum in the upper part of the stratum tunnel. When the effective frozen wall thickness is reached, the maximum vertical displacement of the upper stratum of the tunnel under different seepage velocity (0 m/d, 0.5 m/d, 1.0 m/d, 1.5 m/d and 2.0 m/d) is 43.1 mm, 45.2 mm, 48.2 mm, 51.0 mm and 53.5 mm.

The surface frost heaving displacement curve is shown in Figure 15, which is “normal distribution”; furthermore, the peak of the single peak locates directly above the center of the tunnel when the seepage velocity is 0 m/d, and the position of the peak will “shift” to the direction of water flow with the increase in seepage velocity.

The maximum frost heaving displacement is located above the center of the tunnel under no seepage condition. When the flow velocity reaches 0.5 m/d, it is located 161 mm downstream from the center of the tunnel. When the flow velocity reaches 1.0 m/d, it is located 374 mm downstream from the center of the tunnel. When the flow velocity reaches 1.5 m/d, it is located 556 mm downstream from the center of the tunnel, and when the flow velocity reaches 2.0 m/d, it is located 968 mm downstream from the center of the tunnel. These data show that the existence of seepage leads to the thickness of frozen wall downstream being greater than that upstream, which leads to the deviation of the maximum frost heaving positive on the ground.

5.3. Effect of Tunnel Buried Depth on Frost Heaving Displacement of Ground

In this section, selecting the flow velocity of 1.0 m/d as the invariant, three buried depth of tunnel (11 m, 13 m, 15 m) are considered, to explore the distribution law of stratum frost heaving displacement under the condition of groundwater seepage.

The vertical displacement distribution law of seepage stratum and the vertical displacement curve of frost heaving on the surface after the frozen wall reaches the effective thickness are shown in Figure 16 and Figure 17. On the one hand, it is not difficult to find that, under the same seepage velocity, the maximum frost heaving of stratum is inversely proportional to the buried depth of tunnel. On the other hand, when the buried depth is 11 m, the maximum frost heaving of surface is located at 368 mm from downstream to the center of tunnel; when the buried depth is 13 m, it is located at 384 mm from downstream to the center of tunnel, and the maximum is 38.1 mm; when the buried depth is 15 m, the vertical displacement of surface frost heaving reaches the maximum at 372 mm from downstream to the center of the tunnel, and the maximum is 35.2 mm.

At the same time, the effect of tunnel buried depth on the vertical deformation of ground frost heaving under the action of seepage has the characteristics of regional differences. The frost heaving displacement of the surface is greatly affected by the buried depth of the tunnel within 9.2 m upstream and 12.1 m downstream of the tunnel center, and it decreases with the increase in buried depth.

6. Conclusions

In this paper, the development and distribution law of the freezing temperature field and frost heave displacement field of seepage stratum during the freezing construction period are studied by the numerical simulation method with a horizontal freezing engineering of subway tunnel as the research background. The main research contents and conclusions are as follows:

(1) A thermo–hydro–mechanical coupling numerical model is established using the finite element method. Specifically, the numerical simulation analysis software COMSOL is selected to realize the hydro–thermal coupling in frozen stratum under seepage condition. The numerical simulation analysis software ABAQUS is selected to realize the thermo–mechanical coupling, considering the ice–water phase transition and orthotropic deformation characteristics of frozen–thawed soil. The HyperMesh tool is used to divide the grid of models in COMSOL and ABAQUS to ensure the consistency of the grid, and the calculation results of the former are directly input into the latter program.

(2) Under the action of seepage, the frost heaving displacement of stratum is positively correlated with seepage velocity when the frozen wall reaches the effective thickness. It is worth point out that the maximum vertical displacement of frost heaving is 45.2 mm at the seepage velocity of 0.5 m/d, and the maximum vertical displacement of frost heaving is 53.5 mm at 2.0 m/d; furthermore, the frost heaving displacement of surface also increases from 30.8 mm to 39.6 mm.