## 1. Introduction

As an effective temporary ground improvement technique, artificial ground freezing (AGF) has been widely adopted in geotechnical engineering [

1,

2], including departure and reception of shield tunnel, tunnels connecting passage in Metro, mine shaft sinking and municipal engineering, etc. It has advantages of strong stratum adaptability, good sealing performance, high strength and little influence on the surrounding environment. In ground improvement engineering, the development of the frozen soil wall determines its average temperature, thickness, physical and mechanical properties. These indexes reflect the strength and stability of the frozen soil wall, which is directly related to the scheduling management of projects. In the process of artificial freezing, the temperature evolution of soil is a transient heat conduction problem including ice–water phase change, latent heat release, internal heat source, moving boundary and irregular geometric boundary. The soil temperature distribution is also affected by the interaction of freezing pipes. Therefore, the forming process of frozen soil wall is very complicated, and a better understanding of heat transfer mechanisms is essentially important, which can provide necessary technical guarantees for the implementation of artificial freezing projects.

Based on the steady heat conduction theory, a large number of scholars have proposed the analytical methods of temperature evolution with a single pipe, single or double row pipes [

3,

4,

5]. However, these methods are limited to simplified boundary conditions and idealized initial conditions. Besides, the theoretical formula is too complicated for engineering application. In recent years, numerical methods have been widely applied in heat conduction problems with phase change. Singh [

6] studied the flow and heat transfer characteristics of a phase transition, melting problem. Santos [

7] applied the finite element method to predict freezing times of mushrooms. Farrokhpanah [

8] introduced a new smoothed particle hydrodynamics (SPH) method to model the heat transfer with phase change considering the latent heat released (absorbed) during solidification (melting). Furenes [

9] used the event location algorithm in the finite difference method for phase-change problems.

When compared with the traditional numerical methods, the lattice Boltzmann method (LBM) enjoys advantages of both macroscopic and microscopic approaches [

10,

11]. It has clear physical conception, easy programming, high computational efficiency and is easy to apply for complex domains [

12,

13,

14]. So, LBM has been explored to deal with heat conduction problems with phase change. Miller [

15] proposed a simple model for the liquid-solid phase change based on the lattice Boltzmann method with enhanced collisions. Jiaung [

16] firstly developed an enthalpy-based lattice Boltzmann model for simulating solid/liquid phase change problem governed by the heat conduction equation. Huber et al. [

17,

18,

19] improved this model, and used it to couple thermal convection and phase change of single-component systems. Eshraghi [

20] developed a new variation to solve the heat conduction with phase change by treating implicitly the latent heat source term. Huang [

21] proposed a new lattice Boltzmann model to treat the latent heat source term by modifying the equilibrium distribution function. Sadeghi [

22,

23] proposed a three-dimensional Boltzmann model to study the film-boiling phenomenon. Chatterjee [

24] extended the lattice Boltzmann formulation to simulate three-dimensional heat diffusion coupled with solid–liquid phase change. Li [

25] presented a three-dimensional multiple-relaxation-time lattice Boltzmann model for the solid–liquid phase change based on the enthalpy conservation equation. The above studies mainly neglected the change of thermal diffusivity for simplifying the calculation. However, the thermal diffusivity of liquid water is only 1/9 of ice, and the water content of artificial frozen soil is generally high, so it is necessary to consider the change of thermal diffusivity during the forming process of artificial frozen soil wall.

In this paper, the enthalpy approach is applied to treat the latent heat source term in the energy equation, the adjustable thermal diffusivity is utilized to simulate the change of thermophysical parameters, and a thermal lattice Boltzmann model is proposed to simulate the forming process of artificial frozen soil wall. The model is subsequently tested with the solid–liquid phase change of pure substance in semi-infinite space. Finally, the forming process of an artificial frozen wall is simulated when the four freezing pipes are arranged in a square, the evolution of the freezing front and the temperature distribution are analyzed during the artificial freezing process, which provides the theoretical basis for the design and construction of practical engineering.

## 4. Results and Discussion

In practical engineering, the freezing pipes are usually arranged in a rectangular (or diamond) shape. In this paper, the four freezing pipes arranged in a square are selected as an example, which is shown in

Figure 6. The development of frozen soil wall and temperature distribution are studied during the freezing process. The dimension of physical model is 4.0 m × 4.0 m. The spacing of freezing pipes is 1.2 m, and the outer diameter is 0.12 m. To ensure the mesh accuracy of the freezing pipes, the entire domain is divided into a lattice of 1000 × 1000 grid cells. The temperature of freezing pipes

${T}_{0}$ is kept at −30 °C, the freezing temperature

${T}_{f}$ is 0 °C, and the initial temperature of soil

${T}_{i}$ is 10 °C. The thermophysical parameters of soil are shown in

Table 1. The freezing pipes are set as the constant temperature and the four side boundaries of the model are thermally insulated.

The temporal evolutions of frozen zone are presented in

Figure 7, and the time–history curves of temperature at the points

A,

B,

C,

D and

O are shown in

Figure 8. It can be seen that the temperature at the point

A, which is closer to the freezing pipe, has larger thermal gradient, the soil freezes quickly, the latent heat has little influence on it, and the time–history curve is smooth. The temperature at the points

C,

D and

O, which is farther from the freezing pipe, has a similar temperature evolution trend. The time–history curves show strong multistage, and for point

C it can be divided into four stages as shown in

Figure 8: (1) Cooling: the temperature drops rapidly at the early stage of artificial freezing, and reaches 0 °C in about 10 days. (2) Phase change: when the temperature drops to 0 °C, soil begins to freeze and releases the latent heat. The further away from the freezing pipe, the slower the energy transfers, and the longer the persistent time of phase change stage. (3) Partly frozen: the temperature descends faster in this stage, because there is larger temperature gradient and the thermal diffusivity of frozen soil is greater than that of unfrozen soil. (4) Completely frozen: the temperature evolution is mainly affected by the thermal diffusivity of frozen soil in this stage, the overall trend is relatively stable. For the point

B, the distance from freezing pipe is moderate, the temperature is somewhere in between, and shows insignificant multistage.

Figure 9 shows the temperature distribution in the main section. Under the action of freezing pipes, the soil temperature decreases rapidly in 10 days, and there is funnel-shaped distribution around the freezing pipes. In about 20 days, the frozen soil wall overlaps in the main section, after that the temperature drops rapidly until soil completely frozen between double rows of freezing pipes in about 40 days, and then the temperature decrease rate slows down gradually.

Figure 10 shows the temperature distribution in the intersection. The distance from the freezing pipes is relatively farther, so the temperature development in the intersection is obviously slower than that in the main section. In about 30 days, most of soil has been frozen in the intersection, then the temperature decreases rapidly, and the stable frozen soil wall forms in about 40 days.

The time–history curves of freezing front between two freezing pipes are present in

Figure 11, which show that the closer the spacing of freezing pipe is, the faster the freezing front develops, but the whole difference is not significant. Before the frozen soil wall overlapped, the spacing of freezing pipes has little effect on the evolution of freezing front.

Figure 12 shows the thickness evolution of frozen soil wall at the point

C along the

x direction. The thickness develops faster at the early stage of frozen soil wall overlapped, and the developing speed gradually slows down as time goes on. In general, the thickness evolution has a similar tendency, but for the different freezing pipe spacing, there is a significant difference in overlapping time at the point

O. When the spacing is 1.0 m, 1.2 m, 1.4 m, respectively, the required time for soil freezing is 8 day, 11 day and 16 day from point

C to point

O.

Figure 13 shows the relationship between the overlapping time and the spacing of freezing pipes. The overlapping time of frozen soil wall is greatly influenced by the spacing. The freezing times at the point

C and point

O increase with the increasing of the spacing, and there is a power function relationship between them. Therefore, at the design stage of artificial freezing engineering, the spacing of freezing pipes should be decided according to the overlapping time of artificial frozen soil wall.

## 5. Conclusion

(1) Based on the enthalpy method, a lattice Boltzmann model is proposed to simulate the heat conduction problem with phase change. The model is applied to test the solid–liquid phase change of pure substance, and the results show that the evolution of both temperature distribution and solid–liquid interface are in good agreement with the analytical solutions, and the mesh resolution has little effect on the numerical results.

(2) The temperature evolution of soil is associated with the distance from freezing pipe. When it is closer to the freezing pipe, the time–history curve of temperature is smoother, which is less affected by the latent heat. While it is farther, the time–history curve shows strong multistage, which can be divided into four stages: cooling, phase change, partly frozen, and completely frozen.

(3) Due to the effect of freezing pipes, the soil temperature in the main section decreases rapidly, there is funnel-shaped distribution around the freezing pipes, and the frozen soil wall is overlapped in about 20 days. The temperature development in the intersection is obviously slower compared with that in the main section. In about 30 days, most of soil has been frozen in the intersection, then the temperature decreases rapidly, and the stable frozen soil wall forms in about 40 days.

(4) The spacing of the freezing pipes has a significant influence on the overlapping time of artificial frozen soil wall, and there is a power function relationship between them. But it has little effect on the evolution of freezing front and the thickness of frozen soil wall.