## 1 Introduction  
Artificial ground freezing (AGF) is a soil stabilization technique, which is used to freeze the water, rock, and soil in a formation by circulating a low-temperature refrigerant (usually low-temperature brine or liquid nitrogen) in the freezing tubes embedded in the rock and soil layer. The frozen wall with certain strength and low permeability can resist the earth pressure of groundwater, isolate the groundwater, and ensure the safe implementation of civil engineering projects [1]. This method is environmentally friendly and can be adapted for complex engineering conditions [2]. In the 1880s, this method was proposed for soft soil tunneling, mining, and excavation of underground regions. Since the 1950s, it has been widely used in the United States, Australia, China, and other countries [3]. In recent years, owing to its unique advantages, the AGF method has been increasingly used in the excavation of tunnels in regions rich in water and soft soil [4; 5; 6; 7; 8], deep foundation pit excavation [9; 10], and other projects. Because coal mine shafts generally have a large depth, the topsoil is thick, and the geological and hydrological conditions of the topsoil are complex, so the freezing method is also widely used in coal mine shafts and has received considerable research attention. [PERSON] et al. [11; 12; 13; 14] derived the analytical solution of the steady-state temperature field under different arrangements of freezing pipes, which greatly promoted the theoretical application of the freezing method. [PERSON] et al. [15] analyzedthe development law of the temperature field of frozen wall in the deep alluvial multicircle holes and proposed a new numerical model for the freezing temperature field. The feasibility of the numerical model was verified through field measurement. Based on the actual engineering conditions of the Banji coal mine, [PERSON] et al. [16] studied the application of the differential control freezing technology in the repair of damaged wellbore in deep alluvium by using numerical simulation from the perspective of temperature field development and distribution law. To understand the temporal and spatial evolution law of the frozen wall temperature field in the deep alluvial layer, it is essential to examine the characteristic surface temperature, average temperature, and thickness of the frozen wall of the multi-turn tubes in the deep alluvial layer [17; 18; 19]. In the construction of coal mine shafts using AGF method, if the groundwater velocity is greater than 5 m/d, the influence of the groundwater velocity on the freezing temperature field should be considered in the construction design of the freezing method and the analysis of the temperature field [20; 21; 22]. [PERSON] et al. [23] studied the development law of the freezing temperature field of an expansive clay layer in the Yangcun Mine by using a large-scale physical model which can guide engineering applications. Based on the study of the freezing temperature field, the frozen wall is generally regarded as a temperature-dependent material, and its mechanical properties have been examined. Some researchers considered the temperature field of the frozen wall to be homogeneous and analyzed the elastoplasticity of the frozen wall to obtain the distribution of the displacement field and stress field under the action of external loads [24; 25; 26]. However, some scholars regarded the frozen wall as a non-homogeneous functionally graded material along the radial direction, simplified the radial temperature field of the frozen wall, and conducted an elastoplastic analysis of the frozen wall considering the excavation unloading effect to obtain the distribution law of stress field and displacement [27; 28; 29; 30]. When the AGF method is used to excavate the shaft, due to the complexity of the deep soil geological environment and the diversity of soil properties, it is necessary to study the freezing temperature field of the deep soil.