## 3 Analysis of Temperature Field of Frozen Wall
### Numerical Results and Discussion
#### 3.3.2 Average Temperature and Thickness of Effective Frozen Wall  
The effective frozen wall represents the original frozen wall in each layer of this study minus part of frozen soil removed by shaft excavation and the retained frozen wall. Due to the random deflection of the freezing holes, the thickness of the frozen wall in different directions is not uniform, so the thickness in four different directions is taken in this study, and the average thickness of the four directions is taken as the research index of the frozen wall thickness. The average temperature of the frozen wall is the weighted average of the temperature in its area. By extracting the effective frozen wall thickness and average temperature data from the numerical model, the variation in the effective frozen wall thickness and average temperature with time is obtained, as shown in Figure 6.  
(1) Average temperature of frozen wall  
It can be seen from Figure 6 that the average temperature of the frozen wall can be roughly divided into five stages. The first stage is from 40 to 60 d, during which the average temperature of the frozen wall decreases rapidly with the freezing time. The second stage is from 60 to 80 d; during this period, the average temperature of the frozen wall decreases slightly with the freezing time and generally remains stable. The average temperature of the frozen wall increases with the freezing time from 80 to 96 d, and it decreases with the freezing time from 96 to 109 d. It again increases with the freezing time from 109 to 120 d. The freezing process can be divided into active freezing period and maintenance freezing period. After the on-site freezing begins, the freezing unit pumps the continuously cooled brine through the circulation system to the freezing pipe embedded in the soil. By circulating the cold brine to cool the formation, the water in the formation is frozen, and a closed frozen wall is formed around the working face to resist the water and soil pressure from the formation. According to the brine temperature measured on site, after freezing for 20 to 30 d, the brine temperature in the main freezing holes and auxiliary holes reaches the designed low temperature level. Further, the brine temperature remains basically stable and then enters the active freezing stage (80 d before freezing). According to Figure 3, after freezing for 80 d, the brine temperature circulating in the auxiliary holes is in a state of constant fluctuation, so 80-120 d is the maintenance freezing stage. In this stage, the average temperature of the frozen wall rises and falls with the temperature of brine circulating in the frozen hole and shows the same variation trend. Therefore, the average temperature of the frozen wall rises (drops) as the temperature of brine circulating in the freezing holes increases (decreases). According to the above analysis,  
Figure 6: Variation in the thickness and average temperature of effective frozen wall. in the active freezing stage, the average temperature of the frozen wall decreases rapidly after overlap and then remains stable after reaching a certain level. In the maintenance freezing stage, the average temperature of the frozen wall changes with the temperature of the circulating brine and the two temperatures are synchronized. For the same freezing time, the average temperature of fine sand is the lowest, followed by that of sandy clay, and the average temperature of clay is the highest. In the active freezing period, the average temperature of the fine sand frozen wall is 0.2-1.0 and 0.5-2.5 \({}^{\circ}\)C lower than that of the sandy clay layer and clay layer, respectively. The difference decreases with the freezing time. During the maintenance freezing period, the average temperature of the fine sand frozen wall remains basically the same as that of the sandy clay layer, but it is 0.3-0.5 \({}^{\circ}\)C lower than that of clay frozen wall.