## 5 Analysis of Numerical Results
### Spatiotemporal Evolution Law of the Temperature Field before Exaxation
#### 5.2.2 Average Temperature and Thickness of the Effective Freezing Wall  
The effective frozen wall represents the frozen wall retained after removing a part of the frozen soil cleared by shaft excavation from the original frozen wall. The average temperature of the frozen wall is the weighted average of the soil temperature below the freezing point in its area [27,28]. As shown in Figure 10a, the average temperature of the frozen walls in both soil layers decreases significantly with the freezing time. The frozen wall of the fine sand layer maintains a lower average temperature compared with that of the medium sand layer, attributed to its lower initial ground temperature and the impact of an external heat source on the frozen wall of the medium sand layer, resulting in a higher temperature than the fine sand layer. During the freezing process, the average temperature of the frozen walls in both the soil layers decreased as time progressed, with the frozen wall of the fine sand layer showing a roughly negative linear correlation with time. The thermal trend in the medium sand layer can be broadly divided into three stages: 1 During the first 50 days of freezing, the average-temperature change trends in the frozen walls of both the soil layers were nearly identical, with average-temperature decline rates of 0.0588 \({}^{\circ}\)C/day and 0.0601 \({}^{\circ}\)C/day, respectively. 2 Between 50 and 68 days of freezing, the frozen wall extended to the excavation path, causing the outer frozen soil to come into contact with the surrounding soil layers. This led to a considerable amount of heat exchange between them, consuming the cold energy released by the low-temperature brine and causing the frozen wall temperature of the medium sand layer to decrease gradually. 3 After 68 days of freezing, the average temperature of the frozen wall within the medium sand layer dropped quickly at a rate of 0.1030 \({}^{\circ}\)C/day, while the fine sand layer experienced a reduction rate of 0.0931 \({}^{\circ}\)C/day. Considering the freezing speed, the average temperature reduction rate of the frozen wall in the medium sand layer during this stage was higher than that of the frozen wall in the fine sand layer. This is because the thermal conductivity of the medium sand layer was higher after freezing compared with that of the fine sand layer, leading to a more efficient heat transfer and a quicker temperature drop. After freezing for 88 days, the frozen walls of the two soil layers had average temperatures of \(-\)13.99 \({}^{\circ}\)C and \(-\)12.11 \({}^{\circ}\)C, resulting in an overall average temperature of \(-\)12.81 \({}^{\circ}\)C.  
Figure 9: Overall shape and distribution of the frozen wall after 88 days of freezing. Figure 10b shows the relationship between the effective frozen wall thickness and freezing time. The thickness of the effective frozen wall exhibited an upward exponential trend. In the initial stages of the frozen wall closure, the thickness of both layers of the frozen wall increased rapidly, with the increase rate gradually decreasing over time. This can be explained by the fact that the frozen wall expanded to both sides simultaneously at the beginning of the closure. When it extended to the excavation path, the outer frozen soil came into contact with the surrounding soil layers, thereby decreasing the growth rate of the frozen wall thickness. Under the same freezing time conditions, the frozen wall formed by the fine sand layer was thicker than that formed by the medium sand layer. The complete intersection times of the frozen walls in the fine and medium sand layers were 24 days and 33 days, respectively. At these intersection times, the average effective frozen wall thicknesses of the two soil layers were 4 m and 3.8 m, respectively. Notably, the thickness of the frozen wall in the fine sand layer had reached the design thickness of the frozen wall of the side wall (4 m). After 88 days of freezing, the effective frozen wall thickness of the two soil layers reached 5.22 m and 4.85 m, respectively.