# Formation of a Freezing Wall Around a Vertical Shaft Under Localized Freezing
## 4 Results and Discussion  
### Evolution of the Freezing Wall During the Active Freezing Period  
#### 4.1.1 Morphology of the Freezing Wall Under Typical Parameters  
The temperature field evolution under typical parameters (freezing temperature \(T_{f}=-30\)\({}^{\circ}\)C, initial stratum temperature \(T_{0}=24\)\({}^{\circ}\)C, and shaft lining thickness \(H=2\) m) is illustrated in Figure 9. As localized freezing occurs around the shaft lining, a low-temperature zone is formed, and the thermally conductive adhesive extracts heat from the concrete shaft lining and the surrounding strata. Due to the thermal transfer, a freezing front forms and expands into the clay stratum, forming a freezing curtain covering the outer periphery of the shaft lining. The thickness of the freezing wall reaches 1.2 m on the 123 rd day, meeting the slotting criteria. The freezing wall expands gradually from the inner to the outer part. This growth pattern provides ample drainage pathways for unfrozen water, preventing the formation of unfrozen water pockets between the freezing wall in the clay and the shaft lining and significantly reducing the risk of frost-heave-induced damage.  
A 90-day (d) monitoring period was adopted in the numerical study. The evolution of the freezing wall thickness is illustrated in Figure 10. During the initial freezing period, the freezing wall had a rounded semi-rectangle shape, with a flat central section and rounded transitions at both ends. As the freezing duration increased, the freezing wall expanded radially. The flat central portion gradually diminished, and the shape changed to an arch. Notably, the vertical span of the freezing wall exceeded the vertical dimension of the freezing pipes after 180 days of freezing when an external thermal insulation layer was applied. Subsequently, we used two monitoring paths to analyze the thickness and temperature distributions of the freezing wall and compare its evolution under different influencing factors. Monitoring Path 1 was positioned at the geometric center of the slotting zone, and Monitoring Path 2 was located 2 m inward from the upper boundary of the freezing pipe array (Figures 5 and 10).  
As illustrated in Table 3, the freezing wall thickness along Monitoring Path 1 was 0.74 m, 1.89 m, 2.78 m, and 3.48 m at 90, 180, 270, and 360 days after freezing, respectively. The corresponding thicknesses along Monitoring Path 2 were 0.44 m, 1.27 m, 1.87 m, and 2.33 m. The growth rates and average temperatures of the freezing wall for both monitoring paths are presented in Figure 11. The growth rate of the thickness increased and decreased during the freezing period. The average daily growth rates were 0.0082 m/d, 0.0105 m/d, 0.0103 m/d, and 0.0097 m/d for Monitoring Path 1 and 0.0049 m/d, 0.0071 m/d, 0.0069 m/d, and 0.0065 m/d for Monitoring Path 2 after 90, 180,  
Figure 9: Nephogram of the temperature field evolution for typical parameters.  
270, and 360 days, respectively. The average temperature of the freezing wall declined continuously, with temperatures of \(-\)5.42 \({}^{\circ}\)C, \(-\)8.86 \({}^{\circ}\)C, \(-\)10.41 \({}^{\circ}\)C, and \(-\)11.18 \({}^{\circ}\)C for Monitoring Path 1 and \(-\)4.18 \({}^{\circ}\)C, \(-\)6.79 \({}^{\circ}\)C, \(-\)7.78 \({}^{\circ}\)C, and \(-\)8.49 \({}^{\circ}\)C for Monitoring Path 2 in the same time intervals.