# Formation of a Freezing Wall Around a Vertical Shaft Under Localized Freezing
## 4 Results and Discussion
### Evolution of the Freezing Wall During the Slotting Period
#### 4.2.2 Influence of Slotting Duration on the Evolution of the Freezing Wall  
The temperature field nephogram for different slotting durations and a slot height of 1.5 m is illustrated in Figure 21. As the slotting duration increased, the exposed concrete shaft lining in the slotting zone warmed due to convective heat transfer. The affected area expanded toward the adjacent freezing wall near the shaft lining. The upper and lower freezing pipes transferred the cooling capacity to the freezing wall through the low-temperature shaft lining. The excavation did not affect the temperature field at the outer periphery of the freezing wall, and the freezing wall expanded outward.  
The temperature evolution in Monitoring Path 1 for different slotting durations is illustrated in Figure 22. As the slotting duration increased, the horizontal range of the warming zone adjacent to the slotting zone increased. The upper and lower freezing pipes near the slotting zone extracted heat from the concrete shaft lining and the surrounding clay stratum. Due to the combined effects of convective heat transfer in the slotting zone and the cooling caused by the freezing pipes, a turning point (Figure 22) occurred on the temperature curve at 1.21-1.22 m from the outer edge of the outer shaft lining. When the distance from the outer edge of the outer shaft lining was less than the value of the turning point, the temperatures increased with the slotting duration. In contrast, the temperature decreased at positions larger than this value.  
Figure 21: Nephogram of the temperature field for different slotting durations after 15 days.  
Figure 22: Temperature evolution in Monitoring Path 1 for different slotting durations.  
As the slotting duration increased, the horizontal range of the temperature rise adjacent to the slotting zone increased. The horizontal ranges of the warming zone were 0.61 m, 0.81 m, 0.97 m, and 1.04 m for slotting durations of 5 d, 10 d, 15 d, and 20 d, respectively. The final temperatures in the regions of temperature increase were below \(-4\)\({}^{\circ}\)C (\(<\)\(-2\)\({}^{\circ}\)C). Although a temperature increase was observed in the freezing wall adjacent to the shaft lining during slotting, the freezing wall never thawed, and the frozen zone expanded outward.  
As depicted in Figures 22 and 23, the freezing wall expanded outward during slotting at a freezing temperature of \(-2\)\({}^{\circ}\)C. The rate of increase in the thickness of the freezing wall increased and decreased with the slotting duration, reaching the maximum value on the 15 th day of slotting. Conversely, the average temperature increased during the slotting period. The freezing wall thicknesses were 1.29 m, 1.31 m, 1.32 m, and 1.31 m after 5 days, 10 days, 15 days, and 20 days, respectively, at a slot height of 1.5 m, and the average temperatures were \(-6.86\)\({}^{\circ}\)C, \(-6.03\)\({}^{\circ}\)C, \(-5.65\)\({}^{\circ}\)C, and \(-5.43\)\({}^{\circ}\)C. The thickness of the freezing wall was 0.09 m, 0.11 m, 0.12 m, and 0.11 m larger, and the average temperature was 0.18\({}^{\circ}\)C, 1.01 \({}^{\circ}\)C, 1.39 \({}^{\circ}\)C, and 1.61 \({}^{\circ}\)C higher after slotting than before slotting.