## 3 Numerical Calculation Analysis
### The Effect of the Thermal Conductivity of Unfrozen Soil  
The temperature evolution curves of the interlayer (\(I_{1}\)) and outside (\(O_{1}\)) walls of the shaft wall with different thermal conductivity (\(\lambda\)) and freezing time (\(t\)) are plotted in Figure 16a. The increase in the thermal conductivity of unfrozen soil could reduce the temperature in the interlayer and outside wall position by enhancing the cooling energy transfer throughout the stratum. Meanwhile, the variance in the temperature in the interlayer and outside wall position is relatively smaller when the thermal conductivity of unfrozen soil increases from 1.2 W\(\cdot\)m\({}^{-1.\circ}\)C\({}^{-1}\). to 1.5 W\(\cdot\)m\({}^{-1.\circ}\)C\({}^{-1}\). For example, the temperature in these two positions decreases by approximately 0.5 \({}^{\circ}\)C and 1 \({}^{\circ}\)C when the thermal conductivity of unfrozen soil increases by 0.1 W\(\cdot\)m\({}^{-1.\circ}\)C\({}^{-1}\) (180 d). Figure 16: Variance in temperature and thickness under different thermal conductivities of unfrozen soil and freezing time: (**a**) temperature in interlayer and outside wall of shaft, (**b**) temperature and thickness in the inside and outside of frozen soil wall and its average or sum value with thermal conductivities of unfrozen soil of 1.2 W\(\cdot\)m\({}^{-1.\circ}\)C\({}^{-1}\) and 1.5 W\(\cdot\)m\({}^{-1.\circ}\)C\({}^{-1}\). As expected, the temperature difference magnitude inside (\(I_{2}\)) and outside (\(O_{2}\)) the frozen soil wall induced by the increase in the thermal conductivity of unfrozen soil reaches approximately 0.6 \({}^{\circ}\)C\({}^{-1}\), while the sum thickness (\(\delta_{0}\)) of the frozen soil wall increases by 0.34 m when the thermal conductivity of unfrozen soil increases from 1.2 W\(\cdot\)m\({}^{-1}\)\({}^{\circ}\)C\({}^{-1}\) to 1.5 W\(\cdot\)m\({}^{-1}\)\({}^{\circ}\)C\({}^{-1}\) (180 d), as shown in Figure 16b. The higher thermal conductivity of unfrozen soil can clearly enhance the transfer of cooling energy from the low-temperature region to the high-temperature region, reducing the temperature and forming the frozen soil wall at a higher speed when the flow rate and temperature in the freezing tube are fixed. In addition, the average temperature (\(\overline{T}\)) and thickness in the inside position are always below those of the outside position, while the temperature and thickness difference between these two positions increases when the thermal conductivity of unfrozen soil increases. The wider outer space provides better conditions for the formation of the frozen wall. The heat transfer efficiency would be improved when the thermal conductivity of unfrozen soil increases; thus, the conditions for the temperature reduction in the outside frozen soil wall become more favorable, the temperature increases, and then the thickness obviously increases. The temperature gradually decreases with the increase in freezing time, while the freezing region around the freezing tube increases with an increase in thermal conductivity of unfrozen soil, as shown in Figure 17a,b. The second low-temperature region with a thermal conductivity for unfrozen soil of 1.5 W\(\cdot\)m\({}^{-1}\)\({}^{\circ}\)C\({}^{-1}\) gradually increases from a small semicircle to a large ellipse, which occurs throughout the whole model and is more than that of 1.2 W\(\cdot\)m\({}^{-1}\)\({}^{\circ}\)C\({}^{-1}\). In addition, the temperature of the freezing interface decreases from 0.5 \({}^{\circ}\)C to \(-\)8 \({}^{\circ}\)C (below 0 \({}^{\circ}\)C) as the freezing time increases when the thermal conductivity of unfrozen soil is 1.2 W\(\cdot\)m\({}^{-1}\)\({}^{\circ}\)C\({}^{-1}\), and this temperature decreases to approximately \(-\)14 \({}^{\circ}\)C at 180 d when the thermal conductivity of unfrozen soil increases to 1.5 W\(\cdot\)m\({}^{-1}\)\({}^{\circ}\)C\({}^{-1}\). The cross-loop of the frozen soil wall is completed in 180 d under ventilation conditions.