## 3 Numerical Calculation Analysis
### The Effect of Soil Moisture Content  
Figure 18a shows the temperature evolution curves of the interlayer (\(I_{1}\)) and outside (\(O_{1}\)) wall of the shaft wall with different soil moisture contents (\(\omega\)) and freezing time. The temperature in the interlayer and outside wall positions gradually increase with an increase in soil moisture content, while the temperature variance magnitude between these two positions also increases when the soil moisture content increases from 10% to 30%. The increases in soil moisture content enhance the energy requirements for the reduction in unit temperature per unit volume of soil, particularly boosting the energy required for the latent heat of the phase transition; thus, the temperature of the shaft increases when the soil moisture content increases. In addition, the effect of the variation in the soil moisture content on the shaft wall is less than that of the change in the freezing tube parameters; the temperature in these two positions ranges from 12.5 \({}^{\circ}\)C and 3.7 \({}^{\circ}\)C to 13.7 \({}^{\circ}\)C and 5.9 \({}^{\circ}\)C, respectively, when the soil moisture content increases from 10% to 30% (the saturation ranges from approximately 33% to 100%). The temperatures (\(T\)) of the frozen soil wall's inside (\(I_{2}\)) and outside (\(O_{2}\)) positions both show increasing trends with the soil moisture content of 10% and 30%, while the time needed for the soil moisture content of 30% to freeze exceeds 30 days, as shown in Figure 18b. The thickness (\(\delta\)) of the frozen soil wall with a soil moisture content of 30% gradually increases after 30 days, and the sum (\(\delta_{0}\)) thickness of this material is less than that of 10% soil moisture content. The increasing energy required for the latent heat of the phase transition with a higher soil moisture content would delay the formation of the frozen soil wall and then decrease its thickness. This result indicates that the soil moisture content plays an important role in the initial freezing generation under ventilation conditions. Figure 17: Temperature contours within 30 d, 90 d and 180 d: (**a**) thermal conductivity of unfrozen soil of 1.2 W\(\cdot\)m\({}^{-1}\)\(\cdot\)\(\circ\)C\({}^{-1}\), (**b**) thermal conductivity of unfrozen soil of 1.5 W\(\cdot\)m\({}^{-1}\)\(\cdot\)\(\circ\)C\({}^{-1}\). The temperature contours in Figure 19a,b show that the temperature of the freezing interface decreases from \(-5.6\)\({}^{\circ}\)C to \(-13\)\({}^{\circ}\)C (below \(0\)\({}^{\circ}\)C) as the freezing time increases when the soil moisture content is 10%, and this temperature decreases to \(-8\)\({}^{\circ}\)C at 180 d when the soil moisture content increases to 30%. The cross-loop of the frozen soil wall is completed in 180 days, even when the soil moisture content increases to 30%, indicating that the change in soil moisture content would not have an obvious effect on the cross-loop. In addition, the freezing temperature range around the freezing tube dramatically increases when the soil moisture content decreases, which also proves that the soil moisture content plays an important role in the initial freezing generation. Figure 18: Variance in temperature and thickness under different soil moisture contents 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 soil moisture content of 10% and 30%. position is no more than 3 \({}^{\circ}\)C, while the gap in the interlayer position is approximately 4 \({}^{\circ}\)C when the freezing time is the same (within 180 d). The increase in shaft wall thicknesses will prevent the loss of cooling energy and thus contribute to the freezing effect due to the lower heat transfer efficiency of the shaft wall, which will reduce the temperature of the whole shaft wall.