## 3 Numerical Calculation Analysis
### The Effect of the Freezing Tube Pitch  
Figure 12a shows the temperature evolution curves of the interlayer (\(I_{1}\)) and outside (\(O_{1}\)) wall of the shaft wall with different freezing tube pitches (\(L_{d}\)) and freezing time (\(t\)). The temperatures in the interlayer and outside wall position rapidly decrease with an increase in freezing time, while this trend becomes less obvious when the freezing tube pitch increases. Meanwhile, the temperature difference between the interlayer and outside wall position shows a significant decrease when the freezing tube pitch is shortened. The temperature in these two positions ranges from 16.1 \({}^{\circ}\)C and 10 \({}^{\circ}\)C to 12.5 \({}^{\circ}\)C and 3.8 \({}^{\circ}\)C, respectively, when the freezing tube pitch decreases from 2.5 m to 1.3 m (within 180 d). The increasing freezing tube pitch would offer more cooling capacity and a higher heat exchange efficiency, which greatly contributes to the reduction in the shaft temperature. Meanwhile, the temperature in the interlayer and outside wall would be closer due to the rapid heat exchange and cold transfer in the inside wall of the shaft. Therefore, the value of the freezing tube pitch has a greater impact on the temperature of the interlayer and outside wall of the shaft under ventilation conditions.  
The temperature (\(T\)) and thickness (\(\delta\)) evolution curves inside (\(I_{2}\)) and outside (\(O_{2}\)) the frozen soil wall and its average (\(\overline{T}\)) or sum value (\(\delta_{0}\)) with different freezing tube pitches are shown in Figure 12b. The decreasing trend of temperature and the increasing trend of the thickness of the frozen soil wall can be obtained when the freezing tube pitch is 1.3 m, while these two parameters are unable to achieve the desired freezing effect. The sum thickness (\(\delta_{0}\)) of the frozen soil wall with a freezing tube pitch of 2.5 m is approximately 0 m with the freezing time increasing by 150 freezing days, and the frozen soil wall temperature of 0 \({}^{\circ}\)C cannot be exceeded. This value only increases to 0.31 m when the freezing time increases to 180 days. In contrast, the sum thickness of the frozen soil wall with the freezing tube pitch of 1.3 m increases to 2.81 m when the freezing time attains 180 days. The contours in Figure 13a,b clearly show the trend of temperature variation. The temperature of the freezing interface decreases from \(-\)5.6 \({}^{\circ}\)C to \(-\)19 \({}^{\circ}\)C (below 0 \({}^{\circ}\)C) as the freezing time increases when the freezing tube pitch is 1.3 m; however, this temperature only reaches 2.2 \({}^{\circ}\)C (above 0 \({}^{\circ}\)C) when the freezing tube pitch increases to 2.5 m, indicating that the frozen soil wall did not complete the cross-loop in 180 days under ventilation when the freezing tube pitch exceeded a certain threshold. In addition, the frozen area with a freezing tube pitch of 2.5 m is very limited, which may be because the excessive freezing tube pitch disperses the cooling volume and cannot form an effective cooling capacity transfer route, and the formation temperature cannot exceed the freezing temperature due to the considerable loss of cooling capacity, resulting in an unsatisfactory freezing effect and failure of the cross-loop. Therefore, it is important to control the deflection of the freezing tube as much as possible and reduce the freezing tube pitch to create the cross-loop of the frozen soil wall in the expected time under ventilation conditions.