## 2 Mathematical Model of the Freezing Temperature Field and Related Physical Parameters
### Calculation Procedure  
The boundary temperature of the far field at 30 m radius is the original ground temperature, and the temperature of each unit of the soil layer at the beginning of freezing is the original ground temperature (19.5 \({}^{\circ}\)C), which is a fixed value due to its relatively lower effect on the formation and development characteristics of frozen soil wall. The inner surface temperature and air temperature of the shaft wall are 25 \({}^{\circ}\)C, the initial temperature of the soil is 19.5 \({}^{\circ}\)C, the temperature of saline water is \(-\)30 \({}^{\circ}\)C in the active freezing period (0-60 d), and the temperature of saline water is \(-\)25 \({}^{\circ}\)C in the maintenance freezing period (61-180 days), according to the actual measured temperature over 12 months. At the convective heat transfer boundary, the temperature of the air and shaft wall is 25 \({}^{\circ}\)C. The temperature at the outer wall of the freezing tube is the temperature value of saline water, and the whole freezing time is 180 d.  
The following two scenarios are considered for the calculation scheme and parameters:  
(1) The influence of various influencing factors on the development laws of the frozen soil wall is studied when the boundary of convection heat transfers between the air in the well, and the shaft wall is selected in the case of ventilation in the well. The value ranges of all parameters were selected, and shown in Tables 2 and 3: \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value Ranges** & **Number of Values** &
\begin{tabular}{c} **Value in the East** \\ **Ventilation Shaft** \\ \end{tabular} \\ \hline Convective heat transfer coefficient & 5, 10, 20, 30, 40, 100 & 6 & 25 \\ Layout radius of freezing tube (\(R_{d}\))/m & 5, 5, 5.7, 5.9, 6.1, 6.3 & 5 & 6.5 \\ Tube pitch (\(L_{d}\))/m & 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, & 13 & 1.45 \\ Diameter of freezing tube (\(D\))/mm & 140, 159 & 2 & 140 \\ Thermal conductivity of unfrozen soil & 1.2, 1.3, 1.4, 1.5 & 4 & 1.4 \\ (\(\lambda_{1}\))/W\(\cdot\)m\({}^{-1.5}\)/c\({}^{-1}\) & 1.44, 1.56, 1.68, 1.8 & 4 & 1.68 \\ Thermal conductivity of frozen soil & 10, 15, 20, 25, 30 & 5 & 20 \\ (\(\lambda_{2}\))/W\(\cdot\)m\({}^{-1.5}\)/c\({}^{-1}\) & 1.44, 1.56, 1.68, 1.8 & 4 & 1.68 \\ Moisture content (\(\omega\))/\% & 10, 15, 20, 25, 30 & 5 & 20 \\ The thickness of the wall (\(\delta\))/m & 1.1, 1.5 & 2 & 1.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Value ranges of each parameter in the single-factor calculation. Figure 5: Geometric model. (2) To further study the effect of the main factors on the development laws of the frozen soil wall, a full combination calculation of these four factors was studied based on the single-factor calculation results. The value ranges of all parameters were selected and shown in Tables 4 and 5, including 384 sets of numerical experiments: \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Value Ranges** & **Number of Values** & **Step Size** \\ \hline Layout radius of freezing tube (\(R_{d}\))/m & 6.25, 6.5, 6.75, 7.0 & 4 & 0.25 \\ Tube pitch (\(L_{d}\))/m & 1.2, 1.4, 1.6, 1.8, 2.0, 2.2 & 6 & 0.2 \\ Thermal conductivity of unfrozen soil & 1.2, 1.4, 1.6, 1.8 & 4 & 0.2 \\ \((\lambda_{1})\)/W\(\cdot\)m\({}^{-1.\circ}\)C\({}^{-1}\) & 15, 20, 25, 30 & 4 & 5 \\ \hline \end{tabular}
\end{table} 
Table 4: Value ranges of each parameter in the full combination calculation.