## 3 Finite Element Modeling  
Based on the Thermal module of ADINA, a numerical model was developed and its geometry size was based on the actual dimensions of the tunnel. This model simulates the real freezing situation by aligning the freezing tubes with the actual layout of the freezing tubes. In order to reasonably describe the evolution of the temperature field in the soil, the model employs a transient thermal conductivity double-tunnel model with phase changes since the temperature changes with time during the freezing process are accompanied by phase changes of water and ice at the same time. The energy generated by latent heat during the ice-water phase change is simplified to the change in heat capacity and thermal conductivity of the soil in the frozen and unfrozen regions. Therefore, this item is not written in the governing equations. The governing equation used in this study is shown below. For the frozen soil  
\[C_{\rm f}\frac{\partial T_{\rm f}}{\partial t}=\frac{\partial}{\partial_{\rm x }}\left({\rm k}_{\rm f}\frac{\partial T_{\rm f}}{\partial_{\rm x}}\right)+ \frac{\partial}{\partial_{\rm y}}\left({\rm k}_{\rm f}\frac{\partial T_{\rm f }}{\partial_{\rm y}}\right)+\frac{\partial}{\partial_{\rm z}}\left({\rm k}_{ \rm f}\frac{\partial T_{\rm f}}{\partial_{\rm z}}\right) \tag{1}\]  
For the unfrozen soil  
\[C_{\rm u}\frac{\partial T_{\rm u}}{\partial t}=\frac{\partial}{\partial_{\rm x }}\left({\rm k}_{\rm u}\frac{\partial T_{\rm u}}{\partial_{\rm x}}\right)+ \frac{\partial}{\partial_{\rm y}}\left({\rm k}_{\rm u}\frac{\partial T_{\rm u }}{\partial_{\rm y}}\right)+\frac{\partial}{\partial_{\rm z}}\left({\rm k}_{ \rm u}\frac{\partial T_{\rm u}}{\partial_{\rm z}}\right) \tag{2}\]  
where \(C_{\rm u}\) is the volumetric heat capacity of the soil; T is temperature; \({\rm k}_{\rm f}\) is the thermal conductivity of the soil. A densely meshed model can accommodate more cells per unit interval, which can improve the accuracy of the calculations. Thus, in this numerical model, the mesh density \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \multirow{2}{*}{**Hole Type**} & **Drill Hole** & **Hole Depth** & **Positioning** & **Elevation** & **Perforated** & **Total Hole** & **Depth (m)** \\  & **Number** & **(m)** & **Angle (\({}^{\circ}\))** & **Angle of** & **Horizontal** & **Hole** \\  & & **(m)** & **Angle (\({}^{\circ}\))** & **Correlation (\({}^{\circ}\))** & **Angle (\({}^{\circ}\))** & **Depth (m)** \\ \hline \multirow{4}{*}{Temperature} & C1\(\sim\)C2 & 2.0 & 0 & 0 & 0 & 4.0 \\  & C3 & 2.0 & 40 & 31 & 0 & 2.0 \\ \cline{1-1} measurement & C4 & 3.0 & \(-\)44 & \(-\)18.4 & 0 & 3.0 \\ \cline{1-1} holes & C5\(\sim\)C6 & 2.0 & \(-\)21 & 0 & 0 & 4.0 \\  & C7\(\sim\)C8 & 2.0 & \(-\)44 & 0 & 0 & 4.0 \\ \hline \multirow{4}{*}{Pressure relief hole} & X1 & 2.0 & & 0 & 0 & 2.0 \\  & X2 & 2.0 & & 0 & 0 & 2.0 \\ \cline{1-1}  & X3 & 2.0 & & 0 & 0 & 2.0 \\ \cline{1-1}  & X4 & 2.0 & & 0 & 0 & 2.0 \\ \hline Total & & & & & & 25.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of depth and pitch (elevation) angle of temperature measurement and pressure relief holes. was set at 0.5 m near the frozen tube area, with complex shape and severe temperature changes and 1.5 m at the boundary of the frozen tube, in order to reduce the number of calculations and ensure higher accuracy. After free division, the model consisted of 1,455,790 cells after meshing in a 4-node meshing mode with a merging tolerance of 0.0001. As the temperature is time-dependent, in the finite element calculation, the temperature-time curve was discretized into load steps. Each load step consisted of multiple sub-steps, and each sub-step used an iterative algorithm. In this study, 1 day was used as a time step to divide, resulting in a period of 40 days of active freezing. Based on the strata survey report and the actual brine cooling scheme that was employed in the project, the model calculations were performed by entering the soil material parameters into the corresponding module of the software.