# Three-Dimensional Numerical Modeling of Artificially Freezing Ground in Metro Station Construction
## 3 Methodology
### Calculation Model and Parameter Selection  
According to the freezing hole arrangement scheme, the intersection point of the upper row tunnel axis and the longitudinal section axis of the connecting passage was set as the coordinate origin, with a dimension of the \(X\)-axis direction (longitudinal length) \(\times\)\(Y\)-axis direction (lateral width) \(\times\)\(Z\)-axis direction (vertical height) = 15.33 \(\times\) 10 \(\times\) 10 m. The 4-node meshing format was selected. The mesh near the freezing pipe area (frozen soil curtain) was dense, and the mesh was sparse away from the area, with a total of 1,464,495 elements. Figure 4 schematically shows the geometry and meshed model, using a transient thermal model with phase transition. Adiabatic boundary conditions were set on the outer boundary of the domain were set on the outer boundaries of the domain.  
Table 4 shows the material parameters of the model obtained by the frozen soil tests [27; 28; 29]. In geometric modeling, the freezing pipe body is subtracted from the overall model by Boolean operation, leaving the surface of the freezing pipe as the thermal load boundary and the brine temperature as the boundary load. From the field-measured data, the process of brine cooling during the active freezing period is shown in Table 5. According to the cooling plan, the freezing time step was 40 d, and the length of each step was 24 h.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Soil Layer**} & \multirow{2}{*}{**Density/(kg.m \({}^{-3}\))**} & \multirow{2}{*}{\
\begin{tabular}{c} **Water** \\ **Content/\%** \\ \end{tabular} } & \multicolumn{3}{c}{
\begin{tabular}{c} **Thermal** \\ **Conductivity/(kJ-m\({}^{-1}\) d-\({}^{-1.5}\)C-1)** \\ \end{tabular} } & \multicolumn{3}{c}{**Specific Heat/(kJ-kg\({}^{-1.5}\)C-1)**} & \multicolumn{3}{c}{**Phase Change**} & \multicolumn{1}{c}{**Freeze**} \\  & & & & **Unfrozen Soil** & **Frozen Soil** & **Frozen Soil** & **Heat(\(\times\)108/m\({}^{*}\))** & **Range(\({}^{-1}\)C)** \\ \hline Silty & \multirow{2}{*}{\(1180\)} & \multirow{2}{*}{\(16.8\)} & \multirow{2}{*}{\(118\)} & \multirow{2}{*}{\(179\)} & \multirow{2}{*}{\(1.53\)} & \multirow{2}{*}{\(1.61\)} & \multirow{2}{*}{\(1.20\)} & \multirow{2}{*}{\([-1,0]\)} \\  & & & & & & & & \\ \end{tabular}
\end{table}
Table 4: Thermal parameters of soils.  
Figure 4: Mesh size and geometric size of the numerical model. (**a**) mesh and geometric size of the model; (**b**) distribution of freeze pipes.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Time/d**} & \multirow{2}{*}{**0**} & \multirow{2}{*}{**1**} & \multirow{2}{*}{**5**} & \multirow{2}{*}{**10**} & \multirow{2}{*}{**15**} & \multirow{2}{*}{**20**} & \multirow{2}{*}{**30**} & \multirow{2}{*}{**40**} \\  & & & & & & & & \\ \hline Temperature/\({}^{\circ}\)C & 18 & \(-2.5\) & \(-30\) & \(-30\) & \(-30\) & \(-30\) & \(-30\) & \(-30\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Freezing time plan for brine temperature.  
As shown in Figure 3, the drilling characteristics of the temperature measurement holes, C1-C7, are listed in Table 6. According to the layout of the temperature measuring holes, 7 paths and the analysis points on the corresponding positions of the numerical model were taken for the present study in order to analyze the temperature field distribution of the frozen soil curtain of the connecting passage. The analysis points on the 7 paths were set every 0.25 m.