## 4 Analysis of Simulation Results  
### Temperature Field Calculation Results  
In any freezing project, understanding the development of the temperature field is crucial for analyzing the freezing effect. Thus, the temperature field changes in the spatial section of freezing for 10, 20, 30, and 45 days were extracted, and freezing temperature iso-surface diagrams were constructed, as shown in Figure 4. As seen in Figure 4, to avoid the adverse effects of freezing caused by rapid heat exchange, the cooling pipe maintains the same temperature as the initial ground temperature at the beginning of cooling. On the first day of freezing, a single-hole temperature development mode is observed with the freezing pipes all reduced to below 18 \({}^{\circ}\)C. By the 10 th day of freezing, the temperature of the freezing pipes reaches the critical phase change temperature of 0 \({}^{\circ}\)C. By the 20 th day of freezing, the temperature of the freezing pipes reaches \(-\)18 \({}^{\circ}\)C, and that of the surrounding \begin{table}
\begin{tabular}{c c c} \hline \hline
**Type of Parameter** & **Parameter** & **Value** \\ \hline Soil density (kg/m\({}^{3}\)) & 1930 \\ Water density (kg/m\({}^{3}\)) & 1000 \\ Ice density (kg/m\({}^{3}\)) & 920 \\ Thermal conductivity of soil (W/(m\({}^{\circ}\)C)) & 1.22 \\ Soil thermophysical parameters & Thermal conductivity of water (W/(m\({}^{\circ}\)C)) & 0.63 \\ Thermal conductivity of ice (W/(m\({}^{\circ}\)C)) & 2.31 \\ Specific heat capacity of soil (J/(kg\({}^{\circ}\)C)) & 1530 \\ Specific heat capacity of water (J/(kg\({}^{\circ}\)C)) & 4200 \\ Specific heat capacity of ice (J/(kg\({}^{\circ}\)C)) & 2100 \\ Latent heat of phase transition (J/kg) & 334,720 \\ \hline Soil mechanical parameters & Elastic modulus of unfrozen soil (MPa) & 40 \\ Angle of internal friction in unfrozen soil (\({}^{\circ}\)) & 120 \\ Poisson’s ratio for unfrozen soil & 30 \\ Poisson’s ratio for frozen soil & 0.23 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calculation parameters for the most unfavorable soil layer. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Time/d** & 0 & 10 & 20 & 30 & 45 \\ \hline
**Temperature/\({}^{\circ}\)C** & 18 & 0 & \(-\)20 & \(-\)28 & \(-\)28 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Brine cycle freezing tube cooling program. Figure 3: Schematic diagram of the geometric mode (**a**) and meshing (**b**). soil reduces to below the permafrost temperature of \(-1\)\({}^{\circ}\)C, initiating the formation of the freezing curtain. Individual unconnected areas appear in the weak freezing zone. By the 30 th day of freezing, the temperature of the freezing pipes reaches the lower limit of the freezing temperature at \(-28\)\({}^{\circ}\)C. The soil in the weak freezing zone reaches temperatures below \(-20\)\({}^{\circ}\)C, and the soil in the strong freezing zone reaches temperatures below \(-25\)\({}^{\circ}\)C, resulting in the complete formation of the freezing curtain. By the 45 th day of freezing, the freezing curtain significantly thickens, with the thickness of the frozen soil in the strong freezing zone significantly exceeding that in the weak freezing zone. The length of the frozen pipe in this horizontal freezing project is the same, so the \(Z=5\) m section was analyzed to understand the characteristic changes over time. Specifically, the \(-1\)\({}^{\circ}\)C and \(-10\)\({}^{\circ}\)C isotherms for the \(Z=5\) m section were extracted after 15, 22, 25, 28, 30, and 45 days of freezing, as shown in Figure 5. Based on this figure, when the freezing pipe brine temperature reaches \(-10\)\({}^{\circ}\)C on the 15 th day of freezing, the \(-1\)\({}^{\circ}\)C and \(-10\)\({}^{\circ}\)C isotherms expand outward from the freezing pipe in concentric circles. In the weak freezing zone, the cold volume is more dispersed, causing the \(-1\)\({}^{\circ}\)C isotherms to be scattered. Conversely, the \(-1\)\({}^{\circ}\)C isotherms link more quickly in the strong freezing zone, indicating a faster intersection of the circles in the strong freezing zone.