# Key Issues in Using the Freezing Method in Power Tunnel Rehabilitation Projects
## 4 Analysis of Numerical Simulation Results  
To enhance understanding of the impact of varying seepage velocities on the three-dimensional temperature field of the freezing method, three sections--X, Y, and Z--were chosen for analysis of temperature field cloud isothermograms. Given the significance of the Z section in the study, a more detailed analysis of its numerical simulation results was performed. The precise locations of these sections in the three directions are illustrated in Figure 6.  
\begin{table}
\begin{tabular}{c c} \hline
**Parameter Name** & **Parameter Value** \\ \hline Soil density (kg/m\({}^{3}\)) & Unfrozen ground 1920 \\  & Frozen ground 1880 \\ Soil thermal conductivity (W/(m\(\cdot\)K)) & Unfrozen ground 1.18 \\  & Frozen ground 1.79 \\ Specific heat of soil (J/(kg\(\cdot\)K)) & Unfrozen ground 1.53 \\  & Frozen ground 1.61 \\ Soil permeability coefficient (m/s) & Unfrozen ground 0.56 \(\times\) 10\({}^{-4}\) \\ Soil porosity & Frozen ground 1.91 \(\times\) 10\({}^{-30}\) \\ Density of ice (kg/m\({}^{3}\)) & 0.4 \\ Thermal conductivity of water (W/(m\(\cdot\)K)) & 0.55 \\ Thermal conductivity of ice (W/(m\(\cdot\)K)) & 2.14 \\ Specific heat of water (J/(kg\(\cdot\)K)) & 4180 \\ Specific heat of ice (J/(kg\(\cdot\)K)) & 2100 \\ Latent heat of phase transition of water ice (KJ/(kg\(\cdot\)K)) & 334 \\ \hline \end{tabular}
\end{table}
Table 1: Relevant parameter values.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline Time/d & 0 & 5 & 10 & 20 & 30 & 40 & 50 & 60 \\ \hline Temperature/\({}^{\circ}\)C & 18 & \(-\)15 & \(-\)28 & \(-\)28 & \(-\)28 & \(-\)28 & \(-\)28 \\ \hline \end{tabular}
\end{table}
Table 2: Salt water cooling program.  
Freezing Temperature Field and Permafrost Curtain Development Pattern in the Z-Section 4.1.1. Z-Section Without Considering the Seepage Flow Scenario  
When determining a scenario without seepage in the soil body, the head difference between the upstream and downstream boundaries of the model was set to 0 m, focusing solely on the heat transfer process within the porous medium. The temperature field and isotherm maps at \(-1\)\({}^{\circ}\)C and \(-10\)\({}^{\circ}\)C from the numerical simulation results without seepage are depicted in Figure 7. As illustrated in Figure 7, at the onset of the freezing period (1 d), the minimal cold generated by the freezing pipe was not propagated to distant areas. The isotherm map indicates that the permafrost curtain was not yet formed in the immediate vicinity of the freezing pipe. By the 10 th day of freezing, the cold produced by the freezing pipe, which had been accumulating over time, began to transfer to warmer areas. At this stage, the frozen area adjacent to the shaft side and parts of the upper side of the \(-1\)\({}^{\circ}\)C isotherm exhibited partial intersection with the circle, gradually defining the outline of the frozen soil curtain. By the 23 rd day of freezing, it was evident that the majority of the frozen area had reached the freezing temperature. At this point, the \(-1\)\({}^{\circ}\)C isotherm of the entire frozen area had completed its circular intersection, with the frozen curtain nearly fully formed. The outer side of the \(-10\)\({}^{\circ}\)C isotherm had completed the circular intersection, while the inner side showed incomplete circles, likely due to the less densely packed freezing pipes on the inner side. After 26 days of active freezing, all \(-10\)\({}^{\circ}\)C isotherms had completed their circular intersections, and the overall permafrost curtain had been fully closed. As the freezing period continued to lengthen, the permafrost curtain entered a stable development phase, expanding uniformly in all directions.