## 5 Discussion  
### Effect of Groundwater seepage on Freezing Temperature Field  
The temperature range of cloud diagram is set as \(-26\)\(\sim\)\(-1\)\({}^{\circ}\)C (i.e., the freezing temperature of soil is \(-1\)\({}^{\circ}\)C) to characterize the thickness of frozen wall in numerical simulation. Then, it can be found that the development of frozen front is uneven due to the influence of seepage. Therefore, a unified measure is established in numerical simulation by defining the thickness of frozen wall at different locations AB, BC, DE and EF which correspond to \(L_{\text{AB}}\), \(L_{\text{BC}}\), \(L_{\text{DE}}\) and \(L_{\text{EF}}\), respectively, when the construction requirements are met. As shown in Figure 9, \(L_{\text{AB}}\) is the thickness from the center of freezing pipe to the upstream outer front, \(L_{\text{BC}}\) is the thickness from the center of the freezing pipe to the upstream inner front, \(L_{\text{DE}}\) is the thickness from the center of the freezing pipe to the downstream inner front and \(L_{\text{EF}}\) is the thickness from the center of the freezing pipe to the upstream outer front. The frozen wall formed at three groundwater seepage velocities for the given boundary conditions are shown in Figure 10. The closure time of frozen wall at 0 m/d is only 9 d, while it is 24 d at 1.0 m/d and 61 d at 2.0 m/d. The faster the seepage is, the more cooling capacity generated by the upstream freezing pipes is brought to the downstream, and the later the upstream frozen wall closes, that is, the longer the closure time of frozen wall is. Similarly, the time required to reach design frozen wall thickness for construction at different seepage velocities is also different, as shown in Figure 10. The time required to reach design frozen wall thickness at 0 m/d, 1.0 m/d and 2.0 m/d is 41d, 83d and 121d, respectively. Therefore, it will take more time to complete the freezing engineering in the seepage stratum with a large flow velocity. The temperature field at different seepage velocities is shown in Figures 11 and 12. Figure 9: Custom frozen wall thickness. Figure 11: Initial closure of frozen wall: (**a**) 0 m/d; (**b**) 1.0 m/d; and (**c**) 2.0 m/d. Figure 10: Freezing time of frozen wall under different states: \(T_{1}\) is the time for initial closure of frozen wall, and \(T_{2}\) is the time for complete closure of frozen wall. Table 4 shows that when the minimum thickness of the frozen wall is satisfied (i.e., the construction requirements are met), the thickness of the upstream frozen wall \(L_{\text{AB}}\) decreases correspondingly with the increase in seepage velocity, and the thickness of the downstream frozen wall \(L_{\text{DE}}\) and \(L_{\text{EF}}\) increases correspondingly. However, the thickness variation of the upstream frozen wall \(L_{\text{DC}}\) does not vary linearly. This is because with the greater seepage velocity, the heat at the inner front of the upstream frozen wall is carried away, resulting in a change in the thickness of the frozen wall \(L_{\text{BC}}\). Therefore, in the actual project, when the seepage velocity is large enough, sufficient attention should be paid to the upstream freezing wall and corresponding measures should be taken. As shown in Figure 13, the development trend of frozen wall thickness at each position is similar at the same seepage velocity, the thickness \(L_{\text{AB}}\) is much smaller than \(L_{\text{BC}}\), \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Seepage Velocity (m/d)** & \(T_{1}\) **(d)** & \(T_{2}\) **(d)** & \(L_{\text{AB}}\) **(m)** & \(L_{\text{BC}}\) **(m)** & \(L_{\text{DE}}\) **(m)** & \(L_{\text{EF}}\) **(m)** \\ \hline
0 & 9 & 41 & 1.139 & 1.258 & 1.281 & 1.135 \\
1.0 & 24 & 83 & 0.752 & 2.258 & 2.112 & 1.726 \\
2.0 & 61 & 121 & 0.247 & 2.348 & 2.347 & 1.759 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Relevant parameters of frozen wall at different flow velocity. Figure 12: Complete closure of frozen wall: (a) 0 m/d; (b) 1.0 m/d; and (c) 2.0 m/d.