## 4 Results and Discussion  
### Effects of Seepage  
The development of temperature field of four reference cases can be visually seen in Figure 7, which depicts the \(-10\)\({}^{\circ}\)C and \(10\)\({}^{\circ}\)C temperature isotherm curves at four typical freezing durations (i.e., 10 days, 30 days, 50 days and 70 days). It can be seen that the frozen zone develops obviously towards the downstream side affected by seepage flow. The temperature-evolution process of five horizontal monitoring points (\(A\), \(B\), \(C\), \(D\), and \(E\)) under different velocities of seepage are shown in Figures 8 and 9. When there is no seepage, the frozen body extends approximately at an equal speed along the upstream and downstream direction. The temperature curves of two points on the outer side (\(A\) and \(E\)) are almost identical (Figure 8a), and so are the two inner points (\(B\) and \(D\)). Figure 9a indicates the symmetrical development of the temperature field as well. After about 24 days without seepage, the curves of two inner points drop slightly faster than that of two outer points. This is because a stable and closed frozen wall is formed at this time, and the freezing effect of freeze pipes on the internal zone is more concentrated. As shown in Figure 8b, when the seepage velocity is 0.5 m/d, two points at the upstream direction of the corresponding freeze pipes (\(A\) and \(D\)) cool slower than points \(B\) and \(E\). It can be explained that the water flow from upstream releases heat to hinder the freezing effect of pipes, but the passed cold water even helps to cool the downstream zone, which is clearly illustrated by Figure 9a,b. After about 15 days, the curves of the two outer points (\(A\) and \(E\)) drop slower than that of the two inner points (\(D\) and \(B\)), respectively. This is because the frozen wall is initially formed, and the groundwater flow is blocked into the tunnel, but the flow along the outface of the frozen wall, illustrated in Figure 10 (larger arrows represent higher velocities), provides continuous heat energy for the two outer points and makes them freeze slower. Figure 7: Development of the temperature field of reference groups, four typical freezing durations (i.e., 10 days, 30 days, 50 days and 70 days) are selected. When the seepage velocity increases to 1 m/d and 1.5 m/d, even if it is far away from the freeze pipes, the origin (point \(C\)) cools down in a shorter time due to the rapid flow of passed cold water. The overall trend of curves is unchanged but the discrepancy of the freezing speed between the upstream and downstream points becomes more pronounced. Figure 8: Temperature curves of horizontal monitoring points under different velocities of seepage: (**a**) 0 m/d, (**b**) 0.5 m/d, (**c**) 1 m/d, and (**d**) 1.5 m/d. Figure 9: _Cont._  
A circular closed frozen wall with a thickness of 1.5 m is often desired in geotechnical practices [39], so the closed ring formed by monitoring points (except origin) is treated as the desired region for the frozen wall in this research (see Figure 3b). Considering the high energy costs of soil freezing, minimizing the time required to achieve a fully frozen body within the desired zone is of great benefit to the economy. [PERSON] and [PERSON] [43] suggested that freezing the frozen wall to \(-3\)\({}^{\circ}\)C can ensure its strength and stability. Therefore, in this study, when all the sixteen temperature monitoring points drop below \(-3\)\({}^{\circ}\)C, it is considered that a desired frozen wall with enough thickness and strength has been completely formed. However, the presence of ground water flow has considerable influence on the formation of a closed and stable frozen wall. Firstly, the water flow provides continuous heat energy to the soil within the gaps between adjacent freeze pipes, which delay the closure of the frozen wall, as shown in Figure 11. With the increase in seepage velocity from 0 m/d to 1.5 m/d, the adverse effects of water flow are more considerable.