## 4 Results and Discussion
### Effects of Soil Thermal Conductivity  
Figure 12 indicates that the formation of a completed frozen wall is directly related to the soil freezing at points \(A\) and \(D\). The seepage flow delays the expansion of an effective frozen zone in the upstream direction. Another factor affecting the heat conduction is the soil thermal conductivity, \(\lambda_{\rm s}\). [PERSON] et al. [17] found that a lower soil thermal conductivity could result in a slower freezing speed during the AGF process. To make sense of the importance of soil thermal conductivity on soil freezing, the deterministic analyses of soil thermal conductivity without spatial variation are established before the Monte-Carlo simulations. In this part, the value of \(\lambda_{\rm s}\) ranges from 0.9 to 4.9 (unit: W/(m\(\cdot\)K)) and keeps constant in each single simulation. The corresponding temperature curves for points \(A\) and \(D\) are plotted in Figure 13. A similar conclusion can be observed in that the freezing efficiency shows a positive correlation with soil thermal conductivity. Especially when 0.5 m/d seepage flow exists, the cost time cooling point \(A\) to \(-3\,\mathrm{\SIUnitSymbolCelsius}\) even increases from 20 days (\(\lambda_{\rm s}\) = 4.9 W/(m\(\cdot\)K)) to 60 days (\(\lambda_{\rm s}\) = 0.9 W/(m\(\cdot\)K)) (see Figure 13c). This huge discrepancy in freezing efficiency caused by varying soil thermal conductivity should not be ignored. As for the Monte-Carlo simulation groups (\(\mu_{\rm 3s}\) = 2.9 W/(m\(\cdot\)K) and COV = 0.25), 100 times are respectively calculated with each velocity (i.e., 0 m/d, 0.5 m/d, 1 m/d, and 1.5 m/d). Figure 14 summarizes the temperature curves of five horizontal monitoring points, the black curves represent the deterministic results in reference cases where \(\lambda_{\rm s}\)  
Figure 13: Temperature curves of points \(A\) and \(D\) with five different soil thermal conductivities: (**a**) point \(A\), no seepage; (**b**) point \(D\), no seepage; (**c**) point \(A\), 0.5 m/d; (**d**) point \(D\), 0.5 m/d. A target temperature is set at \(-3\,\mathrm{\SIUnitSymbolCelsius}\). For points \(A\) and \(D\), Figure 15 depicts the relationship between temperature and seepage velocity at three typical freezing periods (i.e., 20 days, 40 days, and 60 days). The effects of spatially varied soil thermal conductivity can be reflected by comparing the deterministic result with the mean value of the random field result. In addition, considering the high operating cost of artificial freezing, the unfavorable scenario with the worst freezing efficiency should be given priority by engineers. Therefore, the maximal value of temperature among 100 repetitive simulations is also plotted in Figure 15. It can be observed that the adverse effect of seepage flow on point \(A\) is more evident, when the seepage velocity exceeds 1 m/d, soil at point A cannot descend to the target temperature (\(-3\)\({}^{\circ}\)C) at 40 days. If the seepage velocity is controlled within 1 m/d, the desired frozen wall should be formed within 60 days in most cases. Figure 15: Temperature of Points \(A\) and \(D\) at three typical freezing durations (i.e., 20 days, 40 days, and 60 days) under different seepage velocities: (**a**) point \(A\), 20 d; (**b**) point \(A\), 40 d; (**c**) point \(A\), 60d; (**d**) point \(D\), 20 d; (**e**) point \(D\), 40 d; (**f**) point \(D\), 60 d. The red dashed line signifies the target temperature (\(-3\)\({}^{\circ}\)C). One of the most concerned points for engineers is the formation time of the desired frozen wall, which is a key factor relating to the project schedule and cost in practical engineering. In reference groups, when the seepage velocity is 0 m/d, 0.5 m/d and 1 m/d, the formation time of the desired frozen wall is computed as 22 days, 29 days and 49 days, respectively. However, when the seepage velocity increases to 1.5 m/d, the desired frozen wall has not been formed, even at the end of the simulation. Similar observations were reported in the literature (e.g., [14; 15]).