## 5 Conclusions  
This study developed a coupled hydro-thermal numerical model to investigate the performance of an AGF system in tunnel excavations. Groundwater flow and spatial variation of soil thermal conductivity were considered as influencing factors. The concept of random field is adopted to characterize the spatial variation of soil thermal conductivity. The coupled hydro-thermal process was simulated by a finite element method incorporated with spatially random thermal conductivity. Groups of Monte-Carlo simulations have been carried out to assess the effects of soil heterogeneity on the AGF under different velocities of seepage flow. The model is validated by laboratory test results.  
The simulation results shown that the AGF in tunnel excavation is vulnerable to seepage and spatial variability of soil thermal conductivity, since a small portion of leakage can undermine the closure of a frozen wall. The place most prone to seepage accidents is at the upstream face of the tunnel. The temperature development and formation time of a desired frozen wall from Monte-Carlo simulations are compared to those of deterministic scenarios. The results indicate that the spatial variation of thermal conductivity adversely  
Figure 18: Additional freezing time due to seepage flow and spatial variation in soil thermal conductivity. (RFEA = random finite element analysis).  
affects the freezing performance, especially for scenarios with a velocity of seepage flow higher than \(1\,\mathrm{m}/\mathrm{d}\), where the adverse effect of the spatially varied soil thermal conductivity on freezing efficiency is even more pronounced.  
As a compensation remedy to account for the coupled hydro-thermal effects caused by seepage flow and soil heterogeneity, additional freezing time is required to ensure the formation of a desired frozen wall. Based on random finite element analysis, this study quantitively tabulated the required additional freezing time above the deterministic scenario. Two levels of the additional freezing time are provided, namely the average level and conservative level, aiming to facilitate practitioners in making a rule of thumb estimation in the design for comparable situations.  
The developed model is limited to the study of the hydro-thermal process of the AGF technique; however, the mechanical behaviors of soil freezing are ignored in this model. To obtain more practical recommendations, it is necessary to consider the fundamental mechanisms of the AGF system, as well as the spatial randomness of permeability in future work. In addition, only typical random field parameters were considered in the current study. These limitations will be considered in future work.  
Conceptualization, Y.L. and J.H.; methodology, P.L.; software, P.L. and P.Q.; validation, P.Q. and P.L.; formal analysis, P.Q.; investigation, P.Q.; resources, Y.L.; data curation, P.Q.; writing--original draft preparation, P.Q.; writing--review and editing, P.Q.; visualization, P.L.; supervision, Y.L.; project administration, Y.L.; funding acquisition, J.H. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (Grant Nos. 51879203, 51968019), and the High-Level Talent Project of Hainan Basic and Applied Basic Research Plan (Grant Nos. 2019 RC148, 2019 RC351).  
The authors