# Heat-Flow Coupling Law for Freezing a Pipe Reinforcement with Varying Curvatures
## 7 Conclusions  
In this study, the development and characteristics of the temperature field and seepage field of a single tube and multiple tubes frozen at different arcs were analyzed. Additionally, this study has conducted simulations to investigate the effectiveness of water-stopping conditions in shallowly buried tunnels for both straight tubes and long arc tubes with a ratio of s = 7 under the conditions of water surging. Subsequently, this study has performed a comparative analysis, leading us to the following key conclusions:  
1. During the freezing process, the shape of the curtain is mainly determined by the shape of the freezing monotube; as the freezing process progresses, the upstream cold is carried downstream by the groundwater, and a \"cold channel\" appears in the soil layer.
2. As the long arc ratio s decreases and the curvature of the pipe increases, after freezing with a single pipe in the presence of water flow, the change in the freezing-wall thickness in the curved pipe becomes smaller than that of the straight pipe. This difference is particularly prominent in the upstream section, where change in the freezing wall's thickness in the straight pipe exceeds that of the s = 7 curved pipe by approximately 350 mm at a distance of \(-\)2000 mm from the main surface. A smaller s-value brings the freezing pipe closer to the permeability boundary, resulting in a shorter time for the groundwater phase change and a closer proximity of the upstream seepage-velocity contour to the boundary, ultimately leading to improved effectiveness of the water barrier.
3. In the multi-pipe freezing model, the s = 7 curved pipe has a freezing-wall thickness approximately 120 mm greater than that of the straight pipe at a distance of \(-\)2000 mm from the main surface.
4. The use of artificial freezing of water isolation in the lower step of a shallowly buried tunnel as a solution is safe and reliable and can provide new ideas for similar engineering risks.
5. Under the water-surging condition of this shallowly buried excavation, the most unfavorable permafrost curtain thickness of the freezing of a straight pipe at 58 d can be reached by using the freezing of an arc pipe at 46 d, which effectively shortens the construction period to 12 d and proves the superiority of the freezing of the arc pipe to block water.  
Conceptualization, K.Y. and J.H.; methodology, J.H.; software, [PERSON]; validation, [PERSON] and [PERSON]; formal analysis, [PERSON]; investigation, K.Y.; resources, [PERSON] and [PERSON]; data curation, K.Y.; writing--original draft preparation, K.Y.; writing--review and editing, [PERSON]; visualization, K.Y.; supervision, [PERSON]; project administration, [PERSON]; funding acquisition, [PERSON] and T.W. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the State Key Laboratory for GeoMechanics and Deep Underground Engineering, China University of Mining and Technology (SKLGDUEK2204), the Hainan University Collaborative Innovation Center Project (XTCX2022 STB09), the Hainan Provincial Natural Science Foundation Innovation Research Team Project (522 CXTD511), the High-tech Direction Projectof Key R&D Program of Hainan Provincial Science and Technology Department (ZDYF2021 GXJS020), and the Enterprise Entrusted Project of Hainan University (HD-KYH-2022405).  
Not applicable.