# Heat-Flow Coupling Law for Freezing a Pipe Reinforcement with Varying Curvatures
## 1 Introduction  
Artificial ground freezing is a method of freezing groundwater and soil into a layer of ice in order to control the flow of groundwater and soil or provide solid support to underground works. The method mainly involves injecting a refrigerant into underground boreholes to cool the surrounding soil and water so that it freezes and forms a barrier with a certain degree of strength, which can effectively prevent the flow of underground soil and water into the working platform. Due to rapid economic growth and the extensive construction of underground projects, artificial ground-freezing technology has gained widespread use in tunnel construction. Its adoption is primarily driven by its environmentally friendly nature, controllability, safety features, and other advantageous characteristics [1; 2; 3; 4; 5; 6]. Straight freezing tubes are usually preferred as they have the advantages of simple processing, ease of transportation and construction, etc. [7; 8; 9; 10]. However, the freezing wallshape formed by this traditional straight pipe is more regular and uniform and fails to provide targeted reinforcement to weak areas with soil and water flow. Previous research has primarily focused on altering the layout, spacing, and cross-sectional area of straight freezing tubes to enhance freezing effectiveness [11; 12]. These changes aimed to increase the overall impact. Although arc-freezing single pipes have a more complex structure, demanding different processing techniques and higher material consumption than straight pipes, they result in more balanced structural stress in the frozen soil curtain. This enhances safety. In the case of reinforcing weak areas, arc single pipes offer improved safety and economy compared to merely increasing the quantity using straight pipes. Additionally, they prove to be safer and more cost-effective. Researchers across the world have focused more on the shape of frozen monotubes in terms of cross-sectional dimensions and have drawn many useful conclusions. For example, [PERSON] et al. [13] found that the cross arrangement of elliptical freezing tubes has a more significant heat-transfer effect. [PERSON] et al. [14] modeled elliptical single and double freezing tubes with different length-to-short-axis ratios and found that the larger the length-to-short-axis ratio, the larger the freezing area, and elliptical freezing tubes are more efficient than circular freezing tubes in freezing. Meanwhile, the analogous problem of curvature in the lengthwise direction of freezing tubes has been less studied. [PERSON] et al. [15] studied ring-shaped freezing tubes during a tunnel road expansion project. Their aim was to ensure the construction process's reliability while optimizing the original structure to improve the effectiveness of freezing water. However, they did not conduct a comprehensive investigation into the freezing behavior of curved single pipes and multi-pipes within soil, nor did they explore their advantages over straight pipes. [PERSON] et al. [16] proposed the use of a ring-shaped freezing tube to freeze and reinforce the shield-tail brush replacement effectively. However, they did not thoroughly explore the freezing characteristics of curved single tubes or multiple tubes in the soil or their advantages over straight tubes. Investigating the water-heat coupling evolution between curved single tubes and multi-tubes during freezing is crucial for enhancing freezing-engineering methods. Based on the principles of heat transfer and Darcy's law in porous media, this study establishes a numerical model for arc-frozen single-pipe and multi-tube permafrost and analyzes temperature and seepage fields under different arc single-pipe and multi-tube freezing reinforcements. This paper will compare these results with the more effective curved arc pipes used in shallow buried tunnels with water influx. This analysis aims to assess their water-stopping capabilities and understand temperature field isotherms, freezing-wall thickness, and other distributions.