## 1 Introduction  
The development of underground public spaces is considered a major contributor to urban sustainable development [1]. The selection of auxiliary methods in tunnel construction is particularly important [2]. The method of combining pipe curtains with artificial freezing represents a new tunnel excavation method suitable for water-rich environments, and it was successfully applied for the first time in the North Arch Tunnel of the underground excavation section of the Hong Kong-Zhuhai-Macao Bridge [3].  
Regarding the pipe curtain method, Japan has conducted extensive research in this direction and was the first to propose combining pipe curtain steel pipes with freezing methods for construction [4; 5; 6]. In the construction of the North Arch Tunnel using this method, the team, led by [PERSON] [7; 8; 9], conducted theoretical analysis, indoor physical experiments, studies on the freezing and water sealing effects and composite structural support effects, on-site in situ tests, and other research on the innovative \"pipe curtain freezing method\", providing a technical reference for engineering construction and achieving good practical results. [PERSON] et al. [10] improved the pipe curtain freezing method for the Gongbei Tunnel of the Hong Kong-Zhuhai-Macao Bridge and proposed a new patented technology for the pipe curtain freezing method, studying its cooling effect. Compared to the traditional pipe curtain freezing method, this construction method places the freezing pipes outside the steel pipes of the pipe curtain. The freezing pipes and steel pipes are arranged alternately. According to the structural shape of the underground engineering project being constructed, the supporting structure can be arranged in a rectangular or circular shape or in any other shape.  
However, in the field of civil engineering, the introduction of construction techniques often follows practice before theoretical research. The pipe curtain freezing method is no exception. The existing research works and reports on the pipe curtain freezing method are mainly based on engineering examples, with relatively little theoretical research.  
In frozen engineering in permeable soil layers, the impact of the groundwater flow on frozen soil curtains needs further research. Groundwater seepage carries heat, and the velocity of the groundwater flow plays a decisive role in whether frozen engineering projects can be completed smoothly [11; 12; 13]. In order to conduct in-depth research on the coupling of the seepage field and temperature field, domestic and foreign researchers have used various research methods, such as field measurement, indoor physical model tests, and numerical finite element analysis [14; 15; 16]. Many achievements have been obtained on the issue of water-heat coupling and relevant industry standards have been formulated [17; 18; 19; 20]. However, the research has focused more on mine and rock fissure flows. The study of the temperature field of frozen pipes under seepage conditions often uses numerical simulations combined with physical model experiments to study single-row and multi-row frozen pipes. There is relatively little research on the new pipe curtain freezing method under seepage conditions.  
This study combines the theory of heat transfer in porous media and Darcy's law. Based on the actual engineering of the Gongbei Tunnel, a new pipe curtain freezing model under seepage conditions is established using the COMSOL 6.0 finite element software. The study investigates the influence of different factors on the new pipe curtain freezing method under seepage conditions, aiming to provide a reference for similar projects in the future.