## 1 Introduction  
With the continuous expansion of urban development and construction areas, engineering projects that cross rivers, oceans, and other water bodies have become inevitable [1]. This is particularly true for urban highway construction, where large cross-section trans-river tunnels are often employed to traverse such obstacles. Artificial ground freezing technology has emerged as a significant engineering method for underground projects in recent years. This technology's ability to reinforce soil to facilitate excavation and provide excellent water sealing means it holds substantial engineering value [2; 3; 4; 5]. Artificial ground freezing (AGF) technology utilizes artificial refrigeration techniques, such as brine cycle freezing and liquid nitrogen freezing, to lower the water temperature in a construction soil body [6]. This process transforms the original soil into firm, frozen soil, enhancing its strength and stability and preventing water influx. This technique can form a solid permafrost curtain which supports the safe and efficient construction of underground works [7; 8; 9; 10]. [PERSON] et al. [11; 12] conducted indoor experiments on the AGF technique and proposed a novel freezing method. [PERSON] et al. [13] used indoor permafrost experimental data to derive a multi-field coupled model in order to ensure that AGF modeling can accurately predict the regular changes in the freezing process. In a practical project, [PERSON] et al. [14]used a numerical model to predict the freezing process in a mine tunnel near Cigar Lake, Canada, and analyzed the temperature and displacement data. [PERSON] et al. (2015) described the application of AGF technology in tunnel construction and analyzed the effectiveness and versatility of AGF technology through various freezing construction cases. [PERSON] et al. (2016) studied a tunnel freezing project in Naples, Italy, and assessed the heat transfer process of the AGF technique using numerical simulation. In addition, [PERSON] et al. (2016) carried out a detailed study of the freezing process in soils, studied the effects related to the freezing and expansion of soils, and made several recommendations for the future development of AGF. The excavation of large-section underwater tunnels often involves the employment of a pipe curtain-permafrost combination structure to provide enclosure during construction. This method utilizes high-strength steel pipe curtains for support. In contrast, forming a permafrost curtain between the pipes effectively seals underground water flow channels, playing a crucial role in water sealing ([PERSON] et al., 2017; [PERSON] et al., 2018; [PERSON] et al., 2019). The pipe curtain-permafrost combination structure maximizes bearing and water-sealing capabilities, which has led to it being widely used in underwater tunnel construction. A notable example is the construction of the Hong Kong-Zhuhai-Macao Bridge Gongbei Tunnel, where the successful application of the pipe curtain-permafrost combination structure has significantly advanced the technology. This success has not only enhanced the application scope of the pipe curtain-permafrost combination structure but also led to the development of a new pipe curtain freezing construction technology ([PERSON] and [PERSON], 2018; [PERSON], 2019; [PERSON], 2020). In addition, the soil experiences freezing and expansion effects during the soil freezing process. This process includes in situ freezing and expansion as well as sub-condensation freezing and expansion. The former refers to the conversion of liquid water within the closed pores of the soil into ice. The latter involves the migration of free liquid water within the pores, driven by the temperature gradient difference during the freezing process, which subsequently transforms into ice, causing volume expansion. Reviewing relevant information and previous research shows that the freezing expansion caused by in situ freezing has a comparatively smaller effect than that caused by sub-condensation freezing and expansion.