## 1 Introduction  
The theoretical analysis of artificial freezing temperature fields has been extensively investigated in the ground freezing engineering field, particularly in heat conduction problems, including \"phase transition\", \"hydrothermal coupling\", \"temporal and spatial effect\" and other factors [1; 2]. The currently applied research methods include analytical, experimental and numerical methods [3; 4; 5; 6; 7]. These analytical methods use mathematical and physical equations to establish accurate functional relationships for research problems. The relationships between variables in the function are clear and can be directly solved, and they can be applied by engineers and technicians during the design stage and to evaluate the effect of on-site freezing construction. Therefore, they can be utilized to study artificial freezing temperature fields [8; 9; 10]. However, considering the differences in the number and arrangement of freezing tubes, freezing front movement, water migration and other factors, the analytical method still has some limitations when applied in mathematical solutions, and can only analyze the temperature field of single tube freezing. In academic and engineering fields, many cases [11] have confirmed that the artificial freezing temperature field develops very slowly, at the final stage of the freezing process. Since the freezing tube will have reached the equilibrium state of cooling and heat absorption, the size of the frozen wall almost does not change, and would be very close to the steady-state heat transfer temperature field [12]. Therefore, the steady-state heat transfer model can be used for approximate calculation. Extensive research and derivation have been performed on the analytical solutions of freezing steady-state temperature field. Currently, the commonly used analytical solutions include the single-tube freezing steady-state temperature field [13], the two-tofive-tube equidistant linear arrangement steady-state temperature field [14; 15; 16] presented by [PERSON], the symmetric and asymmetric steady-state temperature fields of linear single-row tubes [17], linear double-row tubes and three-row tubes [18; 19; 20], and the annular single- circle-tube and double- circle-tube temperature field [21; 22]. The aforementioned results are based on the classic Trupak single-tube freezing steady-state temperature field analytical solution formula, obtained through potential function superposition, separation variable method, conformal mapping and other processing methods, combined with practical engineering situations to simplify the corresponding model, and improve its application value in engineering. However, with the continuous and rapid development of urban construction in China, higher requirements have been proposed regarding the formation and functioning of underground engineering structures. For many engineering fields that involve difficult construction, it is challenging to establish strong support methods to handle complex geological conditions. Most analytical solutions also need to be constantly revised or optimized to meet higher requirements such as various geological conditions, accurate and fast calculation of temperature and frozen soil curtain thickness [23; 24; 25]. Therefore, continuous in-depth studies should be performed to establish a theoretical basis that is in line with engineering practices. To overcome the challenges associated with constructing shallow buried and concealed excavation tunnels with large sections in the water-rich soft soil of coastal cities, Chinese experts and scholars have put forward a new tunnel construction method; the freeze-sealing pipe roof method (FSPR) that integrates the pipe roof method (PRM) and the artificial ground freezing method (AGF) [26; 27]. In Figure 1, a plurality of closely arranged large-diameter steel pipes are jacked into the strata at both ends of the tunnel section to form a pipe roof.