## 1 Introduction  
Artificial ground freezing (AGF) is used to temporarily freeze the formation matrix and water together, which can improve the strength of soil and redirect the flow paths of groundwater. This method is widely used for deep excavation, tunnels construction, and strengthening building foundations, especially in soft rocks and shallow soils [1,2]. In general, AGF is realized by circulating refrigerant (generally low-temperature brine or liquid nitrogen) in a closed-loop system, where a double-walled pipe is used [3,4]. In 1880, German engineer [PERSON]. [PERSON] first proposed the artificial freezing method [5]. Recently, this method was generally accepted and applied. The construction of Nanjing Metro Line 2 in China used AGF to treat the inrush of water during excavation, in which brine at \(-20\)\({}^{\circ}\)C, circulating in double-walled vertical pipes with well spacing of 1 m, was used to freeze formations, and construction conditions were realized successfully after 40 days [6]. The Farth subway and undercrossing of the Limmat River in Zurich were constructed using brine at \(-40\)\({}^{\circ}\)C, circulating in horizontal pipes. The comprehensive frozen wall could be formed after 40 days, and the frozen radius of a single well was approximately 1 m [2]. In addition, for the first time, the combination of AGF and pipe-roofing was applied in the Gongbei tunnel that connects the Hong Kong-Zhuhai-Macao Bridge in order to obtain a safer construction environment [7]. In order to profoundly promote the development of AGF, scholars have conducted in-depth research by means of experiments and numerical analysis. Research on AGF mainly focused on the evolution of the induced frozen front and the frozen range. Many workers have studied freezing characteristics under different conditions, which have enriched the database available for application [8; 9; 10; 11]. According to their results, the frozen radius of a single well was usually 0.5-1 m, but several months were required, which meant that the freezing rate was low, with a limited frozen range. Innovatively, [PERSON] et al. used the natural cold source in the cold region to cool the cycle medium [12], which effectively reduced the cost of the freezing process, but the frozen range and freezing rate were not improved. [PERSON] et al. studied the law of artificial ground freezing with groundwater flow, and water flow would undoubtedly weaken the freezing effect and reduce the frozen range on the windward side, but the optimal layout of the freezing pipes was obtained through a numerical analysis [13; 14]. Some mathematical models that consider hydro-thermal coupling characteristics have also been proposed to simulate the underground temperature field [15; 16; 17; 18]. In addition, the frozen top layer of soil by cold air had been tried for mine pit construction during the wintertime in Siberia; then, the next layer of soil was frozen and excavated [19]. In addition, the effect of the water-ice phase transition on water migration in pores has been a research hotspot domestically and abroad [20; 21; 22], and the results show that moisture will move from the hot end to the cold end under a temperature gradient [23]. Meanwhile, the gravity and attraction of porous media should be taken into consideration. In several studies, the way of circulating refrigerant in double-walled pipes was used to freeze formation by heat conduction, as shown in Figure 1a. It is well known that heat conduction is a slow and inefficient heat transfer mode compared to heat convection; so, this will result in a longer freezing time (months are required) and a limited frozen radius (approximately 1 m). Meanwhile, the use of a large amount of refrigerant and a long freezing period has also increased construction costs, which deviates from the energy-efficient track promoted by the international community. Accordingly, in this study, as shown in Figure 1b, we consider directly injecting natural cold gas into the subsurface and then strengthening the freezing rate by heat convection.