## 1 Introduction  
Ground space resources are increasingly tense, and the development of metro tunnels is the general trend. Due to its high automation, fast construction speed, and little influence on ground disturbance, the shield method is widely used in metro tunnels [1]. The experience of shield tunneling construction in the past shows that the construction of the shield tunnel breakthrough working shaft undoubtedly is a high-risk stage [2, 3]. This is especially true for soft soil geological conditions and soil that is rich in water, and improper construction due to the influence of groundwater may cause water gushing and sand gushing at the portal, resulting in a series of consequences, such as economic losses, delays in the construction period, and ground surface settlement. Therefore, the water sealing effect of the support system is even more important [4]. Usually, some conventional construction methods, such as the deep mixing method, grouting method, high-pressure rotary spraying method, and precipitation method are used to strengthen the soil around the working shaft [5]. Generally, when the conventional reinforcement method is completed, and serious sand and water gushing phenomena sometimes appears in the stage of breakthrough working shaft, in order to ensure the safety of the breakthrough stage,artificial ground freezing technology (AGF) is often used for auxiliary reinforcement [6]. Compared with other soil reinforcement methods, the AGF method has the advantages of good reinforcement uniformity, high strength, a good water sealing effect, safety and environmental protections, and the reinforcement process and effect can be monitored during the construction. It has been widely used in the working shaft construction of shield tunnel [7]. At present, the AGF has a rich theoretical basis and engineering practice experience, such as the freezing-sealing pipe roof method applied to the century projects of the Gongbei tunnel of the Hongkong-Zhuhai-Macau Bridge [8]. The key to the successful application of the AGF construction method lies in the stability of the frozen soil curtain. The temperature field is the most critical control index for AGF method. In order to grasp the development of the frozen soil curtain during construction, the temperature monitoring data are usually used to invert the development process of the frozen soil curtain to determine whether the thickness of the frozen soil meets the excavation requirements, but enough in-site monitoring data are not always available to predict and analyze the strength and development of the frozen soil curtain. Therefore, it is necessary to judge the development status of the frozen soil and the rationality of the design in advance through other methods, such as model tests [9; 10] and numerical simulation [5], which can reduce the safety hazards of the AGF construction, so as to scientifically guide the construction of the project [11]. Obviously, the model test can only be aimed at a certain working condition and costs a lot, while the numerical simulation method has a higher applicability. At present, there have been many cases of studying the freezing effect of shield tunnel breakthrough working shafts through numerical simulation [12; 13]. There are two conventional approaches to modeling the heat transfer of the AGF problem, the apparent heat capacity formulation and the enthalpy-porosity formulation [14]. The research on the influencing factors of the freezing effect has been relatively comprehensive. However, in actual construction, because the shield machine will also generate a large amount of heat during the excavation process [15], the freezing effect is not as good as expected, and the freezing strength and construction safety cannot be guaranteed. There are still some risks in AGF construction in some extreme conditions.