## 1 Introduction  
With population growth and economic development, the demand for underground space is stronger and stronger. Therefore, tunnel construction is reaching its peak [1, 2]. For tunnel construction, the shield method is commonly used. In the construction process of the shield tunnel, after the sealed gate is broken, the tunnel's end soil stress distribution changes greatly, which seriously endangers the stability of the end soil. Especially for large-diameter shield tunnels in water-rich sand layers, portal breakage will cause serious engineering accidents easily if the end reinforcement method is unreasonable [3, 4, 5, 6]. Thus, a reasonable tunnel-end reinforcement method is required to avoid the engineering accidents. Many studies have been carried out on tunnel-end soil reinforcement. [PERSON] [7] studied the effects of the length of the tunnel's unlined segment on the instability of the tunnel face in a sand soil layer using centrifugal model tests. [PERSON] and [PERSON] [8] believed that the tunnel face could be maintained in a self-stability state if a positive uniformly distributed force of 10 kPa were provided on the tunnel face. [PERSON] and [PERSON] [9] concluded that the destabilizing load of the tunnel face is linearly related to the tunnel diameter, and is not affected by the tunnel's burial depth and soil density. [PERSON] et al. [10] carried out a theoretical analysis of the portal soil stability, using different computational models, based on the Fuxing East Road river-crossing tunnel in Shanghai, China. It was verified that the stability calculated by the sliding theory was conservative and the soil cohesion \(c\) was a dominant factor in the stability. The three-axis mixing pile and artificial freezing method are commonly used for shield tunnel end reinforcement. The three-axis mixing pile can significantly improve soil strength, and is often used to reinforce the shield's initial end [11]. However, the sealing effect of the three-axis mixing pile is poor, which causes the water and mud inrush to occur in water-rich sand layers. The artificial freezing method can overcome the above shortcomings. The rectangular freezing soil mass formed by the artificial freezing method not only significantly improves the strength of the tunnel end soil, but also can stop water fully. It has strong collapse resistance, and the construction safety can be greatly improved. The artificial freezing method is an ideal shield tunnel end reinforcement technology. It is especially suitable for the high water-pressure sand soil layer [12; 13; 14]. After the 1990s, the application of artificial ground-freezing technology became more widespread. The artificial freezing method was used to effectively avoid water gushing in the Valencia subway [15]. The strength of the frozen soil mass was improved by increasing the water content of the soil in the Dusseldorf central railway station construction [16]. [PERSON] et al. [17] summarized the basic principles and characteristics of the artificial freezing method. Compared with other methods of ground reinforcement, the artificial freezing method is more effective at controlling surface displacement and vault settlement. [PERSON] et al. [18] pointed out that the freezing method cannot be replaced by other methods for tunnel construction in water-rich soft soil layers, due to its good performance. [PERSON] et al. [19] discussed the rectangular freezing soil mass' design, construction, and influences on the surrounding environment based on the Zhangfuyuan Station of Nanjing Metro, and concluded that the rectangular freezing soil mass was efficient for portal reinforcement. The rectangular freezing soil mass would swell during the construction of the tunnel portal, which would have negative influences on the surrounding environment. It was concluded that a reasonable model must be used to predict the ground swelling deformation during the design stage of the tunnel [20; 21].