## 1 Introduction  
Artificial freezing technology is a method of stabilizing the subsoil using an artificial refrigerant to freeze the water in the soil into ice, thereby cementing it to the soil to achieve a water-stopping and reinforcing effect for construction safety. The artificial freezing method has been widely used in the mining and tunneling industry for over 100 years [1, 2, 3]. This technology has been extensively used in the building of metro tunnels [4], connecting passages [5], and shield machine initiation [6]. However, when the tunnel reinforcement method of freezing is adopted, the ground will undergo an obvious frost heave, and there is thus settlement movement [7, 8, 5, 9], which will impact the serviceability performance of surface-sensitive buildings and surrounding pipelines as well as the tunnel construction. The results of soil-freezing experiments indicate that the soil frost heave consists of ground displacement induced by water freezing into ice and fractionated frost heave due to water migration [9, 10]. The studies of [PERSON] [11] and [PERSON] [12] provided important theoretical support for a better understanding of the soil freezing mechanism. The Soviet scholar [PERSON] [13] proved through indoor tests that soil melt settlement is composed of both hot melt settlement from melting ice and the settling of soil that has been compacted as a result of external and internal loads, laying the foundation for subsequent related research. [PERSON] et al. [14] used the change in pore ratio as a criterion to establish a three-field coupled moisture-heat-stress model for frozen ground. However, considering the practicality of the artificially frozen ground heave and settlement movement, how to achieve effective prediction of ground surface movement when using the freezing method for reinforcement becomes a pressing issue that should be of greater concern. To achieve an effective prediction of ground surface movement during freezing construction, many scholars have carried out model tests on the heave and settlement movement of frozen soil. [PERSON] et al. [15] derived and predicted the temperature field and thaw settlement displacement development for the artificial freezing method of thawing using centrifuge experiments. [PERSON] et al. [16] studied the displacement change law of soft soil in Shanghai during the construction process of the freezing method using a self-designed freezing device. Model tests can provide a visual understanding of soil displacement patterns in freezing construction, but it is difficult to obtain more accurate and consistent conclusions with individual model tests. Furthermore, the conditions required to carry out multiple similar model tests are too demanding. Numerical simulation allows researchers to obtain a better solution to this problem by performing multiple calculations on a model to predict soil displacement as a guidance of engineering applications. [PERSON] et al. [17] analyzed the temperature distribution and freezing displacement of horizontal freezing by numerical simulation based on an example of a frozen Shanghai metro tunnel project. Based on the segregation potential approach, [PERSON] et al. [18; 19] implemented a prediction of freezing soil with a two-dimensional thermodynamic coupling model, and applied it to the numerical simulation of freezing construction in an existing tunnel in Kobe, Japan. [PERSON] et al. [20] proposed a frost heave prediction model for construction using the freezing method, a thermo-hydro-mechanical coupled model and applied it to engineering practice. [PERSON] et al. [21] combined indoor experiments and numerical simulations to compare and analyze the displacement change patterns of soils during freezing and thawing. [PERSON] et al. [22] proposed a method to predict the surface displacement during the whole freezing process, and the validity of the method was verified by comparing numerical results with the measured data.