## 3 The Establishment of a Numerical Model  
### Establishment of a Calculation Model of Freezing Body Temperature Field  
To facilitate the calculation, the following assumptions are made for the finite element model:  
* The soil is continuous and homogeneous during freezing;
* When the soil is frozen, the latent heat of phase change is continuously released at the freezing interface;
* Assume that the moisture in the soil is completely frozen when it is cold; that is, the content of unfrozen water is zero [17; 18];
* The freezing temperature load on the freezing hole wall of the model changes with the change in temperature;
* The wall of the freezing tube is negligible relative to the whole model, so the material characteristics of the freezing tube itself are not considered. To analyze the influence of horizontal freezing of the contact channel on the existing tunnel in the upper part, according to the general situation of the project and the symmetry of the contact channel, this paper adopts the FLAC3D finite difference software to establish a structural model, and then realize the three-dimensional temperature field calculation of the frozen body. The position relationship between the frozen tube and the existing tunnel is shown in Figure 2. The total size of the model is 44 m long \(\times\) 15 m wide \(\times\) 35 m high, as shown in Figure 3.  
Figure 1: Location map of the shield tunnel in the Dongbin section. Application steps of FLAC3D in freezing method tunnel excavation calculation:  
* According to the actual situation of the project, the three-dimensional geometric model, including tunnel, frozen soil wall, surrounding soil layer, and adjacent pipeline, is established in FLAC3D. * Meshing ensures that the mesh density of the model is high enough at key locations, such as tunnel boundaries, the interface between the frozen wall, and the surrounding soil, to accurately capture stress concentration and deformation characteristics. * The physical and mechanical parameters, such as elastic modulus, Poisson's ratio, density, cohesion and internal friction angle, are set according to laboratory tests or empirical formulas. * Set the boundary conditions of the model, such as fixed boundaries, rolling boundaries, or free boundaries, to simulate the constraints in real situations. Figure 3: Numerical simulation grid model. Figure 2: Position relationship between freezing tube and existing tunnel. * The freezing temperature field is set, the temperature gradient produced by the freezing tube in the soil is simulated, and the forming process of the frozen soil wall is calculated by a heat conduction equation. * Monitoring the thickness, temperature distribution, and strength changes of the frozen soil wall, evaluating its supporting ability. * According to the predetermined excavation sequence and excavation rate, the material of the tunnel is gradually removed to simulate the excavation process. * In the process of excavation, the stress, strain, and displacement of soil around the tunnel are monitored in real time, especially the longitudinal deformation of adjacent pipelines. * The simulation results are compared with the field-measured data to ensure the accuracy and reliability of the simulation results. * The effects of different excavation rates, freezing temperatures, and other parameters on pipeline settlement are analyzed, and the conclusions mentioned in the abstract are verified. When calculating the model numerically, the vertical tunnel axis direction in the water intake plane is the X axis, the direction along the tunnel axis is Y axis, and the vertical upward direction is the Z axis.