## 3 Numerical Model  
In this project, an ultra-deep caisson is used as the receiving well for water intake pipe jacking. The length of the pipeline from the water source to the caisson is about 1.4 km, and the diameter of the water intake pipe is 1.8 m. In order to jack the water pipe into the water source, it needs to exert a reaction force on the side wall of the caisson. Due to the large diameter and long distance of the water intake pipeline, and the fact that the stratum at the depth of the pipeline is mainly a sand layer, the reaction force acting on the sidewall of the caisson is very large. In order to investigate the overall stability of the caisson, a series of numerical simulations via Plaxis 3D are conducted to analyze the load-displacement characteristics of the caisson under the jacking force. Figure 5 is a schematic diagram of the numerical analysis model for the bearing and deformation characteristics of the caisson under lateral jacking force. In the numerical model, fixed constraints are adopted at the bottom of the model and normal constraints are adopted at the sides. A horizontal force is applied at the inner wall of the caisson, within the same depth with the water intake pipeline. In order to be more consistent with the actual situation, the soil disturbance ring caused by the caisson sinking is considered in the numerical model. A disturbance ring with a different thickness is set around the caisson wall in the numerical model, and the mechanical parameters are reduced for the disturbance layer. In addition, the soil outside the shaft wall used for the jacking force is usually reinforced in the actual engineering to ensure that the caisson will not undergo excessive lateral displacement and inclination (as shown in Figure 5). By comparing the deformation characteristics of the caisson under fan-shaped reinforcement (Figure 5b) with those without reinforcement, the effect of soil reinforcement outside the shaft wall is evaluated and the optimized reinforcement range is proposed. Figure 4: The cone penetration curves of the site. Figure 6 is the numerical analysis model for the open caisson under lateral loads. The size of the numerical model is 200 m \(\times\) 100 m \(\times\) 80 m (length \(\times\) width \(\times\) depth). The caisson depth is 46.5 m, and the caisson diameter is 16 m, which can ensure that the boundary conditions have no effect on the calculation results [19]. In the numerical model, the caisson wall is simulated by the plate structure element with equal thickness to improve the calculation efficiency (the average wall thickness of the caisson structure at different heights in the original design). It should be noted that, due to the small length-diameter ratio of the open caisson, the overall deformation characteristic of the caisson appears as rigid rotation. Simplifying the caisson wall with unequal thickness to the wall with equal thickness will not affect the overall deformation characteristics of the structure. In addition, in the numerical model of this paper, the bottom cover concrete of the open caisson is also simplified as a plate structure element, and the plate thickness is the actual thickness of the bottom cover concrete (as shown in Figure 2). The interactions between the caisson foundation and the surrounding soil are modeled by the interface elements with Mohr-Coulomb failure criterion in Plaxis. These interfaces have properties of friction angle, cohesion, dilation angle, tensile strength, Young's modulus, and Poisson's ratio. The values of interface properties in Plaxis can be set by a reduction factor (\(R_{i}\leq 1.0\)) applied to the soil material when defining soil property values (the default value is \(R_{i}=1.0\), i.e., a fully-bonded interface). Hence, the interface property values are directly related to the mechanical properties of the soil around the interface. In this paper, a reduction factor \(R_{i}=0.7\) is adopted for the caisson-soil interface, which is similar to that for the pile foundation.