## 4 Numerical Simulation and Analysis
### Analysis of Heave Results  
The presence of a significant heave zone, as identified in the previous sections, requires a closer examination of the heave behavior at various soil layers. The following analysis focuses on the heave patterns observed in different soil layers at different depths during the shaft sinking process. Figure 15 shows the heave development curves for four measurement points at various depths (located 3 m from the center of the shaft). These points (CD represents the point position of heave displacement measurement) correspond to the different layers of the soil, which allows us to observe the variation in heave as the shaft deepens. The graph clearly shows that the heave increases with the sinking depth of the shaft. The heave response is most pronounced in the shallow layers of the soil because of their proximity to the excavation. These shallow layers experience the earliest and most significant unloading effects, causing rapid heave in the initial stages of excavation, as observed at measurement points CD-1 and CD-2. A study by [PERSON] et al. [27] on the evolution of soil heave with excavation depth defines the depth at which heave accelerates as the \"primary unloading influence depth.\" Once the excavation reaches the primary depth of influence, the heave continues to develop as a function of the mechanical properties of the soil. This trend was observed when the excavation reached 19 m, at which point the measurement point CD-2 presented the largest heave value. At this depth, the release of stress in the shallow soil layers accumulated, causing the heave to reach its maximum. On the other hand, the deeper soil layers (such as those at CD-3 and CD-4) exhibited less heave due to the stronger mechanical properties of the deeper soil, which achieved greater resistance to the unloading effect. The soil at the pit bottom is more prone to heave because of the unloading caused by excavation, whereas the soil near the shaft edge is effectively restrained by the supporting structure, limiting the development of heave. Therefore, two locations were selected for analysis during the shaft sinking process: the vertical axis LQ-1 along the center of the shaft and LQ-2 along the shaft edge near the riverbank. Figure 15: Development curve of the heave within the pit. Figure 16 shows the variation curves of soil heave along the vertical axis at the shaft center and the shaft edge. The heave values of each soil layer increase gradually as the distance to the excavation surface decreases, and their variation is closely related to the depth of the soil layer. The heave in the upper layers increases after the main influence depth is entered because of their weaker mechanical properties. In contrast, the heave in the deeper layers initially increases slightly and then shows a more pronounced response after a certain depth from the excavation surface is reached. This finding shows that the heave caused by excavation unloading during shaft sinking significantly affects soils only within a certain depth beneath the excavation surface, with minimal impact on deeper soil layers. To explore the soil heave patterns at different excavation depths during shaft sinking, the heave values of the soil within a 25-metre range along the vertical axis of the shaft perpendicular to the river channel were analyzed, as shown in Figure 17. The heave values within the shaft area are relatively large, whereas those at the two sides are smaller. The overall heave trend within the shaft changes from a single-peak \"convex\" shape to a double-peak \"concave\" shape. When the shaft reaches the middle depth (0.4 \(H_{\text{m}}\) to 0.7 \(H_{\text{m}}\)), the soil layers are primarily soft clay, and significant heave occurs at the pit bottom. During the shaft sinking process, when the excavation depth increases rapidly or approaches key soil layers, stress redistribution in the soil triggers significant displacement changes. This variation reflects the nonlinear response of the soil to unloading.