## 3 Simulation Results and Analysis
### Overall Three-Dimensional Displacement Field of the Model  
#### 3.1.1 Overall Z-Direction Displacement Field of the Model  
The model exhibits a pattern where the vertical displacement progressively diminishes in a divergent fashion from the interface of the horsehead roadway and the shaft upwards to the model's summit. This diminishing trend amplifies as the subsidence displacement of the horsehead roadway's roof escalates, thereby extending the sphere of influence upon the overlying rock strata. At roof subsidence displacements of 0.5 m, 1.0 m, and 1.5 m for the horsehead roadway, the resultant Z-direction displacement fields within the model are illustrated in Figure 4.  
At a subsidence displacement of 0.5 m for the horsehead roadway's roof, the Z-direction displacement within the roof region predominantly ranges from 0.3 m to 0.5 m, diverging upward to the model's apex where it tapers off, as depicted by the displacement contour line in Figure 4a. The displacement at the model's top measures approximately 9 cm, correlating to roughly one-fifth of the horsehead roadway's roof subsidence displacement. The Z-direction displacement field's four cross-sections along the Z-axis, illustrated in Figure 4b, are situated at distances of 100 m (the model's top surface), 66 m, 33 m, and 0 m  
Figure 4: Z-direction displacement field of the model (cm).  
(the horsehead roadway's roof) from the model's summit downward. These cross-sections reflect the transition of the Z-direction displacement contour line from an O-shape, evident at the 100 m and 66 m levels, to an X-shape at the 33 m and 0 m levels, indicating a shift in displacement pattern with decreasing distance from the roof. Furthermore, beyond 33 m from the horsehead roadway's roof, the Z-direction displacement contour line's density variation is relatively subdued, and the maximum displacement (\(d_{\text{max}}\)) within this plane is approximately 10 cm.  
With a 1.0 m MSD of the horsehead roadway's roof, the Z-direction displacement at the roof predominantly falls within the range of 0.6 to 1.0 m, projecting divergently upwards and diminishing towards the model's apex, as shown by the displacement contour line in Figure 4c. At this increased displacement level, the model's summit registers a displacement of approximately 19 cm, maintaining the proportionality of about one-fifth the MSD experienced by the horsehead roadway's roof. An analysis of Figure 4 reveals a transformation of the Z-axis displacement contour line within vertical sections: transitioning from an O-shape at the 100 m and 66 m levels to an X-shape closer to the roof, at the 33 m and 0 m levels. Moreover, when compared to a subsidence displacement of 0.5 m, the 1.0 m displacement notably influences the 33 m section, evidenced by increased contour line density in contrast to the 66 m and 100 m sections, with the peak Z-direction displacement surpassing 20 cm.  
When the horsehead roadway's roof undergoes the MSD of 1.5 m, the Z-direction displacement ranges from 1.0 to 1.5 m at the point of the roof. The displacement profile exhibits an initial upward extension of approximately 30 m before diverging and tapering off towards the model's apex, as illustrated by the displacement contour line in Figure 4e. The displacement registered at the model's summit is approximately 25 cm, aligning closely with one-fifth of the horsehead roadway's roof MSD. Observations from Figure 4f indicate that, in the vertical sections, the Z-direction displacement contour line shifts from an O-shape at the 100 m and 66 m levels to an X-shape at the 33 m and 0 m levels as the distance from the horsehead roadway's roof decreases. Moreover, when contrasted with a 0.5 m roof subsidence displacement, the 1.5 m displacement exerts a pronounced effect on the 33 m section. This is characterized by a high relative density of the contour lines compared to the 66 m and 100 m sections, with the peak Z-direction displacement exceeding 25 cm.