## 4 Results and Discussion  
### Results and Discussion of the Numerical Simulation  
The movement characteristics of the overburden under these different injection-extraction ratios at this working face are depicted in Figure 12, based on the results of the numerical simulations. Figure 12: Strata subsidence cloud map under different injection–extraction ratios. According to Figure 12a, the movement of rock layers and surface subsidence without growing displays a funnel-shaped pattern, with the fractured rock layers in the overburden space arranged trapezoidally. Figure 12b-e demonstrate that the fly ash filling body significantly impedes the continuous displacement of rock layers at both the upper and lower interfaces of separation, effectively reducing the subsidence of key strata and their overlying rock layers. Specifically, Figure 12b,c show that at injection-extraction ratios of 0.1 and 0.2, the volume of the compacted filling body is not sufficient to provide effective support for the key strata. Figure 12d illustrates that at an injection-extraction ratio of 0.3, there is significant control over the subsidence of the key strata. This control, combined with the unloading expansion of the overburden, results in a noticeable effect on surface subsidence mitigation. Figure 12e reveals that as the injection-extraction ratio approaches 0.4, close to its ultimate value, the fly ash filling body effectively fills the entire separation space, providing robust support for the key strata and minimizing surface subsidence. In practical engineering applications, maintaining a stable separation space for an extended period becomes challenging due to the continuous advancement of the working face. Additionally, the slurry grouting process typically requires considerable time to complete. Consequently, achieving the ultimate injection-extraction ratio is nearly impossible in real-world scenarios. However, this ratio serves as an essential estimation basis for determining the maximum cement consumption during grouting into overburden separation projects. According to the numerical simulation study, the subsidence curves of key strata and ground surface under different injection-extraction ratios are shown in Figure 13.  
According to Figure 13a, the impact of the injection-extraction ratio on the subsidence of key strata is pronounced, with both the key strata and ground surface experiencing a progressive decrease in subsidence as the injection-extraction ratio increases. Without grouting, the maximum subsidence recorded for the key strata is 4849 mm. However, when the injection-extraction ratio is increased to 40%, this subsidence dramatically reduces to just 235 mm. This indicates that an appropriately managed injection-extraction ratio in grouting into overburden separation projects can effectively support the key strata and significantly reduce overburden subsidence. Additionally, the alignment of numerical simulation results with theoretical calculations confirms the reliability and practical applicability of using the elastic foundation beam theory to address the deflection of key strata. In Figure 13b, the surface subsidence curve mirrors the laboratory test results cited in [35], showcasing a funnel-shaped subsidence pattern across various injection-extraction ratios. This pattern highlights the effectiveness of the grouting process, particularly at an injection-extraction ratio of 40%, where the maximum ground surface subsidence is dramatically reduced to only 76 mm, in stark contrast to the 3485 mm observed in  
Figure 13: Influence of the injection–extraction ratio on the subsidence curve of the key strata and ground surface. (**a**) Comparison between numerical simulation and theoretical results of the subsidence curve for the key strata; (**b**) subsidence curve of the ground surface. unground conditions.