## 2 Methodology  
### Key Strata Deflection Solving Model  
The technology of grouting into overburden separation is a coal mining technique that emphasizes controlling the deflection of key strata [25]. Thus, understanding the changes in the deflection of these key strata is critical for accurately predicting the effects of surface subsidence when this method is applied. Mechanical Model for Key Strata Deflection  
The technology of grouting into overburden separation primarily involves injecting a slurry, composed of fly ash prepared on the surface, into the separation within the overlying strata. By applying grouting pressure, a compacted zone is formed in the central region of the goat, which supports the key strata through the filling body and effectively mitigates surface subsidence [26]. A schematic diagram illustrating the subsidence reduction model induced by grouting is presented in Figure 1.  
As depicted in Figure 1, the stress state of the coal-rock mass beneath the key strata is classified into three zones: a solid coal area, an insufficiently compacted area, and a compacted area, with a symmetrical distribution across the region. For the load-bearing structure created by grouting into overburden separation, the following assumptions are made: (1) The key strata are modeled as an elastic foundation beam that bears self-weight stress \(q\) from the upper interface of the overlying rock layers. (2) The lower coal-rock mass beneath the key strata is considered an elastic foundation, with elastic foundation coefficients \(k_{1}\), \(k_{2}\), and \(k_{3}\) corresponding to the solid coal area, insufficiently compacted area, and compacted area, respectively. (3) The subgrade reaction on the lower interface of the key strata at the solid coal area and the compacted area is assumed to be uniformly distributed as \(q_{1}\) and \(q_{3}\), respectively. Additionally, it is assumed that linear loads \(q_{2}\) exert pressure on the lower interface of the key strata within the insufficiently compacted area. A mechanical model illustrating the deflection of key strata in the grouting-supported structure of overburden is depicted in Figure 2. In this figure, \(W\) denotes the width of the working face; \(U\) represents the width of the compacted area; \(b\) signifies the width of the coal pillar; \(S_{a}\) indicates the thickness of the filling body; \(h_{ic}\) represents the distance between the grouting layer and top plate of the coal seam; \(h_{z}\) corresponds to the depth at which the grouting layer is positioned; and \(\varphi\) symbolizes the fully extracted angle of the working face. Figure 1: Schematic diagram of the grouting-induced subsidence reduction model. An analysis is conducted using the Winkler elastic foundation beam theory to determine the subsidence value for the key strata. Based on the equilibrium condition, a differential equation is derived to describe the deflection of the key strata [27; 28]:  
\[\left\{ \begin{array}{l}\frac{\mathrm{d}^{4}w_{i}(x)}{\mathrm{d}x^{4}}+\frac{k_ {i}}{Et}w_{i}(x)=\frac{q_{i}}{Et}\\ I=\frac{aH^{3}}{12}\end{array} \right. \tag{1}\]  
where \(w_{i}(x)\) is the subsidence value of the key strata; \(E\) is the elastic modulus of the key strata; \(I\) is the cross-sectional moment of inertia, \(\mathrm{m}^{4}\); \(k_{i}\) is the elastic foundation coefficient of the key strata in the solid coal area, insufficiently compacted area, and compacted area; \(q_{i}\) represents the foundation reacting force on the lower interface of the key strata; \(a\) is the width of the key strata; \(h\) represents the thickness of the key strata; \(i\) is the area number, as shown in Figure 2, and solid coal area, insufficiently compacted area, and compacted area are respectively recorded as Area I, Area II, and Area III. Area I is \(0\leq x<b\) and \(b+W\leq x<2b+W\); Area II is \(b\leq x<(b+h_{ic}/\tan\!\rho)\) and \((W+b-h_{ic}/\tan\!\rho)\leq x<W+b\); Area III is \((b+h_{ic}/\tan\!\rho)\leq x<(W+b-h_{ic}/\tan\!\rho)\).