## 2 Methodology
### Determination of Elastic Foundation Coefficient  
The elastic foundation coefficient is a key parameter for constructing the mechanical model of key strata deflection and solving Equation (2). Based on the assumption of the elastic foundation beam, the calculation formula for the elastic foundation coefficient under conditions of grouting into overburden separation is as follows:  
\[k_{i}=\frac{q_{i}}{S_{i}} \tag{5}\]  
where \(q_{i}\) represents the subgrade reaction exerted on the lower boundary of the key strata within section \(i\) and \(S_{i}\) represents the subsidence value of the key strata within section \(i\). As shown in Figure 2, the subgrade reaction within Section 3, as depicted in Figure 2, is equivalent to the pressure of the grout filling below the key strata (grouting layer pressure \(\sigma_{z}\)) [29], which can be calculated using the following equation:  
\[q_{3}=P_{jt}+\gamma_{ji}h_{z} \tag{6}\]  
where \(P_{jt}\) denotes the pressure at the grouting pipe hole; \(\gamma_{ji}\) represents the unit weight of the grouting slurry; \(h_{z}\) corresponds to the vertical depth at which the grouting layer is positioned. According to the condition of mechanical equilibrium, the subgrade reaction on the lower interface of the key strata is equivalent to the overlying load on the upper interface. Consequently, it can be deduced by the subsequent formula:  
\[(2b+W)q=(2b+h_{ic}/\tan\varphi)q_{1}+(W-h_{ic}/\tan\varphi)q_{3} \tag{7}\]  
The subgrade reaction on the lower interface of the key strata in the solid coal area can be obtained by substituting Equation (6) into Equation (7). \[q_{1}=\frac{\gamma h_{z}(2b+W)-\big{(}P_{jt}+\gamma_{ji}h_{z}\big{)}(W-h_{ic} /\tan\varphi)}{2b+h_{ic}/\tan\varphi} \tag{8}\]  
The subsidence of the key strata is obviously equivalent to the compression deformation of the elastic foundation, allowing us to determine the subsidence of the solid coal area \(S_{1}\) through the calculation of the underlying rock layers' compression deformation. The corresponding equation is as follows:  
\[S_{1}=\sum_{u=1}^{v}\frac{\Delta\sigma_{u}h_{u}}{E_{u}} \tag{9}\]  
where \(\Delta\sigma_{u}\) denotes the supplementary stress exerted by the \(u\)-th layer beneath the key strata; \(h_{u}\) represents the thickness of the \(u\)-th layer below the key strata; \(E_{u}\) signifies the elastic modulus of the \(u\)-th layer underlying the key strata; \(v\) corresponds to the number of rock layers situated between the key strata and coal seam. The elastic foundation coefficient of the solid coal area (Section 1) can be obtained by substituting Equations (8) and (9) into Equation (5). \[k_{1}=\frac{\gamma h_{z}(2b+W)-\big{(}P_{il}+\gamma_{jl}h_{z}\big{)}(W-h_{ic}/ \tan\varphi)}{\sum\limits_{u=1}^{v}\frac{\Delta\sigma_{u}h_{u}}{E_{u}}(2b+h_{ ic})/\tan\varphi} \tag{10}\]  
The subsidence value \(S_{3}\) of the key strata in the compacted area is correlated with mining height \(M\), filling body thickness \(S_{a}\), and height of crushing expansion of the caving zone \(H_{s}\). \[S_{3}=M-H_{s}-S_{a} \tag{11}\]  
The height of crushing expansion of the caving zone \(H_{s}\) can be calculated by  
\[H_{s}=H_{c}(k_{p}-1) \tag{12}\]  
where \(H_{c}\) denotes the height of the caving zone; \(k_{p}\) represents the residual bulking coefficient of the caving zone rock layer. The empirical formula for calculating the height of the caving zone is given by \(H_{c}=100\Sigma M/(c_{1}\Sigma M+c_{2})\). According to the definition of bulking coefficient, the expression for residual bulking coefficient within the caving zone can be formulated based on strain, as follows:  
\[k_{p}=\frac{1-\varepsilon_{c}}{1-\varepsilon_{\max}} \tag{13}\]  
where \(\varepsilon_{c}\) represents the stress of the broken rock mass in the caving zone; \(\varepsilon_{\max}\) represents the maximum strain that can occur in the caving zone rock mass.