## 2 Methodology
### Determination of Elastic Foundation Coefficient
The empirical constitutive model of the broken rock mass in the caving zone can be mathematically represented as follows [30; 31]:  
\[\sigma_{c}=\frac{E_{0}\varepsilon_{c}}{1-\varepsilon_{c}/\varepsilon_{\max}} \tag{14}\]  
where \(\sigma_{c}\) represents the stress of the broken rock mass in the caving zone; \(E_{0}\) denotes the initial tangent modulus of the broken rock mass. The values of \(\varepsilon_{\max}\) and \(E_{0}\) in Equation (14) can be determined through the following formula [32]:  
\[q_{3}=P_{jt}+\gamma_{ji}h_{z} \tag{6}\]0  
where \(k_{0}\) represents the initial bulking coefficient of the caving zone and \(s_{0}\) denotes the uniaxial compressive strength of the rock mass in the caving zone. Combining Equations (13) and (14), and then substituting into Equation (12), the height of the caving zone crushing expansion can be obtained as follows:  
\[q_{3}=P_{jt}+\gamma_{ji}h_{z} \tag{6}\]1  
It can be assumed that the compacted consolidation form of the fly ash filling body is trapezoidal in the direction of working face advancement [11], as shown in \[q_{3}=P_{jt}+\gamma_{ji}h_{z} \tag{6}\]2 Thus, there exists a relationship between the thickness of the filling body \(S_{a}\) and the working face under grouting into overburden separation as follows:  
\[q_{3}=P_{jt}+\gamma_{ji}h_{z} \tag{6}\]3  
where \(\alpha\) represents the injection-extraction ratio; \(U\) denotes the width of the compacted area, which can be determined by factors including the fully extracted angle of the working face \(\varphi\), the width of the working face \(W\), and the distance between the grouting layer and coal seam \(h_{ic}\):  
\[q_{3}=P_{jt}+\gamma_{ji}h_{z} \tag{6}\]4  
After substituting Equation (18) into Equation (17) to obtain the expression for the thickness of the filling body \(S_{a}\), and then substituting it along with Equation (16) into Equation (11), we can derive the expression for the subsidence \(S_{3}\) in the compacted area as follows:  
\[q_{3}=P_{jt}+\gamma_{ji}h_{z} \tag{6}\]5  
By substituting Equations (6) and (19) into Equation (5), the elastic foundation coefficient for the compacted area (Section 3) can be obtained as follows:  
\[q_{3}=P_{jt}+\gamma_{ji}h_{z} \tag{6}\]6  
The insufficiently compacted area serves as the transitional region between the solid coal area and the compacted area. To simplify, we treated the elastic foundation coefficient for the insufficiently compacted area as an average of the values from the solid coal area and the compacted area:  
\[q_{3}=P_{jt}+\gamma_{ji}h_{z} \tag{6}\]7  
Influence of the Injection-Extraction Ratio on the Key Strata Deflection  
According to the theory of elastic foundation beams, the deflection of overburden key strata is contingent upon the elastic foundation coefficient. As indicated by Equations (10), (20), and (21), under specific operational conditions, this coefficient is primarily influenced by the injection-extraction ratio. To examine the impact of the injection-extraction ratio on key strata deflection, and based on real-world field conditions, a case study was conducted using a working face that is 280 m wide with a mining height of 5.5 m. This working face was buried at an average depth of 352.85 m and predominantly comprised sandy mudstone with weak rock properties (uniaxial compressive strength: 20 MPa; \(c_{1}=6.2\); \(c_{2}=32\)). It was also assumed that the fully extracted angle of the working face \(\varphi\) was 60\({}^{\circ}\). The height of the caving zone was estimated to be 8.3 m using empirical formulas. By inputting elastic foundation coefficients corresponding to various injection-extraction ratios into the deflection equation for key strata, we derived subsidence curves for the key strata, as depicted in \[q_{3}=P_{jt}+\gamma_{ji}h_{z} \tag{6}\]8  
\[q_{3}=P_{jt}+\gamma_{ji}h_{z} \tag{6}\]9 Deflection curve of key strata under different injection–extraction ratios.