# Risk Identification Method and Application of Roof Water Inrush Under Multi-Working Face Mining
## 2 The Theory of Key Strata and Full Mining Disturbance
### Critical Dimension Calculation  
The void space created by mining in the working face is subsequently filled by the swelling characteristics of the roof strata. Swelling occurs only in the strata within the caving and fractured zones, while the strata in the bending and subsiding zones do not experience volumetric changes. When the swelling coefficient of the caving overburden approaches the residual swelling coefficient, the height of the water-conducting fracture zone reaches its maximum [28].  
The void space \(\Delta_{k}\) of the \(k\)-th key stratum is as follows:  
\[\Delta_{k}=M-\sum_{i=1}^{n}h_{i}(K_{i}-\ 1) \tag{5}\]  
In the equation, \(M\) is the mining height (m); \(h_{i}\) is the height of the \(i\)-th overlying stratum above the coal seam (m); \(K_{i}\) is the residual swelling coefficient of the \(i\)-th overlying stratum above the coal seam (\(i=1,2,\ldots,n\)); and \(n\) is the overlying stratum number of the \(k\)-th key stratum.  
The \(k\)-th key stratum prior to the initial fracture can be modeled as a simply supported beam. This beam experiences bending deformation under the combined effects of the overlying strata and its own weight. Utilizing the mechanical model of a simply supported beam subjected to a uniform load, as illustrated in Figure 4, the maximum deflection is calculated as follows:  
\[\omega_{\max}=\frac{5q_{k}{l_{k}}^{4}}{384E_{k}{l_{k_{\max}}}} \tag{6}\]  
In the equation, \(l_{k}\) is the overhanging distance of the \(k\)-th key stratum (m); \(l_{k}\) is the moment of inertia of the cross-sectional area of the \(k\)-th key stratum with respect to the neutral axis (m\({}^{4}\)).  
Figure 4: Simply supported beam under uniform load mechanical model.  
The fracture of the \(k\)-th key stratum in the overlying strata must simultaneously satisfy the following two conditions [35; 36]: (1) The overhanging distance of the \(k\)-th key stratum must exceed its ultimate span at the time of initial fracture, implying that the dimensions of the working face must be sufficiently large. (2) The height of the free space below the \(k\)-th key stratum must be greater than the maximum deflection of the key stratum, as described by the following equation:  
\[\left\{\begin{array}{rl}l_{k}>&s_{k}\\ \frac{5q_{k}l_{k}^{4}}{384E_{k}I_{k}}<\Delta_{k}\end{array}\right. \tag{7}\]  
Therefore, when the thickness of the coal seam, as well as the lithology and structure of the overlying strata, are fixed, the ultimate span of the \(k\)-th key stratum must satisfy the following condition:  
\[\frac{5q_{k}s_{k}^{4}}{384E_{k}I_{k}}<\ M-\ \sum_{i=1}^{n}h_{i}(k_{i}-1) \tag{8}\]  
Based on the above analysis, when the coal seam mining thickness and the lithology and structure of the overlying strata are fixed, for the \(k\)-th key stratum of rock, if its maximum span is \(s_{k}\), and the following condition is met,  
\[\frac{5q_{k}s_{k}^{4}}{384E_{k}I_{k}}=M-\ \sum_{i=1}^{n}h_{i}(k_{i}-1) \tag{9}\]  
then the value of \(s_{k}\) represents the critical span at which the overlying strata experience sufficient mining-induced disturbances. When the exposure distance \(l_{k}\) of the \(k\)-th key stratum exceeds \(s_{k}\), the overlying strata are fully disturbed; when \(l_{k}\) is less than \(s_{k}\), the overlying strata experience insufficient disturbance.  
By solving Equations (2) and (9) simultaneously, the critical dimension of the working face for the fracture of the stratum is as follows:  
\[\text{L}=\sqrt[4]{\frac{384E_{k}I_{k}}{5q_{k}}\Big{[}M-\ \sum_{i=1}^{n}h_{i}(k_{i}-1) \Big{]}}+2\sum_{i=1}^{n}h_{i}\text{cot}\,\alpha \tag{10}\]  
Clearly, the critical dimension of the working face for the fracture of the overlying strata is not only influenced by the intrinsic mechanical properties of the rock, but also closely related to the overburden load and the fragmentation characteristics of the underlying rock mass [37].