# Study on the Co-Evolution Mechanism of Key Strata and Mining Fissures in Shallow Coal Seam Mining
## 4 Analysis of Mining Fissure Propagation Characteristics and Evolution Parameters
### Analysis of Mining Fissure Evolution Parameters
From the force characteristics of the fissure tip unit body and the expression of Formula (44) at the same depth, we can obtain  
\[\sigma=\frac{(\sigma_{x}+\Delta\sigma_{x})+\sigma_{z}}{2}+\frac{(\sigma_{x}+ \Delta\sigma_{x})-\sigma_{z}}{2}\cos 2\beta_{\rm c} \tag{45}\]  
Simultaneous Formulas (40), (42), and (45) can obtain the distance \(r_{\rm II}\) of the type II fissure tip unit as  
\[r_{\rm II}=\frac{K_{\rm II}^{2}\Big{[}\sqrt{2}\cos\frac{\beta_{\rm c}}{2} \Big{(}1-\sin\frac{\beta_{\rm c}}{2}\sin\frac{3\beta_{\rm c}}{2}\Big{)}+\cos \frac{\beta_{\rm c}}{2}\sin\frac{\beta_{\rm c}}{2}\Big{(}1+\Big{(}1+\cos\frac{ \beta_{\rm c}}{2}\cos\frac{3\beta_{\rm c}}{2}\Big{)}\!\cos 2\beta_{\rm c}\Big{)} \Big{]}^{2}}{2\pi{\rm c}^{2}} \tag{46}\]  
From Formula (46), it can be seen that this formula is the relational formula between the stress intensity factor \(K_{\rm II}\) of the fissure tip of the unit body and the fracture angle \(\beta_{\rm c}\) of the rock formation. Similarly, the unit body stress and rock formation failure angle can be calculated using Formulas (19) and (22), respectively. Therefore, only the stress intensity at the tip of the fissure should be an unknown parameter. The type II mining fissures formed by the mining of the working face are the result of mutual dislocation between the strata in the same layer. According to the continuous distribution theory of dislocation in fracture mechanics, the calculation expression of the stress intensity factor \(K_{\rm II}\) at the fissures tip of the unit body is  
\[K_{\rm II}=\tau_{f}\sqrt{\pi a} \tag{47}\]  
Substituting Formula (47) into Formula (46), we can obtain the evolution height of the upward type II mining fissure under the condition of polar coordinates and transform it into Cartesian coordinates to obtain the evolution height \(H_{\rm sr}\) of the upward mining fissure as  
\[H_{\rm sr}=\frac{\tau_{f}^{2}\pi a\Big{[}\sqrt{2}\cos\frac{\beta_{\rm c}}{2} \Big{(}1-\sin\frac{\beta_{\rm c}}{2}\sin\frac{3\beta_{\rm c}}{2}\Big{)}+\cos \frac{\beta_{\rm c}}{2}\sin\frac{\beta_{\rm c}}{2}\Big{(}1+\Big{(}1+\cos\frac {\beta_{\rm c}}{2}\cos\frac{3\beta_{\rm c}}{2}\Big{)}\!\cos 2\beta_{\rm c}\Big{)} \Big{]}^{2}}{2\pi{\rm c}^{2}}\cos\beta_{\rm c} \tag{48}\]  
To predetermine the evolution characteristics of mining fissures, the penetration of upward and downward mining fissures has become one of the evaluation criteria. From the depth of the downward type I mining fissures and the development height of the upward type I and type II mining fissures, the determination formula of mining fissure penetration can be obtained as  
\[\left\{ \begin{array}{l}H_{\rm c}+H_{\rm st}\geq H_{\rm gd}\\ H_{\rm c}+H_{\rm sr}\geq H_{\rm gd}\end{array} \right. \tag{49}\]  
where \(H_{\rm gd}\) is the buried depth of the key strata, m.  
It can be seen from the above formula that when the sum of the up and down mining fissures exceeds the burial depth of the key strata, it indicates that the mining fissures have penetrated the surface and are prone to water inrush, sand inrush, and air leakage. On the contrary, the working face will not produce the above disaster phenomenon.