# Numerical Modeling of Engineering-Scale Jointed Coal Mass and Confining Pressure Effect
## 5 Potential Application in Estimation of Abutment Pressure Distribution Ahead of Working Face  
### Stress Distribution in Coal Seam  
After coal seam mining, the overburden load in the gob is transferred to the coal seam ahead of the working face, forming an abutment pressure zone [46; 47; 48], as displayed in Figure 13. The elevation zone arises due to stress redistribution caused by the deformation and movement of the coal seam and rock strata after coal mining. The vertical stress \(\sigma_{vx}\) in the zone ahead of the working face gradually increased under the effect of the self-gravity stress of the overlaying rock layer from the initial vertical stress \(\sigma_{v}\), and then started to decline until it reached the ultimate load-bearing stress \(K_{1}\sigma_{v}\) of the coal mass. It finally fell to the residual stress \(K_{2}\sigma_{v}\) at the coal wall. According to the stress distribution, the zone ahead of the working face can be divided into the in situ stress zone, the vertical stress elevation zone, and the vertical stress reduction zone. The stress distribution in the vertical stress elevation zone is as follows:  
\[\sigma_{vx}=(K_{1}-1)\sigma_{v}e^{(L_{m}-x)/a_{m}}+\sigma_{v} \tag{4}\]  
where \(K_{1}\) is the peak stress concentration factor, its value is generally between 1 and 4, and it is dimensionless; \(L_{m}\) is the distance between the peak stress point and the coal wall, with a unit of \(m\); \(x\) is the distance between the stress distribution point and the coal wall, with a unit of \(m\); and \(a_{m}\) is the fitting constant, which is related to mining and geological factors.  
Since the coal wall mined at the working face was in an unconfined state, the horizontal stress in the coal seam in the zone ahead of the working face was in a decreasing state until  
Figure 12: The angle of main fissure under different confining pressures.  
Figure 13: Stress distribution of longwall top-coal caving.  
it dropped to 0 at the coal wall. The horizontal stress distribution approximately obeyed a negative exponential relationship:  
\[\sigma_{hx}=\sigma_{h}\Big{(}1-e^{-\beta_{m}x}\Big{)} \tag{5}\]  
where \(\beta_{m}\) is the fitting constant, which is related to mining and geological factors.  
The horizontal stress distribution could be directly derived by Equation (5) with the given fitting constant \(\beta_{m}\), while the vertical stress distribution remained un-derivative for \(L_{m}\) with the given fitting constants \(\alpha_{m}\) and \(K_{1}\). The magnitude of \(L_{m}\) was related to the location of the critical failure of the coal mass, and the magnitude of \(\sigma_{vx}\) at \(x=L_{m}\) was \(K_{1}\sigma_{v}\). Combined with the Mohr-Coulomb criterion, the vertical and horizontal stresses of the coal seam at the critical failure satisfied the following equation:  
\[\left\{\begin{array}{l}\sigma_{1}=\frac{1+\sin\varphi}{1-\sin\varphi}\sigma_ {3}+\frac{2C\cos\varphi}{1-\sin\varphi}\\ \sigma_{1}=\sigma_{vx},\sigma_{3}=\sigma_{hx}\end{array}\right. \tag{6}\]  
where \(C\) and \(\varphi\) are the cohesion and internal friction angle of the coal mass in MPa and \({}^{\circ}\), respectively. Combination of Equations (4)-(6) yielded the \(L_{m}\):  
\[L_{m}=-\frac{1}{\beta_{m}}\ln\left[1-\frac{\left(K_{1}\sigma_{v}-\frac{2C\cos \varphi}{1-\sin\varphi}\right)}{\sigma_{h}\left(\frac{1+\sin\varphi}{1-\sin \varphi}\right)}\right] \tag{7}\]  
The above equation confirmed that \(L_{m}\) was related to \(C\) and \(\varphi\). These two variables are inherent mechanical parameters of the coal mass that can be calculated from the peak strengths of the loaded specimens with different confining pressures.