# Numerical Modeling of Engineering-Scale Jointed Coal Mass and Confining Pressure Effect
## 4 Effect of Confining Pressure on Jointed Coal Mass Under Different Joint Distributions
### Quantification Method of Joint Distribution  
To describe the joint distribution pattern, this paper introduced the fissure degree \(K\) (the unit is m\({}^{-1}\)) to describe the distribution density. The \(K\) refers to the number of joints per unit length on a sampling line in the jointed coal mass, and the calculation formula is as follows:  
\[K=N/L_{m} \tag{1}\]  
where \(N\) is the number of joints on the sampling line, which is dimensionless; \(L_{m}\) is the total length of the sampling line, with a unit of m.  
When joints in different directions were distributed in coal masses, several sets of sampling lines in different directions could be set up to calculate the \(K\) in each direction. For example, Figure 5 displays two groups of joints \(\underline{J}_{a}\) and \(\underline{J}_{b}\). The average spacing of each group of joints in the normal direction was \(\overline{d}_{a}\) and \(\overline{d}_{b}\). On this basis, the average spacing of the horizontal sampling line \(X\) of the two groups of joints could be further calculated as \(\overline{d}_{a}/\sin\theta_{a}\) and \(\overline{d}_{b}/\sin\theta_{b}\), respectively. Then, the fissure degrees \(K_{a}\) and \(K_{b}\) of the two groups of joints could be determined by the following equation:  
\[K_{a}=\sin\theta_{a}/\overline{d}_{a},\ K_{b}=\sin\theta_{b}/\overline{d}_{b}, \tag{2}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Parameters** & **Unit** & **Magnitude** \\ \hline Normal stiffness & (GPa) & 22.90 \\ Shear stiffness & (GPa) & 2.95 \\ Friction coefficient & (/) & 0.57 \\ Tensile strength (MPa) & (MPa) & 0.2 \\ Cohesion & (MPa) & 0.2 \\ \hline \end{tabular}
\end{table}
Table 4: Numerical simulation parameters of direct shear test on the joint.  
Figure 4: Numerical matching of coal direct shear test.  
when the joints have \(n\) directions, the \(K\) of the \(n\)-th joint on the sampling line \(X\) was \(\sin\theta_{n}/\overline{d}_{n}\). The fissure degree on the sampling line \(X\) could be determined by summing the fissure degrees of the joints in each direction:  
\[K_{X}=K_{a}+K_{b}+\cdots+K_{n} \tag{3}\]  
To characterize the fissure distribution of the model as much as possible, the sampling line was set to be the diagonal of the uniaxial compression model, with an angle of 63.4\({}^{\circ}\) with the horizontal direction, as shown in Figure 6. With reference to the statistical results of the joint parameters described above, the fissure degree of the bedding and cleats on the sampling line \(Y\) was calculated as \(K_{Y}=8.26\) m\({}^{-1}\). To establish the relationship between the fissure degree and the uniaxial compressive strength as well as the elastic modulus of the coal mass, this paper took the initial joint spacing at the 1714 working face as the basis, and synchronously enlarged (or narrowed) its bedding spacing and cleat spacing by a factor of 0.50, 0.75, 1.00, 1.25, and 1.50, respectively, and the corresponding \(K_{Y}\) values were 33.04 m\({}^{-1}\), 14.68 m\({}^{-1}\), 8.26 m\({}^{-1}\), 5.28 m\({}^{-1}\), and 3.67 m\({}^{-1}\) respectively. Five groups of models are displayed in Figure 7. The model parameters and loading conditions were kept consistent with the calibrated simulation described above.