## 2 Fracture Propagation Model of CBM Reservoirs with Different Lithologies
### Mathematical Formulation and Solution Technique
 \tag{4}\]  
where \(\sigma_{x,ij}\) is the induced stress component along the \(x\) axis, MPa; \(\sigma_{y,ij}\) is the induced stress component along the \(y\) axis, MPa; \(\tau_{xy,ij}\) is the shear stress component, MPa; \(N\) is the number of fracture elements; \(p_{net}\) is the net fracture pressure, MPa; \(a\) is the fracture half-length, \(m\); \(l_{m}\), \(1_{1,m}\), and \(l_{2,m}\) are the distances from the spatial point \((i,j)\) to the fracture midpoint and its two ends, m; and \(\theta_{m}\), \(\theta_{1,m}\), and \(\theta_{2,m}\) are the angles between the spatial point \((i,j)\) and the fracture midpoint and its two ends. In hydraulic fracture propagation studies, the maximum circumferential stress theory is commonly used to predict fracture propagation. According to this theory, when the circumferential stress at the tip of a hydraulic fracture exceeds the critical circumferentialstress, the fracture begins to extend [13; 14]. Considering that there may be \(M\) mineral particles at the fracture node, the fracture toughness of the rock cannot be calculated by the fracture toughness of a single mineral. In this study, based on the principle of superposition, we apply a weighted fracture toughness when multiple minerals are present at the fracture tip. As a result, fractures at the interface of different minerals are influenced by the properties of each mineral. Generally, hydraulic fracturing creates type I (opening-mode) fractures. Combined with the influence of various minerals, the critical circumferential stress \(\sigma_{\sigma}\) of fracture propagation can be expressed as  
\[\sigma_{\sigma(i,j)}=\frac{1}{\sqrt{2\pi r}}\cdot\sum_{k=1}^{M}\left(K_{IC(i, j)}/M\right) \tag{5}\]  
where \(\sigma_{\sigma(i,j)}\) is the critical circumferential stress at the \((i,j)\) position, MPa; \(K_{IC(i,j)}\) is the rock fracture toughness of different lithologies at the \((i,j)\) position, MPa\(\cdot\)m\({}^{0.5}\); and \(M\) is the number of different mineral nodes around the fracture tip. Considering the differences in mechanical characteristics of different minerals, the mechanical conditions for the extension of fractures in CBM reservoirs with different minerals are  
\[\sigma_{\theta(i,j)}-\sigma_{\sigma(i,j)}=\cos\frac{\theta_{(i,j)}}{2}(K_{1(i,j)}\cos^{2}\frac{\theta_{(i,j)}}{2}-\frac{3}{2}K_{2(i,j)}\sin\theta_{(i,j)} )-\sum_{k=1}^{M}\left(K_{IC(i,j)}/M\right)>0 \tag{6}\]  
where \(\sigma_{\theta(i,j)}\) is the circumferential stress in the direction of the fracture tip (\(i\), \(j\)), MPa; \(K_{1(i,j)}\) and \(K_{2(i,j)}\) are type I and type II fracture stress intensity factors in the \((i,j)\) direction, MPa\(\cdot\)m\({}^{0.5}\); and \(\theta_{(i,j)}\) is the angle between the fracture tip \((i,j)\) direction and the fracture axis. Fracture propagation is random in hydraulic fracturing [15]. During fracturing, fracture morphology is not the same. In the fracturing process, the fracture propagation satisfies a certain probability function \(F\), and the fracture propagation is controlled by the fractal index \(\gamma\). When the fracture tip encounters minerals, the fractal index is also different due to the difference in mineral properties. Furthermore, when the fracture reaches an interface between multiple minerals, its propagation probability is influenced by the combined properties of those minerals. Similar to the treatment of fracture toughness, the superposition principle is used to weight \(M\) minerals at the interface, and the comprehensive fractal index \(\overline{\gamma}\) is obtained. The fracture probability of the fracture tip \((i,j)\) is the proportion of the residual stress intensity of the point to the total residual stress intensity of all the fracture tips. The total residual stress intensity should take into account the sum of all single-mineral residual stress intensity and all mineral-interface residual stress intensity.