## 2 Fracture Propagation Model of CBM Reservoirs with Different Lithologies
### Mathematical Formulation and Solution Technique
The fracture probability of the fracture tip \((i,j)\) can be expressed as  
\[P(i,j)=\frac{\frac{1}{M}\cdot\sum\limits_{k=1}^{M}\left(\sigma_{\theta(i,j)}- \sigma_{\sigma(i,j)}\right)^{\overline{\gamma}}}{\sum\limits_{k=1}^{N}\left( \sigma_{\theta(i,j)}-\sigma_{\sigma(i,j)}\right)^{\gamma_{1}}+\sum\limits_{k =1}^{D}\left(\sigma_{\theta(i,j)}-\sigma_{\sigma(i,j)}\right)^{\overline{ \gamma}}} \tag{7}\]  
where \(P(i,j)\) is the fracture probability of node \((i,j)\); \(N\), \(D\) is the number of nodes that can form a fracture in a single mineral; \(G\) is the number of nodes that can form a fracture at the mineral interface; \(M\) is the number of different mineral nodes around the fracture tip; \(\gamma_{i}\) is the fractal index of a single mineral; and \(\overline{\gamma}\) is the comprehensive fractal index of node \((i,j)\), which can be expressed as  
\[\overline{\gamma}=\frac{1}{M}\sum\limits_{k=1}^{M}\gamma_{(i,j)} \tag{8}\]  
The propagation direction of hydraulic fracture has a certain random distribution characteristic, and at the same time satisfies the probability distribution. According to the probability distribution (Equation (7)), the sum of the probabilities of all possible propagation points around the fracture tips is 1. The probability value of each point is related to its circumferential stress and the critical stress of fracture initiation calculated from the lithological characteristics at that point, as p1 to \(p_{N}\) shown in Figure 2. In the simulation of fracture propagation, a computer program is used to automatically generate a random number between 0 and 1, and the direction of fracture propagation is determined according to the random number. As shown in Figure 2, if the random number \(R\) is within the range of \(\sum\limits_{k=1}^{K-1}p_{k}\) and \(\sum\limits_{k=1}^{K}p_{k}\), the new fracture will propagate along the direction of node \(K\)th. Considering the influence of three different minerals on hydraulic fracture morphology, a hydraulic fracture propagation model under different mineral conditions is shown in Figure 3. In this model, the reservoir is divided into several equivalent nodes, with each node representing a mineral type. The coalbed methane reservoir is thus depicted as a network of interconnected mineral nodes: black nodes represent clay, yellow nodes represent quartz, blue nodes represent pyrite, while red nodes and lines indicate fractures formed by hydraulic fracturing. Hydroraulic fracture propagation needs to consider the stress interference between cracks and the influence of different minerals on the determination of fracture propagation direction. The main calculation process includes (a) dividing model nodes according to actual reservoir size and mineral content; (b) parameter assignment, mainly setting the geological parameters and physical parameters of the node; (c) setting perforation position and updating geological parameters; (d) distinguishing between single-mineral nodes and multi-mineral interface nodes and modifying the fracture toughness and comprehensive fractal index of fractures; (e) calculation of the circumferential stress of mineral nodes at the fracture tip; and (f) calculation of the fracture probability to determine the next fracture propagation node. The process of solving the model is shown in Figure 4.  
Figure 3: Hydraulic fracture propagation model with different lithologies. Figure 2: The direction distribution of fracture propagation based on random function and probability distribution.