## 2 Fracture Propagation Model of CBM Reservoirs with Different Lithologies
### Mathematical Formulation and Solution Technique  
Coal rock primarily consists of three components: minerals, organic matter, and fracture pores [8]. These components significantly influence the morphology of hydraulic fractures formed during fracturing. In this section, we develop a fracture propagation model that incorporates the complex lithological characteristics of CBM reservoirs, specifically accounting for the effects of different mineral components on fracture propagation. This study accounts for the heterogeneity of geological parameters and uses dynamic logging data to determine rock mechanical properties. To obtain accurate values for parameters such as Young's modulus and Poisson's ratio, which are essential for calculating in situ stress, specific data must be collected from logging measurements. These include shear wave transit time, compressional wave transit time, and reservoir density at specified locations. Based on these parameters, the Poisson's ratio and elastic modulus of the reservoir can then be expressed as follows:  
\[E_{d,i}=\frac{4\xi_{i}-3\xi_{i}\xi_{i}}{\xi_{i}-\xi_{i}}\quad,\quad v_{d,i}= \frac{2\xi_{i}-\xi_{i}}{2\left(\xi_{i}-\xi_{i}\right)} \tag{1}\]  
where \(E_{d,i}\) is the [PERSON]'s modulus of the CBM reservoirs, GPa; \(v_{d,i}\) is [PERSON]'s ratio; and \(\xi_{i}\) and \(\xi_{i}\) are defined as  
\[\xi_{i}=\frac{\rho_{b,i}}{\left(\Delta t_{s}\right)^{2}}\quad,\quad\xi_{i}= \frac{\rho_{b,i}}{\left(\Delta t_{c}\right)^{2}} \tag{2}\]  
where \(\rho_{b,i}\) is the density of the CBM reservoirs, kg/m\({}^{3}\); \(\Delta t_{s}\) is the shear wave travel time, \(s\); and \(\Delta t_{c}\) is the compressional wave travel time, \(s\). Considering the rock as a porous elastic medium, the maximum and minimum principal stresses of the rock can be expressed as [12]  
\[ \begin{array}{l}\sigma_{h}=\frac{v_{d,i}}{1-v_{d,i}}(\sigma_{v}-\alpha p_{s} )+\xi_{h}\frac{E_{d,i}H}{1+v_{d,i}}+\frac{\alpha_{T}E_{d,i}\Delta T}{1-v_{d,i} }+\alpha p_{s}+\Delta\sigma_{h}\\ \sigma_{h}=\frac{v_{d,i}}{1-v_{d,i}}(\sigma_{v}-\alpha p_{s})+\zeta_{h}\frac{ E_{d,i}H}{1+v_{d,i}}+\frac{\alpha_{T}E_{d,i}\Delta T}{1-v_{d,i}}+\alpha p_{s}+ \Delta\sigma_{H}\end{array}  \tag{3}\]  
where \(\sigma_{h}\) and \(\sigma_{H}\) are the minimum and maximum horizontal principal stresses, MPa; \(\sigma_{v}\) is the vertical stress, MPa; \(p_{s}\) is the pore pressure, MPa; \(\alpha\) is the effective stress coefficient; \(\alpha_{T}\) is the coefficient of thermal expansion; \(\Delta T\) is the temperature difference, K; and \(\xi_{H}\) and \(\zeta_{h}\) are the strain coefficients. When fractures are present in the formation, induced stress affects the further propagation of the fractures. The induced stress at any arbitrary point in space can be obtained using the superposition principle and fracture state parameters. \[ \begin{array}{l}\sigma_{x,ij}=\sum\limits_{k=1}^{N}p_{net,k}[\frac{l_{m}}{ \sqrt{1_{m}l_{2,m}}}\cos\frac{2\theta_{m}-\theta_{1,m}-\theta_{2,m}}{2}-2]- \sum\limits_{k=1}^{N}p_{net,k}\frac{l_{m}}{a}(\frac{a^{2}}{\sqrt{1_{m}l_{2,m} }})^{3/2}\sin\theta_{m}\sin(\frac{3}{2}(\theta_{1,m}+\theta_{2,m}))\\ \sigma_{y,ij}=\sum\limits_{k=1}^{N}p_{net,k}[\frac{l_{m}}{\sqrt{1_{m}l_{2,m} }}\cos\frac{2\theta_{m}-\theta_{1,m}-\theta_{2,m}}{2}-2]-\sum\limits_{k=1}^{N} p_{net,k}\frac{l_{m}}{a}(\frac{a^{2}}{\sqrt{1_{m}l_{2,m}}})^{3/2}\sin\theta_{m} \sin(\frac{3}{2}(\theta_{1,m}+\theta_{2,m}))\\ \tau_{xy,ij}=\sum\limits_{k=1}^{N}p_{net,k}\frac{l_{m}}{a}(\frac{a^{2}}{ \sqrt{1_{m}l_{2,m}}})^{3/2}\sin\theta_{m}\cos(\frac{3}{2}(\theta_{1,m}+\theta _{2,m}))\end{array}