## 2 Materials and Methods
### Modeling Method
#### 2.2.2 Calculating the Rock Matrix Elastic Modulus
\tag{7}\]  
where \(\hat{P}_{i}\) and \(\hat{Q}_{i}\) are geometric factors of the \(i\)th type of empty connected pores added to the rock skeleton, \(K_{dry}\) and \(\mu_{dry}\) are the elastic modulus of dry rock skeleton. Calculating the Stiffness Matrix of Fractured Dry Rock  
Assuming that the rock matrix is an isotropic medium, the stiffness matrix of dry rock skeleton \(C_{0}\) is as follows [40]:  
\[C_{0}=\left[ \begin{array}{cccccc}\lambda_{dry}+2\mu_{dry}&\lambda_{dry}& \lambda_{dry}&0&0&0\\ \lambda_{dry}&\lambda_{dry}+2\mu_{dry}&\lambda_{dry}&0&0&0\\ \lambda_{dry}&\lambda_{dry}&\lambda_{dry}+2\mu_{dry}&0&0&0\\ 0&0&0&0&\mu_{dry}&0\\ 0&0&0&0&0&\mu_{dry}\end{array} \right] \tag{8}\]  
where \(C_{ij}\) is the stiffness element of dry rock skeleton, and the Lame parameter \(\lambda_{dry}=K_{dry}-\frac{2}{3}\mu_{dry}\), where \(K_{dry}\) and \(\mu_{dry}\) can be calculated by Equation (7). As discussed in the previous section, the Xu-2 Member reservoir can be assumed to be an HTI medium. An HTI medium can be considered as a simple combination of fractures and isotropic background rocks. The isotropic background rocks are characterized by the Lame parameter \(\lambda\) and the shear modulus \(\mu\), while the imbedded fractures are characterized by the dimensionless normal and tangential fracture weakness parameters \(\delta_{N}\) and \(\delta_{T}\), respectively [41]. By using these four parameters, expressions for the stiffness matrix of fractured dry rock \(C^{dry}\) can be written as  
\[C^{dry}=\left[ \begin{array}{cccccc}M_{dry}(1-\delta_{N})&\lambda_{dry}(1- \delta_{N})&\lambda_{dry}(1-\delta_{N})&0&0&0\\ \lambda_{dry}(1-\delta_{N})&M_{dry}\Big{(}1-\Big{(}\frac{\lambda_{dry}}{M _{dry}}\Big{)}^{2}\delta_{N}\Big{)}&\lambda_{dry}\Big{(}1-\Big{(}\frac{ \lambda_{dry}}{M_{dry}}\Big{)}^{2}\delta_{N}\Big{)}&0&0&0\\ \lambda_{dry}(1-\delta_{N})&\lambda_{dry}\Big{(}1-\frac{\lambda_{dry}}{M_{ dry}}\delta_{N}\Big{)}&M_{dry}\Big{(}1-\Big{(}\frac{\lambda_{dry}}{M_{ dry}}\Big{)}^{2}\delta_{N}\Big{)}&0&0&0\\ 0&0&0&\mu_{dry}&0&0\\ 0&0&0&0&\mu_{dry}(1-\delta_{T})&0\\ 0&0&0&0&\mu_{dry}(1-\delta_{T})\end{array} \right] \tag{9}\]  
where the normal and tangential weaknesses parameters \(\delta_{N}\) and \(\delta_{T}\) can be calculated from the normal and tangential fracture compliances of the fractured rocks [42]. The P-wave modulus \(M_{dry}=K_{dry}+\frac{4}{3}\mu_{dry}\), can also be calculated by Equation (7). Calculating the Stiffness Matrix of Fractured Saturated Rock  
Assuming that the dry isotropic porous rocks with interconnected vertical fractures contribute little to the porosity of rocks [43], according to the stiffness matrix of fractured dry-rock, we can use the anisotropic fluid substitution equation to substitute a mixture of hydrocarbon and water into the pore system of rock matrix. The anisotropic fluid substitution equation is as follows [44]:  
\[C_{ij}^{sat}=C_{ij}^{dry}+a_{i}a_{j}\frac{K_{o}}{\left(1-\frac{K^{*}}{K_{o}} \right)-\phi_{i}\left(1-\frac{K_{o}}{K_{f}}\right)},\ i,j=1,2,\cdots,6 \tag{10}\]  
\[K^{*}=\frac{1}{9}\sum\limits_{i=1}^{3}\sum\limits_{j=1}^{3}C_{ij}^{dry} \tag{11}\]  
\[\frac{1}{K_{f}}=\sum\limits_{i=1}^{N}\frac{f_{i}}{K_{i}} \tag{12}\]  
\[a_{m}=b_{m}-\frac{\sum\limits_{n=1}^{3}C_{mn}^{dry}}{3K_{o}},m=1,2,\cdots,6 \tag{13}\]  
\[b_{m}=\left\{ \begin{array}{ll}1,&m=1,2,3,\\ 0,&m=4,5,6,\end{array} \right. \tag{14}\]  
where \(C_{ij}^{sat}\) is the stiffness element of fractured saturated rock, \(K_{f}\) is the bulk modulus of the mixed fluid, \(f_{i}\) is the volume fraction of each fluid, and \(K_{i}\) is the bulk modulus of each fluid. The specific derivation and calculation results of the stiffness element of fractured saturated rock are shown in Appendix B.