## 3 Methodology
### Modeling Rock Physics  
Ideally, the rock physics model should describe the underlying physics behind rocks as much as possible and be associated with a geological element such as depositional processes and compaction, as well as provide a reliable estimation of the rock elastic properties directly from petrophysical properties. This study treats rock as a compound material of two fundamental elements: rock matrix and pore fluid. The rock matrix is composed of quartz and clay. Elastic properties of the rock matrix are estimated based on the Voigt-Reuss-Hill average formulation [19]. Once we form the rock matrix, we can establish a dry rock frame by introducing pore space within the rock matrix by implementing a generalized Xu-White model [20]. We can express the formulations as follows:  
\[K_{d}-K_{m}=\frac{1}{3}\big{(}K^{\prime}-K_{m}\big{)}\frac{3K_{d}+4\mu_{m}}{3K _{m}+4\mu_{m}}\sum_{\iota=S,C}\phi_{\iota}T_{i\iota j\bar{\iota}}(\pi_{\iota}) \tag{1}\]  
and  
\[\mu_{d}-\mu_{m}=\frac{(\mu^{\prime}-\mu_{m})}{5}\frac{6\mu_{d}(K_{m}+2\mu_{m}) +\mu_{m}(9K_{m}+8\mu_{m})}{5\mu_{m}(3K_{m}+4\mu_{m})}\ \times\sum_{\iota=S,C}\phi_{\iota}F(\alpha_{\iota}) \tag{2}\]  
The \(K_{d}\), \(K_{m}\), and \(K^{\prime}\) are the bulk moduli of the dry frame, the rock matrix, and the pore inclusion material, respectively, and \(\mu_{d}\), \(\mu_{m}\), and \(\mu^{\prime}\) are the corresponding shear moduli. \(\phi_{\mathrm{s}}\) and \(\phi_{\mathrm{c}}\) are the aspect ratios for the stiff and compliant pores. Meanwhile, \(T_{i\iota j\bar{\iota}}(\alpha_{\iota})\) and \(F(\alpha_{\iota})\) are pore aspect ratio functions derived from the tensor Tijkl that relate the uniform strain field from infinity toward within the elastic ellipsoidal inclusion. In the meantime, we estimate the bulk modulus and density of individual fluid phases using the Batzle-Wang formulation [21]. The bulk modulus of the mixed pore fluid phases (\(K_{f}\)) can be determined using [PERSON]'s equation, known as inverse bulk modulus averaging. [22], and the density of the mixed pore fluid phases (\(\rho_{f}\)) can be estimated using arithmetic averaging of densities of mixed fluids. Next, we use the Gassmann fluid substitution [23] to model the fluid filling process. The formulations are as follows:  
\[K=K_{d}+\frac{\Big{(}1-\frac{K_{d}}{K_{0}}\Big{)}^{2}}{\frac{\phi}{K_{f}}+ \frac{(1-\phi)}{K_{0}}-\frac{K_{d}}{K_{0}}}, \tag{3}\]  
\[\mu=\mu_{d}\, \tag{4}\]  
and  
\[\rho=\phi\rho_{f}+(1-\phi)\rho_{0} \tag{5}\]  
where \(K\), \(K_{d}\), \(K_{f}\), and \(K_{0}\) are the bulk modulus of the fluid-saturated rock, dry rock frame, pore fluid, and rock matrix, respectively, \(\mu\) and \(\mu_{d}\) are the shear modulus of the fluid-saturated rock and dry rock, \(\rho\), \(\rho_{f}\), and \(\rho_{0}\) are the density of the fluid-saturated rock, pore fluid, and rock matrix, respectively, and \(\phi\) is for total porosity. Then, the rock physics model needs to be calibrated with in-situ well logs to ensure that it is relevant to represent the study area. Once calibrated, the model can establish a generalized rock physics template (RPT) and use it as a reference for the next part.