## 2 Materials and Methods
### Modeling Method
#### 2.2.2 Calculating the Rock Matrix Elastic Modulus  
According to the observation of the thin sections, it was found that the pore types of tight sandstone reservoirs mainly include intergranular pores, dissolution pores and micro-fractures. Thus, the total porosity \(\phi_{t}\) of rock can be expressed by the following equation:  
\[\phi_{p}+\phi_{s}+\phi_{f}=\phi_{t} \tag{2}\]  
where \(\phi_{p},\phi_{s},\phi_{f}\) denote intergranular pores, dissolution pores and micro-fractures, respectively. Each type of porosity can be calculated using logging curves [38], and the detailed calculation formula is shown in Appendix A. Assuming that dissolution pores and micro-fractures form connected pores of rocks, and intergranular pores form isolated pores of rocks, the connectivity between pores can be calculated by the following equation:  
\[v_{i}=\frac{\phi_{i}}{\phi_{t}} \tag{3}\]  
\[\xi=\frac{\phi_{con}}{\phi_{t}} \tag{4}\]  
The scale factor \(v\) is the ratio defined by the Equation (3), where \(\phi_{i}\) is the porosity of the \(i\)th type of pore. According to the connectivity between pores, they can be divided into connected pores \(\phi_{con}\) and isolated pores \(\phi_{iso}\), and the connectivity of pores can be described by the connectivity coefficient \(\xi\) based on the Equation (4). Although dissolution and tectonic processes increase the connectivity between pores, there were still a certain proportion of disconnected pores in the rock. Since fluid in disconnected pores has difficulty flowing and substituting, when elastic wave propagates in this kind of porous medium, fluids in pores with different shapes and connectivity will produce different responses. Therefore, in order to satisfy the assumption that all pores are connected in the classical fluid substitution theory, the disconnected pores are regarded as a part of the rock-solid matrix, and the fluid substitution only occurs in the connected pores. Due to the inclusion-based effective medium theory and fluid substitution equation being effectively able to characterize the influence of different pore shapes on the frame modulus, we assumed that the rock matrix is composed of matrix minerals and isolated pores, and adopted the MT model by considering the pore shape and the interaction of adjacent inclusions to couple the matrix minerals with the saturated isolated pores. Thus, the rock matrix modulus can be calculated by the following equation:  
\[\phi_{mat}=\frac{\phi_{t}-\phi_{con}}{1-\phi_{con}} \tag{5}\]  
\[\left\{ \begin{array}{l}K_{o}=\frac{(1-\phi_{mat})K_{uu}+\phi_{mat}\frac{u}{ \mu}}{(1-\phi_{mat})+\phi_{mat}\frac{u}{\mu}}\frac{v_{i}K_{u}P_{i}}{v_{i}P_{i} }\\ \mu_{o}=\frac{(1-\phi_{mat})H_{mat}}{(1-\phi_{mat})+\phi_{mat}\frac{u}{\mu}} \frac{v_{i}Q_{i}}{v_{i}Q_{i}}\end{array} \right. \tag{6}\]  
where the matrix porosity \(\phi_{mat}\) can be calculated by Equation (5). \(P_{i}\) and \(Q_{i}\) are the geometric factors of the \(i\)th type of saturated isolated pores added to the rock matrix, which is the function of the fluid modulus and the pore aspect ratio [39]. \(K_{uv}\) is the bulk modulus of the bound water in the matrix pores, \(K_{o}\) and \(\mu_{o}\) are the rock matrix elastic modulus, and \(K_{ma}\) and \(\mu_{ma}\) are the mixed minerals' elastic modulus. Calculating the Elastic Modulus of Porous Dry Rock  
The elastic modulus of dry rock skeleton is calculated using the MT model by adding dry connected pores into the rock matrix; the calculation equation is as follows:  
\[\left\{ \begin{array}{l}K_{dry}=\frac{(1-\phi_{i})K_{ma}+(\phi_{i}-\phi_{ona}) \sum\limits_{i=1}^{n}v_{i}K_{o}n_{i}}{(1-\phi_{i})+(\phi_{i}-\phi_{ona})\sum \limits_{i=1}^{n}v_{i}P_{i}+\phi_{ona}\sum\limits_{i=1}^{n}v_{i}\hat{P}_{i}}\\ \mu_{dry}=\frac{(1-\phi_{ona})\sum\limits_{i=1}^{n}v_{i}\hat{Q}_{i}+\phi_{ona} \sum\limits_{i=1}^{n}v_{i}\hat{Q}_{i}}{(1-\phi_{i})+(\phi_{i}-\phi_{ona})\sum \limits_{i=1}^{n}v_{i}\hat{Q}_{i}}\end{array} \right.