## 3 Dynamic Adaptive Rock Physics Model (DARPM)
Therefore, [PERSON] and [PERSON] [30] rewrote the theory in the form of fluid saturation and elastic stiffness coefficient matrices as follows:  
\[\frac{c_{13}^{if}+\frac{2}{3}c_{55}^{if}}{K_{k}}=\frac{1+\left[4\mu_{k}\Big{(}K _{f}-K_{k}\Big{)}\big{/}\Big{(}3K_{f}+4\mu_{k}\Big{)}K_{k}\right]S}{1-\left[3 \Big{(}K_{f}-K_{k}\Big{)}\big{/}\Big{(}3K_{f}+4\mu_{k}\Big{)}\right]S} \tag{2}\]  
\[\frac{c_{55}^{if}}{\mu_{k}}=\frac{(1-S)\Big{(}9K_{f}+8\mu_{k}\Big{)}}{9K_{k}+ 8\mu_{k}+S(6K_{k}+12\mu_{k})} \tag{3}\]  
where  
\[S=\Phi_{f}/\Big{(}\Phi_{f}+\Phi_{k}\Big{)} \tag{4}\]  
\[\rho_{if}=\Big{(}\Phi_{k}\rho_{k}+\Phi_{f}\rho_{f}\Big{)}\Big{/}\Big{(}\Phi_{ k}+\Phi_{f}\Big{)} \tag{5}\]  
where \(K\) represents the bulk modulus, \(\mu\) represents the shear modulus. \(S\) represents fluid saturation in the organic matter mixture; \(\rho_{if}\) represents the density of the organic matter mixture; \(\Phi\) represents the volume fraction; \(c^{if}_{ij}\) is the elastic stiffness coefficient matrix of the organic matter mixture; the subscript \(k\) refers to kerogen, and subscript \(f\) refers to fluid. Next, we begin constructing the rock matrix with VTI anisotropic characteristics. The isotropic matrix is composed of various minerals, and the oriented arrangement of clay within the matrix induces VTI anisotropy. Step 3: Apply the HSB (Hashin-Shtrikman boundary) theory to mix various minerals like calcite, dolomite, and quartz to construct a rock matrix with isotropic characteristics [31]. When different components of rocks are mixed without information of geometric details, the allowable narrow range of values is calculated as follows:  
\[K^{\text{HS}\pm}=K_{1}+\frac{f_{2}}{\left(K_{2}-K_{1}\right)^{-1}+f_{1}\Big{(} K_{1}+\frac{4}{3}\mu_{1}\Big{)}^{-1}} \tag{6}\]  
Figure 4:0  
In these equations, \(f\) represents the volume fractions, and the subscripts 1 and 2 refer to different components. Step 4: Consider the layered distribution of the reservoir due to the oriented arrangement of clay. Anisotropic Backus theory [15] can be used to introduce clay into the isotropic rock matrix to create a layered structure, forming an equivalent VTI anisotropic medium model. For a single thin layer of VTI medium, the elastic stiffness coefficient of the equivalent medium after Backus averaging can be written as \(C_{ij}^{\text{-}}\):\[ \begin{split}&\mathbf{C}_{11}^{\epsilon}=\mathbf{C}_{22}^{\epsilon}=\langle\mathbf{C} _{11}\rangle+\Big{\langle}\frac{\mathbf{C}_{13}}{\mathbf{C}_{33}}\Big{\rangle}^{2} \Big{\langle}\frac{1}{\mathbf{C}_{33}}\Big{\rangle}^{-1}-\Big{\langle}\frac{\mathbf{C }_{13}^{2}}{\mathbf{C}_{33}}\Big{\rangle},\\ &\mathbf{C}_{33}^{\epsilon}=\Big{\langle}\frac{1}{\mathbf{C}_{33}}\Big{ }^{-1},\\ &\mathbf{C}_{44}^{\epsilon}=\mathbf{C}_{55}^{\epsilon}=\Big{\langle}\frac {1}{\mathbf{C}_{44}}\Big{\rangle}^{-1},\\ &\mathbf{C}_{66}^{\epsilon}=\langle\mathbf{C}_{66}\rangle,\\ &\mathbf{C}_{12}^{\epsilon}=\langle\mathbf{C}_{12}\rangle+\Big{\langle} \frac{\mathbf{C}_{13}}{\mathbf{C}_{33}}\Big{\rangle}^{2}\Big{\langle}\frac{1}{\mathbf{C} _{33}}\Big{\rangle}^{-1}-\Big{\langle}\frac{\mathbf{C}_{13}^{2}}{\mathbf{C}_{33}} \Big{\rangle}=\mathbf{C}_{11}^{\epsilon}-2\mathbf{C}_{66}^{\epsilon},\\ &\mathbf{C}_{13}^{\epsilon}=\mathbf{C}_{23}^{\epsilon}=\Big{\langle}\frac {\mathbf{C}_{13}}{\mathbf{C}_{33}}\Big{\rangle}\Big{\langle}\frac{1}{\mathbf{C}_{33}} \Big{\rangle}^{-1}.\end{split}  \tag{8}\]  
where \(\mathbf{C}_{ij}\) is the elastic stiffness coefficient of the equivalent medium before Backus averaging. Step 5: Introduce inorganic pores using the anisotropic Krief formula [32]. The bulk modulus and shear modulus of dry rocks can be calculated with empirical formulas:  
\[K_{m}=K_{s}(1-\Phi)^{A/(1-\Phi)} \tag{9}\]  
\[\mu_{m}=K_{m}\mu_{s}/K_{s} \tag{10}\]  
Here, \(A\) is a constant related to the rock type; the value of porosity \(\Phi\) is related to the critical porosity (between 0.4 and 0.6); the subscripts \(m\) and \(s\) refer to the dry rock skeleton model and the solid matrix.