## 2 Materials and Methods
### Multiscale Compositional Model
#### 2.2.2 Sequential Formulation  
To facilitate the use of multiscale formulations later, the sequential formulation is employed to split the entire system of equations into pressure equations and transport equations. Various methods exist to construct the sequential formulation pressure equation based on reservoir numerical models. Using the linear algebraic approach by [PERSON] [54], in Equation (19), considering the physical significance of different elements in the system, the term \(U\) corresponding to \(u\) represents the accumulation term containing the time derivative, while \(V\) corresponding to \(v\) includes the remaining constraint terms, excluding the accumulation term. Consequently, the Jacobian matrix \(J_{e}\) and its Schur complement reduced matrix \(\bar{J}_{u}\) can be reformulated to facilitate the explanation of subsequent transformations  
\[J_{e}=\left[ \begin{array}{cc}\frac{\partial AC}{\partial u}&\frac{\partial AC }{\partial v}\\ \frac{\partial\theta}{\partial u}&\frac{\partial\theta}{\partial v}\end{array}  \right],\ \bar{J}_{u}=\frac{\partial AC}{\partial u}-\frac{\partial AC}{\partial v} \left(\frac{\partial\theta}{\partial v}\right)^{-1}\frac{\partial\theta}{ \partial u}. \tag{21}\]  
where \(AC\) represents the accumulation terms in the system of equations, \(\theta\) represents the constraint equation block, and \(u\) and \(v\) represent the primary and secondary unknowns, respectively. The purpose of the sequential formulation is to decouple the pressure variable from other variables. Referring to the derivation process of the sequential formulation in the black oil model, it achieves this by using a linear combination of the accumulated phases, which has zero derivatives with respect to non-pressure variables. Thus, the pressure variable can be decoupled from the non-pressure variables. Drawing on this concept, from a linear algebra perspective, we construct a linear combination coefficient vector--the weight factor w, which is formulated as follows:  
\[\left(\bar{J}_{u}^{T}\right)^{-1}w=e_{p},\ \left(e_{p}\right)_{i}=\left\{ \begin{array}{ll}1&\text{if $u_{i}$ is a pressure variable}\\ 0&\text{otherwise}\end{array} \right. \tag{22}\]  
Mathematically, the weight factor w aligns with the derivation of the sequential formulation in the black oil model, as both eliminate non-pressure variables through weighted summation. Thus, by applying w, a linear algebraic transformation can be performed on Equation (19) to derive the pressure equation. The pressure equation is derived based on Equations (19) and (22):  
\[-\bar{J}_{p}\Delta u_{p}=\bar{r}_{p},\ \bar{J}_{p}\in R^{n_{e}\times n_{e}},\ \bar{r}_{p}\in R^{n_{e}}. \tag{23}\]  
The coefficients and the right-hand side of Equation (23) are constructed through the following expressions:  
\[\left(\bar{r}_{p}\right)_{j}=\sum_{i=0}^{m-1}w_{j+in_{c}}(\bar{r}_{u})_{j+in_ {c}},\ \left(\bar{J}_{p}\right)_{jk}=\sum_{i=0}^{m-1}w_{j+in_{c}}(\bar{J}_{u})_{j+in_ {c},k}+\frac{\partial w_{j+in_{c}}}{\partial p_{k}}(\bar{r}_{u})_{j+in_{c}}. \tag{24}\]  
The pressure increment is obtained by solving the above equations. Subsequently, the transport equations are solved using the computed pressure increment. Once the pressure Equation (23) has converged, the Darcy velocities for different phases are determined using the multiphase Darcy equation (Equation (13)).