## 3 Results and Discussions  
The Peng-Robinson equation of state was applied to the CO\({}_{2}\), n-butane, and n-decane system to calculate the phase behavior. The model accurately captures the equilibrium between the liquid and vapor phases, particularly under typical reservoir conditions and near-critical points. The generated tie-lines and close-up view depict the distribution of phases in the compositional space. As expected, the Peng-Robinson EoS performs well in identifying two-phase regions, with distinct tie-line slopes representing phase equilibrium. Figure 2 represents the phase behavior and equilibrium constants calculated using the Peng-Robinson equation of state for a CO\({}_{2}\), n-butane, and n-decane mixture under two conditions: near-critical and regular. In Figure 2a, the near-critical conditions at 80 atm and 60 \({}^{\circ}\)C show the tie-lines becoming shorter and denser, indicating the difficulty of distinguishing between the liquid and vapor phases as the system approaches the critical point. The equilibrium constant Ki approaches one, indicating the critical point. In Figure 2b, under regular conditions at 75 atm and 60 \({}^{\circ}\)C, the tie-lines are longer and more evenly spaced, indicating well-defined phase behavior. The equilibrium constant \(K_{i}\) displays smoother transitions across the concentration scan, with CO\({}_{2}\) showing higher \(K_{i}\) values at lower concentrations, while n-decane's heavier nature results in a lower \(K_{i}\). Figure 3 shows the training, validation, and testing performance of the neural network models using 4000 flash calculation samples for the Direct Model, Hybrid Model 1, and Hybrid Model 2. All the models demonstrate an excellent fit, with near-perfect matching between the predicted and actual values. However, the Direct Model (a) requires the highest number of training iterations. While the overall accuracy of all the models is impressive, a more detailed analysis of how each model performs in compositional space will be presented in the following section. The results shown in Figure 4 reveal that while the direct model produced a nearly perfect match during the training, validation, and testing, there were notable issues when examining the model's behavior in compositional space. Specifically, for decane, we observed unphysical fluctuations in the equilibrium constants (\(K_{i}\)) under both near-critical and regular conditions. These fluctuations suggest that the model struggles to maintain consistent physical behavior, particularly for heavier components like decane, likely because the equilibrated \(y_{i}\) values are very small and challenging to capture by the Direct Model. Additionally, a closer examination of the tie-lines shows that they cross each other, which is another indication of unphysical behavior. This crossing of tie-lines should not occur in realistic phase equilibrium scenarios and highlights the limitations of the Direct Model, despite its high accuracy in the training process. These inconsistencies underscore the need for hybrid approaches to ensure both computational efficiency and physical validity. Figure 2: Reference case generated using [PERSON]–[PERSON] equation of state: (**a**) near-critical conditions of 80 atm and 60 \({}^{\circ}\)C; and (**b**) regular conditions of 75 atm and 60 \({}^{\circ}\)C.  
Figure 4: Ternary diagram generated using the direct machine learning model (\(x_{i}\), \(y_{i}\)): (**a**) near-critical conditions of 80 atm and 60 \({}^{\circ}\)C; and (**b**) regular conditions of 75 atm and 60 \({}^{\circ}\)C.  
Figure 3: Machine learning (neural network) training, validation, and testing using 4000 flash calculation samples (pressure [80 atm to 40 atm], temperature [57 \({}^{\circ}\)C to 67 \({}^{\circ}\)C], and overall composition of CO\({}_{2}\)-C4-C10 [0 to 1]) for (**a**) Direct Model, (**b**) Hybrid Model 1, and (**c**) Hybrid Model 2.