## 1 Introduction  
The injection of carbon dioxide into oil and gas reservoirs for long-term storage or enhanced oil recovery is an advanced technique to mitigate greenhouse emissions and improve hydrocarbon recovery rates [1, 2]. When injected, CO\({}_{2}\) can achieve a certain degree of miscibility with oil, thereby complicating the mass and component exchange of CO\({}_{2}\) and hydrocarbons at the displacement front [3, 4]. Therefore, the accurate modeling of CO\({}_{2}\)-oil/gas phase behavior is critical to assessing the success of CO\({}_{2}\) injections. Compositional modeling is needed to capture the CO\({}_{2}\)-hydrocarbon phase equilibria and the flow of multiple phases. Compositional modeling involves an analysis of the phase behavior of fluids based on their pressure, temperature, and overall composition. This process includes determining the phase stability and conducting flash calculations to establish the number and composition of coexisting phases at equilibrium [5]. These calculations are necessary for each discretization block and time step [6] to provide the physical properties for the flow equations. Depending on the space and time discretization, some simulation runs may require millions of phase equilibrium calculations, which can significantly impact the run time [7] and can consume up to 50% of CPU run time [8], due to the iterative nature of phase equilibrium calculations. The solution to a flash problem must satisfy mass and component balances and ensure that each component has the same fugacity in the coexisting equilibrium phases, in order to minimize the Gibbs energy of the system [9]. The Rachford-Rice equation is a well-established method for flash calculations [10], which incorporates the equilibrium ratio or K-value. This value is determined through an iterative process using an equation of state (EoS), with the Peng-Robinson (PR) [11] and the Soave-Redlich-Kwong (SRK) [12] cubic EoSs being the most commonly used in petroleum engineering. Many researchers have developed alternatives to conventional phase equilibrium calculations to speed up phase computations. The reduction method was first attempted by [PERSON] in 1986 [13], which involved reducing the calculation to three nonlinear equations and setting the binary interaction coefficients to zero. Later, reduction methods were developed by [PERSON], [PERSON] et al., [PERSON] et al., and [PERSON] [14; 15]. Another approach developed in an attempt to reduce computational time involved utilizing precalculated tie-lines and was introduced by [PERSON] and [PERSON] [16] and [PERSON] et al. [17]. Recently, machine learning techniques have been used to address computational time challenges in phase equilibrium calculations. [PERSON] and [PERSON] developed a non-iterative solution for phase stability using support vector classifiers trained on EoS-generated data points [8]. They claimed that this solution was as accurate as the thermodynamic model itself. In a subsequent study, [PERSON] and [PERSON] [18] expanded the model to include a split phase by applying an Artificial Neural Network (ANN) to provide the equilibrium coefficients. [PERSON] et al. [19] demonstrated that their algorithm reduced the phase equilibrium calculations by using a relevance vector machine (RVM) and ANN, resulting in a speeding up of up to 90% computational time. Their work was built on [PERSON] and [PERSON] [20] and comprised three stages: supercritical phase determination, phase stability, and split phase problems tailored to CO2 flooding simulations, with two classifiers for the first two stages and an ANN for the splitting step. In another study, [PERSON] et al. (2019) employed an ANN instead of the SVM method to perform both the phase stability tests and phase splitting. An ANN for a stability test is used to predict saturation pressure to identify stable phases, and another ANN provides initial mole fractions and equilibrium ratios for flash calculations. [PERSON] et al.