With the rising global energy demand, shale gas and oil emerge as pivotal resources. Recent innovations utilizing CO₂ as an injectant can effectively enhance shale oil and gas recovery and facilitate CO₂ storage within shale reservoirs. However, low-temperature CO₂ injection may result in the coexistence of three hydrocarbon phases, while the abundant nanopores in shale formations also notably influence the phase behavior of reservoir fluids. To optimize shale oil recovery and CO₂ sequestration in shale formations, it is a prerequisite for precisely capturing the effect of confinement on the phase behavior of reservoir fluids within nanopores during CO₂ injection. In this work, we introduce a novel three-phase vapor-liquid-liquid equilibrium calculation algorithm, which is designed to handle the unique phase behavior challenges presented by CO₂ utilization and storage in shale reservoirs. To improve the robustness and efficiency, the proposed algorithm integrates a trust region-based stability test with a hybrid flash calculation algorithm that combines the Newton-Raphson and trust-region methods. Our thermodynamic model incorporates the capillarity effect and shifts in the critical points due to molecule-wall interactions, which are essential for accurate phase behavior simulation under confinement. Initial validations against experimental bulk phase data show promising results, and further investigations indicate that confinement alters three-phase vapor-liquid-liquid equilibria by suppressing two-phase and three-phase regions and shifting boundaries in the phase diagrams. The proposed algorithm not only advances our understanding of multiphase equilibrium in nanoporous media but also enhances the practicality of CO₂ sequestration and improved oil recovery strategies in shale formations.

In this paper, a mathematical model has been developed to quantitatively examine the effect of viscosity and heterogeneity on dispersion in porous media at the pore scale during miscible flooding processes. More specifically, the Navier-Stokes equation and advection-diffusion equation are coupled with supplementary equations to describe the solvent transport behaviour. Two-dimensional heterogeneous models are numerically developed as a function of porosity and permeability, assuming that the grain sizes satisfy normal distribution. In addition, the performance of miscible hydrocarbon gas injection in heterogeneous porous media is comprehensively evaluated. It is found that a larger aspect ratio (ratio of pore throat size) in the single non-flowing pore model results in a greater asymmetry of the concentration curve. As for single non-flowing pore models and heterogeneous models, the dispersion coefficients increase with the expansion of the non-flowing domain. Both the heterogeneity of porous media and the variable viscosity of the fluid mixture contribute to the asymmetry of the concentration curve in the heterogeneous model. A negative correlation is established between the sorting coefficients of pore throat size and the power-law coefficients. As for slug injection, the injected solvent slug size along the longitudinal direction does not effectively influence the longitudinal length of the mixing zone for a given porous medium and fluids, though the Peclet number and the porosity greatly affect the length and concentration distribution of the mixing zone.