Three-phase equilibrium calculations for water-CO2-hydrocarbon mixtures are required in the compositional simulation of various applications in CO2 storage, geothermal systems, and enhanced oil recovery. The very low solubility of hydrocarbon components in water leads to a special mathematical structure of the problem. Several techniques were suggested, such as the free-water flash (FWF) and the augmented free-water flash (AFWF); in the former, the aqueous phase is pure water, while in the latter only certain components, CO2 or methane for example, are dissolved in the aqueous phase. However, only the first-order successive substitution method was used in the previous published approaches, making them unattractive for compositional simulations in which a significant number of phase equilibrium calculations are performed. In this work, a robust and efficient AFWF method is proposed, using combined successive substitutions-modified Newton iterations. The new method is general, allowing partial solubility of any selected component in the water-rich phase, depending on the specific compositions and operating conditions. A detailed description of second-order methods in a Gibbs energy minimization framework for the general AFWF is presented. In the AFWF, the dimension of the problem and the number of function evaluations (thus the computation time) are significantly reduced. Moreover, it is shown that the augmented method always has better convergence properties than its conventional multiphase flash counterpart, in both first- and second-order methods. The new AFWF method is tested for various hydrocarbon-water-CO2 mixtures and proved to be robust and efficient, systematically outperforming the conventional approach. Unlike in previous AFWF formulations, the number of components soluble in water is not limited, leading to a controlled accuracy with respect to a full three-phase equilibrium, even at high pressures and/or large amounts of CO2.@en