Abstract: A method for calculating a fluid flow in a given underground medium, the method including receiving initial molar densities for components of the fluid; introducing a scalar auxiliary variable r into an inhomogeneous Helmholtz free energy equation F with a Peng-Robinson equation of state; calculating a molar density ni of each component of the fluid based on a discretized scalar auxiliary variable rk; and determining the flow of the fluid based on the calculated molar densities ni.