C1-Continuous relative permeability and hybrid upwind discretization of three phase flow in porous media

•Developed a hybrid upwinding scheme for three-phase flow in the presence of gravity.•Demonstrated that the hybrid scheme is consistent and convergent in nonlinear Newton iterations.•Examined C1 -continuity and consistency in three-phase relative permeabilities.•Proposed the consistency condition for phase transition that improves numerical convergence in phase appearance and disappearance.