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In this work we introduce the development of a three--phase incompressibleNavier--Stokes/Cahn--Hilliard numerical method to simulate three--phase flows,present in many industrial operations. The numerical method is then applied tosuccessfully solve oil transport problems, such as those found in the oil andgas industry. The three--phase model adopted in this work is a Cahn--Hilliarddiffuse interface model, which was derived by Boyer and Lapuerta et al. 2006.The Cahn--Hilliard model is coupled to the entropy--stable incompressibleNavier--Stokes equations model derived by Manzanero et al. 2019. The spatialdiscretization uses a high--order discontinuous Galerkin spectral elementmethod which yields highly accurate results in arbitrary geometries, while animplicit--explicit (IMEX) method is adopted as temporal discretization. Thedeveloped numerical tool is tested for two and three dimensional problems,including a convergence study, a two--dimensional jet, a three--dimensionalannular flow, and realistic geometries like T--shaped pipe intersections.

High--order discontinuous Galerkin approximation for a three--phase incompressible Navier--Stokes/Cahn--Hilliard model