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