  This paper presents the numerical solution of immiscible two-phase flows in
porous media, obtained by a first-order finite element method equipped with
mass-lumping and flux up-winding. The unknowns are the physical phase pressure
and phase saturation. Our numerical experiments confirm that the method
converges optimally for manufactured solutions. For both structured and
unstructured meshes, we observe the high-accuracy wetting saturation profile
that ensures minimal numerical diffusion at the front. Performing several
examples of quarter-five spot problems in two and three dimensions, we show
that the method can easily handle heterogeneities in the permeability field.
Two distinct features that make the method appealing to reservoir simulators
are: (i) maximum principle is satisfied, and (ii) mass balance is locally
conserved.
