On the Strict Hyperbolicity of the Buckley–Leverett Equations for Three-Phase Flow in a Porous Medium
Lars Holden · SIAM Journal on Applied Mathematics · 1990
In this paper the mathematical properties of three-phase flow are discussed. From the properties of the relative permeabilities on the boundary of the three-phase region, it is argued that the system is strictly hyperbolic but with a nonremovable elliptic region in the interior. The nonremovable elliptic region may be reduced to a point (umbilic point) by continuously deforming the relative permeability functions in the interior of the three-phase region. It is well known that elliptic regions lead to severe instabilities in the saturations. Usually it is assumed that the relative permeability of water only depends on the water saturation and that the relative permeability of gas only depends on the gas saturation, whereas the relative permeability of oil depends on all the saturations (Stone’s assumption). With this assumption it is possible to be more precise. If three specified curves intersect at one common point and the isoperms of the relative permeability of oil are concave, then the system is strictly hyperbolic, except at the intersection point, which is an umbilic point. If these three curves do not intersect, then there is at least one elliptic region. This assumption also implies that if the eigenvalues are real, then they are positive. Four different models for generating three-phase relative permeabilities are discussed. All four models are based on two-phase relative permeabilities. It is proved that Stone’s model almost always gives a nonremovable elliptic region.