Mathematical Properties of Three-Phase Flow Equations

The equations for nondiffusive three-phase flow with gravity are examined according to the relative permeability model used. Three different classes of equations are distinguished (1) hyperbolic with a single umbilic point, (2) hyperbolic with multiple umbilic points, and (3) mixed hyperbolic and elliptic. The new equations with multiple isolated umbilic points are obtained when using Corey relative permeabilities that allow inflection points (i.e., extended Corey models). An example in which gravity and extended Corey relative permeabilities appear to give strictly hyperbolic equations is also included. The influences of viscosity, gravity, and relative permeabilities on the location of these umbilic points are shown. The relationship between the umbilic point(s) and the elliptic regions is discussed. The effect of gravity on the size of the elliptic regions is analyzed. It is shown that the size of the elliptic regions can be significantly increased or decreased by changing the gravity to viscous ratio, but without following an obvious systematic pattern.