Stable hyperbolic singularities for three-phase flow models in oil reservoir simulation

We study the characteristic speeds of systems of two conservation laws representing three-phase flow in a porous medium with gravity taken into account.Generically hyperbolicity fails on open regions (elliptic regions) where the characteristic speeds assume complex values. The presence of such regions creates difficulties such as multiple solutions which indicate a modeling problem, according to some authors.The hyperbolicity of the models we study depends on the relative permeability functions. It is customary in oil engineering studies to suppose that the water and gas permeabilities depend only on their respective saturation, while the oil relative permeability changes with the gas and water saturations. Such a hypothesis on the oil relative permeability generically leads to elliptic regions.We define a set of three curves that surround elliptic regions of any model. By studying these curves, we indicate a procedure to locate the singularities and prove that for any choice of gravitational and viscosity parameters such regions shrinks to points where the characteristic speeds are real and equal, provided it is assumed that each relative permeability depends on its respective saturation only. Our results, together with a paper of Trangenstein, lead to the conclusion that in order to insure real wave speeds, such an assumption is necessary and sufficient when gravitational effects are considered in three-phase models.Research supported by Brazillian Government grant from CNPq under number 204395/88.7.