Solutions to the Three-Phase Buckley-Leverett Problem

This paper presents method of characteristics (MOC) solutions to the three-phase Buckley-Leverett problem with and without gravity, for the three classes of equations described by Guzmán and Fayers (1997).12 Thus, solutions for equations with single and multiple umbilic points, and mixed elliptic and hyperbolic equations are constructed. A close relation between the mathematical properties of the equations (in terms of location and quantity of umbilic points) and the physics of flow is described. This relationship can be observed by the effect of gravity on the efficiency of the displacement, the formation of oil and water banks, and the presence of a unique or infinite solution paths within the three-phase flow region for a class of solutions. Elliptic regions did not generate physically inadmissible effects and stable shocks with saturations on opposite sides of elliptic regions were constructed in all the cases examined.