Central schemes for the modified Buckley–Leverett equation

Abstract

In this paper, we extend the second and third order classical central schemes for the hyperbolic conservation laws to solve the modified Buckley–Leverett (MBL) equation which is of pseudo-parabolic type. The MBL equation describes two-phase flow in porous media, and it differs from the classical Buckley–Leverett (BL) equation by including a balanced diffusive–dispersive combination. The classical BL equation gives a monotone water saturation profile for any Riemann problem; on the contrast, when the dispersive parameter is large enough, the MBL equation delivers non-monotone water saturation profiles for certain Riemann problems as suggested by the experimental observations. Numerical results in this paper confirm the existence of non-monotone water saturation profiles consisting of constant states separated by shocks.