Central schemes for the modified Buckley–Leverett equation

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.

► We extend the second and third order classical central schemes to solve the modified Buckley–Leverett equation.
► Numerical results show that the jump locations and plateau heights are consistent with the theoretical calculation.
► Numerical results confirm the existence of non-monotone water saturation profiles consisting of constant states separated by shocks.
► The proposed second and third order schemes achieve the desired accuracy.