Bounded Domain Problem for the Modified Buckley-leverett Equation

Abstract. The focus of the present study is the modified Buckley-Leverett (MBL) equation 1 describing two-phase flow in porous media. The MBL equation differs from the classical Buckley2 Leverett (BL) equation by including a balanced diffusive-dispersive combination. The dispersive term 3 is a third order mixed derivatives term, which models the dynamic effects in the pressure difference 4 between the two phases. The classical BL equation gives a monotone water saturation profile for 5 any Riemann problem; on the contrast, when the dispersive parameter is large enough, the MBL 6 equation delivers non-monotone water saturation profile for certain Riemann problems as suggested 7 by the experimental observations. In this paper, we first show that the solution of the finite interval 8 [0, L] boundary value problem converges to that of the half-line [0,+∞) boundary value problem 9 for the MBL equation as L → +∞. This result provides a justification for the use of the finite 10 interval in numerical studies for the half line problem. Furthermore, we numerically verify that the 11 convergence rate is consistent with the theoretical derivation. Numerical results confirm the existence 12 of non-monotone water saturation profiles consisting of constant states separated by shocks. 13