Bounded domain problem for the modified Buckley-Leverett equation

TL;DR

This paper investigates the modified Buckley-Leverett equation, which models two-phase flow in porous media with dispersive effects, showing convergence of finite interval solutions and confirming non-monotone saturation profiles through numerical schemes.

Contribution

It proves the convergence of finite interval solutions to the half-line problem and extends numerical schemes to solve the pseudo-parabolic MBL equation.

Findings

Finite interval solutions converge to half-line solutions.

Numerical schemes confirm non-monotone saturation profiles.

Non-monotone profiles consist of shocks separated by constant states.

Abstract

The focus of the present study is the modified Buckley-Leverett (MBL) equation describing two-phase flow in porous media. The MBL equation differs from the classical Buckley-Leverett (BL) equation by including a balanced diffusive-dispersive combination. The dispersive term is a third order mixed derivatives term, which models the dynamic effects in the pressure difference between the two phases. 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 profile for certain Riemann problems as suggested by the experimental observations. In this paper, we first show that the solution of the finite interval [0,L] boundary value problem converges to that of the half-line [0,+\infty) boundary value problem for the MBL equation as L-> +\infty. This result provides a justification for the use of the finite interval boundary value problem in numerical studies for the half line problem. Furthermore, we extend the classical central schemes for the hyperbolic conservation laws to solve the MBL equation which is of pseudo-parabolic type. Numerical results confirm the existence of non-monotone water saturation profiles consisting of constant states separated by shocks.