Lattice Boltzmann method assessment for the Buckley–Leverett equation: Extension to dynamic capillary effect

We develop a lattice Boltzmann (LB) approach for solving different formulations of the Buckley–Leverett equation. The proposed model is built upon an existing moments based LB model for high-order nonlinear partial differential equations. In comparison with previous Buckley–Leverett LB models, the current formulation distinguishes itself by representing the third-order mixed partial derivative in both space and time. The proposed model is analyzed using the Chapman–Enskog, showing the model consistency with the inclusion of this high-order derivative term. The model is applied to describe convective and convective–diffusive flows with static and dynamic capillary pressure. The proposed model is applied to describe convective and convective–diffusive flows under both static and dynamic capillary pressure conditions. The obtained results agree well with both the analytical solution and numerical results from other authors. Additionally, numerical convergence rate analyses are performed for spatial, convective, and diffusive scaling, demonstrating first-order accuracy for all Buckley–Leverett formulations.