Numerical resolution of a pseudo-parabolic Buckley-Leverett model with gravity and dynamic capillary pressure in heterogeneous porous media

Abstract

We discuss a numerical approach for the simulation of a hydrodynamic model for immiscible incompressible two-phase flow in heterogeneous porous media to deal with effects of dynamic capillary pressure and gravity. The governing system of equations consists of a nonlinear pseudo-parabolic Buckley-Leverett saturation transport model coupled with an elliptic pressure-velocity problem. Upon certain manipulation of the equations, we rewrite the transport model as a nonlinear elliptic reaction-diffusion problem along with a simpler time-dependent relation to the dynamic capillary pressure. Hybridized mixed finite elements and domain decomposition procedures are used for the spatial discretization of the equations. Proper locally conservative finite volume techniques are used for a balancing discretization of the first-order hyperbolic flux and the dispersive behavior inherent of the pseudo-parabolic model. After discretizing in time with an implicit method, the resulting nonlinear algebraic equations is solved with a sequential time-marching approach and the arising systems of linear equations are solved efficiently with an algebraic multigrid method. Thus, we formulate and test numerically a fully coupled and implicit formulation of the transport problem of pseudo-parabolic nature. We find that the new procedure is accurate, robust, efficient and easy for implementation. We simulated the two-dimensional transport of the full nonlinear modified Buckley-Leverett model with a dynamic capillary pressure model in a variety of flow regimes along with heterogeneous permeability fields and domination of gravity. Our numerical experiments demonstrated the viability of the proposed formulation based on computational simulations of typical flow situations and saturation overshoot.