A locally conservative Eulerian-Lagrangian numerical method and its application to nonlinear transport in porous media

Eulerian-Lagrangian and Modified Method of Characteristics (MMOC) procedure s provide computationally efficient techniques for approximating the soluti ons of transport-dominated diffusive systems. The original MMOC fails to pr eserve certain integral identities satisfied by the solution of the differe ntial system; the recently introduced variant, called the MMOCAA, preserves the global form of the identity associated with conservation of mass in pe troleum reservoir simulations, but it does not preserve a localized form of this identity. Here, we introduce an Eulerian-Lagrangian method related to these MMOC procedures that guarantees conservation of mass locally for the problem of two-phase, immiscible, incompressible flow in porous media. The computational efficiencies of the older procedures are maintained. Both th e original MMOC and the MMOCAA procedures for this problem are derived from a nondivergence form of the saturation equation; the new method is based o n the divergence form of the equation. A reasonably extensive set of comput ational experiments are presented to validate the new method and to show th at it produces a more detailed picture of the local behavior in waterfloodi ng a fractally heterogeneous medium. A brief discussion of the application of the new method to miscible flow in porous media is included.