Numerical modeling of coupled hydrological phenomena using the Modified Eulerian-Lagrangian method

The chapter presents the advantages of using the Modified Eulerian-Lagrangian (MEL) method for solving the transport of extensive quantities in a porous medium. In this modified formulation, the material derivatives are written in terms of modified velocities. These are the velocities at which the various phase and component variables propagate in the domain, along their respective characteristic curves. It is shown that these velocities depend on the heterogeneity of various solid matrix and fluid properties. Accordingly, an extension to the Peclet number is presented that also accounts for governing equations that may be advective dominant with no reference to the fluid velocity or even when this velocity is not introduced. A mathematical analysis proves that for coupled partial differential equations (PDEs), unlike the Eulerian implicit finite difference scheme the MEL method unconditionally guarantees the absent of spurious oscillations. The MEL formulation is demonstrated for a coupled set of PDEs concerning the problem of saltwater intrusion, with heat transfer. A numerical example suggests that the MEL scheme produces better resolution compared to the Eulerian and the Eulerian-Lagrangian ones.