The original ``Modified Method of Characteristics" (MMOC) procedure, which was first published by Douglas and Russell in the early 80's, provides a computationally efficient approach to many transport-dominated diffusion problems. Unfortunately, it does not necessarily preserve critical integral identities as algebraic identities in many applications. In recent work by Douglas, Furtado, Pereira, and Yeh on two-phase, immiscible displacement in porous media (the ``waterflood" problem), where the desired integral identity expresses the conservation of the mass of the water phase, it was found that the error in the mass of the water phase, as computed by a standard form of the MMOC, was large enough to be comparable to that caused by the uncertainty in the physical properties of the porous medium when the geological parameters are modelled as fractals, as suggested by geologists. A variant of the MMOC, called the ``Modified Method of Characteristics with Adjusted Advection" (MMOCAA), was developed about three years ago and has been applied to a number of problems associated with petroleum reservoir analyses and the transport of nuclear contaminants in porous media; this method conserves mass globally (in space) at all time levels and retains all of the computational advantages of the slightly simpler MMOC. However, it does not necessarily preserve mass locally in space, and there are a number of very important physical problems for which local conservation is essential. The methods labelled ELLAM do provide, under many circumstances, the desired local conservation in an Eulerian-Lagrangian setting, but only at a computational cost that is several times that of the MMOC or MMOCAA.
We (this is joint work with Pereira and Yeh) shall introduce a new family of locally conservative Eulerian-Lagrangian methods, denoted here generically by LCELM, that again retain the computational efficiency of the MMOC and MMOCAA techniques. There is a fundamental difference between the MMOC and MMOCAA techniques and the LCELM procedure; MMOC and MMOCAA procedures consider the partial differential system (or, at least, the parabolic-type equations in the system) in nondivergence form and make use of the characteristics associated with the first-order transport part of the system in a fractional step procedure that splits the transport from the diffusive part of the system. In contrast, the LCELM method will relate to the divergence form of the equations and then will split the transport from the diffusion. It is the use of the divergence form that allows the localization of the transport so that the desired conservation property can also be localized.
The MMOC, MMOCAA, and LCELM procedures will be described in detail for a a scalar parabolic equation, along with an indication of the convergence results that have been obtained for the three concepts. Then, the application of these methods to the waterflood problem will be mentioned. Computational experiments on this problem show that the LCELM provides a far better picture of the local behavior of the solution of the waterflood problem in reservoirs having quite inhomogeneous permeability distributions than can be obtained through MMOCAA calculations with equal spatial and temporal discretizations; this better approximation of local behavior shows clearly the (physically correct) bypassing of oil in regions of low permeability. Moreover, the LCELM shows a faster convergence rate of the approximate solution than the MMOCAA, which is in turn much faster than the MMOC. The comparisons of the MMOC, MMOCAA, and LCELM techniques for the waterflood problem validate the new LCELM procedure.