A Local Streamline Eulerian-Lagrangian Method for Two-Phase Flow

A method for solving the saturation equation for two-phase ow is presented. The method may be viewed as an operator splitting method or as an Eulerian-Lagrangian backtracking procedure or as a modiied method of characteristics. For each time step, the method consists of an advection step and a diiusion step. The advection step requires the tracking of streamlines locally around certain integration points. On each streamline we have to solve a nonlinear hyperbolic equation. This is done using a front tracking method. The solution from the tracking step is projected back onto a xed grid, and we nally have to solve a parabolic heat-type equation. 1 INTRODUCTION The numerical solution of advective-diiusive transport problems arise in many important applications in science and engineering, e.g. oil reservoir ow, transport of solutes in ground water and surface water, the movement of aerosols and trace gases in the atmosphere, to mention a few examples. The diiculty of solving such problems, especially if advection dominates, have long been recognized as one of the more challenging tasks in scientiic computing, see (Morton 1996) for an overview. An important class of numerical schemes for solving such problems are the Eulerian-Lagrangian localized adjoint methods (ELLAM) (Celia et al. 1990; Herrera et al. 1993; Wang et al. 1999). These schemes have been successfully applied to linear transport problems of various types, and give a framework for devising Eulerian-Lagrangian type methods that are both mass conservative and able to handle boundary conditions in a fairly systematic way. A more diicult problem is the solution of multi-phase transport processes. This leads to nonlinear advection which greatly complicates the tracking part of the algorithm. Usually this diiculty has been overcome by some kind of linearization, e.g., (Dahle et al. 1995; Douglas et al. 1997). However, such linearizations may put artiicial restrictions on the time steps that are not dictated by the physical processes investigated. The tracking algorithm is a major part of