Numerical Solution of Reservoir Flow Models Based Onlarge Time Step Operator Splitting

During recent years the authors and collaborators have been involved in an activity related to the construction and analysis of large time step operator splitting algorithms for the numerical simulation of multi-phase ow in heterogeneous porous media. The purpose of these lecture notes is to review some of this activity. We illustrate the main ideas behind these novel operator splitting algorithms for a basic two-phase ow model. Special focus is posed on the numerical solution algorithms for the saturation equation, which is a convection dominated, degenerate convection-diiusion equation. Both theory and applications are discussed. The general background for the reservoir ow model is reviewed, and the main features of the numerical algorithms are presented. The basic mathematical results supporting the numerical algorithms are also given. In addition, we present some results from the BV solution theory for quasilinear degenerate parabolic equations, which provides the correct mathematical framework in which to analyse our numerical algorithms. Two-and three-dimensional numerical test cases are presented and discussed. The main conclusion drawn from the numerical experiments is that the operator splitting algorithms indeed exhibit the property of resolving accurately internal layers with steep gradients, give very little numerical diiusion, and, at the same time, permit the use of large time steps. In addition, these algorithms seem to capture all potential combinations of convection and diiusion forces, ranging from convection dominated problems (including the pure hyperbolic case) to more diiusion dominated problems.