Summary: Even though the art of reservoir simulation has evolved through more than four decades, there is still a substantial research activity that aims toward faster, more robust, and more accurate reservoir simulators. Here we attempt to give the reader an introduction to the mathematics and the numerics behind reservoir simulation. We assume that the reader has a basic mathematical background at the undergraduate level and is acquainted with numerical methods, but no prior knowledge of the mathematics or physics that govern the reservoir flow process is needed. To give the reader an intuitive understanding of the equations that model filtration through porous media, we start with incompressible single-phase flow and move step-by-step to the black-oil model and compressible two-phase flow. For each case, we present a basic numerical scheme in detail, before we discuss a few alternative schemes that reflect trends in state-of-the-art reservoir simulation. Two and three-dimensional test cases are presented and discussed. Finally, for the most basic methods we include simple Matlab codes so that the reader can easily implement and become familiar with the basics of reservoir simulation.
For the entire collection see [Zbl 1113.65002].