Numerical Aspects of Flow Based Local Upscaling

We start this thesis by giving an introduction to reservoir simulation and upscaling in particular. The most common upscaling techniques, including power averaging methods and flow based local and global methods [10], are introduced. Hybrid methods and multiscale methods are also included, and we consider both single- and two-phase systems. Upscaling is viewed in the context of a representative elementary volume (REV), and we argue why flow based local methods can be preferable for this purpose. The elliptic diffusion equation is central in flow based upscaling methods as it governs single-phase flow. We present numerical methods to solve it numerically on corner-point grids, which are the industry standard grids. The newly developed mimetic finite difference method (MFDM) [8] has shown to work nicely on such grids [3], and the MFDM is explained in some details in this thesis. For flow based local upscaling methods, three sets of boundary conditions (BCs) are used by the industry, namely fixed, linear and periodic. We give a detailed analysis of the numerical implications of the three sets of BCs, and we discuss correctness and numerical convergence of these. Results from numerical computations on realistic reservoir models show that linear BCs are considerably faster to solve for. Based on this and that periodic BCs seems intuitively most correct, we propose a new representation and implementation of periodic BCs. This approach is based on mortar methods. Numerical calculations show, however, that this method fails both in terms of correctness and in terms of numerical convergence