Preconditioning for Efficiently Applying Algebraic Multigrid in Fully Implicit Reservoir Simulations

Abstract

AbstractFully implicit petroleum reservoir simulations result in huge, often very ill-conditioned linear systems of equations to solve for different unknowns, for example, pressure and saturations. It is well known that the full system matrix contains both hyperbolic as well as nearly elliptic sub-systems. Since the solution of the coupled system is mainly determined by the solution of their elliptic (typically pressure) components, (CPR-type) two-stage preconditioning methods still belong to the most popular approaches to tackle such coupled systems. After a suitable extraction and decoupling, the numerically most costly step in such two-stage methods consists in solving these elliptic sub-systems. It is known that algebraic multigrid (AMG) provides a technique to solve elliptic linear equations very efficiently. The main advantage of AMG-based solvers – their numerical scalability – makes them particularly efficient for solving huge linear systems.Depending on the application, the system’s properties range from simple to highly indefinite. Unfortunately decoupling pressure and saturation related parts may introduce further difficulties. Consequently, in complex industrial simulations, the application of AMG to elliptic sub-systems might not be straightforward. In fact, an important goal in defining an efficient two-stage preconditioning strategy consists in extracting elliptic sub-systems that are suitable for an efficient AMG solution and, at the same time, ensure a fast overall convergence of the two-stage approach.The importance of this will be demonstrated for several industrial cases. In particular, some of these cases are very hard to solve by AMG if applied in a standard way.Preliminary results for a CPR-type coupling of SAMG to CMG’s PARASOL, a variable degree variable ordering ILU preconditioner using FGMRES, are compared to using PARASOL by itself. Alternative preconditioning operators will be presented giving elliptic sub-systems which are not only more suitable for applying AMG efficiently but also help accelerate the CPR-type process. Comparisons with one-level iterative methods will show the acceleration by AMG is highly superior. Finally, a strategy is presented that combines all linear solver parts in one single AMG-iteration. In this sense CPR can be seen as a special case of AMG for systems. This, in turn, yields a – formally – very simple but simultaneously very flexible solution approach.