Preconditioning for Efficiently Applying Algebraic Multigrid in Fully Implicit Reservoir Simulations

Abstract

Fully 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.

AMGCL: An Efficient, Flexible, and Extensible Algebraic Multigrid Implementation

“…Also, the unified initialization interface of AMGCL building blocks means it is easy to reuse existing functionality in order to come up with new preconditioning techniques. Notable examples included into the library codebase are CPR [21] and Schur complement pressure correction [20,35] preconditioners. CPR works best in fully implicit black-oil simulations, and Schur complement pressure correction is well-suited for Navier-Stokes-like problems.…”

Fully implicit hybrid two-level domain decomposition algorithms for two-phase flows in porous media on 3D unstructured grids

“…The constrained pressure residual (CPR) preconditioner is widely used in the community of reservoir simulation [15,38]. This is based on subblocks of the Jacobian matrix and uses an approximate pressure solve such as algebraic multigrid (AMG) [25] to constrain the residual of the full system, and thus is known as a physics-splitting approach.…”

“…However, CPR-AMG may not work well when some source terms corresponding to complex physics destroy the elliptic properties exploited by AMG. In addition, the decoupling process tends to cause a conflict between the convergence of AMG and the convergence of the outer CPR iteration [25].…”

Full Resolution Full Field Simulation Using Localized Computations

“…In the second stage an ILU process is applied onto the updated global system. Empirically, it is observed that the computational complexity of AMG-CPR scales as O (N ␣ ) with 1 Յ ␣ Յ 1.5, where N is the total number of degrees of freedom in the system (Gries et al (2004); Masson et al (2004); Lacroix et al (2003); Stuben (2001)). Combined with the computational costs of the residual and Jacobian matrix evaluation and the thermodynamics, the overall complexity of a nonlinear iteration is generally superlinear in N.…”