Preconditioning for Efficiently Applying Algebraic Multigrid in Fully Implicit Reservoir Simulations
- Sebastian Gries (Fraunhofer Institute for Algorithms and Scientific Computing (SCAI)) | Klaus Stüben (Fraunhofer Institute for Algorithms and Scientific Computing (SCAI)) | Geoffrey L. Brown (Computer Modelling Group Limited) | Dingjun Chen (Computer Modelling Group Limited) | David A. Collins (Computer Modelling Group Limited)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- August 2014
- Document Type
- Journal Paper
- 726 - 736
- 2014.Society of Petroleum Engineers
- 5.5 Reservoir Simulation
- Constrained Pressure residual, preconditioning, decoupling, dynamic rowsumming, Algebraic Multigrid
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- 320 since 2007
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Fully implicit black-oil simulations result in huge, often very ill-conditioned, linear systems of equations for different unknowns (e.g., pressure and saturations). It is well-known that the underlying Jacobian matrices contain both hyperbolic and nearly elliptic subsystems (corresponding to saturations and pressure, respectively). Because a reservoir simulation is typically driven by the behavior of the pressure, contrained-pressure-residual (CPR) -type two-stage preconditioning methods to solve the coupled linear systems are a natural choice and still belong to the most popular approaches. After a suitable extraction and decoupling, the computationally most costly step in such two-stage methods consists in solving the elliptic subsystems accurately enough. Algebraic multigrid (AMG) provides a technique to solve elliptic linear equations very efficiently. Hence, in recent years, corresponding CPR-AMG approaches have been extensively used in practice. Unfortunately, if applied in a straightforward manner, CPR-AMG does not always work as expected. In this paper, we discuss the reasons for the lack of robustness observed in practice, and present remedies. More precisely, we will propose a preconditioning strategy (based on a suitable combination of left and right preconditioning of the Jacobian matrix) that aims at a compromise between the solvability of the pressure subproblem by AMG and the needs of the outer CPR process. The robustness of this new preconditioning strategy will be demonstrated for several industrial test cases, some of which are very ill-conditioned. Furthermore, we will demonstrate that CPR-AMG can be interpreted in a natural way as a special AMG process applied directly to the coupled Jacobian systems.
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