Preconditioning for Efficiently Applying Algebraic Multigrid in Fully Implicit Reservoir Simulations
Bibliographic record
Abstract
Summary 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, constrained-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.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".