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Domain Decomposition Methods in Science and Engineering XXI pp 141–153Cite as

A Multi-Stage Preconditioner for the Black Oil Model and Its OpenMP Implementation

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Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 98))

Abstract

In this paper, we propose a preconditioner for solving large-scale linear systems arising from fully implicit petroleum reservoir simulation. First, we take several analytical and physical considerations into account, and choose appropriate auxiliary spaces (problems) to design a preconditioner in the framework of auxiliary space preconditioning method. Secondly, we present an efficient implementation of the proposed preconditioner in modern multicore computer environment. Finally, we test the efficiency and robustness of the proposed methods with a large benchmark problem.

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Notes

  1. 1.

    The subscript s indicates that these variables are at the standard conditions instead of reservoir conditions.

  2. 2.

    We denote the solution variable as \(x:= [\delta P,\delta S_{w},\delta S_{o}]^{T}\).

  3. 3.

    This data structure is similar to the compressed sparse row (CSR) format, but each nonzero entry is a 3 × 3 sub-matrix in BCSR.

  4. 4.

    The standard GS sweep is applied in each thread, and parallel (simultaneous) updating is used across multiple threads.

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Acknowledgements

The authors appreciate the anonymous referee for his or her suggestions which led to a better presentation of our method. The authors would like to thank RIPED, PetroChina, for providing the modified SPE10 test. Feng is partially supported by NSFC Grant 11201398. Shu is partially supported by NSFC Grants 91130002 and 11171281 and by the Scientific Research Fund of the Hunan Provincial Education Department of China #12A138. Xu is partially supported by NSFC Grant 91130011. Zhang is partially supported by the Dean’s Startup Fund, Academy of Mathematics and System Sciences and by NSFC Grant 91130011.

Author information

Authors and Affiliations

  1. School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, China

    Chunsheng Feng & Shi Shu

  2. Department of Mathematics, The Pennsylvania State University, University Park, PA, USA

    Jinchao Xu

  3. NCMIS & LSEC, Academy of Mathematics and System Sciences, Beijing, China

    Chen-Song Zhang

Authors
  1. Chunsheng Feng
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  2. Shi Shu
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  3. Jinchao Xu
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  4. Chen-Song Zhang
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Corresponding author

Correspondence to Chen-Song Zhang .

Editor information

Editors and Affiliations

  1. INRIA, Rennes, France

    Jocelyne Erhel

  2. Section de Mathématiques, Université de Genève, Genève, Switzerland

    Martin J. Gander

  3. Laboratoire Analyse, Géometrie & Applications, Université Paris XIII, Villetaneuse, France

    Laurence Halpern

  4. INRIA, Rennes, France

    Géraldine Pichot

  5. Laboratoire LMNO, Université de Caen, Caen, France

    Taoufik Sassi

  6. New York University Courant Institute, New York, New York, USA

    Olof Widlund

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Feng, C., Shu, S., Xu, J., Zhang, CS. (2014). A Multi-Stage Preconditioner for the Black Oil Model and Its OpenMP Implementation. In: Erhel, J., Gander, M., Halpern, L., Pichot, G., Sassi, T., Widlund, O. (eds) Domain Decomposition Methods in Science and Engineering XXI. Lecture Notes in Computational Science and Engineering, vol 98. Springer, Cham. https://doi.org/10.1007/978-3-319-05789-7_11

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