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Stable discretization of poroelasticity problems and efficient preconditioners for arising saddle point type matrices

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Computing and Visualization in Science

Abstract

Poroelastic models arise in reservoir modeling and many other important applications. Under certain assumptions, they involve a time-dependent coupled system consisting of Navier–Lamé equations for the displacements, Darcy’s flow equation for the fluid velocity and a divergence constraint equation. Stability for infinite time of the continuous problem and, second and third order accurate, time discretized equations are shown. Methods to handle the lack of regularity at initial times are discussed and illustrated numerically. After discretization, at each time step this leads to a block matrix system in saddle point form. Mixed space discretization methods and a regularization method to stabilize the system and avoid locking in the pressure variable are presented. A certain block matrix preconditioner is shown to cluster the eigenvalues of the preconditioned matrix about the unit value but needs inner iterations for certain matrix blocks. The strong clustering leads to very few outer iterations. Various approaches to construct preconditioners are presented and compared. The sensitivity of the number of outer iterations to the stopping accuracy of the inner iterations is illustrated numerically.

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Acknowledgments

This work was supported by the European Regional Development Fund in the IT4Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070).

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Authors and Affiliations

  1. King Abdulaziz University, Jeddah, Saudi Arabia

    Owe Axelsson

  2. Institute of Geonics AS CR, IT4 Innovations, Ostrava, The Czech Republic

    Owe Axelsson, Radim Blaheta & Petr Byczanski

Authors
  1. Owe Axelsson
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  2. Radim Blaheta
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  3. Petr Byczanski
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Correspondence to Radim Blaheta.

Additional information

Communicated ByGabriel Wittum.

This paper is dedicated to the memory of Richard (Dick) E. Ewing for his friendly manner and concern for other peoples research. The first author was fortunate to meet Dick already when he worked at Mobil Oil, Texas. Later Dick invited him to visit University of Wyoming at Laramie, Wyoming which enabled collaborative research with one of Dick’s coworkers at that time, Panayot S. Vassilevsky. Later meetings with Dick at College Station, Texas, University of Nijmegen, The Netherlands and at conferences in Bulgaria were also always very useful.

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Axelsson, O., Blaheta, R. & Byczanski, P. Stable discretization of poroelasticity problems and efficient preconditioners for arising saddle point type matrices. Comput. Visual Sci. 15, 191–207 (2012). https://doi.org/10.1007/s00791-013-0209-0

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