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Phase-field modeling of a fluid-driven fracture in a poroelastic medium

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Abstract

In this paper, we present a phase field model for a fluid-driven fracture in a poroelastic medium. In our previous work, the pressure was assumed given. Here, we consider a fully coupled system where the pressure field is determined simultaneously with the displacement and the phase field. To the best of our knowledge, such a model is new in the literature. The mathematical model consists of a linear elasticity system with fading elastic moduli as the crack grows, which is coupled with an elliptic variational inequality for the phase field variable and with the pressure equation containing the phase field variable in its coefficients. The convex constraint of the variational inequality assures the irreversibility and entropy compatibility of the crack formation. The phase field variational inequality contains quadratic pressure and strain terms, with coefficients depending on the phase field unknown. We establish existence of a solution to the incremental problem through convergence of a finite dimensional approximation. Furthermore, we construct the corresponding Lyapunov functional that is linked to the free energy. Computational results are provided that demonstrate the effectiveness of this approach in treating fluid-driven fracture propagation.

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Authors and Affiliations

  1. Université de Lyon, CNRS UMR 5208, Université Lyon 1, Institut Camille Jordan, 43, blvd. du 11 novembre 1918, 69622, Villeurbanne Cedex, France

    A. Mikelić

  2. The Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, TX, 78712, USA

    M. F. Wheeler

  3. RICAM, Austrian Academy of Sciences, Altenberger Str. 69, 4040, Linz, Austria

    T. Wick

  4. Fakultät für Mathematik, Lehrstuhl M17, Technische Universität München, 85748, Garching bei München, Germany

    T. Wick

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  1. A. Mikelić
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Correspondence to A. Mikelić.

Additional information

The research by A. Mikelić, M. F. Wheeler and T. Wick was partially supported by ConocoPhillips grant UTA13-001170 AMD 1 and Statoil grant UTA13-000884: WR DTD 2.13.14. A. Mikelić and T. Wick would like to thank Institute for Computational Engineering and Science (ICES), UT Austin for hospitality during their visits in February, June and August 2015 and support through a JT Oden fellowship. M. F. Wheeler was also supported by Aramco grant UTA11-000320; 1ST and T. Wick by the Austrian Academy of Sciences.

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Mikelić, A., Wheeler, M.F. & Wick, T. Phase-field modeling of a fluid-driven fracture in a poroelastic medium. Comput Geosci 19, 1171–1195 (2015). https://doi.org/10.1007/s10596-015-9532-5

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  • DOI: https://doi.org/10.1007/s10596-015-9532-5

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