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Phase-field modeling through iterative splitting of hydraulic fractures in a poroelastic medium

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Abstract

We study the propagation of hydraulic fractures using the fixed stress splitting method. The phase field approach is applied and we study the mechanics step involving displacement and phase field unknowns, with a given pressure. We present a detailed derivation of an incremental formulation of the phase field model for a hydraulic fracture in a poroelastic medium. The mathematical model represents a linear elasticity system with fading elastic moduli as the crack grows that is coupled with an elliptic variational inequality for the phase field variable. The convex constraint of the variational inequality assures the irreversibility and entropy compatibility of the crack formation. We establish existence of a minimizer of an energy functional of an incremental problem and convergence of a finite dimensional approximation. Moreover, we prove that the fracture remains small in the third direction in comparison to the first two principal directions. Computational results of benchmark problems are provided that demonstrate the effectiveness of this approach in treating fracture propagation. Another novelty is the treatment of the mechanics equation with mixed boundary conditions of Dirichlet and Neumann types. We finally notice that the corresponding pressure step was studied by the authors in Mikelić et al. (SIAM Multiscale Model Simul 13(1):367–398, 2015a).

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Authors and Affiliations

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

    A. Mikelić

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

    M. F. Wheeler

  3. Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167, Hannover, Germany

    T. Wick

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  1. A. Mikelić
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A.M. would like to thank Institute for Computational Engineering and Science (ICES), UT Austin for hospitality during his sabbatical stays. The research by M. F. Wheeler was partially supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences through DOE Energy Frontier Research Center: The Center for Frontiers of Subsurface Energy Security (CFSES) under Contract No. DE-FG02-04ER25617, MOD. 005. The work of T. Wick was supported through an ICES Postdoc fellowship, the Humboldt foundation with a Feodor-Lynen fellowship and through the JT Oden faculty research program. Currently T. Wick is supported by the DFG-SPP 1748 program.

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Mikelić, A., Wheeler, M.F. & Wick, T. Phase-field modeling through iterative splitting of hydraulic fractures in a poroelastic medium. Int J Geomath 10, 2 (2019). https://doi.org/10.1007/s13137-019-0113-y

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