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

Autor: Thomas Wick, Mary F. Wheeler, Andro Mikelić
Přispěvatelé: Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Modélisation mathématique, calcul scientifique (MMCS), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS)-École Centrale de Lyon (ECL), Institute for Computational Engineering and Sciences [Austin] (ICES), University of Texas at Austin [Austin], Institute of Applied Mathematics, Leibniz Universität Hannover, Leibniz Universität Hannover [Hannover] (LUH)
Jazyk: angličtina
Rok vydání: 2019
Předmět:
Zdroj: GEM-International Journal on Geomathematics
GEM-International Journal on Geomathematics, Springer, 2019, 10 (1), ⟨10.1007/s13137-019-0113-y⟩
ISSN: 1869-2672
1869-2680
DOI: 10.1007/s13137-019-0113-y⟩
Popis: 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 Mikelic et al. (SIAM Multiscale Model Simul 13(1):367–398, 2015a).
Databáze: OpenAIRE
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