TY - JOUR

T1 - Phase-field modeling of a fluid-driven fracture in a poroelastic medium

AU - Mikelić, A.

AU - Wheeler, M. F.

AU - Wick, T.

N1 - Funding Information: The research by A. Mikelic, 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. Mikelic 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. Publisher Copyright: © 2015, Springer International Publishing Switzerland. Copyright: Copyright 2016 Elsevier B.V., All rights reserved.

PY - 2015/12/1

Y1 - 2015/12/1

N2 - 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.

AB - 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.

KW - Computer simulations

KW - Hydraulic fracturing

KW - Nonlinear elliptic-parabolic system

KW - Phase field formulation

KW - Poroelasticity

UR - http://www.scopus.com/inward/record.url?scp=84954384579&partnerID=8YFLogxK

U2 - 10.1007/s10596-015-9532-5

DO - 10.1007/s10596-015-9532-5

M3 - Article

AN - SCOPUS:84954384579

VL - 19

SP - 1171

EP - 1195

JO - Computational geosciences

JF - Computational geosciences

SN - 1420-0597

IS - 6

ER -