TY - JOUR

T1 - A monolithic phase-field model of a fluid-driven fracture in a nonlinear poroelastic medium

AU - van Duijn, C. J.

AU - Mikelić, Andro

AU - Wick, Thomas

N1 - Funding Information: The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program ‘‘Investissements d’Avenir’’ (grant number ANR-11-IDEX-0007) operated by the French National Research Agency and by the Darcy Center.

PY - 2019/5

Y1 - 2019/5

N2 - In this paper, we present a full phase-field model for a fluid-driven fracture in a nonlinear poroelastic medium. The nonlinearity arises in the Biot equations when the permeability depends on porosity. This extends previous work (see Mikelić et al. Phase-field modeling of a fluid-driven fracture in a poroelastic medium. Comput Geosci 2015; 19: 1171–1195), where a fully coupled system is considered for the pressure, displacement, and phase field. For the extended system, we follow a similar approach: we introduce, for a given pressure, an energy functional, from which we derive the equations for the displacement and phase field. We establish the existence of a solution of the incremental problem through convergence of a finite-dimensional Galerkin approximation. Furthermore, we construct the corresponding Lyapunov functional, which is related to the free energy. Computational results are provided that demonstrate the effectiveness of this approach in treating fluid-driven fracture propagation. Specifically, our numerical findings confirm differences with test cases using the linear Biot equations.

AB - In this paper, we present a full phase-field model for a fluid-driven fracture in a nonlinear poroelastic medium. The nonlinearity arises in the Biot equations when the permeability depends on porosity. This extends previous work (see Mikelić et al. Phase-field modeling of a fluid-driven fracture in a poroelastic medium. Comput Geosci 2015; 19: 1171–1195), where a fully coupled system is considered for the pressure, displacement, and phase field. For the extended system, we follow a similar approach: we introduce, for a given pressure, an energy functional, from which we derive the equations for the displacement and phase field. We establish the existence of a solution of the incremental problem through convergence of a finite-dimensional Galerkin approximation. Furthermore, we construct the corresponding Lyapunov functional, which is related to the free energy. Computational results are provided that demonstrate the effectiveness of this approach in treating fluid-driven fracture propagation. Specifically, our numerical findings confirm differences with test cases using the linear Biot equations.

KW - Hydraulic fracturing

KW - nonlinear poroelasticity

KW - phase field

UR - http://www.scopus.com/inward/record.url?scp=85060948857&partnerID=8YFLogxK

U2 - 10.1177/1081286518801050

DO - 10.1177/1081286518801050

M3 - Article

AN - SCOPUS:85060948857

VL - 24

SP - 1530

EP - 1555

JO - Mathematics and Mechanics of Solids

JF - Mathematics and Mechanics of Solids

SN - 1081-2865

IS - 5

ER -