Details
| Original language | English |
|---|---|
| Article number | 2 |
| Journal | GEM - International Journal on Geomathematics |
| Volume | 10 |
| Issue number | 1 |
| Publication status | Published - 20 Jan 2019 |
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).
Keywords
- Computer simulations, Hydraulic fracturing, Nonlinear elliptic system, Phase field formulation, Poroelasticity
ASJC Scopus subject areas
- Mathematics(all)
- Modelling and Simulation
- Earth and Planetary Sciences(all)
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In: GEM - International Journal on Geomathematics, Vol. 10, No. 1, 2, 20.01.2019.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Phase-field modeling through iterative splitting of hydraulic fractures in a poroelastic medium
AU - Mikelić, A.
AU - Wheeler, M. F.
AU - Wick, Thomas
N1 - Funding Information: 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.
PY - 2019/1/20
Y1 - 2019/1/20
N2 - 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).
AB - 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).
KW - Computer simulations
KW - Hydraulic fracturing
KW - Nonlinear elliptic system
KW - Phase field formulation
KW - Poroelasticity
UR - http://www.scopus.com/inward/record.url?scp=85060958416&partnerID=8YFLogxK
U2 - 10.1007/s13137-019-0113-y
DO - 10.1007/s13137-019-0113-y
M3 - Article
AN - SCOPUS:85060958416
VL - 10
JO - GEM - International Journal on Geomathematics
JF - GEM - International Journal on Geomathematics
SN - 1869-2672
IS - 1
M1 - 2
ER -