Showing papers by "Mary F. Wheeler published in 2015"
TL;DR: In this paper, a primal-dual active set strategy is proposed to enforce crack irreversibility as a constraint, which can be identified as a semi-smooth Newton method, and the active set iteration is merged with the Newton iteration for solving the fully-coupled nonlinear partial differential equation discretized using finite elements.
300 citations
TL;DR: The pressurized phase- field framework is extended to fluid-filled fractures in which the pressure is computed from a generalized parabolic diffraction problem, and the phase-field variable is used as an indicator function to combine reservoir and fracture pressure.
Abstract: The recently introduced phase-field approach for pressurized fractures in a porous medium offers various attractive computational features for numerical simulations of cracks such as joining, branching, and nonplanar propagation in possibly heterogeneous media. In this paper, the pressurized phase-field framework is extended to fluid-filled fractures in which the pressure is computed from a generalized parabolic diffraction problem. Here, the phase-field variable is used as an indicator function to combine reservoir and fracture pressure. The resulting three-field framework (elasticity, phase field, pressure) is a multiscale problem that is based on the Biot equations. The proposed numerical solution algorithm iteratively decouples the equations using a fixed-stress splitting. The framework is substantiated with several numerical benchmark tests in two and three dimensions.
211 citations
TL;DR: In this article, a phase field variational inequality was proposed for a fluid-driven fracture in a poroelastic medium, where the phase field variable was determined simultaneously with the displacement and phase field, and a solution to the incremental problem was established through convergence of a finite dimensional approximation.
Abstract: 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.
142 citations
TL;DR: In this paper, a quasi-static formulation of a phase-field model for a pressurized crack in a poroelastic medium is presented, where the model represents a linear elasticity system with a fading Gassman tensor as the crack grows, coupled with a variational inequality for the phase field variable containing an entropy inequality.
Abstract: In this paper we present a quasi-static formulation of a phase-field model for a pressurized crack in a poroelastic medium. The mathematical model represents a linear elasticity system with a fading Gassman tensor as the crack grows, that is coupled with a variational inequality for the phase-field variable containing an entropy inequality. We introduce a novel incremental approximation that decouples displacement and phase-field problems. We establish convergence to a solution of the quasi-static problem, including Rice's condition, when the time discretization step goes to zero. Numerical experiments confirm the robustness and efficiency of this approach for multidimensional test cases.
105 citations
TL;DR: In this article, a non-planar fracture model in a poro-elastic medium is presented, where the medium in which the fracture is embedded is governed by the standard Biot equations of linear poroelasticity and the flow of the fluid within the fracture are governed by a lubrication equation.
Abstract: We present a non-planar fracture model in a poro-elastic medium. The medium in which the fracture is embedded is governed by the standard Biot equations of linear poro-elasticity and the flow of the fluid within the fracture is governed by the lubrication equation. We establish existence and uniqueness of the linearized coupled system under weak assumptions on the data. Two discretizations of the problem are formulated: one with a continuous Galerkin method and one with a mixed method for the flow in the reservoir and in the fracture. We perform the numerical analysis of the former and we provide an algorithm and numerical experiments for the latter.
75 citations
53 citations
18 citations
01 Apr 2015
TL;DR: In this paper, the phase field-based fracture modeling in heterogeneous porous media is extended to tackle three-dimensional configurations by using a nested Newton loop that combines a primal-dual active set method and a Newton method to solve the nonlinear, fully-coupled PDE system.
Abstract: This work presents recent progress in phase-field-based fracture modeling in heterogeneous porous media. Existing algorithms that have been developed in the last years are extended to tackle three-dimensional configurations. Our solution technique is formulated in terms of a nested Newton loop that combines a primal-dual active set method (required for treating the crack irreversibility) and a Newton method to solve the nonlinear, fully-coupled PDE system. An advanced numerical test demonstrates the capabilities of our method.
17 citations
TL;DR: The results show that selecting the prior model based on the data mismatch can be misleading and highlights the need to evaluate the Bayesian evidence (estimated by the nested sampling algorithm) as a more reliable prior model selection statistics, especially when the amount of calibration data is limited.
16 citations
23 Feb 2015
11 citations
01 Jan 2015
TL;DR: A new enhanced velocity method to directly construct a flux-continuous velocity approximation with multipoint flux mixed finite element method on subdomains is proposed to give an efficient way to perform simulations on multiblock domains with non-matching hexa- hedral grids.
Abstract: This paper proposes a new enhanced velocity method to directly construct a flux-continuous velocity approximation with multipoint flux mixed finite element method on subdomains. This gives an efficient way to perform simulations on multiblock domains with non-matching hexa- hedral grids. We develop a reasonable assumption on geometry, discuss implementation issues, and give several numerical results with slightly compressible single phase flow.
23 Feb 2015
TL;DR: In this paper, the authors present a numerical method for elastic wave propagation that can incorporate fracture discontinuities in the model, based on the interior-penalty Discontinuous Galerkin method and the fractures are incorporated using the linearslip model.
Abstract: We present a numerical method for elastic wave propagation that can incorporate fracture discontinuities in the model. The scheme is based on the Interior-Penalty Discontinuous Galerkin method and the fractures are incorporated using the linearslip model. The method is suitable for simulating the following effects of fractures in the wave field: scattering, phase-shifting, attenuation and the generation of interface waves. The method does not have restrictions on the number of fractures or their orientations, these can be oblique and have intersections. We show numerical examples using models with one fracture and two orthogonal fractures.
01 Jan 2015
TL;DR: An ALE-based method is proposed (ALE – arbitrary Lagrangian-Eulerian) to perform full 2D computations and a 1D model is derived that approximates the 2D solution by integrating over the thickness of the channel.
Abstract: In this study, reaction-induced boundary movements in a thin channel are investigated. Here, precipitation-dissolution reactions taking place at the boundaries of the channel resulting in boundary movements act as a precursor to the clogging process. The resulting problem is a coupled flow-reactive transport process in a time-dependent geometry. We propose an ALE-based method (ALE – arbitrary Lagrangian-Eulerian) to perform full 2D computations. We derive a 1D model that approximates the 2D solution by integrating over the thickness of the channel. The boundary movements lead in the limit to clogging when the flow gets choked for a given pressure gradient applied across the channel. Numerical tests of the full 2D model are consulted to confirm the theory.