Phase-field modeling of brittle fracture with multi-level hp-FEM and the finite cell method

The difficulties in dealing with discontinuities related to a sharp crack are overcome in the phase-field approach for fracture by modeling the crack as a diffusive object being described by a continuous field having high gradients. The discrete crack limit case is approached for a small length-scale parameter that controls the width of the transition region between the fully broken and the undamaged phases. From a computational standpoint, this necessitates fine meshes, at least locally, in order to accurately resolve the phase-field profile. In the classical approach, phase-field models are computed on a fixed mesh that is a priori refined in the areas where the crack is expected to propagate. This on the other hand curbs the convenience of using phase-field models for unknown crack paths and its ability to handle complex crack propagation patterns. In this work, we overcome this issue by employing the multi-level hp-refinement technique that enables a dynamically changing mesh which in turn allows the refinement to remain local at singularities and high gradients without problems of hanging nodes. Yet, in case of complex geometries, mesh generation and in particular local refinement becomes non-trivial. We address this issue by integrating a two-dimensional phase-field framework for brittle fracture with the finite cell method (FCM). The FCM based on high-order finite elements is a non-geometry-conforming discretization technique wherein the physical domain is embedded into a larger fictitious domain of simple geometry that can be easily discretized. This facilitates mesh generation for complex geometries and supports local refinement. Numerical examples including a comparison to a validation experiment illustrate the applicability of the multi-level hp-refinement and the FCM in the context of phase-field simulations

Multiscale simulation of injection-induced fracture slip and wing-crack propagation in poroelastic media

In fractured poroelastic media under high differential stress, the shearing of fractures and faults and the corresponding propagation of wing cracks can be induced by fluid injection. Focusing on low-pressure stimulation with fluid pressures below the minimum principal stress but above the threshold required to overcome the fracture's frictional resistance to slip, this paper presents a mathematical model and a numerical solution approach for coupling fluid flow with fracture shearing and propagation. Numerical challenges are related to the strong coupling between hydraulic and mechanical processes, the material discontinuity the fractures represent in the medium, the wide range of spatial scales involved, and the strong effect that fracture deformation and propagation have on the physical processes. The solution approach is based on a multiscale strategy. In the macroscale model, flow in and poroelastic deformation of the matrix are coupled with the flow in the fractures and fracture contact mechanics, allowing fractures to frictionally slide. Fracture propagation is handled at the microscale, where the maximum tangential stress criterion triggers the propagation of fractures, and Paris' law governs the fracture growth processes. Simulations show how the shearing of a fracture due to fluid injection is linked to fracture propagation, including cases with hydraulically and mechanically interacting fractures

Diffuse interface modeling and variationally consistent homogenization of fluid transport in fractured porous media

We critically assess diffuse interface models for fluid transport in fractured porous media. Such models, often called fracture phase field models, are commonly used to simulate hydraulic stimulation or hydraulic fracturing of fluid-saturated porous rock. In this paper, we focus on the less complex case of fluid transport in stationary fracture networks that is triggered by a hydro-mechanical interaction of the fluid in the fractures with a surrounding poroelastic matrix material. In other words, fracture propagation is not taken into account. This allows us to validate the diffuse interface model quantitatively and to benchmark it against solutions obtained from sharp interface formulations and analytical solutions. We introduce the relevant equations for the sharp and diffuse, i.e. fracture phase field, interface formulations. Moreover, we derive the scale-transition rules for upscaling the fluid-transport problem towards a viscoelastic substitute model via Variationally Consistent Computational Homogenization. This allows us to measure the attenuation associated with fluid transport on the sub-scale. From the numerical investigations we conclude that the conventional diffuse interface formulation fails in predicting the fluid-transport behavior appropriately. The results even tend to be non-physical under certain conditions. We, therefore, propose a modification of the interpolation functions used in the diffuse interface model that leads to reasonable results and to a good approximation of the reference solutions

Stabilized mixed formulation for phase-field computation of deviatoric fracture in elastic and poroelastic materials

In the numerical approximation of phase-field models of fracture in porous media with the finite element method, the problem of numerical locking may occur. The causes can be traced both to the hydraulic and to the mechanical properties of the material. In this work we present a mixed finite element formulation for phase-field modeling of brittle fracture in elastic solids based on a volumetric-deviatoric energy split and its extension to water saturated porous media. For the latter, two alternative mixed formulations are proposed. To be able to use finite elements with linear interpolation for all the field variables, which violates the Ladyzenskaja-Babuska-Brezzi condition, a stabilization technique based on polynomial pressure projections, proposed and tested by previous authors in fluid mechanics and poromechanics, is introduced. We develop an extension of this stabilization to phase-field mixed models of brittle fracture in porous media. Several numerical examples are illustrated, to show the occurrence of different locking phenomena and to compare the solutions obtained with different unstable, stable and stabilized low order finite elements