Original articles
A phase field approach to pressurized fractures using discontinuous Galerkin methods

https://doi.org/10.1016/j.matcom.2016.11.001Get rights and content

Abstract

Subsurface fractures play an important role in many modern energy technologies (e.g. geothermal energy, fracking, nuclear waste management). Real world experiments concerning fracture propagation are usually expensive and time consuming, therefore numerical simulations become more and more important in this area. The main challenge for numerical methods is the evolving domain. Standard finite element (FE) methods require remeshing to resolve the crack surface once a fracture starts propagating. To overcome this problem we use a phase field approach to regularize the crack surface. Thereby we consider quasi static evolution in fluid filled media. For the one-dimensional case Γ-convergence of the approximating functional to the potential energy of the system is shown. Based on this model we propose a discontinuous Galerkin (DG) formulation for the displacement. This takes into account displacement jumps at the crack surface. Numerical experiments compare our method with a standard FE approach.

Introduction

In recent years new technologies like enhanced geothermal energy induced a risen interest in numerical simulations of subsurface fracture processes. Thereby special attention must be paid to the formation and evolution of fractures and fracture networks. Taking cracks into account in numerical simulations gives rise to a big challenge. Evolving fractures can form complicated, even changing topologies. Not only can triple points and tips occur, cracks can also join. Classical finite element methods need an exact representation of the crack surface by the mesh to include these cracks into the simulation. In the quasi static case this leads to expensive remeshing at every time step, once the crack propagates. To overcome this difficulty new techniques that avoid remeshing are developed. One possible approach is to enrich the function space locally to take the crack into account. The most well known variant of this approach is the eXtended Finite Element Method (X-FEM) introduced by Moës et al. in  [27]; it has many similarities to the Partition of Unity method  [2]. The X-FEM method does not require the mesh to resolve the crack, instead new basis functions are introduced in elements cut by a crack. These methods have been successfully applied to crack propagation and for a recent overview we refer to  [18], [17]. That method is very sensitive to the choice of enrichment functions, in particular the type of enrichment employed at the tip, as the stresses are singular at the tip. In particular a correct approximation of crack speed and crack direction is challenging  [5]. Also the treatment of topological changes, like branching and merging of cracks is not straight forward and requires additional treatment  [12].

These difficulties can be avoided by describing the crack implicitly. An alternative approach to simulate crack propagation was introduced by Francfort and Marigo  [16]. It uses the idea of an implicit crack description and by this can easily handle topological changes and avoid problems arising from the crack tip modeling. Starting from Griffith’s criterion  [20] crack propagation is formulated as an energy minimization problem. The resulting energy strongly resembles the Mumford–Shah Functional from image segmentation  [28]. For numerical simulations this functional was regularized by Ambrosio and Tortorelli  [1] using the concept of Γ-convergence.

The main idea is to introduce a phase field φ that regularizes the crack. As the regularization parameter ε converges to 0, the phase field converges in the sense of Γ-limits to the sharp crack. Thereby one gains an implicit representation of the crack. This does not only avoid the problem of remeshing, but can also handle topological changes implicitly. Neither the forming of triple points, the joining of cracks nor the determination of propagation velocity and direction need any extra attention. The works of Francfort et al.  [16] and Bourdin et al.  [9] only treat the case, where fractures are driven by linear elasticity. Mikelić et al.  [25] then extended their approach by including pressure driven propagation.

This paper is based on the model proposed by Mikelić et al.  [25]. Section  2 will introduce the energy formulation, starting with Griffith’s criterion. Afterwards the total energy is derived. Section  3 gives a proof of Γ-convergence in a one-dimensional setting, which is to our knowledge the first proof for the phase field model for fluid filled fractures. In the following Section  4 a discontinuous Galerkin (DG) discretization of the phase field model is derived. In the solver we employ a split approach, similar to  [26]. In contrast to them we use a DG formulation for the displacement to take into account displacement jumps along the crack surface. Inequality constraints on the phase field are handled using a non-smooth multigrid solver. Finally in Section  5 we present some numerical experiments followed by a conclusion in Section  6.

Section snippets

Griffith’s criterion

Following the principle of minimal total potential energy the final equilibrium state of a material is broken, if it can be reached by a process only decreasing the total potential energy of the system. According to A. Griffith  [20] this energy consists of two parts. First the potential energy of the bulk in some domain ΩRn, Eb(u,C), depending on the displacement u and the crack C. Second some surface energy that describes the necessary work against cohesive forces between the molecules where

Γ-convergence result

In the following we prove Γ-convergence in the one dimensional setting on the interval Ω=(a,b), following the proof of Braides  [11, Theorem 8.1]. We hope that the proof carries over to two and three dimensions applying slicing properties as in  [13], [14], but this is still work in progress.

Let us first recall the definition of Γ-limits  [11].

Definition 3.1

A function f:XR{} is said to be the Γ-limit of a sequence fj:XR{} if for all xXf(x)lim infjfj(xj) holds for every sequence (xj) converging to x

Discretization

Using the principle of free variations for (5) we derive the Euler–Lagrange equations for our problem. 0=Ω(φ2+kε)su:Csw+(1α)φpdivw+φpwdxΓN(τ+pn)wdswV0=Ωφψsu:Csu+(1α)ψpdivu+ψpudx+GcΩ1ε(φ1)ψ+εφψdxψQ.

Similar to Mikelić et al.  [26] we use a split approach to discretize (18), (19). Inside of a fixed-point iteration we solve for the phase field and the displacement in turns. In the following the two discretizations are described in more detail. To ensure sufficient resolution

Numerical examples

To show the capability of the method to cover a wide range of crack propagation problems, it is applied to four different examples. First we will show an example with pure mode I loading and zero pressure. For the chosen configuration the time when propagation starts is given analytically. We will show convergence of our solution to this point with respect to kε and η. Next, two classical benchmark examples are shown and compared to existing literature, namely the single edge notched tension

Conclusion

In this paper the phase field model for fluid filled, pressure driven crack propagation by  [25] was adapted to a discontinuous Galerkin setting. Thus continuity requirements along a crack were weakened.

The usage of the phase field regularization in the presence of additional pressure terms in the bulk energy was justified by a proof of the Γ-convergence in the one-dimensional setting. Numerical examples confirm the suitability of our formulation. In two test cases good agreement with an

Acknowledgment

The authors would like to thank Caterina Zeppieri for fruitful discussions regarding the Γ-convergence proof.

References (30)

  • L. Ambrosio et al.

    Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence

    Comm. Pure Appl. Math.

    (1990)
  • I. Babuska et al.

    The partition of unity method

    Internat. J. Numer. Methods Engrg.

    (1997)
  • P. Bastian et al.

    A generic grid interface for parallel and adaptive scientific computing. Part II: implementation and tests in DUNE

    Computing

    (2008)
  • T. Belytschko et al.

    Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment

    Internat. J. Numer. Methods Engrg.

    (2003)
  • M.A. Biot

    Theory of elasticity and consolidation for a porous anisotropic solid

    J. Appl. Phys.

    (1955)
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