Enrichment strategies and convergence properties of the XFEM for hydraulic fracture problems

In two recent papers Gordeliy and Peirce (2013) investigating the use of the Extended Finite Element Method (XFEM) for modeling hydraulic fractures (HF), two classes of boundary value problem and two distinct enrichment types were identified as being essential components in constructing successful XFEM HF algorithms. In this paper we explore the accuracy and convergence properties of these boundary value formulations and enrichment strategies. In addition, we derive a novel set of crack-tip enrichment functions that enable the XFEM to model HF with the full range of power law rλ behavior of the displacement field and the corresponding rλ−1 singularity in the stress field, for 12≤λ<1. This novel crack-tip enrichment enables the XFEM to achieve the optimal convergence rate, which is not achieved by existing enrichment functions used for this range of power law. The two XFEM boundary value problem classes are as follows: (i) a Neumann to Dirichlet map in which the pressure applied to the crack faces is the specified boundary condition and the XFEM is used to solve for the corresponding crack width (P→W); and (ii) a mixed hybrid formulation of the XFEM that makes it possible to incorporate the singular behavior of the crack width in the fracture tip and uses a pressure boundary condition away from it (P&W). The two enrichment schemes considered are: (i) the XFEM-t scheme with full singular crack-tip enrichment and (ii) a simpler, more efficient, XFEM-s scheme in which the singular tip behavior is only imposed in a weak sense. If enrichment is applied to all the nodes of tip-enriched elements, then the resulting XFEM stiffness matrix is singular due to a linear dependence among the set of enrichment shape functions, which is a situation that also holds for the classic set of square-root enrichment functions. For the novel set of enrichment functions we show how to remove this rank deficiency by eliminating those enrichment shape functions associated with the null space of the stiffness matrix. Numerical experiments indicate that the XFEM-t scheme, with the new tip enrichment, achieves the optimal O(h2) convergence rate we expect of the underlying piece-wise linear FEM discretization, which is superior to the enrichment functions currently available in the literature for 12<λ<1. The XFEM-s scheme, with only signum enrichment to represent the crack geometry, achieves an O(h) convergence rate. It is also demonstrated that the standard P→W formulation, based on the variational principle of minimum potential energy, is not suitable for modeling hydraulic fractures in which the fluid and the fracture fronts coalesce, while the mixed hybrid P&W formulation based on the Hellinger-Reissner variational principle does not have this disadvantage.