A modified extended finite element method for fluid-driven fractures incorporating variable primary energy loss mechanisms

A coupled extended finite element method (XFEM) is presented here for modeling propagation of fluid-driven fractures in different regimes including toughness- and viscosity-dominated regimes. The extended finite element method allows to model growth, and coalescence of arbitrary discontinuities (fractures) without requiring the mesh to conform to discontinuities nor significant refinement near the fractures. Fluid-driven fractures propagation is a coupled, nonlinear and non-local problem with moving boundary conditions. The proposed method is based on the extended finite element method with modifications to incorporate variable stress singularity at the crack tips for the transition between toughness-dominated and viscosity-dominated regimes. These modifications consist of enriched functions that are initially inspired by the asymptotic analytical solutions. The standard extended finite element approximation is enriched by adding near tip asymptotic solutions just for displacements, however the proposed method introduces a consistent enriched function for fluid pressure calculations close to the fracture tips to catch the singularity. Additionally, a technique is presented to remove singularity issue for required numerical integrations. Green's functions concept is proposed here to expedite calculations. To circumvent violation of partition of unity and parasitic terms in the approximation space induced by the blending elements at the edge of the enriched domain, the ramp function is utilized to improve the convergence rate. Stress intensity factors are calculated using a new contour integral method that can handle cases with different tip singularities. The proposed technique is verified with the cases that have analytical solutions. Some examples are presented to show the advantages of this technique in comparison to the regular XFEM.