Implicit level set schemes for modeling hydraulic fractures using the XFEM

We describe two novel XFEM schemes for modeling fluid driven fractures both of which exploit an implicit level set algorithm (ILSA) for locating the singular free boundary that occurs when the fluid and fracture fronts coalesce. Both schemes use the mixed P&W XFEM formulation developed in Gordeliy and Peirce (2013) [1] to incorporate the singular asymptotic solution in the fracture tips. The proposed level set strategy also exploits the asymptotic solution to provide a robust procedure to locate the free boundary, which is not restricted to symmetric growth of the fracture geometry or to a particular mode of propagation. The versatility of the ILSA-XFEM scheme is demonstrated by sampling different asymptotic behaviors along the so-called MK edge of parameter space (Detournay, 2004) [2] by making use of a universal asymptote (Garagash, 1998 [3]; Garagash and Detournay, 2000 [4]). The two ILSA-XFEM schemes differ in the enrichment strategies that they use to represent the fracture tips: a scheme with full tip enrichment and a simpler, more efficient, scheme in which the tip asymptotic behavior is only imposed in a weak sense. Numerical experiments indicate that the XFEM-t scheme, with full tip enrichment, achieves an O ( h 2 ) asymptotic convergence rate, while the XFEM-s scheme, with only signum enrichment to represent the crack geometry, achieves an O ( h ) asymptotic convergence rate.