Coupling schemes for modeling hydraulic fracture propagation using the XFEM

Abstract

We describe coupled algorithms that use the Extended Finite Element Method (XFEM) to solve the elastic crack component of the elasto-hydrodynamic equations that govern the propagation of hydraulic fractures in an elastic medium. With appropriate enrichment, the XFEM resolves the Neumann to Dirichlet (ND) map for crack problems with O(h2) accuracy and the Dirichlet to Neumann (DN) map with O(h) accuracy. For hydraulic fracture problems with a lag separating the fluid front from the fracture front, we demonstrate that the finite pressure field makes it possible to use a scheme based on the O(h2) XFEM solution to the ND map. To treat problems in which there is a coalescence of the fluid and fracture fronts, resulting in singular tip pressures, we developed a novel mixed algorithm that combines the tip width asymptotic solution with the O(h2) XFEM solution of the ND map away from the tips. Enrichment basis functions required for these singular pressure fields correspond to width power law indices λ>12, which are different from the index λ=12 of linear elastic fracture mechanics. The solutions obtained from the new coupled XFEM schemes agree extremely well with those of published reference solutions.