Implicit level set schemes for modeling hydraulic fractures using the XFEM. (English) Zbl 1286.76048
Summary: 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 [E. Gordeliy and A. Peirce, “Coupling schemes for modeling hydraulic fracture propagation using the XFEM”, Comput. Methods Appl. Mech. Eng. 253, 305–322 (2013; doi:10.1016/j.cma.2012.08.017)] 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 [E. Detournay, “Propagation regimes of fluid-driven fractures in impermeable rocks”, Int. J. Geomech. 4 (1), 1–11 (2004; doi:org/10.1061/(ASCE)1532-3641(2004)4:1(35))] by making use of a universal asymptote [D. Garagash, Near-tip processes of fluid-driven fractures. Minneapolis: University of Minnesota (PhD Thesis) (1998); D. Garagash and E. Detournay, J. Appl. Mech. 67, No. 1, 183–192 (2000; Zbl 1110.74448)]. 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.
MSC:
| 76D08 | Lubrication theory |
| 74F10 | Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) |
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 74R10 | Brittle fracture |
Citations:
Zbl 1110.74448Software:
XFEMReferences:
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