Testing and Review of Various Displacement Discontinuity Elements for LEFM Crack Problems

The numerical modeling of hydraulic fractures in unconventional reservoirs presents signifi‐ cant challenges for field applications. There remains a need for accurate models that field personnel can use, yet remains consistent to the underlying physics of the problem [1]. For numerical simulations, several authors have considered a number of issues: the coupling be‐ tween fracture mechanics and fluid dynamics in the fracture [2], fracture interaction [3-5], proppant transport [6], and others [7-9]. However, the available literature within the oil and gas industry often ignores the importance of the crack tip in modeling applications devel‐ oped for engineering design. The importance of accurate modeling of the stress induced near the crack tip is likely critical in complex geological reservoirs where multiple propagat‐ ing crack tips are interacting with natural fractures. This study investigates the influence of various boundary element numerical techniques on the accuracy of the calculated stress in‐ tensity factor near the crack tip and on the fracture profile, in general. The work described here is a part of a long-term project in the development of more accurate and efficient nu‐ merical simulations for field engineering applications. For this investigation, the authors used the displacement discontinuity method (DDM). The numerical technique is applied using constant and higher-order elements. Further, the au‐ thors also applied special crack tip elements, derived elsewhere [10], to capture the square root displacement variation at the crack tip, expected from Linear Elastic Fracture Mechan‐ ics (LEFM). The authors expect that special crack tip elements will provide the necessary flexibility to choose other tip profiles. The crack tip elements may prove instrumental for ef‐ ficient modeling of the different near-tip displacement profiles exhibited by Viscosity-Domi‐ nated or Toughness-Dominated regimes in hydraulic fracture propagation. As others have © 2013 Jo and Hurt; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. shown [1,4,7], the accuracy of tip asymptote is critical in characterizing the stresses in the near-tip region of a propagating fracture. The authors examined the numerically derived stress intensity factor for several crack geo‐ metries with and without higher-order elements and with and without special tip elements, to analytical solutions. As expected, they found that the cases with higher-order elements and special tip elements provide more accurate results than the cases with constant displace‐ ment discontinuity and/or no tip elements. However, the numerical technique developed still proved efficient. These results show that numerical simulators can incorporate proper crack-tip treatments ef‐ fectively. In addition, higher-order elements increase computational efficiency by reducing the number of elements instead of simply increasing the discretization of constant displace‐ ment elements. The accurate modeling of stress intensity factors is necessary to better simu‐ late curved fractures, kinked cracks and interaction between fractures.