**Abstract**

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[5] Peirce, A, & Detournay, E. An Implicit Set Method for Modeling Hydraulically Driv‐ en Fractures, Comput. Methods Appl. Mech. Engrg., (2008). , 197, 2858-2885.

4.0. Minneapolis: Itasca; (2008).

830 Effective and Sustainable Hydraulic Fracturing

4.0. Minneapolis: Itasca; (2008).

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. © 2013 The Author(s). Licensee InTech. This chapter is 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.

This work is a part of a long-term project in the development of more accurate and efficient

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

http://dx.doi.org/10.5772/56445

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To perform the investigation, we used the displacement discontinuity method (DDM), a version of the boundary element method (BEM). The method was developed for, and has been successfully applied to rock mechanics area such as mining engineering [17,18], fracture analysis [19,20], and wellbore stabilities [12]. We have applied DDM here using both constant and higher-order elements. The higher-order elements use a quadratic variation of displace‐ ment discontinuity, and are based on the use of three collocation points over a three-element patch centered at the source element [10], while the constant elements use a constant variation of displacement discontinuity [12]. Details related to the elements are elaborated on in Shou's work [12]. Further, the authors also applied special crack tip elements [10] to capture the square root displacement variation at the crack tip, expected from Linear Elastic Fracture Mechanics (LEFM). The authors expect that special crack tip elements will provide the necessary flexibility to choose other tip profiles. This flexibility will be instrumental for efficient modeling of the different near-tip displacement profiles exhibited by various regimes in hydraulic fracture propagation (e.g., Viscosity-Dominated or Toughness-Dominated [22,23]). As others have shown, the accuracy of tip asymptote is critical in characterizing the stresses in the near-tip

We examined the numerically derived stress intensity factor for three crack geometries with and without higher-order elements and with and without special tip elements, to analytical solutions. The three crack geometries are a pressurized crack orthogonal to the least principle stress, a slanted straight crack, and a circular arc crack. Several authors selected these specific geometries to justify the use of higher-order or specialized boundary elements [24-27]. However, the quantification of the computational efficiency coupled with the accuracy has been limited. Therefore, we present the following analysis that aids in determining the method that provides the most efficient, yet accurate solutions. Accurate and efficient methods are

required for the development of field applications of engineering software packages.

Several other numerical techniques can be implemented within BEM. What we present here is not meant to be a review of possible combinations. We have chosen basic numerical techniques that provide the necessary flexibility to model very complex geometries, yet remain efficient enough for engineering modeling applications. The literature contains numerous examples of refinements to the techniques presented here [24-27]. For example, refinements with respect to the quarter-point method are found in Gray *et al*. [26] and refinements to higherorder elements are suggested by Dong and de Pater [25]. It is expected that implementing more refined methods will increase the efficiency of the numerical calculations. However, this work is primarily concerned with determining the general framework for BEM implementation. The details related to the crack tip elements are available from a number of sources [10,12]. For brevity, this work will only summarize some basic concepts and mathematical formulas of the higher-order elements and the specialized crack tip elements. The next section of this paper describes the general displacement discontinuity method utilizing constant displacement elements. In section 3, the authors summarize the chosen higher order elements. Section 4 defines the special crack tip elements used in this work. Section 5 compares various combina‐

numerical simulations for field engineering applications.

region of a propagating fracture [1].

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.

**Keywords** displacement discontinuity method, higher order elements, crack tip elements
