**Author details**

Hyunil Jo\* and Robert Hurt

\*Address all correspondence to: hyunil.jo@bakerhughes.com

Baker Hughes, Tomball, TX, USA

### **References**

[1] Adachi, A, Siebrits, E, Peirce, A, & Desroches, J. Computer simulation of hydraulic fractures. International Journal Of Rock Mechanics And Mining Sciences (2007). , 44(5), 739-757.

[2] Naceur, K. B, Thiercelin, M, & Touboul, E. Simulation of Fluid Flow in Hydraulic Fracturing: Implications for 3D Propagation. SPE Production Engineering (1990). SPE 16032, 5(2), 133-141.

combinations of higher-order elements coupled with specialized crack tip elements requires

The authors thank Baker Hughes for supporting this research and for permission to publish this paper, Randy LaFollette, for sincere advice and encouragement, and Russell Maharidge,

[1] Adachi, A, Siebrits, E, Peirce, A, & Desroches, J. Computer simulation of hydraulic fractures. International Journal Of Rock Mechanics And Mining Sciences (2007). ,

more complex geometries than presented here.

1 − ν <sup>2</sup>

*c*, *a* Fracture half-length or half height

*h* Fracture length or height

**Acknowledgements**

PhD for considerate reviews.

and Robert Hurt

Baker Hughes, Tomball, TX, USA

44(5), 739-757.

\*Address all correspondence to: hyunil.jo@bakerhughes.com

**Author details**

Hyunil Jo\*

**References**

**Nomenclature**

σ Stress

*u* Displacement

850 Effective and Sustainable Hydraulic Fracturing

*E* Young's modulus ν Poisson's ratio *G* Shear modulus *<sup>E</sup>* ' Plane strain, *<sup>E</sup>*


[14] Olson, J. E. Predicting fracture swarms-- the influence of subcritical crack growth and the crack-tip process zone on joint spacing in rock. Geological Society, London, Spe‐ cial Publications (2004). , 231(1), 73-88.

[28] Pollard, D, & Segall, P. Theoretical displacements and stresses near fractures in rock: with applications to faults, joints, veins, dikes and solution surfaces. In: ATKINSON,

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

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

853

[30] Sneddon, I. N. The Distribution Of Stress In The Neighbourhood Of A Crack In An Elastic Solid. Proceedings Of The Royal Society Of London Series A-Mathematical

B K. Fracture Mechanics of Rock. Academic Press, London, (1987). , 277-350. [29] Valkó, P, & Economides, M. J. Hydraulic Fracture Mechanics. 1 ed: Wiley; (1995).

And Physical Sciences (1946). , 187(1009), 229-260.


[14] Olson, J. E. Predicting fracture swarms-- the influence of subcritical crack growth and the crack-tip process zone on joint spacing in rock. Geological Society, London, Spe‐

[15] Wu, R, Kresse, O, Weng, X, Cohen, C-e, & Gu, H. Modeling of Interaction of Hy‐ draulic Fractures in Complex Fracture Networks. In: SPE Hydraulic Fracturing Tech‐ nology Conference. The Woodlands, Texas, USA: Society of Petroleum Engineers.

[16] Weng, X, Kresse, O, Cohen, C-E, Wu, R, & Gu, H. Modeling of Hydraulic-Fracture-Network Propagation in a Naturally Fractured Formation. SPE Production & Opera‐

[17] NapierJAL. Modelling of fracturing near deep level gold mine excavations using a displacement discontinuity approach. Mechanics of jointed and faulted rock. Ross‐

[18] Peirce, A. P. Napier JAL. A Spectral Multipole Method For Efficient Solution Of Large-Scale Boundary-Element Models In Elastostatics. International Journal For Nu‐

[19] Tada, H, Paris, P. C, & Irwin, G. R. The Stress Analysis of Cracks Handbook. 3 ed:

[20] Exadaktylos, G, & Xiroudakis, G. A G2 constant displacement discontinuity element for analysis of crack problems. Computational Mechanics;, 45(4), 245-261.

[21] Shou, K-J. Napier JAL. A two-dimensional linear variation displacement discontinui‐ ty method for three-layered elastic media. International Journal of Rock Mechanics

[22] Garagash, D. I, & Detournay, E. Near tip processes of a fluid-driven fracture. ASME J

[23] Detournay, E. (2004). Propagation regimes of fluid driven fractures in impermeable

[24] Crawford, A. M, & Curran, J. H. Higher-order functional variation displacement dis‐ continuity elements. International Journal of Rock Mechanics and Mining Sciences &

[25] Dong, C. Y, & De Pater, C. J. (2001). Numerical implementation of displacement dis‐ continuity method and its application in hydraulic fracturing. Computer Methods in

[26] Gray, L. J, Phan, A. V, Paulino, G. H, & Kaplan, T. Improved quarter-point crack tip

[27] Yan, X. Q. A special crack tip displacement discontinuity element. Mechanics Re‐

element. Engineering Fracture Mechanics (2003). , 70(2), 269-283.

cial Publications (2004). , 231(1), 73-88.

tions;26(4):SPE-140253, 368-380.

and Mining Sciences (1999).

Appl Mech (2000). , 67, 183-92.

Geomechanics Abstracts (1982).

rocks. Int. J. Geomechanics , 4(1), 1-11.

Applied Mechanics and Engineering;191:745.

search Communications (2004). , 31(6), 651-659.

manith (ed), Balkema: Rotterdam: (1990). , 709-716.

merical Methods In Engineering (1995). , 38(23), 4009-4034.

American Society of Mechanical Engineering; (2000).

SPE152052

852 Effective and Sustainable Hydraulic Fracturing

[30] Sneddon, I. N. The Distribution Of Stress In The Neighbourhood Of A Crack In An Elastic Solid. Proceedings Of The Royal Society Of London Series A-Mathematical And Physical Sciences (1946). , 187(1009), 229-260.

**Chapter 43**

**Provisional chapter**

**The Impact of the Near-Tip Logic on the Accuracy and**

**The Impact of the Near-Tip Logic on the Accuracy**

**Convergence Rate of Hydraulic Fracture Simulators**

Brice Lecampion, Anthony Peirce, Emmanuel Detournay, Xi Zhang, Zuorong Chen, Andrew Bunger, Christine Detournay, John Napier, Safdar Abbas, Dmitry Garagash and

We benchmark a series of simulators against available reference solutions for propagating plane-strain and radial hydraulic fractures. In particular, we focus on the accuracy and convergence of the numerical solutions in the important practical case of viscosity dominated propagation. The simulators are based on different propagation criteria: linear elastic fracture mechanics (LEFM), cohesive zone models/tensile strength criteria, and algorithms accounting for the multi-scale nature of hydraulic fracture propagation in the near-tip region. All the simulators tested here are able to capture the analytical solutions of the different configurations tested, but at vastly different computational costs. Algorithms based on the classical LEFM propagation condition require a fine mesh in order to capture viscosity dominated hydraulic fracture evolution. Cohesive zone models, which model the fracture process zone, require even finer meshes to obtain the same accuracy. By contrast, when the algorithms use the appropriate multi-scale hydraulic fracture asymptote in the near-tip region, the exact solution can be matched accurately with a very coarse mesh. The different analytical reference solutions used in this paper provide a crucial series of benchmark tests

The propagation of hydraulic fractures is a highly non-linear fluid-solid interaction problem involving a moving boundary (i.e., the propagating fracture front in the neighboroughood of which the governing equations degenerate). Simulating this class of problem numerically is

and reproduction in any medium, provided the original work is properly cited.

©2012 Lecampion et al., licensee InTech. This is an open access chapter 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 Lecampion et al.; 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 Convergence Rate of Hydraulic Fracture**

**Simulators Compared to Reference Solutions**

**Compared to Reference Solutions**

Emmanuel Detournay, Xi Zhang, Zuorong Chen, Andrew Bunger, Christine Detournay, John Napier, Safdar Abbas, Dmitry Garagash and Peter Cundall

Additional information is available at the end of the chapter

that any successful hydraulic fracturing simulator should pass.

challenging, especially properly tracking the evolving fracture front.

Additional information is available at the end of the chapter

Brice Lecampion, Anthony Peirce,

http://dx.doi.org/10.5772/56212

Peter Cundall

10.5772/56212

**Abstract**

**1. Introduction**

**Provisional chapter**

#### **The Impact of the Near-Tip Logic on the Accuracy and Convergence Rate of Hydraulic Fracture Simulators Compared to Reference Solutions The Impact of the Near-Tip Logic on the Accuracy and Convergence Rate of Hydraulic Fracture Simulators Compared to Reference Solutions**

Brice Lecampion, Anthony Peirce, Emmanuel Detournay, Xi Zhang, Zuorong Chen, Andrew Bunger, Christine Detournay, John Napier, Safdar Abbas, Dmitry Garagash and Peter Cundall Brice Lecampion, Anthony Peirce, Emmanuel Detournay, Xi Zhang, Zuorong Chen, Andrew Bunger, Christine Detournay, John Napier, Safdar Abbas, Dmitry Garagash and Peter Cundall Additional information is available at the end of the chapter

Additional information is available at the end of the chapter 10.5772/56212

http://dx.doi.org/10.5772/56212
