**Author details**

N. Weber1 , P. Siebert2 , K. Willbrand3 , M. Feinendegen2 , C. Clauser3 and T. P. Fries1

1 Chair for Computational Analysis of Technical System, RWTH Aachen University, Aa‐ chen, Germany

2 Institute of Geotechnical Engineering, RWTH Aachen University, Aachen, Germany

3 Institute for Applied Geophysics and Geothermal Energy, E.ON ERC, RWTH Aachen Uni‐ versity, Aachen, Germany

### **References**

=

**Figure 8.** The pressure distribution Π(ξ) (a) and the crack opening profile Ω(ξ) (b) for various values of the fluid fraction

The pressure distribution and the crack opening profile along the dimensionless coordinate *ξ* = *x* / *L* are shown in Figures 8(a) and (b) in the viscosity scaling for fluid fraction values ξ<sup>f</sup>

The XFEM with an explicit-implicit crack description has been applied to a plane-strain hydraulic fracture problem. The crack is described explicitly by a line (2D)/triangular (3D) mesh that is aligned with the interface and implicitly by three level-set functions. The enrich‐ ment functions at the tip can be chosen according to the asymptotic behavior of the hydraulic fracture problem. Depending on the propagation regime the stress singularity can be described either by LEFM or by a singularity, which is weaker than predicted by LEFM. However, in this work a partially filled crack with a significant lag is examined and, therefore, crack propagation is governed by LEFM. The results show a good agreement with the known similarity solutions and can be interpreted as an early-time solution that can be used as a starting point in hydraulic

We acknowledge support for this work from the Federal Ministry for the Environment, Nature

Conservation and Nuclear Safety, Germany (FKZ 0325167).

{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.97, 0.99}.

722 Effective and Sustainable Hydraulic Fracturing

ξ*f* .

**6. Conclusions**

fracture simulations.

**Acknowledgements**


[9] Garagash, D, & Detournay, E. The tip region of a fluid-driven fracture in an elastic medium. Journal of Applied Mechanics, (2000). , 67(1), 183-192.

**Chapter 36**

**An ABAQUS Implementation of**

Additional information is available at the end of the chapter

fractures and with rock layers with different properties.

Zuorong Chen

**Abstract**

**1. Introduction**

http://dx.doi.org/10.5772/56287

**the XFEM for Hydraulic Fracture Problems**

A new finite element has been implemented in ABAQUS to incorporate the extended finite element method (XFEM) for the solution of hydraulic fracture problems. The proposed ele‐ ment includes the desired aspects of the XFEM so as to model crack propagation without explicit remeshing. In addition, the fluid pressure degrees of freedom have been defined on the element to describe the fluid flow within the crack and its contribution to the crack de‐ formation. Thus the fluid flow and resulting crack propagation are fully coupled in a natural way and are solved simultaneously. Verification of the element has been made by compar‐

Hydraulic fracturing is a powerful technology for enhancing conventional petroleum produc‐ tion. It is playing a central role in fast growing development of unconventional gas and geothermal energy. The fully 3-D numerical simulation of the hydraulic fracturing process is of great importance to understand the complex, multiscale mechanics of hydraulic fracturing, to the efficient application of this technology, and to develop innovative, advanced hydraulic fracture technologies for unconventional gas production. The accurate numerical simulation of hydraulic fracture growth remains a significant challenge because of the strong nonlinear coupling between the viscous flow of fluid inside the fracture and fracture propagation (a moving boundary), complicated by the need to consider interactions with existing natural

> © 2013 Chen; 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,

> © 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,

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

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

ing the finite element results with the analytical solutions available in the literature.

**Keywords** Hydraulic fracture, extended finite element method, internal pressure


**Chapter 36**
