**1. Introduction**

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 fractures and with rock layers with different properties.

© 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, 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.

Great effort has been devoted to the numerical simulation of hydraulic fractures with the first 3D modelling efforts starting in the late 1970s [1-2]. Significant progress has been made in developing 2-D and 3-D numerical hydraulic fracture models [3-15]. Boundary integral equation methods or displacement discontinuity techniques have generally been employed to investigate the propagation of simple hydraulic fractures such as radial or plane-strain fractures in a homogeneous, infinite or semi-infinite elastic medium where the appropriate fundamental solutions are available. The finite element method has been used and is particu‐ larly useful in modelling the hydraulic fracture propagation in inhomogeneous rocks which may include nonlinear mechanical properties and may be subjected to complex boundary conditions. However, the standard finite element model requires remeshing after every crack propagation step and the mesh has to conform exactly to the fracture geometry as the fracture propagates, and thus is computationally expensive.

injected fluid, i.e., there is no lag between the fluid front and the fracture tip. The solution of the problem consists of determining the evolution of the fracture length, as well as the fracture opening, the fluid pressure, and the deformations and stresses inside the domain as functions

An ABAQUS Implementation of the XFEM for Hydraulic Fracture Problems

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

727

The stress inside the domain, **σ**, is related to the external loading **F** and the fluid pressure *p*

Under the assumptions of small strains and displacements, the kinematic equations, which include the strain-displacement relationship, the prescribed displacement boundary condi‐

> **ε**=(∇**u** + (∇**u**)T)/ 2 *on* Ω **u**=0 on Γ*<sup>u</sup>* **<sup>w</sup>**=*u*<sup>+</sup> - *<sup>u</sup>*- on <sup>Γ</sup>*<sup>c</sup>*

, on Γ*<sup>c</sup>*

(1)

(2)

∇ ⋅**σ**=0, on Ω **σ**⋅**n**=**F**, on Γ*<sup>F</sup>*

= - **σ**⋅*n*<sup>+</sup> = - *pn*<sup>+</sup> = *pn*-

of both position and time.

**2.2. Governing equations**

through the equilibrium equations:

where **n** is the unit normal vector.

tions and the crack surfaces separation, read

**Figure 1.** A two-dimensional domain containing a hydraulic fracture

**σ**⋅*n*-

By adding special enriched shape functions in conjunction with additional degrees of freedom to the standard finite element approximation within the framework of partition of unity, the extended finite element method [16-17] (XFEM) overcomes the inherent drawbacks associated with use of the conventional finite element methods and enables the crack to be represented without explicitly meshing crack surfaces, and so the crack geometry is completely independ‐ ent of the mesh and remeshing is not required, allowing for the convenient simulation of the fracture propagation. The XFEM has been employed to investigate the hydraulic fracture problems [18-19].

In this paper, we explore the application of the extended finite element method to hydraulic fracture problems. By taking good advantage of the XFEM and the flexible functionality of user subroutines provided in ABAQUS [20], a user-defined 2-D quadrilateral plane strain element has been coded in Fortran to incorporate the extended finite element capabilities in 2- D hydraulic fracture problems. The user-defined element includes the desired aspects of the XFEM so as to model crack propagation without explicit remeshing. In addition, the extended fluid pressure degrees of freedom are assigned to the appropriate nodes of the proposed elements in order to describe the viscous flow of fluid inside the crack and its contribution to the coupled crack deformation.
