13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- 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. 2. Problem formulation 2.1. Problem definition Figure 1. A two-dimensional domain containing a hydraulic fracture Consider a two-dimensional hydraulically driven fracture cΓ propagating in a homogeneous, isotropic, linear elastic, impermeable medium Ω under plane strain conditions, see Figure 1. The boundary of the domain consists of FΓ on which prescribed tractions F, are imposed, uΓ on which prescribed displacements (assumed to be zero for simplicity) are imposed, and crack faces cΓ subject to fluid pressure. The fracture propagation is driven by injection of an incompressible Newtonian fluid at constant volumetric rate 0Q at a fixed injection point. It is assumed that the fracture propagation is quasi-static, and that the fracture is completely filled with the 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 of both position and time. 2.2 Governing equations The stress field inside the domain, σ, is related to the external loading F and the fluid pressure p through the equilibrium equations: FΓ uΓ cΓ + Ω _ p n r 1x 1y x y 0Q F
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