13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- 5. Numerical Examples The proposed user element together with the structural elements provided in the ABAQUS element library are used to establish a finite element model to investigate a plane strain hydraulic fracture problem in an infinite impermeable elastic medium. The far-field boundary conditions are modelled by using infinite elements. The initial testing of this new element formulation involves using boundary value problems of an imposed fluid pressure and an imposed fracture opening. These problems are used to test for both of the two limiting cases of a toughness-dominated and viscosity-dominated plane-strain hydraulic fracture for which the analytical solutions are available in the literature [21]. (a) (b) Figure 4. Zero-viscosity case: (a) Imposed pressure; and (b) Imposed opening (a) (b) Figure 5. Zero-toughness case: (a) Imposed pressure; and (b) Imposed opening The simulation results for a plane strain toughness-dominated KGD hydraulic fracture are shown in Figure 4. The corresponding analytical solutions for the zero-viscosity case are also shown for comparison. The crack opening is obtained by imposing a given pressure calculated according to the analytical solution [21] on the crack surface of the finite element model. While the fluid pressure is obtained by applying an opening profile calculated from the analytical solution [21] to the crack surface of the finite element model. The results for the zero-toughness case are shown in Figure 5. Only twenty channel elements in total are meshed along the crack length in the finite element model. It can be seen that the XFEM predictions generally compare well with the analytical solutions for crack openings, while for the fluid pressure the XFEM predictions differ from the analytical solutions at the region close to the crack tip. One main reason for the deviation of the predicted fluid pressure from the analytical solutions near the tip is likely to be because the user-defined element is assumed to be cut through by the crack and no tip element is included in the finite
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