13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- Substituting of Eq. (4) into Eq. (6) leads to the governing equation for the fluid flow within the fracture 0 w p k g t x x (7) where 3 12 k w is the permeability. The general form of the governing equation (Eq. (7)) may be expressed as T 0 w p g k (8) where k is the permeability tensor. According to linear elastic fracture mechanics, the criterion that the fracture propagates continuously in mobile equilibrium (quasi-static) takes the form I Ic K K (9) where IK is the mode I stress intensity factor and Ic K the material fracture toughness. At the inlet, the fluid flux is equal to the injection rate, i.e., 0 inlet q Q (10) At the tip, the boundary conditions are given by the zero fracture opening and zero flow conditions, tip tip 0 w q (11) The above equations constitute the complete formulation that can be used to predict the evolution of the hydraulic fracture. 3. Weak form and FEM discretization The weak form of the equilibrium equation is given by T T T T T 0 t c c c c c c dΩ dΩ dΓ dΓ dΓ ε σ u b u t u p u p (12) where b is the body force, t is the applied traction on the boundary tΓ , u is an arbitrary virtual displacement and ε is the corresponding virtual strain, which is related to u through the strain operator S as ε S u. For the fluid pressure on the crack surfaces, we define c c c c c p p p p p p n n n . (13) The crack opening displacement w is given by T c c c w n u u , or c c c c w n u u n . (14) Then the weak from of the equilibrium equation can be expressed in a more compact form as T T T T 0 t c dΩ dΩ dΓ dΓ ε σ u b u t w p . (15) The weak form of the governing equation for the fluid flow within the fracture can be written as T T 0 c p w p g dΓ k (16) which, after integration by parts and substitution of the boundary conditions describe above, yields T T T 0 c c c p wdΓ p pdΓ p gdΓ k (17) Consider the coupled problem discretized in the standard (displacement) manner with the displacement vector u approximated as
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