13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- , on , on , on F c Ω Γ p p Γ σ 0 σ n F σ n σ n n n (1) where n is the unit normal vector. The kinematic equations include the strain-displacement relationship, the prescribed displacement boundary conditions and the definition of the separation between the two surfaces of the crack. Under the assumptions of small strains and displacements, the kinematic equations read T 1 on 2 on on u c Ω Γ Γ ε u u u 0 w u u (2) where u is the displacement, w is the separation between the two faces of the crack, and ε is the strain. The stress field insider the domain is expressed in terms of the isotropic, linear elastic constitutive law as: : σ D ε (3) where D is Hooke’s tensor. Figure 2. Fluid flow within crack The fluid flow within the crack is modelled using lubrication theory, given by Poiseuille’s law 3 12 w p q x (4) where is the dynamic viscosity of the fracturing fluid, q, the flow rate inside the crack per unit extend of the crack in the direction of x, is equal to the average velocity v times the crack opening w (see Figure 2), i.e., q x v x w x (5) The fracturing fluid is considered to be incompressible, so the mass conservation equation for the fluid may be expressed as 0 w q g t x (6) where the flux discontinuity x g is taken as positive if fluid is leaving the fracture. It can be interpreted as a source density outside the fracture, which accounts for fluid exchange between the fracture and the surrounding medium (e.g. porous rock). w q q dq dx x z v z v dx q q dq w
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