13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- a d w d a d w d '( ', 0) '( ', 0) '( ', 0) *( ') (0 ' 2 ) ' ' ' ' '( ', 0) '( ', 0) '( ', 0) *( ') (0 ' 2 ) ' ' ' ' u x u x u x u x x l x x x x v x v x v x v x x l x x x x ∂ + ∂ + ∂ + ∂ + + = <= <= ∂ ∂ ∂ ∂ ∂ + ∂ + ∂ + ∂ + + = <= <= ∂ ∂ ∂ ∂ (2) where, ' iu and ' iv are displacement components. (2) The inner stress p is exerted at the edge of artificial fracture. The direction is perpendicular to the fracture surface. a, d, w, a, d, w, ( ,0) ( ,0) (,0) , ( ) ( ,0) ( ,0) (,0)0, ( ) yy yy yy xy xy xy x x x p a x a x x x a x a σ σ σ σ σ σ + + = − =< <= + + = − =< <= (3) Based on inclusion theory, combined with the equations (2) and (3), we can obtain four Cauchy singular integral equations. After solving, the stress field at any point of the inclusion reservoirs can be expressed after a series of coordinate conversion. At the same time, we can deduce the stress intensity factor at the tip of artificial fracture and inclusions. When the artificial fracture and line inclusions are disjoint, the stress intensity factor at the tip of the artificial fracture can be obtained by the following equations. I I 2 ( ) 2( ) ( ) lim 1 2 ( ) 2( ) ( ) lim 1 x a x a K a a x g x K a a x g x μ κ μ κ →− → − = + + =− − + (4) where, a is the half-length of artificial fracture, m; IK is the stress intensity factor of model I crack, MPa m ⋅ ; κis the elastic constant of rock matrix, and for plane strain problem, 3 4 κ υ = − , where, υ is Poisson ratio; ( ) g x is the dislocation density function of the artificial fracture at ~ , 0 x a a y =− + = , ( ) ( , 0) (,0) (< ) y y g x u x u x a x a x ∂ ⎡ ⎤ = + − − − < ⎣ ⎦ ∂ (5) In the local coordinate system, the stress intensity factor at the tip of line inclusions can be expressed as follows. I ' 0 I ' 2 1 '(0) 2 ' ( ') lim 2( 1) 1 '(2 ) 2(2 ') ( ') lim 2( 1) x x l K x q x K l l x q x κ κ κ κ → → − =− + − = − + (6) where, 2l is the length of the inclusion, m; ( ') q x is the tangential constraint physical stress of the inclusion on rock matrix, ( ') ( ', 0) ( ', 0) xy xy q x x x σ σ = − − + (7) Therefore, when we compare the above results with the critical stress intensity factor IC K , we can judge the extension of the fractures in the inclusion reservoir. From equation (6) and (7), we can
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