13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- Fluid Solver: (i+1) (i+1) (i) T (i) p f n+1 n+1 n+1 n+1 n n n+1 f t + + + t =0 H Sp Sp p Qu uSpf (10) In the above equations, subscripts n and 1 n represent the real time nt and 1 nt , respectively, 1 f n n t t t is the time increment. np and nu are solutions of pressure and displacement at nt . The displacement rate in Eq.(5) is approximated as ( ) 1 1 i n n n f t u u u . During the iterations within one step, the fluid solver and mechanics solver are executed sequentially, with iteration numbers denoted by superscripts i , 1 i (e.g., ( ) 1 i n p , ( 1) ( 1,*) 1 1 , i i n n p u ). The flow solver solves the pressure at once implicitly, while the mechanics solver solves the displacement explicitly through iterations, denoted by superscripts k , 1 k ... (e.g. ( 1, ) 1 i k n u , ( 1, 1) 1 i k n u ). The boxed term in the flow solver is added to stabilize the system. S is an introduced parameter which will be determined in order to satisfy the numerical stability requirement and to achieve an optimal convergence rate. The stabilization term is essentially a term related to the variation of pressure increment during successive iterations, which vanishes when ( 1) ( ) 1 1 i i n n p p and a consistent solution of displacement and pore water pressure is consequently achieved. Without the stabilization term, the solution of Eq. (10) (i.e. ( 1) 1 i n p ) may be unstable or even unattainable, as will be demonstrated by stability analysis in Section 5. The sequence of solver calling varies in different schemes ([4]). The sequential scheme in this study first updates the pore water pressure by calling the flow solver, and then updates the displacement using the mechanical solver. Nested numerical iterations (denoted using subscript n and superscripts i , k ) are required to solve the system. The general procedure is listed as the following. (1). Initialization at the start of time: 0 ini p p , 0 ini u u . (2). Start time integration n=0 (a) Initialization (0) 1 n n p p , (0) 1 n n u u , update 1 u n f and 1 p n f . (b) Start iteration scheme i =0. i. Call flow solver to solve for ( 1) 1 i n p : 1 (i+1) (i) T (i) p n+1 f n+1 n+1 n n n+1 f = t + - t p H SS Sp Quu Spf ii. Call mechanics solver using predictor-corrector integration method. k starts from 0. (1,0) 1 n n u u . A. Compute the predictors: 2 ( 1, 1) ( 1, ) ( 1, ) ( 1, ) 1 1 1 1 ( 1, 1) ( 1, ) ( 1, ) 1 1 1 ( ) 1 2 2 (1 ) i k i k i k i k n n n n i k i k i k n n n t t t u u v u u v v t =10-5 s (pseudo-time step) B. Update ( 1, 1) ( 1, 1) 1 1 i k T s i k n n d K B C u B for nonlinear materials. C. Compute ( 1, 1) ( 1, 1) 1 ( 1) ( 1, 1) ( 1, 1) 1 1 1 1 1 1 i k i k i u i k i k n n n n n n M Q f Cv K u u p D. Compute the correctors: ( 1, 1) ( 1, 1) 2 ( 1, 1) 1 1 1 ( 1, 1) ( 1, 1) ( 1, 1) 1 1 1 ( ) i k i k i k n n n i k i k i k n n n t t u u u v v u
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