1. 1 Numerical Simulation for Flow in 3D Highly Heterogeneous Fractured Media H. Mustapha J. Erhel J.R. De Dreuzy H. Mustapha INRIA, SIAM Juin 2005

2. 2 Outline Fractured media  geometrical model  flow fluid model Mesh requirements for Finite Element Methods  numerical method  mesh generation difficulties New approach for computing flow based on a projection method  main idea and some examples  quality of the mesh and precision of the computed solution Conclusions and future work H. Mustapha INRIA, SIAM Juin 2005

3. 3 Outline Fractured media  geometrical model  flow fluid model Mesh requirements for Finite Element Methods  numerical method  mesh generation difficulties New approach for computing flow based on a projection method  main idea and some examples  quality of the mesh and precision of the computed solution Conclusions and future work H. Mustapha INRIA, SIAM Juin 2005

4. 4 Geometrical model Discrete fracture network. Impervious matrix. Only the fractures are considered. Network = Set of fractures. H. Mustapha INRIA, SIAM Juin 2005

5. 5  Equations  Q = - K*grad (h)  div (Q) = 0  Boundary conditions Flow fluid model Fixed head (Dirichlet) Q.n = 0 (Neumann) H. Mustapha INRIA, SIAM Juin 2005

6. 6 Outline Fractured media  geometrical model  flow fluid model Mesh requirements for Finite Element Methods  numerical method  mesh generation difficulties New approach for computing flow based on a projection method  main idea and some examples  quality of the mesh and precision of the computed solution Conclusions and future works H. Mustapha INRIA, SIAM Juin 2005

7. 7 Numerical method Mixed Hybrid Finite Element Method complete 3D mesh of the network mesh of each fracture with identical intersections (conforming mesh) Global linear system Assembled by all corresponding fractures linear systems Direct linear solver Conforming mesh H. Mustapha INRIA, SIAM Juin 2005

8. 8 Geometrical complexity: fractures and network Origin of the complexity Important number of intersections. Existence of small intersections. Existence of zones containing small angles => need mesh refinement to improve the quality. Our approach To modify the complex configurations What is the simplifications criteria ? What is the loss in precision ? What are the CPU time and memory capacity improvements ? The blue counter is the fracture border. The red lines are the intersections with the cube borders. The black lines are the intersections with the other fractures of the network. H. Mustapha INRIA, SIAM Juin 2005

9. 9 Network properties size : 18 number of fractures: 285 Generation and quality of the mesh for the fracture networks Mesh After refinement 020406080100120140160180 0 4 8 12 16 % Number of angles Angle in degree Distribution of angles H. Mustapha INRIA, SIAM Juin 2005 zoom before refinement

10. 10 Outline Fractured media  geometrical model  flow fluid model Problem presentation  geometrical complexity in two scales: fractures and network  classical mesh generation for fracture networks New approach for computing flow based on a projection method  main idea and some examples  quality of the mesh and precision of the computed solution Conclusions and future works H. Mustapha INRIA, SIAM Juin 2005

11. 11 Main idea Step (1) 2D Projection 3D projection Generalization H. Mustapha INRIA, SIAM Juin 2005

12. 12 Main idea Step (2) 2D projection H. Mustapha INRIA, SIAM Juin 2005

13. 13 Results of our approach Example 1: projection method H. Mustapha INRIA, SIAM Juin 2005

14. 14 Results of our approach Example 2: projection and generation mesh H. Mustapha INRIA, SIAM Juin 2005

15. 15 Generation and mesh quality using the new approach Mesh with projection Network properties size : 18 number of fractures: 285 H. Mustapha INRIA, SIAM Juin 2005

16. 16 Our new approach: Use a projection method as a simple criteria Lead to a reduced configuration Allows to mesh complex fracture networks Questions to address: What is the loss in precision ? What are the CPU time and memory capacity improvements ? Summary H. Mustapha INRIA, SIAM Juin 2005

17. 17 Precision of computed solution L Q h1h1 h2h2 Math formulas: Global equivalent permeability: Average flow across the fractures:, N: number of fractures Average flow across the intersections:, M: number of intersections Errors: EN = EF = EI = H. Mustapha INRIA, SIAM Juin 2005

18. 18 Very small mesh step K ref 45.9017 K app 45.9015 EN4×10 -6 EF1.8×10 -5 EI2×10 -5 Solution obtained with high precision Approximate solution can be used as a reference solution Precision of computed solution With MHFE method Network properties size : 18 number of fractures: 285 H. Mustapha INRIA, SIAM Juin 2005

19. 19 Examples of mesh and computed solution 3 fractures Head 450 fractures Head 1800 fractures h1 h2 H. Mustapha INRIA, SIAM Juin 2005 Network mesh: 285 fractures

20. 20 Precision and numerical error Loss in precision: 5% Good quality of the mesh Reference solution Mesh step : 0.08 H. Mustapha INRIA, SIAM Juin 2005

21. 21 mesh steplinear system sizeCPU time 0.08313 77627.65 0.09277 16919.21 0.11185 61112.12 0.13131 0297.65 0.1598 0255.18 0.1776 7983.98 0.1962 4603 0.2150 5882.36 Memory usage decrease: 80% CPU time decrease: 90% H. Mustapha INRIA, SIAM Juin 2005

22. 22 Outline Fractured media  geometrical model  flow fluid model Problem presentation  geometrical complexity in two scales: fractures and network  classical mesh generation for fracture networks New approach for computing flow based on a projection method  main idea and some examples  quality of the mesh and precision of the computed solution Conclusions and future works H. Mustapha INRIA, SIAM Juin 2005

23. 23 Conclusions Mesh generation of complex fracture networks. Approximate geometry by projection. The loss in precision of computed solution is very small. The improvement in CPU time and memory capacity is very important. Future work Parallel computation. Stochastic experiments. H. Mustapha INRIA, SIAM Juin 2005