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Numerical Simulation for Flow in 3D Highly Heterogeneous Fractured Media

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

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

  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. Conforming mesh 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 H. Mustapha INRIA, SIAM Juin 2005

  8. Geometrical complexity: fractures and network • 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. Our approach To modify the complex configurations • 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. • What is the simplifications criteria ? • What is the loss in precision ? • What are the CPU time and memory capacity improvements ? H. Mustapha INRIA, SIAM Juin 2005

  9. Generation and quality of the mesh for the fracture networks • Network properties • size : 18 • number of fractures: 285 Mesh After refinement zoom before refinement 16 Distribution of angles 12 8 % Number of angles 4 0 0 20 40 60 80 100 120 140 160 180 Angle in degree H. Mustapha INRIA, SIAM Juin 2005

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

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

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

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

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

  16. Summary • 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 ? H. Mustapha INRIA, SIAM Juin 2005

  17. h1 L h2 Q Precision of computed solution 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. Network properties • size : 18 • number of fractures: 285 Very small mesh step With MHFE method • Solution obtained with high precision • Approximate solution can be used as a reference solution Precision of computed solution H. Mustapha INRIA, SIAM Juin 2005

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

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

  21. Memory usage decrease: 80% • CPU time decrease: 90% H. Mustapha INRIA, SIAM Juin 2005

  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. 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