1. CO 2 maîtrisé | Carburants diversifiés | Véhicules économes | Raffinage propre | Réserves prolongées © IFP Écrire ici dans le masque le nom de votre Direction – Écrire ici dans le masque le titre de la présentation – Date de la présentation Cell centered finite volume scheme for multiphase porous media flows with applications in the oil industry International Conference Scaling Up and Modeling for transport and flow in porous media Dubrovnik, Croatia October 13rd-16th 2008 Léo Agelas, Daniele di Pietro, Roland Masson (IFP) Robert Eymard (Paris East University)

2. © IFP 2 Outline Finite volume discretization of compositional models Cell Centered FV discretization of diffusion fluxes on general meshes

3. © IFP 3 Applications Basin simulation Reservoir simulation C02 geological storage simulation

4. © IFP 4 Compositional Models Phases:  = water, oil, gas Components i=1,...N (H 2 O, HydroCarbon species, C0 2,...) Unknowns Thermodynamics laws (EOS): Hydrodynamics laws: present phases absent phases mass conservation of each component pore volume conservation thermodynamic equilibrium Darcy phase velocities

5. © IFP 5 Discretization of compositional models Main constraints Must account for a large range of physics Robustness and CPU time efficiency Avoid strong time step reduction Cell centered FV discretization in space Euler fully or semi implicit schemes in time Thermodynamic equilibrium and pore volume conservation are implicit

6. © IFP 6 Finite Volume Scheme Discretization Discrete conservation law

7. © IFP 7 Discretization of compositional models present phases Component mass conservations Pore volume conservation and thermodynamics equilibrium

8. © IFP 8 Finite Volume discretization of diffusion fluxes Cell centered schemes Linear approximation of the fluxes Consistent on general meshes Cellwise constant diffusion tensors Cheap and robust Compact stencil Coercivity Monotonicity LGR Fault

9. © IFP 9 Reservoir and basin simulation meshes The mesh follows the directions of anisotropy using hexahedra but is locally non orthogonal due to - Faults - Erosions (pinchout) - Wells

10. © IFP 10 CPG faults

11. © IFP 11 Corner Point Geometries Stratigraphic grids with erosions Examples of degenerate cells (erosions) Hexahedra Topologicaly Cartesian Dead cells Erosions Local Grid Refinement (LGR)

12. © IFP 12 Near well discretizations Multi-branch well Hybrid mesh using Voronoi cells Hybrid mesh using pyramids and tetraedra

13. © IFP 13 Cell centered finite volume schemes on general meshes O and L MPFA type schemes Piecewise constant gradient on a subgrid Cellwise constant gradient construction Success (Eymard et al.): symmetric coercive but not compact

14. © IFP 14 Discrete cellwise constant gradient Cellwise constant linear exact gradient center of gravity of the face

15. © IFP 15 Hybrib bilinear form with HFV (Eymard et al.) or MFD (Shashkov et al.)

16. © IFP 16 Elimination of the face unknowns using interpolation success scheme (Eymard et al)

17. © IFP 17 Success scheme: discrete variational form

18. © IFP 18 Success scheme: fluxes Stencil F KL : Fluxes in a general sense between K and L s.t. with

19. © IFP 19 Success scheme Advantages  Cell centered symmetric coercive scheme on general meshes  Increased robustness on challenging anisotropic test cases Drawbacks  Discontinuous diffusion coefficients  Fluxes between cells sharing e.g. only a vertex  Large stencil Non symmetric formulation with two gradients

20. © IFP 20 Consistent gradient interpolation using only neighbors of K

21. © IFP 21 Interpolation Potential u linear in each cell K, L, M Flux continuity at the edges Potential continuity at the edges The scheme reproduces cellwise linear solutions for cellwise constant diffusion tensor Use an L type interpolation (Aavatsmark et al.) using only neighbouring cells of K

22. © IFP 22 A "weak" gradient

23. © IFP 23 Compact cell centered FV scheme: bilinear form with

24. © IFP 24 Compact cell centered FV scheme: discrete variational formulation

25. © IFP 25 Compact cell centered FV scheme: fluxes Stencil of the scheme: neighbors of the neighbors 13 points for 2D topologicaly cartesian grids 19 points for 3D topologicaly cartesian grids with

26. © IFP 26 Convergence analysis Stability of the gradients Coercivity (mesh and K dependent assumption)

27. © IFP 27 Weak convergence property of the weak gradient

28. © IFP 28 Test case CPG 2D CPG meshes of a 2D basin with erosions 2 km 20 km Mesh at refinement level 3 Smooth solution

29. © IFP 29 Test case CPG 2D Solver iterations (AMG preconditioner) L2 error

30. © IFP 30 Test case: Random Quadrangular Grids Mesh at refinement level 1 Smooth solution Domain = (0,1)x(0,1) Random refinement

31. © IFP 31 Test case Random Grid L2 error Solver iterations (AMG preconditioner)

32. © IFP 32 Test case: random 3D Diffusion tensor Smooth solution

33. © IFP 33 Test case random 3D L2 error

34. © IFP 34 Test case random 3D Solver iterations using AMG preconditioner

35. © IFP 35 Test case random 3D L2 error on fluxes

36. © IFP 36 Test case: random 3D aspect ratio 20 zoom

37. © IFP 37 Test case random 3D with aspect ratio 20 L2 error

38. © IFP 38 Conclusions There exists so far no compact and coercive (symmetric) cell centered FV schemes consistent on general meshes Among conditionaly coercive cell centered FV schemes GradCell Scheme exhibits a good robustness with respect to the anisotropy of K and to deformation of the mesh Compact stencil 2 layers of communication in parallel To be tested for multiphase Darcy flow