International Conference Scaling Up and Modeling for transport and flow in porous media

1. 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) Écrire ici dans le masque le nom de votre Direction – Écrire ici dans le masque le titre de la présentation – Date de la présentation

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

3. Applications • Basin simulation • Reservoir simulation • C02 geological storage simulation

4. Compositional Models Phases:  = water, oil, gas Components i=1,...N (H2O, HydroCarbon species, C02, ...) Unknowns Thermodynamics laws (EOS): Hydrodynamics laws: Darcy phase velocities pore volume conservation mass conservation of each component present phases thermodynamic equilibrium present phases absent phases

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. Finite Volume Scheme • Discretization • Discrete conservation law

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

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 Fault LGR

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. CPG faults

11. Corner Point GeometriesStratigraphic grids with erosions • Hexahedra • Topologicaly Cartesian • Dead cells • Erosions • Local Grid Refinement (LGR) Examples of degenerate cells (erosions)

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

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

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

16. Elimination of the face unknowns using interpolationsuccess scheme (Eymard et al)

17. Success scheme: discrete variational form

18. Success scheme: fluxes with Fluxes in a general sense between K and L s.t. Stencil FKL :

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. Consistent gradient interpolation using only neighbors of K

21. Interpolation Use an L type interpolation (Aavatsmark et al.) using only neighbouring cells of K • 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

22. A "weak" gradient

23. Compact cell centered FV scheme: bilinear form with

24. Compact cell centered FV scheme: discrete variational formulation

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

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

27. Weak convergence property of the weak gradient

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

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

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

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

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

33. Test case random 3D • L2 error

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

35. Test case random 3D • L2 error on fluxes

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

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

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