1. Shuyu Sun Earth Science and Engineering program KAUST Presented at the 2009 annual UTAM meeting, 2:05-2:40pm January 7, 2010 at the Sutton Building, University of Utah, Salt Lake City, Utah

2. Energy and Environment Problems

4. Single Phase Flow in Porous Media Continuity equation – from mass conservation Thermodynamic model For impressible fluid (constant density): Still need one more equation

5. Relate velocity with pressure: Darcy's law Experiment by Henry Darcy (1855–1856)

6. Darcy's law Can be derived from the Navier-Stokes equations via homogenization. It is analogous to –Fourier's law in the field of heat conduction, –Ohm's law in the field of electrical networks, –Fick's law in diffusion theory. In 3D:

7. Incompressible Single Phase Flow Continuity equation Darcy’s law Boundary conditions:

8. Transport in Porous Media Transport equation Boundary conditions Initial condition Dispersion/diffusion tensor

9. Numerical Methods for Flow & Transport Challenge #1: Require the numerical method to be: –Locally conservative for the volume/mass of fluid (flow equation) –Locally conservative for the mass of species (transport equation) –Provides fluxes that is continuous in the normal direction across the entire domain. Methods that are not locally conservative without post-processing –Point-Centered Finite Difference Methods –Continuous Galerkin Finite Element Methods –Collocation methods –……

10. Example: importance of local conservation

12. Numerical Methods for Flow & Transport Challenge #2: Fractured Porous Media –Different spatial scale: fracture much smaller –Different temporal scale: flow in fracture much faster Solutions: –Mesh adaptation for spatial scale difference –Time step adaptation for temporal scale difference

13. Example: flow/transport in fractured media

14. Locally refined mesh: FEM and FVM are better than FD for adaptive meshes and complex geometry

15. Example: flow/transport in fractured media CFL condition requires much smaller time step in fractures than in matrix: adaptive time stepping.

16. Numerical Methods for Flow & Transport Challenge #3: Sharp fronts or shocks –Require a numerical method with little numerical diffusion –Especially important for nonlinearly coupled system, with sharp gradients or shocks easily being formed Solutions: –Characteristic finite element methods –Discontinuous Galerkin methods

17. Example: Comparison of DG and FVM Advection of an injected species from the left boundary under constant Darcy velocity. Plots show concentration profile at 0.5 PVI. Upwind-FVM on 40 elementsLinear DG on 40 elements

18. Example: Comparison of DG and FVM Flow in a medium with high permeability region (red) and low permeability region (blue) with flow rate specified on left boundary. Contaminated fluid flood into clean media.

19. Example: Comparison of DG and FVM Advection of an injected species from the left. Plots show concentration profiles at 3 years (0.6 PVI). FVMLinear DG

20. Numerical Method for Flow & Transport Challenge #4: Time dependent local phenomena –For example: moving contaminant plume Solutions: –Dynamic mesh adaptation Based on conforming mesh adaptation Based on non-conforming mesh adaptation

21. Adaptive DG methods – an example Sorption occurs only in the lower half sub- domain, SIPG is used.

22. Adaptive DG example (cont.) Anisotropic mesh adaptation

23. Adaptive DG example (cont.) Estimators using hierarchic bases

24. Adaptive DG example (cont.) L2(L2) Error Estimators

25. A Posteriori Error Estimators Residual based –L2(L2) –L2(H1) Implicit –Solve a dual problem, can give estimates on a target functional –Disadvantages: computational costly and not flexible –Advantages: More accurate estimates Hierarchical bases – Brute-force: difference between solutions of two discretizations (most expensive) – Local problems-based –Advantage: can guide anisotropic hp-adaptivity Superconvergence points-based –Difficult for unstructured and non-conforming meshes

26. A posteriori error estimates Residuals –Interior residuals –(Element-)boundary residuals

27. A posteriori error estimate in L2(L2) for SIPG Proof Sketch: Compare with L2 projection; Cauchy- Schwarz; Properties of cut-off operator; Approximation results; Inverse and Gronwell’s inequalities; Relation of residue and error

28. Dynamic mesh adaptation with DG Nonconforming meshes –Effective implementation of mesh adaptation, –Elements will not degenerate unless using anisotropic refinement on purpose. Dynamic mesh adaptation –Time slices = a number of time steps; only change mesh for time slices. –Refinement + coarsening  number of elements remain constant.

29. Concentration projections during dynamic mesh modification Standard L2 projection used –Computation involved only in elements being coarsened L2 projection is a local computation for discontinuous spaces –This results in computational efficiency for DG –L2 projection is a global computation for CG L2 projection is locally mass conservative –This maintains solution accuracy for DG –Interpolation or interpolation-based projection used in CG is NOT locally conservative

30. Adaptive DG example (quads) L2(L2) Error Estimators for SIPG

31. Adaptive DG example (quads) L2(L2) Error Estimators for SIPG

32. Adaptive DG example (quads) L2(L2) Error Estimators for SIPG

33. Adaptive DG (with triangles) L2(L2) Error Estimators on Triangles Initial mesh

34. Adaptive DG (with triangles) L2(L2) Error Estimators on Triangles T=0.5 T=1.0

35. Adaptive DG (with triangles) L2(L2) Error Estimators on Triangles T=1.5 T=2.0

36. Adaptive DG example in 3D L2(L2) Error Estimators on 3D T=1.5 T=2.0 T=0.1T=0.5 T=1.0

37. ANDRA-Couplex1 case Background –ANDRA: the French National Radioactive Waste Management Agency –Couplex1 Test Case Nuclear waste management: Simplified 2D Far Field model Flow, Advection, Diffusion-dispersion, Adsorption Challenges –Parameters are highly varying permeability; retardation factor; effective porosity; effective diffusivity –Very concentrated nature of source concentrated in space concentrated in time –Long time simulation 10 million years –Multiple space scales Around source / Far from source –Multiple time scales Short time behavior (Diffusion dominated) Long time behavior (Advection dominated)

38. ANDRA-Couplex1 case (cont.) 200k years 2m years

40. Compositional Three-Phase Flow Mass Conservation (without molecular diffusion) Darcy’s Law

41. Numerical Modeling for Flow & Transport Challenge #5: Importance of capillarity –Capillary pressure usually ignored in compositional flow modeling –Even the immiscible two-phase flow or the black oil model usually assumes only a single capillary function (i.e. assuming a single uniform rock)

42. Two-dimensional 400x200m^2 domain Contains a less-permeable (K=1md) rock in the center of the domain while the rest has K=100md. Isotropic permeability tensor used. Porosity = 0.2 Densities: 1000 kg/m^3 (W) and 660 kg/m^3 (O) Viscosities: 1 cp (W) and 0.45 cp (O) Inject on the left edge, and produce on the right edge Injection rate: 0.1 PV/year Initial water saturation: 0.0; Injected saturation: 1.0 Example: Reservoir Description

43. Relative permeabilities (assuming zero residual saturations): Capillary pressure Reservoir Description (cont.) K=100md K=1md

44. Discretization DG-MFE-Iterative Pressure time step: 10years / 1000 timeSteps Saturation time step = 1/100 pressure time step Mesh: 32x64 uniform rectangular grid:

45. Comparison: if ignore capillary pressure … Saturation at 10 years: Iter-DG-MFE With nonzero capPres With zero capPres

46. Numerical Modeling for Flow & Transport Challenge #6: Discontinuous saturation distribution –Saturation usually is discontinuous across different rock type, which is ignored in many works in literature –When permeability changes, the capillary function usually also changes! Solutions: –Discontinuous Galerkin methods

47. Saturation at 3 years Iter-DG-MFE Simulation Notice that Sw is continuous within each rock, but Sw is discontinuous across the two rocks

48. Saturation at 5 years Iter-DG-MFE Simulation Notice that Sw is continuous within each rock, but Sw is discontinuous across the two rocks

49. Saturation at 10 years Iter-DG-MFE Simulation Notice that Sw is continuous within each rock, but Sw is discontinuous across the two rocks

50. Water pressure at 10 years Iter-DG-MFE Simulation (pressure unit: Pa) Notice that Pw is continuous within the entire domain.

51. Capillary pressure at 10 years Iter-DG-MFE Simulation (pressure unit: Pa) Notice that Pc is continuous within the entire domain.

52. Numerical Modeling for Flow & Transport Challenge #7: Multiscale heterogeneous permeability –Fine scale permeability has pronounced influence on coarse scale flow behaviors –Direct simulation on fine scale is intractable with available computational power Solutions: –Upscaling schemes –Multiscale finite element methods

53. Recall: DG scheme for flow equation Bilinear form Linear functional Scheme: seek such that

54. DG on two meshes Fine mesh Coarse mesh

55. Space decomposition Introduce Solution

56. Closure Assumption Introduce Two-scale solution

57. Implementation Multiscale basis functions: For each Multiscale approximation space: Two-scale DG solution

58. Other Closure Options Local problems for solving multiscale basis functions need a closure assumption. In previous derivation, we strongly impose zero Dirichlet boundary condition on local problems. Other options: –Weakly impose zero Dirichlet boundary condition on local problems. –Strongly impose zero Neumann boundary condition on local problems. –Weakly impose zero Neumann boundary condition on local problems. –Combination of zero Neumann and zero Dirichlet.

59. Comparison with direct DG Memory requirement –Direct DG solution in fine mesh: –Multiscale DG solution Computational time –Direct DG solution in fine mesh: –Multiscale DG solution

60. Example Conductivity: Boundary conditions: –Left: p=1; Right: p=0; top & bottom: u=0. Discretization: –R=r=1; –Coarse mesh 16x16; Fine mesh 256x256

61. Example (cont.) Coarse DG solution Brute-force Fine DG solution

62. Example (cont.) Multiscale DG solution Brute-force Fine DG solution

63. Future work Multiscale DG methods for compositional multiple-phase flow in heterogeneous media, Stochastic PDE simulations, Multigrid solver for DG (including p-multigrid), Other future works: –Automatically adaptive time stepping, –Implicit a posteriori error estimators, –Fully automatically hp-adaptivity for DG, –A posteriori estimators for coupled reactive transport and flow.