Comparison of Numerical Schemes for Accurate Long-term Simulation of Contaminant Migration

1. Comparison of Numerical Schemes for Accurate Long-term Simulation of Contaminant Migration Krzysztof Banaś1,2, Steve Bryant2 1ICM, Cracow University of Technology 2TICAM, The University of Texas at Austin

2. Mary F. Wheeler (DIRECTOR) Todd Arbogast Steven Bryant Clint Dawson Rick Dean Eleanor Jenkins Phu Luong Victor Parr Malgorzata Peszynska Béatrice Rivière John Wheeler CSM Researchers + several visitors + 10 graduate students 1999-2000: 5 PhD, 1 MS completed

3. Overview • Experience with ParSSim • First order Godunov • Higher order Godunov • Characteristics-mixed method • Experience with DG research codes

4. ParSSimParallel Subsurface Simulator • Multicomponent • logically rectangular 3D • operator splitting • General biogeochemistry • interior point minimization of free energy • explicit integration of kinetics ODEs • Scalable Parallel • domain decomposition (MPI) • SP2, cluster of PCs, T3E, Workstations • dynamic load balancing 1 flowing phase, N stationary phases

5. ParSSim solution scheme: operator splitting • Solve flow equation • Solve transport equations • Advect • React • rate-limited reactions, mass transfer • thermodynamic equilibrium • Diffuse • Update composition-dependent viscosity, permeability

6. ParSSim Flow Calculation • Single phase Darcy flow • Logically rectangular, cell-centered finite difference, implicit • Glowinski-Wheeler domain decomposition

7. ParSSim Transport Biogeo-chemistry Radionuclide decay Injection/extraction wells Linear sorption • Advection step: solve

8. ParSSim Transport: advection step (1) • Explicit characteristics-mixed method* • Introduce total concentration Ti : • Resulting PDE: • Solve by characteristic tracking: • Extract advected concentrations: • *Arbogast and Wheeler, SIAM J. Numer. Anal. 32 (1995) 404-424

9. ParSSim Transport: advection step (2) • Higher order Godunov* • Solve directly for advected concentrations • Formally 2nd order, improved by postprocessing step • First order Godunov ( nopostprocessing) • *Dawson, SIAM J. Numer. Anal. 30 (1993) 1315-1332

10. ParSSim Transport: reaction step • React the advected concentrations • Radionuclide decay • Solve the PDE… • By explicit integration • Biogeochemistry • Solve the PDE… • By explicit integration Equilibrium reactions handled by free energy minimization

11. ParSSim Transport: diffusion step • Diffuse/disperse the reacted concentrations • Solve the PDE… • Implicitly by

12. Transport scheme comparisons • Couplex1 • Transport scheme benchmarks* • Moving hill • Curvilinear flow • Mixed waste *http://terrassa.pnl.gov:2080/~kash/workshop/bmark.htm

13. Couplex1 Test Case

14. Couplex1 Simulation 129I plume, Higher Order Godunov solution

15. Couplex1 Simulation 129I plume, Higher Order Godunov solution

16. Couplex1 Variant Decrease clay layer diffusion coefficient 1000 times 129I plume, Higher Order Godunov solution

17. Couplex1 Simulation 129I plume, character-istics-mixed method solution

18. Couplex1 Simulation 129I plume, Characteristics Mixed Method solution

19. Couplex1 Variant Decrease clay layer diffusion coefficient 1000 times 129I plume, Characteristics Mixed Method solution Note “hot spot” at clay-limestone boundary

20. Moving Hill (Benchmark 6) Conservative tracer 5050 grid vx = vy NPe(grid)= 102 NCr = 0.1 y x

21. http://terrassa.pnl.gov:2080/~kash/workshop/problems/pcl/prob1/ashok1.htmhttp://terrassa.pnl.gov:2080/~kash/workshop/problems/pcl/prob1/ashok1.htm

22. http://terrassa.pnl.gov:2080/~kash/workshop/problems/pcl/prob1/ashok1.htmhttp://terrassa.pnl.gov:2080/~kash/workshop/problems/pcl/prob1/ashok1.htm

23. Benchmark 6: First order Godunov

24. Benchmark 6: Higher order Godunov

25. Benchmark 6: Characteristics Mixed Method

26. Benchmark 6: Discontinuous Galerkin Method

27. Benchmark 6: Characteristics Mixed Method, NCr=1

28. Benchmark 6: Results First order Godunov Higher order Godunov Characteristics-mixed method Discontinuous Galerkin

29. Benchmark 5 Description 150 x 150 grid zero diffusion solute inlet effluent collection window

30. Benchmark 5: CMM

31. Benchmark 5: DG

32. Benchmark 5: Tracer Effluent Analytical HOG CMM

33. Benchmark 5: DG results Results from current research code, courtesy K. Banas

34. Benchmark 5: Reactive Solute Effluent

35. Benchmark 5: Reactive Solute Effluent Analytical, half order HOG

36. Transport scheme comments • Characteristics-mixed method (CMM) • Minimal numerical dispersion • No CFL constraint (except that arising from domain decomposition) • Good scaling in parallel • Computationally expensive • Higher order Godunov • Locally mass conservative • CFL constraint • More numerical dispersion than CMM • Computationally inexpensive • Discontinuous Galerkin • Even less numerical dispersion than CMM • Subject of current research

37. DG Methods • Unstructured, non-matchinggrids • Element refinements/de-refinements • Local high order of approximations • Arbitrary tensor coefficients and BC • Locally conservative • Error estimators (for hp adaptivity) • Positive definiteness

38. 3D elements of “arbitrary” shape • Tetrahedral • Prismatic • Hexahedral

39. Unstructured grids

40. Non-matching grids

41. Non-matching grids

42. Slope limiting

43. hp-adaptivity • Order of approximation specified element-wise • Isotropic element divisions • Anisotropic element divisions

44. Example of 3D convection

45. Example of 3D convection

46. Example of 3D convection

47. Iterative linear equations solver • Large, sparse, non-symmetric, positive definite systems of linear equations • Single level GMRES as default solver • Multi-level GMRES for elliptic problems • Preconditioning by ILU, domain decomposition • Implementation based on block Gauss-Seidel iterations • Easy parallelization

48. Application to subsurface flows • Single phase flow • Miscible displacement • Two-phase flow • Interfaces with IPARS framework

49. Application to subsurface flow

50. Miscible Displacement Concentration Front