1. Improvement of a multigrid solver for 3D EM diffusion Research proposal final thesis Applied Mathematics, specialism CSE (Computational Science and Engineering)

2. Tom Jönsthövel TU Delft januari 6, 2006 Tutors: Kees Oosterlee (EWI, TUD) Wim Mulder (SIEP, Shell)

3. Overview presentation Introduction: Practical application/problem statement Mathematical model Discretization model equations Short overview Multigrid Problems encoutered with MG solver Possible improvements on MG solver Summary Questions

4. Practical Application Practical goal: Image structures that could host potential reservoirs Providing evidence of presence of hydrocarbons How? Recovering the conductivity profile from measurements of electric and magnetic fields: * Oil/Gas are more resistive than surrounding (gesteente)

5. (3D) Electromagnetic Diffusion 2D example: Oil/Gas? EM source receivers

6. (3D) Electromagnetic Diffusion Amundsen, Johansen & Røsten (2004): A Sea Bed Logging (SBL) calibration survey over the Troll gas field

7. Mathematical model Maxwell equations in presence of current source With, Ohm’s law

8. Maxwell equations Eliminate the magnetic field from the equation: 

9. Maxwell equations Transform equation from time to frequency domain: With  angular frequency. Now, In practice: 

10. Maxwell equations PEC boundary conditions: (Perfectly Electrically Conduction) domain

11. Discretization model equations Step 1: Choose discretization: Finite Integration Technique (Clemens/Weiland ’01)  Finite volume generalisation of Yee’s scheme (1966)  Error analysis for constant-coeffients (Monk & Sülli 1994)  2 nd order accuracy for electric/magnetic field components

12. Discretization model equations Step 2: Placement EM field components (Yee’s scheme):

13. ii) Discretization model equations Step 3: Next steps; discretize all components of main equation: i) with,  iii)

14. Discretization model equations ii) 1st 2nd

15. Discretization curl Stokes

16. Discretization curl

17. Introduce (discreet) residu Goal: solve for r = 0 How? Multigrid solver

18. Overview Multigrid Idea: use BIM for solving Ax=b 1. the error e=x ex -x apr becomes smooth (not small) 2. Quantity smooth on fine grid  approx on coarser grid (e.g. double mesh size) Concl: error smooth after x relaxation sweeps  approx error on coarser grid  Cheaper/Faster

19. Basis MG Pre-smoothing Coarse grid correction: Restriction Compute approximation solution of defect equation - Direct/iterative solver - New cycle on coarser grid Prolongation Post-smoothing

20. Basis MG Important choices: Coarser grids Restriction operator: residu from fine to coarse Prolongation operator: correction from coarse to fine Smoother

21. MG Components Coarser grids:

22. MG Components Restriction: Full weighting

23. MG Components Prolongation: Linear/bilinear interpolation, is transpose of restriction

24. MG Components Smoother: Pointwise smoother Symmetric GS-LEX

25. Introduce Test probleem Artificial eigenvalues problem: On the domain [0,2π]3. This defines the source term Js. Convergence: 10–8

26. Stretching

27. Problems MG Solver σ 0 =10 S/m, σ 1 =1 S/m cellsequidistantStretched (4%) h max MGbih max MGbi 160.39760.4586 320.20870.26118 640.098870.171214 1280.049860.138132

28. Anisotropy 2D anisotropic elliptical equations:

29. Anisotropy Discretization in stencil notation:

30. Anisotropy Error averaging with GS-LEX: If ε → 0, No smoothing effect in x-direction

31. Anisotropy and stretched grid 2D elliptical equations: Simple stretching: Hence:

32. Anisotropy Two possible improvements MG solver: 1.Semi coarsening 2.Line-smoother

33. Semicoarsening

34. Line-smoother Solve all unknowns on line in direction anisotropy simultaneously. Reason: Errors become smooth if strong connected unknows are updated collectively 1,l,m 2,l,m Nx – 1,l,m

35. Preview results Combination line-smoother and semi-coarsening gives good results Factor 5 less MG iterations needed

36. Summary 1.Oil/Gas reservoir? 2.EM diffusion method  Maxwell equations 3.Multigrid solver  Problems when gridstretching used 4.Improvements: 1.Line-Smoother 2.Semi Coarsening 5.Results are obtained  more research for improvement and generalisation, mathematical soundness

37. Questions?