1. © 2011 - IFP Energies nouvelles Renewable energies | Eco-friendly production | Innovative transport | Eco-efficient processes | Sustainable resources Time Space Domain Decomposition for Reactive Transport in Porous Media Anthony MICHEL

2. © 2011 - IFP Energies nouvelles 2 Contributors Florian Haeberlein PhD Student, IFPEN He will defend his PhD next week ( 14/11/2001) Laurence Halpern, Paris 13, LAGA L.Trenty, J.M.Gratien, A.Anciaux, IFPEN M.Kern, INRIA T.Parra, Geochemistry Dpt, IFPEN D.Garcia, J.Moutte, ENSMSE

3. © 2011 - IFP Energies nouvelles 3 Outlook Part1. Motivation CO2 geological storage modeling CO2 reactivity distribution ANR-SHPCO2 Project Part 2. Reactive Transport Modeling Reactive chemical system Local reactive flash model Global reactive transport model Part 3.Time Space Domain Decomposition Subdomains Non linear DD Method Reactive subdomain definition Part 4. Case Studies Case study 1. Laboratory experiment Case study 2. SHPCO2 Use Case

4. © 2011 - IFP Energies nouvelles 4 Motivation Part 1

5. © 2011 - IFP Energies nouvelles 5 CO2 Geological Storage Storage

6. © 2011 - IFP Energies nouvelles 6 CO2 Geological Storage Modelling CO 2 H2OH2O CH 4 CO 2 H2OH2O Ca++ H+H+ Gas Salt Water Rock Texture OH- Na+ HCO3- Cl- Porous Media Geological Storage = Aquifer + Seal 10 km 100 m Connectivity Fe++ Mg++ Chemical System

7. © 2011 - IFP Energies nouvelles 7 CO2 Reactivity - Physical Distribution ( Garcia, 2008 ) CO2 Carbonatation Effects

8. © 2011 - IFP Energies nouvelles 8 CO2 Reactivity – Numerical Distribution Acid Front Reactivity Local time Stepping High Very Low Time step reduction is due to : - Strong non linearities - High species concentration ratios - What else ?? Low

9. © 2011 - IFP Energies nouvelles 9 SHPCO2 Project Simulation Haute Performance du Stockage Géologique de CO 2 ANR-CIS 2007 4 years project From 2008 to 2011

10. © 2011 - IFP Energies nouvelles 10 SHPCO2 Project Structure SP3 SP5 CPU-Time Newton Krylov + Preconditioners SP2 SP1 SP4 Time Space Domain Decomposition Parallel Computing and Load Balancing Real Study Test Case Numerical Models Integration and Coupling

11. © 2011 - IFP Energies nouvelles 11 Real Study Test Case ( Gaumet, 1997) Carbonates Layering

12. © 2011 - IFP Energies nouvelles 12 Real Study Test Case ( Gabalda, 2010) Dogger, Paris Basin Geological Model

13. © 2011 - IFP Energies nouvelles 13 Reactive Transport Modeling Part 2

14. © 2011 - IFP Energies nouvelles 14 Reactive Chemical System T W c q I I S cx x z S cz components primary species secondary species 0 0 0 0 c1c1 c2c2 x1x1 x4x4 x3x3 z1z1 z2z2 q1q1 q2q2 x2x2 q -> S kc *c + S kx *x q <- S kc *c + S kx *x ( Precip ) ( Dissol ) R kin Kinetic Reactions Equilibrium Reactions Phases and Species solid fluid

15. © 2011 - IFP Energies nouvelles 15 Local Reactive Flash Model qq Mass Balance Equations [  w c] + Scx [  w x] + Scz [  z z] = T [  q q] = W Equilibrium Equations ln(x) = ln(Kx) + Sxc [ ln(c)] (  w > 0 ) ln(z) = ln(Kz) + Szc [ ln(c)] or (  z = 0 ) Closure Equations  c +  x = 1 z = 1 q = 1 c q zz ww z x

16. © 2011 - IFP Energies nouvelles 16 Global Reactive Transport Model Mass Balance Equations Closure Equations (X) Constitutive Laws (X) C W T F RT,kin RW,kin

17. © 2011 - IFP Energies nouvelles 17 Fast Upwind Local Reactive Transport Model Mass Balance Equations Closure Equations (X) Constitutive Laws (X) + q out * q in *C in local

18. © 2011 - IFP Energies nouvelles 18 Time Space Domain Decomposition Part 3

19. © 2011 - IFP Energies nouvelles 19 T T+  t    t x  Time Space DD – Continuous Subdomains

20. © 2011 - IFP Energies nouvelles 20 T T+  t   t x Time Space DD – Discrete Subdomains

21. © 2011 - IFP Energies nouvelles 21 T T+  t  t x     B1B1 B 2  21 A 1 u 1 + R 1 (u 1 ) = F 1 B 1 u 1 =   = B 2  21 u 1 Time Space DD – Local Subdomain Problem

22. © 2011 - IFP Energies nouvelles 22 A 1 u 1 + R 1 (u 1 ) = F 1 B 1 u 1 =   = B 2  21 u 1 A 2 u 2 + R 2 (u 2 ) = F 2 B 2 u 2 =   = B 1  12 u 2 A u + R(u) = F Time Space DD – Global Coupled Problem

23. © 2011 - IFP Energies nouvelles 23 U  =  21 u 1* U 2*  =  12 u 2* A 1 u 1 + R 1 (u 1 ) = F 1 B 1 u 1 =    = B 1 u  A 2 u 2 + R 2 (u 2 ) = F 2 B 2 u 2 =   = B 2 u 1* Time Space DD – Classical Nonlinear Solver

24. © 2011 - IFP Energies nouvelles 24 Downwind Sweeping 1 k-1 kk+1 ncell Bk(Ck) = Flux(Ck) in =  C k-1  0 tt Is Fast Upwind RT a Time Space DD Method ?

25. © 2011 - IFP Energies nouvelles 25 - React(cell) = |Rkin|(cell) / Max (|Rkin|(cell)) - D1 = {React (cell) > TolReact } TolReact = 0.4, 0.2 -  react = D2 + NCellOverLap NCellOverLap = 4 - D2 = D1 + NCellSecurity NCellSecurity = 2 High Reactive Zone Security Layer OverLap Reactive Subdomain Definition

26. © 2011 - IFP Energies nouvelles 26 Numerical Efficiency Results Two Species Reactive Transport Classical / Nested / Common … Newton Iterations

27. © 2011 - IFP Energies nouvelles 27 Link with other NL Preconditionners … Cf Jan Nordbotten Talk, Yesterday

28. © 2011 - IFP Energies nouvelles 28 Case Studies Part 4

29. © 2011 - IFP Energies nouvelles 29 Case study 1 – Laboratory Experiment Plug Boundary External Boundary Study Domain Aqueous Solution Fixed pCO2 Core Cement Reacted Cement Reactive Front R2 R1

30. © 2011 - IFP Energies nouvelles 30 Case study 1 – Laboratory Experiment Portlandite + CO2(aq) -> Calcite Wollastonite -> CaO(aq) + Silice[CO2aq] CaOaq + CO2aq ->Calcite Silice -> SiO2aq [CaOaq] Simplified Overall Reaction Scheme

31. © 2011 - IFP Energies nouvelles 31 Case study 1 – Laboratory Experiment Aqueous Species Minerals Reactive Subdomain Movies …

32. © 2011 - IFP Energies nouvelles 32 Case study 2 - SHPCO2 Use Case Trapped Supercritical CO2 Barreers Regional Hydrodynamics

33. © 2011 - IFP Energies nouvelles 33 Case study 2 - SHPCO2 Use Case

34. © 2011 - IFP Energies nouvelles 34 Case study 2 - Reactive Chemical System

35. © 2011 - IFP Energies nouvelles 35 Case study 2 - SHPCO2 Use Case Movies …

36. © 2011 - IFP Energies nouvelles 36 Perspectives Global Solver Efficiency and Robustness Find a robust linear solver and preconditionner Optimize local computations in the reactive flash Improve newton convergence criterias Re-Visit the Fast Upwind Method Compare efficiency of the two methods Improve Efficiency of our Time-Space DD Solver Define good criterias for reactive subdomains Add appropriate metrics for the nested loops

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