Comparison of numerical schemes for multiphase reactive transport

Transcription

1 Énergies renouvelables Production éco-responsable Transports innovants Procédés éco-efficients Ressources durables Comparison of numerical schemes for multiphase reactive transport T. Faney, A. Michel, Q. L. Nguyen and L.Rouvray

2 Outline I. Introduction Motivation References Problem description II. Formulation and Active Set Algorithm III. Splitting Algorithm IV. Results V. Conclusion and Future Work 2

3 I. Motivation Applications in Geosciences : CO2 Sequestration Gas Production and Storage EOR Mineral Diagenesis Other Applications : Batteries Biological processes 3

4 I. Problem Description Multi-Phase Multi-Species System System Sub-System Phase Species

5 I. Problem Description Advection in Porous media

6 I. Problem Description Equilibrium chemical reactions For each reaction r Closure Equations Equilibrium reaction rates are unknown Rates elimination using chemical analysis

7 I. Problem Description Example System Water (H 2 O,OH -,H +,HCO 3-,Ca ++ ), Gas (CO 2g ), Calcite (CaCO 3s ) H 2 O = H + + OH - CO g 2 + H 2 O = HCO H + HCO Ca ++ = CaCO s 3 + H + Formula matrix A 7 13 equations (4 mass balance equations, 6 closure equations, 3 chemical equilibrium equations) 13 variables (P,φ f, φ s,s w,s g,s s,x H2O->CaCO3s )

8 II. Formulation and Active Set Algorithm Discretization Global Implicit Approach (Implicit Euler) Two-points finite volume scheme Issue with phase appearance and disappearance Complementarity conditions Active Set algorithm vs Semi-smooth Newton methods New context variable I I k = context of cell k Equations and variables depend on I k Split active and non-active species, components and equations 8

9 II. Formulation and Active Set Algorithm Example continued I k = {Gas, Calcite} P A P I S I N A N I 9 9 equations (4 mass balance equations, 5 closure equations, 0 chemical equilibrium equations) for cell k 9 variables (P,φ f, φ s,s g,s s,x CO2g,x CaCO3s,N I H2O,N I H+) for cell k

10 II. Formulation and Active Set Algorithm Global non linear equation solver Each active variable is updated at the end of each Newton iteration Context variable also needs to be updated Negative saturation phase disappearance Equilibrium flash phase appearance Equilibrium flash can be any 0D chemistry solver LMA / GEM formulation External or Internal library

11 III. Splitting Algorithm Splitting occurs at different stages Flow Reactive Transport Porosity update P, S α n α j t n t n+1 φ Transport Flash P, S α N n α j

12 IV. Results Active Set SHPCO2 case study 12

13 IV. Results Active Set 13

14 IV. Results Active Set 14

15 IV. Results Active Set 15

16 IV. Results Comparison SES Benchmark Irina Sin and Jerome Corvisier, Ecole des Mines de Paris, Geosciences research center 16

17 IV. Results Comparison Simple benchmark 17

18 IV. Results Comparison T = years Active Set Splitting (Non iterative) Splitting (Iterative) 18

19 IV. Conclusion and Future Work Active Set algorithm to model multi-phase multi-species flow in porous media Comparison with splitting approach Future work involves Non-Linear solver efficiency Flash solver improvement Semi-smooth Newton approach Extension to kinetics and diffusion fluxes 19

20 Thank you 20