Frontiers and Future of Multiphase Fluid Flow Modeling in Oil Reservoirs

1. Frontiers and Future of Multiphase Fluid Flow Modeling in Oil Reservoirs • Shuyu Sun • Earth Science and Engineering program, Division of PSE, KAUST • Applied Mathematics & Computational Science program, MCSE, KAUST • Acknowledge: • Mary F. Wheeler, The University of Texas at Austin • Abbas Firoozabadi, Yale & RERI; Joachim Moortgat, RERI • Mohamed ElAmin, Chuanxiu Xu and Jisheng Kou, KAUST • Presented at 2010 CSIM meeting, KAUST Bldg. 2 (West), Level 5, Rm 5220, 11:10-11:35, May 1, 2010.

2. Energy and Environment Problems

3. Single-Phase Flow In Porous Media

4. Single-Phase Flow in Porous Media • Continuity equation – from mass conservation: • Volumetric/phase behaviors – from thermodynamic modeling: • Constitutive equation – Darcy’s law:

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

6. DG scheme applied to flow equation • Bilinear form • Linear functional • Scheme: seek such that

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

8. DG scheme applied to transport equation • Bilinear form • Linear functional • Scheme: seek s.t. I.C. and

9. Example: importance of local conservation

10. Example: Comparison of DG and FVM Upwind-FVM on 40 elements Linear DG on 20 elements Advection of an injected species from the left boundary under constant Darcy velocity. Plots show concentration profile at 0.5 PVI.

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

12. Example: flow/transport in fractured media Locally refined mesh: FEM and FVM are better than FD for adaptive meshes and complex geometry

13. Example: flow/transport in fractured media

14. Adaptive DG example L2(L2) Error Estimators

15. A posteriori error estimate in the energy norm for all primal DGs Proof Sketch: Relation of DG and CG spaces through jump terms S. Sun and M. F. Wheeler, Journal of Scientific Computing, 22(1), 501-530, 2005.

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

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

18. Two-Phase Flow In Porous Media

19. Two-Phase Flow Governing Equations • Mass Conservation • Darcy’s Law • Capillary Pressure • Saturation Summation Constraint

20. DG-MFEM IMPES Algorithm – Pressure Equ • If incompressible (otherwise treating it with a source term): • Total Velocity: • Pressure Equation: • MFEM Scheme: • Apply MFEM • Two unknown variables: Velocity Ua and Water potential

21. DG-MFEM IMPES Algorithm – Saturation Equ • Solve for the wetting (water) phase equation: • Relate water phase velocity with total velocity: • Saturation Equation (if using Forward Euler): • DG Scheme: • Apply DG (integrating by parts and using upwind on element interfaces) to the convection term.

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

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

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

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

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

27. Multi-Phase Flow In Porous Media

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

29. Example of CO2 injection • Initial Conditions: C10+H2O(Sw=Swc=0.1), 100 bar,160 F. • Inject water (0.1 PV/year) to 2 PV, then inject CO2 to 8 PV. Poutlet= 100 bar • Relative permeabilities: • Quadratic forms except nw=3. • Residual/critical saturations: • Sor = 0.40; Swc= 0.10; Sgc = 0.02 • Sgmax = 0.8; Somin = 0.2 • ; ; ;

30. Example (cont.) MFE-dG 0.1 PVI. MFE-dG 0.5 PVI. MFE-dG 0.2 PVI.

31. Example 3 (cont.) nC10 at 10% PVI CO2 nC10 at 200% PVI CO2

32. Remarks for Multiphase Flow • Framework has been established for advancing dG-MFE scheme for three-phase compositional modeling. In our formulation we adopt the total volume flux approach for the MFE. • dG has small numerical diffusion • CO2 injection • Swelling effect and vaporization • Reduction of viscosity in oil phase • Recovery by CO2 injection > Recovery by C1 > Recovery by N2

33. Modeling of Phase Behaviors for Reservoir Fluid

34. EOS Modeling of Phase Behaviors • PVT modeling: EOS • Peng-Robinson EOS • Cubic-plus-association EOS • Thermodynamic theory • Stability calculation • Tangent Phase Distance (TPD) analysis • Gibbs Free Energy Surface analysis • Flash calculation • Bisection method (Rachford-Rice equation) • Successive Substitution • Newton’s method

35. Gibbs Ensemble Monte Carlo simulation

36. Three Monte Carlo movements in simulation • Particle displacements • Volume Change • Particle Transfer

37. Microstructure from the ab initio calculation The microstructure of the molecular models form the ab initio calculation The nearest neighbor interaction between the Water and Ethane T-shaped pair of water molecules

38. Water-ethane high pressure equilibria at T=523 K Experimental data are from Chemie-Ing. Techn. (1967), 39, 816 EoS: Statistical-Associating-Fluid-Theory (SAFT)

39. Thank You!