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
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), , 2005.

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

17. Adaptive DG example in 3D
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
This is the overview of my presentation.
K=1md

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

24. but Sw is discontinuous across the two rocks
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. but Sw is discontinuous across the two rocks
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. but Sw is discontinuous across the two rocks
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 0.2 PVI.
MFE-dG 0.5 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
Bond length(Å)
Angle(degree)
Hydrogen length(Å)
OH(H2O)
0.9619
OH(H2O) 2
0.9698
1.9321
CH(C2H6)
1.0938
CH(H2O----C2H6)
1.0940
(H2O)
105.06
(H2O) 2
105.28
107.5
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!