Upscaling two-phase flow in naturally fractured reservoirs using homogenization

1. Upscaling two-phase flow in naturally fractured reservoirs using homogenization by Hamidreza SalimiHans Bruining

2. Outline • Introduction [Fractured Reservoirs] • Basic model assumptions • The four scales in fractured media • Hypothesis • Water flooding example • Conclusion

3. Fractured Reservoirs • Importance: Over 20% of the world’s oil reserves • Oil Recovery Factor: 0-30% • Modeling: • Fundamentally different from heterogeneous reservoirs • Matrix-fracture interactions • Large impact of wettability effects • Recovery mechanisms not completely understood (e.g., counter-current imbibition)

4. Basic model assumptions I • Rock matrix provides the storage capacity and fracture provides the main flow path • The network of fractures is continuous as in the Warren and Root’s sugar cube model • Neighboring matrix blocks have no direct flow contact

5. Basic model assumptions II • Incompressible matrix • Water-wet Matrix and fracture

6. The four scales (A) Field scale (I) 10 ft grid block scale (II) 1 ft

7. The four scales (B) 1 ft Grid block scale (II) -> Periodic unit cell scale (III)

8. The four scales (C) Micro-scale (IV)

9. Upscaling techniques are required • Homogenization • Advantages • Based on microscopic equations • Uses no empirical closure relations • Finds many bifurcations; qualitatively different solutions related to different geological situations • Disadvantage • Uses periodicity assumption (cf. REV)

10. Hypothesis • Transfer function model can be validated by comparison to homogenization solutions

11. Homogenization Procedure • Divide the fractured reservoir into periodic unit cell (PUC) • Describe the transport equations on the Micro-Scale • Split the differential operator into a large and small term • Rescale the transport equations • Expand the dependent variables based on ε=l/L power series

12. l Basic idea of homogenization A fractured medium can be approximated as a periodic structure Two distinct length scales L (Grid block scale) Periodic unit cell scale:

13. Fracture transport equation follows from ε0 term Accumulation term Flow in the fracture Interaction between fractures and matrix

14. Dimensionless numbers I : matrix/fracture permeability ratio • The total volumetric flux in fractures, must be larger than the total volumetric flux in matrix blocks

15. Dimensionless numbers II: Viscous over capillary forces • For the capillary pressure to be dominant we require that

16. Dimensionless numbers III: Gravity segregation within matrix block • Gravity segregation in the matrix block occurs if the height of the matrix block is at least of the same order of magnitude as the height of the capillary transition zone, : h-> Sw->

17. Dimensionless numbers IV: Time to flow from inj->prod well <>time to fill matrix block with water

18. Fracture transport equation follows from ε0 term

19. Matrix equation leads to capillary counter-current imbibition

20. Matrix-fracture exchange term

21. Perforated in all vertical grids Perforated only in the top grid 100m 30 m 500 m Reservoir water flooding patternModel developed by Hamidreza Salimi , paid by this project

25. M=μo/ μw

27. Time dependent effects

28. Matrix equation leads to capillary counter-current imbibition

29. Effective saturation always larger than actual saturation (after Barenblatt et al.)

30. Microscopic mechanism of counter current imbibition

32. Transparent cell(master thesis project Shantie Kamoshi)

33. Conclusion I • Gravity effects have been implemented in the homogenization model • Numerical scheme has been implemented • Using homogenization, the transfer function can be described more explicitly • Capillary pressure requires dynamic term • Counter current imbibition description would benefit from experimental observations