1. Upscaling and History Matching of Fractured Reservoirs Pål Næverlid Sævik Department of Mathematics University of Bergen Modeling and Inversion of Geophysical Data (Uni CIPR) March 26, 2015

2. Outline 2 EnKFSimulation

3. Fractured Rocks What makes fractures different from other heterogeneities? 3

4. What makes fractures special? Dual porosity behaviour Scale separation issues Heterogeneities are larger than lab scale Prior information on fracture geometry may be available 4

5. Dual porosity behaviour 5

6. Scale separation issues Large faults and fractures may be impossible to upscale 6

7. Large and small fractures The distinction between «large» and «small» fractures is determined by the size of the computational cell 7

8. Prior fracture information Core samples Well logs Outcrop analogues Well testing Seismic data EM data (?) 8

9. Fracture parameters 9 Roughness Aperture (thickness) Filler material Connectivity Fracture density Clustering Shape Size

10. Common assumptions 10 Fisher distribution of orientations Power-law size distribution

11. Numerical upscaling Flexible formulation Accurate solution Slow Gridding difficulties May not have sufficient data to utilize the flexible formulation 11

12. Analytical upscaling Idealized geometry Fast solution Easy to obtain derivatives Requires statistical homogeneity Difficult to link idealized and true fracture geometry 12

13. Effective permeability 13 p in p out K1K1 K2K2 φ 1 K 1 + φ 2 K 2 p in p out φ 1 K 1 + φ 2 K 2 = A·a·K 1 + K 2 = A·τ + K 2

14. Several fracture sets 14

15. Partially connected fractures 15

16. Percolation theory 16

17. Connectivity prediction 17

18. Connectivity and spacing 18

19. Transfer coefficient 19

20. Summary: Input and output parameters 20 Connectivity f Density A Transmissitivity τ Density A Aperture a Filler material Permeability K Porosity φ Transfer coefficient σ Orientation Shape Size Clustering Roughness

21. History matching of fractured reservoirs 21 EnKF Simulation

22. Integrated upscaling and history matching 22

23. Ensemble Kalman Filter update 23

24. Test problem: Permeability measurement Single grid cell Measured permeability: 200 mD ± 20 mD Expected aperture: 0.2 mm ± 0.02 mm Expected density: 1 m -1 ± 0.2 m -1 Randomly oriented, infinitely extending fractures Cubic law for transmissitivity 24

25. Test problem: Permeability measurement 25

26. Predicted fracture porosity 26

27. Predicted transfer coefficient 27

28. Inverse relation and connectivity 28

29. Predicted connectivity 29

30. Linear fracture upscaling 30

31. Predicted connectivity 31

32. Partially connected fractures 32

33. Predicted connectivity 33

34. Field case: PUNQ-S3 Three-phase reservoir 6 production wells 0 injection wells (but strong aquifer support) Dual continuum extension with capillary pressure Constant production rate 34

35. Field case: PUNQ-S3 2 years of production 2 years of prediction Data sampling every 100 days Data used – GOR – WCT – BHP Assimilation using LM- EnRML 35

36. Data match summary Number of LM-EnRML iterations 01234 Fracture parameters as primary variables BHP 11.073.150.990.460.44 GOR 11.165.541.380.130.35 WCT 3.600.910.900.410.40 Total 9.313.711.110.370.40 Upscaled parameters as primary variables BHP 11.074.784.154.324.42 GOR 11.1610.269.749.629.65 WCT 3.601.261.231.161.21 Total 9.316.576.156.126.17 36

37. 37 BHP, PRO-1 GOR, PRO-12 WCT, PRO-11 Initial ensembleTraditional approachOur approach

38. 38 Permeability Sigma factor Initial ensembleTraditional approach Our approachTrue case

39. 39 Permeability Connectivity Initial ensembleTraditional approach Our approachTrue case

40. Conclusion Fracture upscaling creates nonlinear relations between the upscaled parameters These relations may be lost during history matching, if upscaled parameters are used as primary variables The problem can be avoided by history matching fracture parameters directly 40