Development of a 2-D Black Oil Reservoir Simulator with Unique Grid-Block System

1. Development of a 2-D Black Oil Reservoir Simulator with Unique Grid-Block System Emeline Chong July 7, 2004 Harold Vance Department of Petroleum Engineering

2. Presentation Outline • Motivation • Problem Definition • Objectives • Approach • Program Validation/Evaluation • Conclusions

3. Motivation Diagonal Parallel Not to scale

4. GRID ORIENTATED PARALLEL TO INJECTOR-PRODUCER PAIRS (PARALLEL RUNS) GRID ORIENTATED AT 45 TO INJECTOR-PRODUCER PAIRS (DIAGONAL RUNS)

5. 20 acres Mobility Ratios : M = 0.5 M = 1.0 M = 10.0 10 acres

9. Saturation Distribution at PVinj =1.0 for M=0.5

15. Saturation Distribution at PVinj =1.0 for M=10.0

16. Motivation Brand, Heinemann, and Aziz (1992) – “In general, Grid Orientation Effect cannot be overcome with grid refinement.” (SPE 21228)

17. Motivation Sammon (1991) Chen & Durlofsky (1991) Mattax & Dalton (1990) Wolcott et.al. (1996) Shiralkar (1990) Brand et.al. (1991) Ostebo & Kazemi (1992) Yanosik & McCracken (1979) Shiralkar & Stephenson (1987) Pruess & Bodvarsson (1983) Todd et.al. (1972)

18. Problem Definition Grid orientation effect significantly affects the results of immiscible displacements in reservoir simulation

19. Objectives • Developing a 2-D, 3-Phase reservoir simulator using finite difference formulation • Reducing the grid orientation effects in a grid model

20. Approach 2-D, 3-Phase IMPES finite difference simulator with unique grid model  “Sim2D”

21. 2D,3-Phase Cartesian Grid • Initial Condition • Rock/Fluid Properties Well Model Well Constraints HGB Grid IMPES Matrix Form Matrix Solver Pn+1, Son+1, Swn+1, Sgn+1 Cutback/Saturation Control Program Validation

22. Sim2D Demo

23. Sim2D VB Application

29. Pressure Plots

30. IMPES MethodFinite Difference Equations • Oil • Water • Gas

31. IMPES Steps… • Calculate coefficients of the pressure equation • Calculate solution of the pressure equation implicitly (matrix equation) for: pn+1 • Calculate solution of the saturation equations explicitly for: Son+1, Swn+1, and Sgn+1

32. IMPES Method • 4 unknowns per block: pn+1, Son+1, Swn+1, and Sgn+1 • To find the unknowns, we need one more equation per block: Son+1 + Swn+1 + Sgn+1 = 1 • Assures fluid volumes fit the pore volume

33. IMPES Method • Oil • Water • Gas

34. IMPES Method Summing up all saturation equations:

35. IMPES Method Approximate Vpn+1 on the right hand side using the identity: and a chord slope:

36. IMPES Method Then, where, Likewise,

37. IMPES Method Final equation can now be written as:

38. Hybrid Grid Block (HGB) System

39. I I J J N W E S Hybrid Grid Block (HGB) System NW NE SE SW

40. Hybrid Grid Block (HGB) System

41. Grid Numbering Numbering #1 Example: 18 Grid Blocks

42. Grid Numbering Numbering #2 Example: 18 Grid Blocks

43. Grid Numbering Numbering #3 Example: 25 Grid Blocks

44. Well Model Peaceman Well Model (1983): Δm For square gridblock, where, α = mass species; oil/water ro = effective wellbore radius

45. Well Model for regular polygon (Palagi, 1992): j = neighbor of wellblock i bij = side of polygon dij = distance between gridpoints N = number of equal sides Well Model i

46. Cartesian Sim2D Cartesian Eclipse 100 HGB Sim2D Program Validation

47. Example Case: Two-Dimensional Areal Model Showing Primary Depletion of an Undersaturated Reservoir (One Producer Well, One Injector Well, Isotropic, 2-Phase, Oil/Water)

48. Validation with Eclipse

49. Validation with Eclipse

50. Application ofHGB grid system to Reduce Grid Orientation Error