1. Analysis of Layered Gas Reservoir Performance Using a Quasi-Analytical Solution for Rate and Pressure Behavior I Nengah Suabdi Department of Petroleum Engineering Texas A & M University 9 May 2001

2. Outline Introduction Objectives Assumptions Semi-analytical solutions New Type Curves for Layered Gas Reservoirs Field Application Conclusions

3. Introduction Depletion Performance Analysis: Can single-layer model performance detect layering..?, layer volume..?, or effect of drawdown..? Is a single layer model satisfactory..? Fetkovich, M.J. et.al (1990) –Using numerical simulations Layered-gas reservoir depletion study: Layer-1 Layer-2 No-Crossflow Single layer model..? where : k 1 >k 2..? Single Layer or Equivalent Single Layer Model

4. 1.To provide a quasi-analitycal solution for the depletion performance of a well produced at a common production pressure in a layered gas reservoir. 2. To utilize this quasi-analytical gas flow solution as a mechanism for charac- terizing the performance of layered gas reservoirs. Objectives

5. The proposed analysis techniques will be used to estimate the following properties for a layered gas reservoir system: The permeability ratio (2-layer case). Layer productivity index (J g ) The total original gas-in-place (G). The total flow capacity (kh product). The moveable reserves in each layer. Objectives

6. Schematic diagram of layered reservoir Layer-1 Layer- 2 Layer- 3 Layer- n

7. Assumptions: h1h1 Physical Model h2h2 k2k2 k1k1 No-Crossflow Production is commingled Two-layer (dry) gas reservoir No crossflow in the reservoir Homogeneous (except k layer ) Bounded radial system (pseudosteady-state flow) Production is commingled at a constant BHP Layer-1 Layer-2

8. Gas Diffusivity Equation in terms of :pressure (p), pseudopressure (p p ), and time : is not constant because µ and c t are functions of pressure

9. Plot of the Viscosity-Compressibility Function (Ansah et.al)

10. Semi-Analytical Solutions We can then develop the dimensionless decline rate (q Dd ), pressure (p D ), and cumulative production (G pD ). We consider the "first-order polynomial model" for correlating the curves. This result is given by Ansah, et al. as:

11. The fundamental form of stabilized flow equation is given by Semi-Analytical Solutions Where :

12. Gas MBE for moderate to low pressure reservoirs: Semi-Analytical Solutions Where the dimensionless pressures are defined by:

13. Dimensionless Pressure (p D ) Semi-Analytical Solutions Where :

14. Dimensionless Decline Rate (q Dd ) Semi-Analytical Solutions Where :

15. Dimensionless Cumulative Production (G pD ) Semi-Analytical Solutions Where :

16. In field units, the dimensionless " decline " time is defined as: Semi-Analytical Solutions Where : t = Time, days k j = Permeability ( layer j), md  j = Porosity ( layer j), fraction c ti = Total system compressibility, psia -1 r e = Radius of the external boundary, ft

17. Semi-Analytical Solutions Where : C j = Stabilized flow coefficient layer-j, Mscf/D/psi 2 k j = Permeability ( layer j ), md  j = Porosity ( layer j ), fraction c ti = Total system compressibility, psi -1 p ref = (p i + p wf )/2, psi Gas rate production for each layer (q gj ) in-term of (p/z) 2 is defined as

18. Pressure Depletion Decline Type Curve

22. Vol Layer-1 Vol Layer-2 p wD = 0.1 G

23. Pressure Depletion Decline Type Curve G

24. G

25. G

26. G

27. Depletion Decline Rate Type Curve

28. Rate Depletion Decline Type Curve

32. G pD vs. Dimensionless Decline Time (t Dd )

33. Stabilized Gas Flow Coefficient (c j )

34. vs G pD,t p/z vs G pD,t Function

35. Field Application (p/z vs. G pt Curve Example) Well Beavers 1-11 (Hugoton Field, Kansas, USA)

36. Field Application (p/z vs. G pt Curve Example) Well Beavers 1-11 (Hugoton Field, Kansas, USA)

37. Field Application (p/z vs. G pt Curve Example) Well Beavers 1-11 (Hugoton Field, Kansas, USA)

38. Field Application (p/z vs. G pt Curve Example) Well Beavers 1-11 (Hugoton Field, Kansas, USA)

39. p/z versus Gpt —Cartesian format. Well Beavers 1-11 (Hugoton Field, Kansas, USA) More Permeable Layer Less Permeable Layer G = 24.11 BSCF

40. q g versus prod time —semilog format. Well Beavers 1-11 (Hugoton Field, Kansas, USA)

41. q g versus prod time —log-log format. Well Beavers 1-11 (Hugoton Field, Kansas, USA)

42. G pt versus prod time —semilog format. Well Beavers 1-11 (Hugoton Field, Kansas, USA)

43. G pt versus prod time —log-log format. Well Beavers 1-11 (Hugoton Field, Kansas, USA)

44. Estimate properties of Well Beavers 1-11 -Total original gas-in-place (G)= 24.11 BSCF - The permeability ratio (k 1 /k 2 )= 68 - Total reservoir thickness, (h tot )= 130 ft - Average reservoir radius, (r e )= 3,250 ft - Average area each layer, (A avg )= 761.76 Acres - The total flow capacity, (kh product)= 482 md-ft - The magnitude of wellbore F. Press (P wf )= 20 psia

45. Field Application (Rate type Curve Example) Nelson well (Hugoton Field, Kansas (USA))

46. Field Application (Rate type Curve Example) Gas Well- B

47. 1. We successfully demonstrated the use of a semi- analytical solution for a single-layer gas system for layered gas reservoir cases presented by Fetkovich, et.al (numerical simulations). 2. A two-layer type curve was developed for the analysis of production performance. The single- layer case can not be used to model the 2-layer case. 3. The sensitivity of individual layer properties was investigated, in particular — permeability ratio, layer volumes, and the effect of drawdown. Conclusions

48. Analysis of Layered Gas Reservoir Using Production Data I Nengah Suabdi Department of Petroleum Engineering Texas A & M University 3 February 2001

49. Field Application (Example) Curtis well (Hugoton Field, Kansas, USA) Less Permeable Layer