1. 1 Study of Pressure Front Propagation in a Reservoir from a Producing Well by Hsieh, B.Z., Chilingar, G.V., Lin, Z.S. May 4, 2007

2. 2 Outline Introduction Purpose Basic theory and simulation tool Results and discussions Conclusions

3. 3 Introduction

4. 4 Producing rate and flowing pressure at wellbore P wf (psi) p=p i q (stb/day) q=constant t (hours) t=t i t=0 r rwrw t (hours)t=t i

5. 5 Pressure distribution in reservoir at t = t i P (psi) r (ft) p=p i t=t i q (stb/day) q=constant t (hours) Radius of investigation (r i ) Pressure front Pressure disturbance area ( Drainage area ) Non-disturbed area t=t i r=r w t=0 ri rwrw t=t i r=r i

6. 6 Pressure distribution in reservoir at various times P (psi) r (ft) p=p i t=t 3 q (stb/day) q=constant t=t 2 t=t 1 t (hours) r i 3 r i 1 r i 2 t=t 1 t=t 2 t=t 3 pressure front s

7. 7 t1t1 rwrw r1r1 r2r2 r3r3 t2t2 t3t3 Plane view of pressure fronts at various times

8. 8 Dimensionless Variables Dimensionless radius Dimensionless time Dimensionless pressure

9. 9 Pressure distribution in a reservoir in terms of dimensionless variables PDPD rDrD t D3 t D2 t D1 r iD1 r iD2 r iD3 pressure front s

10. 10 Radius of investigation (r iD ) and time (t D ) The relationship between the dimensionless radius of investigation (r iD ) and the dimensionless time (t D ) is (Muskat, 1934; Tek et al., 1957; Jones, 1962; Van Poolen, 1964; Lee, 1982; Chandhry, 2004, etc.) r iD 2 = α t D where the radius coefficient (α) is a constant and varied in different studies, from 3.18 to 16

11. 11 Literature on radius of investigation equation Author (Year) Radius of investigation equation Definition of radius of investigation or method used Muskat (1934)r iD 2 = 4t D Material balance method Tek et al. (1957)r iD 2 = 18.4t D The fluid flow rate at the radius of investigation is 1% of that flowing into the wellbore Hurst et al. (1961)r iD 2 = 6.97t D Pressure build-up test Jones (1962)r iD 2 = 16t D The pressure drop at the radius of investigation is 1% of pressure drop at the wellbore Van Poolen (1964)r iD 2 = 4t D Y-function of an infinite and a finite reservoir Hurst (1969)r iD 2 = 8t D The analytical van Everdingen and Hurst solution

12. 12 Literature on Radius of Investigation (Cont.) Author (Year) Radius of investigatio n equation Definition of radius of investigation or method used Earlougher (1977)r iD 2 = 3.18t D After van Poolen (1964) Lee (1982)r iD 2 = 4t D The solution of the diffusivity equation for an instantaneous line source in an infinite medium Kutasov and Hejri (1984)r iD 2 = 4.12t D Constant bottom-hole pressure test Johnson (1986)r iD 2 = 7.89t D The radius enclosing a volume in the reservoir accounts for a specified fraction of the cumulative production of 96.1% Chandhry (2004)r iD 2 = 4t D Pressure transient analysis of pressure build-up test

13. 13 Purposes of the study To estimate the propagation of the radius of investigation from a producing well by using both analytical and numerical methods, including variable flow rates case, skin factor, and wellbore storage effect. To estimate the starting time of transient pressure affected by the reservoir boundary to concurrently determine the radius coefficient

14. 14 Basic theory and simulation tool

15. 15 Analytical Solution – Ei solution The analytical solution of the diffusivity equation for a well (line source) producing in an infinite cylindrical reservoir is (van Everdingen and Hurst, 1949; Earlougher, 1977):

16. 16 Numerical Solution of Diffusivity Equation Numerical solutions are also used in this study for the cases that no analytical is available or the comparisons are required. The IMEX simulator (CMG) is used in this study to generate results in numerical simulation.

17. 17 Basic reservoir parameters used in this study

18. 18 The pressure behavior check -- infinite reservoir Even the specific oil reservoir is used in this study, the pressure behavior (dimensionless pressure as function of time and radius) is checked by comparison with analytical solution that exist in the literature

19. 19 The pressure behavior check -- bounded reservoir

20. 20 The pressure behavior check -- bounded reservoir

21. 21 Results and Discussions

22. 22 Definition of pressure front P (psi) r (ft) p=p i t=t i Δp 1 Δp 2 Δp 3 α1α1 α2α2 α3α3 △ p= pressure drop defined at the pressure front α= radius coefficient ● ● ●

23. 23 Definition of pressure front PDPD rDrD t D5 △ p D = the dimensionless pressure drop defined at the pressure front α= radius coefficient α1α1 α2α2 α3α3 Δp D1 Δp D2 Δp D3

24. 24 Pressure front and radius of investigation From Ei solution such as By defining or giving Δp D (or y), the following equation can be derived Note: The radius coefficient (α) is dependent on the criteria defined at the pressure front (the value of Δp D ).

25. 25 Radius coefficients from analytical solution with constant flow rate in an infinite reservoir By defining a small dimensionless pressure value (Δp D ) at the pressure front, the value of r iD 2 /4t D in the Ei solution can be estimated.

26. 26 Radius coefficients from analytical solution with constant flow rate in an infinite reservoir α = 4.00 ( Δp D =1.095*10 -1 ) α = 10.39 ( Δp D =1.095*10 -2 ) α = 51.22 ( Δp D =10 -7 ) α = 59.84 ( Δp D =10 -8 ) α = 71.15 ( Δp D =10 -9 ) α = 26.06 ( Δp D =10 -4 ) α = 34.28 ( Δp D =10 -5 ) α = 42.69 ( Δp D =10 -6 ) α = 17.82 ( Δp D =1.095*10 -3 )

27. 27 Radius coefficients from numerical solution with constant flow rate in an infinite reservoir α = 3.986 ( Δp D =1.095*10 -1 ) α = 10.363 ( Δp D =1.095*10 -2 ) α = 17.799 ( Δp D =1.095*10 -3 )

28. 28 Radius investigation equation from analytical and numerical solution -- constant flow rate case Constant flow rate test (q=100 stb/day) r iD criteria (I) for Δp D =0.1095 r iD criteria (II) for Δp D =0.01095 r iD criteria (III) for Δp D =0.001095 Analytical solution r iD 2 = 4.00t D R 2 =1 r iD 2 = 10.39t D R 2 =1 r iD 2 = 17.82t D R 2 =1 Numerical solution r iD 2 = 3.986t D R 2 =0.9999 r iD 2 = 10.363t D R 2 =0.9999 r iD 2 = 17.799t D R 2 =0.9999 Different criteria for pressure front will obtain different radius coefficient (α) The smaller the Δp D, the larger the radius coefficient (α), i.e., the faster the pressure front propagation.

29. 29 Results and Discussions (2) Effect of variable flow rates

30. 30 Ei solution with superposition – variable flow rate or where

31. 31 ( a) Increasing flow rate (two-rates) test

32. 32 Radius of investigation equations from analytical solution and numerical solution with increasing flow rate test Increasing flow rate test q 1 =100 stb/day q 2 =150 stb/day r iD criteria (I) for Δp D =0.1095 r iD criteria (II) for Δp D =0.01095 r iD criteria (III) for Δp D =0.001095 Analytical solution r iD 2 = 4.101t D R 2 =0.9991 r iD 2 = 10.411t D R 2 =0.9999 r iD 2 = 17.819t D R 2 =0.9999 Numerical solution r iD 2 = 4.082t D R 2 =0.9991 r iD 2 = 10.381t D R 2 =0.9999 r iD 2 = 17.804t D R 2 =0.9999 Note: Radius coefficient(α) increase slightly for smaller ΔpD

33. 33 (b) Decreasing flow rate (two-rates) test

34. 34 Radius of investigation equations from analytical solution and numerical solution with decreasing flow rate test Decreasing flow rate test q 1 =100 stb/day q 2 = 50 stb/day r iD criteria (I) for Δp D =0.1095 r iD criteria (II) for Δp D =0.01095 r iD criteria (III) for Δp D =0.001095 Analytical solution r iD 2 = 3.887t D R 2 =0.9983 r iD 2 = 10.374t D R 2 =0.9999 r iD 2 = 17.814t D R 2 =0.9999 Numerical solution r iD 2 = 3.868t D R 2 =0.9983 r iD 2 = 10.344t D R 2 =0.9999 r iD 2 = 17.799t D R 2 =0.9999 Note: Radius coefficient(α) decrease slightly for smaller ΔpD

35. 35 (c) Middle flow rate increasing test

36. 36 Radius of investigation equations from analytical solution and numerical solution with middle flow rate increasing test Middle flow rate increasing test: q 1 =100 stb/day q 2 =150 stb/day q 3 =100 stb/day r iD criteria (I) for Δp D =0.1095 r iD criteria (II) for Δp D =0.01095 r iD criteria (III) for Δp D =0.001095 Analytical solution r iD 2 = 4.246t D R 2 =0.9981 r iD 2 = 10.494t D R 2 =0.9999 r iD 2 = 17.849t D R 2 =0.9999 Numerical solution r iD 2 = 4.227t D R 2 =0.9981 r iD 2 = 10.464t D R 2 =0.9999 r iD 2 = 17.833t D R 2 =0.9999 Note: Radius coefficient(α) increase for smaller ΔpD

37. 37 (d) Middle flow rate decreasing test

38. 38 Radius of investigation equations from analytical solution and numerical solution with middle flow rate decreasing test Middle flow rate decreasing test: q 1 =100 stb/day q 2 = 50 stb/day q 3 =100 stb/day r iD criteria (I) for Δp D =0.1095 r iD criteria (II) for Δp D =0.01095 r iD criteria (III) for Δp D =0.001095 Analytical solution r iD 2 = 3.698t D R 2 =0.9956 r iD 2 = 10.280t D R 2 =0.9998 r iD 2 = 17.782t D R 2 =0.9999 Numerical solution r iD 2 = 3.679t D R 2 =0.9956 r iD 2 = 10.249t D R 2 =0.9998 r iD 2 = 17.767t D R 2 =0.9999 Note: Radius coefficient(α) decrease for smaller ΔpD

39. 39 The results of the dimensionless radius of investigation at the criterion Δp D = 0.1095 Note: Radius coefficient(α) is affected by rate changes for larger ΔpD

40. 40 The results of the dimensionless radius of investigation at the criterion Δp D = 0.01095 Note: Radius coefficient(α) is slightly affected by rate changes for small ΔpD

41. 41 The results of the dimensionless radius of investigation at the criterion Δp D = 0.001095 Note: Radius coefficient(α) is very slightly affected by rate changes for smaller ΔpD

42. 42 Results and Discussions (3) Effect of skin factor

43. 43 The effect of skin factor to the radius coefficients in simulation studies (constant flow rate test) α = 17.799 (s=0, 2, 5, 8, 10 for Δp D =1.095*10 -3 ) α = 10.363 (s=0, 2, 5, 8, 10 for Δp D =1.095*10 -2 ) α = 3.986 (s=0, 2, 5, 8, 10 for Δp D =1.095*10 -1 ) The radius coefficient (α) is independent of skin factor

44. 44 Results and Discussions (4) Effect of wellbore storage volume

45. 45 The effect of wellbore storage volume (constant flow rate test) α = 17.799 (C D =10 2, 10 3, 10 4, 10 5 for Δp D =1.095*10 -3 ) α = 10.363 (C D =10 2, 10 3, 10 4, 10 5 for Δp D =1.095*10 -2 ) α = 3.986 (C D =10 2, 10 3, 10 4, 10 5 for Δp D =1.095*10 -1 ) Note: Radius coefficient(α) is independent of wellbore storage volume in late time

46. 46 The effect of wellbore storage volume (constant flow rate test) c D =10 3 c D =0 c D =10 5 c D =10 4 c D =10 2 Δp D =0.1095 Note: Radius coefficient(α) is affected by wellbore storage volume in early time

47. 47 Which criteria for defining pressure front is suitable in conjunction with pressure behavior affected by bounded reservoir?

48. 48 Pressure response for a bounded reservoir P Dwf Log (t D ) Infinite reservoir pressure response Bounded reservoir pressure response Dimensionless boundary affecting time, t D * Deviated point rere

49. 49 Boundary affecting time equation From radius of investigation equation, such as –When pressure front reaches boundary then back to the wellbore, i.e., pressure front propagates two-times of external boundary radius ( r iD = 2r eD ), is applied rere (in terms of wellbore radius, r w ) r iD 2 = α t D

50. 50 (a) bounded circular reservoir with r eD =3000 No-flow boundary r e = 1050 ft r w = 0.35 ft

51. 51 Boundary affecting time estimated from radius of investigation equation for the bounded circular reservoir with r eD = 3000 ( I ) ( II ) ( III ) The visually deviated point from type curve analysis

52. 52 (b) Bounded circular reservoir with r eD =1000 No-flow boundary r e = 350 ft r w = 0.35 ft

53. 53 Boundary affecting time estimated from radius of investigation equation for the bounded circular reservoir with r eD = 1000 ( I ) ( II ) ( III ) The visually deviated point from type curve analysis

54. 54 Discussions of radius of investigation equation The study of radius of investigation in an infinite reservoir –Using different criterion Δp D defined at the pressure front, we obtained different radius coefficients (α) which vary from 4 to 71.15 for the pressure front varied from 0.1095 to 10 -9, respectively. The study of boundary effect time in a bounded reservoir –The results of boundary effect time from the radius of investigation equation with radius coefficient (α) of 17.82 (i.e., r iD 2 =17.82t D for Δp D = 0.001095) are consistent with those from the deviated point of pressure type curves of the infinite and bounded reservoirs. The radius of investigation equation should be r iD 2 =17.82t D, where the radius coefficient (α) is 17.82.

55. 55 Conclusions The relationship between the square of the dimensionless radius of investigation and the dimensionless time is linear (r iD 2 = αt D ) for a constant flow rate but not necessarily linear for variable flow rates The radius coefficient (α) in constant flow rate cases varies from 4 to 71.15 as the defined dimensionless pressure at the pressure front of the radius of investigation is varies from 0.1095 to 10 -9, respectively

56. 56 Conclusions (Cont.) The radius of investigation equation is independent of skin factor. The wellbore storage effect affects the propagation of the radius of investigation only at an early time and depending on the size of wellbore storage volume. The radius coefficient (α) is 17.82 and should be used in the equation of radius of investigation.

57. 57 Thank you for your attention

58. 58