May 4, 2007

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

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

3. Introduction

4. Producing rate and flowing pressure at wellbore q=constant q (stb/day) r rw t (hours) t=0 t=ti p=pi Pwf (psi) t=ti t (hours)

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

6. Pressure distribution in reservoir at various times q=constant q (stb/day) t (hours) t=t1 t=t2 t=t3 ri 1 ri 2 ri 3 p=pi t=t1 t=t2 t=t3 P (psi) pressure front s r (ft)

7. Plane view of pressure fronts at various times t1 t2 r1 rw r2 t3 r3

8. Dimensionless Variables Dimensionless radius Dimensionless time Dimensionless pressure

9. Pressure distribution in a reservoir in terms of dimensionless variables PD pressure front s tD3 tD1 tD2 riD1 rD riD2 riD3

10. Radius of investigation (riD) and time (tD) • The relationship between the dimensionless radius of investigation (riD) and the dimensionless time (tD) is (Muskat, 1934; Tek et al., 1957; Jones, 1962; Van Poolen, 1964; Lee, 1982; Chandhry, 2004, etc.) riD2 = α tD where the radius coefficient (α) is a constant and varied in different studies, from 3.18 to 16

11. Literature on radius of investigation equation

12. Literature on Radius of Investigation (Cont.)

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. Basic theory and simulation tool

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. 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. Basic reservoir parameters used in this study

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. The pressure behavior check-- bounded reservoir

20. The pressure behavior check-- bounded reservoir

21. Results and Discussions

22. Definition of pressure front Δp1 Δp2 Δp3 ● ● ● p=pi α3 ● ● ● α2 α1 P (psi) t=ti △p= pressure drop defined at the pressure front α= radius coefficient r (ft)

23. Definition of pressure front △pD= the dimensionless pressure drop defined at the pressure front α= radius coefficient PD tD5 ΔpD1 ΔpD2 ΔpD3 rD α1 α2 α3

24. Pressure front and radius of investigation • From Ei solution such as • By defining or giving ΔpD (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 ΔpD).

25. Radius coefficients from analytical solution with constant flow rate in an infinite reservoir • By defining a small dimensionless pressure value (ΔpD) at the pressure front, the value of riD2/4tD in the Ei solution can be estimated.

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

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

28. Radius investigation equation from analytical and numerical solution -- constant flow rate case Different criteria for pressure front will obtain different radius coefficient (α) The smaller the ΔpD, the larger the radius coefficient (α), i.e., the faster the pressure front propagation.

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

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

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

32. Radius of investigation equations from analytical solution and numerical solution with increasing flow rate test Note: Radius coefficient(α) increase slightly for smaller ΔpD

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

34. Radius of investigation equations from analytical solution and numerical solution with decreasing flow rate test Note: Radius coefficient(α) decrease slightly for smaller ΔpD

35. (c) Middle flow rate increasing test

36. Radius of investigation equations from analytical solution and numerical solution with middle flow rate increasing test Note: Radius coefficient(α) increase for smaller ΔpD

37. (d) Middle flow rate decreasing test

38. Radius of investigation equations from analytical solution and numerical solution with middle flow rate decreasing test Note: Radius coefficient(α) decrease for smaller ΔpD

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

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

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

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

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 ΔpD=1.095*10-3) α = 10.363 (s=0, 2, 5, 8, 10 for ΔpD=1.095*10-2) α = 3.986 (s=0, 2, 5, 8, 10 for ΔpD=1.095*10-1) The radius coefficient (α) is independent of skin factor

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

45. The effect of wellbore storage volume (constant flow rate test) α = 17.799 (CD=102, 103, 104, 105 for ΔpD=1.095*10-3) α = 10.363 (CD=102, 103, 104, 105 for ΔpD=1.095*10-2) α = 3.986 (CD=102, 103, 104, 105 for ΔpD=1.095*10-1) Note: Radius coefficient(α) is independent of wellbore storage volume in late time

46. The effect of wellbore storage volume (constant flow rate test) ΔpD=0.1095 cD=0 cD=105 cD=104 cD=102 cD=103 Note: Radius coefficient(α) is affected by wellbore storage volume in early time

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

48. Pressure response for a bounded reservoir re Bounded reservoir pressure response Infinite reservoir pressure response PDwf Deviated point Dimensionless boundary affecting time, tD* Log (tD)

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 ( riD= 2reD), is applied riD2 = α tD re (in terms of wellbore radius, rw)

50. (a) bounded circular reservoir with reD=3000 re = 1050 ft rw = 0.35 ft No-flow boundary