高等油層工程 Advanced Reservoir Engineering

1. 高等油層工程Advanced Reservoir Engineering 開課班級: 碩博士班 (2007年秋季) 講授教師： 林再興

2. 目的 • 講授石油及天然氣流體性質，以及生產石油及天然氣所導致的油層壓力變化原理。 • 討論壓力測試分析(井壓測試分析) ，以及生產資料分析(生產遞減曲線分析) 。 • 求得地層參數/預測未來產生產率/計算地層的石油或天然的儲量及蘊藏量。

3. Textbooks and references • (A) Dake , L.P., Fundamentals of Reservoir Engineering, revised edition, Elsevier Scientific B.V., Amsterdam, the Netherlands, 2001. • (B) Ahmed, T., and McKinney, P., Advanced Reservoir Engineering, Gulf Publishing Company, Houston, Texas, 2004 • (B) Craft, B.C., and Hawkins, M.F. , Revised by Terry, R.E. , Applied Petroleum Reservoir Engineering, Second edition., Prentice Hall , Englewood Cliffs, New Jersey, 1991.

4. Textbooks and references • (C) Lee, J., Well Testing, SPE Textbook series, Society of Petroleum Engineers of AIME, Dallas, Texas, 2002. • (D) 林國安等人，石油探採 (第四冊 –油氣生產, Chapter 24 ) ，中國石油股份有限公司訓練教材叢書，中油訓練所，嘉義市， 2004. • (E) Journal papers

5. Advanced Reservoir EngineeringbyAhmed, T., and McKinney, P • Well testing analysis • Water influx • Unconventional gas reservoir • Performance of oil reservoir • Predicting oil reservoir • Introduction to oil fieldeconomics

6. 大 網 • Introduction to reservoir engineering - Gas reservoir - PVT analysis for oil - Material balance applied to oil • The flow equations of single-phase and two-phase flow of hydrocarbon in porous media - Darcy’s law and applications - The basic differential equation in a porous medium • Solutions to the flow equations of hydrocarbon in porous media - Steady and semi-steady states - Unsteady state • Pressure drawdown and buildup analysis for oil and gas wells • Decline curve analysis • Case study

7. Part 1Introduction to Reservoir Engineering • The primary functions of a reservoir engineer: • the estimation of hydrocarbon in place • the calculation of a recovery factor , and • the attachment of a time scale to the recovery • Note: pressure/flow rate information → parameters/future flow rate/future pressure

8. Outlines of Reservoir Engineering • (1) Introduction • Petrophysical properties ( Rock properties) • Fluid properties (gas, water, crude properties) • Calculations of hydrocarbon volumes • Fluid pressure regimes • (2) Gas reservoirs • Calculating gas in place by the volumetric method • Calculating gas recovery factor • Material balance calculation (Depletion & Water drive) • Hydrocarbon phase behavior (gas condensate phase behavior) • The gas equivalent of produced condensate and water • (3) PVT analysis for oil • Definition of the basic PVT parameters • Determination of the basic PVT parameters in the lab. And conversion for field operating conditions.

9. Outlines of Reservoir Engineering – cont. • (4) Material balance applied to oil reservoirs • General form of the material balance equation for a hydrocarbon reservoir (Undersaturated and Saturated reservoir) • Reservoir drive mechanisms • Solution gas drive • Gas cap drive • Natural water drive • (5) Darcy’s law and applications

10. Outlines of Reservoir Engineering – cont. • (6) The basic differential equation for radial flow in a porous medium • Derivation of the basic radial flow equation • Conditions of solution • Linearization of radial flow equation • (7) Well inflow equations for stabilized flow conditions • Semi steady state solution • Steady state solution • Generalized form of inflow equation (for semi steady state)

11. Outlines of Reservoir Engineering – cont. • (8) The constant terminal rate solution of the radial diffusivity equation and its application to oil well testing • Constant terminal rate solution • General Transient flow • Semi steady state flow • Superposition theorem; general theory of well testing • The Matthews, Brons, Hazebroek pressure buildup theory • Pressure buildup analysis techniques • Multi-rate drawdown testing • The effects of partial well completion • After-flow analysis

12. Outlines of Reservoir Engineering – cont. • (9) Gas well testing • - Linearization and solution of the basic differential equation for the radial flow of a real gas • - The Russell, Goodrich, et al. Solution technique • - The Al-Hussainy, Ramey, Crawford solution technique • - Pressure squared and pseudo pressure solution technique • - Non-Darcy flow & determination of the non-darcy coefficient • - The constant terminal rate solution for the flow of a real gas • - General theory of gas well testing • - Multi-rate testing of gas well • - Pressure building testing of gas wells • - Pressure building analysis in solution gas drive reservoirs

13. Outlines of Reservoir Engineering – cont. • (10) Natural water influx • - Steady state model • - Unsteady state model • - The van Everdingen and Hurst edge-water drive • model • - Bottom – water drive model • - Pseudo steady state model (Fetkovich model) • - Predicting the amount of water influx

14. Fluid Pressure Regimes The total pressure at any depth = weight of the formation rock + weight of fluids (oil, gas or water) [=] 1 psi/ft * depth(ft)

15. Fluid Pressure Regimes • Density of sandstone

16. Pressure gradient for sandstone • Pressure gradient for sandstone

17. Overburden pressure • Overburden pressure (OP) = Fluid pressure (FP) + Grain or matrix pressure (GP) • OP=FP + GP • In non-isolated reservoir PW (wellbore pressure) = FP • In isolated reservoir PW (wellbore pressure) = FP + GP’ where GP’<=GP

18. Normal hydrostatic pressure • In a perfectly normal case , the water pressure at any depth • Assume :(1) Continuity of water pressure to the surface • (2) Salinity of water does not vary with depth. • [=] psia • psi/ft for pure water • psi/ft for saline water

19. Abnormal hydrostatic pressure ( No continuity of water to the surface) • [=] psia • Normal hydrostatic pressure c = 0 • Abnormal (hydrostatic) pressure c > 0 → Overpressure (Abnormal high pressure) • c < 0 → Underpressure (Abnormal low pressure)

20. Conditions causing abnormal fluid pressures • Conditions causing abnormal fluid pressures in enclosed water bearing sands include • Temperature change ΔT = +1℉ → ΔP = +125 psi in a sealed fresh water system • Geological changes – uplifting; surface erosion • Osmosis between waters having different salinity, the sealing shale acting as the semi permeable membrane in this ionic exchange; if the water within the seal is more saline than the surrounding water the osmosis will cause the abnormal high pressure and vice versa.

21. Are the water bearing sands abnormally pressured ? • If so, what effect does this have on the extent of any hydrocarbon accumulations?

22. Hydrocarbon pressure regimes • In hydrocarbon pressure regimes • psi/ft • psi/ft • psi/ft

23. Pressure Kick • Assumes a normal hydrostatic pressure regime Pω= 0.45 × D + 15 • In water zone • at 5000 ft Pω(at5000) = 5000 × 0.45 + 15 = 2265 psia • at OWC (5500 ft) Pω(at OWC) = 5500 × 0.45 + 15 = 2490 psia

24. Pressure Kick • In oil zone Po = 0.35 x D + C • at D = 5500 ft , Po = 2490 psi • → C = 2490 – 0.35 × 5500 = 565 psia • → Po = 0.35 × D + 565 • at GOC (5200 ft) Po (at GOC) = 0.35 × 5200 + 565 = 2385 psia

25. Pressure Kick • In gas zone Pg = 0.08 D + 1969 (psia) • at 5000 ft Pg = 0.08 × 5000 + 1969 = 2369 psia

26. Pressure Kick • In gas zone Pg = 0.08 D + C • At D = 5500 ft, Pg = Pω = 2490 psia • 2490 = 0.08 × 5500 + C • C = 2050 psia • → Pg = 0.08 × D + 2050 • At D = 5000 ft • Pg = 2450 psia

27. GWC error from pressure measurement • Pressure = 2500 psia Pressure = 2450 psia • at D = 5000 ft at D = 5000 ft • in gas-water reservoir in gas-water reservoir • GWC = ? GWC = ? • Sol. Sol. • Pg = 0.08 D + C Pg = 0.08 D + C • C = 2500 – 0.08 × 5000 C = 2450 – 0.08 × 5000 • = 2100 psia = 2050 psia • → Pg = 0.08 D + 2100 → Pg = 0.08 D + 2050 • Water pressure Pω = 0.45 D + 15 Water pressure Pω = 0.45 D + 15 • At GWC Pg = Pω At GWC Pg = Pω • 0.08 D + 2100 = 0.45 D + 15 0.08 D + 2050 = 0.45 D + 15 • D = 5635 ft (GWC) D = 5500 ft (GWC)

28. Results from Errors in GWC or GOC or OWC • GWC or GOC or OWC location affecting volume of hydrocarbon OOIP affecting OOIP or OGIP affecting development plans

29. Volumetric Gas Reservoir Engineering • Gas is one of a few substances whose state, as defined by pressure, volume and temperature (PVT) • One other such substance is saturated steam.

30. The equation of state for an ideal gas (Field units used in the industry) p [=] psia; V[=] ft3; T [=] OR absolute temperature n [=] lbm moles; n=the number of lb moles, one lb mole is the molecular weight of the gas expressed in pounds. R = the universal gas constant [=] 10.732 psia∙ ft3 / (lbmmole∙0R) Eq (1.13) results form the combined efforts of Boyle, Charles, Avogadro and Gay Lussac.

31. The equation of state for real gas • The equation of Van der Waals(for one lb mole of gas • where a and b are dependent on the nature of the gas. • The principal drawback in attempting to use eq. (1.14) to describe the behavior of real gases encountered in reservoirs is that the maximum pressure for which the equation is applicable is still far below the normal range of reservoir pressures

32. The equation of state for real gas • the Beattie-Bridgeman equation • the Benedict-Webb-Rubin equation • the non-ideal gas law

33. Non-ideal gas law • Where z = z-factor =gas deviation factor • =supercompressibility factor

34. Determination of z-factor • There are three ways to determination z-factor : • (a)Experimental determination • (b)The z-factor correlation of standing and katz • (c)Direct calculation of z-factor

35. (a) Experimental determination • n mole s of gas • p=1atm; T=reservoir temperature; => V=V0 • pV=nzRT • z=1 for p=1 atm • =>14.7 V0=nRT • n mole of gas • p>1atm; T=reservoir temperature; => V=V • pV=nzRT • pV=z(14.7 V0) • By varying p and measuring V, the isothermal z(p) function can be • readily by obtained.

36. (b)The z-factor correlation of standing and katz • Requirement: • Knowledge of gas composition or gas gravity • Naturally occurring hydrocarbons: primarily paraffin series CnH2n+2 • Non-hydrocarbon impurities: CO2, N2 and H2 • Gas reservoir: lighter members of the paraffin series, C1 • and C2 > 90% of the volume.

37. The Standing-Katz Correlation • knowing Gas composition (ni) •  Critical pressure (Pci) • Critical temperature (Tci) of each component •  ( Table (1.1) and P.16 )  •  Pseudo critical pressure (Ppc) • Pseudo critical temperature (Tpc) for the mixture •  Pseudo reduced pressure (Ppr) • Pseudo reduced temperature (Tpr) •  Fig.1.6; p.17  z-factor

39. (b’)The z-factor correlation of standing and katz • For the gas composition is not available and the gas gravity (air=1) is available. • The gas gravity (air=1) • ( ) •  fig.1.7 , p18 • Pseudo critical pressure (Ppc) • Pseudo critical temperature (Tpc)

40. (b’)The z-factor correlation of standing and katz •  Pseudo reduced pressure (Ppr) • Pseudo reduced temperature (Tpr) •  Fig1.6 p.17 •  z-factor • The above procedure is valided only if impunity (CO2,N2 and H2S) is less then 5% volume.

41. (c) Direct calculation of z-factor • The Hall-Yarborough equations, developed using the Starling-Carnahan equation of state, are • where Ppr= the pseudo reduced pressure • t=1/Tpr Tpr=the pseudo reduced temperature • y=the “reduced” density which can be obtained as the solution of the equation as followed: This non-linear equation can be conveniently solved for y using the simple Newton-Raphson iterative technique.

42. (c) Direct calculation of z-factor • The steps involved in applying thus are: • make an initial estimate of yk, where k is an iteration counter (which in this case is unity, e.q. y1=0.001 • substitute this value in Eq. (1.21);unless the correct value of y has been initially selected, Eq. (1.21) will have some small, non-zero value Fk. • (3) using the first order Taylor series expansion, a better estimate of y can be determined as • where • (4) iterate, using eq. (1.21) and eq. (1.22), until satisfactory convergence is obtained(5) substitution of the correct value of y in eq.(1.20)will give the z-factor. • (5) substitution of the correct value of y in eq.(1.20)will give the z-factor.

43. Application of the real gas equation of state • Equation of state of a real gas • This is a PVT relationship to relate surface to reservoir volumes of hydrocarbon. • the gas expansion factor E, • Real gas equation for n moles of gas at standard conditions •  • Real gas equation for n moles of gas at reservoir conditions •  • > • > surface volume/reservoir volume • [=] SCF/ft3 or STB/bbl

44. Example • Reservoir condition: P=2000psia; T=1800F=(180+459.6)=639.60R; z=0.865 > surface volume/reservoir or SCF/ft3 or STB/bbl

45. (2) Real gas density • where n=moles; M=molecular weight) • at any p and T • For gas • For air

46. (2) Real gas density • At standard conditions zair = zgas = 1 • in general • (a) If is known, then or , (b) If the gas composition is known, then where

47. (3)Isothermal compressibility of a real gas since p.24, fig.1.9

49. Exercise 1.1 - Problem • Exercise1.1 Gas pressure gradient in the reservoir • (1) Calculate the density of the gas, at standard conditions, whose composition is listed in the table 1-1. • (2) what is the gas pressure gradient in the reservoir at 2000psia and 1800F(z=0.865)

50. Exercise 1.1 -- solution -1 • (1) Molecular weight of the gas since • or from • At standard condition