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- 1. Reservoir Engineering 1 Course (1st Ed.)
- 2. 1. Future IPR Approximation 2. Generating IPR for Oil Wells A. Wiggins’ Method B. Standing’s Method C. Fetkovich’s Method 3. Horizontal Oil Well Performance 4. Horizontal Well Productivity
- 3. 1. Vertical Gas Well Performance 2. Pressure Application Regions 3. Turbulent Flow in Gas Wells A. Simplified Treatment Approach B. Laminar-Inertial-Turbulent (LIT) Approach (Cases A. & B.)
- 4. IPR for Gas Wells Determination of the flow capacity of a gas well requires a relationship between the inflow gas rate and the sand-face pressure or flowing bottom-hole pressure. This inflow performance relationship may be established by the proper solution of Darcy’s equation. Solution of Darcy’s Law depends on the conditions of the flow existing in the reservoir or the flow regime. 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 5
- 5. Gas Reservoir Flow Regimes When a gas well is first produced after being shutin for a period of time, the gas flow in the reservoir follows an unsteady-state behavior until the pressure drops at the drainage boundary of the well. Then the flow behavior passes through a short transition period, after which it attains a steady state or semisteady (pseudosteady)-state condition. The objective of this lecture is to describe the empirical as well as analytical expressions that can be used to establish the inflow performance relationships under the pseudosteady-state flow condition. 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 6
- 6. Exact Solution of Darcy’s Equation for Compressible Fluids under PSS The exact solution to the differential form of Darcy’s equation for compressible fluids under the pseudosteady-state flow condition was given previously by: 2013 H. AlamiNia Where Qg = gas flow rate, Mscf/day k = permeability, md ψ–r = average reservoir real gas pseudopressure, psi2/cp T = temperature, °R s = skin factor h = thickness re = drainage radius rw = wellbore radius Reservoir Engineering 1 Course: Gas Well Performance 7
- 7. Productivity Index for a Gas Well The productivity index J for a gas well can be written analogous to that for oil wells as: With the absolute open flow potential (AOF), i.e., maximum gas flow rate (Qg)max, as calculated by: Where J = productivity index, Mscf/day/psi2/cp 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 8
- 8. Steady-State Gas Well Flow In a linear relationship as: Above Equation indicates that a plot of ψwf vs. Qg would produce a straight line with a slope of (1/J) and intercept of ψ–r, as shown in next slide. If two different stabilized flow rates are available, the line can be extrapolated and the slope is determined to estimate AOF, J, and ψ–r. 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 9
- 9. Steady-State Gas Well Flow (Cont.) 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 10
- 10. Darcy’s Equation for Compressible Fluids under PSS Regime Darcy’s equation for compressible fluids under the PSS regime can be alternatively written in the following integral form: Note that (p/μg z) is directly proportional to (1/μg Bg) where Bg is the gas formation volume factor and defined as: 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 11
- 11. Typical Plot of the Gas Pressure Functions vs. P Figure shows a typical plot of the gas pressure functions (2p/μgz) and (1/μg Bg) versus pressure. The integral in previous equations represents the area under the curve between p–r and pwf. Gas PVT data 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 13
- 12. Pressure Regions As illustrated in Figure, the pressure function exhibits the following three distinct pressure application regions: Region III. High-Pressure Region Region II. Intermediate-Pressure Region Region I. Low-Pressure Region 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 14
- 13. Region III. High-Pressure Region When both pwf and p–r are higher than 3000 psi, the pressure functions (2p/μgz) and (1/μg Bg) are nearly constants. This observation suggests that the pressure term (1/μg Bg) in Equation can be treated as a constant and removed outside the integral, to give the following approximation to Equation: Where Qg = gas flow rate, Mscf/day Bg = gas formation volume factor, bbl/scf k = permeability, md 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 15
- 14. Region III. High-Pressure Region, P Method The gas viscosity μg and formation volume factor Bg should be evaluated at the average pressure pavg as given by: The method of determining the gas flow rate by using below Equation commonly called the pressure-approximation method. It should be pointed out the concept of the productivity index J cannot be introduced into above Equation since it is only valid for applications when both pwf and p–r are above 3000 psi. 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 16
- 15. Region II. Intermediate-Pressure Region Between 2000 and 3000 psi, the pressure function shows distinct curvature. When the bottom-hole flowing pressure and average reservoir pressure are both between 2000 and 3000 psi, the pseudopressure gas pressure approach (i.e., below Equation) should be used to calculate the gas flow rate. 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 17
- 16. Region I. Low -Pressure Region, P2 Method At low pressures, usually less than 2000 psi, the pressure functions (2p/μgz) and (1/μg Bg) exhibit a linear relationship with pressure. Golan and Whitson (1986) indicated that the product (μgz) is essentially constant when evaluating any pressure below 2000 psi. 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 18
- 17. Region I. Low -Pressure Region, P2 Method (Cont.) Implementing above observation gives (pressure-squared approximation method): Where Qg = gas flow rate, Mscf/day k = permeability, md T = temperature, °R z = gas compressibility factor μg = gas viscosity, cp It is recommended that the z-factor and gas viscosity be evaluated at the average pressure pavg as defined by: 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 19
- 18. Region I. J Calculation If both p–r and pwf are lower than 2000 psi, the equation can be expressed in terms of the productivity index J as: With Where 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 20
- 19. Laminar Vs. Turbulent Flow All of the mathematical formulations presented thus far in this lecture are based on the assumption that laminar (viscous) flow conditions are observed during the gas flow. During radial flow, the flow velocity increases as the wellbore is approached. This increase of the gas velocity might cause the development of a turbulent flow around the wellbore. If turbulent flow does exist, it causes an additional pressure drop similar to that caused by the mechanical skin effect. 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 22
- 20. PSS Equations Modification (Turbulent Flow) As presented earlier, the semisteady-state flow equation for compressible fluids can be modified to account for the additional pressure drop due the turbulent flow by including the rate-dependent skin factor DQg. The resulting pseudosteady-state equations are given in the following three forms: 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 23
- 21. PSS Equations Modification (Turbulent Flow) (Cont.) First Form: Pressure-Squared Approximation Form Second Form: Pressure-Approximation Form Third Form: Real Gas Potential (Pseudopressure) Form 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 24
- 22. Empirical Treatments to Represent the Turbulent Flow in Gas Wells The PSS equations, which were given previously in three forms, are essentially quadratic relationships in Qg and, thus, they do not represent explicit expressions for calculating the gas flow rate. Two separate empirical treatments can be used to represent the turbulent flow problem in gas wells. 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 25
- 23. Empirical Treatments to Represent the Turbulent Flow in Gas Wells (Cont.) Both treatments, with varying degrees of approximation, are directly derived and formulated from the three forms of the pseudosteady-state equations. These two treatments are called: Simplified treatment approach Laminar-inertial-turbulent (LIT) treatment 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 26
- 24. The Simplified Treatment Approach Based on the analysis for flow data obtained from a large member of gas wells, Rawlins and Schellhardt (1936) postulated that the relationship between the gas flow rate and pressure can be expressed as: Where Qg = gas flow rate, Mscf/day p –r = average reservoir pressure, psi n = exponent C = performance coefficient, Mscf/day/psi2 The exponent n is intended to account for the additional pressure drop caused by the high-velocity gas flow, i.e., turbulence. Depending on the flowing conditions, the exponent n may vary from 1.0 for completely laminar flow to 0.5 for fully turbulent flow. 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 29
- 25. Deliverability or Back-Pressure Equation The performance coefficient C in the equation is included to account for: Reservoir rock properties Fluid properties Reservoir flow geometry The Equation is commonly called the deliverability or back-pressure equation. If the coefficients of the equation (i.e., n and C) can be determined, the gas flow rate Qg at any bottom-hole flow pressure pwf can be calculated and the IPR curve constructed. 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 30
- 26. Deliverability or Back-Pressure Equation (Logarithmic Form) Taking the logarithm of both sides of the Equation gives: This equation suggests that a plot of Qg versus (p–r2 − p2wf) on log-log scales should yield a straight line having a slope of n. In the natural gas industry the plot is traditionally reversed by plotting (p–r2 − p2wf) versus Qg on the logarithmic scales to produce a straight line with a slope of (1/n). 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 31
- 27. Well Deliverability Graph or the Back-Pressure Plot This plot as shown schematically in Figure is commonly referred to as the deliverability graph or the backpressure plot. 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 32
- 28. Calculation of N & C The deliverability exponent n can be determined from any two points on the straight line, i.e., (Qg1, Δp12) and (Qg2, Δp22), according to the flowing expression: Given n, any point on the straight line can be used to compute the performance coefficient C from: 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 33
- 29. Gas Well Testing The coefficients of the back-pressure equation or any of the other empirical equations are traditionally determined from analyzing gas well testing data. Deliverability testing has been used for more than sixty years by the petroleum industry to characterize and determine the flow potential of gas wells. There are essentially three types of deliverability tests and these are: Conventional deliverability (back-pressure) test Isochronal test Modified isochronal test 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 34
- 30. Deliverability Testing These tests basically consist of flowing wells at multiple rates and measuring the bottom-hole flowing pressure as a function of time. When the recorded data are properly analyzed, it is possible to determine the flow potential and establish the inflow performance relationships of the gas well. The deliverability test is out of scope of this course and would be discussed later in well test course. 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 35
- 31. The Laminar-Inertial-Turbulent (LIT) Approach The three forms of the semisteady-state equation as presented earlier in this lecture can be rearranged in quadratic forms for separating the laminar and inertial-turbulent terms composing these equations as follows: 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 37
- 32. Case A. Pressure-Squared Quadratic Form a. Pressure-Squared Quadratic Form With 2013 H. AlamiNia Where a = laminar flow coefficient b = inertial-turbulent flow coefficient Qg = gas flow rate, Mscf/day z = gas deviation factor k = permeability, md μg = gas viscosity, cp The term (a Qg) in represents the pressuresquared drop due to laminar flow while the term (b Q2g) accounts for the pressure squared drop due to inertial-turbulent flow effects. Reservoir Engineering 1 Course: Gas Well Performance 38
- 33. Case A. Graph of the Pressure-Squared Data Above equation can be linearized by dividing both sides of the equation by Qg to yield: The coefficients a and b can be determined by plotting ((p–r^2-pwf^2)/Qg) versus Qg on a Cartesian scale and should yield a straight line with a slope of b and intercept of a. Data from deliverability tests can be used to construct the linear relationship. 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 39
- 34. Graph of the pressure-squared data 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 40
- 35. Case A. Current IPR of the Gas Well Given the values of a and b, the quadratic flow equation, can be solved for Qg at any pwf from: Furthermore, by assuming various values of pwf and calculating the corresponding Qg from above Equation, The current IPR of the gas well at the current reservoir pressure p–r can be generated. 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 41
- 36. Case A. Pressure-Squared Quadratic Form Assumptions It should be pointed out the following assumptions were made in developing following Equation: Single phase flow in the reservoir Homogeneous and isotropic reservoir system Permeability is independent of pressure The product of the gas viscosity and compressibility factor, i.e., (μg z) is constant. This method is recommended for applications at pressures below 2000 psi. 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 42
- 37. Case B. Pressure-Quadratic Form The pressure-approximation equation, i.e., can be rearranged and expressed in the following quadratic form. The term (a1 Qg) represents the pressure drop due to laminar flow, while the term (b1 Q2 g) accounts for the additional pressure drop due to the turbulent flow condition. In a linear form, the equation can be expressed as: 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 43
- 38. Case B. Graph of the Pressure-Method Data The laminar flow coefficient a1 and inertialturbulent flow coefficient b1 can be determined from the linear plot of the equation as shown in Figure. 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 44
- 39. Case B. Gas Flow Rate Determination Having determined the coefficient a1 and b1, the gas flow rate can be determined at any pressure from: The application of following Equation is also restricted by the assumptions listed for the pressure-squared approach. However, the pressure method is applicable at pressures higher than 3000 psi. 2013 H. AlamiNia Reservoir Engineering 1 Course: Gas Well Performance 45
- 40. 1. Ahmed, T. (2006). Reservoir engineering handbook (Gulf Professional Publishing). Ch8
- 41. 1. Turbulent Flow in Gas Wells: LIT Approach (Case C) 2. Comparison of Different IPR Calculation Methods 3. Future IPR for Gas Wells 4. Horizontal Gas Well Performance 5. Primary Recovery Mechanisms 6. Basic Driving Mechanisms

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