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- 1. Reservoir Engineering 1 Course (2nd Ed.)
- 2. 1. PSS Regime A. B. C. D. Average Reservoir Pressure PSS regime for Radial Flow of SC Fluids Effect of Well Location within the Drainage Area PSS Regime for Radial Flow of C Fluids 2. Skin Concept 3. Using S for Radial Flow in Flow Equations 4. Turbulent Flow
- 3. 1. Superposition A. Multiple Well B. Multi Rate C. Reservoir Boundary 2. Productivity Index (PI) 3. Inflow Performance Relationship (IPR)
- 4. Flash Back: Solutions to the Radial Diffusivity Equation The solutions to the radial diffusivity equation appear to be applicable only for describing the pressure distribution in an infinite reservoir that was caused by a constant production from a single well. Since real reservoir systems usually have several wells that are operating at varying rates, a more generalized approach is needed to study the fluid flow behavior during the unsteady state flow period. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 5
- 5. Superposition Theorem The principle of superposition is a powerful concept that can be applied to remove the restrictions that have been imposed on various forms of solution to the transient flow equation. Mathematically the superposition theorem states that any sum of individual solutions to the diffusivity equation is also a solution to that equation. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 6
- 6. Superposition Concept Applications Superposition concept can be applied to account for the following effects on the transient flow solution: Effects of multiple wells Effects of rate change Effects of the boundary Effects of pressure change Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 7
- 7. Effects of Multiple Wells Frequently, it is desired to account for the effects of more than one well on the pressure at some point in the reservoir. The superposition concept states that the total pressure drop at any point in the reservoir is the sum of the pressure changes at that point caused by flow in each of the wells in the reservoir. In other words, we simply superimpose one effect upon the other. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 8
- 8. Appling Superposition: Effects of Multiple Wells Figure shows three wells that are producing at different flow rates from an infinite acting reservoir, i.e., unsteady-state flow reservoir. The principle of superposition shows that the total pressure drop observed at any well, e.g., Well 1, is: Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 9
- 9. Appling Superposition: Effects of Multiple Wells (Cont.) The pressure drop at Well 1 due to its own production is given by the log-approximation to the Ei-function solution presented by: (Qo1=oil flow rate from well 1) The pressure drop at Well 1 due to production at Wells 2 and 3 must be written in terms of the Ei-function solution. The log-approximation cannot be used because we are calculating the pressure at a large distance r from the well, i.e., the argument x > 0.01, or: Fall 13 H. AlamiNia It should also be noted that if the point of interest is an operating well, the skin factor s must be included for that well only. Reservoir Engineering 1 Course (2nd Ed.) 10
- 10. Effects of Rate Change All of the mathematical expressions presented previously require that the wells produce at a constant rate during the transient flow periods. Practically all wells produce at varying rates and, therefore, it is important that we be able to predict the pressure behavior when the rate changes. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 12
- 11. Superposition: Effects of Rate Change For predicting the pressure behavior when the rate changes, the concept of superposition states: “Every flow rate change in a well will result in a pressure response which is independent of the pressure responses caused by other previous rate changes.” Accordingly, the total pressure drop that has occurred at any time is the summation of pressure changes caused separately by each net flow rate change. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 13
- 12. Production and Pressure History of a Multi-Rate Well Consider the case of a shutin well, i.e., Q = 0, that was then allowed to produce at a series of constant rates for the different time periods shown in Figure. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 14
- 13. Pressure Drop of Multi-Rate Well To calculate the total pressure drop at the sand face at time t4, the composite solution is obtained by adding the individual constant-rate solutions at the specified rate-time sequence, or: The above expression indicates that there are four contributions to the total pressure drop resulting from the four individual flow rates. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 15
- 14. Pressure Drop of Multi-Rate Well: 1st Contribution The first contribution results from increasing the rate from 0 to Q1 and is in effect over the entire time period t4, thus: It is essential to notice the change in the rate, i.e., (new rate − old rate), that is used in the above equation. It is the change in the rate that causes the pressure disturbance. Further, it should be noted that the “time” in the equation represents the total elapsed time since the change in the rate has been in effect. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 16
- 15. Pressure Drop of Multi-Rate Well: Other Contributions Second contribution results from decreasing the rate from Q1 to Q2 at t1, thus: Note, however, the above approach is valid only if the well is flowing under the unsteady-state flow condition for the total time elapsed since the well began to flow at its initial rate. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 17
- 16. Effects of the Boundary The superposition theorem can also be extended to predict the pressure of a well in a bounded reservoir. Figure, which shows a well that is located at distance r from the nonflow boundary, e.g., sealing fault. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 19
- 17. Method of Images in Solving Boundary Problems The no-flow boundary can be represented by the following pressure gradient expression: Mathematically, the above boundary condition can be met by placing an image well, identical to that of the actual well, on the other side of the fault at exactly distance r. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 20
- 18. Method of Images Consequently, the effect of the boundary on the pressure behavior of a well would be the same as the effect from an image well located a distance 2r from the actual well. In accounting for the boundary effects, the superposition method is frequently called the method of images. Thus, for a well that is located at distance r from the non-flow boundary, the problem reduces to one of determining the effect of the image well on the actual well. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 21
- 19. Method of Images (Cont.) The total pressure drop at the actual well will be the pressure drop due to its own production plus the additional pressure drop caused by an identical well at a distance of 2r, or: Notice that this equation assumes the reservoir is infinite except for the indicated boundary. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 22
- 20. Extension of the Image Wells Concept The effect of boundaries is always to cause greater pressure drop than those calculated for infinite reservoirs. The concept of image wells can be extended to generate the pressure behavior of a well located within a variety of boundary configurations. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 23
- 21. Effects of Pressure Change Superposition is also used in applying the constantpressure case. Pressure changes are accounted for in this solution in much the same way that rate changes are accounted for in the constant rate case. The superposition method to account for the pressure-change effect is used in the Water Influx. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 24
- 22. Transient Well Testing Detailed reservoir information is essential to the petroleum engineer in order to analyze the current behavior and future performance of the reservoir. Pressure transient testing is designed to provide the engineer with a quantitative analysis of the reservoir properties. A transient test is essentially conducted by creating a pressure disturbance in the reservoir and recording the pressure response at the wellbore, i.e., bottom-hole flowing pressure pwf, as a function of time. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 25
- 23. Pressure Transient Tests The pressure transient tests most commonly used in the petroleum industry include: Pressure drawdown Pressure buildup Multirate Interference Pulse Drill stem Fall off Injectivity Step rate Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 26
- 24. Information Available From a Well Test It has long been recognized that the pressure behavior of a reservoir following a rate change directly reflects the geometry and flow properties of the reservoir. Information available from a well test includes: Effective permeability Formation damage or stimulation Flow barriers and fluid contacts Volumetric average reservoir pressure Drainage pore volume Detection, length, capacity of fractures Communication between wells Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 27
- 25. Well Performance These lectures presents the practical reservoir engineering equations that are designed to predict the performance of vertical and horizontal wells. Also describe some of the factors that are governing the flow of fluids from the formation to the wellbore and how these factors may affect the production performance of the well. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 30
- 26. Production Performance Analysis The analysis of the production performance is essentially based on the following fluid and well characteristics: Fluid PVT properties Relative permeability data Inflow-performance-relationship (IPR) Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 31
- 27. Productivity Index A commonly used measure of the ability of the well to produce is the Productivity Index. Defined by the symbol J, the productivity index is the ratio of the total liquid flow rate to the pressure drawdown. For a water-free oil production, the productivity index is given by: Fall 13 H. AlamiNia Where Qo = oil flow rate, STB/day J = productivity index, STB/day/psi p–r = volumetric average drainage area pressure (static pressure) pwf = bottom-hole flowing pressure Δp = drawdown, psi Reservoir Engineering 1 Course (2nd Ed.) 32
- 28. Productivity Index Measurement The productivity index is generally measured during a production test on the well. The well is shut-in until the static reservoir pressure is reached. The well is then allowed to produce at a constant flow rate of Q and a stabilized bottom-hole flow pressure of pwf. Since a stabilized pressure at surface does not necessarily indicate a stabilized pwf, the bottom-hole flowing pressure should be recorded continuously from the time the well is to flow. The productivity index is then calculated from: Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 33
- 29. Productivity Index Conditions It is important to note that the productivity index is a valid measure of the well productivity potential only if the well is flowing at pseudosteady-state conditions. Therefore, in order to accurately measure the productivity index of a well, it is essential that the well is allowed to flow at a constant flow rate for a sufficient amount of time to reach the pseudosteady-state. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 34
- 30. Productivity Index during Flow Regimes The figure indicates that during the transient flow period, the calculated values of the productivity index will vary depending upon the time at which the measurement s of pwf are made. Productivity index during flow regimes Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 35
- 31. Productivity Index Calculation The productivity index can be numerically calculated by recognizing that J must be defined in terms of semisteady-state flow conditions. Recalling below Equation: Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 36
- 32. Application of Productivity Index Since most of the well life is spent in a flow regime that is approximating the pseudosteady-state, the productivity index is a valuable methodology for predicting the future performance of wells. Further, by monitoring the productivity index during the life of a well, it is possible to determine if the well has become damaged due to completion, workover, production, injection operations, or mechanical problems. If a measured J has an unexpected decline, one of the indicated problems should be investigated. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 37
- 33. Specific Productivity Index A comparison of productivity indices of different wells in the same reservoir should also indicate some of the wells might have experienced unusual difficulties or damage during completion. Since the productivity indices may vary from well to well because of the variation in thickness of the reservoir, it is helpful to normalize the indices by dividing each by the thickness of the well. This is defined as the specific productivity index Js, or: Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 38
- 34. Qo vs. Δp Relationship Assuming that the well’s productivity index is constant: Where Δp = drawdown, psi J = productivity index The Equation indicates that the relationship between Qo and Δp is a straight line passing through the origin with a slope of J as shown in Figure. Qo vs. Δp relationship Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 40
- 35. Inflow Performance Relationship Alternatively, productivity Index Equation can be written as: The above expression shows that the plot pwf against Qo is a straight line with a slope of (−1/J) as shown schematically in Figure. This graphical representation of the relationship that exists between the oil flow rate and bottom-hole flowing pressure is called the inflow performance relationship and referred to as IPR. Qo STB/day Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 41
- 36. Features of the Straight-Line IPR Several important features of the straight-line IPR can be seen in Figure: When pwf equals average reservoir pressure, the flow rate is zero due to the absence of any pressure drawdown. Maximum rate of flow occurs when pwf is zero. This maximum rate is called absolute open flow and referred to as AOF. The slope of the straight line equals the reciprocal of the productivity index. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 42
- 37. Absolute Open Flow Although in practice AOF may not be a condition at which the well can produce, It is a useful definition that has widespread applications in the petroleum industry (e.g., comparing flow potential of different wells in the field). The AOF is then calculated by: Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 43
- 38. IPR For Below Pb (Qo=JΔP) suggests that the inflow into a well is directly proportional to the pressure drawdown and the constant of proportionality is the productivity index. Muskat and Evinger (1942) and Vogel (1968) observed that when the pressure drops below the bubblepoint pressure, the IPR deviates from that of the simple straight-line relationship as shown in Figure. IPR below pb Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 44
- 39. Pressure Dependent Variables Affecting PI Recalling following Equation: Treating the term between the two brackets as a constant c, the above equation can be written in the following form: Fall 13 H. AlamiNia Above equation reveals that the variables affecting the productivity index are essentially those that are pressure dependent, i.e.: Oil viscosity μo Oil formation volume factor Bo Relative permeability to oil kro Reservoir Engineering 1 Course (2nd Ed.) 45
- 40. Schematically Illustration of the Variables as a Function of P Effect of pressure on Bo, μo, and kro Fall 13 H. AlamiNia kro/μoBo as a function of pressure Reservoir Engineering 1 Course (2nd Ed.) 46
- 41. Behavior of Pressure Dependent Variables Above the bubble-point pressure pb The relative oil permeability kro equals unity (kro = 1) and the term (kro/μoBo) is almost constant. As the pressure declines below pb: The gas is released from solution, which can cause a large decrease in both kro and (kro/μoBo). Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 47
- 42. Effect of Reservoir Pressure on IPR Figure shows qualitatively the effect of reservoir depletion on the IPR. Effect of reservoir pressure on IPR Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 48
- 43. Empirical Methods to Predict NL Behavior of IPR Several empirical methods are designed to predict the non-linearity behavior of the IPR for solution gas drive reservoirs. Most of these methods require at least one stabilized flow test in which Qo and pwf are measured. All the methods include the following two computational steps: Using the stabilized flow test data, construct the IPR curve at the current average reservoir pressure p–r. Predict future inflow performance relationships as to the function of average reservoir pressures. Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 49
- 44. Empirical Methods to Generate IPR The following empirical methods that are designed to generate the current and future inflow performance relationships: Vogel’s Method Wiggins’ Method Standing’s Method Fetkovich’s Method The Klins-Clark Method Fall 13 H. AlamiNia Reservoir Engineering 1 Course (2nd Ed.) 50
- 45. 1. Ahmed, T. (2010). Reservoir engineering handbook (Gulf Professional Publishing). Chapter 6 and 7
- 46. 1. Generating IPR for Oil Wells A. Vogel’s Method B. Vogel’s Method (Undersaturated Reservoirs) a. Future IPR Approximation C. Wiggins’ Method D. Standing’s Method E. Fetkovich’s Method

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