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PEH:Inflow and Outflow Performance

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Petroleum Engineering Handbook

Larry W. Lake, Editor-in-Chief

Volume IV - Production Operations Engineering

Joe Dunn Clegg, Editor

Chapter 1 – Inflow and Outflow Performance

Michael L. Wiggins, U. of Oklahoma

Pgs. 1-40

ISBN 978-1-55563-118-5
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The Production System

Understanding the principles of fluid flow through the production system is important in estimating the performance of individual wells and optimizing well and reservoir productivity. In the most general sense, the production system is the system that transports reservoir fluids from the subsurface reservoir to the surface, processes and treats the fluids, and prepares the fluids for storage and transfer to a purchaser. Fig. 1.1 depicts the production system for a single well system. The basic elements of the production system include the reservoir; wellbore; tubular goods and associated equipment; surface wellhead, flowlines, and processing equipment; and artificial lift equipment.

The reservoir is the source of fluids for the production system. It is the porous, permeable media in which the reservoir fluids are stored and through which the fluids will flow to the wellbore. It also furnishes the primary energy for the production system. The wellbore serves as the conduit for access to the reservoir from the surface. It is composed of the drilled wellbore, which normally has been cemented and cased. The cased wellbore houses the tubing and associated subsurface production equipment, such as packers. The tubing serves as the primary conduit for fluid flow from the reservoir to the surface, although fluids also may be transported through the tubing-casing annulus.

The wellhead, flowlines, and processing equipment represent the surface mechanical equipment required to control and process reservoir fluids at the surface and prepare them for transfer to a purchaser. Surface mechanical equipment includes the wellhead equipment and associated valving, chokes, manifolds, flowlines, separators, treatment equipment, metering devices, and storage vessels.

In many cases, the reservoir is unable to furnish sufficient energy to produce fluids to the surface at economic rates throughout the life of the reservoir. When this occurs, artificial lift equipment is used to enhance production rates by adding energy to the production system. This component of the system is composed of both surface and subsurface elements. This additional energy can be furnished directly to the fluid through subsurface pumps, by reducing the backpressure at the reservoir with surface compression equipment to lower wellhead pressure, or by injecting gas into the production string to reduce the flowing gradient of the fluid.

Recognizing the various components of the production system and understanding their interaction generally leads to improved well productivity through analysis of the entire system. As the fluid flows from the reservoir into and through the production system, it experiences a continuous pressure drop (as Fig. 1.1 shows). The pressure begins at the average reservoir pressure and ends either at the pressure of the transfer line or near atmospheric pressure in the stock tank. In either case, a large pressure drop is experienced as the reservoir fluids are produced to the surface. It is the petroleum engineer’s responsibility to use this pressure reduction in an optimal manner. The pressure reduction depends on the production rate and, at the same time, the production rate depends on the pressure change. Understanding the relationship between pressure and production rate is important to predicting the performance of individual oil and gas wells.

To design a well completion or predict the production rate properly, a systematic approach is required to integrate the production system components. Systems analysis, which allows the petroleum engineer to both analyze production systems and design well completions, accomplishes this. This chapter focuses on the flow of reservoir fluids through the production system, particularly inflow performance, which is the reservoir pressure-rate behavior of the individual well, and outflow performance, which is the flow of reservoir fluids through the piping system.

Reservoir Inflow Performance

Mathematical models describing the flow of fluids through porous and permeable media can be developed by combining physical relationships for the conservation of mass with an equation of motion and an equation of state. This leads to the diffusivity equations, which are used in the petroleum industry to describe the flow of fluids through porous media.

The diffusivity equation can be written for any geometry, but radial flow geometry is the one of most interest to the petroleum engineer dealing with single well issues. The radial diffusivity equation for a slightly compressible liquid with a constant viscosity (an undersaturated oil or water) is


The solution for a real gas is often presented in two forms: traditional pressure-squared form and general pseudopressure form. The pressure-squared form is


and the pseudopressure form is


where the real gas pseudopressure is defined by Al-Hussainy, Ramey, and Crawford[2] as


The pseudopressure relationship is suitable for all pressure ranges, but the pressure-squared relationship has a limited range of applicability because of the compressible nature of the fluid. Strictly speaking, the only time the pressure-squared formulation is correct is when the μz product is constant as a function of pressure. This usually occurs only at low pressures (less than approximately 2,000 psia). As a result, it generally is recommended that the pseudopressure solutions be used in the analysis of gas well performance.

Single-Phase Analytical Solutions

Radial diffusivity equations can be solved for numerous initial and boundary conditions to describe the rate-pressure behavior for single-phase flow. Eqs. 1.1 through 1.3 have similar forms, which lends themselves to similar solutions in terms of pressure, pressure-squared, and pseudopressure. Of primary interest to the petroleum engineer is the constant terminal-rate solution for which the initial condition is an equilibrium reservoir pressure at some fixed time while the well is produced at a constant rate. The steady-state and semisteady-state flow conditions are the most common, though not exclusive, conditions for which solutions are desired in describing well performance.

The steady-state condition is for a well in which the outer boundary pressure remains constant. This implies an open outer boundary such that fluid entry will balance fluid withdrawals exactly. This condition may be appropriate when the pressure is being maintained because of active natural water influx or under active injection of fluid into the reservoir. The steady-state solution for single-phase liquid flow in terms of the average reservoir pressure can be written as


The semisteady-state condition is for a well that has produced long enough that the outer boundary has been felt. The well is considered to be producing with closed boundaries; therefore, there is no flow across the outer boundaries. In this manner, the reservoir pressure will decline with production and, at a constant production rate, pressure decline will be constant for all radii and times. This solution for single-phase liquid flow in terms of the average reservoir pressure is


The stabilized flow equations also can be developed for a real gas and are presented in pressure-squared and pseudopressure forms. For steady state, the solutions are




The semisteady-state solutions for gas are




Steady-state or semisteady-state conditions may never be achieved in actual operations. However, these stabilized conditions are often approximated in the reservoir and yield an acceptable estimate of well performance for single-phase flow. In addition, these solutions provide a means to compare production rates for various estimates of rock and fluid properties and well completion options. These relationships are useful as they allow the petroleum engineer the opportunity to estimate production rates before any well completion operations or testing.

Little difference is obtained in estimates of production rates or pressure drops when using the steady-state or semisteady-state solutions and, in practice, many engineers use the semisteady-state solutions. While each solution represents a distinctly different physical system, the numerical difference is minor when compared with the quality of the estimates used for rock and fluid properties, drainage area, and skin factor, as well as accounting for the heterogeneous nature of a reservoir. Dake, [3] Craft, Hawkins, and Terry, [4] and Lee and Wattenbarger[5] provide complete details regarding the development of the diffusivity equations and the associated stabilized-flow solutions.

Gas Well Performance

Early estimates of gas well performance were conducted by opening the well to the atmosphere and then measuring the flow rate. Such "open flow" practices were wasteful of gas, sometimes dangerous to personnel and equipment, and possibly damaging to the reservoir. They also provided limited information to estimate productive capacity under varying flow conditions. The idea, however, did leave the industry with the concept of absolute open flow (AOF). AOF is a common indicator of well productivity and refers to the maximum rate at which a well could flow against a theoretical atmospheric backpressure at the reservoir.

The productivity of a gas well is determined with deliverability testing. Deliverability tests provide information that is used to develop reservoir rate-pressure behavior for the well and generate an inflow performance curve or gas-backpressure curve. There are two basic relations currently in use to analyze deliverability test data. An empirical relationship was proposed by Rawlins and Schellhardt[6] in 1935 and is still frequently used today. Houpeurt[7] presented a theoretical deliverability relationship derived from the generalized radial diffusivity equation accounting for non-Darcy flow effects.

Rawlins and Schellhardt[6] developed the empirical backpressure method of testing gas wells based on the analysis of tests on more than 500 wells. They noted that when the difference between the squares of the average reservoir pressure and flowing bottomhole pressures were plotted against the corresponding flow rates on logarithmic coordinates, they obtained a straight line. This led them to propose the backpressure equation:


where C is the flow coefficient and n is the deliverability exponent. The deliverability exponent is the inverse of the slope of the curve. Once n is determined, C can be obtained by substituting pressure and rate data read directly from the straight-line plot into Eq. 1.11 and solving the resulting relation.

As discussed previously, solutions for gas well performance in terms of pressure-squared are appropriate only at low reservoir pressures. As a result, Rawlins and Schellhardt’s deliverability equation can be rewritten in terms of pseudopressure as


where C and n are determined in the same manner as for Eq. 1.11. The values of n range from 0.5 to 1.0, depending on flow characteristics. Flow characterized by Darcy’s equation will have a flow exponent of 1.0, while flow that exhibits non-Darcy flow behavior will have a flow exponent ranging from 0.5 to 1.0. While the Rawlins and Schellhardt deliverability equation is not rigorous, it is still widely used in deliverability analysis and has provided reasonable results for high-permeability gas wells over the years.

Eqs. 1.11 and 1.12 can be rewritten to facilitate the development of the inflow performance curve. In terms of pressure-squared, the relationship is




in terms of pseudopressure. Once the deliverability exponent is determined from a multirate test and the AOF estimated, Eqs. 1.13 and 1.14 can be applied readily to estimate the rate for a given flowing bottomhole pressure.

Houpeurt developed a theoretical deliverability relationship for stabilized flow with a Forchheimer[8] velocity term to account for non-Darcy flow effects in high-velocity gas production. The resulting relationship can be written in terms of pressure-squared or pseudopressure as




Eqs. 1.15 and 1.16 are quadratic in terms of the flow rate, and the solutions can be written for convenience as shown in Eqs. 1.17 and 1.18.



Jones, Blount, and Glaze[9] suggested Houpeurt’s relationship be rewritten as shown in Eqs. 1.19 and 1.20 to allow the analysis of well-test data to predict deliverability.



A plot of the difference in pressures squared divided by the flow rate or the difference in pseudopressure divided by the flow rate vs. the flow rate yields a straight line on a coordinate graph. The intercept of the plot is the laminar flow coefficient a, while turbulence coefficient b is obtained from the slope of the curve. Once these two coefficients have been determined, deliverability can be estimated from the following relationships in terms of pressure-squared or pseudopressure.




After the coefficients of the deliverability equations have been determined, the relationships can be used to estimate production rates for various bottomhole flowing pressures. This determination of rate vs. pressure is often referred to as the reservoir inflow performance, which is a measure of the ability of the reservoir to produce gas to the wellbore. The inflow performance curve is a plot of bottomhole pressure vs. production rate for a particular well determined from the gas well deliverability equations. Fig. 1.2 depicts a typical gas well inflow performance curve. This curve allows one to estimate the production rate for different flowing bottomhole pressures readily.

Deliverability Test Methods. Several different deliverability test methods have been developed to collect the data for use with the basic deliverability models. These tests can be grouped into three basic categories: tests that use all stabilized data, tests that use a combination of stabilized and transient data, and tests that use all transient data. The basic deliverability test method that uses all stabilized data is the flow-after-flow test. Deliverability test methods that use both transient and stabilized test data include the isochronal and modified isochronal tests. The multiple modified isochronal test consists of all transient test data and eliminates the need for stabilized flow or pressure data.

Flow-After-Flow Tests. Rawlins and Schellhardt[6] presented the basic deliverability test method that uses all stabilized data. The test consists of a series of flow rates. The test is often referred to as a four-point test because many tests are composed of four rates, as required by various regulatory bodies. This test is performed by producing the well at a series of stabilized flow rates and obtaining the corresponding stabilized flowing bottomhole pressures. In addition, a stabilized shut-in bottomhole pressure is required for the analysis. A major limitation of this test method is the length of time required to obtain stabilized data for low-permeability gas reservoirs.
Example 1.1
Table 1.1 provides example flow-after-flow test data, which are analyzed with the Rawlins and Schellhardt and Houpeurt deliverability equations. The traditional Rawlins and Schellhardt analysis requires that the difference in the pressures squared be plotted vs. the flow rate on logarithmic graph paper and a best-fit straight line constructed through the data points. The data should provide a straight-line plot, which serves as the deliverability curve. From this plot, the deliverability exponent, n, is the inverse of the slope of the constructed straight line. Once the deliverability exponent is determined, the flow coefficient, C, can be determined from Eq. 1.11 with a point taken from the straight-line plot. The same approach is used when pseudopressures are used to analyze the data, except that the differences in the pseudopressures are plotted vs. the flow rate and Eq. 1.12 is used to determine C.

Table 1.2 shows the data to be plotted for the Rawlins and Schellhardt analysis, while Figs. 1.3 and 1.4 show the logarithmic plots for the pressure-squared and the pseudopressure analyses, respectively.

Working with the traditional pressure-squared data, draw a straight line through the four data points to yield a slope of 1.54. The deliverability exponent, n, is the inverse of the slope, or 0.651. The flow coefficient, C, can be determined from a point on the straight line. Since the third test point lies on the line, it can be used to determine C using Eq. 1.23 to yield 0.2874 Mscf/D/psia2n.


Once n and C are determined, the deliverability equation can be written and used to determine the AOF and the production rate for any given flowing bottomhole pressure. Eq. 1.24 is the deliverability equation for this particular example well.


The AOF is determined by allowing the flowing bottomhole pressure to be equal to the atmospheric pressure for the current average reservoir pressure of 3,360 psia. In this example, when the atmospheric pressure is assumed to be 14.65 psia, the AOF is 11,200 Mscf/D.

The same approach is used to analyze the data when pseudopressures are used in the analysis. Using Fig. 1.4, the slope of the straight line through the data points is 1.57, yielding an n of 0.637. The flow coefficient, C, is determined to be 0.0269 Mscf/D/(psia2/cp)n from Eq. 1.25 using the third test point.


The resulting deliverability equation is


and the AOF is calculated to be 12,200 Mscf/D using the appropriate pseudopressure values at the current reservoir pressure of 3,360 psia and atmospheric pressure of 14.65 psia.

The difference in the calculated AOF using the pressure-squared approach and the pseudopressure method is noticeable. This variation results from the inclusion of the pressure dependence of the gas viscosity and gas deviation factor in the pseudopressure term. As noted earlier, the pressure-squared approach is only suitable at low pressures, while the pseudopressure method is good for all pressure ranges. Also, the Rawlins and Schellhardt method is not theoretically rigorous, although it is widely used.

The test data can also be analyzed with the Houpeurt approach using both the pressure-squared and pseudopressure approaches. Table 1.3 provides the data to be plotted in the Houpeurt analysis. Fig. 1.5 presents the Houpeurt plot of the pressure squared data, while Fig. 1.6 shows the pseuodpressure data. From Fig. 1.5, one can construct a best-fit line through the data points and determine the slope and the intercept of the line. The slope, b, is 0.0936 psia2/(Mscf/D)2, while the intercept, a, is determined to be 200 psia2/Mscf/D. These deliverability coefficients can be use to develop a deliverability equation after the form of Eq. 1.21 as shown in Eq. 1.27:


The AOF can be estimated for the reservoir pressure of 3,360 psia to be 9,970 Mscf/D.

A similar analysis can be undertaken for the pseudopressure data shown in Fig. 1.6. From this plot, the intercept of the constructed best-fit line is determined to be 10,252 psia2/cp/Mscf/D, while the slope is 5.69 psia2/cp/(Mscf/D)2. These coefficients are used to write the deliverability equation as


From this equation for the current reservoir pressure, the AOF is estimated to be 10,700 Mscf/D. As with the Rawlins and Schellhardt analysis, the AOFs determined by the pressure-squared method and the pseudopressure approach are different because of the pressure dependence of the μz product.

Isochronal Test. Cullendar[10] proposed the isochronal test to overcome the need to obtain a series of stabilized flow rates required for the flow-after-flow test for the slow-to-stabilize well. This test consists of producing the well at several different flow rates with flowing periods of equal duration. Each flow period is separated by a shut-in period in which the shut-in bottomhole pressure is allowed to stabilize at essentially the average reservoir pressure. The test also requires that an extended stabilized flow point be obtained. The test method is based on the principle that the radius of investigation is a function of the flow period and not the flow rate. Thus, for equal flow periods, the same drainage radius is investigated in spite of the actual flow rates.

To analyze the data from an isochronal test, the flow data from the equal flow periods is plotted according to the Rawlins and Schellhardt[6] or Houpeurt[7] methods. These data points are used to determine the slope of the deliverability curve. The stabilized flow point is then used to estimate the flow coefficient, C, for the Rawlins and Schellhardt method or the intercept, a, for the Houpert method by extending the slope of the multirate data to the stabilized flow point.

Example 1.2
Table 1.4 details isochronal test data for a particular well in which the flow periods are one hour in duration. The Rawlins and Schellhardt approach with pressures and the Houpeurt approach with pseudopressures are used to demonstrate the analysis of isochronal test data. Table 1.5 presents the plotting data for both methods. Fig. 1.7 shows the logarithmic plot of the pressure data for the Rawlins and Schellhardt analysis.

A straight line can be constructed through the three transient points to yield a slope of 1.076. The inverse of the slope defines the deliverability exponent, n, which is 0.9294 for this example. The slope through the transient points is extended to the stabilized flow point to depict the deliverability curve. The flow coefficient, C, is calculated from the stabilized flow point,


to be 0.0242 Mscf/D/psia2n. The flow exponent and flow coefficient are used to define the Rawlins and Schellhardt deliverability equation for this well,


which is used to determine the AOF. For an atmospheric pressure of 14.65 psia, the AOF is estimated to be 27,100 Mscf/D. A similar analysis can be undertaken with pseudopressures following the same method described for the pressures squared.

Applying the Houpeurt approach, the transient flow points are used to determine the slope of the best-fit straight line constructed through the data points. This slope is used to determine the intercept from the stabilized flow point. Fig. 1.8 shows the plot of the pseudopressure data for the Houpeurt analysis. From the plot, the slope is determined to be 0.1184 psia2/cp/(Mscf/D)2, which is used to calculate an intercept from the stabilized flow point of 8,814 psia2/cp/Mscf/D as shown in Eq. 1.31.


The deliverability equation can be written in a form similar to Eq. 1.22 to yield Eq. 1.32.


This equation can be used to estimate the AOF of 25,600 Mscf/D for the well or estimate the production rate at any other flowing bottomhole pressure. As the analysis of the flow-after-flow test data showed, the Rawlins and Schellhardt and Houpeurt methods yield different estimates of deliverability.

Modified Isochronal Test. For some low-permeability wells, the time required to obtain stabilized shut-in pressures may be impractical. To overcome this limitation, Katz et al. [11] proposed a modification to the isochronal test by requiring equal shut-in periods. The modified isochronal test is essentially the same as the isochronal test, except the shut-in periods separating the flow periods are equal to or longer than the flow periods. The method also requires the extended stabilized flow point and a stabilized shut-in bottomhole pressure. The modified isochronal test method is less accurate than the isochronal method because the shut-in pressure is not allowed to return to the average reservoir pressure. In the analysis of the collected data, the measured bottomhole pressure obtained just before the beginning of the flow period is used in Eqs. 1.11 and 1.12 or Eqs. 1.19 and 1.20 instead of the average reservoir pressure.

The analysis of the data is exactly the same as that used to analyze the isochronal test data. With the Rawlins and Schellhardt data, the transient flow points are used to construct a best-fit straight line through the data points. The inverse of the slope of this line yields the deliverability exponent, n. The deliverability exponent is then used with the data of the stabilized flow point to estimate the flow coefficient , C, with Eqs. 1.11 or 1.12 depending on whether pressure or pseudopressure data is used. In the Houpeurt analysis, a best-fit straight line is constructed through the transient flow points to yield the slope, b. Once the slope is determined, it is used with the stabilized flow point in the appropriate equation for pressure or pseudopressure (Eqs. 1.19 and 1.20) to determine the intercept, a. Once the flow coefficients are determined by either analysis method, the deliverability equation can be written and used to estimate the AOF and production rates for the well.

Transient Test Methods. The multiple modified isochronal test consists of all transient test data and eliminates the need for stabilized flow or pressure data. The analysis method requires estimates of drainage area and shape along with additional reservoir and fluid property data that are not required with the previous deliverability test methods. As a result, the analysis techniques are more complex than for flow-after-flow, isochronal, or modified isochronal test data. However, the method provides a means to estimate deliverability of slow-in-stabilizing wells and consists of running a minimum of three modified isochronal tests with each test composed of a minimum of three flow rates. To analyze the test data, modifications to the Rawlins and Schellhardt analysis have been proposed by Hinchman, Kazemi, and Poettmann[12] while modifications to the Houpeurt pressure-squared technique have proposed by Brar and Aziz, [13] Poettmann, [14] and Brar and Mattar. [15] These modifications have been extended to the pseudopressure analysis technique by Poe.[16] See the literature[12][13][14][15][16] for complete details on estimating deliverability from transient test data.

Future Performance Methods. The petroleum engineer is required to forecast or predict gas well performance as the reservoir pressure depletes. There are several methods to assist in making these future performance estimates, including the direct application of the appropriate analytical solution to provide estimates of rate vs. pressure for different average reservoir pressures. However, the use of Eqs. 1.7 through 1.10 requires that one estimate rock and fluid properties for the well of interest.

Another technique also requires knowledge of rock and fluid properties by estimating the flow coefficients, a and b, in Houpeurt’s relationships (Eqs. 1.17 and 1.18). When Houpeurt’s method is used in terms of pressure-squared, a and b are




where the non-Darcy flow coefficient


The value of β, the turbulence factor, [17] can be estimated from


When Houpeurt’s relationship is used in terms of pseudopressure, a and b are estimated from




The variables D and β are estimated with Eqs. 1.35 and 1.36. Once the flow coefficients, a and b, are determined at new average reservoir pressures, Eqs. 1.21 and 1.22 can be used to estimate rates for different pressures to generate the inflow performance curve.

Russell et al. [18] studied the depletion performance of gas wells and proposed a technique to estimate gas well performance that was dependent on gas compressibility and viscosity. From this study, Greene[19] presented a relationship to describe the well performance.


In this equation, C1 is a constant that is a function of permeability, reservoir thickness, and drainage area, which can be estimated from a single-point flow test with knowledge of gas compressibility and viscosity. This value is not the same as the flow coefficient C in Eqs. 1.11 and 1.12. C1 will remain constant during the life of the well, assuming no changes in permeability. Once C1 is determined, one can estimate future performance from Eq. 1.39 with the gas compressibility and viscosity estimated at the average bottomhole pressure defined as


A technique that does not require the use of rock and fluid properties assumes that the deliverability exponent, n, remains essentially constant during the life of the well. [20] While this assumption may not be accurate, many gas wells have exhibited behavior such that the deliverability exponent has varied slowly over the life of the well. Under this assumption, future performance can be predicted with the following relationships in terms of pressure-squared and pseudopressure, respectively.



Once the new AOF at the future reservoir pressure has been determined, the inflow performance curve can be constructed with a modified version of the deliverability equation as shown in Eqs. 1.13 and 1.14.

Oilwell Performance

When considering the performance of oil wells, it is often assumed that a well’s performance can be estimated by the productivity index. However, Evinger and Muskat[21] pointed out that, for multiphase flow, a curved relationship existed between flow rate and pressure and that the straight-line productivity index did not apply to multiphase flow. The constant productivity index concept is only appropriate for oil wells producing under single-phase flow conditions, pressures above the reservoir fluid’s bubblepoint pressure. For reservoir pressures less than the bubblepoint pressure, the reservoir fluid exists as two phases, vapor and liquid, and techniques other than the productivity index must be applied to predict oilwell performance.

Inflow Performance. There have been numerous empirical relationships proposed to predict oilwell performance under two-phase flow conditions. Vogel[22] was the first to present an easy-to-use method for predicting the performance of oil wells. His empirical inflow performance relationship (IPR) is based on computer simulation results and is given by


To use this relationship, the engineer needs to determine the oil production rate and flowing bottomhole pressure from a production test and obtain an estimate of the average reservoir pressure at the time of the test. With this information, the maximum oil production rate can be estimated and used to estimate the production rates for other flowing bottomhole pressures at the current average reservoir pressure.

Fetkovich[23] proposed the isochronal testing of oil wells to estimate productivity. His deliverability equation is based on the empirical gas-well deliverability equation proposed by Rawlins and Schellhardt. [6]


and requires a multiple rate test to obtain values of C and n. A log-log plot of the pressure-squared difference vs. flow rate is expected to plot as a straight line. The inverse of the slope yields an estimate of n, the flow exponent. The flow coefficient can be estimated by selecting a flow rate and pressure on the log-log plot and using the information in Eq. 1.44 to calculate C. An IPR can be developed by rearranging Fetkovich’s deliverability equation to obtain Eq. 1.45.


Jones, Blount, and Glaze[9] also proposed a multirate test method in which they attempted to incorporate non-Darcy flow effects. The basic equation to describe the flow of oil is


where a represents the laminar flow coefficient and b is the turbulence coefficient. To use the method, one must obtain multiple rate test information similar to Fetkovich’s method. A plot of the ratio of the pressure difference to flow rate vs. the flow rate on coordinate paper is expected to yield a straight line. The laminar flow coefficient a is the intercept of the plot, while the slope of the curve yields the turbulence coefficient b. Once a and b have been determined, the flow rate at any other flowing wellbore pressure can be obtained by solving


The maximum flow rate can be estimated from Eq. 1.47 by allowing the flowing bottomhole pressure to equal zero.

There are several other two-phase IPR methods available in the literature. Gallice and Wiggins[24] provide details on the application of several of these methods and compare and discuss their use in estimating oilwell performance with advantages and disadvantages.

In certain circumstances, both single-phase and two-phase flow may be occurring in the reservoir. This results when the average reservoir pressure is above the bubblepoint pressure of the reservoir oil while the flowing bottomhole pressure is less than the bubblepoint pressure. To handle this situation, Neely[25] developed a composite IPR that Brown[26] demonstrates. The composite IPR couples Vogel’s IPR for two-phase flow with the single-phase productivity index. The relationship that yields the maximum oil production rate is


The relationships to determine the oil production rate at various flowing bottomhole pressures are


when the flowing bottomhole pressure is greater than the bubblepoint pressure, and


when the flowing bottomhole pressure is less than the bubblepoint pressure. The flow rate at the bubblepoint pressure, qb, used in Eq. 1.50 is determined with Eq. 1.49 where pwf equals pb.

The appropriate J to use in Eqs. 1.48 and 1.49 depends on the flowing bottomhole pressure of the test point. If the flowing bottomhole pressure is greater than the bubblepoint pressure, then the well is experiencing single-phase flow conditions and J is determined by


When the flowing bottomhole pressure is less than the bubblepoint pressure, J is determined from


Once J is determined for the test conditions, it is used to calculate the complete inflow performance curve both above and below the bubblepoint pressure with Eqs. 1.49 and 1.50. The composite IPR is only applicable when the average reservoir pressure is greater than the bubblepoint pressure.

Wiggins[27] presented an easy-to-use IPR for three-phase flow, which is similar in form to Vogel’s IPR. It was based on a series of simulation studies. It yields results similar to two other three-phase flow models[26][28] and is easier to implement. Eqs. 1.53 and 1.54 give the generalized three-phase IPRs for oil and water, respectively.



Example 1.3
Table 1.6 presents data for a multipoint test on a producing oil well used to demonstrate the two-phase IPR methods. The average reservoir pressure for this example is 1,734 psia.

To apply the IPR methods, obtain test information, which includes production rates, flowing bottomhole pressures, and an estimate of the average reservoir pressure. Vogel’s IPR is a single-rate relationship, and the highest test rate is used to demonstrate this IPR. The data obtained at the largest pressure drawdown can be used with Eq. 1.43 to solve for the maximum oil-production rate.


The estimated maximum oil production is 2,065 STB/D. This value is then used to estimate the production rate at other values of flowing bottomhole pressures to develop a complete inflow performance curve. Once again, Eq. 1.43 will be rearranged to calculate the production rate for a flowing bottomhole pressure of 800 psia.


Fetkovich’s IPR requires multiple test points to determine the deliverability exponent n. Table 1.7 shows the test data prepared for plotting. The data are plotted on a logarithmic graph, which is used to estimate the slope of the best-fit straight line through the data. The deliverability exponent n is the inverse of the slope. Once n is determined, Eq. 1.45 can be used to estimate the maximum oil production rate. Fig. 1.9 is the plot of the data that shows the best-fit straight line has a slope of 1.347 yielding an n value of 0.743. The estimated maximum oil production rate is 1,497 STB/D, as Eq. 1.57 shows.


Once the maximum rate is estimated, it is used with Eq. 1.45 to estimate production rates at other flowing bottomhole pressures to develop the inflow performance curve in a manner similar to that demonstrated with Vogel’s IPR. For Fetkovich’s method, the production rate is estimated to be 1,253 STB/D at a flowing bottomhole pressure of 800 psia.

To apply the method of Jones, Blount, and Glaze to this data set, Table 1.8 was prepared and the data plotted on a coordinate graph as shown in Fig. 1.10. The best-fit straight line yielded a slope of 0.0004 psia/(STB/D)2, which is the turbulence coefficient b. The intercept is the laminar flow coefficient and is determined to be 0.23 psia/STB/D. These values are used in Eq. 1.47 to determine the maximum oil production rate of 1,814 STB/D when the flowing bottomhole pressure is 0 psig.


This same relationship is used to estimate the production rate at other flowing bottomhole pressures to generate the inflow performance curve. For a flowing bottomhole pressure of 800 psia, the production rate is estimated to be 1,267 STB/D.

From this example, each of the three methods yielded different values for the maximum oil production rate as well as the production rate at a flowing bottomhole pressure of 800 psia. As a result, production estimates will be dependent on the IPR used in the analysis, and the petroleum engineer should be aware of this concern in any analysis undertaken.

The application of the composite IPR and Wiggins’ IPR is straight-forward and similar to applying Vogel’s IPR. In applying the composite IPR, the appropriate relationship must be used to estimate J because it depends on the flowing bottomhole pressure of the test point. With Wiggins’ IPR, estimates of both oil and water production rates are generated. The inflow performance curve will be developed by adding the estimated oil rates to the water rates to create a total liquid rate.

Future Performance Methods. Once the petroleum engineer has estimated the current productive capacity of a well, it is often desired to predict future performance for planning purposes. Standing[29] was one of the first to address the prediction of future well performance from IPRs. He used Vogel’s IPR with a modified multiphase productivity index to relate current well performance to future performance. Unfortunately, his relationship requires knowledge of fluid properties and relative permeability behavior. This makes Standing’s method difficult to use because one must estimate saturations, relative permeabilities, and fluid properties at a future reservoir pressure.

Fetkovich[23] suggested that Standing’s modified multiphase productivity index ratios could be approximated by the ratio of the pressures. He proposed that the future maximum oil production rate could be estimated from the current maximum production rate with


Fetkovich applied this idea to the use of his IPR. The exponent n in Eq. 1.59 is the deliverability exponent from his IPR; however, Fetkovich’s future performance method has been applied to other IPR methods by allowing the exponent to be one, which provides good results in many cases. This method requires no more information to apply than that obtained for applying the various IPRs. It is important to note that Fetkovich’s method assumes the deliverability exponent does not change between the present and future conditions. Uhri and Blount[30] and Kelkar and Cox[31] have also proposed future performance methods for two-phase flow that require rate and pressure data at two average reservoir pressures.

At the time Wiggins[27] proposed his three-phase IPRs, he also presented future performance relationships for the oil and water phases. These relationships are presented in Eqs. 1.60 and 1.61.



In all cases, once the future maximum production rate is estimated from the current data, inflow performance curves at the future average reservoir pressure of interest can be developed with the IPR of one’s choosing.

Wellbore Flow Performance

The pressure drop experienced in lifting reservoir fluids to the surface is one of the main factors affecting well deliverability. As much as 80% of the total pressure loss in a flowing well may occur in lifting the reservoir fluid to the surface. Wellbore flow performance relates to estimating the pressure-rate relationship in the wellbore as the reservoir fluids move to the surface through the tubulars. This flow path may include flow through perforations, a screen and liner, and packers before entering the tubing for flow to the surface. The tubing may contain completion equipment that acts as flow restrictions such as profile nipples, sliding sleeves, or subsurface flow-control devices. In addition, the tubing string may be composed of multiple tubing diameters or allow for tubing/annulus flow to the surface. At the surface, the fluid must pass through wellhead valves, surface chokes, and through the flowline consisting of surface piping, valves, and fittings to the surface-processing equipment. The pressure drop experienced as the fluid moves from the reservoir sandface to the surface is a function of the mechanical configuration of the wellbore, the properties of the fluids, and the producing rate.

Relationships to estimate this pressure drop in the wellbore are based on the mechanical energy equation for flow between two points in a system as written in Eq. 1.62.


In this relationship, α is the kinetic energy correction factor for the velocity distribution, W is the work done by the flowing fluid, and El is the irreversible energy losses in the system including the viscous or friction losses. For most practical applications, there is no work done by or on the fluid and the kinetic energy correction factor is assumed to be one. Under these conditions, Eq. 1.62 can be rewritten in terms of the pressure change as


This relationship states that the total pressure drop is equal to the sum of the change in potential energy (elevation), the change in kinetic energy (acceleration), and the energy losses in the system. This relationship can be written in the differential form for any fluid at any pipe inclination as


Methods to estimate the pressure drop in tubulars for single-phase liquid, single-phase vapor, and multiphase flow are based on this fundamental relationship.

With Eq. 1.64, the pressure drop for a particular flow rate can be estimated and plotted as a function of rate. In the typical application, the wellhead pressure is fixed and the bottomhole flowing pressure, pwf, is calculated by determining the pressure drop. This approach will yield a wellbore flow performance curve when the pressure is plotted as a function of rate as shown in Fig. 1.11. In this example, the wellhead pressure is held constant, and the flowing bottomhole pressure is calculated as a function of rate. This curve is often called a tubing-performance curve because it captures the required flowing bottomhole pressure needed for various rates.

The following paragraphs summarize the basic approaches for estimating the pressure loss in the tubulars. Complete details of making these calculations are outside the scope of this section.

Single-Phase Liquid Flow

Single-phase liquid flow is generally of minor interest to the petroleum engineer, except for the cases of water supply or injection wells. In these cases, Eq. 1.64 is applicable where the friction factor, f, is a function of the Reynolds number and pipe roughness. The friction factor is most commonly estimated from the Moody friction factor diagram. The friction factor is an empirically determined value that is subject to error because of its dependence on pipe roughness, which is affected by pipe erosion, corrosion, or deposition.

Single-Phase Vapor Flow

There are several methods to estimate the pressure drop for single-phase gas flow under static and flowing conditions. These methods include the average temperature and compressibility method[32] and the original and modified Cullendar and Smith methods. [33],[34] They require a trial-and-error or iterative approach to calculate the pressure drop for a given rate because of the compressible nature of the gas. These techniques are calculation intensive but can be implemented easily in a computer program. Lee and Wattenbarger[5] provide a detailed discussion of several methods used for estimating pressure drops in gas wells.

A simplified method for calculating the pressure drop in gas wells assuming an average temperature and average compressibility over the flow length was presented by Katz et al.[11]






This relationship can be solved directly if the wellhead and bottomhole pressures are known; however, in most applications, one pressure will be assumed and the other calculated. Thus, this method will be an iterative method as the compressibility factor is determined at the average pressure. Eq. 1.65 can be used to calculate the pressure drop for either flowing or static conditions.

Multiphase Flow

Much has been written in the literature regarding the multiphase flow of fluids in pipe. This problem is much more complex than the single-phase flow problem because there is the simultaneous flow of both liquid (oil or condensate and water) and vapor (gas). The mechanical energy equation (Eq. 1.64) is the basis for methods to estimate the pressure drop under multiphase flow; however, the problem is in determining the appropriate velocity, friction factor, and density to be used for the multiphase mixture in the calculation. In addition, the problem is further complicated as the velocities, fluid properties, and the fraction of vapor to liquid change as the fluid flows to the surface due to pressure changes.

Many researchers have proposed methods to estimate pressure drops in multiphase flow. Each method is based on a combination of theoretical, experimental, and field observations, which has led some researchers to relate the pressure-drop calculations to flow patterns. Flow patterns or flow regimes relate to the distribution of each fluid phase inside the pipe. This implies that a pressure calculation is dependent on the predicted flow pattern. There are four flow patterns in the simplest classification of flow regimes: [35] bubble flow, slug flow, transition flow, and mist flow, with a continually increasing fraction of vapor to liquid from bubble to mist flow. Bubble flow is experienced when the liquid phase is continuous with the gas phase existing as small bubbles randomly distributed within the liquid. In slug flow, the gas phase exists as large bubbles separating liquid slugs in the flow stream. As the flow enters transition flow, the liquid slugs essentially disappear between the gas bubbles, and the gas phase becomes the continuous fluid phase. The liquid phase is entrained as small droplets in the gas phase in the mist-flow pattern.

Poettman and Carpenter[36] were some of the earliest researchers to address developing a multiphase-flow correlation for oil wells, while Gray[37] presented an early multiphase correlation for gas wells. A large number of studies have been conducted related to multiphase flow in pipes. Brill and Mukerjee[38] and Brown and Beggs[39] include a review of many of these correlations. Application of the multiphase-flow correlations requires an iterative, trial-and-error solution to account for changes in flow parameters as a function of pressure. This is calculation intensive and is best accomplished with computer programs. Pressure calculations are often presented as pressure-traverse curves, like the one shown in Fig. 1.12, for a particular tubing diameter, production rate, and fluid properties. Pressure-traverse curves are developed for a series of gas-liquid ratios and provide estimates of pressure as a function of depth. These curves can be used for quick hand calculations.

Flow Through Chokes

A wellhead choke controls the surface pressure and production rate from a well. Chokes usually are selected so that fluctuations in the line pressure downstream of the choke have no effect on the production rate. This requires that flow through the choke be at critical flow conditions. Under critical flow conditions, the flow rate is a function of the upstream or tubing pressure only. For this condition to occur, the downstream pressure must be approximately 0.55 or less of the tubing pressure.

For single-phase gas flow, Beggs[40] presents Eq. 1.68, which relates the gas production rate through a choke to the wellhead pressure.


The pressure ratio, y, is the ratio of the downstream pressure to the wellhead pressure. Under critical flow conditions, the pressure ratio is replaced by the critical pressure ratio, yc. The critical pressure ratio is the pressure ratio at which flow becomes critical. This ratio depends on the ratio of the specific heats of the produced gas, as Eq. 1.69 shows.


Empirical equations have been developed to estimate the relationship between production rate and wellhead pressure for two-phase critical flow. These correlations can be presented in a form similar to Eq. 1.70.


Gilbert[41] was the first to present such a relationship based on field data collected from the Ten Section field of California. Ros[42] and Beggs[40] have also presented relationships that are often used. Table 1.9 summarizes the parameters for each equation.

Example 1.4
This example illustrates the use of the multiphase choke equation (Eq. 1.70) to estimate the flowing wellhead pressure for a given set of well conditions. However, this equation can be used to estimate flow rate or choke diameter. The example well is producing 400 STB/D of oil with a gas-liquid ratio of 800 Scf/STB. Estimate the flowing wellhead pressure for a choke size of 12/64 in. with Gilbert’s choke equation.

Use Eq. 1.70 and the proper variable from Table 1.9 to calculate


For these conditions, the estimated flowing wellhead pressure is 1,405 psia. If the Ros choke equation is used, an estimated flowing wellhead pressure of 1,371 psia is calculated. Each of the relationships provides slightly different estimates of the calculated value.

Systems Analysis

Systems analysis has been used for many years to analyze the performance of systems composed of multiple interacting components. Gilbert[41] was perhaps the first to introduce the approach to oil and gas wells but Mach, Proano, and Brown[1] and Brown[26] popularized the concept, which is often referred to as Nodal Analysis™ within the oil and gas industry. The objective of systems analysis is to combine the various components of the production system for an individual well to estimate production rates and optimize the components of the production system.

The flow of reservoir fluids from the subsurface reservoir to the stock tank or sales line requires an understanding of the principles of fluid flow through porous media and well tubulars. As the fluid moves through the production system, there will be an associated pressure drop to accompany the fluid flow. This pressure drop will be the sum of the pressure drops through the various components in the production system. Because of the compressible nature of the fluids produced in oil and gas operations, the pressure drop is dependent on the interaction between the various components in the system. This occurs because the pressure drop in a particular component is not only dependent on the flow rate through the component, but also on the average pressure that exists in the component.

As a result, the final design of a production system requires an integrated approach, since the system cannot be separated into a reservoir component or a piping component and handled independently. The amount of oil and gas produced from the reservoir to the surface depends on the total pressure drop in the production system, and the pressure drop in the system depends on the amount of fluid flowing through the system. Consequently, the entire production system must be analyzed as a unit or system.

Depending on the terminal end of the production system, there is a total pressure drop from the reservoir pressure to the surface, as depicted in Fig. 1.1. If the separator represents the end of the production system, the total pressure drop in the system is the difference between the average reservoir pressure and the separator pressure:


This total pressure drop is then composed of individual pressure drops as the reservoir fluid flows to the surface. These pressure drops occur as the fluid flows through the reservoir and well completion, up the tubing, through the wellhead equipment and choke, and through the surface flowlines to the separator. Thus, the total pressure drop of Eq. 1.72 can be represented by Eq. 1.73.


These individual pressure drops can be divided into yet additional pressure drops to account for restrictions, subsurface safety valves, tubing accessories, etc.

Systems analysis is based on the concept of continuity. At any given point in the production system, there is a particular pressure and production rate associated with that point for a set of conditions. If there is any change in the system, then there will be an associated change in pressure and/or production rate at that same point. This concept allows the production system to be divided at a point of interest for evaluation of the two portions of the system. This evaluation determines the conditions of continuity of pressure and production rate at the division point, which is the estimated producing condition for the system being evaluated.

The approach provides the flexibility to divide the production system at any point of interest within the system to evaluate a particular component of the system. The most common division points are at the wellhead or at the perforations, either at the reservoir sandface or inside the wellbore. The terminal ends of the system will be the reservoir on the upstream end of the system and the separator at the downstream end of the system or the wellhead if a wellhead choke controls the well.

The components upstream of the division point or node comprise the inflow section of the system, while the components downstream of the node represent the outflow section. Once the system is divided into inflow and outflow sections, relationships are written to describe the rate-pressure relationship within each section. The flow rate through the system is determined once the conditions of continuity are satisfied: flow into the division point equals flow out of the division point, and the pressure at the division point is the same in both inflow and outflow sections of the system.

After the division point is selected, pressure relationships are developed for the inflow and outflow sections of the system to estimate the node pressure. The pressure in the inflow section of the system is determined from Eq. 1.74, while the outflow section pressure drop is determined from Eq. 1.75.



The pressure drop in any component, and thus in either the inflow or outflow section of the system, varies as a function of flow rate. As a result, a series of flow rates is used to calculate node pressures for each section of the system. Then, plots of node pressure vs. production rate for the inflow section and the outflow section are made. The curve representing the inflow section is called the inflow curve, while the curve representing the outflow section is the outflow curve. The intersection of the two curves provides the point of continuity required by the systems analysis approach and indicates the anticipated production rate and pressure for the system being analyzed.

Fig. 1.13 depicts a systems graph for a sensitivity study of three different combinations for outflow components labeled A, B, and C. For outflow curve A, there is no intersection with the inflow performance curve. Because there is no intersection, there is no continuity in the system and the well will not be expected to flow with System A. The inflow and outflow performance curves do intersect for System B. Thus this system satisfies continuity, and the well will be expected to produce at a rate and pressure indicated by the intersection of the inflow and outflow curves. System C also has an intersection and would be expected to produce at a higher rate and lower pressure than System B, as indicated by the graph.

The outflow curve for System C has a rapidly decreasing pressure at low flow rates, reaches a minimum, and then begins to slowly increase with increasing rate. This is typical for many outflow curves, which, in some cases, will yield two intersection points with the inflow curve; however, the intersection at the lower rate is not a stable solution and is meaningless. The proper intersection of the inflow and outflow curves should be the intersection to the right of and several pressure units higher than the minimum pressure on the outflow curve.

The effect of changing any component of the system can be evaluated by recalculating the node pressure for the new characteristics of the system. If a change is made in an upstream component of the system, then the inflow curve will change and the outflow curve will remain unchanged. On the other hand, if a change in a downstream component is made, then the inflow curve will remain the same and the outflow curve will change. Both the inflow and outflow curves will be shifted if either of the fixed pressures in the system is changed, which can occur when evaluating the effects of reservoir depletion or considering different separator conditions or wellhead pressures.

Systems analysis may be used for many purposes in analyzing and designing producing oil and gas wells. The approach is suited for evaluating both flowing wells and artificial lift applications. The technique provides powerful insight in the design of an initial completion. Even with limited data, various completion scenarios can be evaluated to yield a qualitative estimate of expected well behavior. This process is very useful in analyzing current producing wells by identifying flow restrictions or opportunities to enhance performance.

Typical applications include estimation of flow rates, selection of tubing size, selection of flowline size, selection of wellhead pressures and surface choke sizing, estimation of the effects of reservoir pressure depletion, and identification of flow restrictions. Other typical applications are sizing subsurface safety valves, evaluating perforation density, gravel pack design, artificial lift design, optimizing injection gas-liquid ratio for gas lift, evaluating the effects of lower wellhead pressures or installation of compression, and evaluating well stimulation treatments. In addition, systems analysis can be used to evaluate multiwell producing systems. Systems analysis is a very robust and flexible method that can be used to design a well completion or improve the performance of a producing well.

Systems Analysis Examples

Examples 1.5 and 1.6 demonstrate the systems analysis approach. Example 1.5 considers the effects of tubing size on gas well performance. Example 1.6 demonstrates the effects of reservoir depletion on the performance of an oil well. Greene, [19] Brown and Lea, [43] and Chap. 4 of Lee and Wattenbarger[5] and Brown[26] provide a series of detailed applications that further exemplify the use of systems analysis for gas and oil wells.

Example 1.5
Analyze a gas well to select an appropriate tubing size. The gas well under consideration is at 9,000 ft with a reservoir pressure of 4,000 psia.

The first step in applying systems analysis is to select a node to divide the system. Initially, the node is selected to be at the perforations to isolate the inflow performance (reservoir behavior) from the flow behavior in the tubing. For this particular case, the well is flowing at critical flow conditions, and, consequently, the wellhead choke serves as a discontinuity in the system, which allows the use of the wellhead pressure as the terminal point for the outflow curve. Once the node point is selected, the pressure relations for the inflow and outflow sections of the system are determined. For this example, Eqs. 1.76 and 1.77 represent the inflow and outflow pressure relationships, respectively.



With these basic relationships, the flowing bottomhole pressure is calculated for different production rates for both the inflow and outflow sections. Table 1.10 presents the inflow performance data while Table 1.11 presents the calculated pressures for three different tubing sizes using a constant wellhead pressure of 1,000 psia. These data are used to construct the inflow and outflow curves in Fig. 1.14 to estimate the production rates and pressures for each tubing size. The intersection of the outflow curves with the inflow curve dictates the estimated point of continuity and the anticipated producing conditions for the analyzed system. For this example, the production rate increases with increasing tubing size, yielding 4,400 Mscf/D for 1.90-in. tubing, 4,850 Mscf/D for 2 3/8-in. tubing, and 5,000 Mscf/D for 2 7/8-in. tubing.

The same well could be analyzed with the wellhead as the system node. This allows the effect of changes in wellhead pressure on well performance to be determined. The new inflow and outflow pressure relationships are


for the inflow curve, and


for the outflow curve. Table 1.12 shows the pressure-rate relationship for both the inflow and outflow curves. Because the wellhead is the node in this analysis, the outflow curve will be constant and equal to the anticipated flowing wellhead pressure.

The data are plotted in Fig. 1.15 and yield the same producing rates and flowing bottomhole pressures that were determined when the flowing bottomhole pressure was used as the node. This is as expected because the choice of a division point or node does not affect the results for a given system. If the wellhead pressure is decreased to 250 psia, the producing rate will change also. This effect is readily determined by constructing a constant wellhead pressure line of 250 psia on the graph and selecting the points of intersection for each tubing size. As observed from the graph, the anticipated production rates increase to 4,950 Mscf/D, 5,200 Mscf/D, and 5,300 Mscf/D for the three tubing sizes by lowering the wellhead pressure.

Example 1.6
Investigate the effects of reservoir depletion of an oil well to estimate producing conditions and consider the need for artificial lift. The well under consideration is producing with a constant wellhead pressure of 250 psia and is controlled by the choke.

Isolate the reservoir performance to visualize the effect of changing reservoir pressure. The flowing bottomhole pressure at mid-perforations is selected as the node and, as the well is producing under critical flow conditions, the wellhead will serve as the terminal end of the system.

The inflow and outflow rate-pressure data is generated with Eqs. 1.76 and 1.77. Table 1.13 provides the inflow performance data for average reservoir pressures of 2,500 psia and 2,000 psia. Table 1.14 shows the tubing-intake data or outflow performance data for a flowing wellhead pressure of 250 psia with 2 7/8-in. tubing. Fig. 1.16 plots this information, which is used to determine the producing conditions at the two reservoir pressures. At an average reservoir pressure of 2,500 psia, the curves intersect at an oil production rate of 380 STB/D and a flowing bottomhole pressure of 1,650 psia. However, there is no intersection or point of continuity between the inflow and outflow performance curves when the reservoir pressure declines to 2,000 psia. This indicates that the well will not flow under these reservoir conditions. On the basis of this analysis, the effects of lowering the wellhead pressure, reducing the tubing size, or installing artificial lift early in the life of the well to enhance its deliverability should be investigated.


This chapter describes the flow of reservoir fluids through the production system and provides methods to estimate oil and gas well deliverability. Analytical and empirical methods that describe fluid flow through the reservoir to the wellbore are presented to assist in predicting the inflow performance of an individual well. A brief overview of flow through circular conduits is used to assist in describing the pressure drop in the production system. Finally, an integrated approach called systems analysis is described to provide the tools needed to estimate well deliverability by integrating the reservoir performance with the tubing performance. These two aspects of the production system must be integrated to determine actual producing conditions.

Systems analysis is an excellent engineering tool for optimizing the design of a new well completion or analyzing the behavior of a current production system. The application of systems analysis requires a thorough understanding of the relationship between flow of reservoir fluids in the subsurface reservoir and fluid flow through the well completion and tubulars to the surface stock tank. Unfortunately, this understanding is often lacking in practice. Inefficient operations may occur because the petroleum engineer does not have a complete understanding of the fluid-flow process or fails to take a comprehensive look at the production system. The proper application of systems analysis provides a basis for determining the interaction of the various components in the production system to optimize the desired production rates for both oil and gas wells.


a = laminar flow coefficient, m2/L5t3, psia2/Mscf/D or m/L4t2, psia2/cp/Mscf/D or mL4/t, psia/STB/D
A1–3 = coefficient in Eq. 1.70
b = turbulence coefficient, m2/L8t2, psia2/(Mscf/D)2 or m/L7t, psia2/cp/(Mscf/D)2 or mL7, psia/(STB/D)2
B = formation volume factor, dimensionless, RB/STB
ct = total compressibility, Lt2/m, psia–1
C = flow coefficient, L3 + 2nt4n–1/m2n, Mscf/D/psia2n or L3 + nt3n–1/mn, Mscf/D/(psia2/cp)n or L3 + 2nt4n–1/m2n, STB/D/psia2n
C1 = flow coefficient in Eq. 1.39, L3 + 2nt4n–2/m2n–1, cp-Mscf/D/psia2n
Cd = discharge coefficient, dimensionless
Cp = specific heat capacity at constant pressure, L2/t2T
Cv = specific heat capacity at constant volume, L2/t2T
d = pipe diameter, L, in.
D = non-Darcy flow coefficient, t/L3, D/Mscf
El = energy loss per unit mass, L2/t2, ft-lbf/lbm
f = friction factor, dimensionless
fM = Moody friction factor in Eq. 1.66, dimensionless
g = gravitational acceleration, L/t2, ft/sec2
gc = conversion factor, dimensionless, 32.2 ft-lbm/lbf-sec2
h = formation thickness, L, ft
J = productivity index, L4t/m, STB/D/psia
k = permeability, L2, md
k = specific heat capacity ratio, Cp/Cv in Eqs. 1.68 and 1.69, dimensionless
L = length, L, ft
M = molecular weight, m, lbm/lbm-mole
n = deliverability exponent, dimensionless
p = pressure, m/Lt2, psia
RTENOTITLE = average bottomhole pressure, m/Lt2, psia
pb = bubblepoint pressure, m/Lt2, psia
pe = external boundary pressure, m/Lt2, psia
pn = node pressure, m/Lt2, psia
pp = gas pseudopressure, m/Lt3, psia2/cp
ppRTENOTITLE = average reservoir pseudopressure, m/Lt3, psia2/cp
pp(pwf) = flowing bottomhole pseudopressure, m/Lt3, psia2/cp
RTENOTITLE = average reservoir pressure, m/Lt2, psia
ps = separator pressure, m/Lt2, psia
psc = standard pressure, m/Lt2, psia
pwf = bottomhole pressure, m/Lt2, psia
pwfs = sandface bottomhole pressure, m/Lt2, psia
pwh = wellhead pressure, m/Lt2, psia
q = flow rate, L3/t, STB/D or Mscf/D
qb = oil flow rate at the bubblepoint pressure, L3/t, STB/D
qg = gas flow rate, L3/t, Mscf/D
qg,max = AOF, maximum gas flow rate, L3/t, Mscf/D
qL = liquid flow rate, L3/t, STB/D
qo = oil flow rate, L3/t, STB/D
qo,max = maximum oil flow rate, L3/t, STB/D
qw = water flow rate, L3/t, STB/D
qw,max = maximum water flow rate, L3/t, STB/D
r = radius, L, ft
re = external drainage radius, L, ft
rw = wellbore radius, L, ft
R = producing gas/liquid ratio, dimensionless, scf/STB
s = skin factor, dimensionless
S = defined by Eq. 1.67, m/L2t
t = time, t
T = temperature, T, °R
Tsc = standard temperature, T, °R
Twh = wellhead temperature, T, °R
v = velocity, L/t, ft/sec
W = work per unit mass, L2/t2, ft-lbf/lbm
y = ratio of downstream pressure to upstream pressure, p1/p2, dimensionless
yc = critical pressure ratio defined by Eq. 1.69, dimensionless
z = gas compressibility factor, dimensionless
Z = elevation, L, ft
α = kinetic energy correction factor, dimensionless
β = turbulence factor, L–1, ft–1
γg = gas specific gravity, dimensionless
Δp = pressure loss, m/Lt2, psia
Δp1 = pressure loss in reservoir, m/Lt2, psia
Δp2 = pressure loss across completion, m/Lt2, psia
Δp3 = pressure loss in tubing, m/Lt2, psia
Δp4 = pressure loss in flowline, m/Lt2, psia
Δpd = change in downstream pressure, m/Lt2, psia
Δpp = difference in pseudopressures, m/Lt3, psia2/cp
ΔpT = total pressure loss, m/Lt2, psia
Δpu = change in upstream pressure, m/Lt2, psia
Δp2 = difference in pressures squared, m2/L2t4, psia
ε = absolute pipe roughness, L, in.
μ = viscosity, m/Lt, cp
ρ = fluid density, m/L3, lbm/ft3
ϕ = porosity, fraction


f = future time
g = gas
o = oil
p = present time
w = water


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SI Metric Conversion Factors

bbl × 1.589 873 E–01 = m3
cp × 1.0* E–03 = Pa•s
ft × 3.048* E–01 = m
ft3 × 2.831 685 E–02 = m3
in × 2.54* E + 00 = cm
lbf × 4.448 222 E + 00 = N
lbm × 4.535 924 E–01 = kg
md × 9.869 233 E–04 = μm2
psi × 6.894 757 E + 00 = kPa
psi2 × 4.753 8 E + 01 = kPa2
°R × 5/9 = K


Conversion factor is exact.