SPE 25458 Generalized Inflow Performance Relationships for Three-Phase Flow Society of Petroleum Engineers M.L. Wiggins, U. of Oklahoma SPE Member Copyright 1993, Society of Petroleum Engineers, Inc. This paper was prepared for presentation at the Production Operations Symposium held in Oklahoma City, OK, U.S.A., March 21-23, 1993. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083·3836, U.S.A. Telex, 163245 SPEUT. ABSTRACT Generalized three-phase inflow performance relationships (IPRs) for the oil and water phases are presented in this paper. These relationships yield adequate estimates of the production-pressure behavior of oil wells producing from homogeneous, bounded reservoirs during boundary-dominated flow. The IPRs are empirical relationships based on linear regression analysis of simulator results and cover a wide range of reservoir fluid and rock properties. Methods to study the effects of changes in flow efficiency and to predict future performance are also presented. INTRODUCTION Predicting the performance of individual oil wells is an important responsibility of the petroleum engineer. Reasonable estimates of well performance allow the engineer to determine the optimum production scheme, design production and artificial lift equipment, design stimulation treatments and forecast production for planning purposes. Each of these items is important to the efficient operation of producing wells and successful reservoir management. When estimating oilwell performance, it is often assumed that fluid inflow is proportional to the difference between reservoir pressure and wellbore pressure. One of the first relationships to be used based on this assumption was the Productivity Index (PI). This straight-line relationship can be derived from Darcy'sllaw for the steady-state flow of a single incompressible fluid and is the ratio of the producing rate to the pressure difference. However, Evinger and Muskat 2 ,3 pointed out that a straight-line relationship should not be expected when multiple phases are flowing in the reservoir. They presented theoretical calculations that showed a curved relationship between flow rate and pressure for two- and three-phase flow. Vogel 4 later developed an empirical inflow performance relationship (IPR) for solution-gas drive reservoirs that accounted for the flow of two phases, oil and gas, in the reservoir based on computer simulation results. The resulting IPR equation is 483 = 1 - 0.2 Pwf - 0.8 (Pwfj2 qo;nax P r P r (1) Fetkovich 5 also presented an empirical inflow performance relationship based on field data that has gained wide acceptance. His relationship, of a form similar to the empirical gas well deliverability equation proposed by Rawlins and Schellhardt 6 , is (2) Both Vogel's and Fetkovich's relations were developed for solution-gas drive reservoirs and are widely used due to their simplicity. In an attempt to extend Vogel's approach to three-phase flow, Brown 7 presented a method proposed by Petrobras for determining the inflow performance of oil wells
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SPE 25458
Generalized Inflow Performance Relationships for Three-Phase Flow
Society of Petroleum Engineers
M.L. Wiggins, U. of Oklahoma
SPE Member
Copyright 1993, Society of Petroleum Engineers, Inc.
This paper was prepared for presentation at the Production Operations Symposium held in Oklahoma City, OK, U.S.A., March 21-23, 1993.
This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society of Petroleum Engineers. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083·3836, U.S.A. Telex, 163245 SPEUT.
ABSTRACT
Generalized three-phase inflow performance relationships (IPRs) for the oil and water phases are presented in this paper. These relationships yield adequate estimates of the production-pressure behavior of oil wells producing from homogeneous, bounded reservoirs during boundary-dominated flow. The IPRs are empirical relationships based on linear regression analysis of simulator results and cover a wide range of reservoir fluid and rock properties. Methods to study the effects of changes in flow efficiency and to predict future performance are also presented.
INTRODUCTION
Predicting the performance of individual oil wells is an important responsibility of the petroleum engineer. Reasonable estimates of well performance allow the engineer to determine the optimum production scheme, design production and artificial lift equipment, design stimulation treatments and forecast production for planning purposes. Each of these items is important to the efficient operation of producing wells and successful reservoir management.
When estimating oil well performance, it is often assumed that fluid inflow is proportional to the difference between reservoir pressure and wellbore pressure. One of the first relationships to be used based on this assumption was the Productivity Index (PI). This straight-line relationship can be derived from Darcy'sllaw for the steady-state flow of a
single incompressible fluid and is the ratio of the producing rate to the pressure difference. However, Evinger and Muskat2,3 pointed out that a straight-line relationship should not be expected when multiple phases are flowing in the reservoir. They presented theoretical calculations that showed a curved relationship between flow rate and pressure for two- and three-phase flow.
Vogel4 later developed an empirical inflow performance relationship (IPR) for solution-gas drive reservoirs that accounted for the flow of two phases, oil and gas, in the reservoir based on computer simulation results. The resulting IPR equation is
483
~ = 1 - 0.2 Pwf - 0.8 (Pwfj2 qo;nax Pr Pr
(1)
Fetkovich5 also presented an empirical inflow performance relationship based on field data that has gained wide acceptance. His relationship, of a form similar to the empirical gas well deliverability equation proposed by Rawlins and Schellhardt6 , is
(2)
Both Vogel's and Fetkovich's relations were developed for solution-gas drive reservoirs and are widely used due to their simplicity.
In an attempt to extend Vogel's approach to three-phase flow, Brown7
presented a method proposed by Petrobras for determining the inflow performance of oil wells
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SPE 25458
producing water. The method uses a constant PI for the water production and adds it to a Vogel relation for the oil production to obtain a composite inflow performance relationship. Sukarno8 proposed a method derived from computer simulation of three-phase flow. This method resulted from nonlinear regression analysis of the generated simulator results and is based on the producing water cut and total liquid flow rate. The resulting relationship is a quadratic whose coefficients are functions of water cut. As of yet, no one has addressed the problem of predicting future performance or studied the effect of a skin region around the wellbore during three-phase flow.
In this paper, generalized inflow performance relationships are presented for three-phase flow in bounded, homogeneous reservoirs. The proposed IPRs are compared with other three-phase methods currently available. The methods presented are based on homogeneous reservoirs where gravity and capillary effects are negligible. Methods are also presented for predicting performance when reservoir conditions change from test conditions. This includes predicting future performance due to depletion and predicting performance when changes occur in the skin region near the wellbore.
GENERALIZED IPRs
Wiggins, Russell and Jennings9
recently proposed an analytical IPR for threephase flow in bounded reservoirs. An advantage of the analytical IPR is that one can develop an IPR specific to a particular reservoir and its operating conditions. The major disadvantage, however, is that it requires knowledge of relative permeability and reservoir fluid properties and how they behave with pressure. This is not a large obstacle if relative permeability and pressurevolume-temperature data are available for the reservoir of interest, along with an idea of the average reservoir pressure and water saturation. With this information, one can develop the required mobility function profiles from the current reservoir pressure to near-
484
zero flowing pressure. This profile can then be used to develop the analytical IPRs for the oil and water phases.
Unfortunately, we do not always have reliable relative permeability or fluid property information. In this case, the analytical IPR is only of academic interest in our operations. To overcome this problem, generalized three-phase IPRs similar to Vogel's were developed and are presented here. The resulting IPR equations are based on regression analysis of simulator results covering a wide range of relative permeability information, fluid property data and water saturations.
Development of Simulator Results
To develop the generalized equations to predict inflow performance, IPR curves were generated from simulator results for four basic sets of relative permeability and fluid property data. Each set of data was used to generate simulator results from irreducible water saturation to residual oil saturation. Sixteen theoretical reservoirs were examined from initial pressure to the minimum flowing bottomhole pressure. Table 1 presents the range of reservoir properties used in the development of the generalized IPRs.
Simulator results were obtained for a radial flow geometry and constant oil rate production. Maximum oil and water production rates were estimated at each stage of depletion from the simulator results at a minimum flowing bottomhole pressure of 14.7 psia. If the flowing bottomhole pressure did not reach this minimum during the simulation, the maximum rate was estimated from the production information available and then checked by rerunning the simulator.
Figs. 1 and 2 present typical oil and water inflow performance curves for Case 3 with an initial water saturation of 20% at several stages of depletion. These curves have the same characteristic concave shape noticed by Vogel in his research. The curves were normalized by dividing each point of
SPE 25458 MICHAEL L. WIGGINS 3
infonnation by the maximum rate and average pressure at the stage of depletion. The resulting IPR curves are presented in Figs. 3 and 4. The individual curves are now almost indistinguishable and can be represented by a single curve. The simulator results from all cases studied were normalized in this manner.
IPRs
To develop the generalized three-phase IPRs, the production rate ratios were regressed on the pressure ratios. A linear regression model of the fonn
(3)
was used to fit the infonnation. The statistical analysis was performed using the linear regression procedure available in the SAS SystemlO, a general purpose software system for data analysis.
The resulting generalized IPRs are
... (4) and
... (5)
Figs. 5 and 6 present the simulator infonnation for all cases studied with the resulting IPR equations. Statistical infonnation is presented in Tables 2 and 3. Overall, the average absolute error was 4.39% for the oil IPR and 6.18% for the water IPR indicating the generalized curves should be suitable for use over a wide range of reservoir properties if the reservoir is producing under boundarydominated flow conditions.
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Comparison with Other Methods
In order to test their reliability, the generalized IPRs were compared with the three-phase IPR methods of Brown and Sukarno. Brown's method was proposed by Petrobras and is based on developing a composite IPR curve. The composite curve is generated by using Vogel's IPR for the oil phase and coupling it with a straight-line PI for the water phase. Sukarno's method is based on nonlinear regression analysis of simulator results. Both methods differ from the generalized three-phase IPR method presented in this paper in that they couple the water and oil rates. The proposed method assumes we can treat each phase separately.
To evaluate the three methods, infonnation presented by Sukarno in his Tables 6-24 to 6-26 was selected for comparison purposes. This information was generated by Sukarno using a simulator and was not used in the development of the proposed method. It was felt that these cases would give an unbiased indication of the reliability of the proposed IPRs.
Tables 4-6 present the results of this analysis. All three methods yield similar estimates of producing rates, indicating the generalized three-phase IPRs yield suitable results. The maximum difference between the simulator results and the generalized IPR is 3.98% for the oil phase and 7.08% for the water phase. This analysis shows that any of the three methods appear suitable for use during boundary-dominated flow; yet, the proposed method is much simpler to use without yielding any degree of reliability. Based on simplicity, the generalized IPRs are recommended for use in applying to field data.
PERFORMANCE PREDICTIONS WHEN RESERVOIR CONDITIONS CHANGE
The generalized IPRs presented in the previous section are useful in allowing the petroleum engineer to calculate the pressure and production behavior of an oil well given the necessary test infonnation. The resulting
4 GENERALIZED INFLOW PERFORMANCE RELATIONSHIPS FOR THREE-PHASE FLOW
SPE 25458
estimates of flowing pressure or production rates assume that there is no change in reservoir conditions from those under which the well test was made. This is fine for many situations where one desires to estimate the effect of changing the flowing pressure on the production rate, or the effect on the flowing pressure if the rate is changed.
There are times, however, when the engineer desires to estimate the pressureproduction behavior under reservoir conditions that are different from those at which the well test was conducted. The two primary conditions of interest are changes in flow efficiency and at different stages of reservoir depletion. Changes in flow efficiency are of interest when one is considering a stimulation treatment to remove damage or improve permeability near the wellbore. The effects of depletion are encountered in predicting future performance at an average reservoir pressure less than the test pressure. In this section, we will look at using test data to predict well performance when reservoir conditions have changed.
Changes in Flow Efficiency
Flow efficiency can be defined as the ratio of the measured production rate to the ideal production rate. The ideal production rate is that rate which would be observed at the measured well bore pressure if skin equals zero. In equation form, this reduces to
T. In- 3
Tw 4 (6) E, = T. 3 In- - - +$
Tw 4
This definition of flow efficiency allows the ratio of the maximum production rates with and without skin to be written as
(7)
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Eqs. 6 and 7 can be used with well test information to study the effects of changes in flow efficiency.
To utilize the proposed method, one would estimate the maximum oil and water production rates from the generalized threephase IPRs (Eqs. 4 and 5) and the flow efficiency from Eq. 6 using the skin factor estimated from a transient well test. It should be noted that large errors in estimating the outer boundary radius of the reservoir results in small errors in the flow efficiency. The maximum flow rates for the oil and water phases without skin are then estimated from Eq.7.
Once the maximum flow rates are determined at a flow efficiency of one, Eq. 7 can be used to predict the maximum production rates at a new flow efficiency. Inflow performance curves are then predicted for the well at the new flow efficiency by using the generalized IPRs.
Table 7 presents a comparison of the proposed method to account for changes in skin during three-phase flow to simulator results. The maximum production rates calculated and presented in the table are from selected test information. The resulting error between the calculated maximum rates and simulator rates includes errors in the generalized IPRs and error in the flow efficiency approximation, Eq. 6. As indicated, the proposed method does a good job of estimating the maximum flow rates for the cases studied.
Predicting Future Performance
If we apply the Taylor series approach proposed by Wiggins, Russell and Jennings in developing the analytical IPR, we can write the present maximum flow rate as
(8)
where D is related to the mobility function by
SPE 25458 MICHAEL L. WIGGINS 5
(9)
The subscript p in Eq. 8 indicates present conditions. If we relate the maximum production rate at some future time to the current maximum rate, we obtain
qtl,JrJU1 = Pr,£D]n,-o
qtl,JrJU, Pr,[D]u,~ (10)
where the f subscript refers to future conditions.
Eq. 10 states that the ratio of the maximum production rate at some future reservoir pressure to the current maximum production rate is related to the ratios of the reservoir pressures and the mobility function terms, D. Since the mobility function terms are functions of the average reservoir pressure, Eq. 10 suggests that the production rate ratio can be written as a polynomial in the ratio of average reservoir pressures.
Maximum oil rate ratios versus the average pressure ratios for all the cases studied in this research are presented in Fig. 7. This information appears to follow a quadratic relationship. As indicated, there is some variation between the curves due to relative permeability and fluid property effects; however, there is no great deviation in the curves. This agrees with the information studied in developing the generalized IPR. Fig. 8 presents the same comparison information for the water phase.
The information presented in Figs. 7 and 8 was fit with a linear regression model of the form
(11)
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As the average reservoir pressure decreases, we see a corresponding decrease in the maximum flow rate. When the average reservoir pressure reaches zero, there is physically no flow from the reservoir. Consequently, a linear regression model with no intercept was chosen.
The resulting relationship to predict the future maximum oil rate is
qtl,m&x, -vJ P r ] qtl,JrJU, = O.15376 __ -lpr:
(12)
while the relationship for water is
qW,m&x1 = 0.59245433(Pr/ ] qw,m&x Pr. p p
(13)
The statistical information for this analysis is presented in Tables 8 and 9. The coefficient of determination for the two relationships is greater than 0.9, indicating a good fit of the information. The F -test indicates that the model is adequate to describe the information while the t-test shows the coefficients are significant.
To use the proposed future performance method, one would estimate the maximum production rates from the generalized IPRs (Eqs. 4 and 5). The maximum future production rates can be estimated from Eqs. 12 and 13 at the desired average reservoir pressure. New inflow performance curves at the future depletion stage can be developed by using the generalized IPR equations with the desired reservoir pressure and maximum future production rates.
Tables 10-12 present a comparison of simulator results and future production rates predicted by the proposed future performance
6 GENERALIZED INFLOW PERFORMANCE RELATIONSHIPS FOR THREE-PHASE FLOW
SPE 25458
method. The results presented in these tables indicate that the error increases as we estimate further in time, however, on an absolute basis, the predictions are within reasonable engineering accuracy.
The analysis suggests that care should be taken in estimating future performance over large stages of depletion as the error may increase. This error may not be significant if the absolute difference in production values are small, as indicated by several of the examples. Based on analysis of information used in developing this method, one should exercise caution in predicting future rates at reservoir pressure ratios less than 70%. While estimates at pressure ratios less than 70% may be relatively accurate, they may contain significant errors. It is recommended that initial future performance estimates be updated every six months to one year. This would progressively reduce the uncertainty in earlier estimates as depletion occurs in the reservoir.
APPLICABILITY
The proposed IPRs and methods presented in this research for three-phase flow were developed from analysis of multiphase flow in bounded, homogeneous reservoirs where there is no external influx of fluids into the reservoir, and apply to the boundarydominated flow regime. The methods are limited by the following assumptions: 1) all reservoirs are initially at the bubble point; 2) no initial free gas phase is present; 3) a mobile water phase is present for three-phase studies; 4) Darcy's law for multiphase flow applies; 5) isothermal conditions exist; 6) there is no reaction between reservoir fluids and reservoir rock; 7) no gas solubility exists in the water; 8) gravity effects are negligible; and 9) there is a fully penetrating wellbore. Strictly speaking, the methods cannot be considered correct when other types of reservoir conditions exist, and the engineer should exercise great care in utilizing the proposed methods.
From a practical viewpoint the
488
proposed methods may have limited applicability, since very few reservoirs completely satisfy the assumptions. One might speculate that the methods have merit under less stringent conditions than those under which they were developed. Examples would include: reservoirs that have very limited water influx; reservoirs that initially had no mobile water phase but began producing water due to limited water influx; large reservoirs experiencing water influx where portions of the reservoir are isolated from the influx by producing wells nearer the reservoir boundaries. Other examples might include reservoirs that are relatively thin with respect to the drainage area where gravity effects are negligible, and partially penetrating wells where there is little vertical permeability. These examples are only speculation and further research is required before the proposed methods can be extended to these situations.
CONCLUSIONS
1. Generalized three-phase IPRs have been presented that are suitable for use over a wide range of reservoir properties. The proposed relationships are Vogel-type IPRs that require single point estimates of oil and water production rates, flowing wellbore pressure and average reservoir pressure.
2. The generalized IPRs have been verified using information presented by Sukarno and by comparison to the three-phase methods of Brown and Sukarno. The proposed method yielded results as reliable as these two methods while being much simpler to use.
3. A method has been presented to estimate pressure-production behavior due to changes in flow efficiency. The method appears to yield suitable results with maximum errors between the predictions and simulator results being less than 15% for the cases studied. This error includes errors from the generalized IPR and the definition of flow efficiency.
SPE 25458 MICHAEL L. WIGGINS 7
4. A method has been proposed for predicting future performance that is similar in form to a Vogel-type IPR The method is suggested by the Taylor series expansion of the multiphase flow equations proposed by Wiggins, Russell and Jennings. To the author's knowledge, no one has proposed a method for predicting future performance during three-phase boundary-dominated flow.
NOMENCLATURE
Ef
~ p
Pr Pwf
CIo CIo,rnax
oil formation volume factor, RB/STB flow efficiency, dimensionless relative permeability to oil pressure, psi average reservoir pressure, psi flowing wellbore pressure, psi oil production rate, BOPD maximum oil production rate, BOPD water production rate, BWPD maximum water production rate, BWPD external boundary radius, ft wellbore radius, ft skin factor, dimensionless regression coefficient oil viscosity, cp
REFERENCES
1. Darcy, H.: Les Fontaines Publiques de la Ville de Dijon, Victor Dalmont, Paris (1856) 590-594.
2. Evinger, H.H. and Muskat, M.: "Calculation of Theoretical Productivity Factors", Trans., AIME (1942) 146, 126-139.
3.
4.
5.
6.
7.
8.
9.
10.
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Evinger, H.H. and Muskat, M.: "Calculation of Productivity Factors for Oil-gas-water Systems in the Steady State", Trans., AIME (1942) 146, 194-203. Vogel, J.V.: "Inflow Performance Relationships for Solution-Gas Drive Wells", JPT (Jan. 1968) 83-92. Fetkovich, M.J.: "The Isochronal Testing of Oil Wells", paper SPE 4529 presented at the 1973 SPE Annual Meeting, Las Vegas, NV, Sept. 30 - Oct. 3. Rawlins, E.L. and Schellhardt, M.A: Backpressure Data on Natural Gas Wells and Their Application to Production Practices, USBM (1935) 7. Brown, KE.: The Technology of Artificial Lift Methods, PennWell Publishing Co., Tulsa, OK (1984) 4, 18-35. Sukarno, P.: "Inflow Performance Relationship Curves in Two-Phase and Three-Phase Flow Conditions", Ph.D. dissertation, U. of Tulsa, Tulsa, OK (1986). Wiggins, M.L., Russell, J.E. and Jennings, J.W.: "Analytical Inflow Performance Relationships for ThreePhase Flow in Bounded Reservoirs", paper SPE 24055 presented at the 1992 Western Regional Meeting, Bakersfield, CA, Mar. 30-Apr. 1. Freund, RJ. and Littell, RC.: SAS System for Regression, SAS Institute, Cary, NC (1986).
8 GENERALIZED INFLOW PERFORMANCE RELATIONSHIPS FOR THREE-PHASE FLOW