Inflow Performance Relationship Wiggins, M.L

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

2 GENERALIZED INFLOW PERFORMANCE RELATIONSHIPS FOR THREE-PHASE FLOW

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.

485

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


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)

486

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


(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)

487

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


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.


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.

489

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).


SPE 25458

Table 1. Reservoir Properties

Property Case 2 Case 3 Case 4 CaseS

Porosity 0.18 0.12 0.20 0.24

Permeability 15.0md 1O.0md l00.0md 5O.0md

Height 25 ft 10ft 10ft 25ft

Temperature 150°F 175° F 200° F 200° F

Initial Pressure 2500 psi 3500 psi 1500 psi 2600 psi

Oil Gravity 25.0° API 45.0° API 15.0° API 35.0° API

Gas Gravity 0.6 0.7 0.6 0.7

Water Solids 12.0% 30.0% 15.0% 18.0%

Residual Oil Saturation 0.35 0.10 0.45 0.05

Irreducible Water

Saturation 0.20 0.10 0.30 0.50

Critical Gas Saturation 0.050 0.000 0.025 0.075

Drainage Radius 1085 ft 506 ft 506ft l085ft

WeUbore Radius 0.328ft 0.328 ft 0.328ft 0.328ft

Table 3. SAS Statistics for Water IPR

DEP VARIABLE: QWRATI ANALYSIS OF VARIANCE

SUM OF MEAN SOURCE OF SQUARES SQUARE F VALUE PROB>F

HODEL 2 160.09582759 80.04791379 124348.602 0.0001 ERROR 408 0.26264508 0.0006437379 U TOTAL 410 160.35847267

ROOT MSE 0.02537199 R-SQUARE 0.9984 DEP MEAN -0.564403 ADJ R-SQ 0.9984 C.V. -4.49537

NOTE: NO INTERCEPT TERM IS USED. R-SQUARE IS REDEFINED.

PARAMETER ESTIMATES

STANDARD T FOR HO: PARAMETER VARIABLE OF ESTIMATE ERROR PARAMETER-O PROB>ITI

PRAT PRAT2

1 -0.7222350.009044375 1 -0.284777 0.01139171

-79.855 -24.999

0.0001 0.0001

490

Table 2. SAS Statistics for Oil IPR

DEP VARIABLE: QORATI ANALYSIS OF VARIANCE

SOURCE DF SUM OF

SQUARES MEAN

SQUARE F VALUE PROB>F

MODEL 2 144.14973559 72.07486780 141748.766 0.0001 ERROR 408 0.20745539 0.0005084691 U TOTAL 410 144.35719098

ROOT MSE 0.02254926 R-SQUARE 0.9986 DEP MEAN -0.526888 ADJ R-SQ 0.9986 C.V. -4.2797


VARIABLE DF PRAT 1 PRAT2 1

PARAMETER ESTIMATES

PARAMETER STANDARD ESTIMATE ERROR

-0.519167 0.008038153 -0.481092 0.01012434

T FOR HO: PARAMETER&O

-64.588 -47.518

PROB>ITI 0.0001 0.0001

Table 4. Comparison of Proposed IPR to Other Methods

Using Information in Sukamo's Table 6-24

Test Information: fwf,fsi fr,~i So,BOPD Sw,BWPD

1155 2100 176.31 50.16

Simulator Wiggins Sukarno Brown

fwf,fsi S2:BOPD S",BOPD S",BOPD S",BOPD

1995 23.06 2252 23.21 22.82

1785 66.10 65.48 66.52 65.94

1575 105.88 105.45 10650 10554

1365 142.66 142.44 143.14 141.38

1155 176.31 176.44 176.44 173.20

945 207.oI 207.46 206.41 200.71

735 234.45 235.50 233.05 223.59

525 259.00 260.56 256.35 241.46

315 279.34 282.62 276.31 253.89

0 301.77 310.13 300.00 264.46

Simulator Wiggins Sukarno Brown

fwf,~i SW, BWPD 9w,BWPD Sw,BWPD 9w , BWPD

1995 5.91 5.51 6.02 6.47

1785 17.54 17.47 17.65 18.71

1575 28.85 28.89 28.89 29.94

1365 39.73 39.75 39.71 40.11

1155 50.16 50.06 50.06 49.14

945 60.01 59.81 59.90 56.94

735 69.18 69.02 69.18 63.43

525 77.46 77.67 77.85 68.50

315 84.64 85.77 85.86 72.03

0 92.67 96.89 96.50 75.03




Test Information:

pwf,psi

2527

2261

1995

1729

1463

1197

931

665

399

o

pwf, psi

2527

2261

1995

1729

1463

1197

931

665

399

o

pwf, psi 1463

Simulator

<jo, BOPD

9.98

27.60

44.82

60.67

76.18

90.44

102.81

117.08

124.45

135.44

Simulator

qw,BWPD

6.92

19.62

32.64

45.75

58.29

70.61

82.40

93.92

102.45

112.85

pr,psi 2660

Wiggins

q",BOPD

9.72

28.25

4550

61.46

76.13

89.51

101.61

112.42

121.94

133.81

Wiggins

qw,BWPD

6.43

20.38

33.69

46.35

58.37

69.75

SO.49

90.68

100.02

112.99

qo,BOPD 76.18

Sukarno

q",BOPD

9.96

28.60

45.83

61.68

76.13

89.18

100.84

111.10

119.97

130.66

Sukarno

qw, BWPD

7.12

20.SO

33.92

46.46

58.37

69.62

SO.15

89.92

98.89

110.73

qw, BWPD 58.29

Brown

So,BOPD

9.42

27.47

44.36

59.93

73.95

86.15

96.13

103.37

107.49

112.76

Brown

qw, BWPD

7.22

21.06

34.02

45.95

56.71

66.06

73.71

79.26

82.42

86.46

Table 7. Comparison of Predicted Maximum Production Rates to

Simulator Results in the Presence of Skin

Case 2, 40% Initial Water Saturation

Skin -2 +5

+20

Skin

-2 +5 +20

pwf,psi 1223 937 60

pwf,psi 1223 937 60

Pr,psi 1863 2045 2163

1863 2045 2163

Test

SoBOPD 25.00 20.00 15.00

Test qw,

BWPD 3.98 2.73 1.82

Case 3, 50% Initial Water Saturation

Skin -2 +5

+20

Skin -2 +5 +20

pwf,psi 1047 902 465

?wf,psi 1047 902 465

Pr,psi 2716 2886 2003

p"psi 2716 2886 2003

Test

SoBOPD 200.00 100.00 25.00

Test Sw,

BWPD 183.19 82.60 25.75

Simulator Predicted qo,max, BOPD (5=0) 38.95 48.49 54.85

Simulator qw,max,

BWPD (S=O) 6.15 6:87 7.28

Simulator

qo.max, BOPD (5=0)

200.03 220.95 117.36

qo,max, BOPD (s=O) 40.28 50.82 56.63

Predicted qw,max, BWPD

(5=0) 6.41 6.94 6.88

l'Tedkted qo,max, BOPD (5=0)

191.28 222.39 118.15

Simulator Predkted qw,max, qw,max, BWPD BWPD

(5=0) 177.78 191.55 122.39

(5=0) 175.20 183.69 121.71

Difference, BOPD -1.33 -2.33 -1.78

Difference,

BWPD -0.26 -0.06 0.41

Difference, BOPD 8.75 -1.44 -o.SO

Difference,

BWPD 2.58 7.86 0.68

Percent Error -3.42 4.SO -3.24

Percent Error 4.15 -0.93 5.61

Percent Error 4.38 -0.65 -0.68

Percent Error 1.45 4.10 0.55

491



Test Information: Pwf,psi pr,psi qo,BOPD qw, BWPD

pwf, psi

1488

1328

1168

1008

848

688

528

368

208

o

pwf,psi

1488

1328

1168

1008

848

688

528

368

208

o

1008

Simulator

q",BOPD

1.63

3.82

5.87

7.76

9.52

11.16

12.60

13.91

15.07

16.29

Simulator

qw, BWPD

6.68

15.99

25.01

33.58

41.87

49.71

56.85

6350

6951

75.93

1600

Wiggins

q",BOPD

1.63

3.83

5.87

7.76

9.50

11.08

12.51

13.78

14.89

16.11

Wiggins

qw, BWPD

6.37

15.87

24.92

33.54

41.71

49.44

56.72

6357

69.97

77.64

7.76

Sukarno

q",BOPD

1.65

3.84

5.88

7.76

9.49

11.05

12.45

13.70

14.78

15.95

Sukarno

qw, BWPD

6.77

16.07

25.00

33.54

41.67

49.37

56.61

63.37

69.62

76.94

33.58

Brown

q",BOPD

151

3.62

5.68

7.66

9.53

11.18

12.30

13.18

14.05

15.19

Brown

qw, BWPD

6.51

15.64

24.53

33.09

41.15

48.28

53.13

56.91

60.70

65.62

Table 8. SAS Statistics for Oil Future Performance Relationship

DEP VARIABLE: QORAT ANALYSIS OF VARIANCE

SOURCE DF SUM OF

SQUARES MEAN

SQUARE F VALUE PROB>F

MODEL ERROR

2 74.45855221 234 0.37840779 236 74.83696000

37.22927611 0.001617127

23021.859 0.0001

U TOTAL

ROOT MSE 0.04021352 R-SQUARE 0.9949 DEP MEAN 0.4476028 ADJ R-SQ 0.9949 C.V. 8.984198


VARIABLE DF

PRAT PRAT2

PARAMETER ESTIMATES

PARAMETER ESTIMATE

STANDARD T FOR HO: ERROR PARAMETER-O

0.15376309 0.01896232 0.83516299 0.02247023

8.109 37.168

PROB>ITI

0.0001 0.0001

10 GENERALIZED INFLOW PERFORMANCE RELATIONSIDPS FOR THREE-PHASE FLOW

SPE 25458

Table 9. SAS Statistics for Water Future Performance Relationship

DEP VARIABLE: QWRAT ANALYSIS OF VARIANCE

SUM OF MEAN SOURCE DF SQUARES SQUARE F VALUE PROB>F

MODEL 2 81. 58898652 40.79449326 30719.520 0.0001 ERROR 234 0.31074416 0.001327966 U TOTAL 236 81.89973068

ROOT MSE 0.03644127 R-SQUARE 0.9962 DEP MEAN 0.4943643 ADJ R-SQ 0.9962 C.V. 7.371341


VARIABLE DF

PRAT PRAT2

PARAMETER ESTIMATES

PARAMETER ESTIMATE

STANDARD ERROR

0.59245433 0.01718355 0.36479178 0.0203624

T FOR HO: PARAMETERaO

34.478 17.915

PROB>ITI

0.0001 0.0001

Table 11. Comparison of Simulator Results and Future Performance

Predictions Using Proposed Relationship for Case 3

Test Information: 20% Initial Water Saturation pr,p' psi 90,max,p' BOPD 9w,max,p, BWPD

3172 416.54 49.44

Simulator Calculated Difference Percent qo,max,l, qo,max,(, Difference

pr,f,psi BOPD BOPD BOPD % 2671 317.77 300.60 17.17 5.40 1790 155.83 146.93 8.91 5.72 549 25.07 2151 3.56 14.20

Simulator Calculated DiUerence Percent qw,max,{, qW,max,(, Difference

pr,f, psi BWPD BWPD BWPD % 2671 40.03 37.46 2.58 6.43 1790 24.37 22.27 2.10 8.61 549 6.57 5.61 0.96 14.59

Test Information: 50% Initial Water Saturation pr,p' psi 9o,max,po BOPD 9w,max,p, BWPD

3364 264.73 227,96

Simulator Calculated Difference Percent qo,max,f, qo,max,l, Difference

pr,f' psi BOPD BOPD BOPD % 2945 227.87 205.08 22.79 10.00 1900 107.09 93.52 13.57 12.69 445 11.66 9.25 2,41 20.64

Simulator Calculated DiUerence Percent qW,max,f, qw,rnax,f, Difference

pr,f, psi BWPD BWPD BWPD % 2945 196.23 181.96 14.26 7.27 1900 114.92 102.81 12.11 10.54 445 23.11 19.32 3.79 16.41 492



Test Information: 30% Initial Water Saturation pr.p, psi 90,max,p' BOPD 9w,max,p, BWPD

2375 96.34 0.63

Simulator Calculated DiUerence Percent 90,max,f, qo,max,l, Difference

pr,f,psi BOPD BOPD BOPD % 1886 56.17 62.50 ~.33 -11.28 1447 29.02 38.89 -9.87 -34.04 633 5.86 9.66 -3.81 ~.99

Simulator Calculated Difference Percent qw .. max,(, qW,max,f, DiHerence


Test Information: 40% Initial Water Saturation pr.p, psi 90,max,po BOPD 9w,max,,,, BWPD

2428 76.47 9.59

Simulator Calculated Difference Percent qo,max,f, qo,max,l, DiHerence

pr,f, psi BOPD BOPD BOPD % 2031 47.73 54.52 ~.79 -14.23 1321 1754 25.30 -7.76 -44.23 420 2.36 3.94 -1.59 ~7.51

Simulator Calculated Difference Percent qW,max,(, qw,max,f, Difference

pr,f, psi BWPD BWPD BWPD % 2031 6.82 7.20 -C.38 -5.64 1321 4.21 4.13 0.08 2.01 420 1.24 1.09 0.16 12.48



Test Information: 40% Initial Water Saturation Pr.p, psi 90,max,,,, BOPD 9w,max,,,, BWPD

1333 5151 1.04

Simulator Calculated Difference Percent qo,ma",f, qo,max,f, Difference

pr,f, psi BOPD BOPD BOPD % 1155 37.45 39.16 -1.71 -4.57 926 23.56 26.26 -2.71 -11.49 507 7.99 9.24 -1.24 -15.50

Simulator Calculated DiUerence Percent qw,rr.ax,{, qw,max,l, DiUerence


Test Information: 50% Initial Water Saturation pr,p, psi 9o,max,,,, BOPD 9w,max,p, BWPD

1421 34.39 18.06

Simulator Calculated Difference Percent qo,max,l, qo,max,f, Difference

pr,f,psi BOPD BOPD BOPD % 1244 27.33 26.64 0.69 2.52 1022 18.41 18.66 -C.25 -1.34 596 6.92 7.27 -C.35 -5.12

Simulator Calculated Difference Percent qw,max,f, qw,max,f, DiHerence

pr,f, psi BWPD BWPD BWPD % 1244 14.85 14.42 0.43 2.93 1022 1150 11.10 039 3.43 596 5.65 5.65 0.01 0.12


1000 2000

pwf,psia

3000 4000

Fig. 1. Oil inflow performance curves for Case 3, 20% Swi, at several stages of depletion generated from simulator results.

1.0 0

0

% 0

0.8 00

cDo /1

~ 0.6 'b

e DC

g. t5% ......

0 0" 0.4 0

00 0

& 0.2

, \

0.0 0.0 0.2 0.4 0.6 0.8 1.0

pwf/pr

Fig. 3. OilIPR curves for Case 3, 20% Swi.

493

Fig. 2 Water inflow performance curves for Case 3, 20% Swi, at several stages of depletion generated from simulator results.

1.0 0

0

°cP 0.8

0

0 00

0

l;l 0.6 0

~ a:t

0 ...... ~ [JJ 0" 0.4 Bo

0

Cu::J 0.2 IC

QJ c

~ 0.0

0.0 0.2 0.4 0.6 0.8 1.0

pwf/pr

Fig. 4. Water IPR curves for Case 3, 20% Swi.

12 GENERALIZED INFLOW PERFORMANCE RELATIONSIDPS FOR THREE-PHASE FLOW

SPE 25458

1;l

~ "-S-

1.0 • Simulator Results

0.8

0.6

0.4

0.2

qo/qomax -1.0000 - 0519167 (Pwf/pr) - 0.481092 (pwf/prl"2 0.0

0.0 0.2 0.4 0.6 0.8

pwf/pr

1.0

Fig. 5. Comparison of simulator results with generalized oil IPR.

1.0

• Simulator Results

0.8 - Pmposed Relation

y - 0.15376309 x + D.83516299 ><"'2

Po. x e 0.6

g. "-] • Ii.

~ 0.4 'II'" C7' .......

. F'alf. • •

.. . 0.2

.. . ..

O.o~~~-'----~-----r----~ __ r-~-----r--------i 0.0 0.2 0.4 0.6 0.8

pr,f/pr,p

Fig. 7. Comparison of simulator results to proposed method for determining future performance for the oil phase.

1.0

494

Po. 1;l' Ii. ~ 0' "-..... 1;l

l

1.0

0.8

1;l 0.6

t "-

~ 0.4

0.2


qw / qwmax - 1.0000 - 0.722235 (pwf/pr) -0.284m (pwf/prl"2

O'O+-----~_r----~----r-~----,_----~_,----~~ 0.0 0.2 0.4 0.6 0.8 1.0

pwf/pr

Fig. 6. Comparison of simulator results with generalized water IPR.

1.0


- Proposed Relation

0.8

Y - 059245433x + 0.364?9178x"2

0.6

0.4

0.2

o.o~~~-r----~~-----------'------~-r------~ 0.0 0.2 0.4 0.6 0.8 1.0

pr,f/pr,p

Fig. 8. Comparison of simulator results to proposed method for determining future performance for the water phase.

Inflow Performance Relationship Wiggins, M.L

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