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A comparison of oil-water slip velocity models used for production log interpretation
Journal of Petroleum Science and Engineering, 8 ( 1992 ) 181-189 Elsevier Science Publishers B.V., Amsterdam
181
A comparison of oil-water slip velocity models used for production log interpretation A.D. Hill Department of Petroleum Engineering, The University of Texas at Austin, Austin, TEX 78712, USA
(Received February 10, 1992; revised version accepted April 15, 1992 )
ABSTRACT Hill, A.D., 1992. A comparison of oil-water slip velocity models used for production log interpretation. J. Pet. Sci. Eng., 8: 181-189.
The accuracies of five slip velocity models that can be used to interpret production logs in oil-water flow have been examined using the flow loop data of Davarzani and Miller. Optimal values of the constants used in the models were determined for each model; the abilities of each model to predict oil and water volumetricflowrates from measuredvalues of water holdup and mixture velocitywere then tested. The empirical model of Nicolas and Witterholt performedthe best of the modelstested.
Introduction The production o f excessive a m o u n t s of water is a pervasive problem in oil production wells, and knowing the location and rate of water entries into a well is crucial in planning remedial action for a particular well or for general reservoir management purposes. Water entry locations are most c o m m o n l y detected through the use o f production logging instruments such as spinner or basket flowmeters, density tools, and capacitance tools. This means, however, o f detecting water entries is somewhat indirect. With these logs, flow properties are measured at a n u m b e r o f discrete locations in the well, or are recorded continuously throughout the well. The log responses at discrete depth locations are then interpreted to obtain the volumetric flow rates o f oil and water at these locations. (Note: In m a n y wells, Correspondence to: A.D. Hill, Department of Petroleum Engineering, The University of Texas at Austin.
gas will also be flowing as a free phase, but this paper will be limited to the case o f oil-water flow only. ) Finally, the rates and locations o f water entries are determined from the differences in the wellbore flow rates. The most c o m m o n approach taken, when logging oilwater flow, is to measure a fluid velocity and either fluid density or fluid capacitance. The velocity measured is assumed to be the average total velocity which is equal to the sum of the oil and water volumetric flow rates divided by the cross-sectional area of the wellbore. F r o m the density or capacitance log, the fraction o f the pipe occupied by water (the water holdup) is inferred. With these measurements, it is then necessary to independently obtain a third parameter, the slip velocity, in order to calculate the volumetric flow rates o f each phase. Thus, the accuracy o f the log interpretation will depend on the accuracies o f the logging measurements themselves and on the accuracy of the m e t h o d used to estimate the slip velocity. In this paper, we ex-
0920-4105/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
182 amine several m e t h o d s or correlations for estimating slip velocity in oil-water flow. Using the oil-water flow data of Davarzani and Miller ( 1983 ), the accuracy of each m e t h o d in interpreting oil and water volumetric flow rates is evaluated. The results show that the empirical model of Nicolas and Witterholt (1972) is the best available slip velocity model for interpreting production logs in oil-water flow.
Production log interpretation in oil-water flow Consider the hypothetical oil well producing oil and water as shown in Fig. 1. I n a s m u c h as water is being p r o d u c e d from the intermediate zone, opposite this zone and above it, the flow stream will consist of both oil and water. Below Zone B, there will likely be some water present that has fallen down from Zone B (Zhu and Hill, 1988); however, this significant complication is not usually considered in production log interpretation and will not be considered here. In any case, the only mobile phase below Zone B is oil. To determine the production rate of water from Zone B, we would m a k e p r o d u c t i o n logging measurements just above and just below
Zone A
Zone B
Zone C
Fig, 1. A wellbore with oil and water entries.
A,D.HILL the zone (locations I and 2 in Fig. 1 ). F r o m the difference in the interpreted water flow rates at locations 1 and 2, the Zone B production rate is obtained. Considering now location 1, we assume that the m e a n velocity, Vm, and the average density, Pro, have been measured with, e.g., a basket flowmeter and a g r a d i o m a n o m e t e r TM*. F r o m these measurements, we can calculate (Hill, 1990) the water holdup from: YwP m - - f l o Pw--Po
( 1)
and the oil and water volumetric flow rates as: qo=A [ l -Yw] [ V~Vw+ Vm]
(2)
and:
qw=Ayw[Vm-Vs( 1 -Yw) ]
(3)
In Eqs. 2 and 3, A is the wellbore cross-sectional area and we have introduced a parameter, vs, the slip velocity, which m u s t be independently estimated. The slip velocity is a representation of the holdup p h e n o m e n o n in two-phase flow; i.e., the fact that the in-situ fractions of each phase in the pipe are not, in general, equal to the input fractions because the lighter phase is moving at a higher velocity than the denser phase. The slip velocity in oil-water flow is likely to be a complex function of fluid properties, pipe diameter, water holdup, and the flow rates of each phase. As the calculated volumetric flow rates of oil and water d e p e n d linearly on the slip velocity, its accurate estimation is crucial to accurate production log interpretation.
Slip velocity models Several different models of oil-water slip velocity have been used to interpret production logs, including assuming the slip velocity to be constant, empirical models based on lab, TM is trademark of Schlumberger Well Services. (See Chilingarian et al., 1987. Surface Operations in Petroleum Production, I. Elsevier, Amsterdam, 821 pp. )
A COMPARISON OF OIL-WATER SLIP VELOCITY MODELS USED FOR PRODUCTION LOG INTERPRETATION
oratory data and theoretically-based models of two-phase flow. In this paper, five slip velocity models are examined; a description of each model follows. (1) Constant slip velocity. The simplest model of slip velocity is to assume that it is constant, independent of the flow rate of each phase and of the holdup. In production log interpretation, sometimes the slip velocity is estimated in the upper part of the well based on knowledge of the surface flow rates, then assumed constant throughout the well. (2) Nieolas-Witterholt model. Based on a series of experiments in 4- and 6-inch I.D. by 50-ft long pipes, Nicolas and Witterholt (1972) suggested the following empirical model of slip velocity: Vs-Ywn voo
(4)
where 1/2<~n~2 and: I"
. "11/4
voo= 1.53L~ ]
(5)
Equation 5 is the Harmathy (1960) equation for the terminal rise velocity of a bubble. A comparison of the slip velocity with the experimental data presented in the Nicolas and Witterholt paper showed the model with n = 1 best fit the data. Thus, in this paper, the Nicolas-Witterholt model is Eq. 4 with n = 1. (3) Drift-flux model (Zuber and Findlay, 1965 ). Zuber and Findlay postulated a theoretical model of two-phase slip behavior that is referred to as the drift-flux model. Derived for gas-liquid bubble flow, but theoretically applicable to any upwards, vertical, two-phase flow, the drift flux model accounts for two mechanisms of slip. First, because the lighter phase is thought to be more concentrated in the center of the pipe where the velocity is highest, the light phase will be moving fast relative to the mixture as a whole. Second, there will be a relative velocity difference between the phases because of the density difference.
183
These assumptions lead to the following equation for the in-situ average velocity of the lighter phase (oil in this case):
vo=Covm+Voo
(6)
Zuber and Findlay found that the concentration profile correction factor, Co, ranged from 1.0 to 1.5 for air-water flow. voo is again defined by Eq. 5. The slip velocity for the driftflux model is: vs-
(Co-1)Vm+Voo Yw
(7)
Notice the distinct contrast between the Nicholas-Witterholt model, in which the slip velocity depends linearly on Yw,and the drift-flux model, in which slip velocity is inversely proportional to water holdup. (4) Hasan-Kabir (1988) model. Hasan and Kabir presented a model of oil-water flow which was based on the drift-flux model. For bubble flow, the Hasan-Kabir model for slip velocity is: v~-
(Co-
1)Vm
Yw
q-vooyw(1-yw)
(8)
where Co and v~o are defined the same as for the drift-flux model. (5) Modified drift-flux model. A modification to the drift-flux model that we might expect to be applicable to high water holdup cases is obtained by substituting the in-situ average water velocity, Vw, for the mixture, Vm. Theoretically, this implies that, in the absence of a drift velocity (voo=0), the effect of the concentration distribution is to make the oil velocity a factor of Co times the continuous phase velocity (Vw) rather than the mixture velocity. Thus, the modified drift-flux model is:
vo=Covw+voo
(9)
The slip velocity for the modified drift-flux model is:
184
v,-
A.D. HILL
(Co-1)Vm+Vo~
(10)
yw+Co(1-yw)
Davarzani-Miller oil-water flow data Davarzani and Miller ( 1983 ) conducted an extensive set of oil-water flow experiments in which the water holdup was measured over a range of oil and water flow rates. Ninety-one individual tests were performed in a 6½ inch I.D. by 30-ft long flow loop. The water holdup was measured both with a ?-ray densitometer and by suddenly shutting in the test section on some tests. The holdups measured by the two methods agreed within 1%. Because Davarzani and Miller desired a broad range of water holdup values, their data does not have a uniform matrix of oil and water flow rates. Instead, there is a significantly greater number of tests with high oil rate relative to the water rate than vice versa. Davarzani and Miller's test conditions are summarized in Table 1. Inasmuch as the oil and water flow rates are known and the water holdup has been measured, the slip velocity can be calculated for each of Davarzani and Miller's tests. From the definition of slip velocity as the difference between the oil and water in-situ average velocities, we have:
v,=
;[
qo
(1-y,~)
(11)
or, in terms of superficial velocities (as the flow rates were reported by Davarzani and Miller):
Vso v s - 1-yw
(12)
The slip velocities measured in the Davarzani-Miller tests as functions of water holdup are shown in Fig. 2. Note that there appears to be a significant difference in slip velocity behavior for values of yw below about 0.4 compared with the behavior for Yw above 0.4. Below about 0.4, there is considerable scatter in the data and slip velocity appears to generally decrease with increasing Yw; above 0.4, the data is much smoother and slip velocity is increasing with increasing Yw. It is likely that several factors are contributing to this difference. First, the lower the water holdup, the larger the errors in the water holdup measured by the y-ray densitometer. Inasmuch as the holdup is used to calculate slip velocity, errors in the measured holdup will give errors in slip velocity (large errors at low Yw, because Ywis in the denominator). Second, below yw=0.4, the external phase is oil, not water; it is not surprising that the oil velocity behavior is different when oil is the external phase. Some of these factors can be observed by examining the slip velocity as a function of holdup for a constant mixture velocity (Figs. 3, 4 and 5). For a mixture velocity of 29.4 ft/ min, the slip velocity decreases with increasing water holdup as seen in Fig. 3. With higher mixture velocities, however, of 58.2 or 87 It/ min (Figs. 4 and 5 ), the slip velocity is apparently increasing with Yw, and there is consid-
TABLE 1 Davarzani-Miller oil-water flow data Oil: kerosene Water: fresh tap water Oil flow-rate range: Water flow-rate range: Input water-fraction range: Water holdup range:
Vsw Yw
p=0.78 g/cm3; /z=2 cP p = l . 0 0 g / c m 3 ; ~=1 cP 0.0115 ft3/s ( 177 b/d)-0.435 ft3/s (6700 b / d ) 0.0023 ft3/s (35 b/d)-0.0553 ft3/s (850 b / d ) 0.025-0.802 0.029-0.928
A COMPARISON OF OIL-WATER SLIP VELOCITY MODELS USED FOR PRODUCTION LOG INTERPRETATION
30
18 5
30
[]
20 =_
>
20
m
10
0
i
0,0 -10
i 0.2
O
i 0.4
i 0.6
i 0.8
0.2
Yw
1.0
Fig. 5. Slip velocityfor mixture velocityof 87 ft/min.
Yw
Fig. 2. Measured slip velocities from Davarzani-Miller experiments. 25
rn m
20'
0.1
high Ywdata because it is likely more accurate. Fortunately, in production logging, we are more often concerned with situations with high water holdup.
[]
Comparison of slip velocity models
15'
10'
5
0
i
0.3
0.2
0.4
Yw
Fig. 3. Slip velocityfor mixture velocityof 29.4 ft/min. 25
20 ¸
15-
10
5
0 0.0
i
i
i
0.1
0.2
0.3
0.4
Yw
Fig. 4. Slip velocityfor mixture velocityof 58.2 ft/min. erable scatter in the data, particularly at 87 ft/min. In considering slip velocity models, we will consider the low Yw and high Yw data separately. More emphasis will be placed on the
The slip velocity models considered here contain one or two constants (in most cases Co and v~ ). Some of the models, such as the driftflux model, include suggested values or equations for obtaining these constants. O p t i m u m values for the constants, however, can be obtained directly from the Davarzani-Miller data. Following the approach suggested by Zuber and Findlay for testing the drift-flux model, by making appropriate plots for each model, the constants can be obtained and the validity of the model tested. For example, for the driftflux model, a plot of Vo versus Vmshould yield a straight line with a slope of Co and a y-intercept of v~ (Eq. 6 ). For each model, plots such as this have been made and the constants determined; the models are then used to calculate the oil and water volumetric flow rates using the values of the constants obtained from the plots. ( 1 ) Constant slip velocity. For this model: Vo
Vm
Yw-Yw
~-vs
(13)
Thus, a plot of Vo/Yw versus Vm/Ywshould yield a straight line with a slope of 1 and a y-
186
A.D.
intercept equal to the slip velocity. Figure 6 shows this plot for the low yw data. The slope is very near 1 and the apparent slip velocity is 15.2 ft/min. Recall, however, that the slip velocity actually varied considerably at low Yw; the good fit of the data shown in Fig. 6 is due in large part to the fact that, for a few points, Vo and Vm are high and nearly equal, whereas Yw is very small, making the quantities Vo/Yw and Vm/Ywlarge and almost equal. Thus, the data for very low Yw tends to force a straight line with a slope of I. This effect will occur with several of the models. The goodness of the fit obtained in plots such as Fig. 6 is not a good measure of the model's validity; instead, these plots are simply a means of estimating the constants in each model. For the high yw data (Fig. 7 ), the fit is not as good, the slope is 0.85 instead of l, and the apparent slip velocity is 19.0 ft/min. This
HILL
value of slip velocity will be used to calculate the oil and water volumetric flow rates for comparison with the known values. (2) Nicolas-Witterholt model. For n = 1, in this model: Vo
Vm
(14)
y2w-y~, Fv~
Plots of Vo/ffwversus Vm/Yewfor the low and high Ywdata are shown in Figs. 8 and 9. In both cases, good fits are obtained and the slope is very near 1, as predicted. For the low yw data, vo~is 66.5 ft/min, whereas for the high Ywdata, v~ is 22.4 ft/min. The theoretical value of voo for the kerosene and water used in the Davarzani and Miller experiments is 29.8 f t / m i n (from Eq. 5 ). Particularly for the high water
200000 =
,
_=
,
.
5000 =
.
.
=
.
4000
>, 10oo0o
3000 > 2000
0
1000
100000 vm
0 0
1000
2000
3000
4000
5000
vm/yw
/
200000
y~
Fig. 8. Test for Nicolas-Witterholt model, low water holdup.
Fig. 6. Test for constant slip model, low water holdup. 300 1 O0
8o
>
. / y = 22.3?3.
].0020x
R^2
= 0.99?
-:~ 200
.
6O 100
40 0 20 2~0
w 40
,
f
60
80
! rqn
Vm/yw
Fig. 7. Test for constant slip model, high water holdup.
~ 100
i 200
300
,,% / y,~,
Fig. 9. Test for Nicolas-Witterholt model, high water holdup.
A COMPARISON
OF OIL-WATER
SLIP VELOCITY
MODELS
USED
FOR
PRODUCTION
~2o ]
LOG
18 7
INTERPRETATION
700
m
600 •
y = 108.56 + 0.78719x
R^2 = 0.418
500 -
~
60
A
~L
400 -
~
300'
4O 200'
20 100
0
i 0
20
40
60
•
80
i
, 100
•
100
•
,
.
,
200
120
.
.
300
400
500
~y~- y~)
Vm(ft/min) Fig. 13. Test for H a s a n - K a b i r model, high water holdup.
Fig. 10. Text for drift-flux model, low water holdup.
5000
50
y = 16.690 + 0.61349x R^2 = 0.551
y = 11.088 + 1.1914x
J
R^2 = 1.000
4000
40 3000 -
3o v
~-
2000
20 1000 10 0
g
1000
I '0
2'0 Vm
30
(ft/min)
3000
4000
5000
vnVyw
40
Fig. 1 1. Test for drift-flux model, high water holdup.
2000
Fig. 14. Test for modified drift-flux model, low water holdup.
200000
I O0
y = 85.725 + 0.83512x
R*2 =
1.000
y = 18.986 + 0.85290x R^2 = 0.963 J 8O
L 100000.
4o 2O
0 100'000
200000
2~ Fig. 12. Test for H a s a n - K a b i r model, low water holdup.
holdup data, the Nicolas-Witterholt slip velocity model appears to work well. (3) Drift-flux model. As shown by Zuber and Findlay, a plot of Vo versus Vm should be a straight line with a slope of Co and a y-intercept of voo if the drift-flux model holds. The surprising results for this model are shown in Figs. 10 and 11. A very good fit is obtained for
A
8~
s~
lOO
v~v Fig. 15. Test for modified drift-flux model, high water holdup.
the low Yw data, though Co is 0.97 and v~o is 4.5 ft/min, both significantly lower than the predicted values of 1.2 and 29.8. The fact that Co is very close to 1 indicates that the oil is distributed uniformly across the pipe, which is reasonable because oil is the continuous phase
188
A.D. HILL
TABLE 2 Average relative errors (%) in oil and water volumetric flow-rate predictions
Constant slip Nicolas-Witterholt Drift-flux Hasan-Kabir Modified drift-flux
Low water holdup O'~ < 0.4 )
High water holdup (yw> 0.4 )
Error in qo
Error in q~
Error in q,,
Error in q,,
4.0 6.8 3.7 12.5 3.6
23.0 34.6 26.7 127 23.1
12.1 3.4 14.~ 20.5 11.4
27. I 6.4 38.1 42.3 25.3
at low Ywin these experiments. For the high Yw data (Fig. 11 ), the fit is poor, Co is 0.61, and vow,is 16.7 ft/min. Comparing this result with the Nicolas-Witterholt result in Fig. 9, it appears that the slip velocity varies linearly with Yw, not inversely, as predicted by the drift-flhx model. (4) Hasan-Kabir model. Solving for the insitu oil velocity for this model yields:
Uo
Coum - q-v.~
fw_yw-yw_yw
(15)
The appropriate plots to test this model for low and high Yw are shown in Figs. 12 and 13. The plot for low Yw is not very meaningful, for the reasons described for the constant slip model. For high Yw (Fig. 13), the fit is poor, the slope Co is 0.79, and vo~ is 108.6 ft/min, much higher than the theoretical value of 29.8 ft/min. (5) Modified drift-flux model. For this model:
Vo[Yw+Co(l-yw)] Covm - - t-v~, Yw Yw
(16)
Inasmuch as Co appears on both sides of this equation, it must be estimated to calculate the term on the left-hand side of the equation; this term can then be plotted against Vm/Yw. These plots for the high and low Ywdata are shown in Figs. 14 and 15, using an estimated Co of 1.2. For the high Yw data, the fit is good, v~ is 18.7 ft/min, and Co is 0.98, suggesting a uniform oil concentration. Changing the estimated
value of Co over a reasonable range did not improve the data fit or the agreement between the estimated and the curve fit values of Co. Using the constants found from Figs. 6-15, the slip velocity can be calculated for each oilwater flow condition with each model. Then, for each slip velocity model, we can calculate the oil and water volumetric flow rates with Eqs. 2 and 3, using the measured holdup and mixture velocity. A comparison of the calculated values with the known oil and water volumetric flow rates provides a test of each model's applicability to oil-water flow holdup and slip velocity prediction. The average relative errors (in %) in the prediction of the oil and water flow rates for each model are shown in Table 2. Again, the data for yw>0.4 is more accurate and generally of more interest for production logging applications. For the high Yw data, the NicolasWitterholt model is clearly the best of those tested here in modeling oil-water flow behavior, as it yielded the lowest average errors in predicting both the oil and water flow rates and by a large margin. Of the other models, the errors increased in the following order: modified drift-flux model, constant slip velocity, driftflux model, and Hasan-Kabir model. For all models, the predictions shown here are better than would be obtained without optimizing the constants for this data set. For example, using the Nicolas-Witterholt model for the high Ywdata, if the theoretically based value of 29.8 f t / m i n is used for v~, the average rela-
A COMPARISON OF OIL-WATER SLIP VELOCITY MODELS USED FOR PRODUCTION LOG INTERPRETATION
tive error in oil flow rate is 39% and the average relative error in water flow rate is 92%.
Conclusions For the vertical flow of a low viscosity oil (kerosene) and water, we conclude the following: (1) The Nicolas-Witterholt model is the best available model of oil-water slip velocity behavior. For the Davarzani-Miller data with a water holdup greater than 0.4, the NicolasWitterholt model predicted oil and water flow rates with average errors of 3.4% for oil rate and 6.4% for water rate. This implies that slip velocity depends linearly on water holdup as assumed in the Nicolas-Witterholt model, not inversely, as assumed in the drift-flux model. (2) All models were improved by determining the constants in the model ( Co and voo) directly from a fit of the Davarzani-Miller data. (3) The apparent bubble rise velocity is significantly lower than that predicted by the Harmathy equation.
Nomenclature
A
Co g qo qw n
Vo Vs
Vm
Vw
pipe cross-sectional area concentration profile constant acceleration of gravity oil volumetric flow rate water volumetric flow rate exponent in Nicolas-Witterholt correlation in-situ average oil velocity (qo/ A(1-yw)) slip velocity mixture velocity ( ( qo + qw) /A ) in-situ average water velocity (qo/ Ayw)
V~w v~ Yw
189
oil superficial velocity (qo/A) water superficial velocity (qw/A) bubble rise velocity water holdup
Greek letters zip density difference between water and oil t7 oil-water interfacial tension mixture density Pm oil density Po water density Pw
Acknowledgements The author thanks Chevron Oil Field Research Co. for support of this work.
References Davarzani, M.J. and Miller, A.A., 1983. Investigation of the flow of oil and water mixtures in large diameter vertical pipes. Soc. Prof. Well Log Anal., 24th Annu. Logging Symp., Calgary, Alta., 14 pp. Harmathy, T.Z., 1960. Velocity of large drops and bubbles in media of infinite or restricted extent. AICHE J., 6:281-288. Hasan, A.R. and Kabir, C.S., 1988. A new model for twophase oil/water flow: Production log interpretation and tubular calculations. Soc. Pet. Eng., 63rd Annu. Tech. Conf. Exhib., Houston, Tex., SPE 18216, 14 pp. Hill, A.D., 1990. Production logging: Theoretical and Interpretive Elements. SPE Monogr., 14:90-111. Nicolas, Y. and Witterholt, E.J., 1972. Measurements of multiphase fluid flow. Soc. Pet. Eng. AIME, 47th Annu. Fall Meeting, Houston, Tex., SPE 4023, 8 pp. Zhu, D. and Hill, A.D., 1988. The effect of flow from perforations on two-phase flow: Implications for production logging. Soc. Pet. Eng., 63rd Annu. Tech. Conf. Exhib., Houston, Tex., SPE 18207, 9 pp. Zuber, N. and Findlay, J., 1965. Average volumetric concentration in two-phase flow systems. Trans. ASME, J. Heat Transfer Ser. C, 87: 453-468.
181
A comparison of oil-water slip velocity models used for production log interpretation A.D. Hill Department of Petroleum Engineering, The University of Texas at Austin, Austin, TEX 78712, USA
(Received February 10, 1992; revised version accepted April 15, 1992 )
ABSTRACT Hill, A.D., 1992. A comparison of oil-water slip velocity models used for production log interpretation. J. Pet. Sci. Eng., 8: 181-189.
The accuracies of five slip velocity models that can be used to interpret production logs in oil-water flow have been examined using the flow loop data of Davarzani and Miller. Optimal values of the constants used in the models were determined for each model; the abilities of each model to predict oil and water volumetricflowrates from measuredvalues of water holdup and mixture velocitywere then tested. The empirical model of Nicolas and Witterholt performedthe best of the modelstested.
Introduction The production o f excessive a m o u n t s of water is a pervasive problem in oil production wells, and knowing the location and rate of water entries into a well is crucial in planning remedial action for a particular well or for general reservoir management purposes. Water entry locations are most c o m m o n l y detected through the use o f production logging instruments such as spinner or basket flowmeters, density tools, and capacitance tools. This means, however, o f detecting water entries is somewhat indirect. With these logs, flow properties are measured at a n u m b e r o f discrete locations in the well, or are recorded continuously throughout the well. The log responses at discrete depth locations are then interpreted to obtain the volumetric flow rates o f oil and water at these locations. (Note: In m a n y wells, Correspondence to: A.D. Hill, Department of Petroleum Engineering, The University of Texas at Austin.
gas will also be flowing as a free phase, but this paper will be limited to the case o f oil-water flow only. ) Finally, the rates and locations o f water entries are determined from the differences in the wellbore flow rates. The most c o m m o n approach taken, when logging oilwater flow, is to measure a fluid velocity and either fluid density or fluid capacitance. The velocity measured is assumed to be the average total velocity which is equal to the sum of the oil and water volumetric flow rates divided by the cross-sectional area of the wellbore. F r o m the density or capacitance log, the fraction o f the pipe occupied by water (the water holdup) is inferred. With these measurements, it is then necessary to independently obtain a third parameter, the slip velocity, in order to calculate the volumetric flow rates o f each phase. Thus, the accuracy o f the log interpretation will depend on the accuracies o f the logging measurements themselves and on the accuracy of the m e t h o d used to estimate the slip velocity. In this paper, we ex-
0920-4105/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
182 amine several m e t h o d s or correlations for estimating slip velocity in oil-water flow. Using the oil-water flow data of Davarzani and Miller ( 1983 ), the accuracy of each m e t h o d in interpreting oil and water volumetric flow rates is evaluated. The results show that the empirical model of Nicolas and Witterholt (1972) is the best available slip velocity model for interpreting production logs in oil-water flow.
Production log interpretation in oil-water flow Consider the hypothetical oil well producing oil and water as shown in Fig. 1. I n a s m u c h as water is being p r o d u c e d from the intermediate zone, opposite this zone and above it, the flow stream will consist of both oil and water. Below Zone B, there will likely be some water present that has fallen down from Zone B (Zhu and Hill, 1988); however, this significant complication is not usually considered in production log interpretation and will not be considered here. In any case, the only mobile phase below Zone B is oil. To determine the production rate of water from Zone B, we would m a k e p r o d u c t i o n logging measurements just above and just below
Zone A
Zone B
Zone C
Fig, 1. A wellbore with oil and water entries.
A,D.HILL the zone (locations I and 2 in Fig. 1 ). F r o m the difference in the interpreted water flow rates at locations 1 and 2, the Zone B production rate is obtained. Considering now location 1, we assume that the m e a n velocity, Vm, and the average density, Pro, have been measured with, e.g., a basket flowmeter and a g r a d i o m a n o m e t e r TM*. F r o m these measurements, we can calculate (Hill, 1990) the water holdup from: YwP m - - f l o Pw--Po
( 1)
and the oil and water volumetric flow rates as: qo=A [ l -Yw] [ V~Vw+ Vm]
(2)
and:
qw=Ayw[Vm-Vs( 1 -Yw) ]
(3)
In Eqs. 2 and 3, A is the wellbore cross-sectional area and we have introduced a parameter, vs, the slip velocity, which m u s t be independently estimated. The slip velocity is a representation of the holdup p h e n o m e n o n in two-phase flow; i.e., the fact that the in-situ fractions of each phase in the pipe are not, in general, equal to the input fractions because the lighter phase is moving at a higher velocity than the denser phase. The slip velocity in oil-water flow is likely to be a complex function of fluid properties, pipe diameter, water holdup, and the flow rates of each phase. As the calculated volumetric flow rates of oil and water d e p e n d linearly on the slip velocity, its accurate estimation is crucial to accurate production log interpretation.
Slip velocity models Several different models of oil-water slip velocity have been used to interpret production logs, including assuming the slip velocity to be constant, empirical models based on lab, TM is trademark of Schlumberger Well Services. (See Chilingarian et al., 1987. Surface Operations in Petroleum Production, I. Elsevier, Amsterdam, 821 pp. )
A COMPARISON OF OIL-WATER SLIP VELOCITY MODELS USED FOR PRODUCTION LOG INTERPRETATION
oratory data and theoretically-based models of two-phase flow. In this paper, five slip velocity models are examined; a description of each model follows. (1) Constant slip velocity. The simplest model of slip velocity is to assume that it is constant, independent of the flow rate of each phase and of the holdup. In production log interpretation, sometimes the slip velocity is estimated in the upper part of the well based on knowledge of the surface flow rates, then assumed constant throughout the well. (2) Nieolas-Witterholt model. Based on a series of experiments in 4- and 6-inch I.D. by 50-ft long pipes, Nicolas and Witterholt (1972) suggested the following empirical model of slip velocity: Vs-Ywn voo
(4)
where 1/2<~n~2 and: I"
. "11/4
voo= 1.53L~ ]
(5)
Equation 5 is the Harmathy (1960) equation for the terminal rise velocity of a bubble. A comparison of the slip velocity with the experimental data presented in the Nicolas and Witterholt paper showed the model with n = 1 best fit the data. Thus, in this paper, the Nicolas-Witterholt model is Eq. 4 with n = 1. (3) Drift-flux model (Zuber and Findlay, 1965 ). Zuber and Findlay postulated a theoretical model of two-phase slip behavior that is referred to as the drift-flux model. Derived for gas-liquid bubble flow, but theoretically applicable to any upwards, vertical, two-phase flow, the drift flux model accounts for two mechanisms of slip. First, because the lighter phase is thought to be more concentrated in the center of the pipe where the velocity is highest, the light phase will be moving fast relative to the mixture as a whole. Second, there will be a relative velocity difference between the phases because of the density difference.
183
These assumptions lead to the following equation for the in-situ average velocity of the lighter phase (oil in this case):
vo=Covm+Voo
(6)
Zuber and Findlay found that the concentration profile correction factor, Co, ranged from 1.0 to 1.5 for air-water flow. voo is again defined by Eq. 5. The slip velocity for the driftflux model is: vs-
(Co-1)Vm+Voo Yw
(7)
Notice the distinct contrast between the Nicholas-Witterholt model, in which the slip velocity depends linearly on Yw,and the drift-flux model, in which slip velocity is inversely proportional to water holdup. (4) Hasan-Kabir (1988) model. Hasan and Kabir presented a model of oil-water flow which was based on the drift-flux model. For bubble flow, the Hasan-Kabir model for slip velocity is: v~-
(Co-
1)Vm
Yw
q-vooyw(1-yw)
(8)
where Co and v~o are defined the same as for the drift-flux model. (5) Modified drift-flux model. A modification to the drift-flux model that we might expect to be applicable to high water holdup cases is obtained by substituting the in-situ average water velocity, Vw, for the mixture, Vm. Theoretically, this implies that, in the absence of a drift velocity (voo=0), the effect of the concentration distribution is to make the oil velocity a factor of Co times the continuous phase velocity (Vw) rather than the mixture velocity. Thus, the modified drift-flux model is:
vo=Covw+voo
(9)
The slip velocity for the modified drift-flux model is:
184
v,-
A.D. HILL
(Co-1)Vm+Vo~
(10)
yw+Co(1-yw)
Davarzani-Miller oil-water flow data Davarzani and Miller ( 1983 ) conducted an extensive set of oil-water flow experiments in which the water holdup was measured over a range of oil and water flow rates. Ninety-one individual tests were performed in a 6½ inch I.D. by 30-ft long flow loop. The water holdup was measured both with a ?-ray densitometer and by suddenly shutting in the test section on some tests. The holdups measured by the two methods agreed within 1%. Because Davarzani and Miller desired a broad range of water holdup values, their data does not have a uniform matrix of oil and water flow rates. Instead, there is a significantly greater number of tests with high oil rate relative to the water rate than vice versa. Davarzani and Miller's test conditions are summarized in Table 1. Inasmuch as the oil and water flow rates are known and the water holdup has been measured, the slip velocity can be calculated for each of Davarzani and Miller's tests. From the definition of slip velocity as the difference between the oil and water in-situ average velocities, we have:
v,=
;[
qo
(1-y,~)
(11)
or, in terms of superficial velocities (as the flow rates were reported by Davarzani and Miller):
Vso v s - 1-yw
(12)
The slip velocities measured in the Davarzani-Miller tests as functions of water holdup are shown in Fig. 2. Note that there appears to be a significant difference in slip velocity behavior for values of yw below about 0.4 compared with the behavior for Yw above 0.4. Below about 0.4, there is considerable scatter in the data and slip velocity appears to generally decrease with increasing Yw; above 0.4, the data is much smoother and slip velocity is increasing with increasing Yw. It is likely that several factors are contributing to this difference. First, the lower the water holdup, the larger the errors in the water holdup measured by the y-ray densitometer. Inasmuch as the holdup is used to calculate slip velocity, errors in the measured holdup will give errors in slip velocity (large errors at low Yw, because Ywis in the denominator). Second, below yw=0.4, the external phase is oil, not water; it is not surprising that the oil velocity behavior is different when oil is the external phase. Some of these factors can be observed by examining the slip velocity as a function of holdup for a constant mixture velocity (Figs. 3, 4 and 5). For a mixture velocity of 29.4 ft/ min, the slip velocity decreases with increasing water holdup as seen in Fig. 3. With higher mixture velocities, however, of 58.2 or 87 It/ min (Figs. 4 and 5 ), the slip velocity is apparently increasing with Yw, and there is consid-
TABLE 1 Davarzani-Miller oil-water flow data Oil: kerosene Water: fresh tap water Oil flow-rate range: Water flow-rate range: Input water-fraction range: Water holdup range:
Vsw Yw
p=0.78 g/cm3; /z=2 cP p = l . 0 0 g / c m 3 ; ~=1 cP 0.0115 ft3/s ( 177 b/d)-0.435 ft3/s (6700 b / d ) 0.0023 ft3/s (35 b/d)-0.0553 ft3/s (850 b / d ) 0.025-0.802 0.029-0.928
A COMPARISON OF OIL-WATER SLIP VELOCITY MODELS USED FOR PRODUCTION LOG INTERPRETATION
30
18 5
30
[]
20 =_
>
20
m
10
0
i
0,0 -10
i 0.2
O
i 0.4
i 0.6
i 0.8
0.2
Yw
1.0
Fig. 5. Slip velocityfor mixture velocityof 87 ft/min.
Yw
Fig. 2. Measured slip velocities from Davarzani-Miller experiments. 25
rn m
20'
0.1
high Ywdata because it is likely more accurate. Fortunately, in production logging, we are more often concerned with situations with high water holdup.
[]
Comparison of slip velocity models
15'
10'
5
0
i
0.3
0.2
0.4
Yw
Fig. 3. Slip velocityfor mixture velocityof 29.4 ft/min. 25
20 ¸
15-
10
5
0 0.0
i
i
i
0.1
0.2
0.3
0.4
Yw
Fig. 4. Slip velocityfor mixture velocityof 58.2 ft/min. erable scatter in the data, particularly at 87 ft/min. In considering slip velocity models, we will consider the low Yw and high Yw data separately. More emphasis will be placed on the
The slip velocity models considered here contain one or two constants (in most cases Co and v~ ). Some of the models, such as the driftflux model, include suggested values or equations for obtaining these constants. O p t i m u m values for the constants, however, can be obtained directly from the Davarzani-Miller data. Following the approach suggested by Zuber and Findlay for testing the drift-flux model, by making appropriate plots for each model, the constants can be obtained and the validity of the model tested. For example, for the driftflux model, a plot of Vo versus Vmshould yield a straight line with a slope of Co and a y-intercept of v~ (Eq. 6 ). For each model, plots such as this have been made and the constants determined; the models are then used to calculate the oil and water volumetric flow rates using the values of the constants obtained from the plots. ( 1 ) Constant slip velocity. For this model: Vo
Vm
Yw-Yw
~-vs
(13)
Thus, a plot of Vo/Yw versus Vm/Ywshould yield a straight line with a slope of 1 and a y-
186
A.D.
intercept equal to the slip velocity. Figure 6 shows this plot for the low yw data. The slope is very near 1 and the apparent slip velocity is 15.2 ft/min. Recall, however, that the slip velocity actually varied considerably at low Yw; the good fit of the data shown in Fig. 6 is due in large part to the fact that, for a few points, Vo and Vm are high and nearly equal, whereas Yw is very small, making the quantities Vo/Yw and Vm/Ywlarge and almost equal. Thus, the data for very low Yw tends to force a straight line with a slope of I. This effect will occur with several of the models. The goodness of the fit obtained in plots such as Fig. 6 is not a good measure of the model's validity; instead, these plots are simply a means of estimating the constants in each model. For the high yw data (Fig. 7 ), the fit is not as good, the slope is 0.85 instead of l, and the apparent slip velocity is 19.0 ft/min. This
HILL
value of slip velocity will be used to calculate the oil and water volumetric flow rates for comparison with the known values. (2) Nicolas-Witterholt model. For n = 1, in this model: Vo
Vm
(14)
y2w-y~, Fv~
Plots of Vo/ffwversus Vm/Yewfor the low and high Ywdata are shown in Figs. 8 and 9. In both cases, good fits are obtained and the slope is very near 1, as predicted. For the low yw data, vo~is 66.5 ft/min, whereas for the high Ywdata, v~ is 22.4 ft/min. The theoretical value of voo for the kerosene and water used in the Davarzani and Miller experiments is 29.8 f t / m i n (from Eq. 5 ). Particularly for the high water
200000 =
,
_=
,
.
5000 =
.
.
=
.
4000
>, 10oo0o
3000 > 2000
0
1000
100000 vm
0 0
1000
2000
3000
4000
5000
vm/yw
/
200000
y~
Fig. 8. Test for Nicolas-Witterholt model, low water holdup.
Fig. 6. Test for constant slip model, low water holdup. 300 1 O0
8o
>
. / y = 22.3?3.
].0020x
R^2
= 0.99?
-:~ 200
.
6O 100
40 0 20 2~0
w 40
,
f
60
80
! rqn
Vm/yw
Fig. 7. Test for constant slip model, high water holdup.
~ 100
i 200
300
,,% / y,~,
Fig. 9. Test for Nicolas-Witterholt model, high water holdup.
A COMPARISON
OF OIL-WATER
SLIP VELOCITY
MODELS
USED
FOR
PRODUCTION
~2o ]
LOG
18 7
INTERPRETATION
700
m
600 •
y = 108.56 + 0.78719x
R^2 = 0.418
500 -
~
60
A
~L
400 -
~
300'
4O 200'
20 100
0
i 0
20
40
60
•
80
i
, 100
•
100
•
,
.
,
200
120
.
.
300
400
500
~y~- y~)
Vm(ft/min) Fig. 13. Test for H a s a n - K a b i r model, high water holdup.
Fig. 10. Text for drift-flux model, low water holdup.
5000
50
y = 16.690 + 0.61349x R^2 = 0.551
y = 11.088 + 1.1914x
J
R^2 = 1.000
4000
40 3000 -
3o v
~-
2000
20 1000 10 0
g
1000
I '0
2'0 Vm
30
(ft/min)
3000
4000
5000
vnVyw
40
Fig. 1 1. Test for drift-flux model, high water holdup.
2000
Fig. 14. Test for modified drift-flux model, low water holdup.
200000
I O0
y = 85.725 + 0.83512x
R*2 =
1.000
y = 18.986 + 0.85290x R^2 = 0.963 J 8O
L 100000.
4o 2O
0 100'000
200000
2~ Fig. 12. Test for H a s a n - K a b i r model, low water holdup.
holdup data, the Nicolas-Witterholt slip velocity model appears to work well. (3) Drift-flux model. As shown by Zuber and Findlay, a plot of Vo versus Vm should be a straight line with a slope of Co and a y-intercept of voo if the drift-flux model holds. The surprising results for this model are shown in Figs. 10 and 11. A very good fit is obtained for
A
8~
s~
lOO
v~v Fig. 15. Test for modified drift-flux model, high water holdup.
the low Yw data, though Co is 0.97 and v~o is 4.5 ft/min, both significantly lower than the predicted values of 1.2 and 29.8. The fact that Co is very close to 1 indicates that the oil is distributed uniformly across the pipe, which is reasonable because oil is the continuous phase
188
A.D. HILL
TABLE 2 Average relative errors (%) in oil and water volumetric flow-rate predictions
Constant slip Nicolas-Witterholt Drift-flux Hasan-Kabir Modified drift-flux
Low water holdup O'~ < 0.4 )
High water holdup (yw> 0.4 )
Error in qo
Error in q~
Error in q,,
Error in q,,
4.0 6.8 3.7 12.5 3.6
23.0 34.6 26.7 127 23.1
12.1 3.4 14.~ 20.5 11.4
27. I 6.4 38.1 42.3 25.3
at low Ywin these experiments. For the high Yw data (Fig. 11 ), the fit is poor, Co is 0.61, and vow,is 16.7 ft/min. Comparing this result with the Nicolas-Witterholt result in Fig. 9, it appears that the slip velocity varies linearly with Yw, not inversely, as predicted by the drift-flhx model. (4) Hasan-Kabir model. Solving for the insitu oil velocity for this model yields:
Uo
Coum - q-v.~
fw_yw-yw_yw
(15)
The appropriate plots to test this model for low and high Yw are shown in Figs. 12 and 13. The plot for low Yw is not very meaningful, for the reasons described for the constant slip model. For high Yw (Fig. 13), the fit is poor, the slope Co is 0.79, and vo~ is 108.6 ft/min, much higher than the theoretical value of 29.8 ft/min. (5) Modified drift-flux model. For this model:
Vo[Yw+Co(l-yw)] Covm - - t-v~, Yw Yw
(16)
Inasmuch as Co appears on both sides of this equation, it must be estimated to calculate the term on the left-hand side of the equation; this term can then be plotted against Vm/Yw. These plots for the high and low Ywdata are shown in Figs. 14 and 15, using an estimated Co of 1.2. For the high Yw data, the fit is good, v~ is 18.7 ft/min, and Co is 0.98, suggesting a uniform oil concentration. Changing the estimated
value of Co over a reasonable range did not improve the data fit or the agreement between the estimated and the curve fit values of Co. Using the constants found from Figs. 6-15, the slip velocity can be calculated for each oilwater flow condition with each model. Then, for each slip velocity model, we can calculate the oil and water volumetric flow rates with Eqs. 2 and 3, using the measured holdup and mixture velocity. A comparison of the calculated values with the known oil and water volumetric flow rates provides a test of each model's applicability to oil-water flow holdup and slip velocity prediction. The average relative errors (in %) in the prediction of the oil and water flow rates for each model are shown in Table 2. Again, the data for yw>0.4 is more accurate and generally of more interest for production logging applications. For the high Yw data, the NicolasWitterholt model is clearly the best of those tested here in modeling oil-water flow behavior, as it yielded the lowest average errors in predicting both the oil and water flow rates and by a large margin. Of the other models, the errors increased in the following order: modified drift-flux model, constant slip velocity, driftflux model, and Hasan-Kabir model. For all models, the predictions shown here are better than would be obtained without optimizing the constants for this data set. For example, using the Nicolas-Witterholt model for the high Ywdata, if the theoretically based value of 29.8 f t / m i n is used for v~, the average rela-
A COMPARISON OF OIL-WATER SLIP VELOCITY MODELS USED FOR PRODUCTION LOG INTERPRETATION
tive error in oil flow rate is 39% and the average relative error in water flow rate is 92%.
Conclusions For the vertical flow of a low viscosity oil (kerosene) and water, we conclude the following: (1) The Nicolas-Witterholt model is the best available model of oil-water slip velocity behavior. For the Davarzani-Miller data with a water holdup greater than 0.4, the NicolasWitterholt model predicted oil and water flow rates with average errors of 3.4% for oil rate and 6.4% for water rate. This implies that slip velocity depends linearly on water holdup as assumed in the Nicolas-Witterholt model, not inversely, as assumed in the drift-flux model. (2) All models were improved by determining the constants in the model ( Co and voo) directly from a fit of the Davarzani-Miller data. (3) The apparent bubble rise velocity is significantly lower than that predicted by the Harmathy equation.
Nomenclature
A
Co g qo qw n
Vo Vs
Vm
Vw
pipe cross-sectional area concentration profile constant acceleration of gravity oil volumetric flow rate water volumetric flow rate exponent in Nicolas-Witterholt correlation in-situ average oil velocity (qo/ A(1-yw)) slip velocity mixture velocity ( ( qo + qw) /A ) in-situ average water velocity (qo/ Ayw)
V~w v~ Yw
189
oil superficial velocity (qo/A) water superficial velocity (qw/A) bubble rise velocity water holdup
Greek letters zip density difference between water and oil t7 oil-water interfacial tension mixture density Pm oil density Po water density Pw
Acknowledgements The author thanks Chevron Oil Field Research Co. for support of this work.
References Davarzani, M.J. and Miller, A.A., 1983. Investigation of the flow of oil and water mixtures in large diameter vertical pipes. Soc. Prof. Well Log Anal., 24th Annu. Logging Symp., Calgary, Alta., 14 pp. Harmathy, T.Z., 1960. Velocity of large drops and bubbles in media of infinite or restricted extent. AICHE J., 6:281-288. Hasan, A.R. and Kabir, C.S., 1988. A new model for twophase oil/water flow: Production log interpretation and tubular calculations. Soc. Pet. Eng., 63rd Annu. Tech. Conf. Exhib., Houston, Tex., SPE 18216, 14 pp. Hill, A.D., 1990. Production logging: Theoretical and Interpretive Elements. SPE Monogr., 14:90-111. Nicolas, Y. and Witterholt, E.J., 1972. Measurements of multiphase fluid flow. Soc. Pet. Eng. AIME, 47th Annu. Fall Meeting, Houston, Tex., SPE 4023, 8 pp. Zhu, D. and Hill, A.D., 1988. The effect of flow from perforations on two-phase flow: Implications for production logging. Soc. Pet. Eng., 63rd Annu. Tech. Conf. Exhib., Houston, Tex., SPE 18207, 9 pp. Zuber, N. and Findlay, J., 1965. Average volumetric concentration in two-phase flow systems. Trans. ASME, J. Heat Transfer Ser. C, 87: 453-468.