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Prediction of Multiphase Horizontal Pipe Flow A Reassessment Part I Evaluation of Two-Phase Correlations Part II Pressure Drop Prediction in Three-Phase Gas-Oil-Water Flows.

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Dev. Chem. Eng. Mineral Process. 14(3/4), pp. 567-584, 2006.
Prediction of Multiphase Horizontal Pipe
Flow: A Reassessment
Part I: Evaluation of Two-Phase Correlations
Part 11: Pressure Drop Prediction in Three-Phase
Gas-Oil-Water Flows
P.L. Spedding", E. Benard and G.F. Donnelly
School of Aeronautical Engineering, Queen 's University Belfast, Ashby
Building, Stranmillis Road, Belfast BT9 5AH, Northern Ireland, UK
~~
~~~
~
~
~
~
~~
Prediction of two-phase pressure drop using published correlations and a wide data
base showed that no single correlation could predict across the whole range of
possible flow patterns. Particular diflculty was found in attempting to predict the
intermittent SI and BTS patterns. However, successful prediction was achieved for the
St + R W,St + R W + D, D and A + R Wpatterns.
Two-phase pressure drop predictive correlations were extended to the three-phase
situation. Detailed evaluation identified three correlations which gave reasonable
prediction of intermittent flow pressure drop. An adaptation is presented that
enhances the prediction of these correlations for three-phase flow. The separated
three-phase patterns could not be predicted to an acceptable standard using these
types of correlations.
Introduction
Prediction of frictional pressure drop and hold-up in three-phase gas-oil-water
horizontal flow has proven to be difficult. One approach adopted has been to use
models developed for two-phase gas-liquid flow and attempt to adapt them to the
three-phase case. One problem with this approach is that the predictive performance
of the two-phase models themselves has not been encouraging when tested against
reliable data collected over a wide range of conditions. For example, homogeneous
and separated flow models have been shown to be of limited value for prediction of
pressure drop [I]. Consequently only empirical two-phase correlations have been
adapted for three-phase prediction. An added problem is that computer codes are used
for three-phase prediction and, while being convenient, little consideration is given to
their basis which often depends on adaptation of two-phase correlations. Indeed very
little work on basic modelling has been reported in recent years.
* Author for correspondence.
567
P. L. Spedding, E. Benard and G.F. Donnelly
Spence and Spedding [ 11 tested over 50 pressure drop correlations using a limited
set of two-phase data [2-51 and found no method predicted pressure drop for all
possible flow regimes within an average o f f 15% and a spread of -+ 30%. However
prediction was achieved for the F+D [6-101, A+RW [7] and D [6-101 flow patterns by
the correlations [6-101. Ferguson and Spedding [I 11 widened the scope of the study
to include more correlations and for a wider data base [4, 5, 121 and provide the
recommendations in Table 1.
Table 1. Prediction of pressure drop in horizontal two-phase pipe jlow using
empirical correlations and excluding models based on momentum balance.
Not all flow regimes were predicted, the notable exceptions being the intermittent
S1 type flows and the St, SttR regimes. A number of the correlations that were
successfi~lcame from a similar basis such as those by Dukler-Wicks-Cleveland [19]
and Nguyen-Spedding [15]. Not all of the two-phase correlations could easily be
adapted to apply to the three-phase situation which usually was achieved by treating
the liquid phase as a single entity.
In Part I of this work reliable two-phase gas-liquid data [4, 5, 12, 21-24], over a
wider range of conditions than was used to derive Table 1, were employed to test the
reliability of two-phase correlations that were adaptable to prediction of three-phase
pressure drop. Only a selection of the detailed results are presented in order to
illustrate the general trends.
Anticipating the results, some models have proved to be successfi~lin prediction
of pressure drop for particular two-phase flow regimes. These correlations were
subsequently adapted to predict gas-oil-water three-phase pressure drop by treating
the liquid phase as a single entity. Some workers [25, 261 have claimed that adapted
two-phase correlations under-estimate the pressure loss of three-phase systems.
However, Fayed and Otten [27] adapted the Dukler-Wicks-Cleveland [19] and Beggs568
Prediction of Multiphase Horizontal Pipe Flow: A Reassessment. Parts I and 11
Brill [28] two-phase correlations and found that the adaptations over predicted the SI
and Fro flows and under predicted the B flow for three-phase. Schlichting [29]
adapted the Lockhart-Martinelli [6] correlation to three-phase pressure drop although
Hall [30] claimed the result was of limited value. Other workers [3 1, 321 have also
suggested that the current methods of predicting three-phase pressure drop by
adoption of two-phase correlations were inadequate, and a more rigorous approach
was required based on the momentum balance approach.
In Part I1 of this work, reliable three-phase gas-oil-water [24,30, 33-37] data were
employed to test the validity of a number of two-phase correlations that were
adaptable to predict pressure drop in three-phase flow. While the liquid phases have
been treated as a single entity the Brinkman [38] and Pal-Rhodes [39] models were
used to estimate the viscosity of the combined liquid phase. Again only a selection of
the detailed results are presented in order to illustrate general trends.
Part I: Evaluation of Two-Phase Correlations
Lockhart-Martinelli Correlation
Prediction of two-phase pressure drop using the Lockhart-Martinelli correlation [6]
was mediocre with a wide scatter. Most data were bounded within a *40% spread but
with a likelihood of over prediction. Modifications of the correlation [16-181 gave
only marginal improvement in performance. Figure 1 shows that for the 0.0935 m i.d.
data of Hand [I21 the prediction deteriorated as the liquid hold-up rose at low gas
rates. In general, the S1 and A+BTS regimes were very poorly predicted. Also the
0.0252 m i.d. data of Andritsos [22] and the 0.026 m i.d. data of Donnelly [24] were
underpredicted by about -40%. However, the high-pressure 0.0525 m i.d. data of
Andrews [21] were overpredicted in excess o f f 60%. The findings of Spence and
Spedding [l], albeit from a more limited data base, that the Lockhart-Martinelli [6]
correlation predicted pressure drop in the F+D and D regimes were not supported by
this work particularly at high and low diameters.
The prediction of liquid hold-up using the Lockhart-Martinelli [6] correlation
deteriorated with an increase in pipe diameter as illustrated by Figures 2a and 2b,
although they generally followed the findings of Spedding [40] that useful hold-up
prediction was obtained with the St+RW, St+IW and A+BTS flow regimes. However,
the latter flow pattern was not predicted for pressure drop using this correlation.
Dukler- Wicks-ClevelandCorrelation
Use of this correlation requires simultaneous prediction of hold-up. The correlation of
Eaton et al. [41] provided the most favourable overall prediction for the DuklerWicks-Cleveland [ 191 correlation. Selecting the data of Nguyen [4], predictive results
are presented in Figure 3 where gross overprediction was obtained for the SI and
A+BTS regimes while the separated regimes in the St and A patterns generally were
bounded within a &50% spread with a tendency to underpredict. The overall error
given in Table 2 was thus misleading since the performance of the correlation was
strongly flow regime dependent. These conclusions agree with the findings of Fayed
and Otten [27].
569
P.L. Spedding, E. Benard and G.F.Donnelly
0.3
0.2
0.1
0
0.5
0.4
0.7
0.6
4 u i d Iboldup Rc
Figure 1. Measured to predicted pressure drop against liquid hold-up for the
0.0935 m i.d. data of Hand [I21 using the Lockhart-Martinellicorrelation [6].
1
11
0.8
4
4
0.4w
0.2
4*
0
0
0.2
0.4
0.6
0.8
Me8$urSa~Holdup
1
0
0.2
0.4
0.6
0.8
I
hasuluedLiquidHoldup
Figure 2. Measured against predicted liquid hold-up using the Lockhart-Martinelli
correlation [4] for: (a) 0.0935 m i.d. data of Hand [12]; (b) 0.0454 m i.d. data of
Chen [ 5 ] ,
5 70
Prediction of Multiphase Horizontal Pipe Flow: A Reassessment. Parts I and II
Figure 3. Measured against predicted pressure drop for the 0.0454 m i.d. data of
Nguyen [4] using the Dukler- Wicks-Cleveland(191 plus Eaton [4 I] correlation.
*rr
a
4
4
.
4
Y
4
0
20
40
4
4-4
4
4 4
4
60
8o
I00
120
140
SupasCd OBS Vebcity Vsg (ds)
Figure 4. Measured to predicted pressure drop against superficial gas velocity for
the 0.025 15 m i.d data of Andritsos [22] using the Beggs-BriN[28] correlation.
P.L. Spedding, E. Benard and G.F. Donnelly
Beggs-Brill Correlation
The Beggs-Brill [28] correlation overpredicted the pressure loss with most data and,
as Table 2 shows, performed worse overall than the other correlations. Figure 4 shows
that prediction was poor at low and high gas flow rates in the SI and A regimes
respectively. The high pressure observations of Andrews [2 11 correlated well within a
*20% margin at low gas flow rates in the St type patterns. Nevertheless the
performance of the correlation deteriorated significantly to yield gross overprediction
at higher gas flows in the D and in mist flow regimes. Prediction of liquid hold-up
exhibited a diameter effect as illustrated in Figures 5a and 5b with increasing
underprediction for pipe in excess of 0.05 m i.d.
Friedel Correlation
The Friedel [42] correlation gave significant overprediction of pressure drop with a
wide scatter for all diameters, as evidenced by the data of Chen [5] in Figure 6. Close
examination revealed that the correlation performed well for high quality flows over
the whole range of data. Thus, the A+RW and A+D regimes were bounded within a
*30% margin. The high-pressure data of Andrews [2 11 gave reasonable performance
being within a *40% spread. Intermittent flows were poorly predicted.
Beattie- Whalley Correlation
The Beattie-Whalley [43] correlation performed poorly for both the SI and A+BTS
flows with the pressure drop being mostly underpredicted as illustrated in Table 2.
There was also a noticeable scale-up effect with larger diameter tubes being
overpredicted and small pipes being underpredicted, particularly with St type
separated flows.
Table 2. Average percentage error for two-phase pressure gradient prediction using
various correlations.
Data
Andritsos [22]
Hand r121
Andrews [21]
Ferguson [23]
Nguyen [4]
Chen [51
Donnelly [24]
Andritsos [22]
5 72
I
I
I
m i.d.
I
Dukler [I91
I
Beggs (281
I
I
Beattie[43]
0.09525
56.4
80.4
0.0935
87.7
102.7
73.3
0.0525
0.0508
12.9
39.4
6.5
69.9
-35.0
21.8
0.0454
0.0454
0.0259
0.025 15
24.4
21.0
54.3
25.7
53.3
0.8
-7.9
36.0
-6.6
51.7
75.6
81.4
39.5
1
Prediction of Multiphase Horizontal Pipe Flow: A Reassessment. Parts I and II
0.9
$
0.7
-4
4
4
U
00
0.8 'r
I
-I
4 0.8 40.7 -0.6 -3 0.5 -3 0.4 -3 0.3 --
4
@*
6
644
0
0.2
0.4
a6
MururcdLiquidHoldup
0.8
0
0.2
0.4
0.6
0.8
Mwrumd Liquid Holdup
Figure 5. Measured against predicted liquid hold-up using the Beggs-Brill [28]
correlation: (a) 0.0935 m i.d. data of Hand[l2j; (b) 0.0454 m id data of Chen [5].
Figure 6. Predicted against measured pressure drop for the 0.0454 m id. data of
Chen (53 using the Friedel [42] correlation.
5 73
P. L. Spedding, E. Benard and G.F. Donnelly
Muller-Steinhagen-Heck Correlation
The Muller-Steinhagen-Heck [44] correlation exhibited serious underprediction for
high quality flows over the entire range of data. Figure 7 shows that performance
improved as the liquid hold-up was increased in the intermittent region for S1 and
A+BTS flows.
Olujic Correlation
The Olujic [ 131 correlation exhibited noticeable scale-up difficulties with pressure
drop being underpredicted for diameters under 0.05 m, but improvements in
prediction were achieved as the diameter was raised. Only the St type flow data of
Donnelly [24] were bounded within *30% as illustrated in Figure 8. The level of
confidence for the remaining Donnelly [24] data varied fiom +50-60% for the larger
diameter and *40-50% for pipe under 0.05 m. Predictions of intermittent SI and BTS
flows were particularly poor.
Discussion of Part I
None of the correlations tested gave acceptable prediction of two-phase pressure drop
over the whole range of flow patterns. However, prediction was achieved for specific
flow regimes by some correlations. The Dukler-Wicks-Cleveland [191 correlation
predicted the St+RW+D and D flows; the Beggs-Brill [28] correlation predicted the
St+RW and St+RW+D regimes; the Friedel [42] correlation predicted the D and
A+RW regimes, while the Olujic [13] correlation predicted the St+RW and
St+RW+D patterns. All the correlations except that by Muller-Steinhagen and Heck
[44] had particular difficulty when attempting to handle intermittent SI and BTS flow
patterns. This latter condition of prediction of intermittent flow will prove to be of
interest when the correlations are applied to the three-phase case.
Part 11: Pressure Drop Prediction in Three-Phase
Gas-Oil-Water Flows
Lockhart-Martinelli Correlation
For the Lockhart-Martinelli [6J correlation the predictive results of the three-phase
data of Donnelly [24] are given in Figure 9. There was a wide variation in the WD
region with a tendency to overprediction. The overprediction became more noticeable
following inversion to the OD regime. In addition, the large diameter high-pressure
data of Lunde et al. [36] and the data of Sobocinski [33] were grossly over predicted
with a high degree of scatter.
5 74
Prediction of Multiphase Horizontal Pipe Flow:A Reassessment. Parts I and 11
'I
0
0.1
0.2
0.3
0.4
0.5
0.7
0.6
Meclsund~uidHokl-upRL
Figure 7. Measured to predicted pressure drop against liquid hold-up for the
0.09525 m id. data of Andritsos (221 using the Muller-Steinhagen and Heck [44)
correlation.
Figure 8. Predicted against measured pressure drop using the St 0.0259 m id. data
of Donnelly [24] and the Olujic [I31 correlation.
5 75
P. L. Spedding, E. Benard and G.F.Donnelly
In contrast, the intermittent SI type flow data of Malinowsky [34], Laflin and
Oglesby [35], and Hall [30] gave superior results with the Lockhart-Martinelli [6]
correlation. The data of Malinowsky [34] showed more scatter and a tendency to
under prediction. There was a moderate degree of scatter and under prediction around
the inversion point if the Pal-Rhodes [39] viscosity relation was employed as shown
in Figure 10. Improved performance was achieved if the Brinkman [38] viscosity
relation was used and prediction of the data of Hall [30] was well bounded within
%30%as illustrated in Figure 1 1.
In general, the performance of the Lockhart-Martinelli [6] correlation was found
to be mediocre but improvement was achieved for the intermittent SI type flows when
the Brinkman [38] viscosity relation was used.
Dukler- Wicks-Cleveland Correlation
The adapted Dukler-Wicks-Cleveland 1191 correlation exhibited a low degree of
scatter for the intermittent S1 flow data of Malinowsky [34], Laflin and Oglesby [35],
and Hall [30] with the majority being within a *30% spread using the Brinkman [38]
or Pal-Modes [39] viscosity correlations. The data of Hall [30] in Figure 12 show that
under prediction was in evidence at the inversion point and with some OD flows.
The separated flow data of Lunde et al. [36], Sobocinski [33], and Donnelly [24]
gave a poorer performance with a sizeable portion of the data falling outside the
*30% band. As shown in Table 3, the general performance of the correlation was an
improvement over that for the two-phase case.
Beggs-Brill Correlation
The Beggs-Brill [28] correlation over predicted separate flow data of Sobocinski [33]
and the 0.0259 m i.d. data of Donnelly [24]. These over predictions occurred more
frequently with WD flows. In contrast, the high-pressure data of Lunde et al. [36]
were generally under predicted. Considerable deviation was in evidence at the region
of phase inversion.
Table 3. Average relative percentage error for three-phase pressure loss prediction
using various correlations.
Data
Diameter
m
Lunde r361
Hall [30]
Sobocinski I331
0.1
0.079
0.079
0.0501
Dukler- WicksCleveland [19}
-3.7
12.5
80.0
BeggsBrill 28
-39.5
104.5
57.6
102.2
5 76
BeattieWhaN 43
-70.2
86.5
23.3
1
73.6
I
Prediction of Multiphase Horizontal Pipe Flow: A Reassessment. Parts I and I1
1.5
1
- 3v??
3 1.2
f
3
.z
B
0.9
0.6
:1
7)
3
ii I
0.3
0
I
0
T
0.1
0.2
0.3
0.4
W c W
0.5
0.6
0.7
0.8
0.9
1
Input Water Fraction
Figure 9. Measured to predicted pressure drop against input water fraction for the
0.0501 m i.d. data of Donnelly [24] using rhe Lockhart-Martinelli [6] and Brinkman
[38] correlations.
.
2 -?i
O !
0
I
0.1
I
0.2
0.3
0.4
0.5
0.6
0.7
I
0.8
I
0.9
1
hput Water Fraction
Figure 10. Measured to predicted pressure drop against input water fraction for the
0.031 m i.d. data of Lafllin-Oglesby [35] using the Lockhart-Martinelli [6] and PalRhodes (391 correlarions.
577
1'. I,. S j d d i n g , E. Benard and G.F.Donnelly
0
100
200
300
400
500
600
700
Measured Pressure Gradient (Palm)
Figure 11. Measured to predicted pressure drop for the 0.079 m i.d. data of Hall
[30J using the Lockhart-Martinelli [4] and Brinkman [38] correlations.
Figure 12. Predicted against measured pressure drop for the 0.079 m i d data of
Hall [30] using the Dukler- Wicks-Cleveland [l9] and Brinkman [38] correlations.
5 78
800
Prediction of Multiphase Horizontal Pipe Flow: A Reassessment. Parts !and !I
04
4
_.
I
Figure 13. Predicted against measured pressure drop for the 0.0381 m i.d, data of
Malinowsky [34] using the Beggs-Brill[28] and Brinkman [38] correlations.
0
Figure 14. Predicted against measured pressure drop for the 0.0381 m i.d. data of
Malinowsky [34] using the Friedel [42] and Pal-Rhodes [39] correlations.
5 79
P.L. Spedding,E. Benard and G.F. Donnelly
Significant scatter was obtained with the intermittent S1 flow data [30, 34, 351
using the Pal-Rhodes [39] viscosity correlation. The scatter was mostly reduced to a
&30%band as illustrated in Figure 13 by employing the Brinkman [38] viscosity
correlation. However, over prediction was more noticeable with WD flows while
under prediction occurred with OD flows and particularly at the inversion point.
Total liquid hold-up prediction by the Beggs-Brill [28] correlation was poor with
gross under predictions with all data except that of Hall [30].
Friedef Correlation
The Friedel 1421 correlation gave significant over prediction of the separate flow data
of Sobocinski [33] and Donnelly [24] where over estimates were in excess of 70%
and 50% respectively. An acceptable level of prediction within a *20% spread was
achieved for 75% of the data of Lunde et al. 1361.
The intermittent SI flow data [30, 34, 351 generally fell within a *20% spread,
except at the inversion point which was under predicted. Figure 14 shows the
Malinowsky [34] data with the Pal-Rhodes [39] viscosity correlation, which recorded
slightly better results than that obtained with the Brinkman 1381 correlation.
Beattie- Whalley Correlation
The Beattie-Whalley [43] correlation grossly under predicted the separated flow data
of Lunde et a]. [36]. The 0.0501 m i.d. data of Donnelly [42] exhibited a high degree
of scatter as illustrated in Figure 15. All the intermittent Sl flow data [30, 34, 351
displayed acceptable prediction within a *30% spread particularly the Malinowsky
[34] data given in Figure 16. There was a tendency to over predict for WD flows and
under predict around the inversion point and for OD flows. Table 3 indicates that
overall the Beattie-Whalley [43] correlation gave the most acceptable prediction for
the intermittent S1 type flows.
Discussion
Other correlations such as those by Muller Steinhagen-Heck [44] and Olujic [ 131 gave
gross under prediction of pressure drop for the entire range of data.
The most interesting result of this study was that correlations that with two-phase
data gave poor predictions with intermittent type flows when adapted to the threephrase situation gave reasonable agreement. Also, the opposite performance was
registered with separated type flows. The correlations that performed best in
prediction of three-phase pressure drop were the Beggs-Brill [28] Brinkman [38], the
Friedel [42] Pal-Rhodes 1391 and the Beattie-Whalley [43] Brinkman [38]. With all of
these correlations there was a tendency to over predict WD flow and under predict
OD flows and the inversion point. Improvement in prediction for intermittent threephase flow can be achieved if the predicted WD flows are decreased by 15% and the
OD flows increased by lo%, with the inversion point area being consistent with a
50% increase in prediction obtained for the actual inversion point.
The mechanism for successful prediction of three-phase pressure loss by using
adapted two-phase models may be linked to the calming effect of oil on the water
layer.
580
Prediction of Multiphase Horizontal Pipe Flow: A Reassessment. Parts I and I1
1.6 7
04
0
I
4
1
0.1
I
0.2
,
0.3
I
0.4
I
0.5
0.6
t
0.7
I
I
0.8
0.9
1
Input Water Fraction
Figure 15. Measured to predicted pressure drop against input water fraction for the
0.0501 m i.d, data of Donnelly [24] using the Beattie- Whalley [43] and Brinkman
[38] correlations.
3000
Figure 16. Predicted against measured pressure drop for the 0.0381 m i.d. data of
Mdinowsky [34] using the Beattie-Whalley [43] and Brinkman [38] correlations.
58 I
P . L. Spedding, E. Benard and G.F.Donnelly
Conclusions
Prediction of two-phase pressure loss was achieved for the St+RW+D [13, 19, 281,
St-t-RW[13, 281, D [19, 421 and A+RW [42] regimes. All models except that by
Muller-Steinbagen and Heck [44] had difficulty in attempting to predict intermittent
S1 and BTS flows.
Adoption of the two-phase models for prediction of three-phase pressure loss
highlighted three correlations:
1. Beggs-Brill [28] / Brinkman [38]
2. Friedel [42] / Pal-Rhodes [39]
3. Beattie-Whalley [43] / Brinkman [38]
that gave acceptable prediction for the intermittent SI type flows only.
A modification of these adaptations was proposed that allowed more accurate
pressure loss prediction across the whole range of possible liquid compositions.
Prediction of three-phase separate flows was not successful with any model.
Nomenclature
A
BTS
D
F
IW
i.d.
j
Annular flow
Blow through slug regime
Droplet flow
Film flow
Inertial wave flow
Internal diameter
Actual value
LRW
N
RL
RW
s1
St
Vsc
Long roll wave flow
Number
Hold-up
Roll wave flow
Slug flow
Stratified flow
Superficial gas velocity (m/s)
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Appendix
The average percentage error = IOOC
measured j - predicted j
measured j
I
gives a result that has greater emphasis on over prediction, since under prediction
cannot exceed 100%. This can be overcome by use of the average relative percentage
error
measured j - predicted j
= IOOC
O.S(measuredj + predicted j )
1
584
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