transient paper
TRANSCRIPT
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Drift-Flux Modeling of Transient Countercurrent Two-phase Flow in Wellbores
H. Shi1, J.A. Holmes
2, L.J. Durlofsky
1, K. Aziz
1
1Department of Petroleum Engineering, Stanford University, Stanford, CA 94305-2220, USA
2Schlumberger GeoQuest, 11 Foxcombe Court, Wyndyke Furlong, Abingdon, Oxfordshire, OX14 1DZ, UK
Abstract
Drift-flux modeling techniques are commonly used to represent multiphase flow in pipes and wellbores. These
models, like other multiphase flow models, require a number of empirical parameters. In recent publications we
have described experimental and modeling work on steady-state multiphase flow in pipes, aimed at the
determination of drift-flux parameters for large-diameter inclined wells. This work provided optimized drift-flux
parameters for two-phase water-gas and oil-water flows and a unified model for three-phase oil-water-gas flow for
vertical and inclined pipes.The purpose of this paper is to extend this modeling approach to transient countercurrent
flows, as occur in pressure build-up tests when the well is shut in at the surface. The experiments on which the
steady-state models are based also include transient flow data obtained after shutting in the flow by fast acting
valves at both ends of the test section. We first compare predictions from the existing steady-state drift-flux model
to transient data and show that the model predicts significantly faster separation than is observed in experiments. We
then develop a two-population approach to account for the different separation mechanisms that occur in transient
flows. This model introduces two additional parameters into the drift-flux formulation the fraction of
bubbles/droplets in each population and a drift velocity multiplier for the small bubbles/droplets. It is shown that the
resulting model is able to predict phase separation quite accurately, for vertical and inclined pipes, for both water-
gas and oil-water flows. Finally, the model is applied to interpret a well test in which transient countercurrent
wellbore flow effects are important. It is demonstrated that (to be added by Jon).
Keywords: Transient, Drift-flux, Countercurrent, Two-phase, Three-phase, Large diameter, Inclined, Steady state,
Water-gas, Oil-water, Oil-water-gas, Wellbore, Bubble, Shut-in, Phase redistribution, well testing, two-population
model
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2 H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
Introduction
The drift-flux technique is well-suited for modeling
multiphase wellbore flow in reservoir simulators.
This is because the calculation of phase velocities is
relatively simple and efficient and the equations are
continuous and differentiable, as required by
simulators. However, the drift-flux model includes a
number of empirical parameters, which need to be
tuned to the particular conditions being modeled.
Prior to our recent work, the parameters reported
in the literature and used in commercial simulators
were (typically) determined from experimental data
in small-diameter pipes (5 cm or less) and might
therefore not be appropriate for large-diameter
wellbores. In previous publications1,2,3, we described
experimental and modeling work in which we
determined optimized drift-flux parameters
appropriate for large-diameter vertical and deviated
wells. This was based on steady-state in situ volume
fraction data for a variety of water-gas, oil-water and
oil-water-gas flows in a 15 cm diameter, 11 m long
pipe at 8 deviations ranging from vertical to near-
horizontal1. We showed that the optimized
parameters significantly improved in situ volume
fraction predictions for two and three-phase flows2,3
compared to predictions based on parameters derived
from small-diameter experiments.
In this paper we revisit the two-phase experiments
to investigate the ability of the drift-flux formulation
to model the transient flow that occurs after the test
section is closed at both ends by fast-acting valves.
During this period, phases separate through
countercurrent flow. This phenomenon is similar to
the flow that occurs when a well is shut in (as in a
well test), so the ability to model it could improve
numerical well test interpretation procedures. The
drift-flux formulation is capable of modeling
countercurrent flow as it describes the slip between
two fluids as a combination of a profile effect and a
drift velocity. Our previous analysis was for steady-
state cocurrent flow, but by modeling phase
separation we can test the applicability of the drift-
flux formulation to countercurrent flow.
Although steady-state countercurrent flows (for
example, flooding phenomena in countercurrent gas-
liquid annular flow) have been investigated
previously4,5, compared to steady-state cocurrent
flow, relatively few studies involving countercurrent
flow have been conducted. Transient cocurrent flows
have not received very much attention either.
Therefore, not surprisingly, available data for
transient countercurrent multiphase flow in large-
scale systems are essentially nonexistent. Following
is a review of the literature for steady-state
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3 DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
countercurrent and transient cocurrent flows, with
emphasis on large-diameter systems.
Steady-state countercurrent flows. Taitel and
Barnea6 proposed models for three typical (bubble,
slug and annular) vertical gas-liquid countercurrent
flow patterns. An additional flow pattern (semi-
annular) was subsequently reported by Yamaguchi
and Yamazaki7,8 from their experiments with vertical
water-air systems in 4 and 8 cm diameter pipes.
Hasan et al.9 developed a drift-flux model for
vertical countercurrent bubble and slug flow. The
value of the profile parameter C0 (discussed in detail
below) was found to be 2.0 for bubble flow. They
concluded that the Harmathy10 and Nicklin11
correlations for small bubbles and Talyor bubbles
were valid for countercurrent flows. However, these
conclusions were based on experimental data with
maximum mixture velocities of only 0.5 m/s. Kim et
al.12 also found that their experimental data from a 2
cm diameter vertical pipe were well fitted with the
drift-flux model with Nicklins11 correlation.
However, we are not aware of any published studies
validating the Harmathy10 and Nicklin11 correlations
for large-diameter, high flow rate liquid-gas systems.
Inclined countercurrent data are very limited.
Johnston13,14 developed a semi-empirical model for
liquid-gas countercurrent flows for stratified and slug
flow (as occurs in horizontal and near-horizontal
pipes). The pipe diameter was in the range of
5.712.1 cm and the maximum pipe inclination was
xxx from horizontal. Ghiaasiaan et al.15 conducted
vertical and deviated gas-liquid experiments in a 1.9
cm diameter pipe. The deviations were set to be 0,
28-30, and 60-68 from vertical. In an attempt to
apply the drift-flux model for hold up calculations for
slug flow, they adjusted both the profile parameter C0
and the drift velocity Vd for different liquid
viscosities to match their data.
Zhu and Hill16 and Zavareh et al.17 performed oil-
water tests in an 18.4 cm diameter acrylic pipe at
deviations of 0, 5, and 15 from upward vertical.
Ouyang18,19 classified oil-water countercurrent flow
into five categories and developed models to compute
the phase in situ volume fractions and pressure drop.
His model predictions agreed well with the
experimental data from Zhu and Hill16.
Almehaideb et al.20 presented a coupled
wellbore/reservoir model to simulate three-phase oil-
water-gas countercurrent flow in multiphase injection
processes. Both a two-fluid model and a simple
mixture/homogeneous model were implemented for
wellbore flow. This comprehensive model considered
a black-oil system, in which the oil and water phases
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4 H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
are immiscible and gas is soluble in oil.
Transient cocurrent flows. Asheim and Grdal21
used a modified steady-state drift-flux model to
predict holdup in a transient vertical oil-water
system. The pipe used in the experiment was 4.3 cm
in diameter. To investigate the performance of two-
phase transient flow models, Lopez et al.22,23
considered numerical simulations using OLGA
(based on a two-fluid model), TACITE (based on a
drift-flux model) and TUFFP (based on a two-fluid
model) against both laboratory and field data. They
concluded that all three models could match the
transient data from laboratory tests. However, only
OLGA and TACITE were capable of simulating real
transient flows in long, large-diameter pipelines, with
TACITE providing more accurate predictions than
OLGA.
As indicated above, models for transient
countercurrent phase separation are useful for the
interpretation of well tests (the models of
Almehaideb et al.20 and Hasan and Kabir24 can be
applied under limited conditions). However, the
amount of published transient countercurrent data for
small-diameter, vertical pipes is quite limited. To our
knowledge, there has been no published data for
large-diameter, inclined pipe, transient countercurrent
multiphase flows.
In previous studies, when drift-flux models were
applied to countercurrent steady-state or transient
flow, specific flow regimes, such as bubble and slug
flow, were considered. Thus, a comprehensive drift-
flux model for such systems has yet to be presented.
Furthermore, the Harmathy10 correlation, which is
based on the single bubble rise velocity in a stagnant
liquid, is commonly used to calculate drift velocity.
In this type of correlation, all the gas bubbles/oil
droplets are considered to rise at the same velocity. In
practical cases, however, all flow regimes can exist
simultaneously in the wellbore, with more than one
population of bubbles and droplets. We would expect
different drift velocity mechanisms for
bubbles/droplets of different sizes. To apply the drift-
flux concept to transient countercurrent flows,
therefore, it is useful to consider bubbles/droplets of
different sizes, as we will demonstrate below.
This paper proceeds with a brief description of the
experimental setup and some sample transient data
for two-phase water-gas and oil-water systems and
three-phase oil-water-gas flows. The drift-flux model
used in this work is then reviewed. It is shown that
predictions of water-gas and oil-water separation
during transient flow are not adequately modeled
using the steady-state drift-flux parameters. A two-
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5 DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
population drift-flux model is then proposed and
evaluated for two-phase flows. Finally, the
application of the transient model to phase separation
in a well during a build-up test is discussed.
Experimental procedure
The detailed experimental work was described in
Oddie et al.1 Sample data for steady-state two-phase
water-gas and oil-water flows, and three-phase oil-
water-gas systems were shown in our previous
modeling work2,3. In this paper we briefly explain the
experimental setup and present representative
transient data, which will be used for the transient
flow model.
Experimental setup. The test apparatus used in this
investigation is an 11 m long inclinable pipe with a
diameter of 15 cm. Experiments were performed with
kerosene, tap water and nitrogen. The viscosity of
the oil is 1.5 cP at 18C and the density is 810 kg/m3.
Tests were conducted with pipe inclinations of 0
(vertical), 5, 45, 70, 80, 88, 90 (horizontal), and
92 (downward 2). Data at 90 and 92 flows were
strongly impacted by end effects1 and were therefore
not used for the determination of model parameters.
The test section, shown schematically in Fig. 1,
was of clear acrylic pipe that could be closed at both
ends with fast-acting valves. These two-valves,
which were normally open, were simultaneously
closed to trap the fluid instantaneously (the incoming
fluids were led to a bypass system to minimize water
hammer). Ten electrical conductivity probes were
installed along the test section to measure in situ
water fraction. The probes were placed perpendicular
to the pipe axis and positioned at 1, 2, 3, 4, 5, 6, 7,
7.75, 9 and 10 m along the test section. These probes
were one source for determining the steady-state in
situ volume fraction. This quantity was also
determined through gamma densitometer
measurements and measurement of the final position
of the interface after the fluids settled to their final
positions. The probes also provided the transient
flow data during phase separation after shut-in.
Transient data. In this study, vertical flows are
emphasized in this study because separation of the
phases is generally the slowest in vertical pipes,
though deviations of 5, 45, 70, 80, 88 are also
considered. The flow rate ranges for the water-gas
tests are: 2.0 m3/h Qw100.0 m3/h and 2.6 m3/h
Qg 72.2 m3 /h. The tests for oil-water flow were
conducted in the range of 2.0 m3/h Qo40.0 m3/h
and 2.0 m3/h Qw 130.0 m3 /h. For oil-water-gas
flow, the data are in the range of 2.0 m3/h Qo40.0
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6 H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
m3/h, 2.0 m3/h Qw40.0 m3/h, and 1.8 m3/h Qg
38.7 m3/h.
Three sets of transient data are shown in Figs. 2-4
to illustrate the probe response with time for vertical
flows of water-gas, oil-water and oil-water-gas,
respectively. The figures show dimensionless water
depth (h/D) with h/D = 0 corresponding to the bottom
of the pipe and h/D = 1 to the top of the pipe. Each
figure represents the probe responses for a particular
set ofQo, Qw, Qg.
Both steady-state pre-shut-in and transient data
for a water-gas test are plotted in Fig. 2. Fig. 2 (a)
shows steady state data over a ten second interval.
The response from each probe varies in time as the
probe is subjected to different flow conditions. The
observed flow pattern for this test is elongated
bubble. The flow is statistically steady and most of
the oscillations are around an h/D value of 0.40.5.
The shut-in water volume fraction ( w ) is 49% for
this case.
Fig.2 (b) shows the electrical probe signals from
the time of shut-in to a time after the phases are
completely settled. The settling time for this case is
around 50 seconds. Since w = 49%, the profiles of
probes 15 reach h/D = 1.0 as they are fully
immersed in water, while probes 610 are totally in
the gas phase. Note that signals from probes 610
register nonzero h/D at the end of the transient. This
nonzero h/D is due to the probe calibration procedure
and provides an estimate of the error associated with
the probe data.
Fig. 3 shows the transient profile of a vertical oil-
water test.The water and oil flow rates are almost the
same for this test (Qo =40.2, Qw =40.4), and the flow
rates are relatively high. For this case, oil and water
were observed to be totally mixed to form a
homogeneous phase. The shut-in water volume
fraction value is 51%, which confirms a
homogeneous flow pattern with the flowing volume
fraction equal to the in situ volume fraction. An
interesting phenomenon is apparent in Fig. 3.
Though the pipe is eventually half filled with water
(water at the bottom and oil at the top), probes 15,
which are eventually immersed in water, reach their
final state more quickly than probes 610, which are
finally immersed in oil. This phenomenon occurs due
to the different behaviors of water-in-oil emulsions
compared to oil-in-water emulsions, as discussed in
Oddie et al.1
An oil-water-gas test is displayed in Fig. 4. The
water and oil flow rates are the same for this test as
for the oil-water test shown in Fig. 3. The flow
pattern here was elongated bubble/slug. The
relatively high gas flow rate (26.2 m3 /h) has very
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7 DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
little effect on the overall flow. Compared with the
oil-water vertical flow case (Fig. 3), the settling time
is almost the same for this three-phase flow case. The
expectation was that the settling time for this three-
phase transient process would be longer due to the
gas entrainment in the oil-water mixture leading to
smaller droplets. Similar setting times may be
observed because of complex emulsion behaviors
that occur for the oil-water system around the phase
inversion point, which for this case is expected to be
around 50% water (based on an analysis of the probe
response). Tight emulsions around the phase
inversion point are more difficult to separate, leading
to longer settling times.
From the sample data discussed above, we can
conclude that transient countercurrent flows are
extremely complicated, especially for oil-water and
oil-water-gas systems. Our goal is to develop a
relatively simple model for these systems that is
consistent with our previous models for steady-state
flow.
Steady-state drift-flux models
The original26 and optimized steady-state drift-flux
models for two-phase water-gas, oil-water and three-
phase oil-water-gas flows have been discussed in
detail in our previous publications2,3. Here, we briefly
review both the original26 and optimized liquid-gas
and oil-water models before illustrating the
performance of the steady-state models for transient
flows. The emphasis here is on vertical flows, though
deviated flows are also considered.
Liquid-gas flow. Zuber and Findly25 correlated
actual gas velocity Vg and mixture velocity Vmusing
twoparameters, C0 and Vd:
dmg
sgg VVCV
V +== 0
(1)
where Vsg is the gas superficial velocity (gas flow rate
divided by total pipe area) and g is the gas in situ
volume fraction. The accuracy of the predicted g
depends on the use of appropriate values for C0 and
Vd.
In the original (Eclipse26) model, C0 generally
varies from 1.0 to 1.2, so we have
2.10.1 0 C (2)
and Vd is computed via:
)(
1
)()1(
0
00
m
CC
VKCCV
g
l
g
og
cgg
d
+
=
(3)
where 0( ) 1.53gK C = when 1g a and
when( ) ( )g uK K = D 2g a . Parameters a1 and a2
are the two gas volume fractions which define the
transition from the bubble flow regime. ( )uK D is the
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8 H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
critical Kutateladze number, which is a function of
the dimensionless pipe diameter D . The dependency
( )uK D on D is given in Shi et al.2 Vc is called the
characteristic bubble rise velocity, which was
determined by Harmathy26, and is the density.
The parameter )(m , where is the deviation
from vertical, is very important for modeling flow in
deviated pipes, as it accounts for the deviation from
vertical through a multiplier to Vd. In the original
model,
( ) 25.0 )sin1()(cos)0( += mm (4)
where .00.1)0( =m
In the optimized model, based on the large
diameter data, the values for both C0 and Vd are
significantly different. The first major difference is
the profile parameter, for which we obtain 0.10 =C .
This lower value of C0 directly leads to a much
higher Vd value. For example, the optimized
deviation effect is
( ) 95.021.0 )sin1()(cos)0( += mm (5)
and for vertical liquid-gas flow, . Thus
the optimized V
85.1)0( =m
d value is 1.85 times higher than the
original Vdfor vertical liquid-gas flow.
Oil-water flow. The general form of the drift-flux
model applied to oil-water flows is:
0o lV C V V d = + (6)
where Vo is the in situ oil velocity and Vl is the liquid
mixture velocity. The original value for 0C is in the
same range as C0 for liquid-gas flows:
2.10.1 0 C (7)
anddV is calculated by
27,
)()1(53.1 mVV nocd = (8)
where, as before, cV is also determined by the
Harmathy17
correlation, except that the gas in the
correlation is replaced by oil.
In the original oil-water model27, 0.2=n and
0.1)0( =m for vertical flow. The optimized
parameters for oil-water flow are2: 0.10 =C ,
0.1=n and 07.1)0( =m . Unlike for the liquid-gas
flow, the optimized value of is not much
different from its original value of 1.0. However, the
value of the exponent n is reduced from 2.0 to 1.0.
Compared with the original model, this makes
)0(m
dV
decrease linearly and much more rapidly with
increaseing o .
During transient flow after shut-in, there is no net
flow, so Vm= 0. Hence there is no effect of the profile
parameters C0 or 0C and the gas or oil velocity
depends only on the drift velocity. Therefore, the key
to modeling the transient process is to model the drift
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9 DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
velocity accurately.
Comparison with experimental observations.
Eclipse26 applies the same drift-flux models (the
original steady-state models) for both steady-state
and transient multiphase flow. This is based on the
assumption that transient flow can be represented by
a sequence of steady-state flows. One of the
objectives of our work is to test the validity of this
assumption.
We proceed by identifying two interfaces for two-
phase flows. The gas interface is the interface
between the pure gas and the mixture of gas and
liquid. Similarly, the liquid interface is defined as the
interface between the pure liquid and the mixture of
gas and liquid. Therefore, during the transient
process, the gas interface moves down and the liquid
interface moves up. The two interfaces meet when
the phases are completely separated.
Liquid-gas vertical flow. Fig. 5 shows a sample
comparison of experimental data with predictions for
vertical water-gas flow. Both the original and
optimized steady-state models are considered. In this
case the volume of gas and water in the system is
almost the same. We see that the original model25
predicts the speed of the gas interface height
reasonably well, but the predicted speed of the water
interface is higher that that observed. The optimized
model predicts even higher velocities for both the gas
and water interfaces. This is perhaps surprising, since
the optimized model is more accurate for steady-state
predictions.
Oil-water vertical flow. A sample comparison for
model predictions with experimental data for vertical
oil-water flow is illustrated in Fig. 6. Again the
volume of the two fluids in the system is about the
same. As in the previous case, the speed of the water
interface with the original model is much higher than
that observed in the experiment. Furthermore, the
optimized model yields even higher velocities for
both oil and water interfaces.
An explanation for the disagreement between
transient experiments and steady-state model
predictions can be offered by considering the drift-
flux model parameters. For liquid-gas systems,
0.10 =C for the optimized model, i.e., there is no
profile slip. Hence )(m , the Vd multiplier, must
increase accordingly, and for vertical flow, it is
almost twice the value as in the original model.
Therefore the optimized model predicts much faster
settling.For oil-water flow, the major reason for the
prediction of faster separation by the optimized
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10 H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
model compared to the original model is the
reduction in the exponent n from 2.0 to 1.0.
From these comparisons of experimental data
with model predictions, we see that our steady-state
models do not fully capture the mechanics of
countercurrent transient flows. These findings are
consistent with earlier work by King et al.28, who
tried to capture the characteristics of transient slug
flows. They conducted water-air tests in a 36 m long,
7.6 cm diameter stainless steel horizontal pipe. The
experimental results demonstrated that generally
transient slug flow cannot be modeled by the quasi-
steady-state approach. In order to overcome the
limitations of the sequence of steady-states approach,
we will now consider a two-population model.
Two-population model
Our water-gas transient experiments show thatsome
small gas bubbles are entrained in the water and
move with the water phase at the beginning of the
settling process. Similarly, for oil-water flow, some
small water droplets are entrained in oil and move up
with the oil phase at the beginning of the separation.
These small bubbles/droplets separate from the phase
in which they are entrained later in the separation
process.
As illustrated in Fig. 7, the model of drift-flux
velocity used in both steady-state models (original
and optimized) does distinguish between large and
small bubbles/droplets. Fig. 7 (a) shows that a linear
interpolation is used to connect the bubble flow
regime and liquid flooding curve2 over the range
. Since bubble size increases with21 aa g
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11 DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
system. In fact, in reality there will exist a
distribution of bubble sizes, with the smaller bubbles
having even lower drift velocity4. The dashed line in
Fig. 7 (b) illustrates this.
From Fig. 7, we see that by shifting a1 and a2, we
can potentially represent both steady-state and
transient flows using one drift-flux model. The two-
population model discussed below is a unified model
for steady-state and transient flows. This unification
is especially important for reservoir simulation, in
which a smooth transition between steady-state and
transient flows is required.
Model development. Based on our observations of
steady-state and transient flows we can conclude that
in the separation of water and gas, two processes
occur. First, large gas bubbles separate from the gas-
water mixture, and next the entrained small gas
bubbles separate from the water. This can be modeled
by dividing the total gas fraction into two parts,
corresponding to large bubbles and small bubbles:
gSgLg += (9)
where subscript L represents the large bubbles and S
the small bubbles.
We can apply the drift-flux model, Eq. (1), for
large and small bubbles separately. With the
assumption that there is no profile slip for small
bubble separation ( ) due to the large bubble
separation with the mixture, a general drift-flux
model is obtained (see Appendix A for details):
0.10 =SC
dSgSdL
gL
gLg
m
gL
LgLg
gg
VV
VC
V
+
+
=
1
)1(
]1
)1)(1(1[
0
(10)
Here is the profile parameter for the separation
of large bubbles from the mixture of small bubbles
and liquid. and define the drift velocity of
large bubbles and small bubbles respectively. This
equation reduces to the original form when there is
only one kind of bubble and there is no profile slip
for small bubbles.
LC0
dLV dSV
Two-population model for oil-water systems. The
two-population oil-water model is similar to the
liquid-gas model, but the mechanisms involved in
oil-water separation are different. Specifically, large
water droplets move down while the small water
droplets entrained in the oil move up with the oil
phase. This is also consistent with the observation by
Zhu and Hill16 and Zavareh et al.17. In addition, the
entrained small water droplets further separate from
the oil.
We divide the water droplets into two
populations:
wSwLw += (11)
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12 H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
and apply the oil-water drift-flux model, Eq. (8), to
both settling processes with the assumption that the
profile slip of small droplets is 1.0 due to the
disruption of large water droplets separating with the
mixture. The resulting two-population model for oil-
water separation is (see Appendix A):
))(1( 0 dSdLmLwSwLmww VVVCVV ++= (12)
where is the profile parameter for the separation
of large water droplets from the mixture of oil and
water. and represent the drift velocity of the
oil when separating with large and small water
droplets respectively.
LC0
dLV dSV
We note that the two-population model described
here represents a considerable simplification of the
true transient process, in which a continuous
distribution of bubble or drop sizes presumably
exists. Nonetheless, as shown below, this model does
appear to capture the key transient effects observed in
the experiments. This is likely because the two
populations of bubble/drop sizes (and corresponding
adjustable parameters) represent, in some sense, an
appropriate sampling of the true continuous
distribution.
Results and discussion
To implement the two-population model, we
introduce two additional adjustable parameters.
These are the fraction f of large bubbles/droplets to
the total bubbles/droplets in the system and the drift
velocity multiplier mS for small bubbles/droplets
(where VdS = mSVdL( )). These parameters
depend, in general, on the shut-in holdup, though in
many cases constant values suffice. Using the two-
population model with these two parameters, we can
achieve close matches to the transient experimental
data. In the following figures, the model results are
shown in terms of interface height. Predictions by the
optimized steady-state parameters are also shown.
0=g
Vertical water-gas flow. For all water-gas cases, a
single set of optimized parameter values
(independent of g and w) was determined:
3.0== ggLf and . These values indicate
that most (70%) of the gas bubbles in the water-gas
systems are small bubbles.
3.0=Sm
The water-gas results are illustrated in Figs. 7-9.
Each figure corresponds to a particular value of
(as indicated in the figure). The first example is
for a relatively low ( ). We see from Fig.
8 that the optimized steady-state model predicts very
fast separation, while the new two-population model
matches the data much more closely. Fig. 9 shows
similar results for a gas volume fraction of 0.32.
g
g 18.0=g
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13 DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
The amount of water and gas in the system is
about the same for the last example displayed in Fig.
10. The results from the steady-state models for this
case were presented in Fig. 4. Here the movement of
the gas interface is predicted by the two-population
model to be too slow at the beginning of the
separation but overall the results for both the gas and
water interfaces are in reasonable agreement with the
experiments.
Vertical oil-water flow. The tuning of the two
parameters and is more complicated for the
oil-water system than for the water-gas system. The
optimized value for is found to be 0.2 for all of the
oil-water transient data. However, in contrast to the
water-gas system, a single value for could not be
obtained. This is a result of the formation of oil-water
emulsions.
f Sm
f
Sm
1 Furthermore, small droplet behavior can
be very different from small bubble behavior29.
The oil-water model results are shown in Figs. 10-
12. We see that for low oil fractions the new model
represents the data very well, as shown in Fig. 11.
The data in Fig. 12 were also shown in Fig. 5 along
with steady-state model predictions. Again the match
between the experimental data and model predictions
is very close. In this case, the value is very small
(
Sm
03.0=Sm ). We attribute this to our expectation
that the phase inversion point is around 50% for this
oil-water system (the fine oil and water droplets
separate very slowly around the phase inversion
point). Table 1 gives Sm values for seven oil-water
tests. It clearly demonstrate that reaches a
minimum at around
Sm
w = 50%. Accurate results are
also obtained in the case of high oil fraction, as
shown in Fig. 13.
Deviated two-phase flows. We now briefly consider
the applicability of the two-population model to
deviated wells. For these cases, we use the )(m
determined in the steady-state optimizations (Eq. (8)
for liquid-gas systems).
Results for water-gas and oil-water systems are
shown in Figs. 14 and 15 respectively. For liquid-gas
flow, we present an example at a 5 deviation. We
select this deviation because the settling process for
our water-gas tests is very fast at the higher
deviations (recall that there is no data available
between 5 and 45). For the oil-water system,
however, the settling time for a deviation of 45 (as
considered in Fig. 15) is long enough to illustrate the
results. As displayed in Figs. 14 and 15, transient
data for both deviated water-gas and oil-water
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14 H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
systems are represented very well by the two-
population models. We again emphasize that the
models in this case are consistent with the steady-
state models, as )(m is the same in both cases.
Application to well testing
(Jons contribution)
* Why phase redistribution can be important
* Hallmarks of phase redistribution
* Simulation results
* What tweaks to d-f are necessary to match the
observations
Conclusions and recommendations
From this study, we can draw the following
conclusions:
The drift-flux model is well suited for steady-
state concurrent flows as well as transient
countercurrent flows in wellbores and pipes.
Experimental data from large-diameter pipes
suggest that wellbore transient flow cannot be
represented by a series of steady-state flows.
Experimental observations show that gas exists
as large and small bubbles during the settling
process for water-gas flow. In oil-water
separation, water exists as large and small water
droplets.
A new unified two-population drift-flux model
was developed for transient two-phase flows.
The model reduces to the steady-state model in
appropriate limits. The model predictions match
transient experimental data reasonably well for
both vertical and deviated water-gas and oil-
water flows.
Application to well testing (Jons contribution)
A concern with this model (or many wellbore flow
models) is that the model parameters are based on
transient data collected in a relatively short pipe (11
m). In addition, the disturbances caused by the fast-
acting valves may not represent actual conditions in
the field. It is therefore possible that the model
parameters may require tuning for specific
applications. This can only be gauged by testing the
model against other experimental data sets, which are
not currently available. Even though the model
parameters may require tuning for a particular
application, it is still reasonable to expect that the
two-population model presented here (or a very
similar model) can be used to represent transient
countercurrent wellbore flows.
Acknowledgments
The support from Schlumberger and the other
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15 DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
industrial affiliates of the Stanford Project on the
Productivity and Injectivity of Advanced Wells
(SUPRI-HW) is greatly appreciated.
Nomenclature
a1 = drift velocity ramping parameter
a2 = drift velocity ramping parameter
a3 = gas effect parameter
A = profile parameter term, value in bubble/slug
regimes for liquid-gas flows
A = profile parameter term for oil-water flows
B = profile parameter term, gas volume fraction
at which C0 begins to reduce
B1 = profile parameter term, oil volume fraction
at which begins to reduce0C
B2 = profile parameter term, oil volume fraction
at which falls to 1.00C
Co = profile parameter
D = pipe internal diameter
f = fraction of large bubbles/droplets
g = gravitational acceleration
Ku = Kutateladze number
L = test section length
m = drift velocity multiplier for water-gas flows
m = drift velocity multiplier for oil-water flows
mS = drift velocity multiplier for small buubbles
Sm = drift velocity multiplier for small water droplets
n = drift velocity exponent for oil-water flows
Q = volumetric flow rate
V = velocity
Vc = characteristic velocity for liquid-gas flows
cV = characteristic velocity for oil-water flows
Vd = gas-liquid drift velocity
dV = oil-water drift velocity
Vm = mixture velocity
Vs = superficial velocity
Subscripts
g = gas
l = liquid
L = large bubbles/droplets
m = mixture
o = oil
S = small bubbles/droplets
w = water
Greek
= in situ fraction or holdup
= interfacial tension/surface tension
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16 H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
= density
= deviation from vertical
References
1. Oddie, G., Shi, H., Durlofsky, L.J., Aziz, K., Pfeffer,
B. and Holmes, J.A.: Experimental Study of Two and
Three Phase Flows in Large Diameter Inclined Pipes,
Int. J. Multiphase Flow, (2003) 29, 527-558.
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L.R., Alkaya, B. and Oddie, G.: Drift-Flux Modeling
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(March, 2005) 10,24-33.
3. Shi, H., Holmes, J.A., Diaz, L.R., Durlofsky, L.J.,
Aziz, K.: Drift-Flux Parameters for Three-Phase
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McGraw-Hill, New York, 1969.
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6. Taitel, Y., and Barnea, D.: Counter Current Gas-
Liquid Vertical Flow, Model for Flow Pattern and
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637-647.
7. Yamaguchi, K. and Yamazaki, Y.: Characteristics of
Coutercurrent Gas-Liquid Two-Phase Flow in Vertical
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8. Yamaguchi, K. and Yamazaki, Y.: Combined Flow
Pattern Map for Cocurrent and Countercurrent Air-
Water Flows in Vertical Tubes, J. Nucl. Sci.
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9. Hasan, A.R., Kabir, C.S., and Srinivasan, S.:
Countercurrent Bubble and Slug Flows in a Vertical
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10. Harmathy, T.Z.: Velocity of Large Drops and
Bubbles in Media of Restricted Extent, AIChEJ
(1960) 6, 281-290.
11. Nicklin, D. J., Wilkes, J. O. and Davidson, J.F.: Two-
Phase Film Flow in Vertical Tubes, Trans. Inst.
Chem, (1962) 40, 61-68.
12. Kim, H.Y., Koyama, S. and Mastumoto, W.: Flow
Pattern and Flow Characteristics for Counter-current
Two-phase Flow in a Vertical Round Tube with Wire-
coil Inserts,Int. J. Multiphase Flow, (2001) 27, 2063-
2081.
13. Johnston, A.J.: An Investigation into Stratified Co-
and Countercurrent Two-Phase Flow, SPEPE(Aug.
1988) 393-399.
14. Johnston, A.J.: Controlling Effects in Countercurrent
Two-Phase Flow, SPEPE(Aug. 1988) 400-404.
15. Ghiaasiaan, S.M., Wu, X., Sadowski, D.L., and Abdel-
Khalik, S.I.: Hydrodynamic Characteristics of
Counter-Current Two-Phase Flow in Vertical and
Inclined Channels: Effect of Liquid Properties, Int. J.
MultiphaseFlow, (1997) 23, 1063-1083.
16. Zhu, D., and Hill, A.D.: The Effect of Flow from
Perforations on Two-Phase Flow: Implications for
Production Logging, SPE paper 18207 presented at
the 1988 SPE Annual Technical Conference and
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17 DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
Exhibition, Houston, TX, 2-5 October.
17. Zavareh, F., Hill, A.D. and Podio, A.: Flow Regimes
in Vertical and Inclined Oil/Water Flow in Pipes,
SPE paper 18215 presented at the 1988 SPE Annual
Technical Conference and Exhibition, Houston, TX,
2-5 October.
18. Ouyang, L.B.: Mechanistic and Simplied Models for
Countercurrent Flow in Deviated and Multilateral
Wells, SPE paper 77501 presented at the 2002 SPE
Annual Technical Conference and Exhibition, San
Antonio, TX, 29 Sept2 Oct.
19. Ouyang, L.B.: Mechanistic and Simplied Models for
countercurrent flow in deviated and multilateral
wells, Petroleu Sci.& Tech, (2003) 21, 2001-2020.
20. Almehaideb, R.A., Aziz, K. and Pedrosa, O.A.: A
Reservoir/Wellbore Model for Multiphase Injection
and Pressure Transient Analysis, SPE paper 17941
presented at the 1989 SPE Middle East Oil Technical
Conference and Exhibition, Manama, Bahrain, 11-14
March.
21. Asheim, H. and Grodam, E.: Holdup Propagation
Predicted by Steady-State Drift Flux Models, Int. J.
MultiphaseFlow, (1998) 24, 757-774.
22. Lopez, D., Dhulesia, H., Leporcher, E. and Duchet-
Suchaux, P.: Performances of Transient Two-Phase
Flow Models, SPE paper 38813 presented at the 1997
SPE Annual Technical Conference and Exhibition,
San Anitonio, TX, 5-8 October.
23. Lopez, D. and Duchet-Suchaux, P.: Performances of
Transient Two-Phase Flow Models, SPE paper 39858
presented at the 1998 International Petroleum
Conference and Exhibition of Mexico, Villahermosa,
3-5 March.
24. Hasan, A.R. and Kabir, C.S.: Modeling Changing
Storage During a Shut-in Test, SPEFE(1994) 9, 279-
284.
25. Zuber, N. and Findlay, J.A.: Average Volumetric
Concentration in Two-Phase Flow Systems, J. Heat
Transfer, Trans. ASME, (1965) 87, 453-468.
26. Schlumberger GeoQuest, ECLIPSE Technical
Description Manual, 2001.
27. Hasan, A.R. and Kabir, C.S.: A Simplified Model for
Oil/Water Flow in Vertical and Deviated Wellbores,
SPE Prod. & Fac. (February 1999) 56-62.
28. King, M.J.S., Hale, C.P., Lawrence, C.J., and Hewitt,
G.F.: Characteristics of Flow Rate Transients in Slug
Flow,Int. J. MultiphaseFlow, (1997) 24, 825-854.
29. Pal, R.: Pipeline Flow of Unstable and Surfactant-
Stabilized Emulsions, AIChE J.(1993) 39, 1754-
1764.
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18 H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
Fig. 1: Schematic of the test section of the flow loop
inlet outlettemperatureelectrical probes
differentialpressure pressure
gammadensitometer
valvevalve
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20 H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
Fig. 3. Oil-water data for =0, Qo=40.2 m3/h, Qw=40.4m
3/h (w=51%).
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21 DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
Fig. 4. Oil-water-gas data for =0, Qo=40.2 m3/h,Qw=40.4 m
3/h, Qg=26.2 m
3/h (w=44%, o=42%).
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22 H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
0
1
2
3
45
6
7
8
9
10
11
0 10 20 30 40 50 60 70 8Time (s)
Interfac
eHeight(m)
0
Experiment_gas
Original_gas
Optimized_gas
Experiment_water
Original_water
Optimized_water
Fig. 5. Water-gas interface height for =0, Qw=2.0 m3/h,Qg=60.2 m
3/h (w=49%).
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23 DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
0
1
2
3
4
5
6
7
8
9
10
11
0 100 200 300 400 500 600 700 800Time (s)
Interfa
ceheight(m)
Experiment_oil
Original_oil
Optimized_oil
Experiment_water
Original_water
Optimized_water
Fig. 6. Oil-water interface height for =0, Qw=40.4 m3/h,Qo=40.2 m
3/h (w=51%).
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24 H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0g
Vd
large bubbles
smallbubbles
a a
(a) Original drift velocity for liquid-gas system
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
g
Vd
a2a1
small bubbles
smaller bubbles
(b) Small bubble drift velocity for liquid-gas system
Fig. 7. Drift velocity mechanism in two-population forliquid-gas system
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25 DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
0
1
2
3
45
6
7
8
9
10
11
0 10 20 30 40 50 60 70 80 90 100
Time (s)
Interfac
eHeight(m)
Experiment_gas
Optimized_gas_ss
Optimized_gas_t
Experiment_water
Optimized_water_ss
Optimized_water_t
Fig. 8. Water-gas interface height for =0, Qw=2.0 m3/h,Qg=11.4 m
3/h (w=82%).
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26 H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
0
1
2
3
4
5
6
7
8
9
10
11
0 10 20 30 40 50 60 70 80 90 100
Time (s)
Interfa
ceHeight(m)
Experiment_gas
Optimized_gas_ss
Optimized_gas_t
Experiment_water
Optimized_water_ss
Optimized_water_t
Fig. 9. Water-gas interface height for =0, Qw=2.0 m3/h,Qg=28.6 m
3/h (w=68%).
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28 H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
0
1
2
3
45
6
7
8
9
10
11
0 100 200 300 400 500 600 700
Time (s)
Interfac
eHeight(m)
Experiment_oil
Optimized_oil_ss
Optimized_oil_t
Experiment_water
Optimized_water_ss
Optimized_water_t
Fig. 11. Oil-water interface height for =0, Qw=100.0 m3/h,Qo=40.2 m
3/h (w=72%).
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29 DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
0
1
2
3
45
6
7
8
9
10
11
0 100 200 300 400 500 600 700 800
Time (s)
InterfaceHeight(m)
Experiment_oil
Optimized_oil-ss
Optimized_oil-t
Experiment_water
Optimized_water_ss
Optimized_water_t
Fig. 12. Oil-water interface height for =0, Qw=40.4m
3/h, Qo=40.2 m
3/h (w=51%).
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30 H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
0
1
2
3
45
6
7
8
9
10
11
0 100 200 300 400 500 600 700 800
Time (s)
Interfac
eHeight(m)
Experiment_oil
Optimized_oil-ss
Optimized_oil_t
Experiment_water
Optimized_water_ss
Optimized_water_t
Fig. 13. Oil-water interface height for =0, Qw=2.0 m3/h,Qo=10.0 m
3/h (w=27%).
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31 DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
0
1
2
3
4
5
6
7
8
9
10
11
0 10 20 30 40 50 60 70 8
Time (s)
InterfaceHeight(m)
0
Experiment_gas
Optimized_gas_ss
Optimized_gas_t
Experiment_water
Optimized_water_ss
Optimized_water_t
Fig. 14. Water-gas interface height for =5, Qw=10.1 m3/h,Qg=58.8 m
3/h (w=52%).
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32 H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
Fig. 15. Oil-water interface height for =45, Qw=100.0m
3/h, Qo=40.2 m
3/h (w=72%).
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33 DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
TABLE 1SUMMARY OF PARAMETER Sm
FOR OIL-WATER SYSTEMS
w 0.27 0.51 0.60 0.72 0.82 0.85 0.93
Sm 0.05 0.03 0.07 0.10 0.50 0.80 0.95
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34 H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
Appendix A
Derivation of two-population drift-flux
models
Liquid-gas flow. Because of small gas bubbles that
are entrained in the water phase, while the overall,
gas is rising a mixture of water-gas is sinking.
This system could be model with two populations
of bubbles: large bubbles with volume fraction of
gL , and small bubbles with volume fraction of gS .
gSgLg += (A-1)
The fraction ofgL and gS depends on the relative
densities of large and small bubbles.
Since large bubbles separate from the mixture of
liquid and entrained small bubbles, we first apply the
drift-flux model to large bubbles:
(A-2)dLmLgL VVCV += 0
The total mixture velocity is:
mSgLgLgLm VVV )1( += (A-3)
where VmS is the mixture velocity of the small
bubbles and liquid. From Eqn (A-2) and (A-3), we
obtain,
dL
gL
gL
m
gL
LgL
mS VVC
V
=
11
1 0 (A-4)
In this small bubble and liquid mixture, the small
bubbles travel with a velocity VgS, which can also be
computed by drift-flux model:
dSgSmSSgSgSgS VVCV += 0 (A-5)
Assuming that there is no profile slip for small
bubbles since the profiles are disrupted by large
bubbles, , Eqn (A-5) becomes:0.10 =SC
dSmSgS VVV += (A-6)
The mixture velocity for small bubbles and liquid
can be written as:
lggSSgmSgLVVV )1()1( += (A-7)
where Vl is the liquid velocity. We can rearrange the
above expression for Vl:
gS
g
gS
mS
g
gL
l VVV
=
11
1 (A-8)
and combining Eqn (A-4), (A-6) and (A-8), to obtain:
gS
g
gS
dL
gL
gL
m
gL
LgL
l VVVC
V
=
111
1 0 (A-9)
For the liquid-gas system we:
lgggm VVV )1( += (A-10)
where Vg is the average gas velocity of both large
bubbles and small bubbles . By combining Eqn (9)
and (10), we can obtain the general two-population
model for liquid-gas flow:
dSgSdLgL
gLg
m
gL
LgLg
gg
VV
VC
V
+
+
=
1
)1(
]1
)1)(1(1[
0
(A-11)
Oil-water flow. Our experiments show that water
entrained in the oil phase, and the water-in-oil
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35 DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
dispersions/emulsions separated much slower than
pure phases. Therefore, we can assume that in the
overall system mixture of oil and small water
droplets rises while large water droplets sink.
Similarly to the treatment of the liquid-gas
system, let there be two populations of water
droplets: large water droplets with volume fraction of
, and small dropllets with volume fraction of
.
wL
wS
wSwLw += (A-12)
The fractions and depends on the relative
densities fluid properties and flowing conditions.
wL wS
Since large water droplets separate from a mixture
of oil and entrained small water droplets, we first
apply the drift-flux model to the system of the rising
oil-water mixture and sinking large water droplets:
(A-13)dommLom VVCV += 0
where Vom is the in situ velocity of the mixture of oil
and the small droplets, and Vdom is the drift velocity
of the mixture.
In the rising mixture, the velocity of pure oil can
be determined from:
doomSo VVCV += 0 (A-14)
For an oil-water system, we have the following
relationship:
owwwm VVV )1( += (A-15)
where Vw is the average water velocity of both large
water droplets and small water droplets .
Combining Eqn (A-12), (A-13), (A-14) and (A-
15), and assuming that the profile slip for the oil and
small water droplets system is disrupted by large
water droplets ( 0.10 =SC ) we obtain the following
two-population model for oil-water flow:
))(1( 0 dSdLmLwSwLmww VVVCVV ++=
(A-16)