Drift-Flux Modeling of Transient Countercurrent Two-phase Flow in Wellbores
H. Shi1, J.A. Holmes2, L.J. Durlofsky1, K. Aziz1
1
Department of Petroleum Engineering, Stanford University, Stanford, CA 94305-2220, USA
2
Schlumberger 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 watergas 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
2
H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
Introduction
In this paper we revisit the two-phase experiments
The drift-flux technique is well-suited for modeling
to investigate the ability of the drift-flux formulation
multiphase wellbore flow in reservoir simulators.
to model the transient flow that occurs after the test
This is because the calculation of phase velocities is
section is closed at both ends by fast-acting valves.
relatively simple and efficient and the equations are
During
continuous and differentiable, as required by
countercurrent flow. This phenomenon is similar to
simulators. However, the drift-flux model includes a
the flow that occurs when a well is shut in (as in a
number of empirical parameters, which need to be
well test), so the ability to model it could improve
tuned to the particular conditions being modeled.
numerical well test interpretation procedures. The
this
period,
phases
separate
through
Prior to our recent work, the parameters reported
drift-flux formulation is capable of modeling
in the literature and used in commercial simulators
countercurrent flow as it describes the slip between
were (typically) determined from experimental data
two fluids as a combination of a profile effect and a
in small-diameter pipes (5 cm or less) and might
drift velocity. Our previous analysis was for steady-
therefore not be appropriate for large-diameter
state cocurrent flow, but by modeling phase
wellbores. In previous publications1,2,3, we described
separation we can test the applicability of the drift-
experimental and modeling work in which we
flux formulation to countercurrent flow.
parameters
Although steady-state countercurrent flows (for
appropriate for large-diameter vertical and deviated
example, flooding phenomena in countercurrent gas-
wells. This was based on steady-state in situ volume
liquid
fraction data for a variety of water-gas, oil-water and
previously4,5, compared to steady-state cocurrent
oil-water-gas flows in a 15 cm diameter, 11 m long
flow, relatively few studies involving countercurrent
pipe at 8 deviations ranging from vertical to near-
flow have been conducted. Transient cocurrent flows
horizontal1.
optimized
have not received very much attention either.
parameters significantly improved in situ volume
Therefore, not surprisingly, available data for
fraction predictions for two and three-phase flows2,3
transient countercurrent multiphase flow in large-
compared to predictions based on parameters derived
scale systems are essentially nonexistent. Following
from small-diameter experiments.
is a review of the literature for steady-state
determined
optimized
We
showed
drift-flux
that
the
annular
flow)
have
been
investigated
3
DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
3
countercurrent and transient cocurrent flows, with
liquid-gas countercurrent flows for stratified and slug
emphasis on large-diameter systems.
flow (as occurs in horizontal and near-horizontal
pipes). The pipe diameter was in the range of
Steady-state countercurrent flows. Taitel and
5.7−12.1 cm and the maximum pipe inclination was
Barnea6 proposed models for three typical (bubble,
xxx° from horizontal. Ghiaasiaan et al.15 conducted
slug and annular) vertical gas-liquid countercurrent
vertical and deviated gas-liquid experiments in a 1.9
flow patterns. An additional flow pattern (semi-
cm diameter pipe. The deviations were set to be 0°,
annular) was subsequently reported by Yamaguchi
7,8
and Yamazaki
from their experiments with vertical
water-air systems in 4 and 8 cm diameter pipes.
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
9
Hasan et al. 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
10
Harmathy
and
Nicklin
and the drift velocity Vd for different liquid
viscosities to match their data.
Zhu and Hill16 and Zavareh et al.17 performed oilwater tests in an 18.4 cm diameter acrylic pipe at
11
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
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
12
al. also found that their experimental data from a 2
cm diameter vertical pipe were well fitted with the
drift-flux
model
with
11
Nicklin’s
correlation.
However, we are not aware of any published studies
10
validating the Harmathy
11
and Nicklin
correlations
for large-diameter, high flow rate liquid-gas systems.
Inclined countercurrent data are very limited.
13,14
Johnston
developed a semi-empirical model for
experimental data from Zhu and Hill16.
Almehaideb et al.20 presented
a
coupled
wellbore/reservoir model to simulate three-phase oilwater-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
4
H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
multiphase flows.
are immiscible and gas is soluble in oil.
In previous studies, when drift-flux models were
Transient cocurrent flows. Asheim and GrØdal21
applied to countercurrent steady-state or transient
used a modified steady-state drift-flux model to
flow, specific flow regimes, such as bubble and slug
predict holdup in a transient vertical oil-water
flow, were considered. Thus, a comprehensive drift-
system. The pipe used in the experiment was 4.3 cm
flux model for such systems has yet to be presented.
in diameter. To investigate the performance of two-
Furthermore, the Harmathy10 correlation, which is
phase transient flow models, Lopez et al.22,23
based on the single bubble rise velocity in a stagnant
considered
numerical simulations using OLGA
liquid, is commonly used to calculate drift velocity.
(based on a two-fluid model), TACITE (based on a
In this type of correlation, all the gas bubbles/oil
drift-flux model) and TUFFP (based on a two-fluid
droplets are considered to rise at the same velocity. In
model) against both laboratory and field data. They
practical cases, however, all flow regimes can exist
concluded that all three models could match the
simultaneously in the wellbore, with more than one
transient data from laboratory tests. However, only
population of bubbles and droplets. We would expect
OLGA and TACITE were capable of simulating real
different
transient flows in long, large-diameter pipelines, with
bubbles/droplets of different sizes. To apply the drift-
TACITE providing more accurate predictions than
flux concept to transient countercurrent flows,
OLGA.
therefore, it is useful to consider bubbles/droplets of
As
indicated
above,
models
for
transient
countercurrent phase separation are useful for the
drift
velocity
mechanisms
for
different sizes, as we will demonstrate below.
This paper proceeds with a brief description of the
of
experimental setup and some sample transient data
Almehaideb et al.20 and Hasan and Kabir24 can be
for two-phase water-gas and oil-water systems and
applied under limited conditions). However, the
three-phase oil-water-gas flows. The drift-flux model
amount of published transient countercurrent data for
used in this work is then reviewed. It is shown that
small-diameter, vertical pipes is quite limited. To our
predictions of water-gas and oil-water separation
knowledge, there has been no published data for
during transient flow are not adequately modeled
large-diameter, inclined pipe, transient countercurrent
using the steady-state drift-flux parameters. A two-
interpretation
of
well
tests
(the
models
5
DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
population drift-flux model is then proposed and
ends with fast-acting valves. These two-valves,
evaluated
the
which were normally open, were simultaneously
application of the transient model to phase separation
closed to trap the fluid instantaneously (the incoming
in a well during a build-up test is discussed.
fluids were led to a bypass system to minimize water
for
two-phase
flows.
Finally,
hammer). Ten electrical conductivity probes were
Experimental procedure
installed along the test section to measure in situ
The detailed experimental work was described in
water fraction. The probes were placed perpendicular
Oddie et al.1 Sample data for steady-state two-phase
to the pipe axis and positioned at 1, 2, 3, 4, 5, 6, 7,
water-gas and oil-water flows, and three-phase oil-
7.75, 9 and 10 m along the test section. These probes
water-gas systems were shown in our previous
were one source for determining the steady-state in
modeling work2,3. In this paper we briefly explain the
situ volume fraction. This quantity was also
experimental
determined
setup
and
present
representative
through
gamma
densitometer
transient data, which will be used for the transient
measurements and measurement of the final position
flow model.
of the interface after the fluids settled to their final
positions.
Experimental setup. The test apparatus used in this
The probes also provided the transient
flow data during phase separation after shut-in.
investigation is an 11 m long inclinable pipe with a
diameter of 15 cm. Experiments were performed with
Transient data. In this study, vertical flows are
kerosene, tap water and nitrogen. The viscosity of
emphasized in this study because separation of the
the oil is 1.5 cP at 18°C and the density is 810 kg/m3.
phases is generally the slowest in vertical pipes,
Tests were conducted with pipe inclinations of 0°
though deviations of 5°, 45°, 70°, 80°, 88° are also
(vertical), 5°, 45°, 70°, 80°, 88°, 90° (horizontal), and
considered. The flow rate ranges for the water-gas
92° (downward 2°). Data at 90° and 92° flows were
tests are: 2.0 m3/h ≤ Qw ≤ 100.0 m3/h and 2.6 m3/h ≤
strongly impacted by end effects1 and were therefore
Qg ≤ 72.2 m3/h. The tests for oil-water flow were
not used for the determination of model parameters.
conducted in the range of 2.0 m3/h ≤ Qo ≤ 40.0 m3/h
The test section, shown schematically in Fig. 1,
and 2.0 m3/h ≤ Qw ≤ 130.0 m3/h. For oil-water-gas
was of clear acrylic pipe that could be closed at both
flow, the data are in the range of 2.0 m3/h ≤ Qo ≤ 40.0
5
6
H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
m3/h, 2.0 m3/h ≤ Qw ≤ 40.0 m3/h, and 1.8 m3/h ≤ Qg ≤
register nonzero h/D at the end of the transient. This
38.7 m3/h.
nonzero h/D is due to the probe calibration procedure
Three sets of transient data are shown in Figs. 2-4
to illustrate the probe response with time for vertical
and provides an estimate of the error associated with
the probe data.
flows of water-gas, oil-water and oil-water-gas,
Fig. 3 shows the transient profile of a vertical oil-
respectively. The figures show dimensionless water
water test. The water and oil flow rates are almost the
depth (h/D) with h/D = 0 corresponding to the bottom
same for this test (Qo =40.2, Qw =40.4), and the flow
of the pipe and h/D = 1 to the top of the pipe. Each
rates are relatively high. For this case, oil and water
figure represents the probe responses for a particular
were observed to be totally mixed to form a
set of Qo, Qw, Qg.
homogeneous phase. The shut-in water volume
Both steady-state pre-shut-in and transient data
fraction
value
is
51%,
which
confirms
a
for a water-gas test are plotted in Fig. 2. Fig. 2 (a)
homogeneous flow pattern with the flowing volume
shows steady state data over a ten second interval.
fraction equal to the in situ volume fraction. An
The response from each probe varies in time as the
interesting phenomenon is apparent in Fig. 3.
probe is subjected to different flow conditions. The
Though the pipe is eventually half filled with water
observed flow pattern for this test is elongated
(water at the bottom and oil at the top), probes 1–5,
bubble. The flow is statistically steady and most of
which are eventually immersed in water, reach their
the oscillations are around an h/D value of 0.4–0.5.
final state more quickly than probes 6–10, which are
The shut-in water volume fraction ( α w ) is 49% for
finally immersed in oil. This phenomenon occurs due
this case.
to the different behaviors of water-in-oil emulsions
Fig.2 (b) shows the electrical probe signals from
the time of shut-in to a time after the phases are
compared to oil-in-water emulsions, as discussed in
Oddie et al.1
completely settled. The settling time for this case is
An oil-water-gas test is displayed in Fig. 4. The
around 50 seconds. Since αw = 49%, the profiles of
water and oil flow rates are the same for this test as
probes 1–5 reach h/D = 1.0 as they are fully
immersed in water, while probes 6–10 are totally in
the gas phase. Note that signals from probes 6–10
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
7
DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
7
little effect on the overall flow. Compared with the
review both the original26 and optimized liquid-gas
oil-water vertical flow case (Fig. 3), the settling time
and
is almost the same for this three-phase flow case. The
performance of the steady-state models for transient
expectation was that the settling time for this three-
flows. The emphasis here is on vertical flows, though
phase transient process would be longer due to the
deviated flows are also considered.
oil-water
models
before
illustrating
the
gas entrainment in the oil-water mixture leading to
smaller droplets. Similar setting times may be
Liquid-gas flow. Zuber and Findly25 correlated
observed because of complex emulsion behaviors
actual gas velocity Vg and mixture velocity Vm using
that occur for the oil-water system around the phase
two parameters, C0 and Vd:
inversion point, which for this case is expected to be
Vg =
around 50% water (based on an analysis of the probe
V sg
αg
= C 0V m + Vd
(1)
phase
where Vsg is the gas superficial velocity (gas flow rate
inversion point are more difficult to separate, leading
divided by total pipe area) and α g is the gas in situ
to longer settling times.
volume fraction. The accuracy of the predicted αg
response).
Tight emulsions
around
the
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
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
1.0 ≤ C 0 ≤ 1.2
relatively simple model for these systems that is
consistent with our previous models for steady-state
(2)
and Vd is computed via:
flow.
Vd =
(1 − α g C 0 ) C 0 K (α g ) Vc
α g Co
Steady-state drift-flux models
ρg
ρl
K (α g ) = 1.53 C0
⋅ m(θ )
(3)
α g ≤ a1
and
+ 1 − α g C0
The original26 and optimized steady-state drift-flux
where
models for two-phase water-gas, oil-water and three-
K (α g ) = K u ( Dˆ ) when α g ≥ a2 . Parameters a1 and a2
phase oil-water-gas flows have been discussed in
when
are the two gas volume fractions which define the
2,3
detail in our previous publications . Here, we briefly
transition from the bubble flow regime. Ku ( Dˆ ) is the
8
H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
critical Kutateladze number, which is a function of
Vo = C0′ Vl + Vd′
(6)
the dimensionless pipe diameter D̂ . The dependency
where Vo is the in situ oil velocity and Vl is the liquid
K u ( Dˆ ) on D̂ is given in Shi et al.2 Vc is called the
mixture velocity. The original value for C0′ is in the
characteristic bubble rise velocity, which was
same range as C0 for liquid-gas flows:
determined by Harmathy26, and ρ is the density.
1.0 ≤ C 0′ ≤ 1.2
The parameter m(θ ) , where θ is the deviation
from vertical, is very important for modeling flow in
(7)
and Vd′ is calculated by27,
Vd′ = 1.53Vc′ (1 − α o ) n m ′(θ )
(8)
deviated pipes, as it accounts for the deviation from
vertical through a multiplier to Vd. In the original
where, as before, V c′ is also determined by the
Harmathy17 correlation, except that the gas in the
model,
m(θ ) = m(0)(cos θ ) 0.5 (1 + sinθ ) 2
(4)
correlation is replaced by oil.
In the original oil-water model27, n = 2.0 and
where m(0) = 1.00 .
m ′(0) = 1.0
for
vertical
flow.
The
optimized
oil-water
flow
are2:
C0′ = 1.0 ,
In the optimized model, based on the large
parameters
for
diameter data, the values for both C0 and Vd are
significantly different. The first major difference is
the profile parameter, for which we obtain C 0 = 1.0 .
n = 1.0 and m ′(0°) = 1.07 . Unlike for the liquid-gas
flow, the optimized value of m′(0) is not much
This lower value of C0 directly leads to a much
different from its original value of 1.0. However, the
higher Vd value. For example, the optimized
value of the exponent n is reduced from 2.0 to 1.0.
Compared with the original model, this makes Vd′
deviation effect is
m(θ ) = m(0)(cosθ ) 0.21 (1 + sinθ ) 0.95
(5)
and for vertical liquid-gas flow, m(0) = 1.85 . Thus
the optimized Vd value is 1.85 times higher than the
original Vd for vertical liquid-gas flow.
decrease linearly and much more rapidly with
increaseing α 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 C0′ and the gas or oil velocity
Oil-water flow. The general form of the drift-flux
depends only on the drift velocity. Therefore, the key
model applied to oil-water flows is:
to modeling the transient process is to model the drift
9
DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
velocity accurately.
9
reasonably well, but the predicted speed of the water
interface is higher that that observed. The optimized
observations.
model predicts even higher velocities for both the gas
Eclipse26 applies the same drift-flux models (the
and water interfaces. This is perhaps surprising, since
original steady-state models) for both steady-state
the optimized model is more accurate for steady-state
and transient multiphase flow. This is based on the
predictions.
Comparison
with
experimental
assumption that transient flow can be represented by
a sequence of steady-state flows. One of the
Oil-water vertical flow. A sample comparison for
objectives of our work is to test the validity of this
model predictions with experimental data for vertical
assumption.
oil-water flow is illustrated in Fig. 6. Again the
We proceed by identifying two interfaces for two-
volume of the two fluids in the system is about the
phase flows. The gas interface is the interface
same. As in the previous case, the speed of the water
between the pure gas and the mixture of gas and
interface with the original model is much higher than
liquid. Similarly, the liquid interface is defined as the
that observed in the experiment. Furthermore, the
interface between the pure liquid and the mixture of
optimized model yields even higher velocities for
gas and liquid. Therefore, during the transient
both oil and water interfaces.
process, the gas interface moves down and the liquid
An explanation for the disagreement between
interface moves up. The two interfaces meet when
transient
experiments
and
steady-state
model
the phases are completely separated.
predictions can be offered by considering the driftflux model parameters. For liquid-gas systems,
Liquid-gas vertical flow. Fig. 5 shows a sample
C 0 = 1.0 for the optimized model, i.e., there is no
comparison of experimental data with predictions for
profile slip. Hence m(θ ) , the Vd multiplier, must
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
increase accordingly, and for vertical flow, it is
almost twice the value as in the original model.
Therefore the optimized model predicts much faster
25
almost the same. We see that the original model
predicts the speed of the gas interface height
settling. For oil-water flow, the major reason for the
prediction of faster separation by the optimized
10
H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
model compared to the original model is the
velocity used in both steady-state models (original
reduction in the exponent n from 2.0 to 1.0.
and optimized) does distinguish between large and
From these comparisons of experimental data
small bubbles/droplets. Fig. 7 (a) shows that a linear
with model predictions, we see that our steady-state
interpolation is used to connect the bubble flow
models do not fully capture the mechanics of
regime and liquid flooding curve2 over the range
countercurrent transient flows. These findings are
a1 < α g < a2 . Since bubble size increases with α g at
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 quasisteady-state approach. In order to overcome the
limitations of the sequence of steady-states approach,
we will now consider a two-population model.
this range, the small bubbles will have a drift velocity
equal to the value at the bottom of the ramp, and the
large bubbles will have a drift velocity equal to the
value at the top of the ramp.
The values of a1 and a2 are optimization
parameters in our steady-state modeling procedure.
The original values of a1 and a2 were 0.2 and 0.4
respectively, based on the work of Zuber and
Findlay25. However, our steady-state optimization
results provide a1 = 0.06 and a2 = 0.21. The steady-
Two-population model
Our water-gas transient experiments show that some
state flow experiments confirmed that elongated
bubble flow occurred at α g ≈ 0.12 . This implies that
small gas bubbles are entrained in the water and
in most of our steady-state experiments the gas is in
move with the water phase at the beginning of the
large bubbles.
settling process. Similarly, for oil-water flow, some
However, when the pipe is shut-in, the quickly
small water droplets are entrained in oil and move up
closing valves cause disturbances which break the
with the oil phase at the beginning of the separation.
large bubbles into small bubbles. Thus the effective
These small bubbles/droplets separate from the phase
values of a1 and a2 should increase, and we could
in which they are entrained later in the separation
assume that all of the bubbles are small during the
process.
transient process. The solid line in Fig. 7 (b) displays
As illustrated in Fig. 7, the model of drift-flux
the situation when there are only small bubbles in the
11
DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
11
system. In fact, in reality there will exist a
bubble separation ( C 0 S = 1.0 ) due to the large bubble
distribution of bubble sizes, with the smaller bubbles
separation with the mixture, a general drift-flux
having even lower drift velocity4. The dashed line in
model is obtained (see Appendix A for details):
Fig. 7 (b) illustrates this.
α g V g = [1 −
From Fig. 7, we see that by shifting a1 and a2, we
+
(1 − α g )(1 − α gL C 0 L )
1 − α gL
(1 − α g )α gL
can potentially represent both steady-state and
1 − α gL
]V m
(10)
V dL + α gS V dS
transient flows using one drift-flux model. The two-
Here C 0 L is the profile parameter for the separation
population model discussed below is a unified model
of large bubbles from the mixture of small bubbles
for steady-state and transient flows. This unification
and liquid. V dL and V dS define the drift velocity of
is especially important for reservoir simulation, in
large bubbles and small bubbles respectively. This
which a smooth transition between steady-state and
equation reduces to the original form when there is
transient flows is required.
only one kind of bubble and there is no profile slip
for small bubbles.
Model development. Based on our observations of
steady-state and transient flows we can conclude that
Two-population model for oil-water systems. The
in the separation of water and gas, two processes
two-population oil-water model is similar to the
occur. First, large gas bubbles separate from the gas-
liquid-gas model, but the mechanisms involved in
water mixture, and next the entrained small gas
oil-water separation are different. Specifically, large
bubbles separate from the water. This can be modeled
water droplets move down while the small water
by dividing the total gas fraction into two parts,
droplets entrained in the oil move up with the oil
corresponding to large bubbles and small bubbles:
phase. This is also consistent with the observation by
α g = α gL + α gS
(9)
Zhu and Hill16 and Zavareh et al.17. In addition, the
where subscript L represents the large bubbles and S
entrained small water droplets further separate from
the small bubbles.
the oil.
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
We
divide
the
water
droplets
into
two
populations:
α w = α wL + α wS
(11)
12
H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
and apply the oil-water drift-flux model, Eq. (8), to
These are the fraction f of large bubbles/droplets to
both settling processes with the assumption that the
the total bubbles/droplets in the system and the drift
profile slip of small droplets is 1.0 due to the
velocity multiplier mS for small bubbles/droplets
disruption of large water droplets separating with the
(where VdS = mSVdL( α g = 0 )). These parameters
mixture. The resulting two-population model for oil-
depend, in general, on the shut-in holdup, though in
water separation is (see Appendix A):
many cases constant values suffice. Using the two-
α wV w = V m − (1 − α wL − α wS )( C 0′ LV m + V dL′ + V dS′ )
(12)
population model with these two parameters, we can
where C 0′ L is the profile parameter for the separation
achieve close matches to the transient experimental
of large water droplets from the mixture of oil and
data. In the following figures, the model results are
water. V dL′ and V dS′ represent the drift velocity of the
shown in terms of interface height. Predictions by the
oil when separating with large and small water
optimized steady-state parameters are also shown.
droplets respectively.
We note that the two-population model described
here represents a considerable simplification of the
Vertical water-gas flow. For all water-gas cases, a
single
set
of
optimized
parameter
values
true transient process, in which a continuous
(independent of αg and αw) was determined:
distribution of bubble or drop sizes presumably
f = α gL α g = 0.3 and m S = 0.3 . These values indicate
exists. Nonetheless, as shown below, this model does
that most (70%) of the gas bubbles in the water-gas
appear to capture the key transient effects observed in
systems are small bubbles.
the experiments. This is likely because the “two
The water-gas results are illustrated in Figs. 7-9.
populations” of bubble/drop sizes (and corresponding
Each figure corresponds to a particular value of
adjustable parameters) represent, in some sense, an
α g (as indicated in the figure). The first example is
appropriate
for a relatively low α g ( α g = 0.18 ). We see from Fig.
sampling
of
the
true
continuous
distribution.
8 that the optimized steady-state model predicts very
fast separation, while the new two-population model
Results and discussion
To implement the two-population model, we
introduce two additional adjustable parameters.
matches the data much more closely. Fig. 9 shows
similar results for a gas volume fraction of 0.32.
13
DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
The amount of water and gas in the system is
( m S′ = 0.03 ). We attribute this to our expectation
about the same for the last example displayed in Fig.
that the phase inversion point is around 50% for this
10. The results from the steady-state models for this
oil-water system (the fine oil and water droplets
case were presented in Fig. 4. Here the movement of
separate very slowly around the phase inversion
the gas interface is predicted by the two-population
point). Table 1 gives m S′ values for seven oil-water
model to be too slow at the beginning of the
tests. It clearly demonstrate that m S′ reaches a
separation but overall the results for both the gas and
water interfaces are in reasonable agreement with the
minimum at around α w = 50%. Accurate results are
also obtained in the case of high oil fraction, as
experiments.
shown in Fig. 13.
Vertical oil-water flow. The tuning of the two
parameters f ′ and m S′ is more complicated for the
Deviated two-phase flows. We now briefly consider
the applicability of the two-population model to
oil-water system than for the water-gas system. The
deviated wells. For these cases, we use the m(θ )
optimized value for f ′ is found to be 0.2 for all of the
determined in the steady-state optimizations (Eq. (8)
oil-water transient data. However, in contrast to the
water-gas system, a single value for m′S could not be
obtained. This is a result of the formation of oil-water
1
emulsions. Furthermore, small droplet behavior can
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
29
be very different from small bubble behavior .
The oil-water model results are shown in Figs. 1012. 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
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
with steady-state model predictions. Again the match
considered in Fig. 15) is long enough to illustrate the
between the experimental data and model predictions
results. As displayed in Figs. 14 and 15, transient
is very close. In this case, the m S′ value is very small
data for both deviated water-gas and oil-water
13
14
H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
systems are represented very well by the two-
•
A new unified two-population drift-flux model
population models. We again emphasize that the
was developed for transient two-phase flows.
models in this case are consistent with the steady-
The model reduces to the steady-state model in
state models, as m(θ ) is the same in both cases.
appropriate limits. The model predictions match
transient experimental data reasonably well for
Application to well testing
both vertical and deviated water-gas and oil-
(Jon’s contribution)
water flows.
* Why phase redistribution can be important
•
Application to well testing (Jon’s contribution)
* Hallmarks of phase redistribution
* Simulation results
A concern with this model (or many wellbore flow
* What tweaks to d-f are necessary to match the
models) is that the model parameters are based on
observations
transient data collected in a relatively short pipe (11
m). In addition, the disturbances caused by the fast-
Conclusions and recommendations
acting valves may not represent actual conditions in
From this study, we can draw the following
the field. It is therefore possible that the model
conclusions:
parameters
•
The drift-flux model is well suited for steady-
applications. This can only be gauged by testing the
state concurrent flows as well as transient
model against other experimental data sets, which are
countercurrent flows in wellbores and pipes.
not currently available. Even though the model
Experimental data from large-diameter pipes
parameters may require tuning for a particular
suggest that wellbore transient flow cannot be
application, it is still reasonable to expect that the
represented by a series of steady-state flows.
two-population model presented here (or a very
Experimental observations show that gas exists
similar model) can be used to represent transient
as large and small bubbles during the settling
countercurrent wellbore flows.
•
•
process
for
water-gas
flow.
In
may
require
tuning
for
specific
oil-water
separation, water exists as large and small water
Acknowledgments
droplets.
The support from Schlumberger and the other
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
m′ =
drift velocity multiplier for oil-water flows
mS =
drift velocity multiplier for small buubbles
m S′ =
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
Vc′ =
characteristic velocity for oil-water flows
regimes for liquid-gas flows
Vd =
gas-liquid drift velocity
A′ =
profile parameter term for oil-water flows
Vd′ =
oil-water drift velocity
B =
profile parameter term, gas volume fraction
Vm =
mixture velocity
at which C0 begins to reduce
Vs =
superficial velocity
=
B1 =
B2 =
profile parameter term, oil volume fraction
at which C′0 begins to reduce
Subscripts
profile parameter term, oil volume fraction
g
=
gas
at which C0′ falls to 1.0
l
=
liquid
L
=
large bubbles/droplets
m
=
mixture
o
=
oil
S
=
small bubbles/droplets
w
=
water
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
Greek
α
=
in situ fraction or holdup
σ
=
interfacial tension/surface tension
15
16
H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
ρ
=
density
θ
=
deviation from vertical
Water Flows in Vertical Tubes”, J. Nucl. Sci.
Technol., (1984) 21, 321-327.
9.
Hasan, A.R., Kabir, C.S., and Srinivasan, S.:
“Countercurrent Bubble and Slug Flows in a Vertical
References
System”, Chem. Engng Sci. (1994) 49, 2567-2574.
1.
Oddie, G., Shi, H., Durlofsky, L.J., Aziz, K., Pfeffer,
10. Harmathy, T.Z.: “Velocity of Large Drops and
B. and Holmes, J.A.: “Experimental Study of Two and
Bubbles in Media of Restricted Extent”, AIChEJ
Three Phase Flows in Large Diameter Inclined Pipes”,
(1960) 6, 281-290.
Int. J. Multiphase Flow, (2003) 29, 527-558.
11. Nicklin, D. J., Wilkes, J. O. and Davidson, J.F.: “Two-
2.
Shi, H., Holmes, J.A., Durlofsky, L.J., Aziz, K., Diaz,
Phase Film Flow in Vertical Tubes”, Trans. Inst.
L.R., Alkaya, B. and Oddie, G.: “Drift-Flux Modeling
Chem, (1962) 40, 61-68.
of Two-Phase Flow in Wellbores”, SPE Journal,
12. Kim, H.Y., Koyama, S. and Mastumoto, W.: “Flow
(March, 2005) 10, 24-33.
Pattern and Flow Characteristics for Counter-current
3.
Shi, H., Holmes, J.A., Diaz, L.R., Durlofsky, L.J.,
Two-phase Flow in a Vertical Round Tube with WireAziz, K.: “Drift-Flux Parameters for Three-Phase
coil Inserts”, Int. J. Multiphase Flow, (2001) 27, 2063Steady-State Flow in Wellbores”, SPE Journal, (June,
2081.
2005) 10, 130-137.
13. Johnston, A.J.: “An Investigation into Stratified Co-
4.
Wallis, G. B.: One Dimension Two-Phase Flow,
and Countercurrent Two-Phase Flow”, SPEPE (Aug.
McGraw-Hill, New York, 1969.
1988) 393-399.
5.
Zabaras, G.J. and Dukler, A.E.: “Countercurrent Gas14. Johnston, A.J.: “Controlling Effects in Countercurrent
liquid Annular Flow Including the Flooding Modeling
Two-Phase Flow”, SPEPE (Aug. 1988) 400-404.
State”, AIChEJ (1988) 34, 389-396.
15. Ghiaasiaan, S.M., Wu, X., Sadowski, D.L., and Abdel-
6.
Taitel, Y., and Barnea, D.: “Counter Current GasKhalik,
S.I.:
“Hydrodynamic
Characteristics
of
Liquid Vertical Flow, Model for Flow Pattern and
Counter-Current Two-Phase Flow in Vertical and
Pressure Drop”, Int. J. Multiphase Flow, (1983) 9,
Inclined Channels: Effect of Liquid Properties”, Int. J.
637-647.
Multiphase Flow, (1997) 23, 1063-1083.
7.
Yamaguchi, K. and Yamazaki, Y.: “Characteristics of
16. Zhu, D., and Hill, A.D.: “The Effect of Flow from
Coutercurrent Gas-Liquid Two-Phase Flow in Vertical
Perforations on Two-Phase Flow: Implications for
Tubes”, J. Nucl. Sci. Technol., (1982) 19, 985-996.
Production Logging”, SPE paper 18207 presented at
8.
Yamaguchi, K. and Yamazaki, Y.: “Combined Flow
the 1988 SPE Annual Technical Conference and
Pattern Map for Cocurrent and Countercurrent Air-
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”,
presented at the
1998
International
17
Petroleum
Conference and Exhibition of Mexico, Villahermosa,
3-5 March.
SPE paper 18215 presented at the 1988 SPE Annual
24. Hasan, A.R. and Kabir, C.S.: “Modeling Changing
Technical Conference and Exhibition, Houston, TX,
Storage During a Shut-in Test”, SPEFE (1994) 9, 279-
2-5 October.
284.
18. Ouyang, L.B.: “Mechanistic and Simplied Models for
25. Zuber, N. and Findlay, J.A.: “Average Volumetric
Countercurrent Flow in Deviated and Multilateral
Concentration in Two-Phase Flow Systems”, J. Heat
Wells”, SPE paper 77501 presented at the 2002 SPE
Transfer, Trans. ASME, (1965) 87, 453-468.
Annual Technical Conference and Exhibition, San
Antonio, TX, 29 Sept–2 Oct.
26. Schlumberger
GeoQuest,
ECLIPSE
Technical
Description Manual, 2001.
19. Ouyang, L.B.: “Mechanistic and Simplied Models for
27. Hasan, A.R. and Kabir, C.S.: “A Simplified Model for
countercurrent flow in deviated and multilateral
Oil/Water Flow in Vertical and Deviated Wellbores”,
wells”, Petroleu Sci.& Tech, (2003) 21, 2001-2020.
SPE Prod. & Fac. (February 1999) 56-62.
20. Almehaideb, R.A., Aziz, K. and Pedrosa, O.A.: “A
28. King, M.J.S., Hale, C.P., Lawrence, C.J., and Hewitt,
Reservoir/Wellbore Model for Multiphase Injection
G.F.: “Characteristics of Flow Rate Transients in Slug
and Pressure Transient Analysis”, SPE paper 17941
Flow”, Int. J. Multiphase Flow, (1997) 24, 825-854.
presented at the 1989 SPE Middle East Oil Technical
29. Pal, R.: “Pipeline Flow of Unstable and Surfactant-
Conference and Exhibition, Manama, Bahrain, 11-14
Stabilized Emulsions”, AIChE J.(1993) 39, 1754-
March.
1764.
21. Asheim, H. and Grodam, E.: “Holdup Propagation
Predicted by Steady-State Drift Flux Models”, Int. J.
Multiphase Flow, (1998) 24, 757-774.
22. Lopez, D., Dhulesia, H., Leporcher, E. and DuchetSuchaux, 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
18
H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
electrical probes temperature
inlet
valve
differential
pressure
gamma
densitometer pressure
outlet
valve
Fig. 1: Schematic of the test section of the flow loop
19
DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
(a)
(b)
Fig. 2. Water-gas data for θ=0°, Qw= 40.4 m /h, Qg= 58.0 m /h
(αw=52%).
3
3
19
20
H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
Fig. 3. Oil-water data for θ=0°, Qo=40.2 m /h, Qw=40.4
3
m /h (αw =51%).
3
21
DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
Fig. 4. Oil-water-gas data for θ=0°, Qo=40.2 m /h,
3
3
Qw=40.4 m /h, Qg=26.2 m /h (αw =44%, αo=42%).
3
21
Interface Height (m)
22
H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
11
10
9
8
7
6
5
4
3
2
1
0
Experiment_gas
Original_gas
Optimized_gas
Experiment_water
Original_water
Optimized_water
0
10
20
30
40
50
60
70
80
Time (s)
Fig. 5. Water-gas interface height for θ=0°, Qw=2.0 m /h,
3
Qg=60.2 m /h (αw =49%).
3
23
DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
11
Interface height (m)
10
Experiment_oil
9
Original_oil
8
7
Optimized_oil
6
5
Experiment_water
4
Original_water
3
2
Optimized_water
1
0
0
100
200
300
400
500
600
700
800
Time (s)
Fig. 6. Oil-water interface height for θ=0°, Qw=40.4 m /h,
3
Qo=40.2 m /h (αw =51%).
3
23
24
H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
0.7
large bubbles
0.6
Vd
0.5
small
bubbles
0.4
0.3
0.2
0.1
a
a
0.0
0.0
0.1
0.2
0.3
0.4
0.5
αg
0.6
0.7
0.8
0.9
1.0
(a) Original drift velocity for liquid-gas system
0.7
0.6
0.5
Vd
0.4
small bubbles
0.3
0.2
smaller bubbles
0.1
a1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
a2
1
αg
(b) Small bubble drift velocity for liquid-gas system
Fig. 7. Drift velocity mechanism in two-population for
liquid-gas system
DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
Interface Height (m)
25
11
10
9
8
7
6
5
4
3
2
1
0
Experiment_gas
Optimized_gas_ss
Optimized_gas_t
Experiment_water
Optimized_water_ss
Optimized_water_t
0
10
20
30
40
50
60
70
80
90
100
Time (s)
Fig. 8. Water-gas interface height for θ=0°, Qw=2.0 m /h,
3
Qg=11.4 m /h (αw =82%).
3
25
26
H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
11
Experiment_gas
Interface Height (m)
10
9
Optimized_gas_ss
8
7
Optimized_gas_t
6
5
Experiment_water
4
3
Optimized_water_ss
2
Optimized_water_t
1
0
0
10
20
30
40
50
60
70
80
90
100
Time (s)
Fig. 9. Water-gas interface height for θ=0°, Qw=2.0 m /h,
3
Qg=28.6 m /h (αw =68%).
3
27
DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
11
Interface Height (m)
10
Experiment_gas
9
Optimized_gas_ss
8
7
Optimized_gas_t
6
5
Experiment_water
4
3
Optimized_water_ss
2
1
Optimized_water_t
0
0
10
20
30
40
50
60
70
Time (s)
Fig. 10. Water-gas interface height for θ=0°, Qw=2.0
3
3
m /h, Qg=60.2 m /h (αw =49%).
80
27
28
H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
11
Interface Height (m)
10
Experiment_oil
9
Optimized_oil_ss
8
7
Optimized_oil_t
6
5
Experiment_water
4
Optimized_water_ss
3
2
Optimized_water_t
1
0
0
100
200
300
400
500
600
700
Time (s)
Fig. 11. Oil-water interface height for θ=0°, Qw=100.0 m /h,
3
Qo=40.2 m /h (αw =72%).
3
Interface Height (m)
29
DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
11
10
9
Experiment_oil
8
Optimized_oil-ss
7
6
Optimized_oil-t
5
4
Experiment_water
3
Optimized_water_ss
2
1
Optimized_water_t
0
0
100
200
300
400
500
600
700
Time (s)
Fig. 12. Oil-water interface height for θ=0°, Qw=40.4
3
3
m /h, Qo=40.2 m /h (αw =51%).
800
29
30
H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
11
Interface Height (m)
10
Experiment_oil
9
Optimized_oil-ss
8
7
Optimized_oil_t
6
5
Experiment_water
4
3
Optimized_water_ss
2
Optimized_water_t
1
0
0
100
200
300
400
500
600
700
800
Time (s)
Fig. 13. Oil-water interface height for θ=0°, Qw=2.0 m /h,
3
Qo=10.0 m /h (αw =27%).
3
Interface Height (m)
31
DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
11
10
9
8
7
6
5
4
3
2
1
0
Experiment_gas
Optimized_gas_ss
Optimized_gas_t
Experiment_water
Optimized_water_ss
Optimized_water_t
0
10
20
30
40
50
60
70
80
Time (s)
Fig. 14. Water-gas interface height for θ=5°, Qw=10.1 m /h,
3
Qg=58.8 m /h (αw =52%).
3
31
32
H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
Fig. 15. Oil-water interface height for θ=45°, Qw=100.0
3
3
m /h, Qo=40.2 m /h (αw =72%).
33
DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
TABLE 1—SUMMARY OF PARAMETER
m ′S
FOR OIL-WATER SYSTEMS
αw
0.27
0.51
0.60
0.72
0.82
0.85
0.93
m ′S
0.05
0.03 0.07
0.10
0.50
0.80
0.95
33
34
H. SHI, J.A. HOLMES, L.J. DURLOFSKY, K. AZIZ
α gS V gS = α gS C 0 S V mS + α gS V dS
Appendix A
(A-5)
Derivation of two-population drift-flux
models
Assuming that there is no profile slip for small
Liquid-gas flow. Because of small gas bubbles that
bubbles, C 0 S = 1.0 , Eqn (A-5) becomes:
are entrained in the water phase, while the overall,
gas is rising a mixture of water-gas is sinking.
bubbles since the profiles are disrupted by large
V gS = V mS + V dS
(A-6)
The mixture velocity for small bubbles and liquid
This system could be model with two populations
of bubbles: large bubbles with volume fraction of
can be written as:
(1 − α gL )V mS = α g S V gS + (1 − α g )Vl
(A-7)
α gL , and small bubbles with volume fraction of α gS .
where Vl is the liquid velocity. We can rearrange the
α g = α gL + α gS
(A-1)
The fraction of α gL and α gS depends on the relative
above expression for Vl:
Vl =
1 − α gL
1−α g
V mS −
α gS
1−α g
(A-8)
V gS
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:
and combining Eqn (A-4), (A-6) and (A-8), to obtain:
Vl =
1 − α gL C 0 L
1 − α gL
α gL
Vm −
1 − α gL
V dL −
α gS
1−α g
V gS
(A-9)
For the liquid-gas system we:
V gL = C 0 LV m + V dL
(A-2)
The total mixture velocity is:
V m = α g V g + (1 − α g )V l
(A-10)
where Vg is the average gas velocity of both large
V m = α gLV gL + (1 − α gL )V mS
(A-3)
bubbles and small bubbles . By combining Eqn (9)
where VmS is the mixture velocity of the small
and (10), we can obtain the general two-population
bubbles and liquid. From Eqn (A-2) and (A-3), we
model for liquid-gas flow:
obtain,
V mS =
1 − α gL C 0 L
1 − α gL
Vm −
α gL
1 − α gL
α g V g = [1 −
V dL
(A-4)
+
(1 − α g )(1 − α gL C 0 L )
1 − α gL
(1 − α g )α gL
1 − α gL
]V m
(A-11)
V dL + α gS V dS
In this small bubble and liquid mixture, the small
bubbles travel with a velocity VgS, which can also be
Oil-water flow. Our experiments show that water
computed by drift-flux model:
entrained in the oil phase, and the water-in-oil
35
DRIFT-FLUX MODELING OF TRANSIENT MULTIPHASE FLOW IN WELLBORES
dispersions/emulsions separated much slower than
where Vw is the average water velocity of both large
pure phases. Therefore, we can assume that in the
water droplets and small water droplets .
overall system mixture of oil and small water
droplets rises while large water droplets sink.
Combining Eqn (A-12), (A-13), (A-14) and (A15), and assuming that the profile slip for the oil and
Similarly to the treatment of the liquid-gas
small water droplets system is disrupted by large
system, let there be two populations of water
water droplets ( C 0′ S = 1.0 ) we obtain the following
droplets: large water droplets with volume fraction of
two-population model for oil-water flow:
α wL , and small dropllets with volume fraction of
α wS .
α w = α wL + α wS
(A-12)
The fractions α wL and α wS depends on the relative
densities fluid properties and flowing conditions.
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:
Vom = C 0′ LV m + V dom
(A-13)
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:
Vo = C 0′ S Vom + V do
(A-14)
For an oil-water system, we have the following
relationship:
V m = α wV w + (1 − α w )Vo
(A-15)
α wV w = V m − (1 − α wL − α wS )( C 0′ LV m + V dL + V dS )
(A-16)
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