Transient Paper

8/9/2019 Transient Paper

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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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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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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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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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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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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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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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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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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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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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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


14/35


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


15/35


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


16/35


= 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.

2. Shi, H., Holmes, J.A., Durlofsky, L.J., Aziz, K., Diaz,

L.R., Alkaya, B. and Oddie, G.: Drift-Flux Modeling

of Two-Phase Flow in Wellbores, SPE Journal,

(March, 2005) 10,24-33.

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Aziz, K.: Drift-Flux Parameters for Three-Phase

Steady-State Flow in Wellbores, SPE Journal, (June,

2005) 10,130-137.

4. Wallis, G. B.: One Dimension Two-Phase Flow,

McGraw-Hill, New York, 1969.

5. Zabaras, G.J. and Dukler, A.E.: Countercurrent Gas-

liquid Annular Flow Including the Flooding Modeling

State,AIChEJ(1988) 34, 389-396.

6. Taitel, Y., and Barnea, D.: Counter Current Gas-

Liquid Vertical Flow, Model for Flow Pattern and

Pressure Drop, Int. J. Multiphase Flow, (1983) 9,

637-647.

7. Yamaguchi, K. and Yamazaki, Y.: Characteristics of

Coutercurrent Gas-Liquid Two-Phase Flow in Vertical

Tubes,J. Nucl. Sci. Technol., (1982) 19, 985-996.

8. Yamaguchi, K. and Yamazaki, Y.: Combined Flow

Pattern Map for Cocurrent and Countercurrent Air-

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

System, Chem. Engng Sci. (1994) 49, 2567-2574.

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


17/35


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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Fig. 1: Schematic of the test section of the flow loop

inlet outlettemperatureelectrical probes

differentialpressure pressure

gammadensitometer

valvevalve


19/35


20/35


Fig. 3. Oil-water data for =0, Qo=40.2 m3/h, Qw=40.4m

3/h (w=51%).


21/35


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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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%).


23/35


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%).


24/35


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


25/35


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


3/h (w=82%).


26/35


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


3/h (w=68%).


27/35


28/35


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


3/h (w=72%).


29/35


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%).


30/35


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


3/h (w=27%).


31/35


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


3/h (w=52%).


32/35


Fig. 15. Oil-water interface height for =45, Qw=100.0m

3/h, Qo=40.2 m

3/h (w=72%).


33/35


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


34/35


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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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)

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