comparison among the homogeneous no-slip model, beggs...

IV Journeys in Multiphase Flows (JEM 2017) March 27-31, 2017, São Paulo, SP, Brazil

Copyright © 2017 by ABCM Paper ID: JEM-2017-0038

COMPARISON AMONG THE HOMOGENEOUS NO-SLIP MODEL,

BEGGS & BRILL AND MECHANISTIC MODELS FOR THE PREDICTION

OF LIQUID HOLDUP AND TOTAL PRESSURE GRADIENT

Thiago Manzioni Faria Ruella Universidade Santa Cecília, Rua Oswaldo Cruz, 277 - Boqueirão, Santos – SP. 11045-907

[email protected]

Alexandre Trabulsi Nasser Universidade Santa Cecília, Rua Oswaldo Cruz, 277 - Boqueirão, Santos – SP. 11045-907

[email protected]

Eric Giuliano Palmiero Universidade Santa Cecília, Rua Oswaldo Cruz, 277 - Boqueirão, Santos – SP. 11045-907

[email protected]

Thauãn de Sá Gonçalves Universidade Santa Cecília, Rua Oswaldo Cruz, 277 - Boqueirão, Santos – SP. 11045-907

[email protected]

Victor Suman Guiráo Universidade Estadual de Campinas, Cidade Universitária Zeferino Vaz – Barão Geraldo, Campinas – SP, 13083-970

Universidade Santa Cecília, Rua Oswaldo Cruz, 277 – Boqueirão, Santos – SP, 11045-907

[email protected]

Luciene de Arruda Bernardo Universidade Santa Cecília, Rua Oswaldo Cruz, 277 – Boqueirão, Santos – SP, 11045-907

[email protected]

Abstract. In order to design and operate oil production systems optimally, it is essential to rigorously anticipate the

behavior of two-phase flow in pipelines.

The present work aims to validate the results of the total pressure gradients (sum of the gravitational, friction and

acceleration pressure gradient) and liquid holdup obtained through the homogeneous no-slip model, the empirical

method proposed by Beggs & Brill, the mechanistic model proposed by Shoham (2006) for bubble flow, the

mechanistic model developed by Ansari et al. (1994) for slug flow and the mechanistic model developed by Alves et al.

(1991) for annular flow. For this, input data were considered, and then the described models were plotted in electronic

spreadsheets by using macros and programming in Visual Basic for Applications (VBA). These models were then

validated through the values obtained of liquid holdup and total pressure gradient from the OlgaS correlation of the

software PIPESIM®. With the elaboration of this work, it became clear the importance of estimating the pressure

gradient in a system. Any improvement or calibration that can be obtained by analyzing the best modeling can result in

extraordinary profits in oil production systems. The individualized description of the components of the total pressure

gradient enables to develop or operate more efficiently in the wells, such as injection of friction reducers.

Keywords: multiphase-flow, homogeneous no-slip model, Beggs & Brill, bubble flow, slug flow, annular flow,

mechanistic models, liquid holdup, total pressure gradient.

1. INTRODUCTION

In order to design and operate oil production systems optimally, it is essential to rigorously anticipate the behavior

of two-phase flow in pipelines. For this, several empirical correlations were developed, making possible the prediction

of liquid holdup determination, which is the relationship between the volume of a segment of tube occupied by liquid

and the total volume of that segment of tube, as well as the pressure gradient. However, empirical correlations are

restricted due to simplifications of the complex physical mechanisms involved in the flow of two or more phases.

From the 1970s, mechanistic models have been developed (Zhang et al., 2003). Based on physical phenomena, these

allow verification and improvement through limited experimental data and, by incorporating the physical phenomena

mailto:[email protected]

mailto:[email protected]

T. M. F. Ruella et al. Comparison among the homogeneous no-slip model, Beggs & Brill and mechanistic models

and the important flow variables, enable the use of different operating conditions (Shoham, 2006). In general, in a

mechanistic model, first it is identified the flow pattern to then calculate the liquid holdup and the pressure gradient.

This work will briefly present the homogeneous no-slip model, the classical empirical correlation, generally used in

field applications of Beggs & Brill, the mechanistic model proposed by Shoham (2006) for bubble flow, the

mechanistic model proposed by Ansari et al. (1994) for slug flow and the mechanistic model proposed by Alves et al.

(1991) for annular flow. The values obtained from these models, when using the same input data for the determination

of the liquid holdup and the pressure gradient will be validated through the results obtained from the OlgaS correlation

of the PIPESIM® software.

2. ANALYZED MODELS

2.1 Homogeneous No-Slip Model

The homogeneous flow theory provides the simplest technique for study and analysis of multiphase flow, which is

treated as a pseudo-fluid. For that, a suitable average of the properties is adopted and the equations used in single-phase

flow are applied (Wallis, 1969).

The required average properties are: thermodynamic properties, transport properties and velocity. These pseudo

properties are the weighted average of the values obtained and not necessarily the properties of one of the phases. In

order to determine the proper values for a property, it often starts with very complex equations that are rearranged, to

resemble single-phase flow equations (Wallis, 1969). This model is widely used and recognized as an already

consolidated method, but cannot be applied in all cases, since many conditions and adaptations are made.

To develop this model, some assumptions must be made, under which it is possible to explain the model:

Steady-state one-dimensional flow.

The two phases are well mixed and are in equilibrium.

No slippage occurs between the phases.

Both phases are compressible.

The pipe cross-sectional area is not constant and can vary along the axial direction.

Mass transfer occurs between the phases, and the quality varies along the pipe (Shoham, 2006).

This method considers that both phases are present at any flow point, and that the velocity and temperature values of

the mixture, the gas phase and the liquid phase are the same. Another hypothesis, which is considered the most rigorous

of the Homogeneous No-Slip Model is the fact that the in-situ liquid holdup is the no-slip liquid holdup (Shoham,

2006).

2.1.1 Liquid Holdup and Pressure Drop Calculation

The liquid holdup (no-slip liquid holdup) is calculated from the equation:

(1)

Being:

: no-slip liquid holdup; : liquid flow rate (m³/s); : gas flow rate (m³/s);

: surface velocity of the liquid (m³/s);

: surface velocity of the gas (m³/s).

The total pressure gradient is obtained from the sum of the friction, gravitational and acceleration pressure gradients,

obtained by the following equations:

2.1.1.1 Frictional Pressure Gradient

The frictional pressure gradient is given by:

)

(2)

Being:

IV Journeys in Multiphase Flows (JEM 2017)

)

frictional pressure gradient (Pa/m);

d: pipe diameter (m);

: Fanning friction factor;

no-slip mixture density (kg/m³);

mixture velocity (m/s);

G: mass flow of the mixture (kg / m²).

2.1.1.2 Gravitational Pressure Gradient

The gravitational pressure gradient is given by:

)

(3)

Being:

)

: gravitational pressure gradient (Pa/m);

pipe angle (º).

g : gravity (m/s²)

: mixture density (kg/m³);

2.1.1.3 Accelerational Pressure Gradient

The accelerational pressure gradient is given by:

)

,

*

( )

+-

(4)

Being:

) accelerational pressure gradient (Pa/m);

2.1.1.4 Total Pressure Gradient

The total pressure gradient is given by:

)

)

)

(5)

2.2 Beggs & Brill

The classical empirical correlation, generally used in field applications of Beggs& Brill, uses flow pattern maps and

considers slippage between phases. For each flow pattern there is a different correlation for liquid holdup and friction

factor.They can be applied to any pipe angle.

2.2.1 Liquid Holdup and Pressure Drop Calculation

For the calculation of liquid holdup, the flow patterns are first defined according to the following framework:

a. Segregated if:

(6)


Where:

(7)

b. Transition if:

(8)

c. Intermittent if:

(9)

d. Distributed if:

(10)

Then, the horizontal liquid holdup (0º) must be calculated. The value adopted will be the highest obtained

through the following solution:

( º)= (

) (11)

Table 1. Standard values for each flow regime.

Flow Pattern Ap Bp Cp

Segregated 0,98 0,4846 0,0868

Intermittent 0,845 0,5351 0,0173

Distributed 1,065 0,5824 0,0609

Source: Beggs& Brill adapted (1973).

If the pattern found was transitional, the liquid holdup will be calculated as follows:

(12)

So that:

(13)

(14)

(15)

(16)

The next step is to correct the ( ) found by the angle correction factor ( ).This step is very important because

without it, there is no adaptation to the vertical flow.

( ) (17)

Being:

( ) ( ) (18)

(( ) (

) ) (19)

√

(20)

Table 2. Values of standardized variables for each condition.


Flow Pattern d e f g

Upward Segregated 0,011 -3,7680 3,5390 -1,6140

Upward Intermittent 2,960 0,305 -0,4473 0,0978

Upward Distributed Uncorrected, C=0 =1

Downward All 4,700 -0,3692 0,1244 -0,5056

Source: Beggs& Brill adapted (1973).

The value of can be obtained through the Moody diagram, or the Colebrook-White equation, after obtaining the

Reynolds number:

(21)

Where:

(22)

(23)

(24)

Being:

: viscosity of the mixture (lb/ft.s);

: no-slip mixture density (lb/ft³);

: density of the mixture with slippage (lb/ft³).

The variable "y" must be found, in order to establish a criterion for the continuity of the problem:

( ) (25)

When y <1 or y> 1,2:

( )

( ) ( ) ( ) (26)

When 1 <and <1.2:

( ) (27)

Found the value of “s”, should be used to calculate :

(28)

Finally, it is possible to calculate the pressure gradient:

(

( )) (29)

2.3 Mechanistic Models

A mechanistic modeling is the product of the development of a physical model that approaches real phenomena,

along with the application of mathematical equations. In general, in a mechanistic model, first it is identified the flow

pattern to then calculate the liquid holdup and the pressure gradient. Taitel et al. (1980) developed a map for the

prediction of transition limits between flow patterns that occur in ascending vertical flow. In their work "Modeling flow

pattern transitions for steady upward gas-liquid flow in vertical pipes" the authors suggest physical mechanisms for the

transitions among the various flow patterns, modeling each transition based on the mechanism in which it occurs.


Figure 1. Adapted map of the transitions limits for vertical ascending flow.

Source: Taitel et al. (1980).

Once the flow pattern is known, the total pressure gradient and liquid holdup are calculated by the existing mechanical

methods for each flow pattern.

2.3.1 Dispersed Bubble Flow Modeling

The mechanistic modeling for dispersed bubble flow can be considered as solution, the same homogeneous model

mentioned in item 2.1 of this paper. In the homogeneous model it is not considered physical interaction between each

phase. In the dispersed bubble regime, these slip effects are negligible and their phases behave as a monophasic pseudo-

fluid. This consideration is sufficient to justify and adopt the homogeneous model as a model capable of calculating

satisfactorily the pressure gradient for dispersed bubble flow.

2.3.2 Bubble Flow Modeling

This flow pattern is characterized for its gaseous form be uniformly distributed as discrete bubbles in a continuous

liquid phase. In the case of vertical flow, the distribution of the bubbles is approximately homogeneous in the tubular

section. This flow typically occurs at low liquid flows with little gas presence. It is characterized by slippage between

gas and liquid phases, resulting in large values of liquid holdup (SHOHAM, 2006).

2.3.2.1 Liquid Holdup and Pressure Drop Calculation

Following what is proposed by Shoham (2006) for bubble flow pattern, the liquid holdup can be found by the

equation:

( )

( )

(30)

The equation 31 is an implicit equation for the in-situ gas voids fraction (α) for bubble flow. The equation is solved

by trial and error for the value of α. A suitable initial value for α can be obtained through the solution of Eq. 32, which

is a quadratic equation.

( ) (31)

Once the liquid holdup is determined, it is possible to predict the pressure gradient in a simple manner.

2.3.2.1.1 Frictional Pressure Gradient

The frictional pressure gradient is given by:

)

(32)


Being:

friction factor of the mixture.

2.3.2.1.2 Gravitational Pressure Gradient

The gravitational pressure gradient is given by:

) (33)

2.3.2.1.3 Total Pressure Gradient

The total pressure gradient is given by:

)

)

)

(34)

2.3.3 Slug Flow Modeling

In this pattern, most of the gas is located in large bubbles with bullet-shape that have a diameter almost equal to the

diameter of the pipe. They move uniformly upward and are sometimes referred to as Taylor bubbles. Taylor bubbles are

separated by gaps of continuous liquid containing small bubbles of gas. Between the Taylor bubbles and the pipe wall,

the liquid flows downwardly in the form of a thin falling film. This pattern has low flow rate and the boundaries

between gas and liquid are well defined (SHOHAM, 2006).

Figure 2.Slug-flow schematic.

Source: Brill, Mukherjee adapted (1999).


According to the Ansari et al. Mechanistic model for slug flow pattern, the liquid holdup can be obtained by the

following equation:

𝑽𝑻𝑩 Taylor bubble rise velocity (m / s);

𝑽𝒈𝑻𝑩 Gas velocity in the Taylor bubble (m / s);

𝑯𝒈𝑻𝑩 Liquid-slug void fraction;

𝑽𝑳𝑻𝑩 Liquid film velocity with downward flow (m / s);

𝜹𝑳 Thickness of the liquid film around the Taylor bubble (m);

𝑯𝑳𝑳𝑺 Liquid holdup on the body of the liquid slug;

𝑽𝑳𝑳𝑺 Liquid rise velocity in the body of the liquid slug (m / s);

𝑽𝒈𝑳𝑺 Speed of ascension of the small bubbles in the body of the

liquid slug (m / s);

𝑳𝑻𝑩 Length of the Taylor Bubble (m);

𝑳𝑳𝑺 Comprimento da Golfada de líquido (m);

𝑳𝑪 Length of the Taylor bubble cap (m);

𝑳𝑺𝑼 Length of the slug unit (liquid slug + Taylor bubble) (m).


( )

(35)

2.3.3.1.1 Frictional Pressure Gradient

The friction pressure losses are assumed to occur only in the liquid slug, being neglected along the Taylor bubble.

Therefore, the friction component of the pressure gradient is:

(

)

( ) (36)

Being:

density of the liquid slug (kg / m³);

friction factor in the liquid slug;

ratio of the Taylor bubble / liquid film zone.

2.3.3.1.2 Gravitational Pressure Gradient

For fully developed slug, the gravitational pressure gradient component that occurs across a slug unit is given by:

(

)

[( ) ] (37)


For fully developed slug flow, Ansari et al. (1994) assumed that the acceleration component of the pressure gradient

can be neglected, so the total pressure gradient of a slug unit is given by:

(

)

(

)

(

) (38)

2.3.4 Churn Flow Modeling

Churn flow does not have an exclusive modeling in the literature. In this model, there is a natural tendency of the

flow along the pipe (considering vertical and upward flow) to present similar behavior as the slug flow. Therefore, in

this work, the region of the flow pattern map for churn flow was considered as part of the slug flow pattern for the

calculation of the pressure gradient.

2.3.5 Annular Flow Modeling

The annular flow type for vertical pipes occurs in cases of high gas flow and is characterized by the continuity of the

gas phase in the center of the pipe. The liquid phase moves upwardly, partly as liquid film and partly as droplets,

entrained in the gas core. The gas velocity, therefore, should be high enough to carry the liquid. The thickness of the

liquid film around the tube is practically uniform (SHOHAM, 2006).

The Alves et al. Model(1991), which is the model used in this work for annular flow, allows the visualization and

calculation of physical details that are very relevant for the equation of the scenarios, among them: velocity distribution,

liquid film thickness in the pipe walls, fraction of the gas core, and finally the pressure gradient. The applicability

comprises horizontal vertical and inclined tubes.


According to Alves et al. (1991), the liquid holdup can be found by the equation:

(

)

(39)

Where:


(40)

And

( ) (41)

which is the entrainment fraction, defined as the fraction of liquid flow that is entrained in the gas core in form of

droplets.


The total pressure gradient can be obtained by the gradient of the gas core or liquid film. In this case, the same

pressure gradient of the gas core was chosen, as it is shown in the equation:

)

)

( ) (42)

3. METHODOLOGY

For the development of this work, as previously mentioned, the values obtained from the liquid holdup and the total

pressure gradient were compared using the methods described in the previous topics and validated by the OlgaS

correlation of the software PIPESIM®. The input values used in both the mentioned models and PIPESIM® are

described next.

3.1 Boundary Conditions and Experimental Data

- Water and air flow;

- Water considered as incompressible fluid;

- Isothermal system at ambient temperature = 20 ° C = 293,15 K;

- Vertical upward flow;

- Specific gravity of fresh water = 1;

- Specific gravity of air = 1;

- Smooth tube (without roughness);

Table 3. Experimental data.

EXPERIMENTAL DATA

d (m) 0,092456

P (Pa) 1013250

(kg/m³) 998,21

(kg/m³) 12,05

(N/m) 0,0728

(Pa.s) 0,001002

(Pa.s) 0,0000182

g (m/s²) 9,81

Le camera (m)( ) 25

le/d 270,4 (1)

Le camera = Observation point.

With these data, it was possible to plot the graph of Taitel et al (1980) adapted for this study in the software Excel®

as well as generate the flow pattern map in the software PIPESIM®. Through points chosen in the different regimes in

the flow patterns map, it was calculated the liquid holdup and the total pressure gradient for the respective mechanistic


model for each flow pattern. It is important to note that the points selected according to the liquid and gas surface

velocities are in the same regions in the flow pattern maps showed in figures 3 and 4.

Table 4. Types of regimes and superficial velocities of the phases.

Cases Flow Pattern Surface velocity of the

liquid, (m/s)

Surface velocity of the

gas, (m/s)

A Dispersed bubble 10 0,01

B Bubble 1 0,05

C Slug 0,1 1

D Annular 0,1 10

Figure 3. Flow pattern map - Adapted Taitel et al (1980). Figure 4. Flow pattern map - PIPESIM®

3.2 Modeling methods

All methods described in this study were plotted in electronic spreadsheets by using macros and programming in

Visual Basic for Applications (VBA), facilitating the achievement of the results. The data was also insert in the

software PIPESIM®.

Figure 5. All methods plotted in electronic spreadsheets.

0,01

0,1

1

10

100

0,01 0,1 1 10 100

Vsl

(m

/s)

Vsg (m/s)

Flow pattern map - Adapted Taitel et al. (1980)

Bubbles Slug

Churn

Annular

Dispersed

Bubbles

0,01

0,1

1

10

100

0,01 0,1 1 10 100

Vsl

(m

/s)

Vsg (m/s)

Flow pattern map - PIPESIM®

Bubbles

Slug

Dispersed Bubble

Annular

Case A

Case B

Case C Case D

Case A

Case B

Case C Case D


4. PRESSURE DROP AND HOLDUP VALIDATION

The mechanistic models presented in this work, the homogeneous no-slip model and the classical correlation

generally used in field applications (Beggs & Brill) were implemented in Excel® and the results were validated by the

OlgaS flow correlation of the software PIPESIM®. This section presents the results obtained.

4.1 Validation of the results obtained by the homogeneous no-slip model, Beggs & Brill and mechanistic

models for the case A ( m/s e 0,01 m/s), which is in the dispersed bubble region in the flow

patterns maps, with the results obtained by the OlgaS correlation of the software PIPESIM®

Figure 6 shows graphically the comparison of liquid holdup and the total pressure gradients.

Figure 6. Validation of liquid holdup and the total pressure gradients for case A.

As can be observed in Figure 6, the difference in the values obtained by the homogeneous model, Beggs & Brill,

mechanistic model proposed by Shoham (2006) for bubble flow, and mechanistic model proposed by Ansari et al.

(1994) for slug flow was of 1% when compared with the OlgaS correlation, both for liquid holdup as for the total

pressure gradient values. This minimal difference can be explained by the fact that the presence of gas is extremely

small, whose slip effects between phases are practically negligible, resulting in a natural simplification for calculations

and resolutions. The mechanistic model proposed by Alves et al. (1991) for annular flow showed not to be applicable in

cases where the surface velocity of gas is less than 6 (m/s). This surface velocity of gas corresponds to the transition

line from slug to annular in the flow pattern map proposed by Taitel et al. (1980), adapted for this case study. By using

the liquid and gas surface velocities of case A in the annular model, it generated results that were incompatible with

reality, such as negative density and positive pressure gradients.


models for the case B ( m/s e 0,05 m/s), which is in the bubble region in the flow patterns

maps, with the results obtained by the OlgaS correlation of the software PIPESIM®.

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

99% 100% 100% 100%

0,000

99,9%

Liquid Holdup (%)

0,00

2000,00

4000,00

6000,00

8000,00

10000,00

12000,00

14000,00

16000,00

18000,00

9782,78 9779,42 9788,49 9783,79

0,00

9795,33

6376,32 6510,90 6385,00 6381,84

0,00

6463,03

0,83

0,00

0,0196737

Pressure Gradient (Pa/m)

dP/dL Gravitational dP/dL Friction dP/dL Acceleration

Total 16159,92

Total 16290,32

Total

16173,50

Total 16165,63

Total NA

Total 16258,38

NA


0,00

1000,00

2000,00

3000,00

4000,00

5000,00

6000,00

997,69

2792,50

5548,65

3018,68

0,000

2.498,94 13,34

17,41

66,27

35,34

0,000

0

0,06

0,000

0,032249


dP/dL Gravitational dP/dL Friction dP/dL Acceleration

Total 2498,94

NA

Total 3054,02

Total 5614,92

Total 2809,91

Total 1010,09

Figure 7 shows graphically the comparison of liquid holdup and the total pressure gradients.

Figure 7. Comparison of liquid holdup and the total pressure gradients for case B.

According to the graphs, for the case B, the liquid holdup calculated through the homogeneous no-slip model

presented 1% of deviation, Beggs & Brill correlation 4%, the mechanistic model proposed by Shoham (2006) for

bubble flow presented 2% and the Ansari et al. (1994) model for slug flow presented 1% of deviation when compared

to the OlgaS correlation of the software PIPESIM®.

When comparing the pressure gradients, the homogeneous no-slip model, the Beggs & Brill correlation, the

mechanistic model proposed by Shoham (2006) for bubble flow and the mechanistic model proposed by Ansari et al.

(1994) for slug flow presented 2% of deviation in relation to the OlgaS correlation. Once again the mechanistic model

proposed by Alves et at. (1991) for annular flow showed not be applicable for surface velocities of gas below 6 m / s in

this case study.


models for the case C ( m/s e 1,0 m/s) which is in the slug region in the flow patterns maps,

with the results obtained by the OlgaS correlation of the software PIPESIM®.

Figure 8 shows graphically the comparison of Liquid Holdup and the total pressure gradients.

Figure 8. Comparison of liquid holdup and the total pressure gradients for case C.

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%95%

100% 98% 97%

0%

96%

Liquid Holdup (%)

0,00

1000,00

2000,00

3000,00

4000,00

5000,00

6000,00

7000,00

8000,00

9000,00

10000,00

Dispersedbubbles

Beggs & Brill MechanisticBubbles

MechanisticSlug

MechanisticAnnular

OlgaS

9331,76 9789,09 9630,37 9455,41

0,00

9.425,40

105,06 129,64 106,03 104,09

0,00

114,7

0,24

0,00

0


Dp/Dl Gravitational Dp/Dl Friction dP/dL Acceleration

Total 9437,07

Total 9918,73

Total 9736,40

Total 9559,50

Total NA

Total 9540,10

0%

10%

20%

30%

40%

50%

60%

9%

28%

56%

37%

0%

29%

Liquid Holdup (%)

NA

NA


0,00

1.000,00

2.000,00

3.000,00

4.000,00

5.000,00

6.000,00

7.000,00

8.000,00

Dispersedbubbles

Beggs &Brill

Bubbles Slug Annular OlgaS

213,99 643,69

4.246,30

635,43 132,15 690,869 148,83

237,39

2.807,22

349,78 632,93

192,454 0,41

1,02964


dP/dL Gravitational dP/dL Friction dP/dL Accelaration

The case C showed the highest deviations. According to the graphs, the liquid holdup calculated through the

homogeneous no-slip model presented 20% of deviation, the Beggs & Brill correlation 1%, the mechanistic model

proposed by Shoham (2006) for bubble flow 27% and the mechanistic model proposed by Ansari et al. (1994) for slug

flow presented 8% of deviation when compared to the OlgaS correlation.

When comparing the pressure gradients, the homogeneous no-slip model presented 40,42% of deviation, the Beggs

& Brill correlation 12,44%, the mechanistic model proposed by Shoham (2006) for bubble flow 124,69% and the

mechanistic model proposed by Ansari et al. (1994) for slug flow presented 22,21% of deviation. These differences can

be explained, due to the complexity of the calculations of the variables present in a slug regime. The mechanistic model

of slug, proposed by Ansari et al. (1994), is the only one that considers such variables, making a physical analysis

currently considered one of the most complete. The Beggs & Brill correlation, however, presented results even closer to

those obtained through the OlgaS correlation, showing good resolution flexibility even in this scenario. The

homogeneous no-slip model and the mechanistic model for bubble flow proposed by Shoham (2006) showed

unsatisfactory values both for the liquid holdup as well as for the pressure gradient.


models for the case D ( m/s e 10 m/s), which is in the annular region in the flow patterns

maps, with the results obtained by the OlgaS correlation of the software PIPESIM®.

Figure 9. Comparison of liquid holdup and the total pressure gradients for case D.

For the case D, the liquid holdup calculated through the homogeneous no-slip model presented 5% of deviation, the

Beggs & Brill correlation 1%, the mechanistic model proposed by Shoham (2006) for bubble flow 37%, the

mechanistic model proposed by Ansari et al. (1994) for slug flow presented 13% and the mechanistic model developed

by Alves et al. (1991) for annular flow 5% of deviation when compared to the OlgaS correlation.

When comparing the pressure gradients, the homogeneous no-slip model and the mechanistic model proposed by

Shoham (2006) for bubble flow presented unsatisfactory results, with more than 40% of deviation when compared to

the OlgaS correlation. All the other methods showed satisfactory results.

It is important to note in this case that, despite the results obtained by the model proposed by Alves et al. (1991) for

annular regime be satisfactory when considering the total pressure gradient, when analyzing the portions of the

components of gravitational and friction gradient pressure separately, it is observed a reversal in the results, where the

largest portion obtained by the OlgaS correlation refers to the gravitational pressure gradient and in the model of Alves

et al. (1991) concerns to the friction pressure gradient.

Therefore, it is not possible to specify the reasons for the inversion in the OlgaS software solution, because it is a

black box model in which only the result can be accessed, not the formulas used in its resolution.

0%

5%

10%

15%

20%

25%

30%

35%

40%

45%

1%

5%

43%

19%

1%

6%

Liquid Holdup (%)

Total 884,35 Total

765,08 Total

985,21

Total 7054,01

Total 881,10

Total 363,23


5. CONCLUSIONS

In relation to the obtained data, the homogeneous no-slip model and the mechanistic model proposed by Shoham

(2016) for bubble flow, showed an increase in the deviation of the results by increasing the superficial velocity of gas.

Thus, its application is limited to superficial velocities which are in the regions of dispersed bubble and bubbles, in the

flow pattern map proposed by Taitel et al. (1980).

The Beggs & Brill correlation demonstrated robustness to the proposed case study, justifying its great applicability

in the industry whose scenarios are similar to the one studied in this study. It showed consistent and low divergence

results in all the studied cases, when compared to those obtained by the OlgaS correlation of the PIPESIM® software,

both in the liquid holdup as well as for the pressure gradient. Its resolution is simple and with few steps, achieving

satisfactory results.

The mechanistic model developed by Ansari et al. (1994) for slug regime, when considering the case C, which is in

the slug region in the flow pattern map, was the model that presented the greatest deviation comparing to the others

mechanistic models of this study when considering the region that they were developed for. The Ansari et al. (1994)

model, presented 22.21% of deviation for the total pressure gradient when validated by the OlgaS correlation of the

PIPESIM® software. The deviation can be explained by the complexity of the calculations of the variables considered

in the slug regime. In addition, the proximity of the case C with the transition line from slug to annular flow, in the flow

patterns map generated by the correlation OlgaS and showed in figure 4, could imply in mathematical smoothing,

whose methods are not accessible, possibly causing an increase in the divergences found.

The model developed by Alves et al. (1991) for annular flow, has shown satisfactory results in situations of high gas

flow within the annular region verified by the flow pattern map, and was not applicable for cases that use superficial

velocities of liquid and gas outside this region. The attempt to use it for scenarios that the case was in the dispersed

bubbles, bubbles and slug region in the flow patter map has been frustrated, resulting in values incompatible with

reality, such as negative density and positive pressure gradients.

In general, all models studied showed better results in scenarios with lower surface velocities of gas. The increase in

the gas flow rate was a complication in the precision of the calculations of the models, especially in those that present

more pronounced physical simplifications. It is also worth highlighting the high complexity of the mechanistic models,

so that the attention of these models to the physical phenomena of the flow are fundamental to guarantee their accuracy.

With the elaboration of this work, it became clear the importance of estimating the pressure gradient in a system.

Any improvement or calibration that can be obtained by analyzing the best modeling can result in extraordinary profits

in oil production systems. With the individualized description of the components of the total pressure gradient, it is

possible to develop or operate more efficiently in the wells, such as injection of friction reducers.

6. REFERENCES

Alves, I.N.; Caetano, E.F.; Minami, K.; Shoham, O. Modeling Annular Flow Behavior for Gas Wells. ASME. Chicago,

435 p. 1991.

Ansari, A.M.; Sylvester N.D.; Sarica, C.; Shoham, O.; Brill, J.P. A Comprehensive Mechanistic Model for Upward

Two-Phase Flow in Wellbores. SPE Annual Technical Conference and Exhibition, New Orleans, 152 p. 1994.

Beggs, H. D.; Brill, J. P. A Study of Two-Phase Flow in Inclined Pipes. Reprint Series JPT. Richardson, Texas: Society

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Sylvester, N.D. A Mechanistic Model for Two-Phase Vertical Slug Flow in Pipes. ASME JERT, 1987.

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Thomas, J. E. Fundamentos da Engenharia de Petróleo. 2nd Edition. Interciência: Petrobras, Rio de Janeiro. 271 p.

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Zuber, N.; Hench, J. Steady-State and Transient Void Fraction of Bubbling Systems and Their Operating Limits.

General Electric Report, Schenectady, New York, 1962.

7. RESPONSIBILITY NOTICE

The authors are the only responsible for the printed material included in this paper.

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