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
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:
)
Being:
(2)
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:
)
2.2
)
)
(5)
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)
T. M. F. Ruella et al.
Comparison among the homogeneous no-slip model, Beggs & Brill and mechanistic models
Where:
(7)
b.
Transition if:
(8)
c.
Intermittent if:
(9)
d.
Distributed if:
(10)
Then, the horizontal liquid holdup
through the following solution:
( º)=
(
(0º) must be calculated. The value adopted will be the highest obtained
)
(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.
IV Journeys in Multiphase Flows (JEM 2017)
Flow
Upward
Upward
Upward
Downward
Pattern
d
e
Segregated
0,011
-3,7680
Intermittent
2,960
0,305
Distributed
Uncorrected, C=0
All
4,700
-0,3692
Source: Beggs& Brill adapted (1973).
f
3,5390
-0,4473
0,1244
g
-1,6140
0,0978
=1
-0,5056
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.
T. M. F. Ruella et al.
Comparison among the homogeneous no-slip model, Beggs & Brill and mechanistic models
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 pseudofluid. 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)
IV Journeys in Multiphase Flows (JEM 2017)
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).
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).
Figure 2.Slug-flow schematic.
Source: Brill, Mukherjee adapted (1999).
2.3.3.1 Liquid Holdup and Pressure Drop Calculation
According to the Ansari et al. Mechanistic model for slug flow pattern, the liquid holdup can be obtained by the
following equation:
T. M. F. Ruella et al.
Comparison among the homogeneous no-slip model, Beggs & Brill and mechanistic models
(
)
(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)
2.3.3.1.3 Total Pressure Gradient
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.
2.3.5.1 Liquid Holdup and Pressure Drop Calculation
According to Alves et al. (1991), the liquid holdup can be found by the equation:
(
Where:
)
(39)
IV Journeys in Multiphase Flows (JEM 2017)
(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.
2.3.5.1.1 Total Pressure Gradient
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)
(kg/m³)
998,21
(N/m)
0,0728
(Pa.s)
(Pa.s)
0,001002
12,05
0,0000182
g (m/s²)
9,81
( )
Le camera (m)
(1)
25
le/d
270,4
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
T. M. F. Ruella et al.
Comparison among the homogeneous no-slip model, Beggs & Brill and mechanistic models
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
A
B
C
D
Dispersed bubble
Bubble
Slug
Annular
Surface velocity of the
liquid,
(m/s)
10
1
0,1
0,1
Flow pattern map - Adapted Taitel et al. (1980)
Flow pattern map - PIPESIM®
100
100
Dispersed
Bubbles
10
Vsl (m/s)
Vsl (m/s)
Dispersed Bubble
Case A
10
Case A
Churn
Case B
1
0,01
0,1
1
10
Slug
Bubbles
100
0,01
Bubbles
Case B
1
0,1
Annular
Case C
0,01
1
Case C
0,1
10
100
Case D
0,1
Case D
Vsg (m/s)
Figure 3. Flow pattern map - Adapted Taitel et al (1980).
3.2
Surface velocity of the
gas,
(m/s)
0,01
0,05
1
10
Slug
Annular
0,01
Vsg (m/s)
Figure 4. Flow pattern map - PIPESIM®
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.
IV Journeys in Multiphase Flows (JEM 2017)
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.
Liquid Holdup (%)
99%
100%
100%
Pressure Gradient (Pa/m)
100%
99,9%
100%
Total
16159,92
18000,00
90%
70%
14000,00
60%
12000,00
50%
10000,00
40%
8000,00
30%
10%
0%
Total
16165,63
Total
16173,50
0,83
6000,00
NA
0,000
6376,32
9782,78
Total
16258,38
0,0196737
16000,00
80%
20%
Total
16290,32
6510,90
9779,42
6385,00
9788,49
6463,03
6381,84
9795,33
9783,79
4000,00
Total
NA
2000,00
0,00
0,00
dP/dL Gravitational
dP/dL Friction
dP/dL Acceleration
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.
4.2
Validation of the results obtained by the homogeneous no-slip model, Beggs & Brill and mechanistic
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®.
T. M. F. Ruella et al.
Comparison among the homogeneous no-slip model, Beggs & Brill and mechanistic models
Figure 7 shows graphically the comparison of liquid holdup and the total pressure gradients.
Liquid Holdup (%)
100%
95%
100%
98%
Pressure Gradient (Pa/m)
97%
Total
9437,07
96%
0,24
10000,00
90%
Total
9736,40
Total
9918,73
Total
9540,10
Total
9559,50
129,64
106,03
104,09
0
114,7
9789,09
9630,37
9455,41
9.425,40
105,06
9000,00
80%
70%
8000,00
60%
7000,00
50%
6000,00
40%
5000,00
30%
9331,76
4000,00
20%
3000,00
10%
NA
0%
Total
NA
2000,00
0%
1000,00
0,00
0,00
Dispersed
bubbles
Beggs & Brill
Mechanistic
Bubbles
Dp/Dl Gravitational
Mechanistic
Slug
Dp/Dl Friction
Mechanistic
Annular
OlgaS
dP/dL Acceleration
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.
4.3
Validation of the results obtained by the homogeneous no-slip model, Beggs & Brill and mechanistic
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.
Pressure Gradient (Pa/m)
Liquid Holdup (%)
6000,00
Total
5614,92
66,27
56%
60%
5000,00
50%
37%
40%
4000,00
30%
Total
3054,02
Total
2809,91
29%
28%
Total
2498,94
35,34
3000,00
17,41
5548,65
0,032249
0
20%
9%
NA
10%
2000,00
0%
0%
Total
1010,09
0,06
1000,00
2792,50
3018,68
2.498,94
13,34
NA
997,69
0,000
0,00
dP/dL Gravitational
dP/dL Friction
Figure 8. Comparison of liquid holdup and the total pressure gradients for case C.
dP/dL Acceleration
IV Journeys in Multiphase Flows (JEM 2017)
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.
4.4
Validation of the results obtained by the homogeneous no-slip model, Beggs & Brill and mechanistic
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®.
Pressure Gradient (Pa/m)
Liquid Holdup (%)
8.000,00
Total
7054,01
43%
45%
7.000,00
40%
35%
6.000,00
30%
5.000,00
2.807,22
25%
19%
Total
881,10
4.000,00
20%
3.000,00
15%
10%
5%
6%
5%
1%
1%
2.000,00
1.000,00
0%
0,00
4.246,30
Total
363,23
Total
985,21
Total
884,35
Total
765,08
1,02964
0,41
148,83
213,99
237,39
643,69
Dispersed
bubbles
Beggs &
Brill
dP/dL Gravitational
349,78
635,43
Bubbles
192,454
632,93
132,15
Slug
dP/dL Friction
Annular
690,869
OlgaS
dP/dL Accelaration
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.
T. M. F. Ruella et al.
Comparison among the homogeneous no-slip model, Beggs & Brill and mechanistic models
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.
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7. RESPONSIBILITY NOTICE
The authors are the only responsible for the printed material included in this paper.