Probabilistic approach of an oil-water flow pattern map for pipelines from -90° to 90°

inclination.

Diana Estefanía Urbano Caguasango. Chemical Engineering Department, Universidad de los Andes, Bogotá

Colombia.

Abstract:

A probabilistic approach to two-phase liquid-liquid (oil-water) flow patterns is presented. An

experimental dataset of 8073 points was compiled and processed in order to determine an

acceptable data pool. The experimental data was then validated by means of OLGA Multiphase

Toolkit and used to generate the probabilistic flow pattern maps. Considering that different

dispersion patterns were taken into account, a probability surface was generated for each flow

pattern, with the aim of simplifying the reading. Furthermore, a tool was developed to assist the

recognition of probabilities.

Keywords: Liquid-Liquid pipe flow, flow pattern map, probabilistic classification, recognition

techniques.

1. Introduction

Flow of two immiscible liquids is related to many processes in the chemical and Oil & Gas

industries. Liquid - liquid flow is characterized by a low density ratio and a viscosity ratio that can

be of many orders of magnitude [1]. It is not only affected by inertia, viscous, pressure and

interfacial forces, but also by the inclination of the pipe, the wetting characteristics of the pipe, the

pre-wetting component and the operation conditions such as temperature and superficial velocities.

The interaction of these components leads to specific geometric distributions of the phases in a pipe,

which are known as flow patterns [2].

Flow patterns are commonly identified by visual observation, moreover, in recent years high speed

cameras and conductivity probes among others, have been used to support these observations [3].

Figure 1.a illustrates the most common flow patterns identified for horizontal flow, which are

stratified flow (ST), stratified flow with mixture in the interface (ST & MI), dispersion of oil in

water (DO/W), oil in water emulsion (O/W), dispersion of water in oil (DW/O), dispersion of oil in

water (DO/W) and water in oil emulsion (W/O) [4]. For vertical pipes, annular flow, slug flow,

dispersion of oil in water, emulsion of oil in water, churn flow, dispersion of water in oil and

emulsion of oil in water for vertical pipes, as it is shown in Figure 1.b [1, 5, 6, 7, 8]. Finally, for

inclined pipes the interaction of the parameters described above will change with the leaning, as a

consequence, the flow patterns will be similar to those for vertical or horizontal pipes depending on

the degree of inclination. As it is noted by Brauner (2002) [1], stratified flow usually disappears

from a 30° inclination.

Figure 1 (a) Horizontal flow patterns based on Trallero (1997) [4], (b) Vertical flow patterns according to Flores (1997) [6], Atmaca (2007) [7], Ghosh (2011) [8], among others.

Taking into account that each flow pattern results in different flow characteristics, such as

velocities, pressure drops, mixture viscosity and holdups, understanding the behavior of liquid-

liquid flow is crucial to a proper design of separations facilities, multiphase pumps and artificial lift

methods related to oil extraction [4, 9], thus flow pattern maps or simulation tools are required.

Flow pattern maps are constructed based on a set of experimental data points; however, they are

limited to the conditions under which the experiments were carried. The most commonly used are

those of Charles (1961) [10], Malinowsky (1975) [11], Oglesby (1979) [12], Arirachakaran et al.

(1989) [13], Trallero et al. (1997) [4] and Nädler and Mewes (1995) for horizontal pipes, those of

Trallero (1995) [14], Flores (1997) [6] and Alkaya (2000) [15] for vertical pipes and those of

Flores (1997) [6], Brauner (2001) and Rodriguez et al. (2005) [16] for inclined pipes.

Even if many attempts have been made to clearly stablish the boundaries where the transition from

one pattern to another takes place, several authors have found that a combination of flow patterns

can occur, moreover, varying the experimental conditions affects the probability to find a specific

flow pattern [17] [18]. In view of that, the present work uses a probabilistic-based approach to

determine the classification of the regimes for two liquid flow in straight pipes, using experimental

data from a literature review and OLGA™ simulations to validate those registers.

2. Literature review

Two-phase liquid-liquid flow is defined as the simultaneous flow of two immiscible liquids in a

pipe [7]. Despite its frequency in the industry, liquid-liquid flow has received less attention than

gas-liquid multiphase flow; nevertheless, the effect that the second liquid presence has over the

pressure drop and over other system characteristics is crucial and could compromise the safety and

the economic advantages of transporting a liquid-liquid flow [2].

The decrease of oil reservoirs, higher profits related to the increased production from mature oil

fields among others, has compelled the industries to maximize the use of oil wells [17]. To enable

higher oil extraction, water is introduced and as a consequence, the gradient pressure is lower and

the pumping is easier. In recent years water cuts as high as 0.9 have been operated [19].

Accordingly, in recent years the number of studies who try to throw light upon the nature of the

interaction oil-water has increased.

Unlike gas-liquid systems, in a liquid-liquid flow the density difference between the phases is

relatively low, the viscosity ratio can vary in a wide range and the interfacial chemistry is more

complex [1]. These differences do not allow a direct application of the results obtained for gas-

liquid systems to liquid-liquid systems and so, the importance of its study increases [7].

Understand the distribution of the phases in a liquid-liquid two-phase flow contributes to the

prediction of parameters as pressure drop, the inversion point, liquid holdup or the mixture

viscosity. An accurate prediction of the fluid behavior allows a correct equipment sizing and

therefore, safer and cheaper operations [2, 20].

Each phase distribution is known as a flow pattern and has some unique characteristics;

furthermore, it depends of the interaction between system conditions (inclination of the pipe,

pressure, temperature), physical properties of the fluids (density, viscosity, surface tension) and

forces as inertia, turbulent and shear forces [7, 21, 22].

Depending on the flow condition, the flow pattern can vary from fully segregated flow to fully

dispersed flow. There is not a consensus of the flow patterns classification, their boundaries or their

names. According to Brauner (2002) [1], for horizontal pipes there are four basic prototypes:

1. Stratified layers with either smooth or wavy interface.

2. Large slugs, elongated or spherical of one liquid in the other.

3. Dispersed: Droplets of one into the other.

4. Annular flow: one forms the core and the other the annulus.

Figure 2 Four basic prototypes according to Brauner (2002) [1]. Stratified flow with smooth interface (1a), Stratified

flow with wavy interface (1b), Large slugs (2), Dispersed flow (3), Annular flow (4).

However, in many cases the observed flow pattern consists of a combination of the basic prototypes

[1]. Below, there is an explication of the basic flow patterns that he establishes and the necessary

conditions to make a transition between them:

2.1 Horizontal pipes

When the flow has a low velocity, gravity is the dominant force leading to a stratified flow with

either smooth or wavy interface. As the velocity increases, the interface becomes disturbed and

some droplets can be entrained to the other phase near the interface.

While the flow rates continue to increase, the entrainment process does it too, until finally there is a

three layer structure. Two (the lighter and the heavier), where the phase is continuous, and another

placed between them, where there is a concentrated layer of drops. These three layers structure will

be broken when the flow rate becomes sufficiently high, and two courses are possible.

For a sufficiently high water flow rate, the oil phase becomes discontinuous, giving way to oil in

water dispersion or emulsion. Similarly, for a high oil flow rate, the water phase becomes

discontinuous, resulting in water in oil dispersion (Dw/o) or emulsion. Sometimes, the continuous

phase will change from water to oil or vice versa depending of an abrupt change of the pressure

drop [1].

Depending on the flow conditions, an annular flow can be created. It is usually encountered in

systems with low density differential and small tube diameters. When the annulus flow rate

increases, the core flow will be broken to form slugs or will be forced to disperse in the annulus

flow. Both liquids, water and oil, could form the core or the annulus layer. Due to pressure drop

reduction, it is desired that water forms the annulus flow, however, it is not so common.

According to Brauner (2002) [1], when the water is the continuous phase, the oil viscosity have a

minor effect on the flow patterns. However, the oil viscosity it affects the location of the phase

inversion, and leads to a lower water input required to invert the dispersion.

2.2 Vertical pipes

As it is shown in Figure 3, here are six typical flow patterns in vertical flow. Three are water

dominant and three are oil dominant. The flow patterns where water is the continuous phase are: i)

dispersion of oil in water, ii) emulsion or very fine dispersion of oil in water and iii) churn flow. For

oil dominated flow patterns, the three most common flow patterns are: i) water in oil churn flow, ii)

dispersion of water in oil and iii) emulsion or very fine dispersion of oil in water. Annular flow in

vertical pipes can only be observed for very viscous oils [6].

Figure 3 From Flores 1997 [6]

In vertical flow, churn flow is characterized by irregular shapes of continuous oil phase and

continuous water phase. As the mixture velocity increases the droplets size decreases until finally

homogeneous droplets are formed [1].

For low superficial velocities there are large bubbles of oil in a continuous water phase. When the

water velocity increases the bubbles break and the oil in water emulsion takes place. The same

happens when the oil velocity is the one that increases, transforming the flow pattern into a water in

oil emulsion [23].

According to Jing-yu et al. [2010], the main difference between upward and downward vertical

flow for the same superficial velocities is the transition from water dominated flow patterns to oil

dominated flow patterns takes place under a lower superficial oil velocity, which is due to the fact

that the slip velocity in downward flow is larger for the same input conditions.

2.3 Inclined pipes

Atmaca (2007) [7] states that the difference of flow patterns encountered in inclined pipes

differences from those found in horizontal pipes because of the effect of gravity force. In inclined

pipes this force has two components, normal to and parallel to the pipe axis. The normal component

promotes the separation of the phases, while the parallel component will have a different effect

depending on the direction of the flow [7]. For downward flow, water velocity will be higher than

for horizontal flow, while for upward flow it will be lower.

Brauner (2002) states that stratified flow can be found until a 30° inclination. However, taking into

account the low density difference, to solve the interface behavior it is necessary to consider the

wetting properties of both liquids and the surface tension.

As Yusuf et al. (2012) [18] establishes, there is some literature that tries to explain the effect of pipe

geometry over flow patterns; however, there is few literature that tries to explain the effect of oil

viscosity over flow patterns.

Authors as Russell et al. (1959) [24], Oglesby (1979) [12], Arirachakaran et al. (1989) [13], Trallero

(1995) [14], Alkaya (2000) [15] and Mckibben et al. (2000) [25] have found that the viscosity has

an important effect over flow patterns, pressure drop and liquid holdup [2]. Yusuf et al. (2012)

studied the effect of oil viscosity in flow patterns in horizontal pipes by studying each flow pattern

separately. The authors divided their results by stratified, bubbly, annular, dual continuous,

dispersion of oil in water, and dispersion of water in oil flows, and compared their experimental

results with those founded in the literature.

In stratified flows, the effect of viscosity depends on the oil velocity. For low oil velocities, a higher

oil viscosity leaded to a bigger stratified flow zone, this is, higher water velocities were needed to

make a transition to a different flow pattern. On the contrary, for high oil velocity, the water

velocity needed to make a transition to dual continuous flow was lower as the oil viscosity

increased [18].

According to the results of Angeli and Hewitt (2000) and Raj et al. (2005), there is no annular flow

for low viscous oils. As the viscosity increases, the probability of observing this flow pattern is

higher, as a higher oil viscosity leads to a higher chance for intermittent flows to form [18]. Finally,

Yusuf et al. (2012) observed that as the oil viscosity increased, the transition to dispersed flow (oil

in water or water in oil) was slower, as the shear force needed to break the bubbles was higher.

Alwahaibi et al. (2014) use the result of Yusuf et al. (2012) to study the effect of the pipe diameter.

They concluded that there is a significant influence over the flow patterns and the pressure drop, as

the boundaries shift substantially [26]. However, as Angeli and Hewitt (2000) remarked, it is also

important to take into account the wetting characteristics of the pipe and the initial configuration of

the phases, as it can affect significantly the development of the flow pattern [27].

If there is a consensus between all the authors here quoted, is that further investigation is required,

and tools able to explain in a more objective way the patterns that are obtained under certain

conditions must be developed, for low and high viscous fluids.

3. Datasets

In this work, an experimental and a synthetic dataset were used. The first one was obtained by a

literature review of liquid - liquid flows and the second one was generated by means of OLGA

2014.3 Multiphase toolkit.

3.1 Experimental Dataset

The experimental dataset consists of 8073 registers and was created based on the report of 42

authors, as shown in Table 1. These authors carried on experiments for different liquid-liquid

superficial velocities (i.e. the volumetric flow rate or each phase divided by the total cross-sectional

flow area), pipe lengths, diameters and orientations. However, it is noted that most of the

experiments where carried for horizontal inclinations (46.94 %). Is worth to be mentioned that only

straight pipes were covered.

Table 1 Experimental Data

Author # Data

points Fluids Diameter [m] L/D

Vs1 water

[m/s]

Vs1 oil

[m/s]

Pipe

inclination [°]

Roughness

[m]

Abduvayt

(2004) [28] 216

Water -

Kerosene 0.1040 1153.8 0.02 - 1.61

0.01 -

1.55

- 3, - 0.5, 3,

90 4.50E-05

Abubakar

(2015) [29] 150

Water - Oil

(Shell Tellus S2

V 15)

0.0306 326.8 0.01 - 1.35 0.01 - 1.35 0 1.00E-04

Alkaya

(2000) [15] 282 Water - Oil 0.0508 415.9 0.02 - 1.71 0.02 - 1.7 - 5, - 1,0,1,5 1.00E-04

Al - Wahaibi

(2014) [26] 196

Water - Mineral

Oil 0.0190 421.1 0.12 - 1.68 0.06 - 1.71 0 1.00E-04

Al - Yaari

(2009) [30] 102 Water - Oil 0.0254 393.7 0.5 - 3.0 0.1 - 0.9 0 1.00E-04

Angeli

(1996) [31] 179

Water -

Kerosene 0.0254 393.7 0.09 - 2.65 0.09 - 2.66 - 0.5 4.50E-05

Atmaca

(2007) [7] 306

Water - Mineral

Oil 0.0508 271.7 0.02 - 1.78 0.02 - 1.75

0,1,2, - 1, - 2, -

5 1.00E-07

Ayello

(2008) [32] 299

Water - Model

Oil 0.1000 250.0 0.01 - 0.22 0.47 - 2.51 0,5,45,90 4.60E-05

Bannwart

(2004) [33] 885

Water - Heavy

Crude Oil 0.0284 88.0 0.01 - 2.45 0.02 - 0.51 0.9 1.00E-07

Bannwart

(2012) [34] 16

Water - Heavy

Oil 0.0620 4838.7 0.03 - 0.08 0.15 - 0.63 90 1.00E-07

Cai (2012) [35] 100 Water -

Paraffinic Oil 0.1000 140.0 0.02 - 0.22 0.5 - 2.51 0 4.50E-05

Castro

(2011) [36] 144 Water - Oil 0.0260 461.5 0.02 - 1.26 0.02 - 1.23

- 20, - 10, 0,

10, 20 1.00E-07

Dasari

(2013) [37] 536

Water - Lube

Oil 0.0250 120.0 0.1 - 1.07 0.02 - 1.24 0 1.00E-04

Du (2012) [38] 103 Water - White

Oil 0.0200 120.0 0.18 - 1.5 0.25 - 4.06 90 4.50E-05

Elseth

(2001) [39] 84 Water - Oil 0.0563 186.5 0.06 - 2.41 0.07 - 1.98 0 1.00E-04

Flores

(1997) [6] 484

Water - Refined

mineral oil 0.0508 305.9 0.04 - 1.32 0.04 - 1.3 45, 60, 75, 90 1.00E-04

Foroughi

(2010) [40] 102

Water -

Silicone Oil 0.0003 288.0 0.002 - 0.75 0. - 0.02 0 1.00E-04

Ghosh

(2010) [5] 102

Water - Lube

Oil 0.0120 208.3 0.07 - 1.52 0.07 - 0.78 - 90 1.00E-07

Ghosh

(2011) [8] 191

Water -

Lubricating

Oil/Kerosene

0.0120 83.3 0.07 - 1.96 0.07 - 1.21 - 90 1.00E-07

Ismail

(2015) [41] 72

Water - Waxy

crude oil 0.0508 1311.0 1E- 4 - 0.01 1E-4 - 0.01 0 4.50E-05

Kumara

(2009) [19] 504

Water - Exxsol

D60 oil 0.0560 267.9 0.24 - 1.51 0.02 - 0.98 - 1, 0, 1, 5 1.00E-04

Li (2010) [42] 46

Water -

limpidity

mineral oil

0.0500 400.0 2E-4 - 0.09 6E-4 –5E-3 0 1.00E-04

Liu (2008) [43] 124 Water - Oil 0.0261 371.6 0.07 - 0.64 0.05 - 0.96 0 4.50E-05

Lovick

(2004) [17] 100 Water - Oil 0.0380 210.5 0.07 - 2.7 0.07 - 2.41 0 4.50E-05

Lum

(2006) [44] 201 Water - Oil 0.0380 210.5 0.06 - 2.25 0.06 - 2.26 - 5, 5, 10 4.50E-05

Mukhaimer

(2015a) [45] 127

Water/Salty

Water -

Kerosene

0.0225 355.6 0.06 - 1.91 0.08 - 2.1 0 5.00E-06

Nadler

(1997) [46] 79

Water - Mineral

White Oil 0.0590 813.6 0.02 - 1.31 0.02 - 1.45 0 2.01E-02

Poesio

(2008) [47] 43

Water - High

viscosity oil 0.0210 428.6 0.40 - 1.09 0.23 - 0.41 0 1.00E-07

Rodriguez

(2005) [16] 153 Water - Oil 0.0828 181.2 0.02 - 2.57 0.02 - 3.03 0, - 2, - 5,5 4.60E-05

Rodriguez

(2006) [48] 102 Water - Oil 0.0828 181.2 0.02 - 2.62 0.02 - 3.03 1.5,1,2 4.50E-05

Rodriguez

(2011) [49] 33 Water - Oil 0.0260 396.2 1.00 - 3.00 0.2 - 1.00 0 1.00E-07

Soleimaini

(1997) [50] 351 Water - Oil 0.0254 381.9 0.05 - 3.15 0.08 - 3.15 0 4.50E-05

Souza

(2013) [51] 331

Water - Heavy

Oil 0.0260 461.5 0.02 - 2.99 0.02 - 1.01 0 1.00E-07

Tan (2015) [52] 68 Water - White

Oil 0.0500 300.0 0.04 - 2.06 0.07 - 2.0 0 4.60E-05

Trallero

(1997) [4] 250

Water - White

Oil 0.0501 310.0 0.01 - 1.79 0.01 - 1.62 0 1.00E-04

Vielma

(2008) [9] 154

Water - Refined

mineral oil 0.0508 415.9 0.02 - 1.81 0.02 - 1.75 0 5.00E-06

Wang

(2010) [53] 161

Water - Mineral

Oil 0.0254 2047.2 0.02 - 0.24 0. - 1.03 0 1.00E-07

Xu (2008) [54] 29 Water - White

Oil 0.0500 260.0 0.07 - 3.15 0.02 - 2.02 0 1.00E-04

Xu (2010) [55] 95 Water - Diesel

Oil 0.0200 350.0 0.05 - 1.67 0.04 - 0.8 0 1.00E-04

Yao

(2009) [56] 74

Water - Crude

Oil 0.0257 2023.3 0.05 - 0.76 0.1 - 0.88 0 4.60E-05

Yusuf

(2012) [18] 283

Water - Mineral

Oil 0.0254 255.9 0.10 - 1.94 0.06 - 1.8 0 1.00E-04

Zhai

(2015) [57] 216 Water - Oil 0.0200 60.0 0.11 - 2.24 0.1 - 2.59 0 1.00E-04

Total range 8073 - 3E-4 - 0.104 83.3-

4838.7 2E-4 – 3.15 0.01 - 4.06 -90 - 90 1E-7- 1E-4

1 Superficial Velocity

The distribution of experimental data over different parameters is shown from Figure 4 to Figure 6.

According to the authors, the inner diameter of the pipe used in the experiments corresponds with

the diameters used in the Oil and Gas Industry, which can vary from 50.8mm to 200mm.

Figure 4 Distribution of experimental data based on inner diameter

In figure 5 the number of experimental data reported by pipe inclination is shown. It can be seen

that a high proportion of the experimental data found in the literature corresponds to horizontal

pipes, and in a lower degree, to vertical pipes. For inclined pipes the number of data is considerably

lower even though in the Oil industry inclinations in the range of ±5° is common [15].

Figure 5 Distribution of experimental data by pipe inclination

In Figure 6, the distribution of the experimental data based on the pipe material where the flow

patterns were developed is shown. It is notorious that the pipe material used is in general, similar to

the one used in industry [58].

Figure 6 Distribution of experimental data by pipe material

As it is evidenced in figure 7, the recollected database covered a wide range of reported

experiments. It should be noted, the variation of the parameters in figure 7, i.e. Oil viscosity,

Interfacial tension and Oil density, affects the results. Thus, extrapolation should not be done.

Figure 7 Distribution of experimental data by oil viscosity (a), interfacial tension (b), oil density (c).

Based on the data compilation, the range for each parameter that has to be taken into account is

reported in table 2.

Table 2 Ranges for experimental and synthetic datasets.

Parameter Experimental

Minimum Maximum

Pipe inner diameter [m] 0.25 0.106

Pipe angle [°] -90 90

Wall Roughness [m] 1.00E-07 1.00E-04

Pipe Length [m] 0.072 300

Oil superficial velocity [m/s] 1.00E-04 4.06

Water superficial velocity [m/s] 1.00E-04 3.15

Pressure [Pa] 101300 353648

Oil density [kg/m3] 780 988.02

Degree API 49.54 11.43

Water density [kg/m3] 980 1071.8

Oil viscosity [Pa s] 0.0012 2

Water viscosity [Pa s] 0.35 1.2

Tension [N/m] 0.013 0.05

3.2 Synthetic Dataset:

Due to the uneven distribution of experimental data, it was intended to expand the data pool with

synthetic registers. Taking into account the high precision of OLGA Multiphase Toolkit 2014.3 in

the prediction of liquid hold-up, it was considered as the proper software for the task. However, the

liquid - liquid flow patterns used by OLGA were more general that those reported in the literature,

which are intended to use in this report. Furthermore, they did not allowed to study the behavior of

the dispersion phenomena. As a consequence, it was decided to not use OLGA Multiphase Toolkit

as synthetic data generator, but only as a tool to validate experimental data and to complete partial

reported data.

4. Data validation

Considering that the data pool has been created based on different authors’ results, a filter has to be

developed, in order to recognize incoherent registers and minimize their effect over the results.

4.1 Experimental Data

To determine the set of the experimental data that should be used based on a tolerable error, the

same combination of parameters was simulated with OLGA Multiphase Toolkit and compared with

their corresponding experimental dataset by means of the water hold up.

Figure 8 Ratio between simulated and experimental water holdup within ±30% error

In figure 8 the relation between the reported water hold up and the simulated result is presented. It

can be seen that the tendency is to overestimate the water hold up. However, as 7033 registers did

not report it, the implementation of pattern recognition techniques to classify those registers as

acceptable or non-acceptable was necessary. To this end, K-nearest neighbors (KNN) pattern

recognition technique [59], Quadratic discriminant analysis (QDA) [60] and Support Vector

Machines (SVMs) [60] were tested.

KNN technique assumes that the data is distributed in space, so it is possible to have a notion of

distance. Thus, it classifies new data based on the class to which the k nearest neighbors belong.

QDA divides the space into a quadratic surface, classifying the observation into the group with the

smaller squared distance. Finally, SVMs construct a hyperplane in space giving the maximum

separation between classes, and classify new data based on the closest one.

To determine the tolerable error and the patterns recognition technique that should be implemented,

the performance of each classifier was analyzed for all the error span. As figure 9 shows, KNN had

the best performance, with a correct classification rate of 79.71 % under a tolerable error of 29 %.

Thus, the acceptable experimental data pool was then composed by all the registers whose error was

below 29 %. This data pool attempts to include as much available registers as it can without

compromising the reliability of results. As a result 6435 registers are used.

Figure 9 Performance of classifiers for different tolerable errors

4.2 Synthetic Data

In an attempt to expand the data pool, registers who did not reported the liquid hold up were

compared by means of the flow pattern with those registers that were already classified as

acceptable data. When the reported flow pattern and the flow patterns that was given by the

corresponding classification was the same, it was assumed that the observation was correct, and

OLGA Multiphase Toolkit was used to complete the liquid hold up register information.

It should be said that in order to compare the flow patterns, it was not possible to use the 60 flow

patterns reported in the literature, as the quantity made it quite extensive. It was observed that most

of the times, it was the name that the authors used the one that differ from each other, nevertheless,

the description and characteristics of the flow were the same. In view of that, they were organized

in 11 groups, based on Trallero [4], Flores [6] and Atmaca [7] classification.

Once the classification was made, the final data pool consisted of 2775 registers. It should be

noticed that the difficulty of generating synthetic data to fill the universe and the uneven

distribution of the available data will have a significant effect over the flow pattern maps, and the

results should be analyzed based on those limitations. As an example, in figure 10 the acceptable

data is represented for a pipe with 5° inclination, and the gaps are quite significant.

Figure 10 Acceptable data for an inclination of 5°

In spite of the limitations listed above, it is possible to recognize regions were more than one flow

pattern has been reported, thus, it should be possible to determine their main location and transitions

zones. To accomplish it, a probabilistic approach was taken.

5. Probabilistic Flow pattern maps generation

In order to identify the distribution of flow patterns across the map, a mesh of 250000 elements

distributed by 500 rows and 500 columns was created foe every pipe inclination. Then, the posterior

probability related to Bayes’ formula was used to determine the probability of occurrence for a

specific flow pattern as a function of two phase’s velocities. Bayes posterior probability is

calculated by means of Equation 1 [61].

𝑃(𝑇𝑖|𝐷) =𝑃(𝐷|𝑇𝑖)𝑃(𝑇𝑖)

∑ 𝑃(𝐷|𝑇𝑘)𝑃(𝑇𝑘)𝑛𝑘=1

[Eq. 1]

Where:

P(Ti|D) is the posterior probability, which means the probability that the theory Ti is true given

that the data D has been observed.

P(Ti) is the prior probability of the theorem Ti being true. It was calculated as 𝑇𝑖/𝑇𝑡𝑜𝑡𝑎𝑙 based

on the available data.

𝑃(𝐷|𝑇𝑖)

∑ 𝑃(𝐷|𝑇𝑘)𝑃(𝑇𝑘)𝑛𝑘=1

Is the updating term. Where ∑ 𝑃(𝐷|𝑇𝑘)𝑃(𝑇𝑘)𝑛𝑘=1 is the probability of Ti

being false.

Once the posterior probability was calculated for each element in the mesh, surfaces representing

the probability of occurrence for each flow pattern were generated, as show in Figure 11. It can be

noted that some curves present non - smooth surfaces in particular regions, which is due to the lack

of experimental information in those places

Figure 11 Probability surfaces for -5° inclination

With the results it is possible to analyze the effect of pipe inclination over flow patterns. However,

figure 5 has to be taken into account, as it gives an estimation of the reliability of each map.

According to Hanafizadeh et al. [62], non-stratified patterns are dominant for upward flows and

stratified flows are dominant in downwards flows. These observations were supported by the results

obtained, as for pipes with a negative inclination the dominant patterns are stratified (ST) and

stratified with mixture in the interface (ST&MI), and for positive inclined pipes, the dominant

patterns consisted of the dispersion of one phase into the other. The variation of stratified flow

according to pipe inclination is shown in figure 12.

Figure 12 Effect of pipe inclination over ST&MI flow pattern

In general, the results agree with the literature, with slightly differences for 5° inclination, as the

quantity of accepted data was lower. For example, in figures 13 and 14 the transition between

stratified flow and stratified with mixing in the interface for horizontal pipes and slightly inclined

a) -0.5° inclination

b) -1° inclination

c) 5° inclination

pipes is presented, and it corresponds with those found in the literature. The results for other

inclinations are in the appendix A.

Figure 13 horizontal pipe

Figure 14 1° inclined pipe

It should be noted that the probability of stratified flow with mixing in the interface for very

inclined pipes is small, and as none of the consulted references reported it, this pattern disappears

after 5°, which is not true.

After the probability surfaces were generated for each flow pattern and pipe leaning, an attempt of

creating one flow pattern map for each inclination, with all the probabilities functions included, was

made. However, as many dispersion patterns were used, the data overlapped and difficult the

reading of probabilities. In order to overcome these problems, a tool was developed, which offers

different plotting options and calculates all probabilities for new observations. In this way, it is

possible to know the pattern distributions in a more comprehensive way.

6. Developed tool

The probabilistic flow pattern map generator, based on pipe inclination was developed using Matlab

2015a®. First, an Excel file with the two phases’ velocities, inclination and diameter of the pipe is

loaded. Then, the data must be analyzed in order to determine which recognition technique and

tolerable error is chosen, and finally, the preferred inclination must be selected, which belongs to

those available in the dataset. Moreover, if the loaded dataset includes different pipe diameters, a

range can be selected, thus, the effect of the diameter over flow patterns could be studied.

7. Conclusions and future work

A probabilistic tool is developed to generate flow pattern maps based on experimental data.

Different recognitions techniques are implemented to recognize acceptable data under a tolerable

error, which are used afterwards as the base to flow pattern maps generation. These functions are

integrated via a developed tool and furthermore, the possibility to study the effect of pipe diameter

over the flow patterns is presented.

As many dispersion patterns are taken into account, one map per inclination was insufficient to

provide all the information, a decision to generate figures for each pattern was taken and a new

function was included in the tool. The new function allows the recognition of the same input data in

all the different figures and returns the probabilities associated with them. As a consequence, the

transitions boundaries are treated as zones.

An extension of the database is necessary to make the tool more precise, as many registers could not

be used because of the lack of liquid holdup information and synthetic data did not include the same

level of disaggregation of flow patterns. Moreover, the developed tool could be updated, to study

the effect of viscosities, pipe roughness and interfacial surface among others over the flow pattern

transitions.

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Appendix A

Results for pipes deviated from de horizontal by -5°, -3°, -2°, -0.5° 0°, 2°, 3°, 5°, 90° are

here presented, as those inclinations were characterized by the presence of acceptable data

once the data was validated.

A.1) Results for a -5° inclined pipe

A.2 Results for -3° inclined pipe

A.3 Results for -2° inclined pipe

A.4 Results for -0.5° inclined pipe

A.5 Results for horizontal pipe

A.6 Results for 2° inclined pipe

A.7 Results for a 3° inclined pipe

A.8 Results for a 5° inclined pipe

A.9 Results for a 90° inclined pipe