Pressure gradient and phase inversion correlations analysis for oil-water flow in horizontal pipes

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Pressure gradient and phase inversion correlations analysis for oil-water flow in horizontal pipes

Laura Prieto Saavedra

Chemical Engineering Department, Universidad de los Andes, Bogotá, Colombia. 2016. Abstract

Many authors have developed models and specific correlations for the calculation of pressure gradient and phase inversion, but in practice the most useful, accurate and no so complex form of predicting properties is urgently needed. Therefore, a statistical and physical analysis of correlations involved in pressure gradient and phase inversion calculation is presented, using a literature experimental dataset of 2287 points. Two pressure gradient models for stratified flow, one for dispersed flow and one for core-annular flow, are evaluated, with thirty-seven friction factor and twenty-one mixture viscosity correlations. The results showed, combinations with average errors between twenty-five and seventy-one percent, depending on flow pattern. Furthermore, the relation between pressure gradient and phase inversion was studied using twenty-six phase inversion point correlations, the calculated pressure gradient and the experimental data of four authors. Depending on oil viscosity different correlations were found to be better than others. The relation between pressure gradient and phase inversion point calculations, was found only for intermediate mixture velocities.

Keywords: oil-water flow, pressure gradient, phase inversion, friction factor, mixture viscosity.

1. Introduction

Liquid-liquid flow is defined as the simultaneous flow of two immiscible liquids in pipes [1]. According to Park (2016) [2], this flow is frequently seen in different applications such as transport in the petroleum industry, emulsification and two-phase reactions and separation in process industries. However, the study of this flow system has been increasing with the years, as a consequence of its importance during oil extraction from oil wells. Usually during the process of extraction, a significant amount of water comes with the oil in pipelines or in some cases water is introduced for easier transportation [3]; this has important consequences in profit and is specifically related with the changes in flow characteristics that are yet not fully understood.

Oil-water flow is characterized by large momentum transfer capacity, small buoyancy effects, low free energy at interface and small dispersed phase droplet size [4]. Since oil properties can be diverse, the viscosity ratio can vary on many orders of magnitude, but usually a low difference between densities is observed [1]. In addition, the main difference between single phase flow and two phase flow is the presence of flow regimes (i.e. how the two phases are distributed), oil-water flow patterns are determined by different properties such as input flow velocities, how they are introduced, fluid properties, interfacial tension, pipe material (i.e. Roughness), pipe inclination, among others [5].

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Flow patterns have been widely studied and authors have given them a huge number of different names. However, the most common flow patterns identified for horizontal flow are the ones specified by Trallero (1997) and showed in Figure 1 [4].

Figure 1. Horizontal flow patterns defined by Trallero (1997) [4].

When the superficial velocities of both, oil and water, are low, the flow is dominated by gravity and the phases are segregated with a smooth interface; this pattern is known as stratified flow (ST) [4]. An increase in superficial velocities causes the appearance of interfacial waves with possible entrainment of drops at one side or both sides of the interface, leading to the stratified flow with mixture at the interface (ST&MI) [6]. When the forces associated with the motion are not enough to maintain all the droplets suspended and some of them eventually settle a dispersion of oil in water and water (Do/w & w) can be obtained. In addition, at sufficiently high water velocity, the entire oil phase becomes discontinuous in a continuous water phase resulting in an oil in water (Do/w) emulsion; although, when the oil is the phase with high velocity the water phase is completely disperse in the oil phase resulting in a water in oil (Dw/o) emulsion. These two emulsions may coexist obtaining a dispersion of water in oil and oil in water (Dw/o & Do/w) [6].

Another pattern that can be obtained under some specific conditions is the annular-core configuration also known as core-annular flow (CAF) shown in Figure 2. In this pattern, the viscous liquid, forms the core phase which is then surrounded and lubricated by an immiscible low viscosity liquid, as the annular phase [6]. Both, water and oil can be the annular phase, but commonly water as annular phase is encountered. This pattern is common when the two phases have equal densities or when one of the phases has a very high viscosity [5]. In addition, it is of special interest, since, if stable core flow can be maintained, the pressure drop is almost independent of oil viscosity and only higher than for flow of water alone at the mixture flow rate [6]. These are the seven flow patterns used in the present work.

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Figure 2. Core-annular flow described by Brauner (2002) [6].

Since each of the flow patterns named are the result of different flow characteristics, the understanding of the liquid-liquid behavior is essential in order to make a proper design of separation facilities, pumps, pipes and in general all methods related to oil extraction [1, 4]. Through experimental studies, authors have determined how changes in the spatial distribution of the phases can have a significant impact on pressure drop; this makes determination of flow regimes and consequently, quantification of pressure drop across the pipe very important tasks.

Usually, flow pattern determination is made experimentally through visual observations and lastly, new equipment technologies have been also developed [7]; in the case of pressure drop, it has been measured by experimental methods. Trying to make a non-experimental way of specifying these two factors, many authors created flow pattern maps based on their experimental studies and for pressure gradient, a few correlations have been developed according to flow pattern type.

In order to understand better the behavior of both things, a special phenomenon was encountered in a specific flow pattern and with very punctual changes in pressure gradient; this was defined as phase inversion. According to Ismail (2015) [3], phase inversion occurs when the mixture of fluids changes their continuity and it happens in a dispersed flow type, bringing important changes in pressure gradient through the pipeline. Most experimental studies in this topic have been carried in stirred tanks and just a few in pipes. Arirachakaran et al (1989) [8], Nädler and Mewes (1997) [9] and De et al (2010) [10], have noticed that the maximum pressure gradient takes place at the inversion point, i.e. phase fraction at which change of continuous phase takes place [5].

The occurrence of phase inversion, can be either beneficial or not, depending on the application where the two liquid-liquid flow is given [5]. In a pipeline, significant changes on pressure gradient and rheological properties are given, making it difficult to improve pipe design and operating conditions on the transportation of these kind of mixtures. A few models have been developed in order to predict the inversion point or at least the ambivalent region, i.e. a range of phase fractions over which either phase can be continuous, but a wide error between the prediction and experimental results was obtained [5].

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In view of that, this work presents a statistical analysis of correlations developed for the prediction of pressure gradient and phase inversion using experimental data from literature; an analysis of pressure gradient results and their relation to phase inversion is also performed.

2. Prediction of pressure gradient and phase inversion

The understanding of oil-water flow in pipes is crucial, and the prediction of properties and phenomena such as pressure gradient and phase inversion, is necessary. Keeping in mind the objective of the present work, in the following sections, pressure gradient and phase inversion existing studies and correlations are discussed in detail. Furthermore, the relation of pressure gradient with flow patterns and phase inversion is presented.

2.1. Pressure gradient

As mentioned before, pressure gradient in pipes depends on flow patterns and flow rates; although some authors as Atmaca et al (2008) [1] and Sridhar et al (2011) [11], have stated that pressure drop is mainly influenced by oil viscosity [3]. In general, oil-water pressure drop has a greater magnitude than single-phase pressure gradient of water alone [12]. Many authors have studied pressure gradient in horizontal and inclined pipelines with different conditions, fluids and experimental sections, and all have agreed that changes in pressure gradient are greatly related to the phase inversion phenomena and to flow pattern transitions [3].

A few methods have been introduced in order to accurately predict pressure gradient; for some of them the results were satisfactory for certain flow pattern and failed to work with others. From stratified flow, dispersed flow to core-annular flow has been studied. This is mainly because of the relation between pressure loss in an oil-water pipeline and the shear between the fluid and the pipe wall that is clearly affected by the distribution of the phases [5, 13]. Other elements that contribute to pressure gradient are pipe wettability and roughness, which means material of the pipe is also an important factor that must be taken into account. Pipe roughness restricts a liquid from moving smoothly and because of that, a pressure gradient is obtained during liquid-liquid flow [3].

For the calculation of pressure gradient, many factors should be taken into account. Usually, densities, viscosities and velocities of the fluid are needed; a friction factor is also taken into account, which includes Reynolds numbers and pipe roughness. For some models, mixture viscosity is also another variable that must be calculated.

Friction factor correlations have been widely studied since 1840 with Hagen and Poiseuille [14] and a great amount of modifications and new approximations have been presented. Similarly, for the calculation of mixture viscosity many correlations have been developed since Einstein’s first equation in 1906 [15]. A review of all of these equations can be found in Table A.1 and Table A.2 in Appendix A.

Moreover, a review of the pressure gradient prediction equations is presented in Table 1. As it can be seen, the friction factor correlations and in some cases the mixture viscosity calculation are needed. The statistical analysis made in the present work takes into account

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all possible combinations between the correlations without forgetting the principal objective: pressure gradient calculation.

Table 1. Pressure gradient existing correlations.

Author Correlation Developed for

(Used for) Arirachakaran et al. (1989) [3, 8] dp dz = Po Pc (dp dz)fo

+Pw Pc

(dp dz)fw

(Eq. 1)

(dp dz)f

= fρv 2

2gcD

(Eq. 2)

Stratified flow (ST, ST&MI)

Brauner (2002)

[6]

dP dz= 2fm

ρmJm2

D − ρmgsinβ (Eq. 3)

Dispersed flow (Do/w, Dw/o, Do/w & w, Dw/o

& Do/w)

Brauner (2002)

[6]

−Ac( dP

dz) ∓ τiSi+ ρcAcg sin β = 0 (Eq. 4) −Aa(

dP

dz) − τaSa± τiSi+ ρaAag sin β = 0 (Eq. 5)

Core annular flow (CAF) Elseth (2001) [16]

−A0( dp

dz) − τOSO+ τOWSOW+ ρOAOgsinθ = 0 (Eq. 6)

−Aw( dp

dz) − τWSW− τOWSOW+ ρWAWgsinθ = 0 (Eq. 7)

Stratified flow (ST, ST&MI)

In the Arirachakaran et al. (1989) model, the pressure gradient is calculated for each phase separately and a correlation is proposed. The assumptions used in this model are: smooth interface, not relative motion, no mass transfer between the phases and no net shear force at interface [8]. For the dispersed flow Brauner (2002) model, the pressure gradient is calculated using mixture properties, which imply the use of mixture viscosity correlations. This model neglects a possible difference between the in situ velocities of the two liquid phases, making the calculation of the water holdup easier [6]. In the case of the other two models, Brauner (2002) for CAF flow and Elseth (2001) for stratified flow, the calculation of the pressure gradient is made based on a momentum balance for the core and the annulus, or for the oil and water phase, respectively. In both cases, the holdup must be solved iteratively and the interfacial shear stress is taken into account. The geometry plays a very important role in this models [6, 16].

2.2. Phase inversion

A large number of authors have studied and tried to define the phase inversion phenomena. This phenomenon is commonly observed in dispersed liquid-liquid mixtures and in pipe flow or stirred vessels. According to the flow patterns already described, there are two type of dispersions: oil in water dispersion and water in oil dispersion; these are defined according to phase fraction and initial conditions [5]. According to Ngan (2010) [5], this special phenomenon is found when the mixture undergoes changes in phase distribution as phase fraction reaches certain critical values. This phase fraction is defined as phase inversion point.

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In Figure 3, the mechanism of phase inversion can be graphically seen. As the flow rate increases, one of the phases may be broken up into dispersed droplets in the continuum of the other phase. If the concentration of this phase is gradually increased, the phase will become closely packed and at some point (phase inversion point) the drops will coalesce and the phase continuity will switch [5].

Figure 3. Phase inversion mechanism of oil-water flow by Ismail (2015) [3].

As the phase inversion is achieved, a complete change of properties and behavior of the mixture occurs. That is the reason why, a deeper knowledge and an exact prediction of phase inversion point is not an easy task but is absolutely important in the industry. A good understanding could lead to better control and prediction of pump power required to trasnport the fluids across the pipe, increasing productivity and minimizing economic losses [3]. Even though, the prediction of phase inversion referred most of the times to a single point, some results have indicated the existence of an ambivalent region or a range of fractions over which either phase can be continuous. This is supported by various authors, whose measurements have indicated that inversion may not occur simultaneously across the whole pipe cross section, leading to a transitional region where phase inversion begins and completes [3, 17, 18, 19].

The effect of different parameters such as, viscosity, pressure gradient, velocity, phase distribution, pipe diameter and material, wettability, drop size and interfacial tension in phase inversion has been studied for liquid-liquid pipe flow [5]. Many authors as, Martinez et al (1988) [20], Angeli (1996) [21], Nädler and Mewes (1997) [9] and Soleimani et al. (1997), have found that significant changes in pressure gradient are caused by phase inversion, but the changes are still not accurately described. An example of this can be seen in Ioannou (2006) experimental studies; he found, using Exxsol D80 and water, that phase inversion occurs at the peak of the pressure gradient as seen in Figure 4 [23]. Although, as already said, this result has not always been obtained.

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Figure 4. Pressure gradient measurement of an Exxsol D80/water system obtained by Ioannou (2006) [5].

In Table 2, the main phase inversion correlations developed are summarized. These correlations are the ones used in the present work.

Table 2. Phase inversion existing correlations.

Author Correlation

Yeh et al. (1964) [24] fw,inv =

1

1 + √_μμo w

(Eq. 8)

Frechou (1986) [25] (1 − fw,inv) = (1 + ( μw μo )

2 3 )

−1

(Eq. 9)

Arirachakaran et al. (1989)

[24] fw,inv= 0.5 − 0.1108 log (

μO μW

) (Eq. 10)

Nädler and Mewes (1997) [24]

fw,inv=

1

1 + (μo μw)

0.208 (ρo

ρw)

0.625 _{(Eq. 11)}

Brauner and Ullmann (2002) [24]

fw,inv=

1

1 + (_μμo w)

0.4 (_ρρo

w)

0.6 _{(Eq. 12)}

Ngan (2010) [5]

It is a graphical method. The viscosities of the two types of dispersions, water continuous and oil continuous, are calculated with one of the various models available. They are then plotted together against water fraction and the point where the two plots intercept is considered the phase inversion

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3. Dataset

The experimental dataset used, consists of 2287 registers from 27 experimental studies of different authors in horizontal pipes, as shown in Table 3 .These authors used different fluids, superficial velocities, pipe lengths, diameters and materials.

Table 3. Experimental data used in the present study.

Author Data Fluids Vsw [m/s] Vso [m/s]

Wall roughness

[m]

D [m] L [m] L/D

Abubakar (2015)

[26]

86

Water-Mineral Oil (Shell Tellus S2

V15)

0.02-1.35 0.02-1.35 1.00E-04 0.0306 12 392.2

Al-Yaari (2009)

[27]

85 Water-Kerosene

(SAFRA D60) 0.1-2.78 0.1-2.79 1.00E-04 0.0254 10 393.7

Angeli (2000)

[28]

66 Water-Kerosene

(EXXSOL D80) 0.11-2.65 0.43-2.65

1.00E-05/7.00E-05 0.024/0.0243 9.5/9.7 395.8/

399.1

Castro (2011) [29]

6 Water-Oil 0.015 0.03-0.15 1.00E-07 0.026 12 461.5

Dasari (2014)

[30]

77 Water - Lubricating

Oil 0.19-1.06 0.11-1.19 1.00E-04 0.0254 1 39.4

Elseth (2001) [16]

159 Water-Oil (Exxsol

D-60) 0-2.99 0-3 4.50E-05 0.0563 10.2 181.3

Kathibi (2015)

[31]

9 Tap water- Mineral

Oil 0.02-0.7 0.01-0.9 5.00E-05 0.06 16 266.7

Kumara (2009) [32]

69 Water-Oil

(EXXSOL D60) 0.001-1.51 0.001-1.49 4.60E-05 0.056 15 267.9

Kurban (1997)

[33]

75 Water-Oil

(EXXSOL D80) 0.11-2.62 0.43-2.62

1.00E-04/4.50E-05 0.024/0.0243 9.5/9.7 395.8/

399.1

Laflin (1976)

[34]

44 Water-N° 2 Diesel

fuel 0.2-0.9 0.5-1.09 1.00E-04 0.0381 5.7 152

Liu (2008)

[35]

41 Water-Diesel Oil 0.028-0.514 0.05-0.63 4.50E-05 0.026 11,1 429.4

Lovick (2004)

[36]

96 Water-Oil

(EXXSOL D140) 0.001-3 0.001-3 4.50E-05 0.038 8 210.5

Mukhaimer (2015) [12] [37] 22 Water/Salty water-Kerosene (Safrasol 80)

0.005-2.384 0.005-2.384 5.00E-06 0.0225 8 355.6

Nadler (1997)

[9]

401 Water-Mineral Oil

(Shell Ondina 17) 0.001-1.5 0-1.494 1.00E-04 0.059 48 813.6

Oglesby (1979)

[38]

238 Water-Oil 0.07-2.71 0.19-3.19 5.00E-05 0.0411 5.7 140.9

Rodriguez (2011)

[39]

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Schümann (2016)

[40]

53

Tap Eater- Oil (Exxsol D80/Primol

352)

0.04-0.90 0.05-0.9 5.00E-06 0.1 25 250.0

Shi (2015)

[41]

32 Water- Oil (CYL

680/CYL 1000) 0.01-1 0.06-0.57 1.00E-04 0.026 1.7 66.5

Soleimani (1999)

[42]

101 Water-Oil

(EXXSOL D80) 0-3 0-3 4.50E-05 0.0243 9.7 399.2

Souza (2013)

[43]

150 Water-Heavy Oil 0.02-2.99 0.02-1 1.00E-07 0.026 6.1 234.6

Tan (2015)

[44]

29 Water-Mineral

White Oil 0.05-0.66 0.39-0.79 4.60E-05 0.05 2 40.0

Trallero (1995)

[13]

24 Water-Oil (Crystex

Af-M) 0.01-1.82 0.01-1.59 1.00E-04 0.0501 15.5 310.2

Valencia (2003)

[45]

42 Tap Water-Oil

(Purolube 150) 0.13-1.13 0.19-1.26 5.00E-06 0.253 4 16.0

Vielma (2008)

[46]

90 Water-Refined

mineral Oil 0.02-1.80 0.05-1.75 1.00E-04 0.0508 21.1 415.9

Wang (2010) [47]

85 Water-Mineral Oil 0.01-0.8 0.01-0.39 4.60E-05 0.0254 2 78.7

Yao (2009)

[48]

59 Water-Crude Oil 0.05-0.792 0.068-0.901 4.60E-05 0.026 4 153.8

Yusuf (2012)

[49]

115 Water-Mineral Oil 0.15-2.56 0.14-2.27 1.00E-04 0.0254 6.5 255.9

Total 2287

Since a variety of flow patterns names are found in literature, in order to homogenize the data and classify it in the seven flow patterns already described, the tool developed by Urbano (2015) “The probabilistic flow pattern map generator” is used [50]. The distribution of experimental data over different parameters is shown in Figure 5 and Table 4.

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a.) b.)

c.) d.)

e.) f.)

Figure 5. Data distribution. a.) Flow pattern b.) Pipe inner diameter c.) Oil viscosity d.) Oil density e.) Interfacial Tension f.) Pipe Material.

24,4% 28,6% 3,7% 7,6% 10,7% 22,1% 3,1% Do/w Do/w&Dw/o Do/w&w Dw/o ST ST&MI A 0 200 400 600 800 1000 1200

0.024 0.025 0.04 0.06 0.1 Greater

N o . o f D a ta Po in ts

Inner diameter [m]

0 200 400 600 800 1000 1200

0,01 0,1 0,5 2

Oil Viscosity [Pa*s]

0 100 200 300 400 500 600 700 800 900 1000

805 831 857 883 909 935 961

Oil Density [kg/m^3]

0 100 200 300 400 500 600 700 800 900

0,02 0,03 0,04 0,05

Interfacial Tension [N/m]

8,3% 5,1% 20,6% 10,6% 55,4% Glass PVC Stainless Steel Carbon Steel Acrylic

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11 Table 4.Range of important parameters of database.

Parameter Experimental

Minimum Maximum

Pipe inner diameter [m] 0.0225 0.253

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

Pipe length [m] 1 48

Length/Diameter [-] 16 813

Water superficial velocity [m/s] 1.00E-04 3.00 Oil superficial velocity [m/s] 1.00E-04 3.19

Water density [kg/m3_] ₉₈₃ ₁₀₄₃

Oil density [kg/m3_] ₇₈₀ ₉₅₈

Water viscosity [Pa*s] 3.55E-04 1.00E-02

Oil viscosity [Pa*s] 1.57E-03 5.00

Tension [N/m] 0.0017 0.045

It is important to mention, that the initial database had 11106 registers for horizontal flow observations; however, just 2287 reported the experimental pressure gradient. In these terms, the use of OLGA Multiphase Toolkit 2014.3 was contemplated to complete the registers. In order to validate the results given by the software, it was decided to compare rigorously the experimental pressure gradient data that was reported with the one calculated through OLGA for the same registers. This was made for all data points, and also for each one of the flow patterns to find possible right predictions. The results were not satisfactory since high errors and lower R-squared values were obtained during the comparison, this can be seen in Figure B.1, Appendix B. For this reason, it was decided to use only the registers that report experimental pressure gradient in order to minimize the effects over the results of the present study.

4. Methodology

The software used in this work is Matlab R2015a. Two different codes are made, one for pressure gradient analysis and the other for phase inversion analysis, based on the results of the first code. The first code has the following structure: data importation, data classification through flow pattern, calculations, statistics and results, as shown in Figure 6. On the other hand, the structure of the code made for phase inversion predictions can be found in Figure 7.

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Figure 6. Structure of Matlab code for pressure gradient calculations.

The pressure gradient calculation part is divided by model. Each model has different inputs, which means not all the combinations of friction correlations, mixture viscosity correlations and pressure gradient models are evaluated. The number of combinations depend on the model definition. As an example, the Arirachakaran correlation calculates a friction factor for each one of the phases separately, which means no mixture viscosity correlation is used in that case. On the other hand, for Brauner correlation, mixture viscosity correlations are used as well as friction factor correlations, obtaining more different combinations to evaluate. Taking this into account, all possible combinations within these limitations are evaluated.

Figure 7. Structure of Matlab code for phase inversion point predictions.

For the phase inversion point predictions, only the experimental data that report phase inversion points for specific systems is used. This, in order to compare the phase inversion correlations results, with the experimental phase inversion point. This part also uses the pressure gradient results, since the relation between pressure gradient and phase inversion is evaluated.

4.1. Statistics

Since the principal objective is to find the most practical, accurate and non-complex correlation combination, descriptive statistic and specific model comparison criteria are used. Here, the experimental data reported in the database and the one calculated for each pattern, as described before, are compared and analyzed.

Data flow pattern classification Data import

from Excel to Matlab ST ST&MI Do/w Dw/o Do/w & w Do/w & Dw/o CAF Pressure gradient calculation: all possible combinations depending on the model (friction factor and mixture viscosity

correlations) Statistical criterions and validation of the results Results: graphics and tables Classification of data depending on author and mixture velocity Data import with calculated pressure gradient Phase inversion point calculation through different correlations Graphic of calculated pressure gradient data against water cut Graphic with predicted phase inversion points

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13 Table 5. Statistical parameters used in the present work.

Statistical parameter Equation

Average Percent Error (%) [51]

ei=

dPdzcalc− dPdzexp dPdzexp

(Eq. 13)

E1, % =1 n ∑ ei

n

i=1

∗ 100 (Eq. 14)

Absolute Average Percent Error (%)

[51] E2, % =

1 n ∑|ei|

n

i=1

∗ 100 (Eq. 15)

Percent Standard Deviation (%)

[51] E3, % = ∑ √

(ei∗ 100 − E1)2 n − 1 n

i=1

(Eq. 16)

Average Error (Pa/m) [51]

eii= dPdzcalc− dPdzexp (Eq. 17)

E4 =1 n ∑ eii

n

ii=1

(Eq. 18)

Absolute Average Error (Pa/m)

[51] E5 =

1

n ∑|eii| n

ii=1

(Eq. 19)

Standard Deviation (Pa/m)

[51] E6 = ∑ √

(eii− E4)2 n − 1 n

ii=1

(Eq. 20)

Relative Performance Factor (-)

[52] FRP = ∑

|Ei| − |Ei,min| |Ei,max| − |Ei,min| 6

i=1

(Eq. 21)

R-squared [53]

a = ∑ dPdzexp∗ dPdzcalc n

i=1

(Eq. 22)

b =1

n ∑ dPdzexp n

i=1

∑ dPdzcalc n

i=1

(Eq. 23)

c = ∑ dPdzexp2 n

i=1

−1

n(∑ dPdzexp n

i=1

) 2

(Eq. 24)

d = ∑ dPdzcalc2 n

i=1

−1

n(∑ dPdzcalc n

i=1

) 2

(Eq. 25)

R2₌(a − b) 2

(c ∗ d) (Eq. 26)

Akaike Information Criterion [14] AIC = n ln (∑ eii2 n

i=0

) + 2K (Eq. 27)

The average percentage error (E1) and the average error (E4) indicate the agreement between calculated and measured data. Positive values imply overprediction and negative values underprediction. The absolute average percentage error (E2) and the absolute average error (E5), represent the general percentage error of the calculations. The percent standard deviation (E3) and standard deviation (E6) indicate the scatter of the error in respect to their corresponding average error. The three first terms are based on percentage error rather than relative pressure error, which means relative small pressure error that experience a small pressure gradient may give a large percentage error even though the pressure error itself is

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not that far from the actual measurements. In order to make the statistics independent of the magnitude of pressure gradient the other three identical statistical parameters are defined [51, 52].

In addition, the relative performance factor (FRP) created by Ansari et al (1994) [52] is a composite error factor with descriptive statistic criteria in it. The minimum and maximum possible values of FRP are 0 and 6, corresponding to the best and worst prediction performance. The value of R-squared is a statistical measure of how close the data are to the real values and in general, the higher the value, the better the model fits the data. On the other hand, the Akaike Information Criterion for a given data set has no meaning by itself, although the value can be interpreted if it is compared with the AICs of a series of models based on the same data set (observations) with the same dependent variables. Since this criterion takes into account the variables used by the model, a complexity comparison is being made, the lower the AIC most appropriate the model [54].

5. Results and analysis

In this section results are presented in order. First pressure gradient model analysis, followed by phase inversion results.

5.1. Pressure gradient

Since each pressure gradient model was defined for a specific flow pattern, only the possible combinations within these models are evaluated. Since statistical analysis was made to evaluate the performance of each combination, trying to find the most practical, accurate and non-complex model, only the five best combinations with their statistics and graphical results are presented.

Stratified flow (ST)

For this pattern, 40 possible combinations were found to be valid for the 244 experimental data points. The graphical and statistical results of the best five combinations are presented in Table 6 and Figure 8.

Table 6. Statistical results for ST. Pressure Gradient Model Friction Factor Correlation E1 (%) E2 (%) E3 (%) E4 (Pa/m) E5 (Pa/m) E6

(Pa/m) FRP R

2 _AIC

Elseth (2001) (Eq. 6)

Danish et al. (2011) (Eq. 84)

-40.89 42.83 69.26 -66.53 68.15 186.89 0.011 0.64 202.20

Elseth (2001) (Eq. 6)

Drew et al. (1936) (Eq. 32)

-42.96 44.70 67.73 -69.13 70.59 189.99 0.013 0.67 202.90

Arirachakaran (1989) (Eq. 1) Churchill (1977) (Eq. 43)

-9.35 64.09 931.55 -1043.89 1807.90 39244.85 2.86

E-07 0.16 5563.17

Arirachakaran (1989) (Eq. 1)

Morrison (2013)

(Eq. 90)

-10.62 64.60 930.30 -1044.34 1808.20 39243.28 3.01

E-07 0.16 5563.17

Arirachakaran (1989) (Eq. 1) Wood (1966) (Eq. 37)

-64.28 67.30 362.45 -2036.54 2044.74 49070.68 7.88

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Based on the statistical parameters results, there is a tendency to underestimate the pressure gradient value in all combinations. As it can be seen, the average error is between 42 and 45% for combinations using Elseth (2001) model and between 64 and 69% for models using the Arirachakaran (1989) model. The main difference between these two models is that the first one takes into account the interfacial shear stress between the phases, while the second one is a correlation that calculates pressure gradient for each phase separately. The statistic parameters are better for the first two combinations. The standard deviation is lower and the R-squared showed a better response of the data. The relative performance factor (FRP) and the AIC, agreed that the two first combinations are much better than the other ones in terms of parameters and basic statistics.

a.) b.)

c.) d.)

e.)

Figure 8. Graphical results for the best combinations in ST within ±40% error. a.) Elseth and Danish et al. b.) Elseth and Drew et al. c.) Arirachakaran and Churchill d.) Arirachakaran and Morrison e.) Arirachakaran

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The graphical results presented, confirm the underestimation of the pressure gradient in all cases agreeing to the statistical results presented. In addition, a very similar behavior between the two first options can be seen. The two friction factor correlations were developed for either turbulent or laminar regime, although the one developed by Drew et al. (1936) is not an explicit equation as the one developed by Danish et al. (2011). This made the use of the first correlation slightly more complicated than the second one. It is important to mention, that this pattern is not characterized by a higher-pressure gradient, which affects the calculation of a few combinations due to the fact, that some friction factor correlations were developed specifically for turbulent flow, i.e. higher Reynolds numbers. In addition, this pattern was defined as with a smooth interface which agreed with the assumptions made by the Elseth (2011) model. The error encountered for the first combination, is still significant, but it represents a good first approach to the calculation of pressure gradient for stratified flow regimes.

Stratified flow with mixture at the interface (ST&MI)

For this pattern 505 experimental data points, 44 possible combinations were evaluated using the two models for stratified flow. The summary of the results of the best five combinations can be seen in Table 7 and Figure 9.

Table 7.Statistical results for ST&MI.

Pressure Gradient Model Friction Factor Correlation E1 (%) E2 (%) E3 (%) E4 (Pa/m) E5 (Pa/m) E6 (Pa/m) FRP

*10-4 R2 AIC

Elseth (2001) (Eq. 6)

Drew et al. (1936) (Eq. 32)

-2.88 57.29 1279.57 -2209.68 2330.42 80459.71 5.85 0.10 11980.34

Arirachakaran (1989) (Eq. 1) Brkic b (2011) (Eq. 83)

-59.53 65.16 555.50 -2357.88 2405.77 80236.09 6.92 0.05 11990.03

Arirachakaran (1989) (Eq. 1) Brkic a (2011) (Eq. 81)

-61.30 66.06 530.60 -2372.44 2413.58 80406.89 7.00 0.05 11991.65

Arirachakaran (1989) (Eq. 1) Avci & Karagoz (2009)

(Eq. 79)

-59.61 66.54 568.51 -2348.48 2421.15 80332.14 6.96 0.03 11994.55

Arirachakaran (1989) (Eq. 1) Wood (1966) (Eq. 37)

-58.84 67.64 603.91 -2336.62 2403.74 79512.84 6.86 0.07 11982.31

In this case, the average error according to the results lies between 57 and 68%. The pressure gradient values are being underpredicted in all combinations. It is found that the best combination uses Elseth (2001) model, based on momentum balance equations, while the other four use the Arirachakaran (1989) model. The characteristic of this pattern, which can have a possible mixing at the interface complicates the calculation taking into account that both models assumed a smooth flat interface and take it as a straight line for the geometry parameters needed. Although the first model gives a lower value of average error, it gives a very high standard deviation which can significantly affect the pressure gradient results.

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However, this combination is classified by the FRP, R-squared and the AIC as the best one in terms of parameters and precision.

All the friction factor correlations obtained for the Arirachakaran (1989) combinations are explicit simple equations, which make them none so complex to use. However, the Drew et al. (1936) correlation combined with the Elseth (2001) model gives better results. This correlation is not explicit but with the pressure gradient model used, is found to be the best one in terms of parameters.

a.) b.)

c.) d.)

e.)

Figure 9. Graphical results for the best combinations in ST&MI within ±50% error. a.) Elseth and Drew et al. b.) Arirachakaran and Brkic b c.) Arirachakaran and Brkic a d.) Arirachakaran and Avci & Karagoz e.)

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The graphical results presented, confirm the underestimation of the pressure gradient in all cases agreeing to the statistical results presented, but the first combination shows a lower tendency to underestimate the value compared to the others. This pattern is characterized by lower pressure gradient values but already higher than those for stratified flow regime. It is clearly seen, that the mixing at the interface which is not being taken into account in the models affects the results. As a first approach a good result is obtained, however the evaluation of dispersed flow models combined with stratified ones could be a reasonable next step.

Dispersion of oil in water and water (Do/w & w)

In this case, for the 84 experimental data points 777 combinations were found to be valid for this pattern; the five best combinations according to statistical parameters are summarized in Table 8 and Figure 10.

Table 8.Statistical results for Do/w & w.

Pressure Gradient Model Mixture Viscosity Correlation Friction Factor Correlation E1 (%) E2 (%) E3 (%) E4 (Pa/m) E5 (Pa/m) E6 (Pa/m) FRP

*10-5 R2 AIC

Brauner (2002) (Eq. 3) Taylor (1932) (Eq. 94) Drew & Generaux (1936) (Eq. 32)

-5.27 25.38 234.06 -256.97 323.83 4197.93 2.09 0.024 1506.66

Brauner (2002) (Eq. 3) Barnea & Mizrahi (1975) (Eq. 110) Drew & Generaux (1936) (Eq. 37)

-6.66 25.38 231.64 -260.73 324.88 4195.63 2.16 0.024 1506.86

Brauner (2002) (Eq. 3) Einstein (1906) (Eq. 93) Drew & Generaux (1936) (Eq. 37)

-4.80 25.50 235.70 -255.60 324.19 4200.95 2.08 0.024 1506.65

Brauner (2002) (Eq. 3)

Yaron & Gal-Or (1972) (Eq. 107) Drew & Generaux (1936) (Eq. 37)

-4.37 25.60 237.56 -255.14 325.07 4208.29 2.07 0.024 1506.79

Brauner (2002) (Eq. 3) Pal 1 (2001) (Eq. 114) Drew & Generaux (1936) (Eq. 37)

-4.38 25.71 237.16 -252.64 324.12 4197.36 2.06 0.023 1506.48

Based on the statistical results, the best five combinations obtain a very good result with an average error between 25 and 26%. It can be seen that all the combinations tend to underestimate the pressure gradient; although, all have the same R-squared and a similar value for the AIC and FRP criterions.

For all combinations, the Brauner (2002) model developed for dispersed flow is the best option in this case; this taking the oil as the dispersed phase in water. In the case of the friction factor correlations, the Drew & Generaux (1936) one is found to be the best one among all the other ones. This correlation is an implicit and simple equation for the calculation of the friction factor, and it happens to be very similar to the ones proposed by Prandtl (1935) and Colebrook (1939), which are commonly used. In the case of the mixture viscosity calculation, simple equations as the ones by Taylor (1932) and Einstein (1906) are found in the best five

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combinations. Even if the other ones tend to be slightly more complex to solve they also show a good result combining them with the pressure gradient model and respective friction factor correlation.

a.) b.)

c.) d.)

e.)

Figure 10. Graphical results for the best combinations in Do/w & w within ±25% error. a.) Brauner, Taylor and Drew et al. b.) Brauner, Barnea et al. and Drew et al. c.) Brauner, Einstein and Drew et al. d.) Brauner,

Yaron et al. and Drew et al. e.) Brauner, Pal and Drew et al.

In the graphical results, is clearly seen that for lower pressure gradient values the fit of the combination of model and correlations is better. The five combinations evaluated, tend to have the same prediction with some points with high values of pressure gradient, which could be a mistake during data classification. Is interesting how this pattern can be modeled as an

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oil in water dispersion, since no specific model has being developed for it and it is not a total dispersion one phase in the continuum of the other. However, the fact that is a dispersion is being reflected in the calculation of mixture properties, which takes into account the mixture of the continuous and dispersed phase, but also the volume fraction of the dispersed phase. A really good approach of the values is obtained, but it is also important, to note that this is the dispersed flow pattern with less number of data points, which affect the statistical results. Dispersion of oil in water (Do/w)

For this pattern, the total number of valid combinations was of 765 for the 557 experimental data points. The results of the best five combinations according to statistic criteria are presented in Table 9 and Figure 11.

Table 9.Statistical results for Do/w.

*10-9 R2 AIC

Brauner (2002) (Eq. 3) Eiler (1962) (Eq. 103)

Danish et al (2011) (Eq. 84)

4.97 27.29 656.78 -117.61 618.62 15126.10 5.31 0.65 11209.42

Brauner (2002) (Eq. 3) Taylor (1932) (Eq. 94)

3.94 27.94 673.58 -131.44 633.60 15419.61 5.67 0.64 11233.60

Brauner (2002) (Eq. 3) Einstein (1906) (Eq. 93)

5.44 28.69 692.79 -93.19 646.10 15582.00 6.31 0.63 11248.15

Brauner (2002) (Eq. 3) Dougherty & Krieger (1959) (Eq. 102) Drew & Generaux (1936) (Eq. 37)

-13.29 28.77 613.27 -576.39 763.63 18298.74 1.53 0.60 11420.85

Brauner (2002) (Eq. 3) Pal 1 (2001) (Eq. 114)

8.32 30.15 733.68 -31.96 661.02 15705.09 7.57 0.63 11264.76

Based on the statistical parameters results the best five combinations of correlations have average errors lower than 30%. Fourth combinations out of five, tend to overestimate the pressure gradient similarly. In general, a similar standard deviation, FRP values, R-squared and AIC values are encountered.

According to the results, there is no doubt that the best pressure gradient model for this pattern is the one proposed by Brauner (2002), which was specially developed for this pattern and was applied using oil as the dispersed phase and water as the continuous phase. For the friction factor correlation, is Danish et al. (2011) explicit equation the best in four cases. This result is consistent with the fact that as this correlation was proposed it was found that it works for all practical ranges, in laminar and turbulent flow [55], this is not a characteristic of all the friction factor correlations studied. On the other hand, the implicit correlation from Drew et al. (1936) appears again within the best results. In terms of mixture viscosity correlations, the ones obtained to be the best in this case, are characterized by their simplicity in calculation; again, Taylor (1962), Eiler (1962) and Einstein (1906), correlations obtained the best results calculating this mixture property.

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a.) b.)

c.) d.)

e.)

Figure 11. Graphical results for the best combinations in Do/w within ±25% error. a.) Brauner, Eiler and Danish et al. b.) Brauner, Taylor and Danish et al. c.) Brauner, Einstein and Danish et al. d.) Brauner,

Dougherty & Krieger and Drew et al. e.) Brauner, Pal and Danish et al.

As it can be seen in the graphical results, the fourth combination has a tendency to underestimate the value, which agrees with the statistical result. However, in the other four cases the overprediction is not evident since this pattern has a significant amount of data points. The results were expected since the Brauner (2002) model was specially developed for this pattern and mixture viscosity correlations can be clearly defined with oil as the dispersed phase and water as the continuous one.

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Dispersion of water in oil (Dw/o)

In the case of this pattern, for the 173 experimental data points, 774 possible combinations were encountered. The statistical and graphical results of the best five combinations are presented in Table 10 and Figure 12.

Table 10.Statistical results for Dw/o.

*10-5 R2 AIC

Brauner (2002) (Eq. 3) Dougherty & Krieger (1959) (Eq. 102)

29.18 46.23 667.37 -83.27 951.69 12913.46 0.10 0.11 3631.67

Brauner (2002) (Eq. 3) Barnea & Mizrahi (1975) (Eq. 110)

34.93 50.37 741.47 58.07 1004.81 13216.68 0.53 0.11 3635.96

43.73 54.08 701.55 180.01 1022.87 12965.33 1.30 0.10 3638.83

49.45 59.04 776.95 321.47 112.21 14056.60 2.70 0.10 3646.52

50.26 60.64 858.52 296.30 1099.22 14356.68 2.72 0.11 3641.69

Taking into account the statistical results presented, almost the same mixture viscosity and friction factor correlations as in the previous pattern are found. However, the statistical results are different, obtaining an error between 56 and 61%. On one hand, the five combinations showed a similar deviation and a tendency to overestimate the value. On the other hand, the R-squared does not really vary between the options. In terms of the AIC criterion and the FRP, the best combination is found to be the first one.

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a.) b.)

c.) d.)

e.)

Figure 12. Graphical results for the best combinations in Dw/o within ±40% error. a.) Brauner, Dougherty et al and Danish et al. b.) Brauner, Barnea et al and Danish et al. c.) Brauner, Dougherty et al. and Drew et al.

d.) Brauner, Barnea et al and Drew et al. e.) Brauner, Taylor and Danish et al.

The graphical results reflect the statistical results discussed, the overestimation and error. It is interesting how almost the same combinations were encountered for the two dispersion patterns; however, the results are clearly better when the dispersed phase is water and not the oil. This could be attributed to the mixture viscosity correlations, which in this case work better when the continuous phase has a lower viscosity than the dispersed one. The prediction of pressure gradient for this pattern is not as expected, but it is the best approach until now.

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For the reason of phase inversion, this pressure gradient model should be studied and improved.

Dispersion of water in oil and oil in water (Dw/o & Do/w)

In this case, 669 combinations were found possible for the 653 experimental data points of this pattern. The best five combinations are presented in Table 11and Figure 13.

Table 11.Statistical results for Do/w & Dw/o.

*10-8 R2 AIC

Brauner (2002) (Eq. 3) Dougherty & Krieger (1959) (Eq. 102)

5.82 48.43 1264.17 -503.87 937.74 28919.17 0.48 0.03 14693.82

18.04 51.97 1359.26 -404.11 946.45 29003.22 0.58 0.02 14696.20

Brauner (2002) (Eq. 3) Barnea & Mizrahi (1975) (Eq. 110)

66.17 90.64 2423.14 71.14 1240.14 31369.05 1.81 0.01 14761.52

73.12 95.06 2353.74 108.22 1251.10 31120.66 1.97 0.01 14752.50

74.74 95.46 2457.39 170.33 1279.98 31859.02 2.14 0.02 14755.82

Since this pattern does not really have a specific model for itself, but it is clear that it is a dispersion, the Brauner (2002) model was evaluated. In this case, the best five combinations were encountered using the model with the dispersed phase as water and the continuous one as oil. The average error for the two best combinations is between 48 and 52%; while the other combinations have a significant error above 90%. According to the results, there is a tendency to overpredict the values. A really high standard deviation value is found, however, the first and the second combinations are really similar according to the FRP, R-squared and AIC values. Again, the two friction correlations found to be the best ones, are the ones developed by Danish et al. (2011) and Drew & Generaux (1936).

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a.) b.)

c.) d.)

e.)

Figure 13. Graphical results for the best combinations in Do/w & Dw/o within ±45% error. a.) Brauner, Dougherty et al. and Danish et al. b.) Brauner, Dougherty et al. and Drew et al. c.) Brauner, Barnea et al. and

Danish et al. d.) Brauner, Barnea et al. and Drew et al. e.) Brauner, Taylor and Danish et al.

In the plots, the overestimation can be easily identified. It is interesting how the best result is found as a dispersion of water in oil, however, this pattern need a model for itself since it is not entirely true that the regime is oil dominant. The results obtained are a good first approach, but a better model or way to include both dispersions in the calculations is needed. Core-annular flow (CAF)

For this pattern, 38 possible combinations were evaluated for the 71 experimental data points. The statistical and graphical results of the five best combinations can be found in Table 12 and Figure 14.

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26 Table 12.Statistical results for CAF.

Pressure Gradient Model

Mixture Viscosity Correlation

Friction Factor Correlation

E1 (%)

E2 (%)

E3 (%)

E4 (Pa/m)

E5 (Pa/m)

E6 (Pa/m)

FRP *10-8

R2

*10-3 AIC

Brauner CAF (2002) (Eq. 4)

-

Morrison (2013) (Eq. 90)

25.94 71.33 580.37 -1076.39 2340.00 23239.33 0.96 4.48 1540.83

-

Blausius (1913) (Eq. 29)

26.29 74.72 597.30 -1065.52 2403.25 24212.35 1.06 2.84 1541.37

-

Hagen & Poiseuille (1840) (Eq. 28)

-76.11 76.11 107.98 -2677.84 2677.84 20294.91 2.26 1.34 1545.25

-

Wood (1966) (Eq. 37)

30.74 78.07 654.83 -1091.99 2325.55 23510.65 1.03 3.18 1540.27

-

Brkic a (2011) (Eq. 81)

35.15 82.16 673.32 -1009.87 2412.27 24042.41 1.10 3.39 1540.91

It is evident, that the amount of data for this pattern is significantly lower as for the other patterns, the average error of the evaluated combinations was between 70 and 90%. This results were obtained taking the oil phase as the core and the water phase as the annulus, which is the most common result due to viscosities.

Four out of the five combinations overestimate the pressure gradient value, while the others underestimate it. However, the third one obtained the lowest standard deviation. The R-squared is really low for all the combinations which shows that it does not really fit the data. Even though, the FRP and AIC criterion show a similar result for all combinations. The friction factor correlations obtained are simple explicit equations, such as the one by Wood, Blausius and Hagen & Poiseuille, which were almost the first approaches for the prediction of the friction factors.

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a.) b.)

c.) d.)

e.)

Figure 14. Graphical results for the best combinations in CAF within ±70% error. a.) Brauner CAF and Morrison. b.) Brauner CAF and Blausius. c.) Brauner CAF and Hagen & Poiseuille. d.) Brauner CAF and

Wood. e.) Brauner CAF and Brkic a.

The graphical results agree to the statistical results already discussed. The underestimation of the value by the third combination is evident. It is important to mention, that the probabilistic tool used to classify the data does not have this pattern include; therefore, the classification was made taking into account the pattern reported by the authors which can affect the results encountered. It is necessary to extend the database for this pattern in order to make a stronger analysis.

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5.2. Phase Inversion

For this analysis, just the data of a few authors was used. The data was chosen, according to viscosities, densities, number of data points and wide range of mixture viscosity data. All this in order, to analyze how the phase inversion prediction changes, depending on fluid properties and mixture velocities. The other determining criteria, was that the author reported the experimental value or range of phase inversion obtained during their experiments. All the correlations, showed in Table 2 were used. In the case of the last method, the one proposed by Ngan (2010), the 21 mixture viscosity correlations listed on Table A.1.2 in Appendix A, were evaluated, and just the five best results among them are showed with the other five first correlations results. In the following sections, the results for the data of each author is presented.

5.2.1. Al-Yaari (2009)

For this experimental set up, the author used water, with a density of 998 kg/m3_{and a}

viscosity of 0.001 Pa*s, and Kerosene (Safra D60) with a density of 780 kg/m3 and a viscosity of 0.0015 Pa*s [27].

Figure 15. Predicted phase inversion points with each correlation for Al-Yaari (2009) data. The lines represent the experimental range for phase inversion reported by the author.

According to the results obtained, just two of the models obtained a phase inversion point in the experimental range reported. The other ones, tend to overestimate the point. It is interesting how simple mixture viscosity models, as the ones by Taylor (1932) and Einstein (1906), are in this case the best models predicting the phase inversion point. It is also evident, that the method proposed by Ngan (2010), used with the five best chosen correlations in this case, give better results than the other literature correlations, as concluded by Ngan (2010) [5]. The third best result was obtained using the Pal (2001) correlation, which according to Ngan (2010) [5] it usually obtains a really close value to the range for oils around this viscosity.

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Figure 16. Calculated pressure gradient for different mixture velocities for Al-Yaari (2009) data. The lines represent the experimental range for phase inversion reported by the author.

Since the study of the relation between pressure gradient and phase inversion is also of interest, the results of calculated pressure gradient versus water fraction were also plotted. As it can be seen, pressure is higher with higher mixture velocity, as expected. However, there is a different point for each of them, for the reason of the different degree of mixing for each mixture velocity [27], where a significant change on pressure gradient can be seen. For lower mixture velocities, this point tends to be at higher values of water fraction, as it can be seen for the mixture velocities 1.5 and 2 m/s. This result does not really agree with the experimental data obtained by Al-Yaari (2009) [27], because the change of pressure gradient for this velocities, where the pattern is usually stratified or stratified with mixing at the interface, is not from that magnitude. However, for the highest velocities, 3 and 3.5 m/s, the greatest change of pressure gradient measurement occur at a water fraction of 0.3 approximately. According to Al-Yaari (2009) [27], the phase inversion is assumed to happen at or after a peak of the pressure, which in this case does not exist, since pressure gradient increases until almost a water fraction of 0.6. For the mixture viscosities studied in this case, a clear relation between calculated pressure gradient and phase inversion cannot be established.

5.2.2. Elseth (2001)

In this case, the experimental data was obtained using water with density of 1000 kg/m3 and viscosity of 0.001 Pa*s, and oil (Exxsol D-60), with a density of 790 kg/m3 and a viscosity of 0.016 Pa*s [16]. Very similar properties with the fluids used by Al-Yaari (2009).

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Figure 17. Predicted phase inversion points with each correlation for Elseth (2001) data. The lines represent the experimental range for phase inversion reported by the author.

In this case, the best five mixture viscosities correlations found to be the best ones for the prediction of phase inversion point through Ngan (2010) method, are again the ones obtained before for Al-Yaari (2009). The simplest ones, from Taylor (1932) and Einstein (1906), obtain again a predicted value within the experimental range and the other three close values. The literature correlations, overestimate again the phase inversion point and obtain close values to the upper limit of the experimental range reported.

Figure 18. Calculated pressure gradient for different mixture velocities for Elseth (2001) data. The lines represent the experimental range for phase inversion reported by the author.

As showed, in Figure 18, the pressure gradient increases as mixture velocity increases. For the velocities lower than 1 m/s there is not a specific peak of pressure that can be identified. Although, an increase of pressure is found at high water fractions. For velocities higher than 1 m/s the author reported a water fraction of 0.27 for phase inversion. As it can be seen, for velocities 1.34, 1.5 and 2 m/s, the peak of pressure gradient is found in a higher water fraction, which agrees some results of calculated points through correlations. While for 2.5 and 2.67 m/s the peak is found within the experimental range reported and very close to two best results of correlations. According to these results, it can be concluded that for intermediate

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mixture velocities, the pressure gradient calculation tend to be more accurate and tend to meet the predictions of phase inversion points better. Also, a relation described by Abubakar (2015) [26] is also seen, which affirms that as mixture velocity increases the phase inversion point decreases.

5.2.3. Nadler (1997)

This author did his experiments using water and mineral white oil (Shell Ondina 17) varying the temperature of the experiments between 18 and 30°C, obtaining different fluid properties, showed and classified as follows:

 Low viscosity oil: density 846.26 kg/m3_{and viscosity 0.022 Pa*s. Water: density 995.6}

kg/m3_{and viscosity 0.00078 Pa*s.}

Figure 19. Predicted phase inversion points with each correlation for Nadler (1997) low viscosity oil data. The lines represent the experimental range for phase inversion reported by the author.

In this case, only four of the predicted points fall out of the experimental range, which tend to slightly overestimate the point. However, really good results were obtained with mixture viscosities correlations and in the case of literature correlations, with the one proposed by Yeh et al. (1964) and Frechou (1986). It is important to mention, then even though this is the low viscosity oil used in Nadler (1977) experiments, this oil has already a higher viscosity comparing it with all the other studies taken into account; which can be reflected in the way that different mixture viscosity correlations are obtained. The previously obtained correlations, are now replaced by slightly more complex and complete correlations as the ones by Vand (1948), Brinkman & Roscoe (1952), Furuse (1972) and Phan Thien & Pham (1997).

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Figure 20. Calculated pressure gradient for different mixture velocities for Nadler (1997) low viscosity oil data. The lines represent the experimental range for phase inversion reported by the author.

According to the pressure gradient versus water fraction plot showed, for the mixture velocity of 1.5 m/s, there is no peak of pressure gradient within the experimental range, however, a fall in the pressure gradient is presented as the water fraction increases. In the case of 0.3 and 0.6 m/s velocities, a peak in pressure can be seen when the water fraction is too low and also when it is too high. In the case of the 0.9 m/s velocity, the peak of pressure gradient is also seen for very low water fractions. These results do not agree with the experimental range of phase inversion identified by the author. In this case, almost for all mixture velocities a stratified pattern is encountered, which make the prediction of pressure gradient difficult and it can be reflected since peaks of pressure gradient calculations should be related to the phase inversion point of dispersed patterns.

 Medium viscosity oil: density 847.45 kg/m3 and viscosity 0.0267 Pa*s. Water: density 997 kg/m3 and viscosity 0.00089 Pa*s.

Figure 21. Predicted phase inversion points with each correlation for Nadler (1997) medium viscosity oil data. The lines represent the experimental range for phase inversion reported by the author.

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For this oil properties, all the ten predictions obtained a very good result. However, the same three literature correlations, by Arirachakaran et al. (1989), Nädler and Mewes (1997) and Brauner und Ullman (2002) do overestimate the value, while the one by Frechou (1986) tends to underestimate it. In addition, again the same mixture viscosity correlations obtained for the low viscosity oil of this study, are obtained with a good prediction within the experimental range.

Figure 22. Calculated pressure gradient for different mixture velocities for Nadler (1997) medium viscosity oil data. The lines represent the experimental range for phase inversion reported by the author.

Since oil viscosity increases, it is expected that the calculated pressure gradient increases too. This result is obtained; however the peak of pressure gradient for high mixture viscosity is again near high water fractions. In the case of the lowest velocities, 0.3 and 0.6 m/s, a peak of pressure gradient is found also towards highest water fractions. The pressure gradient results, do not meet the phase inversion predictions nor experimental values.

 High viscosity oil: density 848.81 kg/m3 and viscosity 0.035 Pa*s. Water: density 998.6 kg/m3 and viscosity 0.001 Pa*s.

Figure 23. Predicted phase inversion points with each correlation for Nadler (1997) high viscosity oil data. The lines represent the experimental range reported by the author.

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For the high viscosity oil of this data, the same result in terms of literature correlations and mixture viscosity correlations was encountered. However, it can be seen that as oil viscosity increases the predictions by the mixture viscosity correlations, the Yeh et al. (1964) and Frechou (1966) correlations, are moved slightly to the left, to lower water fractions. According to this result, as the viscosity increases the predicted phase inversion point decreases.

Figure 24. Calculated pressure gradient for different mixture velocities for Nadler (1997) high viscosity oil data. The lines represent the experimental range reported by the author.

In this case, the pressure gradient values do not show a clear relation with phase inversion ones. However, the calculated pressure gradients values do increase with the increase of the oil viscosity.

5.2.4. Soleimani (1999)

In this case for the experiments, water with a density of 997 kg/m3_{and a viscosity of 0.001}

Pa*s and an oil (Exxsol D80), with density of 801 kg/m3 and a viscosity of 0.0016 Pa*s were used [42].

Figure 25. Predicted phase inversion points with each correlation for Soleimani (1999) data. The lines represent the experimental range reported by the author.

For this case, all the best ten correlations predict a really close value of phase inversion point. In the case of mixture viscosity correlations, the correlation of Guth & Simha (1936),

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Vermuelen et al. (1955) and Pal (2001) appear again. In this case the correlation that best predict the point is the one developed by Pal (2001). The mixture viscosity method tends to have better results than the other literature correlations, however, as said before, the results are very close. The phase inversion experimental range reported is between 0.3 and 0.4, and almost all the models, overestimate the point near to the upper limit of the range.

Figure 26. Calculated pressure gradient for different mixture velocities for Soleimani (1999) data. The lines represent the experimental range reported by the author.

In this case the data was taken experimentally for three different mixture velocities. For this study the oil properties, are very similar to the ones for the oil in the case of Al-Yaari (2009) and Elseth (2001). Again, the intermediate mixture velocity does have a peak of pressure gradient within the experimental range while the other ones, in the case of the lowest mixture velocity tend to be for a higher water fraction and in the case of the highest mixture velocity for a lower water fraction. The peak for the velocity of 2.5 m/s is the one closest to the correlations predictions. Again, a relation between pressure gradient and phase inversion is seen just for intermediate velocities. The fact that phase inversion point decreases as mixture velocity increases can be seen in this case too.

6. Conclusions and future work Pressure gradient

A statistical and physical analysis was made for each combination possible of: pressure gradient models, mixture viscosity correlations and friction factor models. As a result, the best combination for calculating pressure gradient for liquid-liquid flow in each pattern is:

 For ST: Elseth model and Danish et al. friction factor correlation with an average error of 42.8%.

 For ST&MI: Elseth model and Drew et al. friction factor correlation with an average error of 57.2%.

 For Do/w & w: Brauner model, Taylor mixture viscosity correlation and Drew et al. friction factor correlation with an average error of 25.3%.

 For Do/w: Brauner model, Eiler mixture viscosity correlation and Danish et al. friction factor correlation with an average error of 27.2%.