1

Determination of the best correlation for pressure drop prediction in horizontal pipelines

with two-phase Oil-Water flow

Nicolás Rodríguez Díaz. Laura Prieto Saavedra.

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

General Objective: To choose the best performing set of correlations that predict the pressure

loss in Oil-Water pipeline flow

Specific Objectives:

To identify phenomena that might affect the effectiveness of pressure loss models in an Oil-

Water pipeline flow and gather models that predict pressure loss for the different

phenomena identified previously.

To create a Matlab code that predicts the pressure loss in an Oil-Water pipeline for

Stratified, Dispersed and Core-Annular flow pattern models and that validates that the

returned value of the solver used is indeed a solution in addition to catching possible errors.

To gather the most friction factor and viscosity correlations as possible and successfully

integrate them to the code.

To create a Matlab code that calculates statistical data of the aggregate performance of the

correlation sets on a given dataset, compares the statistical data and sorts the correlations by

its performance.

Integrate an input excel dataset with 6388 data points and the ability to export the resulting

statistical data to an organized Excel datasheet.

Abstract:

A program that indicates the most useful correlations to predict the frictional pressure loss in an

Oil-Water pipeline for a set of 6388 data points calculated using OLGA Multiphase Toolkit

2014.3 was developed. 24 viscosity emulsion correlations and 37 friction factor correlations

where gathered, and their performance in pressure loss prediction models for Stratified, Core-

Annular and Dispersed flow patterns were compared under three different methods. It was shown

that the best performing correlations were the Shaikh-Massan-Wagan correlation for Dispersed

and Core-Annular flow, several smooth pipe correlations such as the Goudar-Sonnad, Li-Seem-

Li and Prandtl correlations for Stratified flow, and simple viscosity correlations for Dispersed

flow such as Taylor and Einstein’s correlation. It is suggested that a probabilistic approach is

developed to quantify the odds that a given predictor will yield the best statistical performance

for the predicted pressure loss for a new data point. In addition, it is suggested that the program

is extended for the calculation of pressure gradient data for non-horizontal pipelines, and that non

OLGA Multiphase Toolkit 2014.3 calculated data is used.

Keywords: Biphasic, pressure loss, Oil, Water, Dispersed, Stratified, Annular, Flow, pattern,

friction, viscosity, correlations

1. Introduction

The design of Oil-Water pipelines involves pressure loss calculations. This pressure loss is

calculated using viscosity and friction factor correlations. However, there is a glut of these

correlations from which pipeline designers need to choose from. Therefore, a comparison between

these correlations is made in which the most useful predictor of pressure loss in an Oil-Water

pipeline flow is chosen.

2

During pipeline flow, Oil-Water systems distribute themselves into geometrical arrangements.

These arrangements are called flow patterns and they are known to affect the pipeline’s pressure

loss [1]. For this reason, a most useful correlation will be selected for each flow pattern.

This selection depends on the performance of the calculated pressure loss to the corresponding

experimental data from the literature put together by the University of Tulsa and organized by

Universidad de los Andes. This performance is evaluated using statistical criteria such as the

Akaike Criterion and statistical descriptors.

2. Literature review

The pressure loss in an Oil-Water pipeline is related to the shear between the fluid and the pipe

wall and the geometric arrangement of the phases [2,3]. These arrangements, also called flow

patterns, imply unique interactions between variables such as phase flow rates, pipe diameters,

inclination angles and physical properties such as the phase’s density, viscosity and interfacial

tension [3]. Therefore, the pressure loss model for each flow pattern is described differently [1].

In this work, Stratified Flow/Mixed Flow (STF/MF), Dispersed phase (DF) and Core-annular

flow (CAF) pressure loss models are reviewed (Table A.1 in the annex). This flow patterns are

named under Arirachakaran et al. liquid-liquid heterogeneous flow classification and can be seen

in (Figure 1) [4].

Figure 1 Selected flow patterns for horizontal pipes under Arirachakaran et al’s classification

[4].

The reviewed pressure loss models depend on friction factor correlations (Table A.2 in the

annex) or effective viscosity correlations (Table A.3 in the annex) as well. However, there is no consensus on which correlations should be used to calculate these properties. Therefore, their

performance is evaluated in this paper.

Each effective viscosity correlation in DF is paired with each friction factor correlation available.

The performance of each individual pairing, which is called a ‘correlation set’, will be evaluated,

thus allowing to compare the compounded predicting error of all possible combinations. This

3

assures the knowledge of how the noise propagates when both correlations are used together in

contrast to choosing the best performing correlation individually [5].

The performance of the turbulence friction factor in STF and the CAF are also evaluated in this

paper. In a turbulent regime, the Reynolds number’s influence on the friction factor decreases

while the surface roughness’s effect on the friction factor increases [6]. However, both theories

ignore the surface roughness in pipeline flow. Therefore, better results of both pressure loss

predictions might be found if a correlation that takes into account the pipe’s surface roughness is

used.

3. Methodology

Three stages of methodology were identified in this paper: data recompilation, data discrimination

and data analysis. In the following section each of this stages will be explained.

3.1. The database

The sources used in this paper (Table 1) were selected from the experimental database from the

literature put together by the University of Tulsa and organized by Universidad de los Andes.

Only the sources which reported either the flow patterns or the pressure loss for horizontal

pipelines experiments where included.

The pressure gradient and water input fraction data was manually extracted from the sources using

the Get Data Graph Digitizer application and then organized. Superficial velocities and mixture

velocities were also calculated per the requirements of the models. The following table resumes

the data gathered and organized:

Table 1. Experimental data

Author

#

Data

points

Fluids

ρ [kg/m3] µ [Pa.s]

Water

Fraction

Mixture

Velocity

[m/s]

Pipe

material

Pipe Diameter

[m]

Water Oil Water Oil

Abubakar

(2015) [7] 127

Water-

Mineral Oil

(Shell

Tellus S2

V15)

997 872 0.00089 0.024 0.1-0.9 0.1-1.5 Acrylic 0.0306

Alkaya

(2000) [8] 66

Water-

Refined

Mineral Oil

(Crystex

Oil AF-M)

994 848 0.00072 0.0129 0.01-

0.99 0.05-2.6 Acrylic 0.0508

Al-wahaibi

(2013) [9] 151

Water-

Mineral Oil 998 875 0.00089 0.012

0.06-

0.96

0.17-

3.38 Acrylic 0.019

Al-Yaari

(2009) [10] 61

Water-

Kerosene

(SAFRA

D60)

998 780 0.00089 0.00157 0.35-

0.93 0.6-2.9 Acrylic 0.025

Angeli

(2000) [11] 68

Tap Water-

Kerosene

(EXXOL

D80)

1000 801 0.00089 0.0016 0.09-

0.83

0.21-

1.22

Acrylic /

Stainless

Steel

0.024/0.0243

Atmaca

(2007) [12] 50

Water-

Refined

Mineral Oil

992-

996

836-

843 0.00089 0.018

0.01-

0.99

0.04-

2.49 Acrylic 0.0508

4

(Tulco

Tech 80)

Ayello

(2008) [13] 105

Water-

Crude Oil

(API 39)

998.2 825-

830 0.001

0.002-

0.004

0.006-

0.2

0-57-

2.70

Carbon

Steel 0.1

Bannwart

(2004) [14] 390

Water-

Heavy

Crude Oil

998 925.5 0.00089 0.488 0.01-

0.94

0.05-

2.93 Glass 0.0284

Cai

(2012) [15] 82

Deionized

water -Oil

(LVT200

oil)

998 825 0.000985 0.002 0.01-

0.201

0.51-

2.02

Stainless

Steel 0.1016

De Castro

(2011) [16] 44 Water-Oil 998 828 0.00089 0.3

0.26-

0.98

0.07-

1.33 Glass 0.026

Chakrabarti

(2006) [17] 574

Water-

Kerosene 1000 787 0.00084 0.0012

0.015-

0.98

0.06-

2.70 Acrylic 0.025

Dasari

(2013) [18] 523

Water-

Lube Oil 1000 889 0.00089 0.107

0.12-

0.88

0.11-

2.09 Acrylic 0.025

Elseth

(2001) [19] 84

Water-Oil

(Exxsol D-

60)

1000 790 0.001 0.0016 0.09-0.9 0.02-

2.10 Acrylic 0.056

Fairuzov

(2000) [20] 62

Water-

Crude Oil

(Isthmus)

1000 851 0.00089 0.0059 0.02-

0.49

0.02-

2.10

Stainless

Steel 0.3635

Ismail

(2015) [21] 72

synthethic

formation

Water-

Light

Malaysian

waxy crude

oil

1000 818 0.00089 0.00175 0.1-0.9 0.1-0.8 Stainless

Steel 0.0508

Kumara

(2009) [22] 83

Water-Oil

(EXXSOL

D60)

996 790 0.00089 0.00164 0.02-

0.98

0.27-

2.47

Stainless

Steel 0.056

Li (2010)

[23] 24

Water-

Limpidity

Mineral Oil

1000 850 0.00089 0.05 0.08-

0.96

0.11-

9.66 Plexiglass 0.05

Liu (2003)

[24] 411

Tap Water-

Mechanical

Oil N°46

1000 983 0.00089 0.131 0.019-

0.89

0.04-

2.91

Acrylic

/Stainless

Steel

0.04

Liu (2008)

[25] 124

Water-

Diesel Oil 998 838 0.00097 0.0034 0.06-0.9

0.11-

1.31

Stainless

Steel 0.026

Lovick

(2004) [26] 49

Water-Oil

(EXXSOL

D140)

997 828 0.00089 0.006 0.1-0.9 0.7-3.5 Stainless

Steel 0.038

Mukhaimer

(2015) [27] 106

Water/Salty

water-

Kerosene

(Safrasol

80)

998 781 0.00098 0.0018 0.03-

0.95

0.25-

2.23 PVC 0.0225

Nadler

(1997) [28] 71

Water-

Mineral Oil

(Shell

Ondina 17)

998 988 0.00089 0.035 0.05-

0.98

0.08-

1.63

Acrylic

(perspex) 0.059

Oddie

(2003) [29] 5

Tap Water-

Kerosene 1000 810 0.00089 0.0015 0.1-0.9

0.06-

1.68

Acrylic

(perspex) 0.152

Oglesby

(1979) [30] 238 Water-Oil

991-

1000

853-

873

0.0008-

0.001

0.02-

0.21

0.03-

0.93

0.88-

3.84

Commercial

Steel 0.0411

Pragya

(2014) [31] 89 Water-Oil 1000 889 0.00183 0.107

0.13-

0.84

0.14-

1.95 Acrylic 0.025

Rodriguez

(2005) [32] 38

Water-Oil

(Shell

Vitrea 10)

1060 830 0.0008 0.0075 0.006-

0.99

0.04-

3.62 Steel 0.0828

Rodriguez

(2011) [33] 26 Water-Oil 998 860 0.00089 0.1

0.62-

0.93 1.8-4 Glass 0.026

Simmons

(2001) [34] 18

Potassium

Carbonate

Solution

(25%)-

Kerosene

1166 797 0.0016 0.0018 0.07-

0.62

0.78-

3.41

Stainless

Steel 0.063

5

Soleimani

(1997) [35] 186

Tap Water-

Oil

(EXXSOL

D80)

997 801 0.00089 0.0016 0.06-0.9 0.25-

1.52 Steel 0.0254

Soleimani

(2000) [36] 100

Water-

Kerosene

(EXXOL

D80)

997 801 0.0009 0.0016 0.006-1 2.10-

3.02

Stainless

Steel 0.0243

Sotgia

(2008) [37] 794

Water-

Mineral Oil 1000 900 0.00089 0.9

0.04-

0.92

0.22-

3.58 Plexiglass 0.026

Souza

(2013) [38] 281

Water-

Heavy Oil 998 854 0.00089 0.329

0.08-

0.99

0.04-

3.59 Glass 0.026

Tan (2015)

[39] 54

Water-

Mineral

White Oil

998 841 0.00089 0.025 0.02-

0.96

0.11-

3.03 Acrylic 0.05

Trallero

(1995) [3] 227

Water-Oil

(Crystex

Af-M)

1033-

1037

880-

884

0.0009-

0.001

0.02-

0.03 0-0.99

0.01-

2.90 Acrylic 0.0501

Vielma

(2008) [40] 136

Water-

Refined

mineral Oil

995 852 0.00089 0.015 0.01-

0.96

0.05-

3.55 Acrylic 0.0508

Wang

(2010) [41] 141

Water-

Mineral Oil 995 854 0.000798 0.62

0.09-

0.98

0.09-

9.23

Stainless

Steel 0.025

Wegmann

(2006) [42] 191

Water-

Paraffin-oil

997-

998

818-

820 0.0043 0.0043

0.01-

0.96

0.22-

4.59 Glass 0.0056/0.007

Xu (2008)

[43] 24

Water-

White Oil 998 860 0.00089 0.052

0.06-

0.96

0.23-

2.27 Acrylic 0.05

Xu (2010)

[44] 83

Tap Water-

Diesel Oil 980 830 0.00089 0.003

0.05-

0.97

0.26-

2.07 Acrylic 0.02

Yao (2009)

[45] 62

Water-

Crude Oil 988 947 0.00035 0.0735

0.23-

0.78 0.19-1.2

Stainless

steel 0.0257

Yusuf

(2012) [46] 192

Water-

Mineral Oil 998 875 0.00089 0.012

0.05-

0.97

0.15-

3.72 Acrylic 0.025

Zhai

(2015) [47] 175

Tap water-

Industry

White Oil

(N° 15)

1000 845 0.00065 0.011 0.04-

0.95

0.21-

4.40 Acrylic 0.02

Total 6387

3.2. Data Discrimination and Validation

The data from the selected sources needs to be further discriminated. First, the usefulness of the

data is constrained by the model’s restrictions. For instance, only data with both, a turbulent and

laminar phase, should be selected for Core-Annular flow. The reason for these restrictions is that

only under this conditions are the friction factor correlations needed.

In addition, the current version of the database only displays data from sources that report flow

patterns. Thus, it excludes valuable sources that report solely experimental pressure gradients.

Accordingly, sources reporting pressure loss were gathered and added to the database in order to

increase the number of experimental data available.

As a consequence of the last step, some data does not report both flow patterns and pressure loss.

The data which missed pressure loss data was completed by using OLGA Multiphase Toolkit

2014.3’s synthetic data. However, it was decided not to use the 2193 data points from the 23 different sources that did not report flow patterns. This was decided since the revised Flow Pattern

Prediction Tools were unprecise in terms of the viscosity and holdup, which turned out to be of

importance for the accuracy of the pressure loss models used. Nevertheless, the data was gathered,

6

organized and made available for other projects, and other 6000+ synthetic data points were used

for the present project.

3.3. Data analysis

The analysis of the correlations involves the calculation of pressure loss data and their comparison

to experimental/synthetic reference values.

Matlab coding is used to calculate the pressure loss data. These predictions are then compared to

each other in order to determine which friction factor or effective viscosity correlation gives the

best Pressure Drop Predictions for each Flow Pattern.

Three methods of comparison will be used. The first one is of Ansari et al.‘s relative performance

factor, designed to compared models based on descriptive statistic criteria [48]:

𝐹𝑟𝑝 = (∑|𝐸𝑖| − |𝐸𝑖,𝑚𝑖𝑛|

|𝐸𝑖,𝑚𝑎𝑥| − |𝐸𝑖,𝑚𝑖𝑛|

8

𝑖=1

) [𝐸𝑞. 1]

Here, the, E1 is the average percent error, E2 is the average absolute percent error, E3 represents

the Standard Deviation of the Absolute Percent Error, E4 is the Root Mean Square Average

Percent Error, and E5 to E8 are the non-percentage equivalents of the first four. The minimum

and maximum possible values for Frp are 0 and 6, corresponding to the best and worst prediction

performance, respectively.

The second method used is the Akaike Information Criterion:

𝐴𝐼𝐶 = 𝑛 ∗ 𝑙 𝑛 (∑ ((ℎ𝑙)𝑝𝑟𝑒𝑑,𝑖−(ℎ𝑙)𝑒𝑥𝑝𝑒,𝑖)

2𝑛𝑖=1

𝑛) + 2 ∗ #𝑑𝑒𝑔𝑟𝑒𝑒𝑠 𝑜𝑓 𝑓𝑟𝑒𝑒𝑑𝑜𝑚 [Eq. 2]

The AIC was added to punish correlations that add parameters which fail to increase the

correlation’s accuracy as much as they increase its complexity. In addition, the Akaike Criterion

has been found to be the most effective and efficient approach to select a model compared to F-

Tests and BIC tests [49,50,51,52].

Furthermore, it was ensured that the AIC and Ansari et al.‘s & García et al.‘s proposed criteria

have equal weight in this method [48,53]. This assures that the correlations are accurate for new

data and not just their training sets [5].

Finally, the third method used is Ripley’s Range factor, given by the following equations:

𝑅 = 𝑒𝑥𝑝 (𝑡0.01,𝑛−1𝐸10√1 +1

𝑛 ) [𝐸𝑞. 3]

𝐸9 =1

𝑛∑𝑙𝑛 (

(ℎ𝑙)𝑝𝑟𝑒𝑑,𝑖

(ℎ𝑙)𝑒𝑥𝑝𝑒,𝑖)

𝑛

𝑖=1

[𝐸𝑞. 4]

𝐸10 = √1

𝑛 − 1∑ (𝑙𝑛(

(ℎ𝑙)𝑝𝑟𝑒𝑑,𝑖

(ℎ𝑙)𝑒𝑥𝑝𝑒,𝑖) − 𝐸9)

2𝑛

𝑖=1

[𝐸𝑞. 5]

7

This test is useful when a measure of the prediction level of confidence of a model is required

[53]. It is used in this paper to statistically support that the equation chosen by the first method

indeed is significantly useful.

4. The Code

The Matlab code reads and imports data rows from an external Excel file and stores them in global

variables. The program allows that the Excel file to store data points from different flow patterns

since it first classifies each imported row according to the flow pattern and calculates the pressure

loss of the system using the corresponding model. For each model, the method returns the

predicted pressure loss calculated through of the available correlations or correlation sets. The

amount of correlations is easily expandable or reducible since the correlations are set up in cell

arrays that can be manually deleted or augmented. The predicted pressure loss calculations are set

up on another file, where the pressure loss is also calculated with any of the three available

models.

Figure 2: Graphical representation of the data importing and pressure gradient calculation section

of the code

Next, the results are ratified using a data validation process. In this process, several solvers were

set up for each model, so that in the case that a solver returns an error the program would catch

the error try to solve it from another initial point, or with successively larger number of iterations.

In the case of the CAF model, since the data did not disclose which fluid flowed through the

annulus, both cases were solved and the one with the solution closest to the real value was chosen.

Figure 3: Graphical representation of the data validation section of the code

8

Afterwards, the statistical data for the performance of each correlation over the data points

belonging to the separate flow patterns was calculated. In the case of the AIC calculation, a

function was created where the degrees of freedom of the correlation were linked to its own name.

This way, the value is assigned automatically for whichever correlations are set up.

Subsequently, the correlations are compared to each other and then sorted from best to worst. The

Command window will display the five best performing correlations for each comparison method

with their relative ranking. Similarly, it will export the statistical data to Excel sheets and

accommodate it in a way similar to the tables presented in the Results section of this document.

Figure 4: Graphical representation of the data analysis section of the code

5. Results and discussion

The pressure loss predictions were evaluated for 37 friction factor correlations and 24 viscosity

correlations sorted by flow pattern. The following data were used: 675 Core-annular Flow data

points, 2963 Dispersed Flow data points and 2752 Stratified Flow data points. The five best

performing correlations for each evaluation method were selected, and their statistical parameters

are presented in the following tables. The I.D. correlation corresponds with the identification

number of the correlations in the annexed Table A.1 for Friction factor correlations and Table

A.2 for the Viscosity correlations.

9

Table 2. Performance evaluation of the friction factor correlations using Core Annular flow (L-

T) data.

Friction

Factor

Correlation

I.D FRP rank

Ripley rank

AIC rank 𝑬𝟏(%) 𝑬𝟐(%) 𝑬𝟑(%) 𝑬𝟒(%)

𝑬𝟓 (𝑷𝒂

𝒎)

∗ 𝟏𝟎𝟑 𝑬𝟔((

𝑷𝒂

𝒎)

∗ 𝟏𝟎𝟑 𝑬𝟕 (

𝑷𝒂

𝒎)

∗ 𝟏𝟎𝟑 𝑬𝟖(

𝑷𝒂

𝒎)

∗ 𝟏𝟎𝟑 𝑭𝑹𝑷

Ripley’s Range Factor

AIC∗

𝟏𝟎𝟑

37 1 1 1 -50,40 62,39 48,28 69,82 -19,50 22,33 41,71 46,05 1,80 17,99 14,51

18 3 4 14 -40,05 61,30 56,24 69,06 -24,92 27,69 50,37 56,20 2,13 34,95 14,78

17 4 9 9 -39,90 61,21 56,31 69,03 -24,90 27,68 50,42 56,24 2,13 35,38 14,77

7 14 3 10 -39,71 61,05 56,28 68,89 -24,89 27,68 50,52 56,33 2,13 34,84 14,77

11 25 2 7 -39,76 61,27 56,41 69,03 -24,79 27,61 50,34 56,12 2,14 34,34 14,77

14 2 23 18 -40,28 61,40 56,27 69,21 -25,10 27,89 50,58 56,47 2,13 36,59 14,78

15 5 11 29 -39,86 61,18 56,32 69,02 -24,92 27,71 50,44 56,27 2,13 35,70 14,79

10 23 5 23 -39,75 61,14 56,35 68,98 -24,94 27,74 50,53 56,36 2,13 35,24 14,78

2 24 28 3 -40,85 62,17 56,71 69,91 -24,99 27,79 50,18 56,07 2,14 39,57 14,76

23 29 29 5 -40,26 61,85 56,91 69,73 -24,93 27,75 50,25 56,11 2,15 40,88 14,77

6 31 30 4 -40,24 61,89 56,98 69,77 -24,92 27,75 50,23 56,08 2,15 40,97 14,77

1 33 33 2 -35,58 58,01 56,56 66,83 -22,77 25,97 50,50 55,41 2,21 107,53 14,75

Table 3. Performance evaluation of the friction factor correlations using Stratified flow data.

Friction Factor Correlation I.D

FRP rank

Ripley rank

AIC rank 𝑬𝟏(%) 𝑬𝟐(%) 𝑬𝟑(%) 𝑬𝟒(%)

𝑬𝟓 (𝑷𝒂

𝒎)

∗ 𝟏𝟎𝟑 𝑬𝟔((

𝑷𝒂

𝒎)

∗ 𝟏𝟎𝟑 𝑬𝟕 (

𝑷𝒂

𝒎)

∗ 𝟏𝟎𝟑 𝑬𝟖(

𝑷𝒂

𝒎)

∗ 𝟏𝟎𝟑 𝑭𝑹𝑷

Ripley’s Range Factor

AIC∗𝟏𝟎𝟑

3 7 1 5 184,55 186,99 259,73 318,73 6,62 6,79 7,65 10,1 1,28 3,29 9,21

27 6 2 6 181,56 184,13 259,90 317,14 6,51 6,69 7,55 9,97 1,23 3,30 9,23

23 9 3 11 199,47 201,55 260,09 327,90 7,01 7,16 7,91 1,06 1,44 3,32 9,25

20 12 5 17 202,30 204,39 260,67 330,09 7,03 7,18 7,90 1,06 1,46 3,36 9,26

9 1 33 1 87,50 97,27 164,24 186,14 4,36 4,81 6,17 7,56 0,03 3,64 8,93

35 2 34 2 95,48 105,05 183,65 207,03 4,57 4,98 6,34 7,82 0,17 3,73 8,96

4 4 32 4 104,68 110,96 199,47 225,32 4,76 5,11 6,45 8,03 0,29 3,59 8,98

32 3 35 3 99,37 108,80 200,54 223,86 4,60 5,01 6,35 7,85 0,25 3,82 8,97

28 19 4 35 203,28 205,33 260,70 330,71 7,08 7,22 7,94 1,06 1,48 3,36 9,50

1 5 37 36 74,24 103,94 204,59 217,66 4,63 5,45 14,0 14,7 0,94 20,37 9,58

10

Table 4. Performance evaluation of the viscosity-friction factor correlation pairs using

Dispersed Flow data.

(𝜇 ,f) ID

FRP rank

Ripley rank

AIC rank 𝐸1(%) 𝐸2(%) 𝐸3(%) 𝐸4(%)

𝐸5 (𝑃𝑎

𝑚)

∗ 103

𝐸6((𝑃𝑎

𝑚)

∗ 103 𝐸7 (

𝑃𝑎

𝑚)

∗ 103 𝐸9(𝑃𝑎/𝑚)∗ 103

𝐹𝑅𝑃∗ 10−77

Ripley’s Range Factor

AIC∗103

(2,37) 6 1 37 28,8 77,4 105,6 109,5 -6,94 12,02 93,60 93,86 2,6 12,7 91,55

(7,37) 3 4 56 38,0 83,9 116,6 122,6 -6,48 12,09 94,15 94,37 2,5 13,4 91,58

(20,37) 4 12 122 35,4 81,5 115,4 120,7 -6,55 12,07 94,30 94,53 2,5 13,5 91,64

(1,37) 1 2 146 40,2 85,0 141,6 147,2 -5,43 12,76 97,71 97,86 2,4 13,1 91,78

(7,18) 95 5 62 -13,5 50,4 96,9 97,8 -7,88 12,01 94,70 95,03 2,8 13,5 91,59

(7,11) 124 3 104 -15,1 50,5 96,8 98,0 -7,91 12,03 94,72 95,05 2,9 13,4 91,61

(8,37) 5 246 183 57,2 102,2 192,9 201,2 -4,82 13,52 99,48 99,60 2,5 14,9 91,89

(23,5) 407 38 1 -45,3 52,7 45,2 64,0 -10,52 11,83 92,49 93,09 3,7 13,7 91,47

(23,14) 412 34 3 -45,3 52,7 45,2 64,0 -10,52 11,83 92,49 93,08 3,7 13,6 91,47

(23,7) 394 53 5 -44,6 52,1 45,5 63,7 -10,51 11,82 92,44 93,04 3,7 13,7 91,47

(18,37) 2 249 201 68,4 110,7 244,4 253,8 -3,97 14,12 99,93 100,00 2,5 14,9 91,92

(23,11) 417 54 4 -45,6 53,1 45,2 64,2 -10,53 11,84 92,49 93,09 3,7 13,7 91,47

(23,8) 419 58 2 -45,6 53,2 45,2 64,3 -10,53 11,84 92,51 93,11 3,7 13,8 91,47

For Core-annular flow, the Shaikh-Massan-Wagan correlation [54] (Friction Factor Correlation

I.D. 37) leads the performance in all three comparison methods. In all three of them, it has a

considerable advantage over the following correlation in the ranking. This correlation is an

explicit approximation to the Colebrook-White equation, designed for quicker and best

performance on relative roughness from 10−4 to 0.05 and Reynolds numbers between 104 −

108. The Colebrook-White equation does not appear in the best performing of any of the three

comparison methods, which would suggest that implicit friction factor correlations do not do well

in the CAF model. This is supported in the fact that none of the three implicit correlations (Friction

factor correlation I.D 3,4 and 5) were also not featured in the table.

A graph representing to the overall performance of the Shaikh-Massan-Wagan correlation over

CAF data is shown hereafter:

11

Figure 5: Predicted Pressure Drop vs Experimental Pressure Drop [Pa/m] for the Shaikh-Massan-

Wagan correlation over CAF datapoints

As represented in Figure 5, the absolute average error (E2) of the Shaikh-Massan-Wagan

correlation is of 62%, which is high, however, since it is similarly high for the other correlations

it is possibly an error intrinsic to the solution of the CAF model, or that an amount of the data

points is of mixed flow. The tendency of the average error (𝐸1) is to underpredict the experimental

pressure loss, which means that the predicted frictional losses might be too small. This could be

an effect of the assumption that there is a smooth interface between the annulus and the core. In

fact, the interfacial waviness and instability of core-annular flow are common subjects of study

[55], whose incorporation in the model might help to reduce the error.

For Stratified flow, the most interesting results are those of the Prandtl Correlation, Wood

Correlation, the Goudar-Sonnad Correlation, and Li-Seem–Li correlations (Friction Factor

Correlation I.D 3,9,27,35), which seem to lead the table in at least two comparison methods. The

Prandtl correlation was one of the first correlations developed, designed for turbulent regime in

smooth pipes [6]. The Goudar-Sonnad Correlation was designed for a range from 10−6 to 5 ∗

10−2 [56] and the Li-Seem-Li Correlation is designed for smooth pipes. The Wood Correlation

would be the exception from this group since it is an explicit approximation of the Colebrook-

White equation, however there seems to be a trend for the better performances of smooth pipe

correlations in the Stratified Flow Model. The reason for this can be coupled with the reason why

many of the correlations have high errors once again.

A graphic reference of the correlations’ behaviour is presented hereafter:

12

Figure 6: Predicted Pressure Drop vs Experimental Pressure Drop [Pa/m] general error distribution behavior for the Li-Seem-Li, Prandtl, Wood and Goudar-Sonnad Correlations over a

sample of STF datapoints

One of the suppositions used for the model is that –however turbulent- the interface between the

two fluids is planar. In reality, a smooth planar interface is only obtained in a limited range of low

flow rates [57], and since the Laminar flow correlation (I.D 1) did not seem either to perform very

well in terms of average error 𝐸2 (103%), it is likely that most data was turbulent enough to create

a wavy interface. This would both affect the performance of the model, since the geometry upon

which the model is based changes, and also because a considerable increase in pressure gradient

would be provoked [57]. Additionally, the high speed during turbulency results in a high pressure

gradient, which explains that the models predicts above experimental values. Also, coupling the

high turbulence and the good performance of smooth pipe correlations suggests that friction

against the wall is less important than the friction between phases.

For Dispersed Flow, the Shaikh-Massan-Wagan correlation (Friction Factor Correlation I.D. 37)

leads as the friction factor correlation with the most appearances in the best performing table.

Similarly, Pal’s Frankel & Acrivos Viscosity Correlation Extension [58] (Viscosity Correlation

I.D. 23) seems to be the viscosity correlation with the most appearances on the best performing

table. However, they are not featured together in the best performing table. This fact endorses that

it was a good decision to compare the compounded error of each set of correlations rather than

choosing independently the best friction factor and viscosity correlations and assume they would

perform the best together.

While Pal’s correlation is preferred due to the relationship between its accuracy and the

complexity – as shown in that it takes a place in all of the five best performing correlations as

measured by the AIC-, the Shaikh-Massan-Wagan correlation is preferred in the distribution of

its statistical data– as shown in that it takes a place in all of the six best performing correlations

as measured by the FRP-. It seems however that the set that comprises the Taylor viscosity

13

correlation and the Shaikh-Massan-Wagan friction factor correlation -I.D (2, 37)- is in general

the best performing set.

A graph representing to the overall performance of the Taylor, Shaikh-Massan-Wagan correlation

pair over DF data is shown hereafter:

Figure 7: Predicted Pressure Drop vs Experimental Pressure Drop [Pa/m] for the Taylor, Shaikh-Massan-Wagan correlation pair over DF datapoints

Similarly to the Taylor correlation, other simple viscosity correlations such the Einstein

correlation (Viscosity Correlation I.D 1) and the Brinkman-Roscoe correlation (Viscosity

Correlation I.D 7) make up an important part of the best performing correlations. This suggests

either that the behavior of the oil-water dispersions is simple to model – which seems unlikely

given that non-Newtonian behavior is often seen in oil-water dispersion-, or that the viscosity

equations are not as influential as the friction factor correlations are.

6. Conclusions

The developed Matlab program calculates the pressure gradient for (L-T) Core-annular, Stratified

and Dispersed pipeline flows using different sets of correlations. Additionally, it evaluates the

performance of each set in each data point and compares it to the performance of the other sets in

more than 6388 pipeline flow data points calculated using the OLGA Multiphase Toolkit 2014.3.

The program is set up with 37 different friction factor correlations and 24 emulsion viscosity

correlations which can be used by the CAF, ST and DF models to calculate the pressure gradient

of the data points stored in an input Excel file. The correlations performances are evaluated using

Ansari’s et al. Relative Performance Factor, Ripley’s Range factor and the Akaike Information

Criterion [48,53]. The scores for each evaluation method are sorted and the five best performing

sets of correlations in each flow pattern are listed as an output.

14

The most favorable correlations for the used data were the Shaikh-Massan-Wagan correlation for

Dispersed and Core-Annular flow, several smooth pipe correlations for Stratified flow, and simple

viscosity correlations for Dispersed flow. While studying Dispersed Flow, it was found that the

prediction error can not be modeled as a linear combination of the prediction error of the

individual correlations. Additionally, since the errors where usually high, the code could be used

mainly to study the behavior and pathologies of the models used (i.e flexibility and validity of the

assumptions).

The results also give insight into over and under prediction biases such as the fact that in STF,

friction against the wall is less important than between phases and that thee assumption smooth

interface CAF model generates an intrinsic error in predicting pressure gradient values. Also, the

code endorses that friction factor correlations are often more important than viscosity correlations

in Dispersed Flow.

This program successfully indicates the most useful predictors of pressure loss in an Oil-Water

pipeline flow for a given dataset under three comparison methods. For future work, we propose

that a probabilistic predictor-performance map is developed to quantify the odds that a given

predictor will yield the best performance for a new data point. In addition, it is suggested that the

program is extended for the calculation of pressure gradient data for non-horizontal pipelines.

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7. Annexes

7.1. Existing models and correlations for liquid-liquid flow studies

23

Information on existing correlations of pressure drop, friction factor and mixture viscosity is

summarized from Table A. 1 to Table A. 3.

Hereafter, selected pressure loss models are reviewed.

Table A.1. Pressure drop models

Correlation Description Author

(𝑑𝑃

𝑑𝑍)𝑓

=𝑃0

𝑃𝐶(𝑑𝑃

𝑑𝑍)𝑓0

+𝑃𝑊

𝑃𝐶(𝑑𝑃

𝑑𝑍)𝑓𝑤

[𝐸𝑞. 6]

(𝑑𝑃

𝑑𝑍)𝑓

=𝑓𝜌𝑣2

2𝑔𝑐𝑑 [𝐸𝑞. 7]

For Dispersed flow Arirachakaran et

al. [4, 59]

−𝐴𝑤 (𝑑𝑝

𝑑𝑧) − 𝜏𝑊𝑤

𝑃𝑤 − 𝜏𝑖𝑃𝑖 = 0 [𝐸𝑞. 8]

−𝐴𝑜 (𝑑𝑝

𝑑𝑧) − 𝜏𝑊𝑜

𝑃𝑜 + 𝜏𝑖𝑃𝑖 = 0 [𝐸𝑞. 9]

𝜏𝑊𝑗 = 𝑓𝑗 ∗ 𝜌𝑗

|𝑈𝑗|𝑈𝑗|𝐹𝑗|𝑛𝑗𝑠𝑖𝑔𝑛(𝐹𝑗)

2 [𝐸𝑞. 10]

𝑛𝑗 =

𝑙𝑜𝑔 (𝑓𝑗

+

𝑓𝑗−)

𝑙𝑜𝑔 (𝑅𝑒𝑗

+

𝑅𝑒𝑗−)

[𝐸𝑞. 11]

Stratified Flow with a turbulent phase

Brauner [57]

𝑑𝑃

𝑑𝑍= 2𝑓𝑚

𝜌𝑚𝑈𝑚2

𝐷− 𝜌𝑚𝑔𝑠𝑖𝑛𝛽 [𝐸𝑞. 12]

Dispersed flow for Homogeneous Newtonian flow

Brauner

𝜒2 =𝑓𝑎𝑓𝑐

(𝜇𝑎

𝜇𝑐) �̃�𝑅𝑒𝑎𝑠

0.8 = (𝑑𝑃𝑑𝑧)

𝑎𝑠

(𝑑𝑃𝑑𝑧)

𝑐𝑠

[𝐸𝑞. 13]

For core-annular flow with laminar core and turbulent annulus.

Brauner [55]

This document uses the definition of the friction factor under the Darcy-Weisbach equation:

f =ΔP ∗ 2 ∗ D

LρU2 [Eq. 14]

7.2. Table A.2. Friction factor correlations

ID

Number Correlation Author

1

f =64

Re [Eq. 15] for 𝑅𝑒 ≤ 2100

Hagen & Poiseuille

Laminar flow (1840)

2 f = {

0.316 Re−0.25 for 2100 < Re ≥ 2 ∗ 106

0.184 Re−15 for Re ≥ 2 ∗ 104

[Eq. 16] Blausius

Correlations (1913)

24

3 1

√f= 2.0 log(Re ∗ 2√f) − 0.8 [Eq. 17]

Prandtl Correlation

(1935)

4 1

√f= 3.2 log10(Re√f) + 1.2 [Eq. 18]

Drew and Generaux

(1936) [60]

5 1

√f= −2.0 ∗ log10 [

(ϵD)

3.7+

2.51

Re√f] [Eq. 19]

Colebrook-

White equation (1939) [61]

6 𝑓 =

1

(1.8 𝑙𝑜𝑔10 𝑅𝑒 − 1.5)2 [𝐸𝑞. 20] Konakov

[60, 62]

7

f = 5.5 ∗ 10−3 [1 + (2 ∗ 104 (ϵ

D)+

106

Re)

13

] [Eq. 21] Moody

Correlation

(1947) [63]

8

f = 0.11(68

Re+

ϵ

D)0.25 [Eq. 22]

Altshul Correlation

(1952) [64, 65]

9 𝑓 =

1

4(𝑎 + 𝑏𝑅𝑒−𝑐) [𝐸𝑞. 23]

Where

𝑎 = 0.53 (𝜖

𝐷) + 0.094(

𝜖

𝐷)

0.225[Eq. 24]

𝑏 = 88(𝜖

𝐷)0.44

[Eq. 25]

c = 1.62 (ϵ

D)0.134

[Eq. 26]

Wood

Correlation (1966) [64, 66,

67]

10 1

√f= −2.0 ∗ log10 [

ϵD

3.71+ (

7

Re)

0.9

] [Eq. 27]

Churchill Correlation

(1973) [64, 68]

11 1

√f= −2.0 log10 (

ϵD

3.71+

15

Re) [Eq. 28]

Eck Correlation (1973) [69, 70]

12 1

√f= −2.0 log10 [

(ϵD)

3.715+ (

6.943

Re)

0.9

] [Eq. 29]

Jain Correlation

(1976) [64, 71]

13 𝑓 =

0.25

𝐴2 [𝐸𝑞. 30]

Where

A = log10(ϵ/D

3.7+

5.74

Re0.9) [Eq. 31]

Swamee & Jain (1976) [64, 72]

14 𝑓 = [(

64

𝑅𝑒)12

+ (𝐴 + 𝐵)−3

2]

1

12

[Eq. 32]

Where

Churchill correlation

(1977) [66, 73, 74]

25

A

=

[

0.8687 ln

(

1

0.883(lnRe)1.282

Re1.007 + 0.27 (ϵD)+

110(ϵD)

Re )

] 16

[Eq. 33]

B = (13269

Re)16

[Eq. 34]

15 1

√𝑓= −2. 𝑙𝑜𝑔10 [

(𝜖𝐷

)

3.7065−

5.0452 𝐴

𝑅𝑒] [𝐸𝑞. 35]

Where

A = log10 ((ϵD)

1.1098

2.8257+

5.8506

Re0.8981) [Eq. 36]

Chen

correlations (1979) [66, 73]

16

f = [1.8 ∗ log10 (Re

0.135(ReϵD) + 6.5

)]

−2

[Eq. 37]

Round

Correlation (1980) [60]

17

f = [−2∗ log10((ϵD)

3.7−

5.02

Relog10(

ϵD3.7

+14.5

Re))]

−2

[Eq. 38]

Shacham Correlation (1980) [73]

18 1

√𝑓= −2 𝑙𝑜𝑔10 (

𝜖

3.7 ∗ 𝐷+

5.158 ∗ 𝑙𝑜𝑔 (𝑅𝑒7

)

𝑅𝑒 (1 +𝑅𝑒0.52

29(𝜖𝐷

)0.7

)

) [𝐸𝑞.39] Barr Correlation

(1981) [73, 75]

19 1

√𝑓= −2 𝑙𝑜𝑔10 [(

𝜖𝐷3.7

) −5.02 𝐵

𝑅𝑒] [𝐸𝑞. 40]

Where

𝐴 = 𝑙𝑜𝑔10 (

𝜖𝐷3.7

+13

𝑅𝑒) [𝐸𝑞. 41]

B = log10 (

ϵD3.7

−5.02A

Re) [Eq. 42]

Zigrang & Sylvester

Correlation (1982) [73, 76]

20 1

√f= −1.8 log10 [(

(ϵD)

3.7)

1.11

+6.9

Re] [Eq. 43]

Haaland Correlation

(1983) [73, 77]

21 1

√𝑓= 𝐴 −

(𝐵 − 𝐴)2

𝐶 − 2𝐵 + 𝐴 [𝐸𝑞. 44]

Where

Serghides Correlation

(1984) [73, 78]

26

𝐴 = −2 𝑙𝑜𝑔10 [(𝜖𝐷)

3.7+

12

𝑅𝑒][𝐸𝑞. 45]

𝐵 = −2 𝑙𝑜𝑔10 [(𝜖𝐷)

3.7 +

2.51𝐴

𝑅𝑒] [𝐸𝑞. 46]

C = −2 log10 [(ϵD)

3.7+

2.51B

Re] [Eq. 47]

22 𝑓 = {

𝐶 𝑖𝑓 (𝐶 ≥ 0.018)0.0028 + 0.85𝐶 𝑖𝑓 (𝐶 < 0.018)

[𝐸𝑞. 48]

Where

C = 0.11(68

Re+

ϵ

D)

0.25

[Eq. 49]

Tsal Correlation

(1989) [73, 79]

23

f = [−2 log10(ϵ

3.70+

95

Re0.983−

96.82

Re)]

−2

[Eq. 50]

Manadili Correlation

(1997) [73, 80]

24 1

√𝑓= −2 𝑙𝑜𝑔10 [

(𝜖

𝐷)

3.7065−

5.0272𝐵

𝑅𝑒] [𝐸𝑞. 51]

Where

𝐴 = 𝑙𝑜𝑔10 [(𝜖/𝐷

7.7918)

0.9924

+ (5.3326

208.815 + 𝑅𝑒)

0.9345

] [𝐸𝑞. 52]

B = log10 (

ϵD

3.827−

4.567 A

Re) [Eq. 53]

Romeo-Royo-Monzón

Correlation (2002) [73, 81]

25

1

√𝑓= 0.8686[𝑙𝑛 (

0.4587 𝑅𝑒

(𝐶 − 0.31)𝐶

𝐶+1

)] [𝐸𝑞. 54]

Where

C = 0.124 ∗ Reϵ

D+ ln(0.4587 Re) [Eq. 55]

Sonnad-Goudar Correlation (2006) [56]

26

1

√𝑓= 𝐴 − [

𝐴 + 2 ∗ 𝑙𝑜𝑔 (𝐵𝑅𝑒)

1 + (2.18𝐵 )

] [𝐸𝑞. 56]

Where

A =(0.744 ln(Re) − 1.41)

(1 + 1.32√ϵD)

[Eq. 57]

Buzelli Correlation

(2008) [82]

27

B =ϵ

3.7DRe + 2.51A [Eq.58]

27 1

√𝑓= 𝑎 [𝑙𝑛 (

𝑑

𝑞) + 𝛿𝐶𝐹𝐴] [Eq. 59]

Where

𝑎 =2

𝑙𝑛 (10) [Eq. 60]

𝑏 =𝜖

𝐷

3.7 [Eq. 61]

𝑑 =𝑙𝑛 (10)

5.02𝑅𝑒 [Eq. 62]

𝑠 = 𝑏𝑑 + 𝑙𝑛 (𝑑) [Eq. 63]

𝑞 = 𝑠𝑠

𝑠+1 [Eq. 64]

𝑔 = 𝑏𝑑 + 𝑙𝑛 (𝑑

𝑞) [Eq. 65]

𝑧 =𝑞

𝑔 [𝐸𝑞. 66]

𝛿𝐿𝐴 =𝑔

𝑔+1𝑧 [Eq. 67]

δCFA = δLA (1 +

z2

(g + 1)1 + (z3) (2g − 1)

)[Eq.68]

Goudar-Sonnad Correlation

(2008) [73, 83]

28 f

=6.4

[ln(Re) − ln (1 + 0.01Re(ϵD) ∗ (1 + 10 ∗ √

ϵD)]

2.4 [Eq. 69]

Avci & Karagoz

Correlation (2009) [84]

29 f =

0.2479 − 0.0000947(7 − log10 Re)4

(log10 (ϵ

3.615D +7.366

Re0.9142))2

2

[Eq. 70]

Papaevangelou-Evangelides-Tzimopoulos Correlation (2010) [85]

30

𝑓 = [−2 𝑙𝑜𝑔 (10−0.4343𝛽 +

𝜖𝐷

3.71)]

−2

[𝐸𝑞. 71]

β = ln(Re

1.816 ∗ ln (1.1 Re

ln(1 + 1.1 Re))) [Eq.72]

Brkić Correlation (a)

(2011) [6]

31 𝑓 = [−2 𝑙𝑜𝑔10((

2.18𝛽

𝑅𝑒) + (

𝜖

𝐷 ∗ 3.71)]

−2

[𝐸𝑞. 73]

Where 𝛽 is equation 74

Brkić Correlation (b)

(2011) [6]

28

32 1

√𝑓= 𝐶0 −

1.73718𝐴 𝑙𝑛(𝐶0)

1.73718 + 𝐶0+

2.62122𝐴(𝑙𝑛𝐶0)2

(1.73718 + 𝐶0)3

+3.03568𝐴(𝑙𝑛𝐶0)

3

(1.73718 + 𝐶0)4 [𝐸𝑞. 74]

Where

C0 = 4 log(Re)− 0.4 [Eq. 75]

Danish- Kumar-Kumar

Correlation (2011) [86]

33 f = 1.613[ln(0.234(

ϵ

D)1.1007

−60.525

Re1.1105

+56.291

Re1.0712]−2

[Eq. 76]

Fang Correlation (2011) [87]

34 f = [−1.52 log ((

ϵ/D

7.21)1.042

+ (2.731

Re)

0.9152

)]

−2.169

[Eq. 77]

Ghanbari-Farshad-Rieke

Correlation (2011) [88]

35

f =−0.0015702

ln (Re)+

0.3942031

ln(Re)2+

2.5341533

ln(Re)3 [Eq.78]

Li-Seem-Li Correlation

(2011) [89]

36

f = 4 ∗ ([0.0076(

3170Re )

0.165

1 + (3170Re )

7.0 ] +16

Re) [Eq. 79]

Morrison Correlation

(2013) [69, 90]

37 𝑓 = 0.25 [𝑙𝑜𝑔(

2.51

𝛼𝑅𝑒+

𝜖/𝐷

3.7)]

−2

[𝐸𝑞. 80]

where

𝛼 = [1.14 − 2 ∗ 𝑙𝑜𝑔 (𝜖

𝐷)]

−2

[𝐸𝑞. 81]

Shaikh-Massan-Wagan

(2015) [54]

29

7.3

Table A.3. Mixture viscosity correlations

ID

Number Correlation Author

1 𝜇𝑒

𝜇𝑐= 1 + 2.5𝜙 [𝐸𝑞. 82]

Einstein (1906) [91, 92]

2 𝜇𝑒

𝜇𝑐= 1 + 2.5𝜙𝐴 [𝐸𝑞. 83]

𝐴 = [𝜇𝑐 + 2.5𝜇𝑑

2.5𝜇𝑐 + 2.5𝜇𝑑] [𝐸𝑞. 84]

Taylor (1932) [93, 58]

Not Used

𝜇𝑒

𝜇𝑐= 𝑒𝑥𝑝(𝐾𝜙) [𝐸𝑞. 85]

Richardson (1933) [94]

3 𝜇𝑒

𝜇𝑐= 𝑒𝑥𝑝 [2.5𝐴 ∗ (𝜙 + 𝜙

53 + 𝜙

113 )] [𝐸𝑞. 86]

A = [μc + 2.5μd

2.5μc + 2.5μd] [Eq. 87]

Levinton and

Leighton (1936) [93]

4 𝜇𝑒

𝜇𝑐= 1 + 2.5𝜙 + 14.1𝜙2 [𝐸𝑞. 88] Guth and Simha

(1936) [94, 19]

Not used

𝜇𝑒

𝜇𝑐= 𝐴 ∗ 𝑒𝑥𝑝(𝐾𝜙) [𝐸𝑞. 89] Broughton and

Squires (1938) [94]

5 𝜇𝑒

𝜇𝑐= 𝑒𝑥𝑝(

2.5𝜙

1 − 0.609𝜙)[𝐸𝑞. 90]

Vand (1948) [91]

6 𝜇𝑒

𝜇𝑐= 𝑒𝑥𝑝(

2.5𝜙

1 − 𝐾𝜙) [𝐸𝑞. 91] Mooney (1951) [92,

94, 95]

7 𝜇𝑒

𝜇𝑐= (1 − 𝜙)−2.5 [𝐸𝑞. 92] Brinkman (1952),

Roscoe (1952) [92]

8 𝜇𝑒

𝜇𝑐= [

10(𝐾 + 1) + 3𝜙(5𝐾 + 2)

10(𝐾 + 1) − 2𝜙(5𝐾 + 2)] [𝐸𝑞. 93]

Oldroyd (1953) [58]

9 𝜇𝑒

𝜇𝑐=

1

1 − 𝜙 [ 1 +

1.5𝜙𝜇𝑑

𝜇𝑐 + 𝜇𝑑] [𝐸𝑞. 94] Vermuelen et al.

(1955) [96]

10 𝜇𝑒

𝜇𝑐= (1 −

𝜙

𝜙𝑚𝑎𝑥)−2

[𝐸𝑞. 95] Maron-Pierce (1956)

Not used

𝜇𝑒

𝜇𝑐= (1 −

𝜙

𝜙𝑚𝑎𝑥)

−𝜇𝑖𝜙𝑚𝑎𝑥

[𝐸𝑞. 96] Dougherty & Krieger (1959)

11 𝜇𝑒

𝜇𝑐= [1 + 2.5𝜙(1 − 𝛼𝐸𝜙)−1]2 [𝐸𝑞. 97]

Eiler (1962) [92, 94]

12 𝜇𝑒

𝜇𝑐= [1 + 2.5𝜙 + 10.05𝜙2 + 0.00273𝑒𝑥𝑝(16.6𝜙)] [𝐸𝑞. 98]

Thomas (1965) [91]

30

13 𝜇𝑒

𝜇𝑐= [1 + 0.75

𝜙

𝜙𝑚𝑎𝑥(1 −

𝜙

𝜙𝑚𝑎𝑥)−1

]

2

[𝐸𝑞. 99]

Chong et al. (1971) [92]

14 𝜇𝑒

𝜇𝑐=

1 + 0.5𝜙

(1 − 𝜙)2 [𝐸𝑞. 100]

Furuse (1972)

15 𝜇𝑒

𝜇𝑐

= 1 + 𝜙 [5.5 [4 𝜙

73 + 10 − (

8411)𝜙

23 + (

4𝐾)(1 − 𝜙

73)]

10(1 − 𝜙103 ) − 25𝜙 (1 − 𝜙

43) + (

10𝐾

)(1 − 𝜙)(1 − 𝜙73)

] [𝐸𝑞. 101]

Yaron and Gal-Or (1972)[2,97]

Not used

𝜇𝑒

𝜇𝑐= 𝑒𝑥𝑝(

𝐾1𝜙

1 − 𝐾2𝜙) [𝐸𝑞. 102]

Barnea and Mizrahi (1973) [91]

16 μe

μc= 1 + ϕ[

2(5K + 2) − 5(K − 1)ϕ73

4(K + 1) − 5(5K + 2)ϕ + 42Kϕ53 − 5(5K − 2)ϕ

73 + 4(K − 1)ϕ

103

]

[Eq. 103]

Choi and Schowalter (1975) [2,97]

17 𝜇𝑒

𝜇𝑐= 𝐵[

23

𝐵 +𝜇𝑑

𝜇𝑐

𝐵 +𝜇𝑑

𝜇𝑐

] [𝐸𝑞. 104]

𝐵 = 𝑒𝑥𝑝 [5𝜙𝐴

3(1 − 𝜙)] [𝐸𝑞. 105]

𝐴 = [𝜇𝑐 + 2.5𝜇𝑑

2.5𝜇𝑐 + 2.5𝜇𝑑] [𝐸𝑞. 106]

Barnea and Mizrahi (1975) [93]

Not used 𝜇𝑒

𝜇𝑐= [1 +

𝜙𝐾

1.1884 −𝜙𝐾

]

2.5

[𝐸𝑞. 107]

Pal and Rhodes (1989) [94]

18

(𝜇𝑒

𝜇𝑐)

25[2(

𝜇𝑒

𝜇𝑐) + 5(

𝜇𝑑

𝜇𝑐)

2 + 5(𝜇𝑑

𝜇𝑐)

]

35

= (1 − 𝜙)−1 [𝐸𝑞. 108]

Phan-Thien & Pham (1997) [58,97]

19

(𝜇𝑒

𝜇𝑐)−2/5

[2 (

𝜇𝑒

𝜇𝑐)+ 5(

𝜇𝑑

𝜇𝑐)

2 + 5(𝜇𝑑

𝜇𝑐)

]

−2/5

= (1 − 𝐾0𝜙) [𝐸𝑞. 109]

Pal (2000) [94,97]

20

𝜇𝑒

𝜇𝑐(2𝜇𝑒

𝜇𝑐+ 5𝐾

2 + 5𝐾)

32

= 𝑒𝑥𝑝(2.5𝜙) [𝐸𝑞. 110]

Pal (2001)[2,58]

31

21

μe

μc(2

μe

μc+ 5K

2 + 5K)

32

= [1 +1.25ϕ

1 −ϕϕm

]

2

[Eq. 111] Pal (2001) [58]

22 𝜇𝑒

𝜇𝑐(2𝜂𝑟 + 5𝐾

2 + 5𝐾)

32

= [1 −𝜙

𝜙𝑚]−2

[𝐸𝑞. 114]

Pal (2001) [2,58]

23

μe

μc(2ηr + 5K

2 + 5K)

32

=9

8

[ (

ϕϕm

)

13

1 − (ϕϕm

)

13

]

[Eq. 115] Pal (2001) [2,58]

24 μe

μc(2ηr + 5K

2 + 5K)

32= [1 +

0.75(ϕϕm

)

1 − (ϕϕm

)]

2

[Eq. 11216] Pal (2001) [58]

Not used

μe

μc= 1 + K1ϕ + K2ϕ

2 [Eq. 117] Polynomial 1 [95]

Not used

𝜇𝑒

𝜇𝑐= 1 + 𝐾1𝜙 + 𝐾2𝜙

2 + 𝐾3𝜙3 [𝐸𝑞. 118113]

Polynomial 2 [95]