research article aic applications in coiled tubing … · calculate the fanning friction ... the...

International Journal of Petroleum & Geoscience Engineering (IJPGE)1 (2): 62-81, 2013 ISSN 2289-4713 © Academic Research Online Publisher

Research Article

AIC Applications in Coiled Tubing Hydraulics

Ali S. Shaqlaiha, Ahmed H. Kamelb a Department of mathematics and Information Sciences, Division of Liberal Arts and Life Sciences, University fo North Texas at Dallas, Dallas, TX, USA b Department of Petroleum Engineering, College of Business & Engineering, University of Texas Permian Basin, Odessa, TX, USA * Corresponding author. Tel.: 432 552 2219; fax: 432 552 2174 E-mail address: [email protected]

ARTICLE INFO Accepted: 7 July 2013

A b s t r a c t Coiled tubing (CT) has been used in well drilling, completion, stimulation, wellbore cleanout, and other operations in the petroleum industry where various fluids are pumped under turbulent flow conditions. Accurate estimation of frictional pressure losses when pumping these fluids through coiled tubing has remained a challenge and a milestone for job success. This challenge has triggered the research activities in the field of coiled tubing hydraulics. As a result, there have been extensive studies on flow of Newtonian fluids in coiled tubing and several equations have been published to calculate the Fanning friction factor. Each of which has its own applications and limitations. Applications of these equations without verification may lead to erroneous results, and eventually job failure. Employing the new Akaike Information Criteria or AIC theory, the authors compared and questioned the applicability of these correlations to propose the most accurate one. This study employs the R^2 method and the Information Theory Approach in selecting the best model to predict the Fanning friction factor. It shows that among six different models, the best model to predict the Fanning friction factor is McCann and Islas model and the weakest one is Mashelkar and Devarajan model. A comparison between R^2 and AIC theory is also presented to show the advantages of AIC theory over R^2 method. In conclusion, the authors are introducing a new theory to the oil and gas industry where they believe that employing this theory can resolve various difficulties faced by the researchers in the oil field. This paper represents the first application of AIC theory in the oil and gas theory and helps advances the state-of-the-art in this field. © Academic Research Online Publisher. All rights reserved.

Keywords: Akaike Coiled tubing Friction factor Turbulent flow

1. Introduction

Since the introduction of coiled tubing (CT) to the oil and gas industry, the application has

increased every year. Total number of CT working units reported in 1999 was approximately

760 units. In January 2009, more than 1650 CT units were available worldwide. This number

Ahmed H. Kamel et al./ International Journal of Petroleum & Geoscience Engineering (IJPGE) 1 (2): 62-81, 2013

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increased to more than 1770 units in 2011 and 1811 in 2012 [1]. CT has gained popularity

because of their numerous advantages over the conventional jointed tubing. They are faster,

safer, cost effective, and environmentally friendly. However, there are some difficulties

accompanying CT operations. These difficulties include small tubing diameter, secondary

flow due to centrifugal forces, which increase flow resistance, and lack of adequate

correlations to predict friction pressure specially for non-Newtonian fluids [2]. CT has been

used in different applications in the oil and gas industry. These applications include solids

and sand washing, well unloading, CT drilling, fracturing acidizing, cement squeeze, and

many other [3]. In most of these applications, fluids are pumped through CT under turbulent

flow conditions.

Excessively high friction pressure losses due to small diameter of pipe and secondary flows

generated due to the curvature in coiled tubing tend to limit the pumping capacity of fluids.

Therefore, accurate prediction of friction pressure losses of fluid flowing through coiled

tubing has been a challenge mainly due to the lack of adequate friction loss correlations and

proper understanding of flow complexity in coiled tubing.The complexity of flow in CT

arises from its unique feature, curvature. Centrifugal forces resulted from CT curvature yields

secondary flow that is superimposed on the primary flow in the axial direction. As a result of

secondary flow, friction losses in CT are higher than in conventional straight tubing. It also

causes a delay in the transition from laminar to turbulent flow [4].

Dean [5, 6] has the credit of conducting the first theoretical analysis of fluid flow through

coiled tubing. His work has led many other researchers to develop a good understanding of

this aspect. Dean simplified the governing equations by assuming small curvature ratio, r/R,

and slow flows. He defined the so-called Dean number which is a fundamental non-

dimensional parameter in developing friction correlations for flow in CT. Various definitions

are available but most researchers prefer the following definition.

NDe = NRe�rR� �

12� (1)

Dean number provides a fundamental parameter in developing flow resistance correlations

for flow in curved pipes. It has been found that at low Dean Number, the law of resistance

can be correlated with NDe only. For high Dean Number, both NDe and curvature ratio (r/R)

will be required.


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Recent increase of coiled tubing applications has been driving the research activities of coiled

tubing hydraulics. As a result, there are extensive studies on Newtonian fluid flow in coiled

tubing and several equations are published to calculate Fanning friction factor for Newtonian

fluids flowing in CT. Each has its own applications and limitations. Applications of these

equations without verification may lead to erroneous results, and eventually job failure.

The objective of this paper is to address, compare, and evaluate the accuracy and applicability

the available friction factor correlations for Newtonian fluids in CT, hoping to provide

valuable information for the professional who are interested in coiled tubing hydraulics. A

new technique, AIC for comparing and evaluation the correlations is presented for the first

time in the oil and gas industry. The validity the AIC technique has been very well

established in other disciplines and this paper represents the first attempt to simply introduce

and elaborate the use o AIC in the oil and gas industry. The authors believe that applying this

technique can overcome various problems facing the researchers and professionals working

in the oil and gas industry.

2. Modeling Techniques

Given a set of data, one can fit many models that may be used to predict unknown cases. This

prediction is more accurate when the model is better representing the information presented

by the data rather than the data itself. When having different models, the question is which

model is the best for prediction. In that context, what best means? How one can choose the

model to be used in prediction and make sure that that model gives good prediction for the

values in the model. There are different methods and techniques for selecting methods such

as Bayesian Method and the R2 method [7–10].

In this study, it is assumed that a collection of models is given and there exists an abstract

model 'truth'. Although we do not actually know the truth model, there are available data

consisting of observations of the truth model but accompanied with noise. One well known

way of capturing the data is to use a least squares criterion to determine physical parameters

to fit observations. This amounts to using the so-called R2 criterion as a criterion for model

selection. While this procedure may determine best fit to data, it is well documented that it

does not necessarily produce a model that maximizes information given by the data [7].


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Another technique is the Akaike Information Criterion or AIC. AIC is statistical procedures

that have been developed to choose the model that minimizes the lost information. In this

paper we will use the AIC method to choose the best model for Fanning friction factor. First,

we present the models then a simple background about the R2 Method and the AIC method

and then we state the reasons that make the AIC better approach in choosing the best model.

The results and comparison between the two approaches are presented in the present paper.

3. Fanning Friction Factor Equations

In turbulent flow, the inertia forces are dominating over the viscous forces and the fluid

particles move in irregular paths causing an exchange of momentum from one portion of the

fluid to another, thus increasing eddies and fluctuations which increase friction losses.

Turbulent flow is the practical flow that encounters engineers mostly. In CT, transition from

laminar to turbulent flow occurs at a higher Reynolds number than in straight tubing. Ito [11]

proposed the following empirical correlation for the critical Reynolds number:

𝑁𝑅𝑒𝑐 = 20,000�rR� �

0.32(2)

This equation provided good agreement with experimental results in the range 15 < r/R <

8860. For r/R > 860, the critical Reynolds number for a curved pipe practically coincides

with that for a straight pipe. Srinivasan et al [12] pointed that Ito’s correlation will not give

NRec = 2100 for a straight tube where r/R = 0. Therefore, another correlation was proposed

based on the experiments as follows:

𝑁𝑅𝑒𝑐 = 2,100 �1 + 12�rR� �

0.5�(3)

However, due to the secondary flow effect, the transition behavior on the plots of friction

factor vs. Reynolds number is very gradual and generally, it is very difficult to accurately

identify when the onset of turbulence. If the fluid is non-Newtonian, this transition would be

even more gradual [13, 14].

In turbulent flow, the classical Darcy–Weisbach equation has been used for predicting

friction pressure losses for fluids flowing under turbulent flow conditions. It is a simple

equation to be used. Yet, it involves a crucial and confusing term to be determined. It is the

friction factor. Friction factor is dependent upon various parameters relevant to pipe


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specifications, fluid behavior, and flow regime. Several equations are available to predict the

Fanning friction factor for Newtonian fluids flowing in smooth coiled tubing. No published

correlations are available to quantify the effect of pipe roughness on fluid flow in CT. This

subject is not well understood and still under investigations [15]. The following paragraphs

elaborative review of the available correlations.

3.1. Ito Correlation (1959)

For turbulent flow of Newtonian fluids in smooth coiled tubing, Ito [11] measured the

turbulent frictional pressure losses in smooth curved pipes using water and drawn-copper

tubing at various curvature ratios. He applied the concept of boundary layer approximation to

the turbulent flow of Newtonian fluid in curved pipes. By assuming the 1/7th power velocity

distribution, he obtained the following friction factor correlation for curved pipe:

𝑓 = 14�r

R� �0.5�0.029 + 0.304 �NRe�r

R� ��−0.25

�(4)

The above equation is for Dean number ranges from 0.034 to 300 and curvature ratio between

0.0012 and 0.067.

3.2. Srinivasan Correlation (1970)

Later, Srinivasan et al [12] measured pressure drops of water and fuel oil in both helical tubes

(with constant curvature) and spiral tubes (with variable curvature). Experimental values

were used to develop equations to predict friction factors for laminar, transition, and turbulent

regions. His equation is valid for 0.0097 < r/R < 0.135 and any Dean number higher than

critical Dean number up to 14,000. It has been proven experimentally that, Srinivasan

correlation is superior and more accurate than Ito correlation.

𝑓 =0.084�r R� �0.5

NRe0.2 (5)


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3.3. Mashelkar and Devarajan Correlation (1976)

Mashelkar and Devarajan [16–18], following Ito’s boundary layer approximation approach

for Newtonian fluids, studied flow of non-Newtonian fluids in coiled tubing. They

theoretically analyzed and numerically solved the flow equations for a power-law fluid for

the conditions of both laminar and turbulent flows. An empirical correlation for laminar flow

was developed based on the numerical solutions and another correlation for visco-elastic

fluids in terms of Weissenberg number based on data of flow tests. Mashelkar and Devarajan

did not provide any turbulent friction factor correlation in terms of flow behavior index n as

they did for the laminar. They only obtained numerical solutions for n = 0.5, 0.75, 0.9, and

1.0. Based on his numerical solution, the following equation can be used:

𝑓 =0.07185�r R� �0.5

�NRe�r R� �2�0.16667(6)

3.4. Mishra and Gupta’s Correlation (1979)

Mishra and Gupta [19] investigated pressure drop in coils of various diameters and pitches.

Their data covered laminar and turbulent flows and corresponding empirical correlations

were proposed. Mishra and Gupta have also measured friction factors of non-Newtonian

fluids in coiled pipes and empirical correlations have been obtained. For Newtonian fluids,

the following equation can be used and it valid for the range of 4,500 >NRe> 105.

𝑓 = 0.079𝑁𝑅𝑒0.25 + 0.0075�𝑟 𝑅� �

0.5(7)

3.5. McCann and Islas Correlation (1996)

McCann and Islas [20] conducted full scale flow experiments using coiled tubing without a

weld bead having 1.15-, 2.00-, and 2.375-in. outer diameters on a 98-in. diameter reel. Their

experimental results for water agreed well with values calculated using Srinivasan et al. [12]

correlation. They also, generalized their correlation for turbulent flow to non-Newtonian

fluids and derived a formula that gave pressure losses in excellent agreement with


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experimental data for the fluids (prepared using bentonite and lime) considered in their study.

The generalized equation is below, however, it can be used for Newtonian fluids when

assuming n = 1.0.

𝑓 =1.06a�r R� �0.1

NRe0.8b (8)

Where a and b are given as follows:

a = log10(n)+3.9350

(8a)

b = 1.75−log10(n)7

(8b)

3.6. Willingham and Shah Correlation (2000)

In recent years, the rapid increase of coiled tubing applications in the oil and gas industry has

driven the research activities of coiled tubing hydraulics using full-scale facilities.

Willingham and Shah [21] reported an empirical friction factor correlation based on a series

of flow tests with polymer solutions through 1, 1½, and 2⅜-in. coiled tubing reels. Their

work discussed the experimental investigation of five different polymeric solutions and one

drilling mud in three different coiled tubing sizes. They developed the following correlation:

�𝑓 = A�rR� , μ@511� +

B�d,μ@511�

�NReg(9)

The constant A is a function of the ratio of the coiled tubing curvature ratio, r/R and the

apparent viscosity at 511 s-1 shear rate, µ@511 while B is a constant. It can be used for

Newtonian fluids after proper modifications. However, this equation is excluded from the

analysis since the constants A and B are not published due to confidentiality agreement.

3.7. CTC Correlation (2007)

Following the approach of boundary layer approximation analysis, Zhouand Shah [22]

developed a new friction factor correlation of laminar flow of power law fluids in coiled

tubing. He compared his correlation with Ito correlation (for n = 1) and experimental data and

found an excellent agreement between the new correlation and both Ito correlation and


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experimental data. Similar approach was undertaken for turbulent flow of power law fluids in

coiled tubing employing numerical analysis. A friction factor correlation for turbulent non-

Newtonian fluid flow in coiled tubing was also developed and verified with Ito correlation for

Newtonian turbulent flow and also with limited experimental data. Zhou’s study extended

Mashelkar and Devarajan’s method to a wider range of flow behavior indices. In his study, an

extensive experimental work was performed with polymer-based fluids (xanthan, PHPA,

guar, HPG, and HEC fluids at various concentrations) using the field-scale and lab-scale flow

loops consisting of ½, 1, 1½-, 1¾, and 2⅜-in. diameter CT with various lengths and curvature

ratios. Simplifying the equation when n = 1.0 gives the following:

𝑓 = 0.073297�𝑟 𝑅� �0.5

�𝑁𝑅𝑒�𝑟 𝑅� �1.9971�0.20023(10)

4. Analytical Work

4.1. The Information Theory Approach

In the information theory approach, it is thought of the full reality as a model; say that f has

conceptually an infinite number of parameters. Thus f represents the full truth and might be

conceptually based on a very large number of parameters [7]. This approach starts with what

is so called Kullback-Leibler (K-L) Information (distance) between functions f and g which is

defined for continuous functions as:

I(f,g)= ∫ ln()(

)(θxgxf dx(11)

ln denotes the natural logarithm and the notation I(f,g) denotes the information lost when g is

used to approximate f [23]. In this approach we seek an approximating model that loses as

little as information as possible. In other words we want to minimize I(f,g) over all possible

models g.

Akaike's [24] paper proposed the use of the Kullback-Leibler (K-L) Information (distance) as

a fundamental for model selection. Akaike found what is now called Akaike information

criterion (AIC) which is:


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AIC= -2 log( )ˆ( yl θ + 2k(12)

k is the number of the estimated parameters in the model including σ (standard deviation) and

the intercept and )ˆ( yl θ is the numerical value of the likelihood at its maximum [25]. AIC

gives an estimate of the expected relative distance between the fitted model and the unknown

true model. The value of AIC tells the information lost if we use that particular model to

approximate the truth model and therefore the model with the smallest value of AIC is

considered the best among a set of candidate models. It is useful to define the AIC

differences as Δi=AICi – AICmin, where AICmin is the smallest value of the AIC values for all

the set of candidate models. It is important to note that it is not the absolute size of the AIC

value, it is the relative values and hence the Δivalues are important [23]. Given the data and

the set of m models, Akaike's weight for a model i is defined as:

∑=

∆−

∆−= m

ri

ii

1))2/1exp((

))2/1exp((ω (13)

It is important to note that ωi depends on the entire set of models; therefore if a model is

added or dropped during the analysis, then the ωi must be recomputed for all the models in

the newly defined set. ωi is considered as the weight of evidence in favor of model i being

the best model in the m candidate models. The ratio of evidence of a model g is the Akaike

weight of that model devilled by the Akaike weight of the best model. The ratio of evidence

of a model gives an evidence of a kind of weak or strong support for the best model versus

any other model in the set of candidate models. Such ratios represent the evidence about

fitted a model as to which is better in a k-L in information sense. In particular there is often

interest in the ratio ωi / ωj where model i is the estimated best model and j indexes the rest of

the models in the set [7, 9, 10]. These ratios are not affected by any other model and therefore

it does not depend on the full set of m models, just on models i,j. For a 95% confidence set on

the actual K-L best model, one of the approaches is based on Akaike weights, interpreted as

approximate probabilities of each model being the actual best model, given the data. In this

approach we sum the Akaike weights from largest to smallest until the sum is just <0.95, then

the corresponding subset of models as a type of confidence set on the K-L best models.


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4.2. R2 Method

One of the most famous methods used in model selection is the R2 method. Given data a set

of data, the coefficient of multiple determinations, R2 is calculated as:

R2 =1-SSYSSE (14)

Where SSE and SSY are given as:

SSE= 21

)ˆ( in

i i yy −∑ =(14a)

SSY= 21

)( in

i i yy −∑ =(14b)

yi. is the experimental value, y is the average value of yi. and iy is the corresponding

predicted value of yi.. Under the R2 method, one selects the best model to be the model with

the largest R2[26, 27]. Rencher and Pun [28] found this approach to be very poor. There are

many examples of models with large values of R2 but with very poor approximation of the

truth [7, 9]

4.3. AIC vs. R2

The question now is why do we claim that the AIC approach is a better approach? First, the

AIC is theoretically sound; in other words it is a formula that is mathematically derived and

proved. AIC is derived to minimize the information lost when we use the best chosen model

to approximate the truth model which we usually don't know. Therefore, the value of AIC for

a particular model tells important information about how close is that model to the true

model. Secondly, the AIC gives an order of the models from best to worst and clearly states if

some of the models should be excluded from considering. In particular, a candidate model

with a value of Δ> 10 should not be considered as a good model [7, 9]. Thirdly, the AIC

confidence set of best models, gives a clear cut of the models that should be considered as

one can clearly see the models that are good and the models that have poor fit and therefore

use only the models that lie in the confidence set. Furthermore, Akaike weight allows us to

see how strong is the fit of a model compared to the other models where we can’t see this


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from, like R2 analysis. In fact R2 value doesn't give us a specific way to compare how the fit

of a model compared to other models. Furthermore, AIC takes the number of parameters in

the model under account where R2 analysis doesn't. Taking the number of parameters under

account is important in case of over fitting and also important to the preferred simplified

models. In AIC analysis, all factors are taking into the formula such as the experimental

values, the expected values, the number of parameters used in the model and the standard

deviation which make the AIC analysis more appropriate. Another reason is that the AIC

ranking is more stable than the R2 ranking of the models which means the order of the models

from best to worse. In the AIC analysis, the order stays the same for subsets of the data where

as it doesn't in the R2 analysis [10, 23]. Finally, it is important to realize that when we are

looking for the best model, we are not modelling the data but rather we are trying to model

the information in the data. Indeed, the data contains noise besides information and therefore

the AIC method fits better in this context as it is based on the Kullback-Leibler (K-L)

Information definition.

5. Experimental Data

The frictional pressure losses data was gathered using the small-scale flow loop shown in

Figure 1.

Fig. 1. Schematic of Small-Scale Flow Loop


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It consists of 825 litre tank for fluid mixing and storage. A model 5M Deming centrifugal

pump is used to feed the fluid to 6P10 Moyno progressive cavity pump that can achieve a

flow rate of up to 0.009 m3/s at 4.1×106 Pa. Several full opening ball type valves are used for

the control of fluid flow. Three 1.27 cm OD diameter stainless steel coiled tubing with 0.01,

0.019, and 0.031 curvature ratio are used as flow conduit. This includes the typical range of

curvature ratio encountered in the field application. A differential pressure sensor is used to

measure the frictional pressure losses. An absolute pressure sensor is used to measure the

system pressure (maximum working pressure of the system is 6.9×106 Pa). A MicroMotion®

flow meter (1.27 cm OD) is used to measure the flow rate, density, and temperature of test

fluid. The data are transmitted through a wireless Fluke Hydra data logger system to a

personnel computer and displayed live on the screen.

Water as a Newtonina fluid was circulated in the flow loop where coiled tubing was replaced

to include three different curvature ratios of 0.01, 0.019, and 0.031. Data gathered from the

flow loop included: flow rate, differential pressure across tubing length, fluid density, and

fluid temperature. Data readings in the transition region due to change in flow rate or due to

purging operation and any unstable points were eliminated. Only steady state data points in

the stabilized region were averaged and used for the analysis. Flow rate and pressure drop

values were first converted to conventional Reynolds number and Fanning friction factor,

respectively. These two dimensionless groups were used in the characterization of fluid flow

through tubing.

The dimensionless variables, Fanning friction factor, f and Reynolds number, NReare given as

follows:

𝑓 = d∆P2lρv2

(15)

𝑁𝑅𝑒 = dvρμ

(16)

A comparison between experimental water data and predictions from the six correlations for

three curvature ratios is shown in Figure 2 for coiled tubing with 0.01 curvature ratio and

Figure 3 for coiled tubing with 0.019 curvature ratio. Figure 4 shows the comparison for

coiled tubing with 0.031 curvature ratio.


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6. Results and Discussion

From Figures 2 through 4 for different curvature ratios, it can be seen that both CTC and

Mashelkar and Devarajan correlations predict low values for the Fanning friction factors. The

other correlations give good predictions with relatively low small errors. However, the errors

of using inappropriate correlation can range from 0.2% to about 39%. Therefore, a more

accurate method is vital to select the best model. One of the most common and simple

methods to select the best model is the R2 method. A recently considered method is AIC. So,

in our analysis, both methods are used and a comparison between them is presented to not

only select the best but also to place all the correlations in order, from the best to the worst.

To analyze the given models, R2 values and AIC values for each model are calculated and

compared. As stated earlier for all given models we use the experiential data to find σ for all

models by which we find the value of AIC. We then find the Δ values and the Akaike weights

as described earlier in the paper.

Fig. 2: Comparison between Experimental Water Data and Predictions from Correlations for 0.01 CT


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Fig. 3: Comparison between Experimental Water Data and Predictions from Correlations for 0.019

CT

For further indication about how strong is the prediction of the model compared to the rest of

the model, we calculated the Akaike weights, ω for all the models.

Fig. 4: Comparison between Experimental Water Data and Predictions from Correlations for 0.031

CT


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The table below summarizes the values which will be used in model analyses.

For more analyses, we also calculate the R2 values for all the models. The model with the

largest R2 value was found to be McCann and Islas model with R2 = .928.The R2 value for

Mishra and Gupta is 0.920 whereas the R2 value for the It model equals 0.847. Finally the R2

values for the Mashelkar and Devarajan model and CTC model are found to be 0.02 and 0.0

respectively.R2 values for all the models are summarized in the table below.

By looking at the results above, we see there are two ranking; AIC and R2 as shown in table

3. As stated earlier the AIC ranking is better to use in this context.

Table.1: AIC Parameters.

Model Name k σ log(L) AIC Δ EXP(-.5Δ) ω Srinivasan 3 0.00025 99.7821 -196.564 3.3 0.191878 0.105

Ito 3 0.00031 97.10299 -191.206 8.7 0.013167 0.007 Mishra and Gupta 3 0.00022 100.955 -198.91 0.96 0.620038 0.339

Mashelkar and Devarajan 3 0.0044 65.22376 -127.448 72.41 1.88E-16 0.0 McCann and Islas 3 0.00021 101.4332 -199.866 0.0 1.000169 0.548

CTC 3 0.00128 79.97771 -156.955 42.91 4.81E-10 0.0

Table.2: R2 Values. Model Name R2

Srinivasan 0.909 Ito 0.847

Mishra and Gupta 0.920 Mashelkar and Devarajan 0.022

McCann and Islas 0.928 CTC 0.000

Table.3: Ranking of the Correlation.

Model Name R2 R2 Ranking Akaike’s Weights AIC Ranking Srinivasan 0.909 3 0.1501 3

Ito 0.847 4 0.0072 Poor Mishra and Gupta 0.920 2 0.3397 2

Mashelkar and Devarajan 0.022 5 0.0000 Poor McCann and Islas 0.928 1 0.5478 1

CTC 0.000 6 0.0000 poor

From these tables, we can see that the best model among all the models is McCann and Islas

model and the second best model is Mishra and Gupta model. On the other hand, the weakest

models among the six models are Mashelkar and Devarajan model and CTC model.

Furthermore we see that, according to the AIC ranking, the 95% confidence set of models

contains McCann and Islas model, Mishra and Gupta model and Srinivasan model. In other


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words the best three models that can be considered are these three models in the confidence

set. However, Ito model still can be considered but it is very weak compared to the other

models in the confidence set of models. In fact, it is clear from the Akaike weights for the

models in the table above that McCann and Islas model with the largest Akaike weight of

0.548. A value of 54.8% is considered as the weight of evidence in favor of McCann and

Islas modelbeing the best model in the set of all six models. Both R2 method and AIC method

suggest that only three models should be considered namely McCann and Islas, Mishra and

Gupta and Srinivasan. However the AIC states that McCann and Islas is far better than

Srinivasan model as the ratio between their Akaike weights is 0.55/0.11 = 5 which states that

McCann and Islas model is five times better than the Srinivasan model. There is no doubt

that McCann and Islas model and Mishra and Gupta model are both good models in this

context by both ranking, however McCann and Islas model is better than Mishra and Gupta

model as their Akaike weights ratio is 0.55/0.33=1.7. We clearly see the advantages of the

AIC method over the R2as the AIC gives the set of models that can be considered. In

particular the AIC ranking clearly indicates that the CTC model and the Mashelkar and

Devarajan should not be used in any prediction. The efficiency of AIC methods in this model

selection process is clear. Indeed this method not only ranks the models but also separates the

models that should not be considered. Furthermore, by the Akaike ratios, it is clear how the

selected models should be preferred.

7. Conclusions

This paper used the AIC approach in selecting the best model to predict the Fanning friction

factor. Both AIC and the R2 methods suggest that the best model to predict the Fanning

friction factor is McCann and Islas and the weakest one is the CTC Model. However, the AIC

ranking suggests that Ito, Mashelkar and Devarajan and CTC models should all be excluded

from consideration as their Akaike weights are very negligible. Furthermore the R2values for

both Mashelkar and Devarajan and the CTC models are 0.0 suggesting that these two models

are good predictors for the Fanning friction factor. In summary, the study strongly

recommends using McCann and Islas model if possible as its Akaike weight is about 55%

indicating a strong evidence of its prediction. Also, the present paper summarized the

concepts of both R2 and AIC and showed the advantages of AIC method over R2 method. The


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authors believe that, AIC is a new tool that can best help to select the accurate models and

advances the state-of-the-art of models comparison.

Nomenclature

A, B = Constants in Eq. 9.

a, b = Constants in Eq. 8.

CT = Coiled tubing

d = Tubing inside diameter [cm]

f = Fanning friction factor [dimensionless]

i & j = Any two models in the set

l = Pipe length between pressure ports [m]

m = Number of models in the set

n = Flow behavior index [dimensionless]

NDe = Dean number [dimensionless]

NRe = Reynolds number [dimensionless]

NRec = Critical Reynolds number [dimensionless]

OD = Outside diameter [cm]

Δp = Pressure drop [Pa]

r = Tubing radius [cm]

R = Reel radius [cm]

r/R = Coiled tubing curvature ratio [dimensionless]

SSE = Summation of squared residuals

SSY = Summation of squared errors

v = Average fluid velocity [m/s]

μ = Newtonian viscosity [Pa.s]

μ@511 = Apparent viscosity at 511 s-1 shear rate [Pa.s]

ρ = Fluid density [km/m3]

ω = Akiake weight [dimensionless]

Δ = Information lost compared with the best model


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