A Three-Coefficient Model with Global Optimization for Heavy End Characterization of Gas Condensate PVT Data

Authors

1 Department of Chemical Engineering, Faculty of Petroleum, Gas and Petrochemical Engineering, Persian Gulf University, 7516913897 Bushehr, Iran

2 Department of Petroleum Engineering, Faculty of Petroleum, Gas and Petrochemical Engineering, Persian Gulf University, 7516913897 Bushehr, Iran

Abstract

Characterization of heavy end, as plus fraction, is among the most crucial steps in predicting phase behavior of a hydrocarbon fluid system. Proper selection of single carbon number (SCN) distribution function is essential for heavy end characterization. The SCN distribution function is subject to fluid nature. The exponential distribution function has been and is widely applied to gas condensate plus fractions. More complicated functions are necessary in systems with jumps or discontinuities in successive SCN groups. Thirty fluid samples of a supergiant gas condensate reservoir are analyzed, most of which showing a discontinuity at SCN=10. A three-coefficient model is developed and then applied to determine the distribution function. The plus fraction is divided into three zones, each characterized by an adjustable parameter. A global optimization algorithm is developed and then applied to obtain unique coefficients for complete set of samples. This developed model predicts the experimental data with 10.6% accuracy and is in better agreement with experimental data compared to existing distribution functions. 

Keywords

Main Subjects


1. Introduction

In simple mixtures, (e.g. systems that contain 2-3 components), the individual components are readily identifiable through the routine analysis techniques, while, many fluid systems in nature and in chemical industry are not readily identifiable because their mixture contains too many components, most of which with similar formula and close physical and chemical properties. This makes their analysis and separation difficult , yet impossible to full characterize and analyze for some mixtures (Danesh, 1998; K. S. Pedersen & Christensen, 2007). Petroleum fluids, polymer solutions, edible oil types, and coal tar liquids are examples of these complex mixtures. Compositional variations are gradual in complex mixtures, and composition of individual components may become hard to distinguish in some cases. Distribution functions are applied to describe compositional variations of successive components in these mixtures. Composition of petroleum fluids can be divided into two categories of the ones containing light components separately identify the ones known as "plus fraction", associated with continuous composition and characterized by distribution functions. For the second category distribution of compositions is described through equation (1), (Ahmed, 2007; Du & Mansoori, 1987; Mansoori, Du, & Antoniades, 1989):

(1)

 

where, F(I) is the distribution function and I is an independent variable that can be either molecular weight or normal boiling point. The plus fraction and discrete components are weighted by   and xi mole fractions, respectively. The two-parameter exponential distribution function has been and is widely applied to characterize heavy end in gas condensate systems (Elsharkawy, 2003; K.S. Pedersen, Blilie, & Meisingset, 1992; K.S. Pedersen, Fredenslund, & Thomassen, 1989; Zuo & Zhang, 2000). In this function, the mole fraction of a component has a linear correlation with its molecular weight in a semi-log coordinates system. In cases where jumps are evident in compositions of successive components, usually observed in SCN=8, the two-parameter exponential distribution function is no longer applicable. In these cases, the three-parameter gamma distribution function may describe the compositional distribution of plus fraction better (Whitson, 1983, 1984; Whitson, Anderson, & Soreide, 1990). This function increases calculation steps and requires parameters like molecular weight of plus fraction, which is reported as being uncertain in many cases. Ahmed et al. (Ahmed, Cady, & Story, 1985) proposed a simple model to describe the jump at SCN=8 by analyzing 34 oil and gas condensate fluid samples. They applied two linear functions for describing changes in molecular weight in SCN. The first line describes changes between SCN=7-8, and the second line considers the distribution from SCN=8 to the end. This approach avoids complexities associated with gamma distribution function. Hosein et al. revealed that the jump or discontinuity in SCN compositions is not limited to SCN=8 and may occur again in SCN=13 for some systems (Hosein, McCain, & Jagai, 2012; K.S. Pedersen, Thomassen, & Fredenslund, 1983, 1984). As the exponential and gamma distribution functions predict the normal distribution for compositions of SCN, they may under-predict the composition of SCN=13 and over-predict for SCN=12. To overcome this issue, Hosein et al. proposed a model to characterize the heavy end with four-coefficient (4CM) by putting four linear functions in four divisions of heavy end compositional zones  (Hosein et al., 2012). This approach is an extension of Ahmed et al (Ahmed et al., 1985) method which covers the jumps in both SCN=8 and 13. They applied the 4CM on 20 fluid samples separately and reported average values of model parameters calculated by minimizing the objective function for each one of the 20 samples. The 4CM proposed by Hosein et al. (Hosein et al., 2012) lacks a global optimization on selected samples, therefore, it is expected that applying the same parameters to other gas condensate fluid systems lead to considerable error.

Here, a global optimization procedure is proposed and applied on fluid samples taken from a supergiant gas condensate field to customize the model parameters for the field under study. In the following sections, the gas condensate field is described first, followed by a review of available pressure-volume-temperature (PVT) data. Next, methodology and model development for three-coefficient model (3CM) is developed and solution technique is described. Then, the results are presented and discussed. Concluding remarks appear at the last section of the article.

2. Available PVT Data

The gas-condensate field under study is located at Persian Gulf, for which 87 PVT reports are collected between 1992-2013. A quality control protocol is applied on sampling conditions and PVT analysis of all samples, thirty PVT reports are found to pass the screening criteria reported by (Drohm, Trengove, & Goldthorpe, 1988; Moffatt & Williams, 1998). Tables 1-a and 1-b the properties of validated samples are tabulated in Tables 1-a and 1-b. The sampling depth and temperature were 2700-3450 (m) and 86-110 (°C), respectively and the wells are distributed all over the whole reservoir area. Preliminary PVT studies reveal that reservoir fluid is lean gas condensate.

 

 

 

 

 

 

Table 1-a. Composition of SCN Groups and Properties of C7+ Fractions for Samples 1 to 15 Applied on the Improved Model

SCN

Well Name

W1

W2

W3

W4

W5

W6

W7

W8

W9

W10

W11

W12

W13

W14

W15

7

0.520

0.510

0.480

0.460

0.470

0.440

0.480

0.490

0.450

0.490

0.480

0.707

0.677

0.550

0.503

8

0.510

0.490

0.450

0.430

0.440

0.450

0.420

0.460

0.370

0.400

0.410

0.573

0.560

0.403

0.364

9

0.340

0.330

0.290

0.280

0.290

0.300

0.290

0.300

0.220

0.240

0.254

0.363

0.346

0.226

0.206

10

0.260

0.250

0.220

0.210

0.220

0.230

0.210

0.230

0.190

0.220

0.252

0.373

0.351

0.254

0.221

11

0.180

0.170

0.150

0.150

0.150

0.160

0.150

0.160

0.150

0.180

0.177

0.262

0.250

0.173

0.152

12

0.130

0.130

0.117

0.110

0.106

0.120

0.110

0.130

0.120

0.140

0.135

0.199

0.186

0.136

0.119

13

0.120

0.110

0.100

0.097

0.094

0.110

0.100

0.110

0.100

0.120

0.128

0.189

0.177

0.121

0.107

14

0.090

0.080

0.075

0.072

0.069

0.080

0.070

0.080

0.070

0.090

0.096

0.144

0.136

0.089

0.079

15

0.070

0.070

0.063

0.060

0.057

0.060

0.060

0.070

0.050

0.070

0.076

0.116

0.109

0.072

0.066

16

0.050

0.050

0.045

0.043

0.041

0.040

0.040

0.050

0.040

0.050

0.053

0.085

0.080

0.050

0.047

17

0.040

0.040

0.037

0.035

0.033

0.040

0.030

0.040

0.030

0.040

0.045

0.074

0.070

0.039

0.039

18

0.040

0.030

0.031

0.029

0.028

0.030

0.030

0.030

0.020

0.030

0.038

0.063

0.060

0.031

0.032

19

0.030

0.020

0.025

0.023

0.022

0.020

0.020

0.030

0.020

0.030

0.029

0.051

0.048

0.023

0.026

C20+

0.090

0.070

0.080

0.070

0.064

0.060

0.040

0.070

0.030

0.070

0.090

0.180

0.180

0.040

0.070

Properties of  Plus fraction:

ZC7+

2.380

2.280

2.083

2.070

2.020

2.130

2.050

2.250

2.060

2.390

2.440

3.620

3.450

2.410

2.210

MWC7+

139.126

137.119

138.479

136.540

134.970

135.840

132.860

140.000

136.000

141.000

139.580

143.780

143.980

133.200

136.870

                                             

 

Table 1-b. Composition of SCN Groups and Properties of C7+ Fractions for Samples 16 to 30 Applied on the Improved Model

SCN

Well Name

W16

W17

W18

W19

W20

W21

W22

W23

W24

W25

W26

W27

W28

W29

W30

7

0.480

0.575

0.575

0.533

0.533

0.534

0.575

0.324

0.401

0.385

0.454

0.496

0.552

0.510

0.565

8

0.353

0.424

0.424

0.390

0.390

0.412

0.406

0.301

0.310

0.303

0.371

0.373

0.382

0.393

0.383

9

0.193

0.236

0.236

0.218

0.218

0.233

0.234

0.138

0.165

0.158

0.191

0.216

0.219

0.219

0.223

10

0.216

0.266

0.266

0.251

0.251

0.269

0.265

0.169

0.186

0.185

0.219

0.245

0.246

0.249

0.254

11

0.149

0.177

0.177

0.172

0.172

0.191

0.178

0.121

0.128

0.129

0.153

0.166

0.166

0.171

0.176

12

0.119

0.129

0.129

0.139

0.139

0.156

0.141

0.095

0.101

0.101

0.120

0.130

0.133

0.136

0.144

13

0.100

0.114

0.114

0.121

0.121

0.141

0.122

0.083

0.087

0.089

0.105

0.112

0.118

0.121

0.131

14

0.073

0.084

0.084

0.091

0.091

0.108

0.092

0.061

0.064

0.065

0.078

0.079

0.088

0.090

0.098

15

0.062

0.084

0.084

0.075

0.075

0.091

0.076

0.050

0.051

0.052

0.063

0.062

0.073

0.075

0.083

16

0.042

0.051

0.051

0.054

0.054

0.067

0.055

0.035

0.036

0.036

0.045

0.042

0.053

0.055

0.061

17

0.033

0.041

0.041

0.044

0.044

0.056

0.045

0.028

0.028

0.029

0.036

0.031

0.043

0.045

0.050

18

0.028

0.036

0.036

0.037

0.037

0.048

0.038

0.023

0.023

0.023

0.030

0.024

0.036

0.038

0.043

19

0.021

0.027

0.027

0.030

0.030

0.038

0.030

0.017

0.017

0.017

0.023

0.016

0.029

0.030

0.034

C20+

0.050

0.090

0.090

0.090

0.090

0.120

0.100

0.047

0.042

0.047

0.068

0.030

0.090

0.090

0.110

 Properties of  Plus fraction: 

ZC7+

2.100

2.540

2.540

2.430

2.590

2.650

2.560

1.640

1.800

1.770

2.130

2.200

2.420

2.400

2.540

MWC7+

134.220

136.460

136.460

138.750

140.640

143.680

138.600

136.450

133.710

135.270

137.010

131.350

137.930

139.280

141.230

                                                           

 

 

The composition of successive SCN for W29 and W16 samples. According to this figure and Tables 1-a and 1-b, the compositional jump, or discontinuity, for most samples is observed at SCN=10, rather than at SCN=8 and 13, as studied by Ahmed et al. (Ahmed et al., 1985) and Hosein et al. (Hosein et al., 2012). The average relative error of SCN predictions for all samples by these techniques are tabulated in Table 2, where, the former technique predictions show average relative errors of 7.05-31.04%, while the latter shows average error within 4.45-37.05% range. The large error of SCN prediction by these techniques and the discontinuity at SCN=10 necessitates the development of these models for the samples under study.

 

 

   

(b)

(a)

Figure 1. Molar Distribution of SCN Composition for Samples: (a) W29, (b) W16

Table 2. Average Relative error of SCN Predictions for All Samples by Ahmed et al. (Ahmed et al., 1985) and Hosein et al. (Hosein et al., 2012) Techniques

SCN

(Ahmed et al., 1985)

(Hosein et al., 2012)

C7

25.69

7.17

C8

19.44

37.05

C9

31.04

31.23

C10

10.00

8.42

C11

7.05

4.45

C12

10.44

20.97

C13

23.46

12.96

C14

20.02

16.58

C15

23.84

17.96

C16

14.69

6.51

C17

17.76

8.81

C18

22.93

15.24

C19

23.64

16.38

Average

19.23

15.67

 


3. Modeling

According to Ahmed et al. (Ahmed et al., 1985), composition of each SCN is calculated as a function of molecular weight and a composition of plus fraction as follows:

(2)

 

The Mn, the molecular weight of SCN=n, is introduced by (Ahmed, 2007; Katz & Firoozabadi, 1978; Whitson, 1984). Normally, molecular weight and composition of C7+ fraction can be measured in the laboratory with accuracy. Here, Eq. (2) is applied in calculating the composition of heavier compounds by starting with C7+. Based on the fluid nature and composition the molecular weight of C7+ fraction is segmented into three different segments and a three-coefficient model (3CM) is developed for plus fraction, expressed as Eqs. (3)-(5):

(3)

 

(4)

 

(5)

 

where, Mn+ and Mn are the molecular weights of plus fraction and nth component at SCN=n, respectively. S1, S2, and S3 are the adjustable parameters calculated through global optimization of the overall objective function, OOF, defined by Eq. (6). In this approach, all samples are treated in a simultaneous manner.

(6)

 

The objective function of an individual sample i , OFi, is the sum of absolute relative errors in all SCN groups’ composition in this set of data, expressed through equation (7):

 

(7)

The flowchart of calculations is shown in Fig. (2). A database of experimental data including molecular weight and composition of plus fraction, together with composition of SCN groups is constructed for all samples and is applied as the input of the model. An initial guess is made for model parameters, next, Eqs. (2- 5) are solved for each sample to calculate composition of SCN groups and molecular weight of plus fraction. This procedure is run for C7+ to C20+ and SCN=7 to SCN=19. Composition of the last fraction is calculated through the material balance:

(8)

 

Equation (7) is then applied to calculate the objective function for each sample. This procedure is repeated for all samples and global objective function is calculated using equation (6). The optimization algorithm, Levenberg-Marquardt (Chandler, 1985), is then applied in minimizing the global objective function. The output of the proposed algorithm include optimum model parameters, composition of SCN groups and molecular weight of plus fraction for each SCN group, which are calculated for all samples in a simultaneous manner. Finally, the results of modeling are compared with the two-parameter exponential (K.S. Pedersen, Thomassen, & Fredenslund, 1989), Ahmed et al. (Ahmed et al., 1985), three-parameter gamma distribution function (Whitson, 1983) and 4CM (Hosein et al., 2012) models. Details of these models are given in Appendix.

 

 

 

Figure 2. Flow Chart of Global Optimization of Data based on Marching Technique (Hosein et al., 2012).


4. Results and Discussions

The statistical analysis of average SCN composition for all PVT samples are tabulated in Table 3, where, the standard deviations in average SCN composition is relatively high, indicating a considerable compositional variation in the same reservoir. The wide compositional variations among samples increase flexibility and wider range of model parameters.

Table 3. Statistical Analysis of Average SCN Composition for 30 Samples.

SCN

Average composition

Relative Standard Deviation (%)

7

0.507

15.10

8

0.412

15.56

9

0.246

22.91

10

0.241

17.62

11

0.168

17.40

12

0.130

17.05

13

0.115

19.75

14

0.086

21.54

15

0.071

21.89

16

0.050

23.09

17

0.041

26.29

18

0.034

28.75

19

0.027

30.93

C20+

0.079

45.85

The optimized parameters for the 3CM is given in table 4.

 

Table 4. Parameters of 3CM

SCN

     

Parameter

S1

S2

S3

Value

15.7

10.9

13.3

 

Applying the developed model, the molecular weight trend for plus fraction in the SCN=7-19 range is shown in Fig. (3) for one sample. This trend is divided in three regions of: I, showing the natural trend of molecular weight versus SCN for  and II-III, representing the molecular weight versus SCN for  and  by considering the discontinuities in SCN=9 and 10, respectively.

 

 

(a)

 

(b)

 

(c)

Figure 3. Plus Fraction Molecular Weight Distribution. (a): Region I, , (b): Region II, , (c): Region III,

 

 

The average relative error of composition applying 3CM is calculated for all samples and the results are tabulated in table 5, along with the calculation results for two-parameter exponential (K.S. Pedersen, Thomassen, et al., 1989), Ahmed et al. (Ahmed et al., 1985), three-parameter gamma distribution function (Whitson, 1983) and 4CM (Hosein et al., 2012) models. The average absolute deviation (AAD%) is defined through Eq. (9):

(9)

 

where, L is the total number of samples.

According to Table 5, the developed 3CM yields the minimum ARE% among all models. For the reservoir under study, the AAD% is 10.6% for 30 samples, while the two-parameter exponential (K.S. Pedersen, Thomassen, et al., 1989), Ahmed et al. (Ahmed et al., 1985), three-parameter gamma distribution function (Whitson, 1983) and 4CM (Hosein et al., 2012) models are 37.2%, 20.0%, 12.0% and 16.0%, respectively. The model validity is checked against composition of SCN groups and properties of C7+ fractions for two samples of Trinidad (Hosein & McCain, 2009) and North Sea (Ahmed et al., 1985). The absolute average deviations of SCN compositions for Trinidad and North sea samples are 16.2% and 13.7%, respectively.

The calculated results for the average composition of SCN groups by 3CM (column bars in white) are compared with average experimental SCN compositions (column bars in gray), Fig. (4), where, as observed the tolerances of the model prediction were also shown on each bar. Results indicatein this figure indicate an excellent match between experimental data and model predictions.

 

Table 5. AAD% of SCN Compositions with Different Models

Well Name

Exponential (K.S. Pedersen, Thomassen, et al., 1989)

Ahmed et al. (Ahmed et al., 1985)

4CM (Hosein et al., 2012)

Gama (Whitson, 1983)

This work

W1

40.8

14.0

10.1

8.4

14.1

W2

34.6

14.7

11.0

6.5

10.1

W3

12.9

12.7

8.1

8.6

12.2

W4

11.5

15.2

12.3

9.7

11.4

W5

12.3

15.6

11.1

13.6

13.0

W6

16.9

19.2

14.6

10.0

12.4

W7

13.9

23.0

18.3

11.4

11.7

W8

23.3

15.4

10.8

7.0

10.3

W9

63.7

24.2

21.0

14.2

12.2

W10

15.4

17.7

15.8

10.3

8.2

W11

23.9

19.5

15.4

11.0

10.4

W12

84.8

16.9

13.0

10.1

9.0

W13

75.9

15.5

12.1

9.9

8.7

W14

34.5

30.1

25.6

17.4

12.4

W15

13.7

20.8

16.7

13.7

9.7

W16

32.9

24.3

20.0

13.6

11.3

W17

29.2

18.4

14.9

14.1

10.3

W18

22.0

19.8

15.7

13.6

9.9

W19

22.6

20.8

16.8

12.7

9.4

W20

22.6

17.7

14.5

12.5

7.0

W21

40.9

20.8

17.1

11.8

8.4

W22

30.2

18.4

14.7

11.9

8.6

W23

124.0

24.2

19.8

13.6

12.9

W24

100.6

24.5

20.1

14.1

12.0

W25

96.9

23.5

19.1

12.9

11.0

W26

22.4

21.1

16.8

12.3

9.8

W27

20.1

31.0

25.9

16.6

12.9

W28

20.6

19.9

16.2

13.7

9.7

W29

21.0

20.9

16.9

12.6

9.1

W30

31.5

20.0

17.0

13.4

9.1

Average

37.2

20.0

16.0

12.0

10.6

 

 

 

 

Figure 4. Average Composition of SCN Groups. Solid Fill Column: Experimental Data, No Fill Column: Results of Developed Model

 

   

 

(a)

(b)

 

   

 

(c)

(d)

 

Figure 5. Distribution of Molecular Weight of Plus Fraction Versus SCN for Sample W29 with Different Models: Ahmad et al. (▬ ▪ ▬), 4 CM (▪▪▪▪), this work (▬▬), Experimental Data (▲). (a) Comparison of All Models, (b) Ahmad et al. Model (Ahmed et al., 1985), (c) 4CM (Hosein et al., 2012), (d) this work (3CM).

 

       

 

 

 

The distribution of molecular weight of plus fraction versus SCN for sample W29 with different models is shown in Fig.(5). The results of Ahmed et al. (Ahmed et al., 1985) model,  the 4CM (Hosein et al., 2012), and the developed model are compare with experimental data in this figure, where, the 3CM is in good agreement with experimental data. As observed, it is clear that the Mn+ slope versus SCN changes in Ahmed et al. (Ahmed et al., 1985) to take into account the discontinuity of compositions at SCN=8, which is not the case for fluids of this reservoir. The 4CM (Hosein et al., 2012) and 3CM proposed here show three times and twice changes in original slope of molecular weight profile. Moreover, both Ahmed et al. (Ahmed et al., 1985) and 4CM (Hosein et al., 2012) models overpredict the experimental results, although the 4CM (Hosein et al., 2012) has three shifts in line slope to better match the data.

The effect of this overprediction on phase behavior of the sample is shown in Fig. (6), where, the P-T results are calculated according to Ahmed et al. (Ahmed et al., 1985) and 3CM models for splitting  plus fraction from SCN of 7 to 20.

It is important to note that the cross points, (i.e. boundaries between zones) coincide in this proposed 3CM, which is not the case in Ahmed et al. (Ahmed et al., 1985) and 4CM (Hosein et al., 2012) models. For example, in Ahmed et al. (Ahmed et al., 1985) model, the two linear functions proposed to describe the jump in SCN=8 do not yield the same values of Mn+. Similar results are obtained when the 4CM introduced by (Hosein et al., 2012) is applied. The discontinuity in predicting molecular weight of plus fraction at jump points is resolved in this proposed model by defining Eqs. (3-5) for different zones, as each zone applies the information of plus fraction nearest to it.

 

 

 

 

Figure 6. Effect of Distribution Function on P-T Diagram for Sample W29. Ahmed et al. (▪▪▪▪) (Ahmed et al., 1985), and 3CM (▬▬) Models

 


5. Conclusion

A global optimization is applied to a comprehensive set of gas condensate data set to characterize plus fractions that show discontinuity, or jump, in composition of SCN=10. It is revealed that the nature of plus fraction is important in selecting the proper distribution function. Therefore, sufficient number of PVT samples are necessary to determine the distribution function which yields the best characterization for heavy end. The proposed 3CM is tested on selected gas condensate samples and is revealed that this model can describe the SCN behavior of plus fraction better than its counterparts. The simplicity of this developed model together with global optimization where the data of all samples are applied, yield flexible, efficient and unique adjustable applicable parameters.

 

List of Symbols

A & B

Constants in equation (A-1)

4CM

Four Coefficient Model

AAD%

Absolute Average Deviation percent

F

Distribution function

I

Independent parameter of distribution function

L

Number of sample set

M

Molecular weight

N

Number of SCN groups

OF

Objective Function of each sample (well)

OOF

Overall Objective Function

S

Adjustable parameters of equations (3) to (5)

SCN

Single Carbon Number

x

Component mole fraction

Z

Molar composition of SCN group

 

 

Greek Letter

 

 

Distribution shape

 

Variable in equation (A-3)

 

Minimum molecular weight of the plus fraction

 

 

Superscripts

 

Cal

Calculation

Exp

Experimental

 

 

Subscripts

 

+

Plus fraction

i

Number of samples

j

Number of SCN groups

n

Number of SCN group

Appendix: Distribution Functions

Exponential Distribution function:

Pedersen et al. proposed equation (A-1) to correlate composition and molecular weight of SCN groups (K.S. Pedersen, Thomassen, et al., 1989):

(A-1)

 

 and  are the composition and molecular weight of SCN=i. Parameters A, B are the adjustable parameters, obtained by optimization methods. In this study A= 0.9813 and B= -0.0179.

Gamma Distribution function:

Whitson et al. proposed a three-parameter distribution function to describe the discontinuity in SCN=8, Eq. (A-2) (Whitson, 1983, 1984; Whitson et al., 1990):

(A-2)

 

Γ is the gamma function, and α,  M and  are the distribution shape, molecular weight and minimum molecular weight of the plus fraction, respectively.  is calculated through Eq. (A-3):

(A-3)

 

where, MC7+ is the molecular weight of C7+ plus fraction. By integrating (A-2) the following is yield:

(A-4)

 

The mole fraction of SCN=I, , is calculated through Eq. (A-5):

(A-5)

 

Ahmed et al. (Ahmed et al., 1985) distribution function:

Ahmed et al. (Ahmed et al., 1985) used equation (2) to calculate composition of SCN:

(2)

 

To obtain discontinuity in SCN=8, two different distribution functions, with the same form as Eq. (A-6) and different adjustable parameters, S are applied:

(A-6)

 

The adjustable parameter, S is presented in Table A-1:

Table A-1. Constant of Equation (A-6) Proposed by Ahmed et al. (Ahmed et al., 1985)

SCN

Gas condensate

Oil

n≤ 8

15.5

16.5

n> 8

17

20.1

The four-coefficient model (Hosein et al., 2012):

In this model Eqs. (A-7) and (A-8) are applied to obtain the discontinuities in SCN=8, 13:

(A-7)

 

(A-8)

 

The plus fraction distribution function is divided into four zones, the constants of which are tabulated in Table (A-2).

Table A-2. Constant for equations (A-7) and (A-8) (Hosein et al., 2012)

SCN

n=8

8<n<13

n=13

n>13

S

12.5

16

13

14.5

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