Impact of Lumping Techniques for Fluid ...

Australian Journal of Basic and Applied Sciences, 7(1): 320-333, 2013 ISSN 1991-8178

Impact of Lumping Techniques for Fluid Characterization in Gas Condensate Reservoir 1

Hamidreza moghadamzadeh, 2Hojjatolla Maghsoodloorad, 3Adel zarabpour, 2,4Ali Hemmati, 5Shahin Shahsavari, 6Shadab Shahsavari 1

Islamic Azad University South Tehran Branch, P.O. Box 19585-466, Tehran, Iran School of Chemical Engineering, Iran University of Science and Technology (IUST), Narmak, Tehran, Iran 3 Islamic Azad University Tabas Branch, 4-police University 5 Department of Chemical Engineering, University of Sistan and Baluchestan, Zahedan, Iran 6 Department of Chemical Engineering, Varamin (Pishva) Branch, Islamic Azad University, Tehran, Iran 2

Abstract: Several problems are associated with “regrouping” the original components into a smaller number without losing the predicting power of the equation of state. These problems include (A) How to select the groups of pure components to be represented by one pseudo-component. (B) What mixing rules should be used for determining the EOS constants (pc, Tc, and ω) for the new lumped pseudocomponents? In this paper the results of the lumping methods are compared with the data for gas condensate systems to find accurate lumping method for accurate fluid characterization. For this purpose, Whitson’s lumping scheme, Pedersen lumping scheme, Danesh et al. lumping scheme, Lee et al. lumping scheme, Behras and Stanndler lumping scheme are used as lumping scheme and Lee mixing rules was used for calculation of thermodynamics properties of lumped group. To compare the Impact of each Lumping method, its effect was examined on the phase diagram. The results show that the Whitson lumping scheme (it is grouping the hydrocarbon from C7 to C16+ into 4 groups) can predict the phase diagram of this gas condensate systems fluid with good approximation, the Pedersen lumping scheme (it is grouping the hydrocarbon from C7 to C16+ into 3 groups ) can predict the phase diagram of this gas condensate systems fluid somewhat and the Pedersen lumping schemes and Danesh et al. lumping schemes (it is grouping the hydrocarbon from C7 to C16+ into 4 groups) approximately predicted the same phase diagram and the Lee et al. lumping scheme (it is grouping the hydrocarbon from C7 to C16+ into 3 groups) can predict the phase diagram of this gas condensate systems fluid exactly. Therefore, the Lee at al. lumping scheme has very good accuracy compared to the other methods. Also, All Lumping Scheme has an effect on dew point line of phase diagram. Key words: Lumping, Groping, Pseudoization, Pseudo component, Mixing rule. INTRODUCTION Petroleum is composed of various hydrocarbon and non-hydrocarbon components. For characterizing and simulating the oil reservoir, the reservoir fluid must be characterized. The large number of components is necessary to describe the hydrocarbon mixture for accurate phase behavior modeling frequently burdens EOS calculations. Generally, with a sufficiently large number of pseudo-components used in characterizing the heavy fraction of a hydrocarbon mixture, a satisfactory prediction of the PVT behavior by the equation of state can be obtained. However the cost and computing time can be increased significantly with the increased number of components in the system. Therefore, strict limitations are placed on the maximum number of components that can be used in compositional models and the original components have to be lumped into a smaller number of pseudo components (Ahmed 2010, 1989). Several problems are associated with “regrouping” the original components into a smaller number without losing the predicting power of the equation of state. These problems include How to select the groups of pure components to be represented by one pseudo-component each. Several unique techniques have been published that can be used to address the above lumping problems; notably the methods proposed by Mehra et al, Schlijper, Gonzalez, Colonomos, and Rusinek, Aguilar and McCain, Pedersen et al. (Ahmed 2010, 1989; Hong 1982; Guo and Du 1989; Lee et al. 1981; Montel and Gouel 1984; Al-Meshari 2004), Whitson (Whitson 1983), Lee et al. (Lee et al. 1981), Montel and Gouel (Montel and Gouel 1984), Behrens and Sandler (Behrens and Sandler 1988), Hong (Hong 1982) and Danesh et al. (Danesh et al. 1992). Corresponding Author: Hamidreza moghadamzadeh, Islamic Azad University South Tehran Branch, P.O. Box 19585-466, Tehran, Iran E-mail: [email protected]

320

Aust. J. Basic & Appl. Sci., 7(1): 320-333, 2013

For describing the phase behavior of a fluid mixture, critical pressure, critical temperature, critical volume, acentric factor, molecular weight and binary interaction parameters of component must be specified. Different mixing rules were evaluated for, characterizing the lumped C7+ fraction from its constituent component properties (Ahmed 2010; Hong 1982; Adepoju 2006; Joergensen and Stenby 1995). Impact of these lumping schemes on one system, especially gas condensate systems, isn’t studied. In this work, the accuracy and impact of Whitson, Pedersen, Danesh et al., Lee et al., Behras and stanndler lumping scheme on phase diagram of the gas condensate systems are researched. Lee’s mixing rule is one of these mixing rules which are used in this research. 2. Methods: 2.1 Experiment: A sample of reservoir fluid is prepared from one of gas condensate oil fields. In the laboratory, it composition is determined by using gas chromatography. Also, the properties of heavy oil fraction were measured with the existing standards methods in the laboratory. 2.2 Lee’s mixing rules: There are many mixing rules which obtained by researchers. Lee’s mixing rules is one of useful rules. Lee et al. (1979) (Ahmed 2010) used Kay’s mixing rules as the characterizing approach for determining the properties of the lumped fractions in their suggested regrouping model. To using this rule, normalized mole fraction of the component i in the lumped fraction must be defined. It was defined by:

φi =

zi L

∑z

(1) i

After that, Lee et al. are proposed the following rules for calculation of thermodynamic property of multi component mixture: (2)

γL =

MwL L

∑ φ Mw i

VcL = ∑

(3)

i

φi MwiVci

(4)

MwL

L

Pcl = ∑ φi Pci

(5)

L

TcL = ∑ φi Tci

(6)

L

ω l = ∑ φi ω i

(7)

3. Results: In this paper, Whitson’s lumping scheme, Pedersen lumping scheme, Danesh et al. lumping scheme, Lee et al. lumping scheme, Behras and Stanndler lumping scheme are used. Calculation of the method described in the appendix. A compositional analysis of gas condensate systems fluid described in Table 1 and the molecular weight and the specific gravity of C16+ is equal to 259 and 0.908 respectively. The phase diagram of this gas condensate system is shown in Figure 1. 3.1 Whitson’s lumping scheme: Whitson (Whitson 1983) proposed a regrouping scheme whereby the compositional distribution of the C7+ fraction is reduced to only a few multiple carbon- number (MCN) groups. Whitson suggested that the number of MCN groups is necessary to describe the plus fraction is given by the following empirical rule:

N G = Int [1 + 3.3 log( N − n )]

(8)

321


Not that, for black oil system this number probably can be reduced by one.6 The molecular weights separating each MCN group are calculated from the following expression:

 MwN MwI = Mwn   Mwn

I

 NG  

(9)

Table 1: Composition of reservoir fluid Component C1 C2 C3 i-C4 n-C4 i-C5 n-C5 C6 C7

zi 0.413506 0.0403 0.2153 0.0539 0.0543 0.0515 0.0519 0.1039 0.00347

Component C8 C9 C10 C11 C12 C13 C14 C15 C16+

zi 0.00268 0.00207 0.00159 0.00123 0.00095 0.00073 0.000566 0.000437 0.001671

Fig. 1: Phase diagram of reservoir fluid Components with molecular weight falling within the boundaries of MI–1 to MI are included in the I’th MCN group. By applying the Whitson lumping scheme can see that it is grouping the hydrocarbon from C7 to C16+ into 4 groups. The results of Whitson’s lumping scheme are shown in Table 2. The impact of this scheme on the phase diagram is shown in Figure 2. Thus, this figure shows that the Whitson lumping scheme can predict the phase diagram of this gas condensate systems fluid with good approximation. Table 2: Whitson's lumping scheme results Group I 1 2 3

4

component

zi

C7 C8 C9 C10 C11 C12 C13 C14 C15

0.00347 0.00268 0.00207 0.001596 0.00123 0.00095 0.00073 0.000566 0.000437 0.0016 71

C16+

zI

MwI

γI

Vcl (ft3/lb)

Pcl (psia)

Tcl (R)

ωl

0.00822

105.9

0.746

0.0627

424

1020

0.3076

0.002826

139.7

0.787

0.0628

339.7

1144.5

0.4000

0.002246

172.9

0.814

0.0631

288

1230.6

0.4794

0.002108

248

0.892

0.0637

223.3

1433

0.6531

Whitson's lumping scheme accuracy can be attributed to function of the number of MCN groups (equation 8) and iterative method to find the molecular weight of each group Determination of the number of groups is the excellent characteristics of this approach.

322


Fig. 2: Impact of Whitson’s lumping scheme on phase diagram 3.2 Pedersen Lumping Scheme: Pedersen et al. (Ahmed 2010, 1989; Hong 1982; Guo and Du 1989; Lee et al. 1981; Montel and Gouel 1984; Al-Meshari 2004) suggested the process of grouping the components on the basis of each group containing approximately the same weight fraction (equal weight fraction), which will give all hydrocarbon segments of the C7+ fractions equal importance. The C7+ fraction is divided into three or more groups that, by weight, are equal size approximately. The weight for each pseudo component, Wj, can be calculated as: nc

w j = ∑ zi × Mwi

(10)

i =1

By applying the Pedersen lumping scheme can see that it is grouping the hydrocarbon from C7 to C16+ into 3 groups (3 groups is selected according to some restrictions). The results of Pedersen lumping scheme are shown in Table 3. The phase diagram of this system is under the impact of Pedersen lumping scheme is shown in Figure 3. Therefore, this figure shows that the Pedersen lumping scheme can predict the phase diagram of this gas condensate systems fluid somewhat. Table 3: Pedersen lumping scheme results Group I 1

2

3

component

zi

C7 C8 C9 C10 C11 C12 C13 C14 C15 C16+

0.00347 0.00268 0.00207 0.001596 0.00123 0.00095 0.00073 0.000566 0.000437 0.001671

zI

MwI

γI

Vcl (ft3/lb)

Pcl (psia)

Tcl (R)

ωl

0.00615

100.794

0.73657

0.06277

438.184

1007.2

0.29395

0.00584

136.522

0.78409

0.06279

348.896

1123.4

0.39027

0.003404

222.709

0.86866

0.06353

244.507

1363.8

0.59401

As you can see, a simple relationship exists between the concentration and molecular weight in Pedersen lumping scheme. Choosing the number of groups is based on the limitations and experiences, so, the accuracy of this method would not be high. 3.3 Danesh et al. lumping scheme: Danesh et al. (Danesh et al. 1992) proposed a lumping method based on the concentration and molecular weight of compounds in a mixture. This grouping method arranged the original components in order of their normal boiling point temperatures and grouped together in ascending order to form Np groups so that the values of Σ (zilnMi) for all the groups become nearly equal, (Quasi-equal-weight criterion). That is:

  l  1  n  × ∑ z i × ln Mwi  ≤ 0 ∑ z i × ln Mwi −     i =1  N p  i =1 And

323

(11)


  l +1  1 n ∑ z i × ln Mwi  ≥ 0 ∑ z i ln Mwi −    N p  i =1   i =1 i = 1,2,..., n, l = 1,2,...N p

(12)

Fig. 3: Impact of Pedersen lumping scheme on phase diagram According to the Danesh et al. lumping scheme can see that it is grouping the hydrocarbon from C7 to C16+ into 4 groups. The results of Danesh et al. lumping scheme are shown in Table 4. The phase diagram of this system is under the impact of Danesh et al. lumping scheme is shown in Figure 4. The predicted phase diagram of this gas condensate systems fluid by the Danesh et al. lumping scheme is look like the phase diagram was predicted by Pedersen lumping scheme. Figure 4 show that this lumping scheme can predict the phase diagram somewhat. In this method, choosing the number of groups was done similar to the Pedersen lumping scheme. In Danesh et al. lumping scheme some limitation is defined for choosing the molecular weight of groups, so it would be more accurate than the Pedersen lumping scheme. Table 4: Danesh et al. lumping scheme results Group zI component zi I 1 C7 0.00347 0.00347

Pcl (psia) 453

Tcl (R) 985

ωl

0.727

Vcl (ft3/lb) 0.06289

MwI

γI

96

96

2

C8 C9

0.00268 0.00207

0.00475

113.101

0.75719

0.06261

403.312

1045.59

113.101

3

C10 C11 C12

0.001596 0.00123 0.00095

0.00377

145.045

0.79169

0.06288

330.17

1159.3

145.045

4

C13 C14 C15 C16+

0.00073 0.000566 0.000437 0.001671

0.003404

222.709

0.86866

0.06353

244.507

1363.78

222.709

3.4 Lee et al. lumping scheme: Lee et al. (Lee et al. 1981) devised a simple procedure that can be served as a guideline for lumping the oil fractions. The idea for the procedure lies in the physical reasoning that crude-oil fractious having relatively close physicochemical properties can be represented fairly accurately by a single fraction, the closeness of these properties can be reflected by the slopes of curve when these properties are plotted against some characteristic independent variables. According to the Lee et al. lumping scheme can see that it is grouping the hydrocarbon from C7 to C16+ into 3 groups. The results of Lee et al. lumping scheme are shown in Table 5. The accuracy of this scheme can be seen in the phase diagram in Figure 5. Hence, Figure 5 shows that the Lee et al. lumping scheme can predict the phase diagram of this gas condensate systems fluid exactly.

324


Fig. 4: Impact of Danesh et al. lumping scheme on phase diagram Lee et al. lumping scheme is more complex than lumping schemes which is discussed in this paper but it gives highly accurate answer. On the other hand, this lumping scheme is determined number of groups which is the key points of this method. 3.5 Behras And Stanndler Lumping Scheme: Behrens and Standler (Behrens and Sandler 1988) have been used a semi continuous thermodynamic description to model the C7+ fraction for equation of state (EOS) calculations. The step-by-step procedure will be illustrated. Step 1: Finding the ends of the distribution the ends of the distribution can be calculated by: (13)

A= starting carbon number - 1/2 B= ending carbon number + 1/2 Table 5: Lee et al. lumping scheme results Group component zi I C7 0.00347 1 C8 0.00268 C9 0.00207 C10 0.001596 C11 0.00123 2 C12 0.00095 C13 0.00073 C14 0.000566 C15 0.000437 3 C16+ 0.001671

(14)

zI

MwI

γI

Vcl (ft3/lb)

Pcl (psia)

Tcl (R)

ωl

0.00615

100.794

0.73657

0.06277

438.184

1007.2

0.29395

0.007573

148.237

0.79453

0.0629

331.518

1155.6

0.41792

0.001671

259

0.908

0.0638

215

1467

0.68

The parameters of the semi continuous distribution in this system are A = 7 - 1/2 = 6.5 B =16 + 1/2 =16.5. Step 2 – determining the slope of the distribution The slope of the distribution is found from the average carbon number Cn which is related to the where average MW approximately as −

c

n

=

( MW + 4) 14

(15)

The slope of the distribution, α, is then found from the following equation:

−  (B − A)e − Bα  = c − A +  − Aα  n α − e − Bα  e

1

(16)

325


Fig. 5: Impact of Lee et al. lumping scheme on phase diagram If the upper endpoint were infinity, the term in the brackets would vanish and l/α could be found explicitly. Since the upper endpoint is finite, α is found by successive substitutions. Slope of the distribution: −

c

n

=

(141.25 + 4) = 10.375 14

Then

 6 × e −12.5*α = 10.375 − 6.5 +  −6.5*α α − e −12.5*α e 1

 → α = 0.0011  

Step 3 - Finding integration points and weights The integration range can be calculated by Equation (17).

c = ( B − A)α

(17)

Integration points and weights C = (16.5-6.5)*0.0011 = 0.011 Because C< 0.3 so with this method cannot lump this fluid. Conclusions: By comparing the phase diagram for reservoir fluid (Figure 1), and data that is lumped with some lumping schemes like Whitson’s Lumping Scheme (Figure 2), Pedersen lumping scheme (Figure 3), Danesh lumping scheme (Figure 4), Lee et al. lumping scheme (Figure 5) and also plot all lumping scheme and the phase diagram of reservoir fluid on one diagram (Figure 6) can be seen that:

Fig. 6: All lumping scheme

326


1- With the Pedersen Lumping Scheme and Danesh et al. Lumping Scheme can reach to the approximately same the phase diagram. 2- Lee at al. Lumping Scheme has very good accuracy related to the other methods in characterization of gas condensate reservoir. 3- All Lumping Scheme has an effect on dew point line. REFERENCES Adepoju, O.O., 2007. Coefficient of Isothermal Oil Compressibility- A Study for Reservoir Fluids by Cubic Equation-of-State. VDM Verlag Dr. Mueller e.K Press. Ahmed, T.H., 1989. Hydrocarbon phase behavior, Contributions in petroleum geology & engineering 7. Gulf Pub. Co Press. Ahmed, Tarek H., 2010. Reservoir engineering handbook. Gulf Professional Pub Press. Al-Meshari, A.A., 2004. New strategic method to tune equation-of-state to match experimental data for compositional simulation, Ph.D. thesis, Texas A&M University, Texas Behrens, R.A., and S.I. Sandler, 1988. The Use of Semicontinuous Description To Model the C7+ Fraction in Equation of State Calculations. SPE Reservoir Engineering, 3(3): 1041-1047. Danesh, Ali, Dong-hai Xu, Adrian C. Todd, and U. Heriot-Watt, 1992. A Grouping Method To Optimize Oil Description for Compositional Simulation of Gas-Injection Processes. SPE Reservoir Engineering, 7(3): 343-348. Guo, T.M., and L. and Du., 1989. A New Three Parameter Cubic Equation of State for Reservoir Fluids– Part II. Application to Reservoir Crude Oils. SPE pp: 19373. Hong, K.C., 1982. Lumped-Component Characterization of Crude Oils for Compositional Simulation. In SPE Enhanced Oil Recovery Symposium, Tulsa, Oklahoma Joergensen, M., and E.H. Stenby, 1995. Optimization of Pseudo-component Selection for Compositional Studies of Reservoir Fluids. In 70th Annul Technical Conference and Exhibition. Daltas, Taxas. Lee, S.T., R.H. Jacoby, W.H. Chen, W.E. Culham, and Gulf Research and Development Co., 1981. Experimental and Theoretical Studies on the Fluid Properties Required for Simulation of Thermal Processes. SPE Journal., 21(5): 535-550. Montel, F., and P.L. Gouel., 1984. A New Lumping Scheme of Analytical Data for Compositional Studies. In SPE Annual Technical Conference and Exhibition. Houston, Texas. Whitson, C., 1983. Characterizing Hydrocarbon Plus Fractions. SPE Journal, 23(4): 683-694. Nomanclature: I = 1, 2. . . Ng i = component i Int = integer L = lumped fraction MwC7 = molecular weight of C7 Mwi = Molecular weight MwN+ = molecular weight of the last reported component in the extended analysis of the hydrocarbon system n = number of carbon atoms of the first component in the plus fraction, i.e., n = 7 for C7+ N = number of carbon atoms of the last component in the hydrocarbon system Ng = number of MCN groups Pci = Critical pressure Tci = Critical temperature vci = Critical volume Wj = weight for each pseudo component Zi = mole fraction Фi = weighting factor ωi = Acentric factor Appendix A: A.1 Whitson’s lumping scheme: Determining the appropriate number of pseudo-components forming in the C7+ by using the Whitson’s Lumping Scheme Step 1. Determining the molecular weight of each component in the system:

327


TableA-1: Molecular weight of components component C7 C8 C9 C10 C11 C12 C13 C14 C15 C16+

zi 0.00347 0.00268 0.00207 0.00159 0.00123 0.00095 0.00073 0.000566 0.000437 0.001671

Mwi 96 107 121 134 147 161 175 190 206 259

Step 2. Calculating the number of pseudo-components: For calculation of pseudo-components having N=16 n=7

N g = Int [1 + 3.3 log( N − n)]

Therefore,

N g = Int [1 + 3.3 log(16 − 7)]

N g = Int [4.15] Ng = 4

Step 3. Determining the molecular weights separating the hydrocarbon groups:

 MwN + MwI = MwC 7   MwC 7  259  MwI = 96   96 

I

  

I

Ng

4

Therefore,

MwI = 96(2.698)

I

4

Table A-2: Molecular weight of each group I MWI 1 123.0359 2 157.6857 3 202.0937 4 259.008

• First pseudo-component: The first pseudo-component includes all components with a molecular weight in the range of 96 to 123. This group then includes C7, C8, and C9. • Second pseudo-component: The second pseudo-component contains all components with a molecular weight higher than 123 to a molecular weight of 158. This group includes C10 and C11. • Third pseudo-component: The third pseudo-component includes components with a molecular weight higher than 158 to a molecular weight of 202. Therefore, this group includes C12, C13, and C14. • Fourth pseudo-component: This pseudo-component includes all the remaining components, i.e., C15 and C16+. Table A-3: Molecular percentofeach group Group I component C7 1 C8 C9 2 C10

zi 0.00347 0.00268 0.00207 0.001596

328

zI 0.00822 0.002826


C11 C12 C13 C14 C15

3 4

0.00123 0.00095 0.00073 0.000566 0.000437 0.001671

C16+

0.002246 0.002108

Zl is calculated by the equation of: l

∑z

i

= zl

there are numerous ways to mix the properties of the individual components, all giving different properties for the pseudo-components, the choice of a correct mixing rule is as important as the lumping scheme. The mixing rules can be employed to characterize the pseudo-component in terms of its pseudo physical and pseudo-critical properties. In this Example Using Lee’s mixing rules to determine the physical and critical properties of the four pseudo-components TableA-4: Thermodynamics property of components Group I 1 2 3 4

component

zi

C7 C8 C9 C10 C11 C12 C13 C14 C15 C16+

0.00347 0.00268 0.00207 0.001596 0.00123 0.00095 0.00073 0.000566 0.000437 0.001671

zI

0.00822 0.002826 0.002246 0.002108

γi from appendix 0.727 0.748 0.768 0.782 0.793 0.804 0.815 0.825 0.826 0.908

Mwi from appendix 96 107 121 134 147 161 175 190 206 259

vci from appendix 0.06289 0.06264 0.06258 0.06273 0.06291 0.06306 0.06311 0.06316 0.06325 0.0638*

Pci from appendix 453 419 383 351 325 302 286 270 255 215*

Tci from appendix 985 1036 1058 1128 1166 1203 1236 1270 1304 1467

* Calculated Step 2. Calculate the physical and critical properties of each group by applying Equations (4-23) through (4-29) to give: Table A-5: φi of components Group I component C7 1 C8 C9 C10 2 C11 C12 3 C13 C14 C15 4 C16+

zi 0.00347 0.00268 0.00207 0.001596 0.00123 0.00095 0.00073 0.000566 0.000437 0.001671

zl 0.00822 0.002826 0.002246 0.002108

Table A-6: Thermodynamics property of each group by Whitson’s lumping scheme Group 1 2 3 zI 0.00822 0.002826 0.002246 MI 105.9 139.7 172.9 0.746 0.787 0.814 γI Vcl 0.0627 0.0628 0.0631 Pcl 424 339.7 288 Tcl 1020 1144.5 1230.6 ωl 0.3076 0.4000 0.4794

4 0.002108 248 0.892 0.0637 223.3 1433 0.6531

A.2 Pedersen Lumping Scheme: Table A-7: Molecular weight of components component zi C7 0.00347 C8 0.00268 C9 0.00207 C10 0.00159 C11 0.00123 C12 0.00095 C13 0.00073

Mwi 96 107 121 134 147 161 175

329

φi=zi/zl 0.423171 0.326829 0.252439 0.57 0.439286 0.431818 0.331818 0.257273 0.208095 0.795714

zi Mwi 0.3331 0.2868 0.2505 0.2131 0.1808 0.153 0.1278

ωi from appendix 0.280 0.312 0.348 0.385 0.419 0.454 0.484 0.516 0.550 0.68*


C14 C15 C16+

0.000566 0.000437 0.01671

190 206 259

0.1075 0.09 4.3279

∑ z Mw =2.1753 i

i

Let take 3 groups so wl for each groups is equal to:

1 2.1753 z i Mwi = = 0.7254 ∑ 3 Wl = 3 According to this value of wl groups are Table A-8: Molecular percent of each group Group I component C7 1 C8 C9 C10 2 C11 C12 C13 C14 3 C15 C16+ By using Lee’s mixing rules: Table A-9: φi of components Group I component C7 1 C8

2

3

C9 C10 C11 C12 C13 C14 C15 C16+

zi 0.00347 0.00268 0.00207 0.001596 0.00123 0.00095 0.00073 0.000566 0.000437 0.001671

zi 0.00347 0.00268 0.00207 0.001596 0.00123 0.00095 0.00073 0.000566 0.000437 0.001671

zI 0.00615

0.00584

0.003404

φi=zi/zl 0.564228 0.435772 0.354452 0.27226 0.210616 0.162671 0.214454 0.166275 0.128378 0.490893 0.564228

zI 0.00615

0.00584

0.003404

Table A-10: Thermodynamics property of each group by Behras and Stanndler lumping scheme Group 1 2 zI 0.00615 0.00584 MI 100.7935 136.5223 0.736566 0.784092 γI Vcl 0.062774 0.062787 Pcl 438.1837 348.8955 Tcl 1007.2 1123.4 ωl 0.293945 0.390271

3 0.003404 222.7089 0.868659 0.063528 244.5065 1363.8 0.594009

A.3 Danesh et al. Lumping scheme: Table A-11: Molecular weight of components component zi C7 0.00347 C8 0.00268 C9 0.00207 C10 0.00159 C11 0.00123 C12 0.00095 C13 0.00073 C14 0.000566 C15 0.000437 C16+ 0.001671

Ln(Mwi) 4.5643 4.6728 4.7958 4.8978 4.9904 5.0814 5.1648 5.247 5.3279 5.5568

zi *Ln(Mwi) 0.015838 0.012523 0.009927 0.007788 0.006138 0.004827 0.00377 0.00297 0.002328 0.009285

∑ z ln(Mw ) =0.075396 i

Let take 4 groups, so groups are:

330

i


Table A-12: Molecular percent of each group Group I component 1 C7 C8 2 C9 C10 3 C11 C12 C13 C14 4 C15 C16+ By using Lee’s mixing rules:

zi 0.00347 0.00268 0.00207 0.001596 0.00123 0.00095 0.00073 0.000566 0.000437 0.001671

Table A-13: φi of components Group I component 1 C7 C8 2 C9 C10 C11 3 C12 C13 C14 4 C15 C16+

zI 0.00347 0.00475 0.00377

0.003404

φi=zi/zl 1 0.564211 0.435789 0.421751 0.32626 0.251989 0.214454 0.166275 0.128378 0.490893

zI 0.00347

zi 0.00347 0.00268 0.00207 0.001596 0.00123 0.00095 0.00073 0.000566 0.000437 0.001671

0.00475 0.00377

0.003404

Table A-14: Thermodynamics property of each group by Danesh et al. lumping scheme Group 1 2 3 zI 0.00347 0.00475 0.00377 MI 96 113.1011 145.0451 0.727 0.757193 0.791691 γI Vcl 0.06289 0.062612 0.062882 Pcl 453 403.3116 330.1698 Tcl 985 1045.587 1159.297 ωl 0.28 0.327688 0.41348

4 0.003404 222.7089 0.868659 0.063528 244.5065 1363.779 0.594009

A.4 Lee et al. Lumping scheme: First plot zi vs. Tb and Mwi vs. Tb Zi = 0.2926e-0.012*Tb

Zi vs. Tb

0.004 0.0035 0.003

Zi

0.0025 0.002 0.0015 0.001 0.0005 0 0

100

200

300

400

500

Tb

Fig. A -1: Molecular percent of each components with bubble point temperature

331

600


Mwi vs. Tb 250 200

Mwi

Mwi = 0.7571*Tb - 203.2 150 Mwi = 0.5224*Tb - 95.878 100 50 0 0

100

200

300

400

500

600

Tb

Fig. A -2: Molecular weight of components with bubble point temperature Set property of zi number 1 and property of Mwi number 2

wi =

1 2 −

−

component

z

Mwi

Tb

m i1

m i2

mi1

mi 2

Mi

C7

0.00347

96

366

-4.3454E-05

0.5224

1

0.69

0.845

C8

0.00268

107

390

-3.258E-05

0.5224

0.7498

0.69

0.7199

C9

0.00207

121

416

-2.3848E-05

0.2524

0.5488

0.69

0.6194

C10

0.00159

134

439

-1.8096E-05

0.2524

0.4164

0.69

0.5532

C11

0.00123

147

461

-1.3898E-05

0.7571

0.3198

1

0.6599

C12

0.00095

161

482

-1.0802E-05

0.7571

0.2486

1

0.6243

C13

0.00073

175

501

-8.5996E-06

0.7571

0.1979

1

0.599

C14

0.000566

190

520

-6.8463E-06

0.7571

0.1576

1

0.5788

C15

0.000437

206

539

-5.4505E-06

0.7571

0.1254

1

0.5627

Table A-15: Molecular percent of each group Group I component C7 1 C8 C9 C10 C11 2 C12 C13 C14 C15 3 C16+ By using Lee’s mixing rules: Table A-16: φi of components Group I component C7 1 C8 C9 C10 C11 2 C12 C13 C14

zi 0.00347 0.00268 0.00207 0.001596 0.00123 0.00095 0.00073 0.000566 0.000437 0.001671

zi 0.00347 0.00268 0.00207 0.001596 0.00123 0.00095 0.00073 0.000566

332

zI 0.00615

0.007573

0.001671

zI 0.00615

0.007573

φi=zi/zl 0.564228 0.435772 0.273339 0.209956 0.162419 0.125446 0.096395 0.074739


C15 C16+

3

0.000437 0.001671

0.057705 1

0.001671

Table A-17: Thermodynamics property of components by Lee lumping scheme Group 1 2 zI 0.00615 0.007573 MI 100.7935 148.2374 0.736566 0.794528 γI Vcl 0.062774 0.062897 Pcl 438.1837 331.5179 Tcl 1007.2 1155.6 ωl 0.293945 0.41792 A.5 Behras and Stanndler lumping scheme Staring component Mole fraction (xD) MW

C7 0.0154 141.25

Step 1: The parameters of the semi continuous distribution in this example are A = 7 - 1/2 = 6.5 B =16 + 1/2 =16.5. Step 2: determine the slope of the distribution −

c

n

=

3 0.001671 259 0.908 0.0638 215 1467 0.68

(141.25 + 4) = 10.375 14

Then

 6 × e −12.5*α  = 10.375 − 6.5 +  −6.5*α α − e −12.5*α  e 1

α = 0.0011 Step 3 - Find integration points and weights C = (16.5-6.5)*0.0011 = 0.011 Because C< 0.3 so with this method cannot lump this oil.

333