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Prediction of Minimum Miscibility Pressure Using the UNIFAC Group Contribution Activity Coeﬃcient Model and the LCVM Mixing Rule Ali Haghtalab* and Ali Kariman Moghaddam Department of Chemical Engineering, Reservoir Engineering Group, Tarbiat Modares University, P.O. Box 14115-143, Tehran, Iran S Supporting Information *

ABSTRACT: Minimum miscibility pressure (MMP) is a key design parameter of any gas ﬂooding project that can be measured in the laboratory or estimated by computational methods based on the equation of state (EOS). Application of the EOS for mixtures is attainable by employing simple mixing rules, such as the van der Waals (VDW) ﬂuid mixing rule, which has been applied widely in petroleum industry so far. The objective of this study is to estimate MMP using multiple mixing cells (MMC), developed by Ahmadi and Johns (K. Ahmadi and R. T. Johns, SPE J. 2011, 16, 733−742), with the EOS/GE mixing rule. The linear combination of the Vidal and Michelsen (LCVM) is used as a mixing rule for the modiﬁed Peng−Robinson EOS by applying the UNIFAC activity coeﬃcient equation. Moreover, in the MMP calculations, a group contribution method is used to characterize undeﬁned petroleum fractions. The MMP results are compared with those estimated using the VDW (classic) mixing rule and experimental data of the slim tube apparatus. Using the mPR-LCVM-UNIFAC method and the group contribution characterization of undeﬁned fractions, the results show good agreement with those experimental data that are obtained for oil displacement through the injection of gas mixtures. The ﬁrst attempt to develop EOS/GE mixing rules was proposed by the Huron−Vidal mixing rule, where their basic assumption was to use inﬁnite pressure as the reference pressure.6 Based on their assumption, one can use the GE models, which do not have an explicit combinatorial/free volume term, such as the NRTL equation, or the residual term of UNIQUAC or UNIFAC. The limitation of the Huron−Vidal mixing rule is that it is not capable of using the data available in the DECHEMA collection, which are based on low-pressure VLE data.5 To overcome this limitation, Michelsen and coworkers proposed zero-pressure as the reference pressure, which permits the development of various mixing rules, which are known as MHV1, MHV2 and PSRK.8,9 The drawback of the zero-pressure approach mixing rules is that they have serious problems in regard to predicting the VLE of asymmetric systems, such as a mixture of CO2 or methane with heavy hydrocarbon.5 In 1994, Boukouvalas and co-workers suggested a new mixing rulea linear combination of the Vidal and Michelsen rule, known as LCVMwhich provides successful prediction of the VLE of polar and nonpolar systems at low and high pressure. They coupled this mixing rule with the modiﬁed Peng−Robinson EOS and the original UNIFAC, so that the predicted results of VLE calculation were acceptable

1. INTRODUCTION The miscibility process through gas injection is a common tool that has been used in enhanced oil recovery (EOR) for many years. One of the most important factors in the gas injection process is the determination of minimum miscibility pressure (MMP), in which, at this pressure, the local recovery factor of oil production becomes almost 100%, so that inaccurate prediction of MMP leads to high cost or undesirable oil production.1 The experimental estimation of MMP, such as a slim tube experiment,2 is time-consuming and costly; therefore, researchers have a tendency to apply computational methods based on the equation of state (EOS) and using less-expensive experimental data of swelling test and multicontact test (MCT) that can replace a slim tube.3 Equations of state has been applied for the multicomponent systems by employing mixing rules in order to perform the vapor−liquid equilibrium (VLE) calculation of oil−gas mixtures.4 On the other hand, the van der Waals (VDW) mixing rule is usually applied for symmetric and nonpolar system at high pressure; nevertheless, excess Gibbs energy functions (GE) through activity coeﬃcient are suitable for more-complex polar and asymmetric systems with low pressure. The strength of both approaches could be implemented in a single model as EOS/GE by incorporating an expression for the excess Gibbs energy in a type of EOS/GE mixing rule.5 The various EOS/GE models have been typically developed via two approaches: one that uses inﬁnite-pressure as the reference pressure (e.g., Huron and Vidal6) and another that uses zero-pressure (e.g., Michelsen7). © XXXX American Chemical Society

Received: November 23, 2015 Revised: January 22, 2016 Accepted: February 18, 2016

A

DOI: 10.1021/acs.iecr.5b04447 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research where the previous EOS/GE models (such as MHV2 and PSRK) perform poorly.10 The various combinations of the cubic EOS and GE models have been applied for hydrocarbon and polar systems; detailed reviews of them have been published in the literature.11−13 Haghtalab and Mahmoodi14 combined the LCVM mixing rule with the UNIFAC-NRF group contribution activity coeﬃcient model for the VLE calculation of binary systems of heavy alkanes and light gases. EOS/GE modeling of real oil and gas systems has rarely been performed. On the other hand, petroleum ﬂuids are composed of undeﬁned composition fractions with unknown structure, e.g., heptane plus fractions (C7+), which require a more comprehensive model to predict their ﬂuid phase behavior. Many group contribution methods and correlations have been developed for the characterization of various components and heavy petroleum fractions.4,15 Carreón-Calderón and coworkers proposed a new procedure based on a group contribution method to estimate the critical properties and pseudo-structures of the undeﬁned petroleum fractions using minimization of the Gibbs energy, which presents the ability to be applied for real oil-gas systems.16,17 The computational methods for MMP estimation may be classiﬁed in three major approaches: slim tube compositional simulation, the method of characteristics (MOC), and the multiple mixing cell (MMC) method. In fact, slim tube compositional simulation emulates the ﬂow in porous media through a slim tube experiment.18 The MOC approach has been developed by an analytical solution of oil and gas displacement.19 Based on the MOC approach, for component NC, there are NC − 1 key tielines so that MMP is calculated when one of these key tielines becomes critical (or its length becomes zero).20,21 The MMC (cell-to-cell) method consists of one cell or a series of cells on which phase equilibrium calculation is performed, and this may be applied through a single cell22,23 or multiple cells.24−26 The drawback of slim tube compositional simulation is that MMP estimation may be aﬀected by numerical ﬂuctuation, so can be more time-consuming, compared to the others.27 The main demerit of the MOC approach is its complexity, so it is possible that the calculation converges to the wrong key tieline, as shown by Yuan and Johns.28 Moreover, Ahmadi et al. have indicated that MOC-based algorithms have a potential problem with negative ﬂash calculation and predict MMP with large errors under bifurcation conditions.29 Ahmadi and Johns proposed a new multiple mixing cell, based on the tieline approach, that is fast and reliable.30 To model oil−gas mixtures, previous studies of the MMP calculation have used the VDW (or classic) mixing rule through a cubic EOS, so that their results are not enough accurate. The objective of this study is the prediction of MMP using the MMC method proposed by Ahmadi and Johns,30 with application of the LCVM mixing rule and UNIFAC group contribution activity coeﬃcient model. Furthermore, the characterization of undeﬁned petroleum fractions is carried out by using group contribution methods. The results are obtained for three-component, four-component, and multicomponent gas injection systems, and are compared with the experimental data that were obtained using the slim tube approach.

2. THERMODYNAMICS FRAMEWORK 2.1. Cubic Equation of State. In this study, the Peng− Robinson (PR) EOS that is used for mixtures is written as31 am RT P= − 2 v − bm v + 2vbm − bm 2 (1) where P and T are the pressure and temperature, respectively. R is the universal gas constant, and v presents the molar volume of the mixture. The PR EOS parameters for pure components are deﬁned as follows:31 ⎛ RT ⎞ bi = 0.077796⎜ ci ⎟ ⎝ Pci ⎠ ai = aciαi(T ) ⎛ R2T 2 ⎞ ci ⎟ aci = 0.457235⎜ ⎝ Pci ⎠

(2)

where Tci and Pci are the critical temperature and critical pressure of component i, respectively. In modiﬁcation of PR EOS, the two versions of the alpha function are used, so that, in PR78 EOS, it is written as32 Tri )]2

αi(T ) = [1 + mi (1 −

(3)

⎧ m = 0.37464 + 1.5422ω − 0.26992ω 2 ω ≤ 0.491 i i i i ⎪ ⎪ ⎨ m = 0.3796 + 1.485ω − 0.1644ω 2 ω i i i > 0.491 ⎪ i ⎪ + 0.01667ωi 3 ⎩

(4)

where Tri and ωi denote the reduced temperature and acentric factor of component i, respectively. Alternatively, another version of the alpha function was proposed by Twu et al.33 as αi(T ) = αi0(Tr ) + ωi(αi1(Tr ) − αi0(Tr ))

α0i (T)

(5)

α1i (T)

where the expressions of and are presented as shown below and their values are calculated as shown in Table 1. αi j(T ) = TriN(M − 1) exp(L(1 − TriNM ))

(j = 0, 1)

(6)

Table 1. Alpha Function Given by Twu et al.33 Tr ≤ 1 α parameter

α

L M N

0.125283 0.911807 1.94815

0

Tr > 1 α

1

α

α1

0.401219 4.96307 −0.2

0.024955 1.248089 −0.8

0

0.511614 0.784054 2.81252

2.2. LCVM Mixing Rule. In this work, the Twu et al. version of the PR EOS is used through the LCVM mixing rule as10

am = αbmRT

(7)

E ⎛ λ 1 − λ⎞ g 1−λ + + α=⎜ ⎟ AM ⎠ RT AM ⎝ AV

+ B

⎛ ai ⎞ ⎟ ⎝ biRT ⎠

∑ xi⎜

⎛ ⎞

∑ xi ln⎜ b ⎟ ⎝ bi ⎠ (8)

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bm =

∑ xibi i

(9)

where AV, AM, and λ are the EOS-dependent parameters, with values for PR EOS of 0.623, −0.53, and 0.36, respectively, as given by Boukouvalas et al.10 As one can see, the LCVM mixing rule is suitable for a size-asymmetric system, because such a mixing rule makes it possible to cancel the double combinatorial term, with one stemming from the activity coeﬃcient model and the other coming from the equation of state. The excess Gibbs energy (gE) is expressed using the activity coeﬃcient of the components in mixture as5 gE = RT

n

∑ xi ln γi i=1

(10)

where the UNIFAC activity coeﬃcient model is applied here and the group-interaction parameter for UNIFAC-LCVM (Ψmn) is written as34 ⎛ a + bmn(T − 298.15) ⎞ Ψmn = exp⎜ − mn ⎟ ⎝ ⎠ T

Figure 1. Flowchart for the characterization of the undeﬁned petroleum fractions. N

(11)

∑ υjΔvj =

where the values of parameters anm and bmn are optimized through experimental phase equilibrium data as provided in the literature.35−37 2.3. Characterization of Undeﬁned and Plus Petroleum Fraction. To apply UNIFAC to calculate the activity coeﬃcient of the heavy plus fraction such as C7+ and C20+, one needs the molecular structure of these unknown components. Herein, we used the procedure proposed by Carreon-Calderon and co-workers to overcome this problem.16 Based on this method, the undeﬁned petroleum fraction is being treated as a hypothetical pure component that consists of one set of functional groups as υ = (υ1, υ2, ..., υk), which are noninteger numbers and are determined by minimizing the Gibbs energy of a pseudo-pure liquid at the ﬁxed standard pressure and temperature. It should be pointed out that experimental density determination of heavy plus fraction is performed at the standard temperature and pressure condition of 288.71 K and 0.101325 MPa. As stated by Carreón-Calderón et al.,16,17 the molar Gibbs energy at constant temperature T and pressure P is written as

j=1

ρil

(14)

where, in the above equations, the subscript j denotes the functional group, which belongs to the undeﬁned petroleum fraction i, and N is the total number of functional groups. MW is the molecular weight and ρ represents the liquid density. Δvj presents the group volume increment, which is expressed (against temperature, T) as Δvj = αj + βjT + χj T 2

(15)

where αj, βj, and χj are the parameters of the group contribution method, which are given in the literature.38,39 The critical properties, in terms of the critical values of the functional groups, have been provided by the group contribution approaches, such as the Joback−Reid method40 and the Marreno−Gani method.41 In this work, we employed the Joback−Reid method, as40 Tc =

e g / RT = φil(Tci(υ1, υ2 , ..., υk), Pci(υ1, υ2 , ..., υk), ωi(υ1, υ2 , ..., υk )) (12)

Pc =

To minimize the molar Gibbs energy, one should minimize the fugacity coeﬃcient of this hypothetical liquid phase, as presented on the right side of eq 12. Thus, one can consider the objective function as the fugacity coeﬃcient of a pure liquid component that should be minimized through using the critical properties of undeﬁned petroleum fractions to obtain the noninteger number of the structural groups, i.e., υ = (υ1, υ2, ..., υk). The calculation procedure is presented in Figure 1. As one can see, a Lagrangian multiplier approach is applied, subject to linear constraints related to the molecular weight and liquid molar volume, as

0.584 +

Tb k 0.965 ∑ j = 1 υjtcj

k

− (∑ j = 1 υjtcj)2

(16)

0.1 k

(0.113 + 0.0032nA − ∑ j = 1 υjpci )2

(17)

where the tcj and pci are the contributions to the critical temperature and critical pressure of functional groups, respectively, and nA is the total number of atoms in a given molecule. The parameter Tb denotes the normal boiling temperature of the pure component. Since no normal boiling points are available for undeﬁned petroleum fractions, the several semiempirical correlations, such as the Riazi−Daubert method42 and the Pedersen−Milter−Sorensen method,43 can be used for this purpose. Here, the Sim−Daubert correlation44 was utilized as 1/2.3776 ⎤ ⎡ 5 ⎢⎛ MW × SG0.9371 ⎞ ⎥ Tb = ⎜ ⎟ ⎥⎦ 9 ⎢⎣⎝ 1.4350476 × 10−5 ⎠

N

∑ υj MWj = MWi j=1

MWi

(13) C

(18)

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Figure 2. Procedure of increasing contacts in the Ahmadi and Johns algorithm.30

where SG is the speciﬁc gravity of the undeﬁned petroleum component. Moreover, the acentric factor (ω) is another parameter that is estimated, as45 ω=−

cell mixes with the injected gas with composition (Zg), and the resultant vapor with composition (yi) from earlier stage mixes with an initial oil reservoir composition (Zo). (4) Repeat this procedure through additional cell neighbors until NC − 1 key tielines of y−x are developed. The development of these tielines is continued when the tielines of the three successive neighbor cells present a zero slope. (5) Calculate the length of the key tielines and store the minimum length of the tieline at the given initial pressure. (6) Repeat the above calculation steps by an increase in the initial pressure. The MMP is obtained through minimization the length of the tieline at critical pressure so that this pressure is estimated by extrapolation of the pressure of the two or three last lines against their minimum tieline lengths. This procedure is suitable for the simple systems of oil and gas for which the number of components is not too high. As the number of components in the systems increases, a greater number of contacts are required in order to develop NC − 1 key tielines. Therefore, using the above procedure, the calculation steps become so time-consuming that it is not practical to be applied for some real cases. Ahmadi and Johns proposed a fast method to estimate the MMP with acceptable accuracy,30 so that it can be applied for real gas injection systems. Instead of adding neighbor cells until NC − 1 key tielines are developed, they implemented a speciﬁc procedure for a certain number of contacts (for example, 30 contacts), and the ﬂash calculations are performed by each additional contact to obtain the smallest tieline length. The minimum tieline length is plotted as a function of (1/N)m, where N is the number of contacts and m is a constant between 0.2 and 0.5. Hence, the minimum tieline length (TL∞) was found by extrapolating the length of the tieline at the inﬁnite number of contacts, so that the MMP is obtained at the pressure in which the minimum tielines approaches zero at an inﬁnite number of contacts.30

ln(Pc /0.101315) + f0 (Tbr ) f1 (Tbr )

(19)

where f0 (Tbr ) =

− 5.97616τ + 1.29874τ1.5 − 0.60394τ 2.5 − 1.06841τ 5 Tbr (20)

and f1 (Tbr ) =

− 5.03365τ + 1.11505τ1.5 − 5.41217τ 2.5 − 7.46628τ 5 Tbr (21)

where Tbr denotes the reduced boiling point (which is deﬁned as Tbr = Tb/Tc) and τ = 1 − Tbr. 2.4. Method of MMP Calculation. In this study, the multiple mixing cell (MMC) method that was developed by Ahmadi and Johns,30 was applied so that this approach is actually a combination of MOC with traditional multiple mixing cells.46 Similar to the MOC approach, the criterion to determining the MMP is the pressure at which the length of one of the key tielines, which is deﬁned as the following equation, becomes zero. N

tieline length =

∑ (xi − yi )2 i=1

(22)

Using this approach, the calculation procedure is presented in Figure 2 and summarized by the following steps: (1) Specify an initial pressure essentially below minimum miscibility pressure (for instance, 500 psi). (2) Start with the two original oil and gas cells, each of which is ﬁlled by injection gas (Zg) and oil reservoir (Zo), respectively. Consequently, a ﬂash calculation47,48 is performed to determine the overall equilibrium compositions of liquid(s) (xi) and vapor(s) (yi) phases. (3) Assume that gas moves ahead of oil. Consequently, the resulting liquid with composition (xi) in the equilibrium D

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Figure 3. Gas and oil tieline lengths versus the number of cells for the C5H12 (53 mol %) + C16H34 (47 mol %) system displaced by CO2 at 1300 psi.

Figure 4. Length of the tieline versus pressure for the C5H12 (53%) + C16H34 (47%) system displaced by CO2 using the van der Waals (VDW) mixing rules (solid lines) and the LCVM mixing rules (dashed lines).

3. RESULTS AND DISCUSSION In the procedure of MMP estimation, many ﬂash calculations are performed; therefore, a more accurate ﬂash calculation leads to signiﬁcantly accurate results. The accuracy of a ﬂash calculation is dependent on several factors, such as the type of EOS, the choice of mixing rule, and heavy cut characterization. Haghtalab and Mahmoodi 14 showed that the combination of PR EOS and LCVM-UNIFAC mixing rule (mPR-LCVM-UNIFAC) allows one to provide the accurate phase envelope, which is directly related to ﬂash calculation of the ﬂuid mixtures. On the other hand, Carreón-Calderón et al.16 revealed that their proposed group contribution method (GCM) of characterization successfully predicts some experimental tests such as isothermal vaporization and ﬂash separation test. In this work, we use the combination of the mPR-LCVM-UNIFAC method and GCM characterization, based on the results of these two works, and compared those results with information obtained through the common scenario of MMP estimation in the previous works, i.e., the VDW mixing rule with constant Kij and the characterization relations based on correlation. 3.1. Displacement of Two-Component Oil by CO2 (System 1). Herein, the simple system of a two-component

oil consisting of C5H12 (53 mol %) and C16H34 (47 mol %) displaced by CO2 is examined at a temperature of 323.15 K.49 To apply the UNIFAC activity coeﬃcient model, the C5H12 molecule is considered to consist of two CH3 and three CH2 subgroups, and the same scenario is assumed for the C16H32 molecule. The CO2 molecule is treated as an individual group. Given the fact that the CH3 and CH2 are in the same main group, the interaction parameters between them is set to zero. Thus, this system consists of the three components, so that the two oil and gas tielines control the miscibility development, as shown in Figure 3. Both the oil and gas tielines are calculated using eq 22; however, it should be explained that the oil tieline is the diﬀerence of the gas streams with the fresh oil and the gas tieline is the diﬀerence of the oil streams with the fresh gas. As one can see this ﬁgure displays the oil and gas tielines versus the number of cells at 1300 psi, so that these two tielines are developed following 60 contact numbers. The lengths of the tielines were obtained using the two types of mixing rules: the classical VDW with binary interaction parameters50 and LCVM mixing rules. As can be seen, the LCVM mixing rule produces the longer tieline length at the 1300 psi; however, the shorter tielines are acquired at higher pressure, as shown in Figure 4. This ﬁgure shows the tielines lengths versus pressure, such that E

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gas, and crossover are presented for this system so that they are developed thoroughly, following 125 contact numbers, as depicted in Figure 5. Since the crossover tieline presents the minimum length, the mechanism of miscibility development is the combination of the vaporizing/condensing gas drive. Figure 6 shows the key tielines lengths versus pressure for the system 2. As shown in the ﬁgure, the dashed and solid lines present the results obtained by the LCVM-UNIFAC model and the VDW mixing rule, respectively. Using the LCVM-UNIFAC mixing rule, the length of the crossover tieline is slightly longer than that is calculated by the VDW mixing rule; however it converges to zero length at lower pressure. Using the crossover tieline, the values of the MMP by LCVM-UNIFAC and VDW mixing rules are 1640 ± 5 and 1655 ± 5 psi, respectively, so that a small diﬀerence between these two values and experimental data of slim tube, 1700 psi,46 are observed. 3.3. Determination of the MMP for the Oil with a TwoComponent Gas (System 3). Table 4 presents the components and their composition of oil and gas, which are referenced as system 3.51 This table also shows the values of the binary molecular interaction parameters that are utilized in the VDW mixing rule.50 As previousy mentioned to characterize the undeﬁned C7+, a fraction group contribution method is applied that is based on the procedure suggested by CarreónCalderón and co-workers. In this work, the objective function (eq 12) was minimized through a constrained nonlinear optimization algorithm52,53 and, using the LCVM-UNIFAC model as the results of the fractional values of the structural groups, are given in Table 5. The speciﬁc gravity and molecular weight of the present liquid phase (C7+) are given in the footnote of Table 4. For this system, we applied the fast method of MMP approximation suggested by Ahmadi and Johns,30 and the calculations are performed for a limited number of contacts so that the minimum tieline length is obtained by extrapolation of the length of the tieline to an inﬁnite number of cells. Herein, the crossover tieline, which has the minimum length, was tracked by performing the calculation for 50 contacts. Figure 7 displays the extrapolation of the last 10 points of the crossover tieline length to the inﬁnite number of contacts with m = 0.2 at 2000 psi, so that the value of minimum tieline length at inﬁnite cell (TL∞) was obtained. By changing

the solid and dashed lines present the results, respectively, using VDW mixing rule-PR78 and mPR-LCVM-UNIFAC. Thus, for this system, the oil tieline has become critical so that the mechanism of miscibility achievement is vaporizing gas drive. Finally, the values of 1418 ± 5 and 1361 ± 5 psi are obtained for MMP, using the VDW mixing rule and the LCVMUNIFAC mixing rule, respectively. Note that the experimental value for this system is 1514 psi49 using slim tube; thus, one can conclude that the result of the VDW mixing rule presents a better agreement with the experiment. 3.2. Displacement of Three-Component Oil by CO2 (System 2). The CH4 + C4H10+ C10H22 system was investigated using CO 2 as a displacement gas. The compositions of this oil-gas system and their binary interaction parameters (Kij) are presented in Table 2.30,46 The CH4 and Table 2. Compositions of Oil-Gas System and the Value of Kij in the van der Waals (VDW) Mixing Rule for System 230,46 Composition

Binary Interaction Parameter (Kij)

component

oil (mol %)

gas (mol %)

CH4

C4H10

C10H22

CO2

CH4 C4H10 C10H22 CO2

25 30 45 0

0 0 0 100

0 0.027 0.042 0.1

0 0.008 0.1257

0 0.0942

0

CO2 molecules are considered as the individual main groups, and the C4H10 and C10H22 molecules consist of CH2 and CH3 subgroups. Table 310 represents the structural parameters of Table 3. Structural Parameters of Groups10 group

Rk

Qk

CO2 CH4 CH3 CH2

1.296 1.129 0.9011 0.6744

1.261 1.124 0.848 0.54

volume (Rk) and area (Qk) for the groups and light molecules that are used in the UNIFAC model. Three key tielines of oil,

Figure 5. Tieline lengths of gas oil and crossover versus the number of cells for the CH4 + C4H10 + C10H22 system at 1500 psi. All the key tielines are developed following 125 contact numbers. F

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Figure 6. Key tieline length versus pressure for the CH4 + C4H10 + C10H22 system displaced by CO2, using the VDW and LCVM mixing rules.

Table 4. Oil and Gas Compositions,51 and the Kij Values50 Used in the VDW Mixing Rule for System 3a Binary Interaction Parameter

Composition

a

mixing rule is longer than those obtained from the VDW mixing rule below 2150 psi; however, it becomes critical at lower pressure. Finally, the estimated values of MMP are 2285 and 2456 psi using the LCVM-UNIFAC and VDW mixing rules, respectively. The slim tube result for this system is 2160 psi,51 which shows that the result of the LCVM-UNIFAC model presents better agreement with the experiment. 3.4. Determination of the MMP for the Multicomponent Oil Mixture and a Gas Mixture. In this section, the performance of LCVM-UNIFAC mixing rule was investigated based on the bank of data given by Jaubert and coworkers.54 Herein, three oil-gas injection systems are considered: F1/G1, F2/G2, and F3/G3. The details of these systems are available elsewhere.54 As shown in the Supporting Information, Tables S1−S3 present the obtained results of minimization of eq 12, using nonlinear programming for the undeﬁned petroleum fractions. As one can see, nine suggested functional groups are observed: these functional groups originate from paraﬃnic, naphthenic, and aromatic components, which are the most common components found in petroleum ﬂuids. For the undeﬁned petroleum fractions, the speciﬁc gravity of each hydrocarbon cut from C7 to C20+ is given in the data bank.54 The critical properties of undeﬁned petroleum fractions for the three petroleum ﬂuids are computed by both a group contribution method and the Twu correlation,55 as presented in Tables 6−8. In the LCVMUNIFAC model, the calculated critical properties of the group contribution method was used; for the VDW mixing rule, the Twu55 correlated values of critical properties are applied. As presented in Tables 6−8, the calculated acentric factor reported by Lee and Kesler56 is used for the VDW mixing rule and eq 19 is applied for the LCVM-UNIFAC mixing rule. Note that using the critical properties obtained through the group contribution method into the VDW mixing approach leads to the poorer MMP estimation results. Figures 9−11 demonstrate the minimum tieline length versus pressure for the three petroleum oils. As one can see, the solid lines and dashed lines present the results obtained by using the VDW and LCVM-UNIFAC mixing rules, respectively. For all of the oil-gas systems, the calculated tielines by the LCVMUNIFAC present the shorter tieline lengths, which are critical at lower pressure. The values of MMP calculated using these two mixing rules and in comparison with the slim tube

component

oil (mol %)

gas (mol %)

N2

CO2

N2 CO2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+

0.6 0.58 9.9 2.72 4.98 1.09 4.91 2.16 3.73 4.85 64.48

0 80 20 0 0 0 0 0 0 0 0

0 0.017 0.1767 0.0311 0.0852 0.1033 0.08 0.0922 0.1 0.08 0.08

0 0.0974 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.01

MWC7+ = 206. SGC7+ = 0.8948.

Table 5. Number of Fractional Groups and Their Critical Properties, Calculated for the C7+ Fraction value Functional Groups CH3 CH2 (chain) CH (chain) C (chain) CH2 (cycle) CH (cycle) C (cycle) ACH ACC

6.70 0.48 1.93 0.09 1.40 0.16 2.22 0.26 1.73 Critical Properties

Tc Pc ω

745.13 K 1.892 MPa 0.484

the initial pressures, the minimum tieline lengths were obtained and, consequently, using the two types of LCVM-UNIFAC and VDW mixing rules, the MMP was obtained by extrapolating TL∞ versus pressure, as shown in Figure 8. As can be seen, the minimum tieline length determined by the LCVM-UNIFAC G

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Figure 7. Tieline length versus the inverse of the number of cells and the extrapolation of minimum tieline length to determine the minimum tieline length (TL∞) at 2000 psi for system 3.

Figure 8. Minimum tieline length versus pressure and its extrapolation to zero length to obtain MMP, using the LCVM and VDW mixing rules for system 3.

Table 6. Critical Properties Calculated for Undeﬁned Petroleum Fractions from F1 Group Contribution Method

Twu and Lee−Kesler Correlation

undeﬁned petroleum fractions

Tc (K)

Pc (MPa)

ω

Tc (K)

Pc (MPa)

ω

C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20+

505.8886 529.7808 554.4687 585.3686 614.9442 629.5067 651.9296 667.3034 684.5726 707.6364 721.3846 740.6933 750.3696 768.1841 1039.671

3.3246 3.3299 3.1115 2.8810 2.5854 2.2456 2.1749 1.9072 1.7214 1.7100 1.5160 1.4303 1.3706 1.2257 0.7318

0.2912 0.2925 0.3204 0.3663 0.4076 0.4963 0.5183 0.5297 0.5818 0.6048 0.6460 0.6190 0.6478 0.6952 0.6710

509.3568 529.8075 556.1567 589.5472 622.0337 647.0716 667.2723 685.3847 706.8222 726.2474 743.3706 756.9110 767.4732 787.1110 956.4168

3.0806 3.0885 2.9756 2.7663 2.5383 2.2619 2.1033 1.9896 1.8828 1.8384 1.7084 1.6221 1.5885 1.5081 1.0232

0.2979 0.3183 0.352 0.4018 0.4571 0.5113 0.5550 0.5948 0.6428 0.6839 0.7325 0.7717 0.7991 0.8552 1.4742

The results of the MMP calculation show that using the VDW mixing rule with the adjusted binary interaction parameters for the simple systems with a small number of components, such as light hydrocarbons, with injected gas gives

experimental data are presented in Table 9. As can be seen, the results of the LCVM-UNIFAC model demonstrate better agreement with the experimental data, compared to those obtained usign the VDW mixing rule. H

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Industrial & Engineering Chemistry Research Table 7. Critical Properties Calculated for Undeﬁned Petroleum Fractions from F2 Group Contribution Method

Twu and Lee−Kesler Correlation

undeﬁned petroleum fractions

Tc (K)

Pc (MPa)

ω

Tc (K)

Pc (MPa)

ω

C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20+

497.9900 529.7681 557.1993 582.2904 613.9200 623.6282 638.5312 662.2003 695.4232 712.3924 720.2605 738.4624 757.8412 766.4882 1015.100

3.0669 2.9952 2.7476 2.5508 2.6790 2.4039 2.2208 2.0164 1.9449 1.7897 1.6632 1.6579 1.5316 1.2495 0.6994

0.3108 0.3428 0.3624 0.4148 0.4485 0.4999 0.5361 0.5919 0.6512 0.6659 0.7283 0.7166 0.6404 0.8628 0.6301

507.7011 539.4306 567.8223 596.6875 623.2659 640.6057 658.3965 685.2079 715.9725 732.1205 744.5736 756.1422 767.6045 793.6523 946.2558

3.0460 3.0097 2.8303 2.6240 2.4585 2.3992 2.2423 2.0910 1.9345 1.8105 1.7292 1.6522 1.6108 1.4402 1.0710

0.2979 0.3319 0.3721 0.4182 0.4636 0.4923 0.5301 0.5857 0.6546 0.6983 0.7325 0.7661 0.7963 0.9432 1.4375

Table 8. Critical Properties Calculated for Undeﬁned Petroleum Fractions from F3 Group Contribution Method

Twu and Lee−Kesler Correlation

undeﬁned petroleum fractions

Tc (K)

Pc (MPa)

ω

Tc (K)

Pc (MPa)

ω

C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20+

505.4885 531.5348 567.2609 594.0142 621.4193 635.8740 653.8740 676.7884 683.2269 713.0880 725.8691 736.8939 751.4852 770.8581 958.7727

3.1747 3.1027 3.1342 2.8923 2.6843 2.5566 2.3866 2.2904 1.8416 1.9069 1.6941 1.5719 1.4632 1.4232 0.6873

0.2767 0.3075 0.3360 0.3830 0.4249 0.4557 0.4894 0.4872 0.5053 0.5127 0.5976 0.6403 0.6716 0.6572 0.6149

509.2416 536.3414 568.3315 598.4854 627.1847 643.7816 663.4878 682.1414 698.4913 719.8047 739.6144 752.2872 766.6794 780.1717 916.0732

3.0781 3.2902 3.0003 2.8032 2.4975 2.4241 2.2674 2.1398 2.0770 2.0115 1.8159 1.6758 1.5758 1.5426 1.0665

0.2979 0.3183 0.3654 0.4117 0.4668 0.4955 0.5363 0.5765 0.6099 0.6546 0.7127 0.7550 0.7991 0.8340 1.3480

Figure 9. Minimum tieline lengths versus pressure and their extrapolations to obtain the MMP of the oil F1/G1, using the LCVM and VDW mixing rules.

successfully. Note that characterization of the plus and undeﬁned petroleum fraction plays an important role in modeling the petroleum oil using the UNIFAC group

better results. However, applying the LCVM-UNIFAC mixing rule model allows one to calculate the MMP of multicomponent oil mixture with light to heavy components I

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Figure 10. Minimum tieline lengths versus pressure and their extrapolations to obtain the MMP of the oil F2/G2, using the LCVM and VDW mixing rules.

Figure 11. Minimum tieline lengths versus pressure and their extrapolations to obtain the MMP of the oil F3/G3, using the LCVM and VDW mixing rules.

advantages have been pointed out and can be applied in MMP estimation in future work. As explained previously, the MMP estimation gives poorer results using the group contribution method with the VDW mixing rule, with respect to the present method. In fact, in the VDW mixing rule with constant values of Kij, only bulk properties of undeﬁned petroleum fractions, e.g., the critical properties (Tc, Pc) and acentric factor, are considered for the phase equilibrium prediction, so that the molecular-level properties, e.g., the number of fractional and structure of groups are not considered. However, in the present work using a minimization of Gibbs energy, the bulk properties of undeﬁned fractions are calculated through determination of the number of fractional groups, using molecular weight (MW) and speciﬁc gravity (SG). This means that these values of critical properties are more consistent with the equilibrium state of the system and are reliable. Besides, the interactions of these fractional groups are taken into consideration in a molecular level that is not considered in the Twu and Lee− Kesler correlation applicable in the VDW mixing rule. In other words, in the present method, both the method of undeﬁned fraction characterization and the LCVM mixing rule use group contribution methods that are taken into consideration by the

Table 9. Results Comparison with the Experimental Data

system F1/ G1 F2/ G2 F3/ G3 a

ADa

van der Waals (VDW) mixing rule

ADa

3284 psi

2.46%

3533 psi

10.23%

3408 psi

3349 psi

1.73%

3714 psi

8.98%

5453 psi

5299 psi

2.82%

5869 psi

7.62%

slim tube experimental data

LCVMUNIFAC model of mixing rule

3205 psi

The parameter AD is deﬁned as follows: AD (%) =

|MMPcalculated − MMPexperiment| MMPexperiment

× 100

contribution model, so that inappropriate characterization leads to a serious deviation in MMP calculation. Another modiﬁcation of PR EOS has been developed as PPR78 EOS.57,58 In this EOS, the binary interaction parameters are temperature-dependent and estimated by three elementary groups of paraﬃnic (CPAR), naphthenic (CNAP), and aromatic (CARO) for pseudo-components, so that it is simple and its J

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(3) Jaubert, J.-N.; Avaullee, L.; Pierre, C. Is it still necessary to measure the minimum miscibility pressure? Ind. Eng. Chem. Res. 2002, 41, 303−310. (4) Danesh, A. PVT and Phase Behaviour of Petroleum Reservoir Fluids; Elsevier: Amsterdam, 1998. (5) Kontogeorgis, G. M.; Folas, G. K. Thermodynamic Models for Industrial Applications: From Classical and Advanced Mixing Rules to Association Theories; John Wiley & Sons: Chichester, U.K., 2009. (6) Huron, M.-J.; Vidal, J. New mixing rules in simple equations of state for representing vapour−liquid equilibria of strongly non-ideal mixtures. Fluid Phase Equilib. 1979, 3, 255−271. (7) Michelsen, M. L. A method for incorporating excess Gibbs energy models in equations of state. Fluid Phase Equilib. 1990, 60, 47−58. (8) Michelsen, M. L. A modified Huron−Vidal mixing rule for cubic equations of state. Fluid Phase Equilib. 1990, 60, 213−219. (9) Holderbaum, T.; Gmehling, J. PSRK: A group contribution equation of state based on UNIFAC. Fluid Phase Equilib. 1991, 70, 251−265. (10) Boukouvalas, C.; Spiliotis, N.; Coutsikos, P.; Tzouvaras, N.; Tassios, D. Prediction of vapor-liquid equilibrium with the LCVM model: A linear combination of the Vidal and Michelsen mixing rules coupled with the original UNIF. Fluid Phase Equilib. 1994, 92, 75− 106. (11) Chen, J.; Fischer, K.; Gmehling, J. Modification of PSRK mixing rules and results for vapor−liquid equilibria, enthalpy of mixing and activity coefficients at infinite dilution. Fluid Phase Equilib. 2002, 200, 411−429. (12) Ahlers, J.; Gmehling, J. Development of a universal group contribution equation of state. 2. Prediction of vapor-liquid equilibria for asymmetric systems. Ind. Eng. Chem. Res. 2002, 41, 3489−3498. (13) Voutsas, E.; Louli, V.; Boukouvalas, C.; Magoulas, K.; Tassios, D. Thermodynamic property calculations with the universal mixing rule for EoS/GE models: Results with the Peng−Robinson EoS and a UNIFAC model. Fluid Phase Equilib. 2006, 241, 216−228. (14) Haghtalab, A.; Mahmoodi, P. Vapor−liquid equilibria of asymmetrical systems using UNIFAC-NRF group contribution activity coefficient model. Fluid Phase Equilib. 2010, 289, 61−71. (15) Avaullee, L.; Trassy, L.; Neau, E.; Jaubert, J. N. Thermodynamic modeling for petroleum fluids I. Equation of state and group contribution for the estimation of thermodynamic parameters of heavy hydrocarbons. Fluid Phase Equilib. 1997, 139, 155−170. (16) Carreón-Calderón, B.; Uribe-Vargas, V. n.; Ramírez-Jaramillo, E.; Ramírez-de-Santiago, M. Thermodynamic characterization of undefined petroleum fractions using group contribution methods. Ind. Eng. Chem. Res. 2012, 51, 14188−14198. (17) Carreón-Calderón, B.; Uribe-Vargas, V. n.; Ramírez-de-Santiago, M.; Ramírez-Jaramillo, E. Thermodynamic Characterization of Heavy Petroleum Fluids Using Group Contribution Methods. Ind. Eng. Chem. Res. 2014, 53, 5598−5607. (18) Metcalfe, R.; Fussell, D.; Shelton, J. A multicell equilibrium separation model for the study of multiple contact miscibility in richgas drives. SPEJ, Soc. Pet. Eng. J. 1973, 13, 147−155. (19) Johns, R.; Dindoruk, B.; Orr, F., Jr. Analytical theory of combined condensing/vaporizing gas drives. SPE Advanced Technology Series 1993, 1, 7−16. (20) Wang, Y.; Orr, F. M. Analytical calculation of minimum miscibility pressure. Fluid Phase Equilib. 1997, 139, 101−124. (21) Jessen, K.; Michelsen, M. L.; Stenby, E. H. Global approach for calculation of minimum miscibility pressure. Fluid Phase Equilib. 1998, 153, 251−263. (22) Jensen, F.; Michelsen, M. Calculation of ﬁrst contract and multiple contact minimum miscibility pressures. In Situ 1990, 14, 1− 14. (23) Fazlali, A.; Nikookar, M.; Mohammadi, A. H. Computational procedure for determination of minimum miscibility pressure of reservoir oil. Fuel 2013, 106, 707−711. (24) Jaubert, J.-N.; Wolff, L.; Neau, E.; Avaullee, L. A very simple multiple mixing cell calculation to compute the minimum miscibility

group interaction parameters, which are consistent and compatible. Moreover, combining the group contribution characterization with the classical vdW mixing rule does not show compatibility from a molecular interaction point of view, so that the poorer results of MMP estimation is expected. Therefore, in the present work, the correlations such as Twu and Lee−Kesler that estimate critical properties using molecular weight (MW) and speciﬁc gravity (SG) directly provide better results with the VDW mixing rule, because of the correlative nature of both the characteristics method and the mixing rule. Finally, the combination of the fractional group contribution characterization and the group contribution activity coeﬃcient through LCVM-UNIFAC presented the better results, with regard to the MMP estimation, than the VDW mixing rule, in which the characterization is done by Twu and Lee−Kesler correlations.

4. CONCLUSION In this study, the minimum miscibility pressure (MMP) is estimated by using the multiple mixing cell (MMC) method. The LCVM-UNIFAC mixing rule with Twu’s modiﬁcation of the Peng−Robinson equation of state (EOS) was used for the MMP calculation of several hydrocarbon-gas injection systems. The results were compared with those obtained through the van der Waals (classic) mixing rule and compared with the slim tube experimental data. The vapor−liquid equilibrium (VLE) calculation through the EOS/GE model using the LCVMUNIFAC mixing rule presented a better agreement with experimental data for the real oil-gas systems containing a large number of light to heavy size molecules. Characterization of the undeﬁned petroleum fractions was carried out using the method reported by Carreón-Calderón et al.,16,17 which demonstrates its important role in the thermodynamic modeling; therefore, it has been revealed as an inﬂuential factor on the results of the MMP calculations.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.5b04447. Description of how the activity coeﬃcient of the UNIFAC model is determined; number of fractional groups for undeﬁned petroleum fractions of F1 (Table S1), F2 (Table S2), and F3 (Table S3) (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: (09821)82883313. Fax: (09821)82883381: E-mail: [email protected] Notes

The authors declare no competing ﬁnancial interest.

■

ACKNOWLEDGMENTS The authors are grateful to Kaveh Ahmadi and Bernardo Carreón-Calderón for their helpful suggestions in carrying out of this work successfully.

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REFERENCES

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L

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