A new robust stability algorithm for three phase flash ...

Journal of Natural Gas Science and Engineering 35 (2016) 382e391

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A new robust stability algorithm for three phase flash calculations in presence of water Nasser Sabet a, Hamid Reza Erfani Gahrooei b, * a b

Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta, Canada Petroleum University of Technology, Ahvaz Faculty of Petroleum Engineering, Ahvaz, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 27 May 2016 Received in revised form 23 August 2016 Accepted 27 August 2016 Available online 30 August 2016

Thermodynamic phase equilibrium calculations for systems containing water are inseparable parts of the compositional hydrocarbon reservoir simulation. In this regard, stability analysis of the phases for distinguishing the existing ones at specific pressures and temperatures is a principal part of such calculations. This study is aimed to develop a new stability algorithm with application to three phase flash calculations in the presence of brine. The developed scheme is capable of doing the phase behavior calculations in a more robust way compared to the other available algorithms. Henry's law is utilized to predict the aqueous phase properties, and a new initial guess is provided for three phase flash calculations that assures the convergence of the scheme. It is shown that the proposed procedure is able to handle the systems with high CO2 content while the available schemes in the literature fail. © 2016 Elsevier B.V. All rights reserved.

Keywords: Phase equilibrium Stability algorithm Three phase flash Henry's law CO2-water systems

1. Introduction Hydrocarbon reservoirs are explored mostly in sandstone or carbonate rocks. These rock types were originally formed in aquatic environments, and they were saturated with water before secondary migration. In secondary migration, oil comes through the reservoir rock from the source rock and displaces water (Tissot and Welte, 2013). Water-wet nature of reservoir rocks causes incomplete displacement of water by immigrated oil. As a result, water remains in hydrocarbon reservoirs in the form of a thin layer covering the rock grains or as a trapped phase in small pores and throats. This water saturation is called as connate water saturation (Anderson, 1987a, 1987b). Water may also be present below the hydrocarbon reservoir in the form of an aquifer. In this case, water will invade the reservoir rock after pressure decline (Ahmed, 2006). In addition to the socalled sources of water in hydrocarbon reservoirs, water may also be injected into the reservoir with the purpose of enhancing oil recovery (Willhite, 1986; Green and Willhite, 1998). According to the aforementioned reasons, water is an inseparable component of reservoir fluids and ignoring water in petroleum reservoir fluid

* Corresponding author. E-mail address: [email protected] (H.R. Erfani Gahrooei). http://dx.doi.org/10.1016/j.jngse.2016.08.068 1875-5100/© 2016 Elsevier B.V. All rights reserved.

phase calculations, has inappropriate consequences like inaccurate or even inconsistent results of reservoir simulation. It is worthy to mention that there is no pure water in the reservoir, so taking account of dissolved salts and ions are crucial factors which can affect the behavior of reservoir fluid, since it has a strong effect on gas solubility in the aqueous phase (Li and Nghiem, 1986). Various equations of states (EOS) are used in the petroleum industry for the purpose of modeling petroleum fluid phase behavior (Avlonitis et al., 1994). The most widely used types are cubic EOS like RK (Redlich and Kwong, 1949), SRK (Soave, 1972) and Peng-Robinson (Peng and Robinson, 1976). In the case of reservoir fluids, vapor and liquid hydrocarbon phases are modeled with these EOS in a range of reasonable error, while the predicted behavior of aqueous phase by these EOS is not yet acceptable. Several researchers have made effort to model water-rich (aqueous) phase with the same EOS as vapor and liquid hydrocarbon phases (Heidemann, 1974; Evelein et al., 1976; Peng and Robinson, 1980; Mokhatab, 2003). They found that accurate prediction of aqueous phase behavior is difficult to achieve with the original form of EOS, so they made some modifications in that EOS in order to achieve a higher degree of accuracy in results (Erbar, 1980; Peng and Robinson, 1980; Reshadi et al., 2011). On the other hand, various researchers discarded using EOS for water-rich phase and used other thermodynamic approaches like Henry's law for modeling the aqueous phase behavior and an EOS

N. Sabet, H.R. Erfani Gahrooei / Journal of Natural Gas Science and Engineering 35 (2016) 382e391

for liquid and vapor hydrocarbon phases (Luks et al., 1976; Mehra et al., 1982; Nghiem and Heidemann, 1982; Li and Nghiem, 1986; Carroll and Mather, 1997). Since hydrocarbon components are sparingly soluble in the aqueous phase, Henry's law constraint is satisfied in the case of three phase flash calculation in the presence of water. Moreover, comparison of results with experimental data shows a good prediction of aqueous phase behavior using Henry's law (Mackay et al., 1979; Li and Nghiem, 1986; Altschuh et al., 1999). Nghiem and Heidemann (1982) performed three phase hydrocarbon-brine flash calculation using Henry's law for modeling of water-rich phase. Since there were not any safely generalized correlations for Henry's law constants until that time, they used Henry's constants calculated from experimental data and only considered the solution of carbon dioxide component in water. They also neglected the presence of water in vapor and liquid hydrocarbon-rich phases. Li and Nghiem (1986) proposed a correlation for calculation of Henry's constants with respect to pressure and temperature for various hydrocarbon components of petroleum industry interest. They also considered the effect of brine salinity, using the scaled-particle theory (SPT) to modify Henry's law constants derived for pure water. Lapene et al. (2010) introduced a new three-phase free-water flash method by using modified Rachford-Rice equation. Their proposed algorithm guarantees the convergence of flash calculations. They assumed water as a free phase and neglected the dissolution of hydrocarbon components in aqueous phase. However, they took the water coexistence with hydrocarbon in vapor and liquid hydrocarbon rich phases into consideration. Their algorithm also does not consider the disappearance of water-rich phase, which can happen in low water-content feeds at high temperature or low-pressure ranges. In addition to EOS selection and phase behavior prediction of reservoir fluid in the flash calculation, phase stability analysis is another important issue in this realm. By means of phase stability analysis, one can predict the presence of phases at a certain pressure and temperature. For instance, there may exist only a hydrocarbon-rich vapor phase in a high temperature and low pressure for a low-water-content overall composition (feed). Two main approaches have been introduced to solve the phase stability problem: the stationary points method (classical method) and the direct minimization of the tangent plane distance (TPD) function (Nichita et al., 2002). The stationary points method was developed by Michelsen (1982b) and used by many researchers for multicomponent flash calculation phase stability check (Nghiem et al., 1983; Li and Nghiem, 1986; Nelson, 1987; Gupta et al., 1991; Ballard and Sloan, 2004; Ghosh et al., 2004). Li and Nghiem (1986) proposed a flow diagram of a stepwise phase stability check procedure for three phase flash calculations in the presence of water. Their stability test method was developed for determining the stable phases at specified pressures and temperatures among liquid, vapor and/or aqueous phases. Their proposed procedure fails to detect the phases, accurately, in some cases that will be discussed in results and discussion part. In this work, a new stability algorithm is proposed for three phase flash calculation in the presence of water and its validity is checked by means of performing a flash calculation on two different overall compositions. The proposed stability algorithm identifies the stable phases suitably at different pressures and temperatures even for systems with high CO2 content while other available schemes fail. Additionally, a new general initial guess is provided for three phase flash calculation which guarantees the convergence of the calculations under different situations.

383

2. Theory 2.1. Henry's law As discussed earlier, vapor and liquid phases are modeled using any adequate EOS, but, the water-rich phase is modeled using Henry's law. The reason behind using Henry's law for this phase is that the solubility of hydrocarbon components in water is low (Polak and Lu, 1973; Wasik and Brown, 1973; Li and Nghiem, 1986). Henry's law constant for a sparingly soluble component in water is defined as follows:

HLC ¼

Partial pressure in gas phase Mole fraction in aqueous phase

(1)

Equation (1) can also be expressed in terms of fugacity that is handy in flash calculations:

fi aq ¼ yi aq Hi isw

(2)

Subscripts aq and w denote aqueous phase and water component, respectively. Hi is the Henry's law constant of component i in the aqueous phase. Smith et al. (2001) proposed a differential equation for calculating changes of Henry's law constant with respect to pressure and temperature, as follows:

dðlnðHi ÞÞ ¼

∞ Vm hi v  h∞ i i dP þ dT RT RT 2

(3)

∞ is partial molar volume of component i in the aqueous where Vm i phase at infinite dilution, hi v is the enthalpy of component i in the vapor phase, and h∞ i is the enthalpy of component i in the aqueous phase at infinite dilution. The term hi v  h∞ i is strongly dependent on temperature but there is not any available correlation for ∞ is describing its dependency. It has also been found that, Vm i usually not very sensitive to pressure (Li and Nghiem, 1986). Therefore, for a given temperature, integration of equation (3) from P0 to P gives:


 ∞ P  P0
 Vm i i lnðHi Þ ¼ ln Hi0 þ RT

(4)

where Hi0 is henry's law constant at reference pressure Pi0 . Equation (4) can also be written as:


 V ∞P lnðHi Þ ¼ ln Hi* þ i RT

(5)

where Hi* is defined as below:


 V ∞ P0
 ln Hi* ¼ ln Hi0  m i i RT

(6)

In this study, Equation (6) is used for calculating Henry's law constant from solubility data and Hi* is defined as the reference Henry's law constant. The molar volume at infinite dilution is calculated using the following correlation proposed by Lyckman et al. (1965):

  ∞ Pc i Vm TPc i i ¼ 0:095 þ 2:35 RTc i CTc i where C is the cohesive energy density of water given by:

(7)

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N. Sabet, H.R. Erfani Gahrooei / Journal of Natural Gas Science and Engineering 35 (2016) 382e391


C¼

s Vs h0w  hsw þ Pw m w  RT



fi v ¼ fi l fi aq ¼ fi l

s is defined as saturated vapor pressure of water at temperature T, Pw s s 0 s Vm w is the molar volume of water at Pw and hw  hw is the enthalpy s and T. departure of liquid water at Pw

Gibbs-Duhem equation is used to calculate the fugacity of water component in the aqueous phase (Prausnitz et al. (1998)) as follows:

0 s B fw aq ¼ yw aq Фsw Pw @

ZP

1 Vm w C dpA RT

(9)

s Pw

s and V Fsw is fugacity coefficient of pure water at Pw m w is molar volume of pure water. s ) and enthalpy deIn this study, vapor pressure of water (Pw parture of liquid water (h0w  hsw ) are calculated from FrostKalkward-Thodos and Yen-Alexander, respectively, which are reported in Poling et al. (2001) in the form of:

!



  1 27 Pvpr ln Pvp r ¼ B  1 þ C lnðTr Þ þ 1 Tr 64 Tr2 " lnðPc Þ þ 2:67 lnðTb r Þ þ 27 64 B¼

! 1 Pc Tb2 r

(10)

#

1  T1b r  0:7816 lnðTb r Þ

(12)

Ki v ¼

xi v Фi l ¼ xi l Фi v

Ki aq ¼

i ¼ 1; …; nc

xi aq Ф ¼ il xi l Фi aq

(17)

i ¼ 1; …; nc

(18)

where xim is the mole fraction of component i in phase m (l, v, aq). Using the material balance for component i, equations (19) and (20) can be derived as: nc nc X X ðxi v  xi l Þ ¼ i¼1

ðKi v  1Þzi ¼0 Fl þ Fv Ki v þ Faq Ki aq

i¼1

nc  nc X  X xi v  xi aq ¼ i¼1

 Ki aq  1 zi ¼0 Fl þ Fv Ki v þ Faq Ki aq

(19)



(20)

where zi is the global mole fraction of component i and Fm is the mole fraction of phase m, which should satisfy the following equation:

(21)

The system of 2ncþ3 non-linear equations (19)e(21), must be solved to obtain 2ncþ3 unknowns (Kiaq, Kiv, Fl, Fv, Faq). In this work, the quasi-Newton successive substitution method is used to solve the system of equations. The following equations are used to obtain the initial guess for equilibrium ratios (Li and Nghiem, 1986). ð0Þ

Ki v ¼

   Pc i T exp 5:42ð1 þ ui Þ 1  c i P T

i ¼ 1; …; nc

(22)

(13)

The fugacity coefficient of water Фsw is calculated by Chou equation which is reported by Rowe Jr. and Chou (1970) as follows:



(16)

Fl þ Fv þ Faq ¼ 1 (11)

and also:

0:333   Pvp 7  4:5688 ln 217:6   h0w  hsw ¼ Tc Pvp 1 þ 0:004 ln 217:6

i ¼ 1; …; nc

i¼1

1

C ¼ 0:7816 B þ 2:67

Фsw ¼

(15)

fim represents the fugacity of component i in m (l, v, aq) phase. Subscripts l, v and aq denote liquid, vapor and aqueous phase, respectively. Li and Nghiem (1986) defined the following sets of Kvalues (equilibrium ratios):

2.2. Water component fugacity



i ¼ 1; …; nc

(8)

s Vm w

ð0Þ

Ki aq ¼ Ki v

2.3. Phase equilibrium calculations for L-V-W systems For a three phase system with nc components, the following equations must be satisfied in order the system to be in thermodynamic equilibrium:

isw

(23)

where P is the system pressure, Pc is the critical pressure, u is the acentric factor, T is the system temperature, and Tc is the critical

0:9958 þ 9:68330  105 T 0  6:175  107 T 02  3:08333  1010 T 03 1

It is worthwhile mentioning that one can use the scaled particle theory for taking into account the dissolved salts in reservoir brine in the calculation of Henry's law constants (Reiss et al., 1960; Lebowitz et al., 1965; Pierotti, 1967, 1976).

P Hi

T 0 > 90 T 0 < 90

+

F + F

 (14)

temperature. For the water component, a large value of Kw aq (for example >1000) is chosen which satisfies the two following conditions:

nc
X i¼1

 ð0Þ Ki aq zi > 1

(24)

N. Sabet, H.R. Erfani Gahrooei / Journal of Natural Gas Science and Engineering 35 (2016) 382e391

0 1 nc X z i @ A>1 ð0Þ i¼1 Ki aq

(25)

Convergence rate of three phase flash calculation algorithm is

385

very sensitive to the initial guesses of phases mole fractions. In this work, we used the following approach to obtain the initial guesses for Fl, Fv and Faq. Global mole fraction of the lightest component, the heaviest component, and the water component are assigned to Fl, Fv and Faq, respectively. Then, the values are normalized, so that the summation of Fl, Fv and Faq equals to unity. Performing 3-phase

Fig. 1. Three phase flash calculation flowchart.

386

N. Sabet, H.R. Erfani Gahrooei / Journal of Natural Gas Science and Engineering 35 (2016) 382e391

flash calculation with the introduced initial guess promises the convergence of the calculations in different conditions. Fig. 1 shows the steps needed for the 3-phase flash calculation. In this flowchart, it is assumed that the feed mixture has three phases in the desired temperature and pressure. The mentioned composition dependent EOS parameters in this figure can be found elsewhere in the literature (Sage et al., 1942; Reamer et al., 1944; Culberson and McKetta Jr., 1951; Kobayashi and Katz, 1953; Takenouchi and Kennedy, 1964; Rigby and Prausnitz, 1968; Brady et al., 1982; Li and Nghiem, 1986; Danesh, 1998). “J” is the Jacobian matrix and the numerical derivative is employed to obtain its elements. Moreover, J is defined as the natural logarithm of the equilibrium ratios. The details of the procedure presented in Fig. 3 are mentioned in Appendix A for a sample three component mixture. 2.4. Three phase stability algorithm Given the global mole fractions, the system can be either singlephase, two-phase or even three-phase. Fig. 2 shows the algorithm for LVW flash calculations. It is similar to the algorithm proposed by Li and Nghiem (1986). However, there are some differences between the two algorithms that will be mentioned in the following sections. In this algorithm, the stability tests are actually two-phase flash calculations including a stability test inside them. The output of such flash calculation is that it can predict whether phase A is the stable phase or phase B is the stable one or even the system is a two-phase A-B system at the specified pressure and temperature. The main difference between the newly proposed procedure and the one proposed by Li and Nghiem (1986) is that this algorithm includes an L-V stability test assuming that there is no water

Fig. 3. Three phase flash calculation of the fourth case based on Li and Nghiem (1986) algorithm.

component in the system. This L-V stability test is used only for distinguishing the number and type of phases and not for calculating the equilibrium properties. In the algorithm depicted in Fig. 2, for two-phase stability checks, the method proposed by Michelsen (1982b) is used (See Appendix B). The superiority of the proposed algorithm over Li and Nghiem (1986) algorithm is its ability to correctly predict the behavior of systems with high CO2 content which exist in some processes like CO2 storage (Hassanzadeh et al., 2008; Salari et al., 2011).

Fig. 2. The proposed algorithm for LVW flash calculations.

N. Sabet, H.R. Erfani Gahrooei / Journal of Natural Gas Science and Engineering 35 (2016) 382e391 Table 1 Feed composition of the first case. Component

Composition

C1 nC5 nC10 CO2 H2S H2O

30 15 25 10 10 10

Table 2 Results of three phase flash calculation for the first case. Component

C1 nC5 nC10 CO2 H2S H2O

This study

Li and Nghiem (1986)

xl%

xv%

xaq%

xl%

xv%

xaq%

22.8840 20.2161 35.8888 9.1302 10.8875 0.9938

65.2194 4.6032 0.6622 16.8451 11.2003 1.4698

0.08 0.0000 0.0000 0.2597 0.3370 99.3179

22.8836 20.2157 35.8889 9.1482 10.9055 0.9581

65.2197 4.6033 0.6622 16.8787 11.2189 1.4170

0.0011 0.0000 0.0000 0.0201 0.1413 99.8375

Fl%

Fv%

Faq%

Fl%

Fv%

Faq%

69.2603

21.6879

9.0548

69.2590

21.6973

9.0437

zl

zv

zaq

zl

zv

zaq

0.4387

0.8405

0.0728

0.4387

0.8406

0.0725

3. Results and discussion In order to perform the 3-phase flash calculation for any desired overall composition, the stability of phases must be checked using the newly proposed algorithm (see Fig. 2). The system can be three phase (L-V-W), two phase (L-V, L-W, V-W), or single phase (L, V, W) under the studied pressure and temperature conditions. If the stability algorithm predicts the existence of 3 phases (L-V-W), then three phase flash calculations are done (see Fig. 1). Otherwise, if LV, L-W, or V-W exist, the 2 phase flash approaches are used (Michelsen, 1982a, 1986; Ammar and Renon, 1987). Two cases are analyzed in this work for validating the proposed algorithm. The first composition is mentioned in Table 1. According to Table 2, Results of flash calculation at 10 MPa and 100  C are in complete agreement with the ones obtained by Li and Nghiem (1986) algorithm. Default values in the literature (Sage et al., 1942; Reamer et al., 1944; Culberson and McKetta Jr., 1951; Kobayashi and Katz, 1953; Takenouchi and Kennedy, 1964; Rigby and Prausnitz, 1968; Brady et al., 1982; Li and Nghiem, 1986; Danesh, 1998) are used for binary interaction coefficients and reference Henry's law constants, which are stated in Table 3. The other case is a system with only two existing phases. 1: 0:2; 2:

fw

T  373 K; !

Hi* s aq ð@Pw Þ

0:49852  0:008ðTÞ;     6 103  C 10 ¼ A þ B TðKÞ T2

387

The results of phase flash calculation at 18 MPa and 200  C for the feed composition of the second case (shown in Table 4) are as Table 5. The results shown in Tables 2 and 5 show that the developed model can predict the phase behavior of hydrocarbon mixtures containing water, accurately. As it was mentioned in the theory part, the proposed algorithm has an LV stability of the mixture assuming there is no water component in the mixture. This stability is used to avoid the miscalculation in phase type distinguish. In CO2 storage, it is needed to study the behavior of ternary systems of CO2, H2S, and H2O (Hassanzadeh et al., 2008; Salari et al., 2011). Algorithm proposed by Li and Nghiem (1986) provides anamolous predictions for the phase type in some cases. Table 5 is a mixture of the mentioned components. The results of phase flash calculation at 15 MPa and 60  C for the feed composition of the third case (shown in Table 6) are presented in Table 7. As it can be seen in Table 7, all quantities are approximately the same. The difference is in phase type prediction. Li and Nghiem (1986) algorithm predicts this mixture as a two-phase vaporaqueous system, while the proposed algorithm predicts it as a twophase liquid-aqueous system which is the true state of the mixture in the specified pressure and temperature. One of the problems associated with incorrect phase detection can be the phase properties prediction such as viscosity in which there will be a huge difference if gas viscosity correlations are used instead of oil viscosity correlations. For a better understanding of Li and Nghiem (1986) algorithm shortcoming in correct phase detection, we may consider another overall composition and perform the flash calculation in different pressures at a fixed temperature, the feed composition of the fourth case is shown in Table 8. If we perform three phase flash calculation on this overall composition based on the proposed algorithm by Li and Nghiem (1986) at various pressures at the fixed temperature of 100  C and plot mole fraction of three phases (feed is three phase at this pressures and temperature) it yields Fig. 3. With increasing pressure at a fixed temperature, we expect more liquid and less vapor in three phase region (increase in Fl and decrease in Fv), but the results show an abrupt decrease in liquid mole fraction and increase in vapor mole fraction, which is undoubtedly contradictory to our expectations. On the other hand, if we perform this series of flash calculations with new proposed algorithm, results in Fig. 4, which does not have the malfunction of Li and Nghiem (1986) in phase detection algorithm and its results are reasonable. 4. Conclusion This paper presents a new robust stability algorithm for true detection of stable phases in any desired pressure and temperature for hydrocarbon/water system, in which hydrocarbon phase or phases are modeled with an appropriate equation of state (EOS) and the aqueous phase is modeled using Henry's law. Stepwise

T > 373 K

Table 3 List of binary interaction coefficients between different components and Henry's law constants. C1 C1 nC5 nC10 CO2 H2S H2O

0.0206 0.0522 0.103 0.031 0.4907

nC5

nC10

CO2

H2S

H2O

A2

B

C

0.0206

0.0522 0.0078

0.103 0.125 0.11

0.031 0.095 0.1 0.096

0.4907 0.5 0.45 0.21 0.275

10.9554 16.0045 e 11.3021 10.8393 e

11.3569 16.2281 e 10.6030 9.8897 e

1.17105 2.13123 e 1.20696 1.11984 e

0.0078 0.125 0.095 0.5

0.11 0.1 0.45

0.096 0.2

0.275

388

N. Sabet, H.R. Erfani Gahrooei / Journal of Natural Gas Science and Engineering 35 (2016) 382e391 Table 4 Feed composition of the second case. Component

Composition (%)

C1 nC5 nC10 CO2 H2S H2O

45 10 5 5 5 30

Table 5 Results of two phase flash calculation for the second case. Component

C1 nC5 nC10 CO2 H2S H2O

This study

Li and Nghiem (1986)

xv%

xaq%

xv%

xaq%

56.5210 12.5733 6.2873 6.2384 6.2149 12.1650

0.2527 0.0000 0.0000 0.1900 0.2815 99.2706

56.5459 12.5685 6.2843 6.2695 6.2155 12.1163

0.0483 0.0000 0.0000 0.0577 0.2675 99.6264

Fv%

Faq%

Fv%

Faq%

79.5248

20.4752

79.5639

20.4361

zv

zaq

zv

zaq

0.8409

0.1159

0.8411

0.1154

Table 6 Feed composition of the third case. Component

Composition (%)

CO2 H2S H2O

45 5 50

CO2 H2S H2O

calculation of Henry constants for different components of hydrocarbon, carbon dioxide and hydrogen sulfide in water, in any temperature and pressure, is presented. Additionally, a new initial guess for mole ratios in three phase flash calculation is provided that makes the algorithm convergence faster. Presented results show that the new proposed algorithm eventuates in reasonable, accurate and true results, even in cases that former methods fail to predict acceptable phase detection.

Appendix A In this appendix, the details of three phase flash calculation for a three component system is presented. Consider the following mixture of methane, normal pentane, and water.

Table 7 Results of flash calculation for the third case. Component

Fig. 4. Three phase flash calculation of the fourth case based on the proposed algorithm.

This study

Li and Nghiem (1986)

xl%

xaq%

xv%

xaq%

89.5317 9.7114 0.75689

2.0116 0.4519 97.5365

89.3934 9.8573 0.7493

0.0485 0.0816 99.8698

Fl%

Faq%

Fv%

Faq%

49.1183

50.8817

49.6877

50.3123

zl

zaq

zv

zaq

0.4095

0.1195

0.4094

0.1178

Table 8 Feed composition of the fourth case.

Table A1 Three component mixture sample. Component

Composition (%)

C1 nC5 H2O

40 30 30

This mixture has three phases at a pressure of 10 MPa and temperature of 100  C based on the stability algorithm introduced in Fig. 2. As depicted in Fig. 1, the first step is to find initial guess for equilibrium ratios and mole fractions of phases. Kv and Kaq are calculated based on equations (22) and (23), respectively:

Kv ¼ ½ 6:4987

Component

Composition (%)

CO2 nC5 nC10 H2S H2O

42 3 9 15 31

Kaq ¼ ½ 0:0090

0:0594 0:0000

0:0110  49:3462 

(A.1) (A.2)

J is defined as the natural logarithm of the equilibrium Ratio and its value at the first iteration is:

Jð0Þ ¼ ½ 1:8716 2:8229 4:5118 4:7131 10:7657 3:8989 

(A.3)

N. Sabet, H.R. Erfani Gahrooei / Journal of Natural Gas Science and Engineering 35 (2016) 382e391

For the phase mole fractions, as expressed earlier in this paper, the composition of the lightest component, the heaviest component, and water, are assigned to Fl (¼ 0.3), Fv (¼ 0.4), and Faq (¼ 0.3), respectively. Then, the values are normalized to reach a summation of unity for phase mole fractions and are updated using equations (19) and (20). The updated values are Fl (¼ 0.4622), Fv (¼ 0.2429), and Faq (¼ 0.2949). The next step is to calculate the mole fractions of each component in each phase as shown in Fig. 1 using the calculated values for equilibrium ratios and the phase mole fractions. Using the obtained mole fractions and the PR 1976 equation of state, the fugacity coefficients and the Jacobian Matrix of “g” functions are calculated. Composition dependent EOS parameters for liquid and vapor phase, fugacity coefficients, and the Jacobian Matrix for the first iteration are:

al ¼ 1:5409  107 ;

bl ¼ 73:0701;

Al ¼ 1:6223;

Bl

¼ 0:2355

(A.4)

av ¼ 4:0721  106 ;

bv ¼ 39:1187;

Av ¼ 0:4287;

Bv

¼ 0:1261

Fl ¼ ½ 2:5870 0:0769 1:4070 

(A.5) (A.6)

Fv ¼ ½ 0:9233 0:4097 0:8419 

(A.7)

Faq ¼ ½ 723:9 2815:5 0:0 

(A.8)

2

0:9258 0:0094 0:0000 0:0001 6 0:0422 0:8773 0:0000 0:0001 6 6 0:0332 0:0009 0:9997 0:0001 6 J¼6 6 0:0727 0:0030 0:0000 0:9999 4 0:0135 0:0006 0:0000 0:0000 0:0342 0:0029 0:0000 0:0001

389

solving the following set of equations (Nghiem and Li, 1984) for the primary set of variables (u):


   y Lnðui Þ þ Ln Фi  Lnðxi Þ  Ln Фxi ¼ 0

(B.1)

In equation (B.1), ui is a set of parameters and Ф is fugacity coefficient. The phase will be stable if: n X

ui < 1

(B.2)

i¼1

In the situation where the phase is unstable, the obtained equilibrium ration Kip can be used as a suitable first guess to start equilibrium calculation.

Kip ¼

ui xi

(B.3)

The flowchart of stability analysis for two phase LV system is described in Fig. B1.

3 0:0000 0:0058 0:0000 0:0022 7 7 0:0000 0:0491 7 7 0:0000 0:0058 7 7 1:0000 0:0022 5 0:0000 0:9506 (A.9)

The new value of J can be calculated as:

jð1Þ ¼ ½ 0:9554 1:4702 0:5350 5:6992 10:4968 4:8968  (A.10) The norm of the difference between J and J based on the formula in Fig. 1 is 0.0154 which is larger than the 105 (acceptable error or condition of convergence). The values of phase mole fractions are updated and the calculations are done for another iteration till the convergence criterion is met. The final value of J is: (0)

(1)

j ¼ ½ 0:7470 1:1992 0:4234 5:8653 10:4143 4:7878  (A.11) Accordingly, the phase mole fractions and the composition of each phase are obtained:

xl ¼ ½ 0:3805

0:6112

0:0083 ;

Fl ¼ 0:6322

(A.12)

xv ¼ ½ 0:8031

0:1842

0:0127 ;

Fv ¼ 0:2941

(A.13)

xaq ¼ ½ 0:0011

0:0000

0:9989 ;

Faq ¼ 0:0736

(A.14)

Appendix B In this appendix the details of two phase L-V stability test is presented. Stability analysis of two phase systems with overall composition (z) by means of tangent plate criterion, results in

Fig. B1. Flow Chart for two phase LV stability test based on Michelsen (1982b) (Nghiem and Li, 1984).

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Nomenclature f H h P T R Vm C

Ф K x z F L V W

u Z

Fugacity Henry's law constant Enthalpy Pressure Temperature Universal gas constant Molar volume Cohesive energy density Fugacity coefficient Equilibrium ratio Mole fraction Global mole fraction Phase mole fraction Liquid phase Vapor phase Aqueous phase Acentric factor Compressibility factor

Subscripts aq Aqueous phase l Liquid phase v Vapor phase c Critical w Water component R Reduced i Component i b Boiling point Superscripts ∞ Infinite dilution * Reference s Saturated References Ahmed, T., 2006. Reservoir Engineering Handbook. Gulf Professional Publishing. Altschuh, J., Brüggemann, R., Santl, H., Eichinger, G., Piringer, O.G., 1999. Henry's law constants for a diverse set of organic chemicals: experimental determination and comparison of estimation methods. Chemosphere 39 (11), 1871e1887. Ammar, M., Renon, H., 1987. The isothermal flash problem: new methods for phase split calculations. AIChE J. 33 (6), 926e939. Anderson, W.G., 1987a. Wettability literature survey part 5: the effects of wettability on relative permeability. J. Petrol. Technol. 39 (11), 1,453e451,468. Anderson, W.G., 1987b. Wettability literature survey-part 4: effects of wettability on capillary pressure. J. Pet. Technol. (United States) 39 (10). Avlonitis, D., Danesh, A., Todd, A., 1994. Prediction of VL and VLL equilibria of mixtures containing petroleum reservoir fluids and methanol with a cubic EoS. Fluid Phase Equilib. 94, 181e216. Ballard, A., Sloan, E., 2004. The next generation of hydrate prediction: Part III. Gibbs energy minimization formalism. Fluid Phase Equilib. 218 (1), 15e31. Brady, C.J., Cunningham, J.R., Wilson, G.M., 1982. Water-hydrocarbon Liquid-liquidvapor Equilibrium Measurements to 530F. Gas Processors Association. Carroll, J.J., Mather, A.E., 1997. A model for the solubility of light hydrocarbons in water and aqueous solutions of alkanolamines. Chem. Eng. Sci. 52 (4), 545e552. Culberson, O., McKetta Jr., J., 1951. Phase equilibria in hydrocarbon-water systems III-the solubility of methane in water at pressures to 10,000 psia. J. Petrol. Technol. 3 (08), 223e226. Danesh, A., 1998. PVT and Phase Behaviour of Petroleum Reservoir Fluids. Elsevier. Erbar, J.H., 1980. Predicting Synthetic Gas and Natural Gas Thermodynamic Properties Using a Modified Soave Redlich Kwong Equation of State. Gas Processors Association. Evelein, K.A., Moore, R.G., Heidemann, R.A., 1976. Correlation of the phase behavior in the systems hydrogen sulfide-water and carbon dioxide-water. Ind. Eng. Chem. Process Des. Dev. 15 (3), 423e428. Ghosh, A., Ting, P.D., Chapman, W.G., 2004. Thermodynamic stability analysis and pressure-temperature flash for polydisperse polymer solutions. Ind. Eng. Chem. Res. 43 (19), 6222e6230. Green, D.W., Willhite, G.P., 1998. Enhanced oil recovery. In: Henry, L. (Ed.), Doherty Memorial Fund of AIME. Society of Petroleum Engineers, Richardson, Tex. Gupta, A.K., Raj Bishnoi, P., Kalogerakis, N., 1991. A method for the simultaneous

phase equilibria and stability calculations for multiphase reacting and nonreacting systems. Fluid Phase Equilib. 63 (1), 65e89. Hassanzadeh, H., Pooladi-Darvish, M., Elsharkawy, A.M., Keith, D.W., Leonenko, Y., 2008. Predicting PVT data for CO 2ebrine mixtures for black-oil simulation of CO 2 geological storage. Int. J. Greenh. Gas Control 2 (1), 65e77. Heidemann, R.A., 1974. Three-phase equilibria using equations of state. AIChE J. 20 (5), 847e855. Kobayashi, R., Katz, D., 1953. Vapor-liquid equilibria for binary hydrocarbon-water systems. Ind. Eng. Chem. 45 (2), 440e446. Lapene, A., Nichita, D.V., Debenest, G., Quintard, M., 2010. Three-phase free-water flash calculations using a new Modified RachfordeRice equation. Fluid Phase Equilib. 297 (1), 121e128. Lebowitz, J.L., Helfand, E., Praestgaard, E., 1965. Scaled particle theory of fluid mixtures. J. Chem. Phys. 43 (3), 774e779. Li, Y.K., Nghiem, L.X., 1986. Phase equilibria of oil, gas and water/brine mixtures from a cubic equation of state and Henry's law. Can. J. Chem. Eng. 64 (3), 486e496. Luks, K.D., Fitzgibbon, P.D., Banchero, J.T., 1976. Correlation of the equilibrium moisture content of air and of mixtures of oxygen and nitrogen for temperatures in the range of 230 to 600 K at pressures up to 200 Atm. Ind. Eng. Chem. Process Des. Dev. 15 (2), 326e332. Lyckman, E., Eckert, C., Prausnitz, J., 1965. Generalized reference fugacities for phase equilibrium thermodynamics. Chem. Eng. Sci. 20 (7), 685e691. Mackay, D., Shiu, W.Y., Sutherland, R.P., 1979. Determination of air-water Henry's law constants for hydrophobic pollutants. Environ. Sci. Technol. 13 (3), 333e337. Mehra, R.K., Heidemann, R.A., Aziz, K., 1982. Computation of multiphase equilibrium for compositional simulation. Soc. Petrol. Eng. J. 22 (01), 61e68. Michelsen, M.L., 1982a. The isothermal flash problem. Part II. Phase-split calculation. Fluid Phase Equilib. 9 (1), 21e40. Michelsen, M.L., 1982b. The isothermal flash problem. Part I. Stability. Fluid Phase Equilib. 9 (1), 1e19. Michelsen, M.L., 1986. Simplified flash calculations for cubic equations of state. Ind. Eng. Chem. Process Des. Dev. 25 (1), 184e188. Mokhatab, S., 2003. Three-phase flash calculation for hydrocarbon systems containing water1. Theor. Found. Chem. Eng. 37 (3), 291e294. Nelson, P., 1987. Rapid phase determination in multiple-phase flash calculations. Comput. Chem. Eng. 11 (6), 581e591. Nghiem, L., Heidemann, R., 1982. General acceleration procedure for multiphase flash calculation with application to oil-gas-water systems. In: Proceedings of the 2nd European Symposium on Enhanced Oil Recovery. Nghiem, L.X., Li, Y.-K., 1984. Computation of multiphase equilibrium phenomena with an equation of state. Fluid Phase Equilib. 17 (1), 77e95. Nghiem, L.X., Aziz, K., Li, Y., 1983. A robust iterative method for flash calculations using the Soave-Redlich-Kwong or the Peng-Robinson equation of state. Soc. Petrol. Eng. J. 23 (03), 521e530. Nichita, D.V., Gomez, S., Luna, E., 2002. Phase stability analysis with cubic equations of state by using a global optimization method. Fluid Phase Equilib. 194, 411e437. Peng, D.-Y., Robinson, D.B., 1976. A new two-constant equation of state. Ind. Eng. Chem. Fundam. 15 (1), 59e64. Peng, D., Robinson, D., 1980. Two-and three-phase equilibrium calculations for coal gasification and related processes. Thermodyn. Aqueous Syst. Ind. Appl. 133, 393e414. Pierotti, R.A., 1967. Scaled-particle theory of dilute aqueous solutions. J. Phys. Chem. 71 (7), 2366e2367. Pierotti, R.A., 1976. A scaled particle theory of aqueous and nonaqueous solutions. Chem. Rev. 76 (6), 717e726. Polak, J., Lu, B.C.-Y., 1973. Mutual solubilities of hydrocarbons and water at 0 and 25 C. Can. J. Chem. 51 (24), 4018e4023. Poling, B.E., Prausnitz, J.M., John Paul, O.C., Reid, R.C., 2001. The Properties of Gases and Liquids. McGraw-Hill, New York. Prausnitz, J.M., Lichtenthaler, R.N., de Azevedo, E.G., 1998. Molecular Thermodynamics of Fluid-phase Equilibria. Pearson Education. Reamer, H., Olds, R., Sage, B., Lacey, W., 1944. Compositions of the coexisting phases of n-butaneewater system in the three-phase region. Ind. Eng. Chem. 36, 381e383. Redlich, O., Kwong, J., 1949. On the thermodynamics of solutions. V. An equation of state. Fugacities of gaseous solutions. Chem. Rev. 44 (1), 233e244. Reiss, H., Frisch, H., Helfand, E., Lebowitz, J., 1960. Aspects of the statistical thermodynamics of real fluids. J. Chem. Phys. 32 (1), 119e124. Reshadi, P., Nasrifar, K., Moshfeghian, M., 2011. Evaluating the phase equilibria of liquid waterþ natural gas mixtures using cubic equations of state with asymmetric mixing rules. Fluid Phase Equilib. 302 (1), 179e189. Rigby, M., Prausnitz, J.M., 1968. Solubility of water in compressed nitrogen, argon, and methane. J. Phys. Chem. 72 (1), 330e334. Rowe Jr., A.M., Chou, J.C., 1970. Pressure-volume-temperature-concentration relation of aqueous sodium chloride solutions. J. Chem. Eng. Data 15 (1), 61e66. Sage, B., Reamer, H., Olds, R., Lacey, W., 1942. Phase equilibria in hydrocarbon systems. Ind. Eng. Chem. 34 (9), 1108e1117. Salari, H., Hassanzadeh, H., Gerami, S., Abedi, J., 2011. On estimating the water content of CO2 in equilibrium with formation brine. Petrol. Sci. Technol. 29 (19), 2037e2051. Smith, J., Van Ness, H., Abbott, M., 2001. Chemical Engineering Thermodynamics. McGraw-Hill, NY. Soave, G., 1972. Equilibrium constants from a modified Redlich-Kwong equation of

N. Sabet, H.R. Erfani Gahrooei / Journal of Natural Gas Science and Engineering 35 (2016) 382e391 state. Chem. Eng. Sci. 27 (6), 1197e1203. Takenouchi, S., Kennedy, G.C., 1964. The binary system H2O-CO2 at high temperatures and pressures. Am. J. Sci. 262 (9), 1055e1074. Tissot, B.P., Welte, D.H., 2013. Petroleum Formation and Occurrence. Springer Science & Business Media.

391

Wasik, S.P., Brown, R.L., 1973. Determination of hydrocarbon solubility in sea water and the analysis of hydrocarbons in water-extracts. Int. Oil Spill Conf. Proc. 1973 (1), 223e227. Willhite, G., 1986. Waterflooding. Third Printing-Richardson, TX.