Phase Oil Equilibria of Oil-water-brine_Yaun Kun Li

Phase Equilibria of Oil, Gas and Water/Brine Mixtures from a Cubic Equation of State and Henry’s Law

YAU-KUN LI and LONG X . NGHIEM

Computer Modelling Group, 3512-33 Street N . W . , Calgary, Alberta T2L 2A6

This paper presents the predictions and calculations of the phase equilibria of oil, gas and water/brine mixtures where the oil and gas are modeled by a cubic equation of state and where the gas solubility in the aqueous phase is estimated from Henry’s law. Henry’s law constants for a variety of compounds of interest to the petroleum industry are correlated against pressure and temperature. The scaled-particle theory is used to take into account the presence of salt in the aqueous phase. An extensive match of experimental data is presented, showing the adequacy of the proposed model for Henry’s law constant.

On donne dans ce travail les pridictions et les calculs d’kquilibre de phase des mClanges pktrole, gaz et eau-saumure, 00 le pktrole et les gaz sont reprksentis par une Cquation d’itat cubique et ou la solubilite des gaz dans la phase aqueuse est estimie a I’aide de la loi d’Henry. Les constantes de Henry sont corrClCes pour une large gamme de composis d’intiret pour I’industrie pktroliere, en fonction de la pression et de la tempkrature. On utilise la thiorie des particules ajusties pour tenir compte de la prisence de sel dans la phase aqueuse. On prksente une vaste comparaison de donnies expkrimentales qui montre la validiti du modkle propost5 pour les constantes de la loi d’Henry.

ater exists abundantly in hydrocarbon reservoirs, either W as a fluid originally in place or as an injected fluid from an enhanced oil recovery process. The exclusion of water from phase equilibrium predictions may lead to results which are inconsistent with field observations since large amount of light hydrocarbons and carbon dioxide can be dissolved in the aqueous phase. In addition, the presence of salt in the aqueous phase must also be taken into account because of its strong effect on the gas solubility.

Cubic equations of state (EOS) have been used ex- tensively to model the oil and gas phases in phase equilibrium computations. Several attempts have been made to model the aqueous phase with the same EOS (Heidemann, 1974; Peng and Robinson, 1976b; Evelein et al., 1976). However, these authors found that accurate predictions of the aqueous-phase compositions are difficult to achieve. To overcome these difficulties, special modifications to the EOS parameters are required for the aqueous phase (Erbar et al., 1980; Peng and Robinson, 1980). Furthermore, the inclusion of the salt effect in an EOS is not obvious.

Another approach, which was used by Luks et al. (1976), Heidemann and Prausnitz (1977), Mehra et al. (1982) and Nghiem and Heidemann (1982), consists of using Henry’s law to model the gas solubility in the aqueous phase, and an EOS for the other phase(s). Luks et al. (1976) performed calculations for nitrogen-oxygen- water and air-water mixtures. Heidemann and Prausnitz ( 1977) calculated the compositions of combustion gases consisting of nitrogen, carbon dioxide and water. Mehra et al. 11982) estimated the gas solubility in the aqueous phase from Cysewski and Prausnitz’s (1976) correlation. Since there did not appear to be any safely generalized correlations for Henry’s law constants at this time (O’Connell, 1980), Nghiem and Heidemann (1982) carried out flash calculations for mixtures of crude oil and water with Henry’s law constants determined from experimental data. However, only carbon dioxide was assumed soluble in the aqueous phase, and the presence of water in the hydrocarbon-rich phases was ne-

glected. All the above references did not consider the effect of salt on the gas solubility.

In this paper, correlations for Henry’s law constants with respect to pressure and temperature for a variety of compounds of interest to the petroleum industry are developed using published experimental data. The effect of salt on the gas solubility in the aqueous phase is handled by using the scaled-particle theory (SPT) to modify Henry’s law constants derived for pure water.

A robust algorithm for three-phase flash calculations involving an aqueous phase is also discussed.

Phase equilibria

EQUILIBRIUM EQUATIONS

For a liquid-gas- water system with n, components, the

(1)

necessary conditions for phase equilibrium are:

lnfIv - Inf,[ = 0

lnfiw - Infie = 0

i = I , . . ,n, . . . . . . . . . . . . . and

i = 1, . . ,n, . . . . . . . . . . . . . ( 2 ) wheref,, (m = e , v, w ) is the fugacity of component i in phase m. The subscripts e , v and w denote respectively the liquid, vapor and aqueous phase. In this paper, the term “liquid phase” refers to the dense phase which is not rich in water. The latter is referred to as the aqueous phase.

In phase equilibrium computations, it is convenient to define two sets of K-values (equilibrium ratios) as follows:

(3) K,, = y,,/y,, i = I , . . ,n, . . . . . . . . . . . . . . . . . and

(4) i = I , . . ,n, . . . . . . . . . . . . . . . . . where y,, denotes the mole fraction of component i in phase m.

K,,, = yIw/y,e

486 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 64, JUNE 1986

The material balance on component i requires that

Z , = F , y , , + F , y , , + F,V,,, i = I , . . . , n , . . . . ( 5 )

where z , is the feed composition (global mole fraction), and where F,,, is the mole fraction of phase m.

The mole fraction y,,,, are related to z, and F,,, through the following relationship

~ , , 1 1 = ~ , , , , z , / ~ KlyFq i = I , . . . . n, . . . . . . . . . (6) Y m = e , V , M J

4 = e , V , M '

where,

. . . . . . . . . . . . . . K,, I i = I 11, (7)

The thermodynamic equilibrium equations (Equations I

In K , , + In - In = 0 i = I , . . . , n , . . . (8)

InK,, , + ln+,, , - In+ , , = 0 i = 1 . . . . . n, . . (9) where +,,,, denotes the fugacity coefficient of component i in phase m.

and 2) can also be rewritten as follows:

By definition, the mole fractions satisfy 11,

C - I = o m = e , v , w . . . . . . . . . . . . . . (10)

C F,,, - I = o m = e , V , ~ . . . . . . . . . . . . . ( 1 1 )

By using Equations (6) and (lo), the following equations

1'1

m

can be derived: " I (K ," - 112, n,. ' ( ' I v - '") = I = ' I F1 + F , K , , + F,,,K,,. = 0 . . . (12)

i = I

A three-phase flash calculation corresponds to solving the 2n,. + 3 nonlinear equations (Equations 8,9, 1 I - 13) for the 2n,. + 3 primary unknowns K , , , K,,, F 1 , F , and F,. The mole fractions y,," can be treated as dependent variables from Equation (6). For the components j which are assumed not soluble in the aqueous phase, take K,,, = 0 and remove the corresponding Equation (9) from the equation set. The system of equations can be solved by either Newton's method or the quasi-Newton successive substitution method discussed in Nghiem and Li (1984) and Mehra et al. (1984). Since the aqueous phase is not modeled by an EOS, some modifications to the calculation procedure of Nghiem and Li (1984) are required. These are described in Appendix A. Note that the procedure does not require good initial guesses of the primary variables, nor a prior knowledge of the number of phases in equilibrium.

THERMODYNAMIC MODELS

As discussed earlier, the liquid and vapor phases are modeled by a cubic EOS, and the solubility in the aqueous phase is handled by Henry's law.

Henry's law for a component sparingly soluble in the aqueous phase gives

i f w ....................... (14)

where the bold subscript w denotes the water component, and the regular subscript w denotes the aqueous phase. Hi is

f i w = yIIYHi

the Henry's law constant of component i in the aqueous phase. The variation of Henry's law constant with respect to pressure and temperature follows the differential equation (Smith and Van Ness, 1975)

V,? h , , - hP + - RT?

dT . . . . . . . . . . . d(ln Hi) = -dp RT

where vIm is the partial molar volume of component i in the aqueous phase at infinite dilution, h,, is the enthalpy of component i in the gas phase, and hp is the enthalpy of component i in the aqueous phase at infinite dilution. The term h, , - hp depends strongly on temperature, and no general correlation describing its change with respect to temperature is available. It has also been found that vlx is usually not very sensitive to pressure. Therefore, for a given temperature T , integration of Equation (15) from p" to p gives

. . . . . . . . . . . (16)

where HP is the Henry's law constant at the reference pressure pp. Equation (16) can also be written as

. . . . . . . . . . . . . . . . . . (17)

In HI = In HY + vP(p - pY) / (RT)

In H, = In H,*+ v,"p/(RT)

In H,* = In HP - vlSpy/(RT) . . . . . . . . . . . . . . . . . (18) where

In this study, Equation (17) is used to correlate Henry's law constants from solubility data. H,* is referred to as the reference Henry's law constant. The molar volume at infinite dilution v;" is computed from the correlation of Lyckman et al. (1965) reported by Heidemann and Prausnitz (1977) in the form:

(2) P r i V i = 0.095 + 2.35($) TP . . . . . . . . . . . . . . RTci

where C is the cohesive energy density of water given by:

C = (h, - h i + p",: - RT)/v: . . . . . . . . . . . . . (20)

p i is the saturated vapor pressure of water at the temperature T, v: is the molar volume of water at p: and T , and h i - hz is the enthalpy departure of liquid water at p i and T .

The fugacity of the component water in the aqueous phase can readily be obtained from the fugacities of the solutes using the Gibbs-Duhem equation as in Prausnitz (1969):

0

p v, f W w = yWw+ip i exp [IpL ~ d p ] . . . . . . . . . . . . . . .

where +: is the fugacity coefficient of pure water at p i , and vi is the molar volume of pure water.

In this paper, the vapor pressure of water p i and (h: - h i ) are calculated respectively from the Frost- Kalkwarf- Thodos and the Yen- Alexander equation reported in Reid et a]. (1977). The molar volume vi is estimated from a correlation due to Chou reported in Rowe and Chou (1970). Alternatively, these values can also be obtained from steam tables. The fugacity coefficient is obtained from the following correlation which was found to match reasonably well the data of Canjar and Manning (1967):

+i = 0.9958 + 9.68330 X 10-5T' - 6.1750 X IO-'T'* - 3.08333 X 10-'"Tf3;

T' > 90°F . . . . . . . . . . . . . . . . . . . . . . . . (22a)

(22b) = I ; T' 5 90°F . . . . . . . . . . . . . . . . . . . . . . . .

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 64, JUNE 1986 487

106

Z

z 8

103

W a

1 I I I I 0 50 100 150 200 250

102 I TEMPERATURE, “C

Figure 1 - Variation of the reference Henry’s Law constant, HF, with temperature.

7” = 1.8T - 459.67 (22c) where T is the temperature in K, and 7” the temperature in OF.

. . . . . . . . . . . . , . . . . . . . . .

Henry’s law constants for gas solubility in water

HENRY’S LAW CONSTANTS FROM SOLUBILITY DATA

Henry’s law constants are obtained by matching the predicted results to experimental binary solubility data. The gas phase is modeled with the Peng-Robinson EOS (Peng and Robinson, 1976a). The compositions of the aqueous and gas phases were matched by regressing on the EOS interaction coefficient dij between water and the solute, and the reference Henry’s law constant H:given in Equation (17). Since the Henry’s law constants will be used to predict gas solubilities at reservoir conditions, more emphasis was given to matching the experimental data at pressures higher than 6.9 MPa whenever these are available. Hence the values of H: obtained do not correspond to Henry’s law constants at zero pressure (i.e., p o = 0 in Equations 16 and 18) discussed in the literature.

Figure 1 shows the variation of H? with respect to temperature for a number of compounds. The dashed curves in Figure 1 represent extrapolated values of H?. As depicted in this figure, H: generally exhibits a maximum when plotted versus temperature. Jonah (1983) has found that In (Hi/fs)

Y” I

METHANE-WATER SYSTEM

- - I

MOLE PERCENT OF METHANE IN VAPOR PHASE

‘0

Figure 2 - Vapor-phase composition of methane- water system.

varies linearly with the inverse of the absolute temperature for the systems he investigated. Here, fs denotes the pure- solvent fugacity. For the binary systems considered in this paper, it has been found that In (H:/fi) deviates slightly from the suggested linear behavior, and follows more accu- rately a second order polynomial in l/T. The coefficients of these polynomials as well as the Peng-Robinson EOS interaction coefficients obtained by regression are given in Table 1. The sources of the experimental data used in the regression are also reported. The fugacityf; is obtained by multiplying given in Equation (22) by the water saturation pressure p a . Because of space limitation, the match is presented only for a few binary systems in the following.

Figures 2 and 3 depict the vapor and aqueous phase compositions of the methane- water binary system. Figures 4 to 6 compare the experimental and calculated composition of the propane which exhibits a three-phase separation. Figures 7 and 8 show the experimental and calculated results for the carbon dioxide-water system. For this system it was found that a temperature-dependent interaction coefficient is required to achieve a good match of the experimental data.

The results show that the use of the Peng-Robinson EOS with Henry’s law provides adequate predictions of the behavior of binary systems involving water. Although not shown here, similar predictions were also obtained for the other systems shown in Table 1. It has also been found that Henry’s law is applicable to the systems considered up to a temperature of about 200°C and a pressure around 100 MPa.

MULTICOMWNENT SYSTEMS

For multicomponent systems, it is assumed that the


TABLE I Reference Henry's Law Constants and Water Interaction Coefficients

T: temperature, K fi: fugacity of saturated water, atm H?: reference Henrv's constant, atm

Coefficients Binary interaction

Comoonent A B C coefficient Data used in regression

Methane Ethane Propane n-Butane n-Pentane n-Octane Carbon dioxide

Carbon monoxide Nitrogen

10.9554 13.9485 14.6331 13.4248 16.0045 3 I .943 1 I 1.3021

10.7069 10.7090

11.3569 13.8254 14.4872 13.8865 16.2281 28.6725 10.6030

11.1313 11.4793

1.17105 1.66544 1.78068 1.71879 2.13123 4.37707 1 .20696

I .08920 I .I6549

0.4907 0.491 I 0.5469 0.5080 0.5000 0.4500

0.200 0.2750

Culberson & McKetta (195 I); Olds et al. (1942) Culberson & McKetta (1950); Reamer et al. (1943) Kobayashi & Katz (1953) Reamer et al. (1944, 1952) Gillespie & Wilson ( I 982) Brady et al. (1982) Wiebe & Gaddy (1939); Todheide & Franck (1963);

Gillespie & Wilson (1980) Saddington & Krase (1934); Rigby & Prausnitz (1968)

Takenouchi & Kennedy (1964)

'0.2, T 5 373 K; 0.49852-0.008 (TI , T > 373 K.

80 METHANE - WATER SYSTEM

MOLE PERCENT OF METHANE IN AQUEOUS PHASE

Figure 3 - Aqueous-phase composition of methane- water system.

Henry's law constants developed for binary systems still apply. This is equivalent to assuming that the interaction between the solutes is negligible, which is true at infinite dilution.

The phase behavior of the methane-n-butane- water system was predicted with HT given in Table 1 and the Peng-Robinson EOS. The predictions match very well the experimental data of McKetta and Katz (1948) (Figure 9).

A typical application of the calculation method to a reservoir fluid is given in Appendix B.

Modeling of gas solubility in brines

OUTLINE

In this section the scaled-particle theory (SPT) is used to modify the Henry's law constants developed previously to take into account the presence of salt in the aqueous phase.

HENRY'S LAW CONSTANTS FROM SCALED-PARTICLE THEORY

Only the necessary equations concerning the SPT of fluids are reported in this section. More details on the subject can be found in Reiss et al. (1960), Lebowitz et al. (1965) and Pierotti (1963, 1967, 1976).

The SPT can be used to obtain an expression for Henry's law constant

(23) Hivs - (:;) (Goi) ln-- - + RT .................... RT

where Gbi is the Gibbs energy (work) required to create a cavity in the solvent of suitable size to accommodate a solute molecule, GGi is the molar Gibbs energy resulting from the interaction of the solute molecule with the surrounding solvent molecules, and where v, is the molar volume of pure solvent.

Lebowitz et al. (1965) have derived an expression for Gbi for gas solubility in mixed solvents. When the mixed solvent is brine, this expression takes the form:

0, = ;qw c pj(u;)" n = 1 ,2 ,3 . . . . . . . . . . (25) i


220

200

180

PROPANE- WATER SYSTEM -

- Kobayashi 8 Katz (19531 V 71.1DC A137.8OC

0 104.4’C 160 - 0 87.8’C 0 154.4’C

MOLE PERCENT OF PROPANE IN VAPOR PHASE

Figure 4 - Vapor-phase composition of propane- water system.

220 PROPANE -WATER SYSTEM

200

180

160

140 a

L ; 120 3 m 100 m

80 a

60 Koboyashi 8 Katz (1953) 0 15.6OC A 137.8OC V 7 1 . V C 0 4 5 4 4 0 ~ 40 0 104 4 O C 20

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.C MOLE PERCENT OF PROPANE IN

AQUEOUS PHASE Figure 5 - Aqueous-phase composition of propane - water system.

. . . . . . . . . . . . . a; = up/u, k = i , j (26)

r), = Nu: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (27) u, and a, are respectively the molecular diameter of solute

i and of species j in the solvent, a, is the molecular diameter of pure water; N is the Avogadro’s number; p, is the molar density of species j in the solvent. The summation in Equation (25) is over all species in the solvent, e.g. for gas solubility in NaCl aqueous solutibn, the summation involves the solutes, pure water, Na’ ions and C1- ions. Masterton and Lee (1970) showed that the solutes may be excluded from the summation, and their approach is used in this study. (The term “solutes” in this paper refers to the compounds soluble in the aqueous phase which are also present in the vapor and/or liquid phases).

Shoor and Gubbins (1969) proposed the following expression for the interaction Gibbs energy Gu,:

5 A

PROPANE -WATER

4

d =, 3 - W

3 v)

Vapor a - v) phase w 2 - a a.

0 Kaboyarhi 0 Katz (19531

Calculations

MOLE PERCENT OF PROPANE

I

CRITICAL POINT

Aqueous phase

C 3 - rich liquid phase

Aqueous phase

Vapor phase

J-uulJ 40 60 80 100

TEMPERATURE,OC

Figure 6 - Thrce-phase composition and temperature of propanc- water system.

TABLE 2 Scaled-Particle-Theory Parameters

u, nm e l k , K IO”a, cm’/molecule

Water 0.275 85.3 I ,590 Carbon dioxide 0.332 300.0 2.590 Methane 0.413 160.0 2.700 Nitrogen 0.373 95.0 I .730 Na’ 0.190 147.4 0.210 C1- 0.362 225.5 3.020

NOTE: N = 6.02283 X 10”; p w = 1.84 X lo-’’ (erg-cm)”’ cm; I nm = m.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . El, = (29)

a; = F(a,* + 0,3/2 . . . . . . . . . . . . . . . . . . . . . . . (30)

where p, is the dipole moment of water, a, is the polariza- bility of solute i , k is Boltzmann’s constant, E, is a parameter which characterizes the strength of the dispersion force, and where F is unity. Schulze and Prausnitz (1981) found that better predictions could be obtained by making F temperature dependent, i.e.,

. . . . . . (31) F(T) = 16280/T2 - 141.75/T + 1.2978

with T in K. Schulze and Prausnitz (1981) reported SPT parameters

for the calculation of Henry’s law constant for nine gases in water. Masterton and Lee (1970) tabulated the parameters for nine ions for the prediction of gas solubility in aqueous salt solutions. Values of those parameters which are used in the present study are listed in Table 2. The values a, of the ions shown were calculated from the Mavroyannis- Stephen theory as described in Shoor and Gubbins (1969). Parame- ters for other compounds can be found in Hirschfelder et al. (1964) and Weast and Astle (1979).

Equations (25) and (28) require the molar volume of water and the ions. For aqueous NaCl solutions, these are estimated as follows

p y = (1 - G)f i s /Mu (32)

PNa+ = pel- = 6 f i r / h f N & (33)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

490 THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 64. JUNE 1986

TABLE 3 Solubility of Nitrogen in Aqueous Sodium-Chloride Solution

Mole % nitrogen in aqueous phase

H20 1 rn NaCl 4 m NaCl T , P “C MPa Measured Calculated Measured Calculated Measured Calculated

51.1

102.5

125.0

10.13 20.27 30.40 40.53 50.66 60.80 10.23 20.37 30.60 40.83 50.97 61.20 10.44 20.67 30.90 41.04 51.37 61.61

0.0799 0.1454 0.2017 0.2490 0.2920 0.3350 0.0777 0.1447 0.2005 0.2520 0.2980 0.3370 0.0808 0.1492 0.2047 0.2570 0.3060 0.3510

0.0822 0. I479 0.2027 0.2499 0.2914 0.3282 0.0812 0. I501 0.2106 0.2641 0.31 15 0.3548 0.0903 0.1682 0.2368 0.2975 0.3533 0.4035

0.0593 0.1076 0. I497 0.1860 0.2216 0.2530 0.0603 0.1113 0.1538 0.1920 0.2252 0.2600 0.0632 0. I102 0.1533 0.1883 0.223 I 0.2550

0.0627 0.1130 0.1551 0.1916 0.2239 0.2528 0.0628 0.1162 0.1633 0.2052 0.2426 0.2769 0.0694 0. I298 0. I833 0.2308 0.2749 0.3146

- 0.0500 0.0700 0.0878 0.1034 0.1 179

0.0523 0.073 I 0.0899 0.1047 0.1205

0.0567 0.0740 0.0921 0.1041 0.1227

-

-

0.0296 0.0534 0.0736 0.0912 0.1070 0.1214 0.0306 0.0569 0.0803 0.1012 0.1200 0.1373 0.0336 0.0632 0.0896 0.1133 0.1353 0. I552

Average error (%) 6.77 9.47 12.94

Measurements are from O’Sullivan and Smith (1970).

TABLE 4 Solubility of Methane in Aqueous Sodium-Chloride Solution

Mole % methane in aqueous phase

Hz0 1 m NaCl 4 m NaCl T , P “C MPa Measured Calculated Measured Calculated Measured Calculated

51.1 10.13 0.1427 20.27 0.2279 30.40 0.2870 40.53 0.3340 50.66 0.3730 60.80 0.4090

102.5 10.23 0.1355 20.37 0.2205 30.60 0.2870 40.83 0.3330 50.97 0.3850 61.20 0.4190

125.0 10.44 0.1434 20.67 0.2321 30.90 0.2960 41.04 0.3430 51.37 0.3960 61.61 0.4300 Average error (%)

0.1416 0.2253 0.2822 0.3257 0.361 1 0.3907 0.1269 0.2149 0.2816 0.3348 0.3787 0.4165 0.1389 0.2399 0.3180 0.3808 0.4350 0.481 I 4.10

0.1076 0.1695 0.2138 0.2500 0.2790 0.3070

0.1693 0.2219 0.2570 0.2890 0.3200 0.1058 0.1752 0.2223 0.2600 0.2940 0.3250

-

0.1116 0.1774 0.2224 0.2567 0.2847 0.3084 0.1006 0.1705 0.2235 0.2660 0.301 1 0.3315 0.1097 0.1898 0.2519 0.3021 0.3455 0.3825 6.30

- 0.0805 0.0997 0.1 154 0. I303 0.1444

0.0826 0.1079 0.121 I 0.1319 0.1433

0.0825 0.1005 0. I164 0. I322 0.1438

-

-

0.0576 0.0914 0.1144 0.1320 0.1464 0. I587 0.0528 0.0895 0. I174 0. I396 0. I580 0.1739 0.0572 0.0990 0.1316 0.1579 0. I805 0.1997

20.033

Measurements are from O’Sullivan and Smith (1970).

where M, and MNaCl are respectively the molecular weight of water and NaCl, fiI is the mass density of the aqueous NaCl solution and where 6 is the weight fraction of NaCl in water.

As pointed out by Pierotti (1967), the use of the SPT does Some predictions of gas solubilities in water and/or not require explicit ionization information about the salt brines with the SPT are presented in Schulze and Prausnitz since the latter is included in the solvent density fis. Thus, (1981), Shoor and Gubbins (1969) and Masterton and Lee an accurate value fi, is required. In this study, bs is estimated (1970). However, because of the uncertainties in the from the correlation of Rowe and Chou (1970). SPT parameters (Schulze and Prausnitz, 1981) and because

FITTING SCALE-PARTICLE THEORY HENRY’S LAW CONSTANTS TO EXPERIMENTAL DATA

THE CANADIAN JOURNAL OF CHEMICAL ENGINEERING, VOLUME 64, JUNE 1986 49 I

CARBON DIOXIDE- WATER SYSTEM

25' 50' 75' 100' 150' 200' 250' C 500 Witbe (1942)

- CARBON DIOXIDE- WATER SYSTEM - - 50' 1000 450' 200' 250' C o m Witbe B Gaddy (19391

0 x 0 A 0 v T6dheidt B Franck(1963) A v Taktnouchi 8 Ktnntdy(l96 I - Caiculatianr

Two phase region McKetta, Jr. 6 Katz ( 1 9 4 8 )

.2 Hydrocarbon rich liquid phasm

Vapor pharm

MOLE PERCENTOF WATER IN VAPOR PHASE Figure 7 - Vapor-phase composition of carbon-dioxide- water system.

o A o v TSdheide B Franch (1963) A v Taktnouchi B Ktnntdy(1964) t

I I I I I 1 1 2 3 4 5 6

MOLE PERCENT OF CARBON DIOXIDE IN AQUEOUS PHASE

Figure 8 - Aqueous-phase composition of carbon-dioxde - water system.

of several approximations made in the theory (Masterton and Lee, 1970) there are usually disagreements between predicted and measured values.

In this paper, to reduce these uncertainties, Henry's law constants from the STP are first fitted to the values calculated from Equation (17) for gas solubility in pure water. The adjustable parameter is the molecular diameter ui of the solute which has a strong influence on the SPT predictions (Schulze and Prausnitz, 1981). Subsequently, the SPT parameters of the ions and the density of the aqueous salt solution are used in Equations (24), (25) and (28) to calculate the Henry's law constants in the presence of salt.

For a given pressure and temperature, the determination

40 30

METHANE - n-BUTANE -WATER SYSTEM

Two DhaSe n a i o n - Calcuiatlons

CALCULATED CRITICAL LOCUS

\ Three phase region I

0.1 1 10 MOLE PERCENT OF WATER

Figure 9 - Composition of water in vapor and hydrocarbon-rich liquid phase: methane-n-butane- Water system.

CARBON DIOXIDE IN AQUEOUS NoCl SOLUTION

Tokenouchi 8 Kennedy ( 1 9 6 5 ) "' , zo;c 4.28m

1.09m

O m

--- l - Calculations

4 . 2 8 m 1 . 0 9 m Om 1 . 0 9 m O m

0 I I I I I I I 1 2 3 4 5 6 7 MOLE PERCENT OF CARBON DIOXIDE

IN AQUEOUS PHASE Figure 10 - Solubility of carbon-dioxide and aqueous NaCl solution.

of cri for each solute involves the solution of a nonlinear equation corresponding to the equality of STP Henry's law constant with the Henry's law constant obtained from Equation (17). Since no salt is present, the equation is rela- tively simple and is solved using Newton's method with


TABLE 5 Compositions of Reservoir and Recombined Fluids at Various Water Cuts

Composition" (mol %) Reservoir Reservoir Water cut (%) aqueous-phase oil-phase

Component Oil G 67 89 (mol %) (mol %) compositionb compositionb

N2

c02

CI c2

C, nC4 nCs n C6 C,--C16 C17-Cxi

H20 Molecular weight Bubble point (MPa)

c:,

at 65.5"C

0.27 0.26

30.58 4.61 6.06 4.71 4.33 3.18

28.26 12.10 5.66 -

132.2

12.040

0.29 0.38

31.88 4.58 5.94 4.60 4.23 3.10

27.63 I I .83 5.53 -

129.7

12.743

0.34 0.69

35.70 4.49 5.61 4.31 3.95 2.90

25.80 11.05 5.17 -

122.3

14.939

0.0015 0.0088 0.1310 0.0050 0.0017 o.oO05 o.oO01 O.oo00 O.oo00 O.oo00 O.oo00

99.8514

0.27 0.26

30.27 4.57 6.00 4.66 4.28 3. I4

27.98 I I .98 5.60 0.99

131.1

I 1.970

'Dry basis. hWet basis.

UNSTABLE

STABLE

W- aqueous phase L = liquid phoso V = vopor phase

ST= stability ?*st / 'flash calculation - =in equiilbrlum

Figure I I - Flow diagram for liquid-vapor-water flash calculations.

analytical derivative. The initial guess for ui corresponds to the value reported in the literature (Table 2). Note that ui is treated as an adjustable parameter and therefore the resulting value may not have any physical meaning.

Results

The measurements of O'Sullivan and Smith (1970) on the solubility of nitrogen and methane in aqueous NaCl solutions are matched with the procedure just described. The results are shown in Tables 3 and 4. Both the solubilities of nitrogen and methane decrease as the salt concentration increases (salting out phenomena). The error in the predictions increases with salt concentration and temperature.

The solubility of carbon dioxide in aqueous NaCl solutions is also calculated and compared to the measurements of Takenouchi and Kennedy (1965). Adequate predictions were obtained as shown in Figure 10. Again, the error increases with salt concentration and temperature.

100 15

Figure 12 - Effect of water cut on producing GOR and bubble- point pressures of recombined fluids.

The values of ui for the solute obtained from matching pure-water Henry's law constants are not far from the initial values reported in the literature. For nitrogen, typical values of ui are 0.37321 nm at 10.132 MPa and 51.5"C, and 0.36627 nm at 61.606 MPa and 125°C compared to a value of 0.373 nm shown in Table 2. For methane, the fitted ui is 0.34314 nm at 10.132 MPa and 51.5"C, and 0.34703 nm at 61.606 MPa and 125°C compared to a reported value of 0.413 nm. For carbon dioxide, the values are 0.33076 nm at 1 MPa and lOO"C, and 0.2733 nm at 20 MPa and 250°C compared to a reported value of 0.332 nm.

DISCUSSIONS

The SPT provides a simple and adequate method for correlating Henry's law constants for gas solubility in brines. Note that in the calculations presented above, only the ui of the solutes are treated as adjustable parameters to match the Henry's law constants predicted by Equation (17).


All other STP parameters are given in Table 2. The procedure can be applied to any aqueous salt solution

provided the STP parameters for all ions are available, and the mass density of the solvent is known. In the present study, the density of brine is estimated from Rowe and Chou’s ( 1970) correlation which is developed for pressures up to 34.3 MPa. However, the pressures in the calculations went beyond 125 MPa, and some of the inaccuracies in the predictions may partly be attributed to an inaccurate solvent density value.

Again, the ui of the ions could also be adjusted to give a better fit of the experimental data. However, this was not carried out in this study.

Conclusions

This paper presents a method for predicting the phase behavior of oil, gas and water or brines where the oil and gas are modeled with the Peng-Robinson equation of state and where the gas solubility in the aqueous phase is estimated from Henry’s law constants. The method is simple and gives adequate predictions for pressure and temperature conditions encountered in reservoir engineering studies.

Correlations for Henry’s law constants and Peng- Robinson equation of state interaction coefficients are determined for a variety of compounds of interest to the oil industry. For practical purposes, any hydrocarbon heavier than pentane may be assumed insoluble in the aqueous phase. Satisfactory predictions of gas solubility in pure water up to 200°C and 100 MPa have been obtained.

The scaled-particle theory was used to predict the change of Henry’s law constant due to the presence of salt in the aqueous phase. By adjusting the solute molecular diameter, adequate predictions of solubility in aqueous NaCl solution up to a 4 m concentration have been obtained.

Although the reference Henry’s law constants in Table 1 were determined with the Peng-Robinson equation of state for the vapor or liquid phase, they could be used with any other equation of state. However, the interaction coefficients in Table 1 are only for the Peng-Robinson equation of state.

Appendix A Three-phase flash calculations for liquid-vapor- water

systems

Procedures for three-phase flash calculations where all phases are modeled with an EOS can be found in Heidemann ( 1974). Peng and Robinson (176b), Fussell (1979), Mehraet al. (1982), Michel- sen (1982b). Risnes and Dalen (1984) and Nghiem and Li (1984).

The algorithm described in the following is a modification of the method of Nghiem and Li (1984) to handle three-phase liquid- vapor- water (LVW) flash calculations where the aqueous phase is not modeled by an EOS. The procedure does not require good initial guesses of K-values since the latter are obtained from stability tests using the tangent plane criteripn. Detailed discussions on this subject are presented in Nghiem and Li (1984). Michelsen (1982a) and Baker et al. (1982).

Figure 11 shows a flow diagram of a stagewise procedure for LVW flash calculations. Note that the symbols L and V in the flow chart could be permuted since the first two-phase flash calculation could either be a LW calculation or a VW calculation. When the cubic EOS yields more than one root in the compressibility factor, the one corresponding to the lowest Gibbs energy is selected (Evelein et al., 1976). The stability test of a phase with composition y consists of solving

d ; ( u ) = In ui + In & ( 6 ) - In y , - In $,(y) = 0 . . . . (A.1)

i, = u,/$ U L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A.2)

The +, in Equation (A. I ) are the EOS fugacity coefficients for u,. The phase with composition y is unstable if, at convergence,

i U , > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A.3)

In the stability test of a VW or LW system, it has been found necessary to prevent convergence to a set of u , corresponding to an aqueous phase predicted by an EOS, since the latter has been modeled by Henry’s law. For this purpose the following equation is solved for u instead of Equation (A. I )

i = I , . . . , n , . . . . (A.4)

where y,,,. is the composition of water in the aqueous phase from a two-phase VW or LW calculation.

The algorithm in Figure 1 1 requires the initial guesses K:::,’. The following equations provide a reasonable estimate for K:::?

! , = I

I = I

d: (u ) = d i ( u ) / l u , - yw,,.I = 0

K::,’ = p/H, i f w . . . . . (A.4)

~ j : ! = K , * ~ / H , i z w . . . . . . . . . . . . . . . . . . . . . . . (AS)

if a VW system is expected, or

if a LW system is expected. K,* is Wilson’s K-value given by

K,* = (p<./ /p) exp [5.42(1 - T < , / T ) ( I + w,)] . . . . . . . (A.6)

Even if VW Kj”’ is selected instead of LW K r l for LW flash calculations or vice-versa, convergence to the correct solution will still be achieved. However, the procedure may take longer to reach the solution.

For the water component, a value for Kc,:, is found such that

i Kj:!z, > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A.7a) , = I

and

$ z,/K::! > 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (A.7b) I = I

This could easily be achieved by setting Kr,:, equal to a large number (say > IOOO). Equation (A.7) will guarantee a split of the feed into two phases when the material balance for two-phase flash calculations is solved (Nghiem et at., 1983; Rachford and Rice. 1952).

Appendix B Effect of water on the producing gas-oil ratio

An example illustrating the effect of gas solubility in the aqueous phase on the producing gas-oil ratio (GOR) from a producing well is given in the following.

Water is usually produced together with oil, and the gas obtaincd at surface conditions comes from both the oil and the aqueous phase. Consider a typical reservoir oil (called Oil G) and composition shown in the second column of Table 5 . The oil is in equilibrium with the aqueous phase at reservoir conditions of 14 MPa and 65.6”C. It is assumed that the solubility of hydrocarbons heavier than nCs is negligible. The compositions of the recombined reservoir fluids obtained by blending the oil and gas at surface conditions (0.101 MPa and 15.5”C) at 67% and 89% water cut (water cut = flow rate of water/(flow rate of water + oil)) are also shown in Table 5. The molecular weights and bubble-point pressures for the various fluids, as well as the aqueous phase composition and the water composition in the oil phase are also reported. Figure 12 illustrates the change in the producing GOR with the water cut for the same system. The computed bubble-point pressure of the recombined fluid is also depicted. The results show


that the effect of gas solubility in the aqueous phase on the producing GOR cannot be ignored at high water cut.

Acknowledgement

This work was supported by thc Department of Energy, Mines and Resources of Canada under Contract 01 SG-23294-4-07s I .

Nomenclature

cohesive energy density defined in Equation (20) defined in Equation (A. I ) defined in Equation (A.4) interaction coefficient fugacity of component i in phase m temperaturc-dependent factor defined in Equation (3 1 ) mole fraction of phase m molar Gibbs energy of interaction between the solute i and the solvent molar Gibbs energy for cavity creation molar enthalpy of component i in vapor phase molar enthalpy of component i at infinite dilution molar enthalpy of water at the given temperature and zero pressure molar enthalpy of water at saturation conditions Henry’s law constant of component i reference Henry’s law constant of component i Henry’s law constant of component at a given pressurep; Boltzmann’s constant Wilson’s K-value defined in Equation (A.6) K-value of component i in phase m molality molecular weight number of components Avogadro number (= 6.02283 X lo”‘) pressure pressure where HY is evaluated universal gas constant temperature temperature in degree F variable in stability-test equation composition corresponding to u, molar volume molar volume of component i at infinite dilution mole fraction of component i in phase m feed (global) composition of component i polarization of solute i parameter which characterizes the dispersion strength of component i geometric mean of E, and E, defined in Equation (27) defined in Equation (25) dipole moment of water molar density mass density molecular diameter of compound i dimensionless molecular diameter of compound i defined in Equation (26) defined in Equation (30) fugacity coefficient of component i in phase m acentric factor of component i weight fraction of salt in water

Superscripts

s = saturated (0) = initial guess

Subscripts

i, j. k = indices for component

e = liquid phase m, q = indices for phase s = solvent v = vapor phase w = aqueous phase w = water component

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