High-Pressure Acid-Gas Viscosity Correlation - OnePetro

High-Pressure Acid-Gas Viscosity Correlation G. Galliéro, C. Boned, and A. Baylaucq, University of Pau, and F. Montel, SPE, Total

Summary Acid gases containing hydrogen sulfide (H2S) are often encountered in the petroleum industry. However, reliable experiments on their thermophysical properties in reservoir conditions, on viscosity in particular, are scarce. From a modeling point of view, H2S and carbon dioxide (CO2) are polar compounds and as such are often considered rather difficult to model accurately. In this work, we propose a correlation with a strong physical background based on a corresponding-states (CS) approach to predict the viscosity from the temperature and the density of a large variety of systems for all stable thermodynamic states (gas, liquid, and supercritical). In particular, this correlation is applicable to predict the viscosity of sour/acid-gas mixtures, whatever the thermodynamic conditions. This approach is based on the Lennard-Jones (LJ) fluid model, which has been studied extensively thanks to moleculardynamics (MD) simulations over a wide range of thermodynamic conditions. This fluid model can be extended to deal with polar molecules such as CO2 or H2S without a loss of accuracy. First, we demonstrate that the proposed physically based correlation is able to provide an excellent estimation of the viscosity [with average absolute deviations (AADs) below 5%] of pure compounds, including normal-alkanes, CO2, or even H2S, whatever the thermodynamic conditions (gas, liquid, or supercritical). Then, using a one-fluid approximation and a set of combining rules, the correlation is applied to various fluid mixtures in a fully predictive way (i.e., without any additional fitted parameters). Using this scheme, the deviations between predictions and measurements are as low as those on pure fluids using temperature and density as inputs. The viscosity of natural- and acid-gas mixtures at reservoir conditions is shown to be very well predicted by the proposed scheme. In addition, it is shown that this correlation can also be applied to predict reasonably the viscosity of asymmetric high-pressure mixtures, even in the liquid phase. This physically based approach is easy to include in any simulation software as long as, apart from temperature and density, the only inputs—the molecular parameters of each species—can be estimated from the critical temperature and the critical volume when not known. Introduction Natural gases containing CO2 or H2S, the so-called acid/sour gases, are often encountered in the petroleum industry (Ungerer et al. 2005). Nevertheless, reliable experiments on their thermophysical properties are scarce. In the case of H2S, practically no data exist because of its high toxicity. This lack of information is even more pronounced concerning transport properties, such as viscosity (Galliéro and Boned 2008; Liley et al. 1988; Schmidt et al. 2008), at typical petroleum-reservoir conditions (i.e., high pressures and high temperatures). A predictive, physically based model is highly required as long as high-pressure viscosity is one of the key physical properties for the design of acid-gases disposal. Molecular simulation, which can be considered a “numerical experiment” on a model fluid, is one of the valuable alternatives to experiments for gathering physical information on such systems of

Copyright © 2010 Society of Petroleum Engineers This paper (SPE 121484) was accepted for presentation at the EUROPEC/EAGE Conference and Exhibition, Amsterdam, 8–11 June 2009, and revised for publication. Original manuscript received for review 27 February 2009. Revised manuscript received for review 18 September 2009. Paper peer approved 29 October 2009.

682

interest. Using this simulation technique allows one to obtain exact results on a molecular model representing the fluid studied without any limitations on the state studied. It has been shown that such an approach is efficient in describing the thermophysical properties of fluids, such as acid-gas mixtures (Galliéro et al. 2007; Ungerer et al. 2005). Nevertheless, molecular simulations require a rather long time to obtain results and so cannot yet be easily used routinely from an engineering point of view. In order to circumvent this problem, it is possible to use the molecular-simulation results on a well-defined fluid model to construct a correlation/theory on the basis of this model. If this fluid model is representative of some real fluids, then this correlation/theory can be applied to predict properties of real fluids by use of a CS approach. The LJ fluid model is a simple two-parameters molecular model representing simple nonassociative real fluids fairly well, at least concerning their viscosity (Galliéro et al. 2006a). This fluid model exactly respects, by definition, a CS behavior. Thus, using a large database of MD simulations of the viscosity of the LJ fluid, we have constructed a correlation that is able to represent the MD results accurately (Galliéro et al. 2005a). MD simulations have been performed on gas, liquid, and supercritical states. In addition, by using temperature-dependent molecular parameters, the LJ fluid model can also be used to represent molecules exhibiting a dipole moment, such as H2S. In previous works (Galliéro et al. 2007; Galliéro and Boned 2008), we have shown that such a fluid model is able to provide results on thermophysical properties of H2S consistent with the scarce database and with more-complex molecular models. In this paper, we will show that this correlation developed to describe the viscosity of the LJ fluid is able to provide a very good estimation of the viscosity of a large variety of species, including H2S in particular, whatever the thermodynamic state (gas, liquid, and supercritical), using temperature and density as inputs. In addition, it will be shown that this fully predictive scheme is able to provide a very good estimation (i.e., an AAD below 5%) of the viscosity of various mixtures: acid gas, natural gas, or even more-complex mixtures, such as asymmetric ones without additional fitted parameters. Models and Correlation The Fluid Model. Nonpolar Pure Compounds. To describe the molecule, the LJ fluid model has been employed. In this model, molecules are represented by simple spheres, without internal degrees of freedom, interacting through an LJ potential (Allen and Tildesley 1987). In this nonpolar-fluid model, each species is completely characterized by its molecular weight M and two molecular parameters, the potential depth and the molecular diameter . These molecular parameters, and , are closely related to the critical temperature Tc and the critical molar volume vc. In a first approximation, the molecular parameters of each species can be obtained with

ε=

k BTc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1) 1.2593

and ⎛ v ⎞ = ⎜ 0.305 c ⎟ NA ⎠ ⎝

1/ 3

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

where NA is the Avogadro number and kB is the Boltzmann constant. September 2010 SPE Journal

In this work, Eq. 1 is used systematically to deduce the of each studied compound. For , Eq. 2 should be employed directly only when there is no previously optimized (on viscosity) value available for a given compound. In this work, some optimized values of are provided for some typical compounds of petroleum fluids.

and

Dipolar Compounds. The LJ fluid model does not include polar or association effects. It can be used directly to describe such complex compounds but will be able to provide reasonable results only for a rather limited temperature range (Galliéro et al. 2007; Yoshimura et al. 2009; Zéberg-Mikkelsen et al. 2006). Nevertheless, for a not-too-polar fluid such as H2S, it is possible to use the LJ model combined with an isotropic multipolar potential to take into account dipolar interactions (Galliéro et al. 2007). In that approximation, it is possible to represent a dipolar molecule by a simple LJ model with temperature-dependent molecular parameters. Between Molecule i and Molecule j, these temperature-dependent molecular parameters are

So, by using the vdW1, a mixture is represented by a single pseudocompound having the molecular parameters Mx, x, and x, defined by Eqs. 6 through 8. It should be noted that, instead of Eqs. 9 and 10, the usual Lorentz-Berthelot combining rules (Ungerer et al. 2005) can be used. Nevertheless, we noticed during this work that they generally induce less-accurate predictions of the viscosity than those yielded by Eqs. 9 and 10 if they are used with the scheme proposed in this work. Reduced Variables. In order to apply a CS scheme, it is necessary to define the reduced (dimensionless) variables in which all fluids are equivalent in terms of both thermodynamic and transport properties. In the following, the reduced variables will be noted with a superscript asterisk. For the LJ fluids, this scaling procedure is performed using the molecular parameters Mx, x, and x, representing the studied fluid, and is defined using Eqs. 6 through 10. The reduced thermodynamic variables used as inputs in this work are defined in this frame as

ε ijpol = ε ij F 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3) and

ijpol =

ij F1 6

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

with

⎛ ε 3 + ε 3 ⎞ ε ij = ⎜ ii ii 3 jj jj ⎟ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (10) 2 ij ⎝ ⎠

T* =

k BT * x3 , = , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11) x Mx

where is the mass density. The reduced viscosity is

i2 2j F = 1+ , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5) 12 k BT ij ij6 where i is the dipole moment of Molecule i that is provided for some compounds in Poling et al. (2000). When a dipole moment is known in Debyes, it should be multiplied by 3.162276×10−25 to be used in Eq. 5. Mixtures. The scheme proposed in this work is based on the CS law (i.e., it is assumed that, with adequate scaling that we will discuss in the next subsection, all fluids follow the same trend in the scaled/reduced space). The LJ fluid model applied on a pure fluid respects two-parameter CS behavior exactly. In LJ mixtures, a single set of molecular parameters representing the mixture should be defined to use a CS approach. To do so, a simple van der Waals one-fluid approximation (vdW1) has been used in this work. The vdW1 is known to provide generally reasonable results (Vidal 2003; Galliéro et al. 2005a) even if, in some asymmetric mixtures, it may exhibit deficiencies (Galliéro et al. 2005b, 2006b). In an N-components LJ mixture, the vdW1 is N

M x = ∑ xi Mi , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6) i =1

N

N

x3 = ∑ ∑ xi x j ij3 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7) i =1 j =1

* =

where is the dynamic viscosity. The knowledge of the dependence of * on T * and * is sufficient to define the viscosity of the LJ fluid model exactly. Viscosity Correlation. To use the LJ fluid as a model to predict the viscosity of real fluid, it is necessary to determine the relation between * and the thermodynamic state (using T * and * as inputs). To do so, we assume that the reduced viscosity can be expressed as a sum of two independent contributions:

* ( T * , * ) = 0* ( T * ) + * ( T * , * ) , . . . . . . . . . . . . . . . . . . (13) where 0* represent the low-density viscosity (function of T * only) and * is the residual viscosity (function of T * and *). At the critical point, the viscosity diverges (Vesovic et al. 1990). Nevertheless, in this work, we have not taken into account this contribution because its effect is limited to a very small area close to the critical point (Vesovic et al. 1990). Low-Density Contribution. The first-order approximation of the Chapman-Enskog approach (Poling et al. 2000) applied to the LJ fluid provides an excellent estimation of the low-density reduced viscosity. This relation is

and

0* = N

N

ε x x3 = ∑ ∑ xi x j ε ij ij3 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (8) i =1 j =1

where xi is the molar fraction of Component i. Cross-molecular parameters between different species have to be defined by combining rules. In this work, it is proposed to define the cross-molecular parameters between Compound i and Compound j as ⎛ 3 + 3jj ⎞ ij = ⎜ ii 2 ⎟⎠ ⎝

x2 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12) Mxε x

5 Ac 16 v

T* , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14)

where Ac is a numerical parameter taken equal to 0.95 (Galliéro et al. 2005a) and v is the collision integral that can be estimated with a good precision for 0.3 ≤ T * ≤ 100 by the Neufeld et al. (1972) correlation:

( )

v = a1 T *

a2

*

*

+ a3e a4T + a5e a6T . . . . . . . . . . . . . . . . . . . . . . (15)

1/ 3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

September 2010 SPE Journal

The numerical parameters ai appearing in Eq. 15 are provided in Table 1. 683

TABLE 1—NUMERICAL PARAMETER VALUES IN EQS. 15 A ND 16 i

1

2

3

ai

1.16145

0.14874

0.52487

bi

0.062692

4.095577

8.743269 10

Residual Contribution. MD simulation allows computation of the viscosity of the LJ fluid in dense states. In a previous work (Galliéro et al. 2005a), we performed a large number of MD simulations of the LJ fluid in various thermodynamic states covering gas, liquid, and supercritical conditions (i.e., for states covering from 0.6 ≤ T * ≤ 6 and * ≤ 1.275 with a reduced critical point located at Tc* ≈ 1.3 and c* ≈ 0.3 ). Using the MD-simulations results, we have developed a correlation that is able to provide an excellent estimation of the residual viscosity by employing

(

) (

) (Tb ) (e

* = b1 e b2 − 1 + b3 e b4 − 1 + *

*

5 * 2

b6 *

)

− 1 , . . . . .(16)

in which the numerical parameters bi have been adjusted on MD results and are provided in Table 1. Thus, using Eqs. 13 through 16, one can write the complete LJ viscosity correlation as

* =

(

4.75 *

16 a1T * a2 + a3e a4T + a5e a6T

(

) (

*

)

T*

)

+ b1 e b2 − 1 + b4 e b5 − 1 + *

4

*

(

)

b5 b6* e − 1 , . . . . . . . . (17) T *2

in which all numerical parameters, ai and bi, are provided in Table 1. Application of the Proposed Scheme. In order to apply the proposed scheme to estimate the viscosity of a real fluid, the following steps are needed: • Estimation of the molecular parameters of the compounds using Tables 2 and 3, or Eqs. 1 and 2 (plus Eqs. 3 and 5 if dipolar compounds)

0.7732 6

11.12492

5

6

2.16178 2.542477 10

2.43787 6

14.863984

• Application of the vdW1 (Eqs. 6 through 10) to deduce Mx, x, and x • Calculation of the reduced thermodynamic conditions T * and * using Eq. 11, for which the viscosity is estimated • Computation of * using the viscosity correlation, Eq. 17, and the parameters in Table 1 • Calculation of the viscosity using Eq. 12 in reverse (i.e., using * M xx = ) x2 As an example, we have applied this scheme to a natural gas (Assael et al. 2001) composed of xC1 = 0.8484, xC2 = 0.084, xC3 = 0.005, xN2 = 0.056, and xCO2 = 0.0066. For T = 353.388 K and p = 13.96 MPa, they have measured = 95.5 kg/m3 and = 16.37 10−6 Pa·s. First, the molecular parameters of each compound of this mixture have been taken from Tables 2 and 3. Then, using the vdW1 relations (Eqs. 6 through 10), we have obtained that, x = 1347.7 J/mol, x = 3.692 10−10 m, and Mx = 0.01822 kg/mol. Next, the reduced thermodynamic conditions have been deduced using Eq. 11—T * = 2.18 and * = 0.159. From those reduced thermodynamic conditions, using Eq. 17, we have calculated that * = * M xε x 0.2739. Finally, using = , we have deduced the vis x2 cosity predicted by the correlation. The calculation has given = 16.54 10−6 Pa·s, which is only 1% over the experimental value. Results on Real Fluids Pure Fluids. As a preliminary step, we have tried to obtain optimized molecular diameters of some compounds, being deduced from Eq. 1 and the critical temperature given in Poling et al. (2000). For each compound studied, the value has been adjusted so that the correlation proposed provides viscosities as close as possible to

TABLE 2—OPTIMIZED MOLECULAR PARAMETERS OF SOME NORMAL-ALKANES n-alkane

(10

10

m)

1

(J·mol )

Tmin–Tmax (K)

pmin–pmax (MPa)

AAD (%)

max

(%)

Database

C1

3.6325

1258.1

200–500

0.1–100

1.3

3.6

Lemmon et al. (2007)

C2

4.2093

2015.8

200–500

0.1–60

3.4

8.6


C3

4.6717

2442

200–450

0.1–30

6.5

10.7


C4

5.0741

2806.7

250–450

0.1–30

5.2

8.7


C5

5.4009

3101

303–383

0.1–100

5.5

10.4

Audonnet and Pádua (2001)

C6

5.7051

3351.2

303–348

0.1–250

4.9

9.9 9.4

Oliveira and Wakeham (1992) Baylaucq et al. (1999)

C7

5.9916

3566.4

293–343

0.1–100

4.5

C10

6.7285

4078.1

293–373

0.1–140

3.3

9.6

Canet et al. (2002)

C16

7.9044

4773.3

298–348

0.1–150

5.4

9.9

Tanaka et al. (1991)

TABLE 3—OPTIMIZED MOLECULAR PARAMETERS OF SOME COMPOUNDS OF PETROLEUM INTEREST Compounds

(10

10

m)

1

(J·mol )

Tmin–Tmax (K)

pmin–pmax (MPa)

AAD (%)

max

(%)

Database

H2S +dipole

3.688

2320

200–500

0.1–6

2.9

11.9

H2 S

3.667

2355

200–500

0.1–6

4.5

18.7

Liley et al. (1988)

CO2

3.6641

2007.9

250–500

0.1–100

4.3

8.7

Lemmon et al. (2007) Lemmon et al. (2007)

684

Liley et al. (1988)

N2

3.5734

833.1

200–500

0.1–100

1.3

2.5

O2

3.3269

1020.6

200–500

0.1–80

2.1

4.0


C6 H6

5.1202

3710.7

283–393

0.1–200

4.7

11.5

Dymond et al. (1981)

C7 H8

5.3897

3906.8

293–373

0.1–140

2.6

6.5

Baylaucq et al. (2003)


15

Methane Carbon dioxide Hydrogen sulfide

100*(1-µcorr/µexp)

10 5 0 –5 –10 –15 0

200

400

600

800

1000 1200

–3

Density (kg∙m ) Fig. 1—Deviations provided by the proposed correlation for the major compounds appearing in acid gas.

the data provided in the most reliable recent literature. Temperature and density have been used as inputs. The adjustment has been made by minimizing the maximum absolute deviation max between correlation results and those coming from the literature. Normal-Alkane. By using Eq. 2 to determine the molecular diameter of normal-alkane, the viscosity of long chains may not be predicted correctly. As an example, if Eq. 2 is used for estimating of the C16, the scheme proposed yields an AAD equal to 137% compared with the Tanaka et al. (1991) values (18 data points from T = 298 to 348 K and p = 0.1 to 150 MPa). If the molecular diameter is optimized on the data, the AAD decreases to 5.4%; see Table 2. In fact, in very dense states, the viscosity values are highly sensitive to the estimation of the molecular diameter of long chains. For the C16, the relative difference between deduced from Eq. 2 and the value given in Table 2 is less than 2%. So, using up-to-date viscosity databases provided in the literature, the molecular parameters of various normal-alkanes have been adjusted for an as-large-as-possible range of thermodynamic conditions. All optimized parameters are provided in Table 2. As shown in Table 2, the LJ fluid model is able to provide a very good estimation of the viscosity of normal-alkane over a wide range of thermodynamic conditions, even for rather long chains with at most only one adjustable molecular parameter per species. Other Important Compounds. In acid-gas mixtures, apart from the n-alkane, there is also a need to model H2S and CO2. In addition, because the proposed scheme is dedicated as well to be applicable for a wide range of mixtures of petroleum interest, other usual compounds have been included. Optimized molecular parameters are provided in Table 3. For H2S, the molecular parameters have been taken from Galliéro et al. (2007), the dipole moment being equal to 0.9 Debye. As expected, the less complex (i.e., the more spherical and the less polar) the molecule, the better the viscosity results provided by the correlation; see Tables 2 and 3. Nevertheless, even for quite complex molecules (e.g., toluene), the results provided by the proposed scheme remain very good compared with the simplicity of the molecular model. It should be noted that optimized molecular parameters of some other compounds are also provided in ZébergMikkelsen et al. (2006) and Yoshimura et al. (2009). Concerning C1, H2S, and CO2, which are the most important compounds in acid-gas mixtures, the correlation is able to provide very good results for the whole range of thermodynamic conditions; see Fig. 1. Concerning CO2, because no quadrupolar moment was included in the proposed model (Galliéro et al. 2007), the deviations are larger than for methane. Concerning H2S, care should be taken because the database provided by Liley et al. (1988) may suffer from deviations from the real values as high as 20% (Liley et al. 1988). September 2010 SPE Journal

Simple Binary Mixtures. The application to mixtures of the scheme proposed in this work is straightforward as long as there are no adjustable extra parameters, which is usually the case in most schemes (Poling et al. 2000). Before testing it on a more-complex mixture, the correlation has been applied on three simple dense mixtures for which densities and viscosities have been measured. The first simple mixture studied is composed of C1 and CO2 for three molar fractions of methane, 0.243, 0.464, and 0.755. Densities and viscosities (159 data points) have been measured by Dewitt and Thodos (1966) at temperatures ranging from 323 to 473 K (supercritical) and pressures ranging from 3.3 to 68 MPa. For this mixture, the proposed scheme yields an AAD of 2.3% with a max of 8.6%. The deviations obtained are slightly larger for a low content of methane than for high content, which is consistent with the fact that the correlation performs better for pure methane than for pure CO2; see Tables 2 and 3. As a point of comparison, if the classical Lee, Gonzalez, Eakin (1966) (LGE) approach is employed for estimating the viscosity of this mixture, it yields an AAD of 76% with a max of 591%. This is not surprising as long as the LGE correlation, dedicated to natural gas, is known to fail when nonhydrocarbon compounds are present in a non-negligible amount (Elsharkawy 2004) and in dense states. As will be shown, our scheme does not suffer from these limitations. The second mixture analyzed is composed of C1 and C2 for three molar fractions of methane, 0.345, 0.5, and 0.685. Viscosities and densities (135 data points) have been measured by Diller (1984) at temperatures ranging from 200 to 300 K and pressures ranging from 1.5 to 30 MPa. The correlation yields an AAD of 2.7% with a max of 8.9% when compared with experimental values. As for the previous mixture, deviations generally decrease when the content of methane increases. The third mixture is composed of C1 and N2 and has been measured by Diller (1982) for three molar fractions, 0.317, 0.5, and 0.714. Measurements have been performed at temperatures ranging from 200 to 300 K and pressures from 1.6 to 30 MPa. On that system, the correlation yields an AAD of 2.3% with a max of 8.8% when compared with experimental values. It is interesting to note that, knowing the densities, the proposed correlation is able to provide a very good estimation of the viscosities of these simple mixtures for a large range of thermodynamic states without any extra parameters adjusted on mixture data. In addition, results obtained on these systems are as good as those on pure fluids, indicating that the one-fluid approximation employed here (vdW1) is efficient at least for such simple mixtures. Natural- and Acid-Gas Mixtures. A more interesting case is the one involving natural and acid gases. Nevertheless, to our knowledge only a few sets of acid-gas viscosity for mixtures containing H2S were published (Elsharkawy 2003, 2004). Concerning natural gas, there is a more comprehensive literature, even if there are not many data published for high pressures. In this study, we have selected three sets of recent measurements on natural gases. The first mixture is the one analyzed by Nabizadeh and Mayinger (1999). Measurements have been performed at T = 298 to 400 K and p = 0.1 to 6.5 MPa (52 data points). The second mixture is the one studied in Assael et al. (2001), which covers a temperature range of 240 to 450 K and a pressure range of 0.1 to 14 MPa (40 data points). The third system is composed of three different samples of a natural gas (from the North Sea) studied by Langelandsvik et al. (2007). The temperatures range from 263 to 303 K, and the pressures range from 5 to 25 MPa (45, 34, and 45 data points). All these natural gases have a content of methane greater than 0.8 in molar fraction. As shown by the results in Fig. 2, the correlation is able to provide a very good estimation of all natural gases studied here with the largest deviations for the mixtures studied by Langelandsvik et al. (2007). It should be noted that the mixtures studied by Langelandsvik et al. (2007) contain more compounds than the two others and are less well defined. In fact, it seems that our correlation tends to slightly and systematically underestimate the viscosity of natural gas for the highest pressures (Fig. 2). More precisely, on the mixture of Nabizadeh and Mayinger (1999), we obtain an AAD 685

8



4

2

0

–2

6

4

2

Nabizadeh and Mayinger (1999)

Langelandsvik et al. (2007), samp. 1 Langelandsvik et al. (2007), samp. 2 Langelandsvik et al. (2007), samp. 3

Assael et al. (2001)

–4

0 0 (a)

20

40

60

80 100 120 140 160

0

–3

50

100

Density (kg∙m )

150

200

250

300

350

Density (kg∙m–3 )

(b)

Fig. 2—Deviations provided by the proposed scheme for different natural gases.

Asymmetric Liquid Mixtures. Generally, viscosity prediction is very difficult to achieve when the species involved are very asymmetric (i.e., differ largely in size). This is mainly because the behavior, from the viscosity point of view, of a small molecule surrounded by large molecules is completely different from the symmetric situation (i.e., a large molecule surrounded by small ones). Such asymmetry in the behavior is not easy to tackle through a simple one-fluid approximation. As a first example, we have applied our correlation on a weakly asymmetric quaternary mixture composed of normal-alkane in the liquid state studied by Wu et al. (1998). This mixture is composed of normal alkanes, C7, C8, C11, and C13 in various proportions at ambient pressure and at temperatures varying from 293.15 to 313.15 K (28 data points). As for the previous gaseous systems, the results yielded by the correlation are in good agreement with the measurement, with an AAD of 6.3% and a max of 9%. Interestingly, this deviation is of the order of magnitude of those on the pure compounds, even if in the liquid phase the viscosity varies more abruptly than in the gas phase (case of the previous mixtures studied). In our group, the densities and viscosities of mixtures of methane and toluene have been measured (Baylaucq et al. 2003) for a large range of thermodynamic states, T = 293 to 373 K and p = 0.1 to 140 MPa. This asymmetric mixture has been studied for xC1 = 0.25, 0.37, 0.5, 0.64, and 0.95, and it has been shown in Baylaucq et al. (2005) that the viscosity of such mixture was difficult to predict accurately by conventional approaches. Results provided by our fully predictive scheme are provided in Fig. 3. Results shown in Fig. 3a clearly demonstrate that the scheme proposed is able to provide a reasonable estimation of viscosity, even in a liquid asymmetric mixture without any adjustable

10

20

5

10

0 –5 –10 –15 –20 –25

xC1=0.25 xC1=0.37 xC1=0.5 xC1=0.64 xC1=0.95

200 300 400 500 600 700 800 900 1000 (a)

–3

Density (kg∙m )



of 0.97% with a max of 1.69 % and an AAD of 0.83% with a max of 3.39 % for the mixture of Assael et al. (2001). Concerning the mixtures of Langelandsvik et al. (2007), we obtain an overall AAD of 4.04% with a max of 6.9%. As a point of comparison with usual correlations, for those three mixtures, the LGE approach yields AADs of 2.17, 2.21, and 1.35%, respectively. This means that, even for natural gas, our correlation is able to provide viscosity estimation as good as that of the LGE scheme, which is dedicated specifically to natural gas. Elsharkawy (2003) has provided densities and viscosities of three different acid-gas mixtures containing H2S. The first mixture (Gas 11) is mainly composed of xH2S = 0.23 and xC1 = 0.76 at T = 352.55 K and p = 34.47 MPa. The second one (Gas 13) is principally composed of xH2S = 0.49 and xC1 = 0.45 at T = 322.05 K and p = 17.24 MPa. The third one (Gas 14) is mainly composed of xH2S = 0.7 and xC1= 0.2 at T = 352.55 K and p = 9.4 MPa. The other minor compounds are CO2, N2, and alkanes. They have been taken into account for the calculation. Applying the correlation, we obtained viscosities equal to 0.0317, 0.0362, and 0.0198 mPa·s, respectively; the LGE correlation yields 0.031, 0.036, and 0.018 mPa·s, respectively; whereas the experimental values are 0.03, 0.03, and 0.022 mPa·s, respectively. Thus, even if there is a non-negligible deviation on the second mixture, the values predicted by our scheme are consistent compared with the experimental values. This is not surprising because both methane and H2S are well represented by the fluid model employed in this work; see Tables 1 and 2. Furthermore, as shown by the previous results on the simple mixtures studied, the scheme proposed is very efficient in dealing with mixtures of similar sizes in a purely predictive way.

0 –10 xC0 =0.15

–20

2

xC0 =0.3 2

xC0 =0.5

–30

2

xC0 =0.85 2

–40 550 (b)

600

650

700

750

800

850

900

Density (kg∙m–3 )

Fig. 3—Deviations provided by the proposed scheme for (a) methane/toluene mixture and (b) CO2/decane mixture. 686


parameters. For the system studied here, we obtain an AAD of 5.9% and a max of 26.3%, which is not worse than for more-complex approaches (Baylaucq et al. 2005). When looking more carefully at Fig. 3a, it appears that the results for xC1 = 0.25, 0.5, and 0.64 are excellent; those for xC1 = 0.37 are reasonable; and those for xC1 = 0.95 yield the largest deviations of all for the highest densities. This is somewhat surprising because we expect that results should have been very good for high methane content, because pure methane is well modeled. Such differences may come from intrinsic limitations of the vdW1 (Galliéro et al. 2005b) when there is a small amount of the lightest compound in an asymmetric mixture. The last asymmetric mixture studied is composed of CO2 and C10. The viscosities of this mixture have been measured by Barrufet et al. (1996) for temperatures ranging from 310 to 403 K and pressures up to 12 MPa. As shown in Fig. 3b, the correlation is able to provide reasonable results except for xCO2 = 0.85, for which the deviations reach 31.6%. This trend is consistent with those on the methane/toluene mixture. For that precise mixture (Fig. 3b), the correlation overestimates viscosity for the lowest densities, whereas it is the contrary for the highest densities. From these results, it can be concluded that, even for asymmetric mixtures, the proposed purely predictive scheme is able to provide a good estimation of viscosity whatever the thermodynamic state, at least from an engineering point of view. A very interesting feature of the proposed correlation is that it provides results on mixtures that are generally as good as those on pure fluids without any additional fitted parameters. Nevertheless, it should be noted that such an approach should be taken with great care when dealing with associative compounds, including water. A simple LJ model cannot tackle the very subtle physics of hydrogen bonding (ZébergMikkelsen et al. 2006). Conclusions In this paper, a fully predictive method for natural- and acid-gases viscosity calculation is presented. The model is based on MD simulations in gas, liquid, and supercritical states. Molecular simulations were performed on the LJ fluid to obtain a large, fully consistent database. The correlation, built on this fluid model in reduced variables, is fully predictive and uses temperature and density as inputs. A CS scheme was developed to allow application of the correlation on real pure fluids according to two molecular parameters, the molecular diameter and the potential depth, for many compounds. Application to pure components usually requires only the knowledge of the critical temperature and the critical volume. For complex molecules, the molecular diameter should be adjusted on available experimental viscosity data (knowing the density), and a database of optimized molecular parameters is provided in this work. In addition, by using temperature-dependent molecular parameters, the method was extended to deal with polar compounds, such as H2S. This correlation is able to provide the viscosity of a large variety of compounds, including normal-alkane, H2S, or CO2, with an excellent accuracy (i.e., AADs below 5%) on a very wide range of thermodynamic states. The method was extended to mixtures using the vdW1 without any extra adjustable parameters. This fully predictive method was tested extensively against available data on multicomponent mixtures in gas, supercritical, and liquid states. It is shown that this approach is able to predict the viscosity of natural- and acid-gas mixtures with accuracy as good as that on pure fluids, whatever the composition. This approach is as efficient as the usual LGE correlation on natural gas at low density but can be applied with success also to dense systems and to nonmethane-dominant mixtures (e.g., CO2- or H2S-rich mixtures), contrary to the LGE correlation. Finally, it is shown that this correlation can also be applied to predict reasonably the viscosity of asymmetric high-pressure mixtures even in the liquid phase without any fitted parameter. Nomenclature ai = numerical parameters for the low-density viscosity bi = numerical parameters for the residual viscosity kB = Boltzmann constant, J·mol−1·K−1 September 2010 SPE Journal

Mi N NA p T v xi i v

= molecular weight of Component i, kg·mol−1 = number components in a mixture = Avogadro number = pressure, Pa = temperature, K = molar volume, m3·mol−1 = molar fraction of Component i = dipole moment of Molecule i, Debye = potential depth, J·mol−1 = dynamic viscosity, Pa·s = density, kg·m−3 = molecular diameter, m = collision integral

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Guillaume Galliéro holds a PhD degree from the University of Bordeaux, France. email: [email protected]. After two years in a postdoctoral position at the Technical University of Denmark and the University of Pau, France, he became an assistant professor at the University of Marne la Vallée, France, in 2005. In 2007, Galliero took a full researcher position at the Centre National de la Recherche Scientifique (CNRS), affiliated with the University of Pau. Since November 2009, he has been a full professor in the Laboratory of Complex Fluids at the University of Pau, France. Galliero’s research interests encompass simulation and modeling of thermophysical properties of fluids as well as multiscale/multiphysics study of heat- and mass-transfer simulations of multicomponent mixtures. He is an expert in molecular-simulations techniques. Galliero has published numerous articles in scientific journals and contributed to many international meetings. Christian Boned holds a PhD degree from the University of Pau and Pays de l’Adour. email: [email protected]. From 1967 to 1988, he was an assistant and then an associate professor at the university. Since 1988, he has been a full professor in the Laboratory of the Complex Fluids, associated with CNRS and Total. Boned is the head of the Viscosity, Transport Properties team. His research team focuses primarily on experimental measurements, especially those involving large temperature and pressure intervals. His expertise is in density and viscosity at high pressure. Boned’s research interest encompasses both experimental and theoretical aspects, such as free-volume theory of dynamic viscosity and molecular-dynamics representation of viscosity. He is the coauthor of more than 130 articles published in international journals. Boned is a member of the International Association on Transport Properties and has been co-organizer of several meetings. Antoine Baylaucq holds a PhD degree from the University of Pau and Pays de l’Adour, France, and worked at the university as an assistant professor until August 2009. email: antoine. [email protected]. In September 2009, he joined Total as head of the Fluids and Organic Geochemistry Laboratory. Baylaucq’s areas of research interest are sampling, pressure/ volume/temperature (PVT), fluid properties, fluid behavior, fluid composition, gas injection, heavy oil, oil shale, gas shale, wax, asphaltene, and transport properties. François Montel holds a PhD degree from the Ecole Supérieure de Physique et de Chimie Industrielles in Paris. email: [email protected]. He was also an assistant professor there for three years. Montel then joined Elf in 1978 and became head of the PVT laboratory and an acknowledged technical expert. Montel now works as an expert in fluid thermodynamics for Total Exploration and Production. He has been published numerous times in scientific journals. Montel is an associate professor at the Ecole Nationale Supérieure du Pétrole et des Moteurs in Rueil and a course instructor in applied thermodynamics at Imperial College. His research interests include catagenesis, migration, trapping, alteration, sampling, PVT, fluid properties, fluid distribution, reservoir connectivity, fluid behaviour, equation of state modeling, phase equilibria, gas injection, CO2 sequestration, heavy oil, oil shale, gas shale, wax, asphaltene, hydrates, and sulphur.
