Properties of Oils and Natural Gases 0872015882, 9780872015883, 087201066X

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Contributions

in Petroleum

Geology & Engineering

ci) 5

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( ( (

Contributions

in Petroleum Geology and Engineering

Series Editor: George V. Chilingar, University of Southern California Volume 1: Geologic Analysis of Naturally Fractured

Reservoirs

Oilsand

Volume 2: Applied Open-Hole Log Analysis Volume 3: Underground

Storage of Natural Gas

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Volume 4: Gas Production Engineering

(

Volume 5: Properties of Oils and Natural Gases

(

Volume 6: Introduction

(

Volume 7: Hydrocarbon

(

Volume 8: Gas Reservoir Engineering

to Petroleum Reservoir Analysis

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Phase Behavior

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PORAPAK R HMOLSIEVE13X~

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SPLlT 1: 100 HWCOT

¡ Gas Chromatography

1. To CIO+ using capillary columnchromatography. 2. From ClO to C20+ using a mini distillation apparatus. 10

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Traditionally, compositional data in the oil industry have only been reported to C7 +, the compositional information being mainly based on low temperature fractionate distillation data. This level is inadequate for accurate modeling of the phase equilibrium and physical properties of the hydrocarbon mixtures. In recent years, new methods have been developed for experimentally determining the composition of hydrocarbon mixtures. These methods yield a far more accurate and detailed description of the hydrocarbon systems and are described in this chapter.

18

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The appearance of capillary columns has enabled separation and quantification of many more individual components than prevíously possible. The compositional description of the reservoir fluid is done by analyzing separately samples of the gas and liquid phases, which when recombined in the correct gas to liquid ratio will yield the reservoir fluid composition. The compositional description of the gas phase is carried out in one step: N2, CO2, CI-CIO by capillary column chromatography. The liquid phase compositional description is carried out in two steps:

8

11

For oil and gas mixtures, the phase behavior and physical properties such as densities, viscosities, and enthalpies are uniquely determined by the state of the system, i.e., the temperature, pressure, and the composition. In sirnulating the phase behavior and physical properties of complex hydrocarbon mixtures accurately, it is necessary to have detai!ed and accurate cornpositional information for each mixture.

GAS CHROMATOGRAPHY

7

CP SIL

51--8

Figure 2-1. Typical chromatogram and column configuration. Peak numbers correspond to calibration numbers given in Table 2-1. The analytical conditions for the organic and inorganic analyses are given in Tables 2-2 and 2-3, respectively. From Osjord and Malthe-S0renssen, 1983, reprinted from the Journal of Chromatography.

The chromatographic separation of oxygen, nitrogen, carbon dioxide, and approximately 60 hydrocarbons from C¡ to CIO+ was achieved by using a combination of packed and capillary columns. The sample was injected vía two time-programmed loops into packed and capillary columns with thermal conductivity and flame-ionization detection. The packed columns were molecular sieve 13X, Porapak R, and the capillary columns were Chrompack Sil 5 (fused silica). The hydrocarbons were analyzed by split injection using the capillary column, and temperature programming starting at 30°C, and the permanent gases were analyzed isothermally at 50°C using the packed column in a special compartment outside the main oyen.

12

Propertíes 01 Oíls and Natural Gases

Composítíonal

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Compositional

Properties of Oils and Natural Gases

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46.577 16.326 14.775 2.295 4.810 1.050 1.283 0.728 0.065 0.793 0.159 0.359 0.055 0.573 0.034 0.127 0.049 0.210 0.009 0.001 0.006 0.016 0.012

.

Mole %

28.01 44.01 16.04 30.07 44.10 58.12 58.12 72.15 72.15 86.18 70.14 84.59 100.21 85.95 78.11 88.59 114.2 99.5 92.1 99.7 127.7 125.0 106.2 118.5 140.0

100.000

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0.659 5.652 68.795 12.863 7.938 0.936 1.961 0.345 0.421 0.200 0.022 0.222 0.038 0.099 0.017 0.153 0.007 0.030 0.013 0.050 0.002 0.000 0.001 0.003 0.002 100.000

Gas molecular weight = 23.69 Gas gravity = 0.818

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Molecular Weight

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Nitrogen Carbón dioxide Methane Ethane Propane ;-Butane n-Butane ;-Pentane n-Pentane e, Paraffins e6 Naphthenes Total e7 Paraffins e7 Naphthenes C, Aromatics e, Total ea Paraffins ea Naphthenes ea Aromatics ea Total C, Paraffins eg Naphthenes C, Aromatics C, Total Decanes plus

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Table 2-4 Composition 01 Separator Gas

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Determinations

In this work the oil phase analysis is divided into two parts: 1. To CJO+ using chromatography 2. from CJO to C20+ using distillation

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Source: Osjord et al., (1985).

To properly quantify the individual components, an internal standard of known quantity is added to the sample. In this way one may compensate for column losses that are known to occur from carbon numbers of 15 and higher. The component used as the internal standard is iso-octane, which is normally not present in naturally occurring oils. As for the gas sample analysis, a calibration program enables thorough identification and quantification of the components present between C2 and Cg. Table 2-6 shows a typical example of such an analysis. This detailed an analysis up to C10+ is normally not needed in practice. Therefore, a regrouping of components may be carried out as shown in Table 2-7.

r: Gas Molecular Weight

100.00 23.69

In addition to the weight fractions of the collective groups such as the C/s, one may from such a detailed starting point calculate average rnolecular, weights, average density, critical properties, and PNA distributions for each of the groups up to C10+' See Tables 2-8 and 2-9. Identification of the components in the range CIO to C20 on the basis of the chromatogram in Figure 2-3 is seen to be impractical. Hence the liquid phase analysis is extended beyond CIO+ by a distillation technique.

Liquid Phase Analysis from CIO to C20+

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Figure 3-5. Black oil bubble point determination. 1. Shrinkage factor, Bo. This is a measure of the ratio of the volume of

the hydrocarbon tan k conditions: Bo

=

system at reservoir conditions to the volume at stock

res. m ' oíl/Sm ' stock tank oil

>LD

L

Pressure

P1 - Psat

Oil

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Ps

1: Gas.

1

v

ro

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One Pha se

1 P2 < Psat

P2 < Psat

P2 < Psat

P3 < P2< Psat

Figure 3-6. Schematic representation 01 a differentialliberation experiment.

(3-2)

2. Solution gas/oil ratio, Rs. This is a measure of the volume of gas in

solution at given pressures (see also Equation 3-8): Rs

=

Sm3 gas/Sm ' stock tank oil

(3-3)

3. Density of the reservo ir fluid at different pressures, including the saturation pressure: p

=

kg of Fluid/rn''

4. Real gas deviation factor Z, i.e., compressibility phase at given pressures and temperatures:

Z

=

PV/nRT

45

(3-4)

factor, for the gas

(3-5)

where V is the volume and n the number of moles. A schematic diagram of the differential depletion experiment is shown in Figure 3-6. The equipment is shown in Figure 3-7. The reservo ir fluid is, at constant temperature, brought to a pressure above its saturation pressure. After this, the pressure is brought to a value

Figure 3-7. A PVT system used tor differentialliberation analyses. One sees from left to right: the motorized pump, PVT cel!, and bath gas col!ection system with condensate trapo

46

Oil and Gas Property Measurements

Properties 01 Oils and Natural Gases

below the saturation pressure. The system will then be in the two-phase region. After the system has been equilibrated, the gas phase is removed from the cell at constant pressure. The gas volume is measured with a gasometer and recorded. The new volume of the oil is calculated from the readings on the mercury pump before and after the discharge of the gas. These readings will also give the gas phase volume at the given pressure and temperature. The hydrocarbon system is again brought into the two-phase region, this time at a lower pressure, and the gas discharge procedure is repeated. The pressure step intervals are chosen so that 8-10 stages will occur between the saturation pressure and the atmospheric pressure. At the last stage, care must be taken to discharge al! of the remaining gas, whereas the residual oil is left in the cell. After the last part of the gas is discharged, the cell is closed, its internal pressure is increased in steps, and the resulting oíl volumes are recorded. Normally, one records volumes at each 50 bar up to 300 bar. The plot of these volumes can be extrapolated back to atmospheric pressure, so that a measure of the residual oil volume at the given temperature, which is often that of the reservoir, can be determined. Then the cell temperature is reduced to 15°C, and the oil volume at this temperature is determined as previously explained. This volume is the reference residual volume. The oil is then discharged from the cell, and its density ís determined as explained in Chapter 2. The data recorded in this analysis can be represented as shown in Figure 3-8. Ps and Vs are the saturation pressure and volume, respectively; VRT the liquid volume at one atmosphere and temperature, Ti and V R is the reference liquid volume at 15°C. From these data the shrinkage factor of the system at the saturation pressurernay be calculated: Bo,s

=

VSIVR

i

~';ft¡t M

~

Ps

'"§

~

~

100

Pressure

(3-7)

VIVR

NSTEP =

L+

n~N

V gas,nlV

[Bar J

Figure 3-8. Determination of saturation pressure and volume from a differential depletion experiment.

Bo .s

~ o ~ o

'"

.•,

The solution gas/oil ratio of the reservoir fluid, Rs, at pressure stage N can be obtained from Rs

300

200

100

=

s.

V

(3-6)

The whole Bo-curve (see Figure 3-9) may be calculated as follows: Bo

47

R

(3-8)

1

where NSTEP is the total number of flash stages and V gas,n is the volume of gas (Sm3) liberated at flash stage n. These data can be represented as shown in Figure 3-10.

200

300

Pressuro

[Bar)

Figure 3-9. The differential liberation Bo-factor as function of pressure.

The density of the reservo ir fluid at each pressure reduction step can be determined from the known density at 15°C, the liquid volume, and the volume and molecular weight of the gas. The density versus pressure data may be presented as shown in Figure 3-11. Deviation from ideal gas behavior may be expressed via the compressibility factor, Z, as defined in Equation 3-5. Employing Equation 3-5 twice,

48

Properties of Oils and "fatural Gases Figure 3-10. The solution gas/oil ratio as function of pressures.

1-

Ps

o,

Oil and Gas Property Measurements

GAS( 1) 1st STAGE T1

p l'

V1

cc

49

GAS (ST)

GAS(2) 2nd STAGE

3rd STAGE

P2,T2

Pst' \t

OIL to stock tank Figure 3-12. Three-stage flash separation train.

The two first stages are accomplished in the same PVT cell as was used in the differential liberation experiment, and the last stage is carried out by transferring the second stage Huid to a single flash apparatus (see "Flash Separation and Compositional Analysis," discussed earlier). From this experiment the overall COR, COR at individual stages, compositions of the gas and liquid phases, properties of the stock tank Huid such as density and molecular weight (see Chapter 2), and also the Bo factor of the oil can be obtained. Tables 3-1 and 3-2 show a typicallaboratory result of a separator test experimento

Pressure Figure 3-11. Oensity 01 reservoir fluid as function 01 pressure.

p

Gas Reinjection Related Analysis Ps

Pressure ).»~

i.e., at two different pressures and temperatures moles, will result in: Z¡

=

and for the same number of

P¡V¡T2Z2/P2V2T\

Equation 3-9 unknown, to the be known. Z2 is and it is usually

In arder to understand the reservoir Huid behavior under gas injection processes, two different analyses have been designed: l. Swelling test 2. Slim tube minimum miscibility pressure determination

(3-9)

relates a compressibility factor at condition 1, where it is compressibility factor at condition 2, where it is assumed to normal!y the compressibility factor at ambient conditions, clase to unity. Process Simulation by Flash Separation

A three-stage flash can be performed in arder to furnish support data for process simulation. Such a flash sequence is shown in Figure 3-12. Stage three is the stock tank stage, where the pressure is one atm and the temperature is 15°C. For the other two stages, temperature and pressure can be chosen as desired.

..

Swelling Test. When gas is injected into a reservoir containing an undersaturated oíl, the gas can go into solution. This has the effect of swel!ing the oil, i.e., the volume of oil becomes larger. Simulation of this effect can be performed in an ordinary PVT cell, starting with the original reservoir Huid in question. Injection gas has been compressed into a separate steel bottle. A srnall, known volume of injection gas is transferred into the PVT cell. A new saturation pressure is determined using the techniques already described, and a new saturation volume is recorded. This process is repeated until the saturation pressure of the fluid is equal to the estimated injection pressure of the system. The data from this test can be presented as shown in Figure 3-13. Slim Tube Tests. One method of improving oil recovery involves injection of gas into the oil reservoir. The gas may be nitrogen, carbon dioxíde, Hue gas,

50

Properties

01 Oils

51

Oil and Gas Property Measurements and Natural Gases Table 3-2 Analyses of Separator Gases from Separator Tests

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59

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100.0

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150.0

200.0

250.0

'-.

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300.0

.......! 350.0

I

Calculated Mole %

% Dev.

Measured Mole %

Calculated Mole %

% Dev.

N2

0.01

0.03

-

0.00

0.00

-

CO2

3.72

3.03

- 18.5

0.00

0.03

-

C,

51.92

53.67

3.3

0.00

0.28

-

C2

16.60

16.82

1.3

0.18

0.60

-

C3

12.44

12.75

2.5

0.96

1.82

89.6

C.

8.64

8.65

C5

3.92

3.55

C6

1.44

0.99

- 31.3

6.89

7.31

C7+

1.30

0.51

- 60.8

83.09

79.34

u

0.0

Measured Mole %

-

0.1

:3.21

4.20

30.8

9.4

5.67

6.42

13.2

'00.0

Pressure (bar)

Figure 8-8. Cumulative mole Iractions 01 mixture 9 (molar composition in Table 4-11 removed as a lunction 01 pressure in a constant volume depletion study at T = 82.5°C. The experimental results are shown in Table 8-5.

% Dev. = 100 x (Calculated - Measured)/Measured. (Only stated,for measured mole %'5 above 0.5.)

6.1 -

4.5

140

Properties of Oils and Natural Gases

An almost-perfect agreement with the experimental results is observed, when the molecular weight of the plus-fraction is adjusted to match the measured dew point.

SEPARATOR TEST Table 8-7 shows the results of a two-stage separator test (described in Chapter 3) on mixture 5. The results comprise measured and calculated gas and liquid-phase compositions. The characterized mixture consisted of 10 hydrocarbon fractions. The measured and calculated gas-phase compositions agree very well, whereas larger deviations are seen for the liquid phases.

SUMMARY OF THE RESULTS OF THE PVT-SIMULATIONS The PVT-simulations provide evidence that good agreement with measured PVT-results can be obtained using the SRK equation of state in conjunction with the C7 + -characterizatíon procedure-of Pedersen et al. (presented in Chapter 7). Six hydrocarbon fractions are, in most cases, sufficient to represent the hydrocarbons of a reservoir fluido The procedure described in Chapter 10, for adjusting the molecular weight of a plus fraction against a measured saturation point, is ver y useful for obtaining reasonable results for the gas/liquid ratios of a gas condensate mixture. An adjustment of less than 10% of the molecular weight of the plus fraction does not influence the phase properties significantly.

Chapter 9

Comparison Between Experimental and Predicted Thermodynamic Properties The calculated results of this chapter are all based on the flash and phase envelope algorithms"of Michelsen described in Chapter 6 and on the SRK equation of state (Chapter 5) coupled with the C7 + -characterization procedure of Pedersen et al. (see Chapter 7). The procedure, used in some cases for adjusting the molecular weight of a plus-fraction, is described in Chapter 10. The compositions of the mixtures mentioned are shown in Chapter 4.

OEW ANO BUBBLE POINTS (PHASE ENVELOPES) Table 9-1 presents measured and calculated dew and bubble points for mixtures 1-10. Calculated results, obtained using the standard flash (nonzero binary interaction coefficients for interactions where non-hydrocarbons take part), and 20 hydrocarbon and 6 hydrocarbon fractions, respectively, to represent the mixture, are shown. The simplified two-phase (P,T)-flash (ki¡ = O), described in Chapter 6, is also used to yield results. The last column of Table 9-1 shows the needed adjustment to the molecular weight of the plus-Iractíon (maintaining the measured weight composition), to get agreement with the measured saturation (dew or bubble) point. Results of using a characterized mixture consisting of 6 and 20 hydrocarbon fractions are of comparable quality. Generally, it makes very little difference whether the standard flash or the ki¡ = O flash is used. For all mixtures except mixture 7, an adjustment of less than 10% of the molecular weight of the plus-fraction is sufficient to get agreement with the measured saturation point. In Figure 9-1 the calculated phase envelope of mixture 10 is shown together with four measured saturation points. The characterized mixture consists of 20 hydrocarbon fractions. Figure 9-2 shows, for mixture 3, the results of three different phase envelope calculations corresponding to the results in Table 9-1. 141

Table 9-1

Measured and Calculated Saturation Pressures for 10 Different Petroleum Mixtures (The Molar Compositions

Poo' (bar)

Calculated p••, with 20 CH (bar)

388.0 267.9 239.0 340.3 464.0 274.5 264.7 264.7 281.0 462.8 401.6 303.5 229.6

398.3 259.5 235.9 346.5 463.7 276.5 248.0 279.6 334.5 435.8 412.6 337.3 246.2

-

-

Measured Mixture No.

Temperature (0C)

1 2 3 4 5 6 7 8 9 10 10 10 10

155.0 92.8 71.6 141.0 129.0 93.3 93.4 93.4

%AAD % BIAS

-

82.5 32.2 67.8 125.4 169.4

Notetion: CH: kij

= o:

% Dev.: MW-adjustment: % AAD: % BIAS:

-

4)

Calculated

% Dev.

kij

386.4 263.5 243.6 347.2 458.5 282.1 252.1 272.9 312.2 415.3 387.2 303.5 198.6

2.7 - 3.1 - 1.3 1.8 - 0.1 0.7 - 6.3 5.6 19.0 - 5.8 2.7 11.1 6.7

.

Calculated p••, with

Psa' with 6 CH (bar)

5.0 2.6

-

-

=

O

% Dev.

(bar)

% Dev.

- 0.4 - 1.6 1.9 2.0 - 1.2 2.8 - 4.8 3.1 11.1 - 10.3 - 3.6 0.0 - 13.5

392.4 251.4 233.1 339.5 457.6 274.2 248.0 279.6 330.5 405.1 390.1 324.3 238.1

1.1 - 6.2 - 2.5 - 0.2 -1.4

4.3 - 1.1

-

gr-~~~

g

o

o

-;o

~

Pressure g ~o~

g

N

o~

o o

o

~o

N

o o

- 5.4 9.1 2.6 - 8.3 0.1 - 3.0 18.5 9.6 - 8.8 7.6 - 2.4 - 6.0 - 2.2

5.1 0.2

6.4 0.9

-

(bar)

MWadjustment (%)

- 0.1 - 6.3 5.6 17.5 - 12.5 - 2.9 6.9 3.7

Number of hydrocarbon fractions in the characterized mixture. The kij = O flash described in Chapter 6 has been used on a characterized mixture consisting of 20 CH-fractions. 100 x (Calculated - Measured)/Measured. The adjustment needed of the molecular weight of the plus-fraction (se e Chapter 10) to get agreement with the measured saturation CH-fractions and standard k¡¡-values (see Chapter 7). % Average absol ute deviation. % Average deviation.

Pressure ,~

are Given in Chapter

point using 20

(bar) o o

o o

o o

o

o

o o

o o

-< ru W

::J a. o w C) e

Ol

n

3 'O N

~ ~ r-e-

e -s ro

n

n

Q¡

Q.I

Q¡

n e

n e

n e

Q¡

Q¡

Q¡

r-+

••••.

r+

t'Tl )(

-< ro

'O

ro -s

N

o

ro

3 ro

;;)

ro ro ro :::J o. o.. o. r+

rlg

o

3

'O

•

rt

..,

e ro

N

o o

•

ID a.

g

'O :;y W

o

en

ID ID

•

::J < ID O'

•

'O ID

en

N

o o

~

:3 X

e

ID

o

~ o~ o

~ N

o

o o

n

'" e n

o o

'"

",

x .., '"

'O

3

~ ~ rt

'"

144

Propertíes 01 Oíls and Natural Gases

Comparing

Experimental

and Predicted Thermodynamic

FLASH RESULTS

:> CI)

O

•... •... o

N

11 O"C

c:: lO

o

°o:t

11

lO

(Molar Composition in Table 4-9)

Pressure (bar) 201.1 173.4 138.9 104.4 70.0 35.5 %AAD % BIAS

Measured Liquid Mole Fraction 0.871 0.775 0.681 0.606 0.537 0.479

-

Calculated Liquid Mole Fraction Using 20 CH

% Dev.

0.861 0.788 0.704 0.625 0.550 0.475

-

- 1.1 1.7 3.4 3.1 2.4 - 0.8 2.1 1.5

.t--•.. :B (")-¡~

CI)

o;XU

"5

c::

s

.¡¡¡

o E o

0.855 0.784 0.701 0.624 0.550 0.477 -

- 1.8 1.2 2.9 3.0 2.4 - 0.4

'" CI)

O

;f.

CI) 1/)

'"

DEN SITIES

C\J

C\J

I

ID

co

~ N ~ ~ ~

ll) C\J

I

I

I

I

I

r-, co

u

"C

e

C\J

co

.n N

ll)

ID

ID

co

CI) ~ ~ ~ ll) ,...: ~ o

r-,

(")

ll)

l1i

I

I

CI)

'" :!! CI)

v

CI)

o;X U

"5

e

.¡¡¡

~lOa. a,(ij CII:5

-.c c::

'Inu

X

I

I

I

I

I

I"I~I(' O> o )IOI~IO O'> o t--: ~ r---: ~ ID

T"""

~

f'..

(I)[j, c_

C\J aJ

T"""

~

·

s: ""

200 bar and temperature around 136.1°C (the temperature of the depletion experiment), has almost ídentical gas and liquid phases. Thus, it exhibits near critical behavior. The problems of handling

157

j

ic, «e,

EXAMPLES ON THE DANGERS OF TUNING

01 EOS-Parameters

e6 e7 ea e e1Q el1 g

5.12 1.04 2.35 0.84 1.12 1.36 2.14 2.20 1.43 0.60 0.48

(

Table 10-1 of North Sea Gas Condensate

(

Molecular Weight

Density (g/cm at 15°C, 1 atm

93 107 120 134 147

.743 .753 .776 .790 .795

3)

(table continued

on next page)

(

(~~

158

Tuníng o] EOS-Parameters

Propertíes o] Oíls and Natural Gases

159;;:JY Q: (

-c -o:.

-~~MW¡

E

.•.. a:: °

,

O .•..

e

:8

>
MW¡

e...

al (')

jj.

o u

C\J

en ~ (O

w

'S-

CXl LO

E

C\J

e

(O

o

.¡¡¡

o o.~ a:

o

- 0.6

% Dev. = 100 x (Cale. - Exp.)/Exp. Sauree: Pedersen el al. (1988a).

k¡¡ = 0.00145 MW¡ MW¡

a

I

i!-

u I/J ~

-

•.

'R

I

Q)

Calculated bubble points based on composition with erroneous GOR using k;¡-values different from zero

k¡¡ = 0.00145 MW¡ MW¡

o

I

c:

~

E Experimental bubble point

:> Q) e

(')

e

o

.¡¡¡

Calculated bubble point based on composition with erroneous GOR

165

Tuning 01 EOS-Parameters

Properties 01 Oils and Natural Gases

el lO lO 0.-

~

C\J

(')

ID c:

(l)

-o Ol

Ol ln

Q)!Jl !Jl

•••• :;::

'" -o Ol a.

'" Q)

1,

Vl

~

a; " . "-

o ~

~"

•• o

(J)

166

Properties

01 Oils

Tuning

and Natural Gases

Gani and Fredenslund (1987) have developed a two-part tuning procedure that overcomes some of these problems: l. A sensitivity analysis, to select the parameters to tune, and a feasibility study, to check whether tuning can/should be done 2. If tuning is feasible, it is accomplished by regression; and evaluation of the results takes place While this tuning policy often yields reasonable results, and often will warn the user not to tune, it is also not completely safe. The dashed curve in Figure 10-4 is drawn with kij-values for C1-C7+ computed from this procedure. The reason for the incorrect rise of the bubble-point branch is that the sensitivity of the low-temperature bubble points to the adjusted kifvalues was not taken into account.

No.

Temp. (OC)

Exp. Dew Pt. (bar)

Calc. Dew Pt. (bar)

1 2 3 4 5 6 7 8 9 10

96.6 93.8 118.9 119.7 150.3 154.1 131.5 129.0 136.1 148.9

282.0 235.0 398.0 394.0 381.5 385.5 367.0 464.0 386.4

311.6 235.2 377.1 369.7 396.3 410.2 378.7 454.2 375.6 493.5

542.5

% BIAS

WEIGHT OF THE PLUS-FRACTION TUNING PARAMETER

AS A

As explained in Chapter 2, the compositional analysis for oil and gas rnixtures is most often given in weight units. Conversion to molar compositions requires knowledge of the molecular weights of each component and carbon number fraction. The experimental inaccuracy in determining the molecular weights of the plus-fractions is of the order 5-10%. Deviations in that order of magnitude can considerably influence a calculated dew-point pressure of a gas condensate mixture. An obvious application of this fact is to use the molecular weight of the plus fraction as an adjustablé parameter. Adjustments within the experimental uncertainty can, e.g., be used to match a measured saturation point, as suggested by Pedersen et al. (1988b). Table 10-10 presents experimental results for the dew-point pressures of 10 gas condensate mixtures. Also shown are the dew-point pressures calculated using the characterization procedure of Pedersen et al. as described in Chapter 7. The average absolute deviation between the experimental and the calculated dew-point pressures is 4.9%. To improve the dew-point calculations, Pedersen et al. attempted to treat the molecular weight of the plus fraction as an adjustable parameter, maintaining the measured weight composition. Table 10-10 also shows the adjustments needed in the molecular weights of the plus-fraction to get agreement with the measured dew point. The maximum adjustment required is 12.1 % (gas condensate 10). The molar compositions of this mixture before and after 'thís adjustment are shown in Tables 1O-1l and 10-12. Figure 10-6 shows the measured liquid dropout curve of a constant mass expansion experiment (described in Chapter 3), on gas condensate 1, in comparison with those calculated using:

167

Table 10-10 Experimental and Calculated Dew Point Results for 10 North Sea Gas Condensate Mixtures

% Dev. 10.5 0.1 - 5/3 - 6.2 3.9 6.4 -

%AAD

THE MOLECULAR

01 EOS-Parameters

-

3.0 2.1 2.8 9.0

• Adjustment of MW -

-

e,,. e+ e+ e+ e+ e+ e+ e+ e+ e+

8.8 0.0 8.3 9.6 5.4

1O 1O 1O 20

9.6 9.0 5.6 8.4

1O 20 20 20

12.1

4.9

7.7

0.2

1.1

Plusfraction

20

• % adjustrnent needed in Ihe molecular weighl 01Ihe plus Iraction lo get agreement wilh the measured dew point maintaining Ihe measured weight composition. % dev. delined as in Table 10-7. % AAD = % average absolute deviation. % BIAS = % average deviation. Source: Thomassen et al. (1986).

Table 10-11 Measured Molar Composition of Gas Condensate 10 of Table 10-10

Componenl

Mole %

N2

.39 3.47

e0 e, e e

2

2 3

-c,

n-C,

te,

n-es C6 C7

ea e e g

10

C " C'2

Molecular Weighl

Densily (g/cm') al 15°C, 1 atrn

80.17 6.28 2.75 .43 .88 .31 .35 .54 .72 .88 .55 .33 .24 .20

96 106 118 132 149 163

.743 .761 .779 .787 .788 .799 (table continued

on next page)

168

Propertíes 01 Oils and Natural Gases

Tuning 01 EOS-Parameters

Table 10-11 Continued

Component

Mole %

8.0

Molecular Weight

Density (g/cm3) at 15°C, 1 atm

175 194 203 217 235 247 255 396

.820 .831 .833 .838 .836 .842 .852 .867

I

i

Experimental Results Calculated Based on Measured MW Calculated Based on Adjusted MW

(j)

5

6.0

o

:>

.21 .18 .15 .11 .12 .09 .08 .57

C'3 C'4 C,S C'6 C17 C,S C'9 C2D+

..,

"

(j)

/~~"

4.0

4-

----

.." .

o U

-, -,

.

" -, \

\

::> o-

~ o r>

Source: Thomassen el al. (1986).

.

e

o e,

'"

169

\ \

,.0

\ \ \

Table 10-12 Molar Composltion of Gas Condensate 10 of Table 10-10 After an Adjustment of the Molecular 'Weight of the C20 + -Fraction by 12.1 %

Component

Mole %

N2 CO2 C, C2 C3 i-C4 n-C4

Density (g/cm3) at 15°C, 1 atm

»c, C6 C7 Cs C9 ClO Cll C'2 C,3 C'4 C,S C'6 C17 C,S C'9 C20+ el al. (1986).

,

,

100.0

..\ \

200.0

,\

300.0

I 400.0

Figure 10-6. Liquid dropout curve (constant mass expansion) lor the gas condensate mixture of Tables 10-1 and 10-12.

l. The measured molecular weight of the plus-fraction and 2. The molecular weight of the plus-fraction adjusted to get agreement with the measured dew point.

80.22 6.28 2.75 .43 .88 .31 .35 .54 .72 .88 .55 .33 .24 .20 .21 .18 .15 .11 .12 .09 .08 .51

I

0.0

Pressure (bar)

.39 3.47

ic,

Source: Thomassen

Molecular Weight

0.0

96 106 118 132 149 163 175 194 203 217 235 247 255 444

.743 .761 .779 .787 .788 .799 .820 .831 .833 .838 .836 .842 .852 .867

Figure 10-7 shows measured and calculated results for the compressibi!ity factor of the gas phase liberated during the constant volume depletion experiment. The results indicate that the adjustment of 12.1 % in the molecular weight of the plus-fraction has ver y little influence on the calculated gas phase density. Pedersen et al. find that an accurate molecular weight of the plus-fraction is less important for the bubble point of an oi! mixture than for the dew point of a gas condensate mixture. Adjustment of the molecular weights is, therefore, a less effective tool for oil mixtures than it is for gas condensate mixtures. On the other hand, there is also less need for improvements in the calculations on oíl mixtures, beca use accurate results can in most cases be obtained based on the measured compositions .

RECOMMENDATIONS

REGARDING

TUNING

Agreement between the results of PVT-experiments, and those obtained with a cubic equation of state, may be improved by adjustment of the criti-

170

Properties 01 Oils and Natural Gases Tuning 01 EOS-Parameters

17

/e:"';' ''\()~ ./a:! ~~\ '

'~

-c co

./

- - --==--¡--- -

/

0.9 0.0

100.0

200.0

300.0

Pressure

400.0

500.0

600.0

(bar)

Figure 10-7. Gas phase compressibility factor of the gas condensate mixture of Tables 10-11 and 10-12.

cal temperature, the critical pressure, the acentric factor, or the binary interaction coeHicients. It is, however, unlikely that parameters tuned to one specific property will provide accurate results for other properties. Pararneters determined in a limited pressure and temperature range may lead to highly inaccurate results when considered at pressures and temperatures not included when performing the parameter estimation. Rather than adjusting the equation of state parameters, it is recommended treating the molecular weight of the plus-fraction as an adjustable parameter. The experimental uncertainty on the molecular weight of the plus-fraction is of the order 5-10 %, and it seems, therefore, justified to adjust the molecular weight of the plus fraction (e.g., against a measured saturation point) within these limits.

REFERENCES Coats, K. H. and Smart, G. T., "Application of a Regression-Based EOS May PVT Program to Laboratory Data," SPE Reserooir Engineering, 1986, pp. 297-298. Gani, R. and Fredenslund, Aa., "Therrnodynarnics of Petroleum Mixtures Containing Heavy Hydrocarbons: An Expert Tuning Systern ,' Ind. Eng. Chem . Research, 26, 1987, pp. 1304-1312.

~

/ h7A

Viscosíty

the molecular hard sphere diameter. The following dependence of temperature and the molecular weight may then be derived: 1/

Viscosity

n:

An external shear stress applied to a portion of a fluid will introduce a movement of the molecules of the affected part of the fluid in the direction of the applied shear stress. The moving molecules will interact with the neighboring molecules. These will start moving too, but with a lower velocity than that of the molecules exposed to the stress. The dynamic víscosity, 1/, of a Newtonian fluid is defined as the following ratío for the flowing fluid:

where

(11-1)

Txy= shear stress Vx = velocity of the fluid in the x-direction (the direction of the applied stress) avx/ay = the gradient of Vxin the y-direction (perpendicular to the x-direction) .

=

constant (MW1I2 p~/3/nI6)

~=

T~/6/MW1I2

(11-5)

~

VISCOSITY CORRELATIONS Several methods have been suggested for estimating the viscosity of hydrocarbon mixtures. In the simulation of processes related to oil and gas production, víscosity correlations are needed that are applicable to a wide range of hydrocarbon mixtures and process conditíons. Methods limited to narrow ranges of compositíon and/or temperature and pressure will not be covered here. Only calculatíon procedures that can be used for both gas and liquid phases, giving consistent results for each, are discussed.

The viscosity correlatíon that has the most widespread use in flow models for petroleum mixtures is probably the correlation of Jossi et al. (1962) in the form suggested by Lohrenz et al. (1964). Gas and liquid viscositíes are related to a fourth-degree polynomial in the reduced density, o, = p/ Pe:

(11-2)

n v MW L

[(1/ - 1/')~ + 10-4]1/4 where

OF LOHRENZ ET AL.

GAS THEORY

For a dilute gas, one may derive an approximate analytical express ion for the viscosity, 1/. It is: 1/ = 1/3

(11-4)

p~/3

THE VISCOSITY CORRELATION KINETIC

(11-3)

Though kinetic gas theory is not applicable to the near critical region, this expression plays an important role in víscosity calculations. It is convenient to introduce the viscosity reducing parameter, ~:

OF VISCOSITY

1/ = - Txy/(aVx/ay)

constant (T1I2MW1I2/a2)

on the

a3 is often associated with the critical molar volume, Ve' If it is further assumed that Ve is proportíonal to RTe/Pe, the following expression may be obtained for 1/ at the critical point:

Chapter 11

DEFINITION

=

1/

173

n = number of molecules per unit volume v =' average molecular speed MW = molecular weight L = mean free path between two molecules.

It can be shown (see e.g., Hirschfelder et al., 1954) that the average speed, v, is proportional to (RT/MW)1/2 and the mean free path.to l/(na2), where a is 172

where al = a2 = a3 = a4 = a5 =

0.10230 0.023364 0.058533 - 0.040758 0.0093324

=

al + a2Pr + a3Pr2 + a4Pr3 + a5Pr4

(11-6)

Viscosity

Properties oj Oils and Natural Gases

174

TI' is the low-pressure gas mixture viscosity. t is the viscosity-reducing rameter, which for a mixture is given by the following expression:

t

ll/6[ .1:N

N

= [ .1: 1=1

z¡Tci

l- 1/2[

z¡MW¡1:

1=1

N

l- 2/3

By expressing the viscosity as a function of the density, the calculated viscosity becomes ver y sensitive to the results obtained for the density. Especially for high viscous fluids this may lead to severe errors for the calculated viscosity (see the examples in the Comparison with Experimental Results section) .

pa-

(11-7)

z¡Pe¡

1= 1

CALCULATING

where N is the number of components in the mixture, and Z¡the mole fraction of component i. Pure components have a well-defined Pe, but it is not obvious how to determine the critical density of a complex hydrocarbon mixture. Lohrenz et al. (1964) suggest calculating the critical density of a petroleum fluid as follows: p, = (Ve)-

1 =

(t

1=

(11-8)

(z.Vci) + ZC7+ • VeC7+) - 1 1

where the critical molar volume (in ft3/lb mole) of the C7+-fraction is found from the expression: =

21.573 + 0.015122 MWC7+ - 27.656

X

SGC7+

In that case, compreheflsive viscosity data are only needed for one of the components of the group. That component is then used as reference substance (o), and the viscosity of another component (x) within the group can easily be calculated. In general, the critical viscosity (TIe) will not be known. The inverse of t for which the expression is given in Equation 11-5, may be used as an approximate value. When the corresponding states principle is expressed in terms of p" and not P" it is convenient to express ~ in terms of Ve, instead of Pe' The final expression for the reduced viscosity (Tlr)then becomes:

In this expression, MW is the molecular weight, SG the specific gravity, and the sub-index C7 + indicates that the mentioned property is an average value for the hydrocarbons with 7 and more carbon atoms. The dilute gas mixture viscosity TI*is determined as f.ollows (Herning and Zippener, 1936): . N

N

z¡TI¡*MW¡I/2/1:

¡=

1

(11-10)

Z¡MW¡I/2

¡= 1

For the dilute gas viscosity of the individual components, TI¡~the following expressions are used (Stiel and Thodos, 1961):

TI¡*= 34

(11-13)

Tlr(P, T) = f(p" Tr)

(11-9)

+ 0.070615 MWC7+ x SGC7+

r¡* = 1:

VISCOSITY USING THE CORRESPONDING STATE S THEORY

The viscosity may al terna tivel y be calculated using a modified form of the corresponding states method. The starting point-is the properties at the critical point (e.g., the critical viscosity, Tle,the critical density, Pe, the critical pressure, Pe, and the critical temperature, Te)' Away from the critical point, the properties may be expressed in terms of reduced properties (TI" p" P" T" etc.), which are defined as the properties at the actual conditions, divided by those at the critical point. A group of substances obey the corresponding states principIe with respect to viscosity, if the functional dependence of Tlr on, e.g., o, and T, is the same for all substances within the group.

i*C7-t

V,C7+

175

X

TI¡*= 17.78

10-5

X

~

t¡

T094

(11-11)

for T; < 1.5

n

10-5 ~ (4.58Tr¡ - 1.67)5/8

for Tr¡ > 1.5

~¡

where ~¡ is given by the expression in Equation

11-5.

(11-12)

\

Tlr(P, T) = TI(p, T) (Te)-1/2 (Ve)2/3 (MW)-1/2

(11-14)

The viscosity of component x, at the temperature T, and a pressure where it has the density P, is then found from the expression: Tlr(P, T)

=

(Te,/Teo)1I2

(Vex/Veo)-2/3 (MW,/MWo)I/2

r¡o(Po, To)

where Po = P Peol Pe, T, = T Teo/Te, Tlo= viscosity of the reference substance at temperature sity Po

(11-15)

T, and den-

Propertíes of Oils and Natural Gases

176

Viscosíty

The corresponding states principie, in this simple form, may be used successfully for, e.g., mixtures of hydracarbons of approximately the same chemical structure and molecular weight. Oil and gas mixtures consist of a large number of components, including paraffinic, naphthenic, and aromatic components. For oil mixtures, the molecular weights may range fram 16 (methane), to about 1100 (Cso). It is not to be expected that methane and Cso should both belong to a group where the simple corresponding states principie applies. Very comprehensive experimental viscosity data exist for methane, and methane is therefore a natural choice as the reference substance. To apply the corresponding states principie to the remaining components found in oil and gas mixtures, some modifications must be intraduced, as compared with Equation 11-15. Ely and Hanley (1981) have suggested expressing the deviations from the simple corresponding states principie in terms of two shape factors, e(T" V" w), and (T"V" w), which en ter into Equation 11-16 as follows: 1/x(p, T)

=

(eTex/Tco)1I2 (VexlVco)-213(MWx/MWo)1I2 1/o(po,Tole)

The deviation from the simple corresponding states principIe is expressed in terms of a ratationaI coupling coefficient, Q' (Tham and Gubbins, 1970). The final expression for the viscosity of a given mixture is the following: 1/m¡.(P,T) = (Te,mi.lTeo)-116(Pe,mixIPeo)213(MWmix/MWo)1I2 Q'mix/Q'o1/o(Po, To) P PeoQ'o Po = P Q" c,mix rrux

T,

=

(11-20)

T TeoQ'o T e, mixQ'mix

(11-21)

The critical temperature and the critical molar volume for unlike pairs of molecuIes (i and j) are found using the following formulas:

(11-16)

Tcij = (TciTej)1I2

(11-22)

!

V el).. = 8 (Vel)13+ VeJ)13)3

(11-17)

f(P" Tr)

Vci = RZciT e;lP ci

(11-24)

where Zci is the compressibility factor of component i at the criticaI point. Assuming that Z¿ is the same for all components, Equation 11-23 may be rewritten:

1

Vcij = 8

•

constant

((T~ ) r,

The critical temperature sion:

1/r = 1/~ =

1/1(Te-1I6

p~13MWII2)

j

i

Te,mix=

(11-18)

113

(T)

+ ~ Pel

113)3

(11-25)

of a mixture is found fram the following expres-

1: 1: ZiZjT V 1: 1: Z;ZjVcij i

where the following expression is used for the reduced viscosity:

(11-23)

The critical molar volume of component i may be related to the critical temperature and the critical pressure as follows:

cij

=

(11-19)

"

Vr is the reduced molar volume and w is the acentric factor. Vr (and thereby p) is itself found by a corresponding states model, and the determination of e and therefore involves an iterative procedure. To use the corresponding states principie on a mixture, it is necessary to represent the mixture as a fluid consisting of one hypothetical pure component with a given Te, Pe, and MW. Ely and Hanley adapted the mixing rules of Mo and Gubbins (1976) for Te and P, (the expression for Te is given in Equation 11-27, and the expression for P, in Equation 11-30). The average molecular weight is formed from considerations regarding the radial distribution function for a fluid in non-equilibrium due to the presence of a shear. Baltatu (1982 and 1984) extended the pracedure of Ely and Hanley to cover boiling point fractions of petraleum mixtures also. The critical properties and the molecular weights of these fractions are estimated using the method suggested by Riazi and Daubert (1980). Pedersen et al. (1984 and 1987) presented a corresponding states model for viscosity calcuIations where the reduced viscosity is expressed in terms of the reduced pressure, P" and the reduced temperature, Tr:

n,

177

cf

(11-26)

j

where Z¡and Zjare mole fractions of components i and j, respectively. Using Equations 11-22 and 11-25, Equation 11-26 may be rewritten:

Viscosity

179

Properties o] Oils and Natural Gases

178

T

.

"" 'r' 'T'

=

Thís mixíng rule is deríved empírícally on the basis of available víscosíty data, and assigns (as one may expect) to the heavíer components a relatívely large ínfluence on the mixture víscosíty. The o-parameter of the mixture is found from the expressíon:

p

[(T)113 (T )113 T ZiZj P:: + P:; [Tci Cj]112

j

(11-27)

C,OlIX

1: 1:

3

z.z¡

J

I

[(Tel)113+ (Tej)1131 Pel Pej

amix = 1.000 + 7.378 x 1O-3p;.847

MWmi~S173

(11-34)

and the o-parameter of the reference substance (methane) as follows: For the crítical pressure of a míxture, Pe,mi» the followíng relation is used:

where the constant equals the one enteríng ínto Equatíon 11-25 and Ve,mixis found as follows: Ve,mix=

1: 1: ZiZjVeij i

ao = 1.000

(11-28)

Pe,mix= constant Te,mi)Ve,mix

ti

¡

(11-29)

j

From Equations 11-27, 11-28, and 11-29, the followíng expressíon may be deríved for Pe,mix:

P

.

=

'r' 'T'

8 ""

ZiZj[(T)113 ~ + (T ~ )

113 3 1[TciTej]112 (11-30)

c.nux

(~

i

~f

[(!::)", (!;)"']'l'

:1.

+

~

"i

1,

The míxing rules of Equations 11-27 and 11-30 are those recommended Mo and Gubbíns (1976). The mixture molecular weíght is found as follows:

by

(11-31)

where MWw and MW n are the weíght average and number average molecular weíghts, respectively:

MW w

=

N

N

i~ 1

i~ 1

1: ZiMWr/1:

ZiMWi

(11-32)

N

MWn

=

1: ZiMWi

i~ 1

p;847

(11-35)

The constants and exponents in Equatíons 11-34 and 11-35 are estimated on the basís of experimental viscosity data. In Equations 11-34 and 11-35: p; = Po(T

(11-36)

TeofTe,mix,P Peo/P e,mix)1 Peo

By expressing the viscosity in terms of the reduced pressure, ínstead of the reduced densíty as in Equatíon 11-13, it is possible to perform a direct calculatíon of the viscosíty. The calculations become much símpler, compared wíth those of Ely and Hanley (1981), where the densíty is calculated usíng an íterative procedure. The use of methane as the reference substance presents problems when methane is in a solid form in its reference state. Pedersen and Fredenslund (1987) have suggested a procedure for ímproving the viscosity predictions in those cases. The reference víscosíty correlation is based on the methane viscosity model of Hanley et al. (1975):

~

.i ')

MW mix= 1.304 x 10 - 4(MW~;303- MW~303) + MW n

+ 0.031

(11-33)

1)'(p,T)

=

1)o(T) + 1)l(T)p + ~1)'(p,T)

(11-37)

l~!

where 1)0' 1)1,and ~1)' are functions defined in the above reference. The methane densíty is found using the BWR-equation in the form suggested by McCarty (1974). In the dense liquíd regíon thís expression is mainly governed by the term ~1)' (p, T):

~1)'

(p,T)

=

expfj¡

+ i4/T) [exp [p01(h +

+ 8pOS(is + ~ +

;2)] - 1.0]

i~/2) (11-38)

where in the work of Hanley (1975), the coeffícients j¡-h have the followíng values:

-.*:-

/.

Víscosíty

Properties oj Oils and Natural Gases

180 Ít

=

iz

=

b

=

j4

=

Ís =

j6

=

h=

.,

10.3506 17.5716 - 3019.39 188.730 0.0429036 145.290 6127.68 -

HTAN

=

exp(~ T) - exp( - ~ T) exp(~ T) + exp( - ~ T)

with ~ T TF

=

T - TF freezing point of methane

=

".-

.-

iIl':.

j

COMPARISON WITH EXPERIMENTAL

RESULTS

¡¡¡ f)

~.

is given by: f) =

(11-39)

(p - Pc)/Pc

Pedersen and Fredenslund (1987) have estimated an additional set of coefficients of Equation 11-38 using viscosity data measured at reduced temperatures below 0.4, i.e., corresponding to a methane reference state where methane is in a solid formo The data include viscosities of oil mixtures and oil fractions (Baltatu, 1984). The estimation gives the following results:

,~.

~

[i iil.~;

~ ~ ~\,

~1)"(p,T)

=

expík¡ + k4/T)[exp[po+2

+

+

~.

;~2)

OP"(k, + ~ + ~:)]- 10].

(11-40)

with k¡ = - 9.74602 k2 = 18.0834 k3 = - 4126.66 k, = 44.6055 ks = 0.976544 ~ = 81.8134 k7 = 15649.9

F¡

=

=

1)o(T) + 1)¡(T)p + Fl~1)'(p,T)

HTAN + 1

2 F

2

=

1- HTAN 2

.,

Table 11-1 Molar Composition of the North Sea Oll (Mixture 1) for which Viscosity Data are Shown in Table 11-3

Component

Mole %

N2 CO2 C, C2 C3 ic,

0.41 0.44 40.48

n-C.

ic, n-C,

Continuity between viscosities above and below the freezing point of methane is secured by introducing ~1)" as a fourth term in Equation 11-37: 1)(p,T)

Table 11-3 gives experimental viscosity results (measured as described in Chapter 3) for three North Sea oil mixtures whose compositions are given in Tables 11-1, 4-7, and 11-2. The results are presented graphically in Figures 11-1 through 11-3, in comparison with the results calculated using the method of Pedersen et al. (1984, 1987) and the method of Lohrenz et al. (1964). The latter results were obtained using densities calculated from the

1~'

+ F2~1)"(p,T)

(11-41)

C6 C7 Ca C9 ClO C" C'2 C'3 C,• C,S C'6 Cl?

c., C'9 C20+

Density (g/cm3) at 15°C, 1 atm

Molecular Weight

100 106 121 135 148 161 175 196 206 224 236 245 265 453

0.7294 0.7492 0.7697 0.7861 0.7919 0.8037 0.8191 0.8331 0.8359 0.8429 0.8400 0.8458 0.8575 0.9183

7.74 8.20 1.23 4.22 1.43 2.21 2.83 4.13 4.31 3.13 2.439 1.880 1.674 1.573 1.207 1.232 0.985 0.977 0.911 0.585 6.382

'\

18;;>-----;

182

Properties o] Oils and Natural Gases Viscosity Table 11-2 of the North Sea Oil (Mlxture 1) for which Viscosity Data are Shown in Table 11-3

Molar Composition

1.75

1.5

Component

Density (g/cm') at 15°C, 1 atm

Mole %

Nz COz C, Cz

Molecular Weight

0.33 0.19 35.42

C3 ¡-C,

n-C,

ic,

n-Cs C6 C,.

1.25

~ .::.

\

•

\

'"o

Experimental results Pedersen et al.

\ \

1.0

\

>, +'

3.36 0.90 0.69 0.26 0.26 0.14 0.72 57.73

183

.75

~

> .5

'" "

-----

Lohrenz et al. using SRK density

------

Lohrenz et al. using Peneloux density

"

-- --