Spe 106855

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Copyright 2007, Society of Petroleum Engineers

This paper was prepared for presentation at the 2007 SPE Latin American and CaribbeanPetroleum Engineering Conference held in Buenos Aires, Argentina, 15–18 April 2007.

This paper was selected for presentation by an SPE Program Committee following review ofinformation contained in an abstract submitted by the author(s). Contents of the paper, aspresented, have not been reviewed by the Society of Petroleum Engineers and are subject tocorrection by the author(s). The material, as presented, does not necessarily reflect anyposition of the Society of Petroleum Engineers, its officers, or members. Papers presented atSPE meetings are subject to publication review by Editorial Committees of the Society ofPetroleum Engineers. Electronic reproduction, distribution, or storage of any part of this paperfor commercial purposes without the written consent of the Society of Petroleum Engineers isprohibited. Permission to reproduce in print is restricted to an abstract of not more than300 words; illustrations may not be copied. The abstract must contain conspicuous

acknowledgment of where and by whom the paper was presented. Write Librarian, SPE, P.O.Box 833836, Richardson, Texas 75083-3836 U.S.A., fax 01-972-952-9435.

Abst ract

This paper compares the output of several available empirical

black oil model correlations against compositional model

results. In this process, the limitations of these models becameapparent.

Even acknowledging the imperfections of black model

implementation, it is possible to improve the quality of theoutputs by means of making the definitions consistent and

coherent across the prediction ranges.

A new method is outlined in order to extend the validity of the

models in predicting both reservoir and multiphase flow

simulations.This new method is presented here and will be extended in a

separated paper.

This paper compares the output of several available empirical

black oil model correlations against compositional modelresults. In this process, the limitations of these models became

apparent.Even acknowledging the imperfections of black model

implementation, it is possible to improve the quality of the

outputs by means of making the definitions consistent and

coherent across the prediction ranges.

A new method is outlined in order to extend the validity of the

models in predicting both reservoir and multiphase flowsimulations.

This new method is presented here and will be extended in a

separated paper.

Introduction

The behavior of black oil fluid is commonly inferred from two

PVT laboratory procedures: flash (or separator test) and

differential liberation. Oil formation volume factor and gassolution ratios are calculated as explained by McCain1. On

the other hand, given a particular EOS is possible to obtain

PVT fluid parameters by simulating the same laboratory

procedures or making direct flash calculations at any particular condition.

The traditional calculation method outlined in 1 can be

modified in a simple way to extend the validity of black oimodel correlations by accounting the dew point curve

Negative gas solution ratios indicate liquid vaporization, and

need not to be masked by any correction method. If we follow

definitions literally, Rs diminish towards dew point andreaches a constant negative minimum at dew point and inside

monophasic gas area. Oil formation volume factor can be

lower than unity and in fact should be zero at dew point.

As modern calculations take into account both reservoir andmultiphase wellbore and pipeline calculations, is of paramoun

importance to be able to accurately predict fluid properties in a

wider range of pressure and temperature conditions.The first objective of this paper is to make apparent the

limitations of current PVT laboratory calculations and propose

a revision.

A second objective is to present black oil model standard

correlations phase diagrams together with phase diagramscalculated with EOS and acknowledge the differences and

limitations of empirical correlations.

The third objective is to outline a new mathematical method toimprove black oil correlations.

Definitions

The following definitions extracted from Dake2 will be taken

as references:

- Rs. The solution (or dissolved) gas oil ratio, which is the

number of standard cubic feet of gas which will dissolve in

one stock tank barrel of oil when both are taken down tothe reservoir at the prevailing reservoir pressure and

temperature (units − scf. gas/stb oil).

- Bo. The oil formation volume factor, is the volume in

barrels occupied in the reservoir, at the prevailing pressure

and temperature, by one stock tank barrel of oil plus its

dissolved gas (units – rb (oil + dissolved gas)/stb oil).

- Bg. The gas formation volume factor, which is the volume

in barrels that one standard cubic foot of gas will occupy

as free gas in the reservoir at the prevailing reservoir

pressure and temperature (units − rb free gas/scf gas).

These parameters enable converting fluid volumes at any

conditions to volumes at standard conditions.

SPE 106855

Phase Envelopes From Black-Oil ModelsMiguel H. Schindler, SPE, DeltaP

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Phase Diagrams from EOS

Starting from an equation of state, it is possible to

determine the vapor liquid equilibrium at any condition by

calculating fugacity coefficients and applying flash methods asgiven by Rachford-Rice3.

The output of this method is expressed in molar fractions

of the phases involved, and phase diagrams usually indicate

molar Vapor or Liquid Fractions as in Figure 1; this phaseenvelope of a real black-oil fluid was generated using a

modified Peng-Robinson EOS4.

0 200 400 600 800

Temperatur e (°F)

Bubble VoF=0%

Dew VoF=100%

Retrograde

VoF=25%

VoF=50%

VoF=75%

0

600

1200

1800

2400

3000

Pressure(psia)

Black Oil PVTMolar Fraction Phase Envelope

Figure 1. Molar Fraction Phase Envelope.

In order to be able to compare to standard black oil model

outputs, we need to be able to calculate volumetric fractions.

As molar densities are available when using EOS, volumetric

flash is readily achievable, as shown in Figure 2.

( , )gas mgasV P T VoF ρ = (1)

( )( , ) 1oil moilV P T VoF ρ = − (2)

( , )

( , ) ( , )

gas

gas oil

V P T VoFv

V P T V P T =

+ (3)

0 200 400 600 800

Temperatur e (°F)

Bubble VoFv=0%

Dew VoFv=100%

Retrograde

VoFv=25%

VoFv=50%

VoFv=75%

0

600

1200

1800

2400

3000

Pressure(psia)

Black Oil PVTVolume Fraction Phase Envelope

Figure 2. Volumetric Fraction Phase Envelope.

Phase Diagrams from Black Oil Correlations

Considering definitions from above, volume of free gas andoil, at any pressure and temperature is calculated as:

( )( , )gasV P T GOR Rs Bg= − (4)

( , )oilV P T Bo= (5)

Knowing Eq. (3), we can now generate a phase envelope

plot from any black oil model correlation. Replacing Eqs. (4

and (5) in Eq. (3), we obtain:

( )

( )

GOR Rs BgVoFv

GOR Rs Bg Bo

−=

− + (6)

In Figures 3 to 8 we can see the phase envelope generated

with the compositional model superimposed with envelopes

generated with standard correlations.

0 300 600 900 1200

Temperatur e (°F)

Black Oil PVTRs & Bo from Standing Correlations

0

800

1600

2400

3200

4000

Pressure(psia)

Bubble VoFv=0%

Dew VoFv=100%

VoFv=25%

VoFv 50%

VoFv=75%

Figure 3. Compositional vs Standing5 Correlation.

0 300 600 900 1200

Temperatur e (°F)

Black Oil PVTRs & Bo from Vazquez & Beggs

0

800

1600

2400

3200

4000

Pressure(psia)

Bubble VoFv=0%

Dew VoFv=100%

VoFv=25%

VoFv 50%

VoFv=75%

Figure 4. Compositional vs Vazquez-Beggs6 Correlation.

0 300 600 900 1200

Temperatur e (°F)

Black Oil PVTRs from Lasater

-100

750

1600

2450

3300

4150

Pressure(psia)

Bubble VoFv=0%

Dew VoFv=100%

VoFv=25%

VoFv 50%

VoFv=75%

Figure 5. Composi tional vs Lasater 7 Correlation.

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0 300 600 900 1200

Temperatur e (°F)

Black Oil PVTRs & Bo from GlasO

-100

750

1600

2450

3300

4150

Pressure(psia)

Bubble VoFv=0%

Dew VoFv=100%

VoFv=25%

VoFv 50%

VoFv=75%

Figure 6. Composi tional vs GlasO8 Correlation.

0 300 600 900 1200

Temperatur e (°F)

Black Oil PVTRs & Bo from Petrosky-Farshad

-100

750

1600

2450

3300

4150

Pressure(psia)

Bubble VoFv=0%

Dew VoFv=100%

VoFv=25%

VoFv 50%

VoFv=75%

Figure 7. Compositional vs Petrosky-Farshad9 Correlation.

0 300 600 900 1200

Temperatur e (°F)

Black Oil PVTRs & Bo from Dindoruk-Christman

0

800

1600

2400

3200

4000

Pressure(psia)

Bubble VoFv=0%

Dew VoFv=100%

VoFv=25%

VoFv 50%

VoFv=75%

Figure 8. Compositi onal vs Dindoruk -Christman10

Correlation.

Main Discrepancies

Any of the empirical models plotted above can be adjusted to

reproduce PVT experiments at a given temperature as shown

in Figure 9.However, it is evident that each of the models has a limited

range of validity.

0 1000 2000 3000 4000

Pressure (psia)

PVT Differential Liberation at 60 degCVazquez & Beggs Black Oil Mo del

1.000

1.070

1.140

1.210

1.280

1.350

Bo#0

0.600

0.680

0.760

0.840

0.920

1.000

DensityOil#0(g/cm3)

Bo Real

DensityOil Real (kg/m3)

Rs Real (ft3/bbl)

Figure 9. Laboratory Data vs. Correlation.

Looking at the envelopes presented, we see that:

- None can predict the curvature of the bubble point ahigher temperatures, although Vazquez & Beggs and

GlasO show a less divergent behavior than the others.

- No correlation takes into account the dew point line.

- There is no possibility of predicting critical points.

PVT correlations are needed both for reservoir andmultiphase flow correlations. As pressure and temperature

changes occur during multiphase flow there is a strong need to

use a correlation able to predict the PVT behavior in a broaderange of conditions.

PVT Parameters Calculated From EOS

Having implemented EOS3 flash calculations, it’s possible to

utilize black oil model definitions to calculate Bo and Rs.

0 750 1500 2250 3000

Pressure (psia)

Bo Calculated From EOSModified Peng Robinson

0.00

0.60

1.20

1.80

2.40

3.00

Bo#01

Bo

Bubble Point

Dew Point

Figure 10. Bo calculated from Modified PREOS.

Bo. At the dew point, by definition, the volume of oil is nullso if we need the correlation to predict the dew point, we

should allow Bo to be null at dew point pressure.

( , ) ( , )

( ) ( )

(1 ) /( , )

( ) (1 ) /

P T moil P T oil

o

oil sc moil sc

VoF V P T B

V sc VoF

ρ

ρ

−= =

− (7)

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Figure 10 shows Bo curve calculated from EOS.

Effectively at dew point Bo is null.

Rs. In the same way is possible to apply black oil model

definitions to calculate solution gas from EOS. In this case,

knowing that the black oil model does not handle gas

composition changes, we present a way to construct the Rs

curve is the following:

( , )

( )

gas P T

g

oil sc

V GOR

B Rs

V

−

= (8)

( ) ( ) ( , ) ( , )

( ) ( )

/ /( )

(1 ) /

sc mgas sc P T mgas P T g

sc moil sc

VoF VoF B Rs

VoF

ρ ρ

ρ

−=

− (9)

0 750 1500 2250 3000

Pressure (psia)

Rs Calculated From EOS

Modified Peng Robinso n

Bubble Point

Dew Point

-110

-60

-10

40

90

140

Rs#01

(1/1)

Rs

Figure 11. Rs calculated from Modified PREOS.

It is important to note that being coherent with black oil

model definitions means to enable Rs to be negative. This

makes physical sense considering that in certain conditions,

when taken from standard to PVT conditions the liquidfraction will not dissolve gas but will release it. The maximum

amount of gas that will be released (dew point conditions) can

be calculated considering that all the mass contained in a barrel of oil will be converted to gas, Eq. (10).

90 180 270 360 450

Temperature (°F)

Rs Behavior near Dew Point

0

10

20

30

40

50

DewPoint(psia)

DewPoint

Negative Rs(SC) A

Figure 12. Rs calculated from Modified PREOS.

( )

min

( )

oil sc

gas sc

Rsδ

δ = − (10)

Figure 12 shows Rs curve calculated from EOS

Effectively Rs turns to be negative approaching the dew poin

curve. As fluid conditions change from standard conditions

(SC) to point A, gas is released from liquid fraction. Thissituation is correctly represented as negative Rs.

The free volume of gas will be higher then than the volumeoccupied by free gas at SC when taken to this conditions.

( , ) ( )gasV P T GOR Rs Bg= − ⋅ (11)

Most empirical correlations artificially force Rs to be nul

at 0 pressure or dew point pressure; we conclude that acoherent black oil model should enable Rs to become negative

under the conditions related before.

In Figure 13 showing Rs correlation from Lasater it is

apparent that Rs could have been allowed to descend belowzero.

0 700 1400 2100 2800

Pressure (psia)

Rs calculated fromLasater correlation

-20

110

240

370

500

630

Rs(ft3/bbl)

Temp.PVT= 20 (°C)Temp.PVT= 40 (°C)

Temp.PVT= 80 (°C)Temp.PVT= 100 (°C)

Temp.PVT= 60 (°C)

Figure 13. Rs calculated from Lasater 7 correlation.

Conclusions

Without need of major modifications is possible improve

significantly the performance of black oil models:

- Enabling the models to accurately reflect PVT flash

experiments.

- Oil volume factor should be null at dew points.

- Rs solution gas ratio should became negative when

approaching dew point pressure and kept constant below

dew point pressures.

- Constant saturation lines should curve to converge at

Critical Point.

A new correlation that takes into account these points is

outlined in the Appendix. The details are presented in adifferent paper.

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Acknowledgments

I want to thank Gastón Fondevila for contributing in multiple

technical and presentation aspects of this paper and adjusting

the correlations. I like to thank Javier Schindler and MatíasMachado for implementing and coding standard PVT

correlations and assisted tuning flash calculations using

compositional models.

I would also like to thank Marcelo Crotti from INLAB for hisconstructive and generous help in reviewing and discussing

this work.

Nomenclature

VoF = molar vapor fraction

VoFv = volumetric vapor fraction

ρ moil = molar density of oil

ρ mgas = molar density of gas

Bo = oil volume factor

Bg = gas volume factor

Rs = solution gas oil ratio

GOR = gas oil ratio at scδ oil = density of oil

δ gas = density of gas

References

1. William D. McCain, Jr.: “Analysis of Black Oil PVT ReportsRevisited”, SPE 77386

2. L.P. Dake: “Fundamentals of Reservoir Engineering”, Elsevier ,Developments in Petroleum Science No. 8.

3. Rachford, H.H. Jr., and Rice, J.D., JPT 4, 10, sec. 1, 19; sec. 2, 3(1952).

4. Peng, D.Y., and Robinson, D.B.: "A new two-constant equation

of state". Ind. and Eng. Chem. Fund. 15, 59-64 (1976).5. Standing, M.B., “A Pressure-Volum-Temperature Correlation

for Mixtures of California Oils and Gases”, Drill. And Prod.Prac., API (1947), 275-87.

6. Vazquez, M.E. y Beggs, H.D. (1980), “Correlations for FluidPhysical Property Prediction”, JPT, 5, 968-970.

7. Lasater, J.A.: “Bubble Point Pressure Correlation”, Trans. AIME

(1958) 231, 379, SPE 957.8. GlasO,: “Generalized Pressure-Volume-Temperature

Correlations”, JPT (May, 1980), pp. 785-795, SPE 8016.9. Petrosly, G.E., Jr. and Farshad, F.F.: “PVT Correlations for Gulf

of Mexico Crude Oils”, SPE 26644.10. Dindoruk, B., and Christman, P.G.,: “PVT Properties and

Viscosity Correlations for Gulf of Mexico Oils”, SPE 89030.

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Appendix

A New Black Oil Model Correlation Using ConformalMapping Techniques

Introduction

Following the discussions in this paper, a need arises for a

black oil correlation that could match fluid behavior in a

broader range of conditions.The objective of this correlation will be to reproduce the

phase behavior as shown in Figure 2. The approach selected

consists in exploring if the use of conformal mapping couldassist in improving model performance.

Why Conformal Mapping?

Without attempting to provide a complete mathematical or

physical demonstration, it seems that saturation curves

(constant volumetric vapor fraction curves) never crossexcepting at critical point and at absolute Zero. Every pressure

and temperature point corresponds to only one volume

fraction.

It also seems that is possible to think about a potentialfunction with a VoFv value of 1 at dew point and with value 0

at bubble point. Constant saturation lines will be calculated as

equipotentials.A convenient set of scale factors could translate imaginary

coordinates into temperature and pressure.

Going from potential to Rs

After defining the set of conformal mappings that could obtainthe desired shape of the phase envelope, a correlation is

optimized using non linear regression to relate transformation

parameters to black oil input parameters: Oil Density(sc), Gas

SG and GOR.The inverse transformation is calculated so for any

combination of Temperature and Pressure we obtain a single

value of VoFv.

( )

( )

1

1

BoGOR VoFv VoFv

Bg Rs

VoFv

− −

=−

(A-1)

Bo and Bg are obtained using convenient standardcorrelations.

Conformal Mapping Details

At this point we show a procedure to transform the upper semi

plane into a “phase envelope” shape.For simplicity some rotation and translation functions

needed between transformations are not included.

Without doubt different procedures could be used to reachsame or better results.

The potential value at any point of the semi plane can be

calculated as:

( )( )0arctan y x x

VoFvπ

−= (A-2)

Figure A-1. Conformal Mapping 1.

Applying Eq. (A-3), we can apply the complete upper

semi plane into a circle of radii = 1, as shown in Figure A-2

looking at this figure we realize that CP was displaced anangle α.

( )

( )

i Z iW e

Z i

α −=

+ (A-3)


We have now generated this “balloon” with known valuesof VoFv potential in it’s interior. The next step will be to

deform it to make it match phase envelope’s shapes.

In order to be able to fine tune the correlation, was needed

to include two intermediate functions to deform the circle

shape.The first of these intermediate functions was included to

make the dew point line flat at low pressures and the second

one was to deform the circle into an “ellipse like” or “egg

shape” before applying the arccoth function.

-1.00 -0.50 0.00 0.50 1.00

x

0.00

0.50

1.00

1.50

2.00

2.50

y

BubbleVoFv=0

DewVoFv=1

VoFv=0. 25

VoFv= 0.5

VoFv=0.75

CP

-4 -2 0 2 4

x

-1

0

1

2

3

4

y

Bubble VoFv=0

Dew VoFv=1

VoFv= .25

VoFv= 0.5

VoFv=0.75

α

CP

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The intermediate functions are:

W = Z 2 (A-4)

Whose output is shown in Figure A-3:


W = Z β (A-5)

That turns the figure into an “egg shape”:


A Joukowsky shape, used for airfoil design seemed

promising. But that transformation deals with the outside of

the circle but isn’t continuous in its interior.The idea is to apply the origin into Absolute Zero

temperature (-549.57 ºF, 0 psia) and to apply CP into critical

point coordinates. The function preferred was:

coth( )W arc Z π = (A-6)


This function is equivalent to squeezing the balloon under

a wheel or cylinder as is shown in Figure A-5.

Now applying convenient scale factor is possible to

convert X coordinate into temperature and Y coordinate intoPressure.

PVT Matching

Using Figure 2 as reference we can now optimize the new

correlation parameters in order to match the complete phaseenvelope.

Figure A-6. Conformal Mapping vs Compositional Model.

At this point we can appreciate the potential of this type of

correlation compared to the ones presented in Figures 3 to 8.It is possible to correctly predict both the bubble point

curve and dew point curves. Further investigation will indicate

if the use of pseudo pressures and temperatures can improvethe matching.

As in Figure 9, is now possible to generate Rs, Bo and

Density PVT curves at any temperature:

-700 -350 0 350 700

Temperature (°F)

-250

450

1150

1850

2550

3250

Pressure(psia)

Black Oil PVT Phase Envelope From Conformal Mapping

Bubble VoFv=0%

Dew VoFv=100%

VoFv=25%

VoFv=50%

VoFv=75%

0.00 0.50 1.00 1.50 2.00 2.50x

0.00

0.25

0.50

0.75

1.00

1.25

y

Bubble VoFv=0

Dew VoFv=1

VoFv= 0.25

VoFv= 0.5

VoFv=0.75

-0.80 0.00 0.80 1.60 2.40

X

0.00

0.50

1.00

1.50

2.00

2.50

y

BubbleVoFv=0

VoFv=0.25

VoFv=0.5

VoFv=0.75

Dew VoFv=1

-0.60 -0.30 0.00 0.30 0.60

x

-0.10

0.15

0.40

0.65

0.90

1.15

y

Bubble

VoFv=0

Dew VoFv=1

VoFv=0. 25

VoFv=0. 5

VoFv=0. 75

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0 900 1800 2700 3600

Pressure (psia)

1.000

1.080

1.160

1.240

1.320

1.400

Bo#0

0

150

300

450

600

750

Rs#0

(ft3/bbl)

Bo Real

DensityOil Real (kg/m3)

Rs Real (ft3/bbl)

PVT Differential Liberatio n at 60 degCConform al Mapping Black Oil Model

Figure A-7. Conformal Mapping PVT Curves.

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