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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)
Figure A-2. Conformal Mapping 2.
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:
Figure A-3. Conformal Mapping 3.
W = Z β (A-5)
That turns the figure into an “egg shape”:
Figure A-4. Conformal Mapping 4.
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)
Figure A-5. Conformal Mapping 5.
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.