EXPERIMENTAL AND THEORETICAL DETERMINATION OF HEAVY
OIL VISCOSITY UNDER RESERVOIR CONDITIONS
FINAL PROGRESS REPORT
PERIOD: OCT 1999-MAY 2003
CONTRACT NUMBER: DE-FG26-99FT40615
PROJECT START DATE:
October 1999
PROJECT DURATION:
October 1999 - May 2003
TOTAL FUNDING REQUESTED:
$ 199,320
TECHNICAL POINTS OF CONTACT:
Jorge Gabitto
Prairie View A&M State University
Department of Chemical Engineering
Prairie View, TX 77429
TELE:(936) 857-2427
FAX: (936) 857-4540
EMAIL:[email protected]
Maria Barrufet
Texas A&M University
Petroleum Engineering
Department
College Station TX, 77204
TELE:(979) 845-0314
FAX:(979) 845-0325
EMAIL:barrufet@spindletop.
tamu.edu
EXPERIMENTAL AND THEORETICAL DETERMINATION OF HEAVY
OIL VISCOSITY UNDER RESERVOIR CONDITIONS
DISCLAIMER
This report was prepared as an account of work sponsored by an agency of the United
States Government. Neither the United States Government nor any agency thereof, nor
any of their employees, makes any warranty, express or implied, or assumes any legal
liability or responsibility for the accuracy, completeness, or usefulness of any
information, apparatus, product, or process disclosed, or represents that its use would not
infringe privately owned rights. Reference herein to any specific commercial product,
process, or service by trade name, trademark, manufacturer, or otherwise does not
necessarily constitute or imply its endorsement, recommendation, or favoring by the
United States Government or any agency thereof. The views and opinions of authors
expressed herein do not necessarily state or reflect those of the United States Government
or any agency thereof.
1
EXPERIMENTAL AND THEORETICAL DETERMINATION OF HEAVY
OIL VISCOSITY UNDER RESERVOIR CONDITIONS
ABSTRACT
The USA deposits of heavy oils and tar sands contain significant energy reserves.
Thermal methods, particularly steam drive and steam soak, are used to recover heavy oils
and bitumen. Thermal methods rely on several displacement mechanisms to recover oil,
but the most important is the reduction of crude viscosity with increasing temperature.
The main objective of this research is to propose a simple procedure to predict heavy
oil viscosity at reservoir conditions as a function of easily determined physical properties.
This procedure will avoid costly experimental testing and reduce uncertainty in designing
thermal recovery processes.
First, we reviewed critically the existing literature choosing the most promising
models for viscosity determination. Then, we modified an existing viscosity correlation,
Pedersen et al.1, based on the corresponding states principle in order to fit more than two
thousand commercial viscosity data. We collected data for compositional and black oil
samples (absence of compositional data). The data were screened for inconsistencies
resulting from experimental error. A procedure based on the monotonic increase or
decrease of key variables was implemented to carry out the screening process. The
modified equation was used to calculate the viscosity of several oil samples where
compositional data were available. Finally, a simple procedure was proposed to calculate
black oil viscosity from common experimental information such as, boiling point, API
gravity and molecular weight.
2
EXPERIMENTAL AND THEORETICAL DETERMINATION OF HEAVY
OIL VISCOSITY UNDER RESERVOIR CONDITIONS
TABLE OF CONTENTS
DISCLAIMER
1
ABSTRACT
2
TABLE OF CONTENTS
3
STATEMENT OF WORK
5
TECHNICAL DESCRIPTION
6
INTRODUCTION
6
OBJECTIVES
7
CRITICAL LITERATURE REVIEW
7
Pure Components and Mixtures of Pure Components
7
Semi-theoretical Methods
7
Empirical methods
10
Crude Oil Fractions
13
Semi-theoretical Methods
13
Empirical methods
14
MODIFICATION OF PEDERSEN’S MODEL
15
Model Development
15
Heavy Oil Fraction Characterization
17
True Boiling Point Tests (TBP Tests)
17
Gas Chromatography (GC)
18
Thermodynamic Properties Prediction
19
Whitson’s Lumping Scheme
21
Compositional Oil Samples
22
Results
23
PROCEDURE TO SCREEN CRUDE OIL VISCOSITY DATA
24
Introduction
24
Viscosity Correlations
26
3
Reservoir Fluid Studies for Reservoir Engineering
27
Data Preparation and Data Screening Routine
28
Data Screening Results
30
MODIFICATION OF PEDERSEN’S MODEL FOR BLACK OIL SAMPLES
31
Introduction
31
Viscosity Correlations
31
Model Development
33
Results
36
CONCLUSIONS
38
NOMENCLATURE
39
Greek Letters
39
Subscripts
40
REFERENCES
41
TABLES AND FIGURES
46
APPENDIX
60
4
STATEMENT OF WORK
Under this Statement of Work (SOW), Dr. Jorge Gabitto from the Chemical
Engineering Department at Prairie View A&M University (PVAMU), Dr. Maria Barrufet
from the Petroleum Engineering Department at Texas A&M University (TAMU) and Dr.
Rebecca Bryant from Bio-Engineering International Inc. (BEI) have conducted research
and training in the area of transport and thermodynamic properties determination for
heavy oils. Chevron Oil Company has provided consulting and some heavy oil samples
used in this project.
A research project was proposed to develop theoretical models, computer algorithms,
and measure experimentally transport and thermodynamic properties of heavy oils.
Model evaluation was an important part of the project.
This research involved training of graduate and undergraduate students in state of the
art techniques. Technology transfer of the results generated by the project has been
achieved through Dr. Bryant’s efforts and publications in refereed journals.
Dr. Gabitto acted as coordinator of the research team and he was responsible by most
of the theoretical program. Dr. Barrufet was Co-Principal Investigator. Dr. Bryant
advised the research team, and she was responsible for transferring the project’s findings
to small independent producers.
5
TECHNICAL DESCRIPTION
INTRODUCTION
The viscosity of heavy oils is a critical property in predicting oil recovery. Viscosity
reduction and thermal expansion are the key properties to increase productivity of heavy
oils. Thermal methods are pivotal in successfully producing oils with an API gravity of
less than 20 degrees. These recovery methods may involve steam, hot water injection,
and in-situ combustion2. For improving heavy oil recovery, steam injection has proven to
be the premier approach for both stimulating producing wells and displacing oil in the
reservoir. The amount of high viscosity oil produced by steam methods is increasing
annually throughout the world3.
Modern
reservoir
engineering
practices
require
accurate
information
of
thermodynamic and transport fluid properties together with reservoir rock properties to
perform material balance calculations.
These calculations lead to the determination
(estimation) of the initial hydrocarbons (oil and gas) in-place, the future reservoir
performance, optimal exploration and production schemes, and the ultimate hydrocarbon
recovery. The technical and economic viability of steam flooding processes have been
established by laboratory and field studies of rock formations and crude oils3. Extensive
knowledge of fluid properties is required to properly develop a steam flooding strategy.
Reservoir simulators are routinely used to predict and optimize oil recovery from oil
fields. These simulators require as input properties of the reservoir fluids as a function of
pressure, temperature and composition.
The accuracy of the fluid properties can
decisively affect the results of the simulation. Among the required fluid properties are:
phase densities, phase viscosities, formation volume factors (Bo), and dissolved gas-oil
ratios. The physicochemical properties of the reservoir fluids are a function of the fluids’
composition. These compositions can be determined by experimental analysis such as,
true boiling point essays and gas chromatography.
In many practical cases no
compositional information is present. A practical method to predict reservoir fluids’
viscosities should be able to calculate viscosity of compositional and black oils.
6
OBJECTIVES
The objectives of this research program are to determine viscosity and other required
thermodynamic properties of heavy crude oil mixtures at various temperatures at pressures
and temperatures characteristic of steam flooding processes.
This research program has been divided in several parts. The first part involves a
critical literature review followed by development of a model based on the corresponding
states theory. A modification of Pedersen et al.1 viscosity correlation for compositional
and black oils has been developed. In order to validate the model presented in this work
a screening process for the experimental data to be used is also presented. Finally,
selected experimental data are used to qualify the accuracy of the proposed viscosity
equation both for compositional and black oils.
CRITICAL LITERATURE REVIEW
Viscosity plays an important role in reservoir simulations as well as in determining
the structure of liquids.
Several models for the viscosity of pure components and
mixtures are available in literature, summarized recently by Monnery at. al.4 (1995) and
Mehrotra et al.5.(1996). Good reviews have also been presented by Reid et al.6,7(1977,
1987), Stephan and Lucas8 (1979) and Viswanath and Natarajan9(1989).
However,
petroleum fluids were covered only by Mehrotra et al.5 (1996). Petroleum fluids are
complex fluids, normally of undefined composition that require a characterization
procedure to obtain relevant properties. The available methods can be grouped in two
categories, Semi-theoretical and Empirical methods.
Semi-theoretical methods are
derived from a theoretical framework, but involve parameters experimentally determined.
Empirical methods include a wide variety of equations used throughout the industry
involving constants calculated from experimental data. Characterization procedures for
heavy oil fractions will be presented in a separate section. The next section reviews the
results for pure components and mixtures using semi-theoretical and empirical methods.
Pure Components and Mixtures of Pure Components
Semi-theoretical Methods
7
Semi-theoretical models are based on the principle of corresponding states or can be
considered applied statistical mechanics models such as, the reaction rate theory, hard
sphere theory, square well theory or their modifications. These methods predict viscosity
as a function of temperature and density (volume), requiring a density prediction model
coupled with the viscosity model.
According to the thermodynamic principle of corresponding states, a dimensionless
property of one substance is equal to that of another (reference) substance when both are
Ely and Hanley10 (1981) proposed the
evaluated at the same reduced conditions.
following extended corresponding states model:
µi(ρ,T) = µo( ρ h i,o , T
1/ 2
f i,o
) (M i / M o)
-2/3
1/2
hi,o f i,o
(1)
h i ,o = h i ,o (ρc,i / ρc,o) φi,o
(2)
f i ,o = (Tc,i / Tc,o) θi,o
(3)
where θi,o, and ϕi,o are shape factors depending on the chemical components. Viscosity
calculations require correlations for a reference fluid viscosity and density along with
critical properties values, acentric factor and molar mass. Methane was selected as a
reference fluid because of the availability of highly accurate data. A problem using
methane is its high freezing point (Tr = 0.48), which is well above the reduced
temperatures of other fluids in the liquid state. In order to overcome this difficulty the
authors extrapolated the density correlation for methane and added an empirical
correlation for non-correspondence and extended the viscosity correlation of Henley et
al.11 (1975). Results are satisfactory for n-paraffins with average absolute deviations
(AADs) typically within 5-10%, but are poor for isomeric paraffins and naphtenes with
AADs as high as 55% (Monnery et al.12 , 1991).
Ely13 (1982) modified the Ely-Henley model to partially correct for noncorrespondence between the reference fluid and pure high molar mass fluids, and for size
and mass differences in mixtures. The non-correspondence was addressed by changing
the reference fluid from methane to propane, since propane has the lowest reduced triple
point among paraffins. The predictions calculated using this new model were similar to
those from the Ely-Henley model. In addition to using a better reference fluid Ely14
(1984) developed simpler shape factor correlations.
8
Haile et al.15 (1976), Hwang and Whiting16 (1987) and Monnery et al.12 (1991)
attempted to improve the method by using viscosity as a conformal equation and/or
making empirical modifications to shape factors.
16
method of Hwang and Whiting
For 38 compounds, the modified
(1987) showed significant improvement for branched
alkanes, naphtenes, some aromatics and various polar and associating chemical
compounds with overall AADs of 5.3%. Using general correlations Monnery et al.12
(1991) predicted viscosities of 46 common hydrocarbons with an AAD of 6%.
Pedersen et al.1 (1984) proposed a similar approach for hydrocarbon and crude oil
viscosities:
µx(P,T) = ( T c, x / T c, o )-1/6 ( P c, x / P c, o )2/3 (M i / M o)1 / 2 αTG, x / αTG,o µo( P*o , T*o ) (4),
and
*
To = (T c, x / T c, o ) (αTG,o / αTG, x )
(5)
*
Po = (P c, x / P c, o
(6),
)
(αTG,o / αTG, x )
where αTG is the Tham-Gubbins17 (1971) rotational coupling coefficient.
According to Pedersen et al.1 the problems associated with representing poly-disperse
mixtures (such as crude oils) are associated with the computation of average molar
masses. Their results indicated that larger molecules should make a greater contribution
to viscosity than the smaller ones. The mixture molar mass was calculated empirically
as,
Mmix = Mn + b1 (Mw - Mn)
(7),
where b1 is an empirical constant obtained by fitting experimental data, Mn is the mass
fraction averaged molecular weight and Mw is the molecular weighted averaged
molecular weight. The Tham-Gubbins17 rotational coupling coefficient ( αTG ) was
determined from the molar mass and reduced density.
The mixture viscosity was
calculated from equation (4) with the mixing rules provided for pseudo-critical
properties.
Pedersen and Fredenslund18 (1986) extended Pedersen et al.1 method to mixtures with
Tr below 0.4 (below methane freezing point) by modifying the equations for Mmix and
αTG .
9
Teja et al.19 (1981) modified Lee-Kessler20 (1975) three-parameters corresponding
states method such that a simple reference fluid was not necessarily retained as one of the
references, resulting in,
Z = Zr1 + [(ω - ωr1)/(ωr2 - ωr1)] (Zr2 - Zr1)
(8),
where ω is the acentric factor factor of a single reference fluid made-up of spherical
molecules; ωri , and Zri are the acentric factor and compressibility factor of a nonspherical fluid. In this case the subscripts r1 and r2 refer to two fluids made-up of nonspherical molecules similar to pure compounds of interest or to the main constituents of
the mixture. They applied this approach to viscosity,
ln (µ ξTR)r = ln (µ ξTR)r1 + [(ω - ωr1)/(ωr2 - ωr1)] (ln (µ ξTR)r2 - ln (µ ξTR)r1)
(9),
where ξTR = V c2 / 3 / ( M T c )(1 / 2 ) . They tested the method for 6 non-polar + non-polar
mixtures with the two components comprising the binary as the reference components
and a fitted interaction parameter. The method correlated experimental data with an
AAD of 0.7%.
Teja and Thurner21 (1986) restated the Teja-Rice22 (1981) viscosity method in terms
of Pc instead of Vc. They adopted the mixing rules of Wong et al.23 (1984) with
essentially the same results.
Aasberg-Petersen et al.24 (1991) proposed a method based on the Teja-Rice22 method
with the reducing parameter in terms of critical pressure and molar mass as the third
parameter instead of molar mass. The method was tested for high pressures up to 70
MPa. The AAD was 7.4% for several binary mixtures and 6.4% for the crude oil data of
Pedersen et al.(1984)1.
Empirical methods
The Andrade25 (1934) equation, first proposed by de Guzman26 (1913), is given by,
ln µ = A + B/T
(10).
For many liquids equation (10) has been applied from the freezing to the boiling
points.
It does not include the effect of pressure, which has resulted in several
modifications. A third parameter has added to obtain the Vogel27 (1921) equation,
10
ln µ = A + B/(T + C)
(11).
Values for A, B and C have been published28 for liquid hydrocarbons within given
temperatures ranges. Several methods have been published to generalize the values of
the constants in order to give predictive capabilities to equation (11). Several authors,
Thomas, Joback, Orrick and Erbar (Reid et. al.6, 7), used group contributions methods to
calculate the values of A, B and C. Another approach is to calculate the values of
equation (11) constants by fitting experimental data for a large number of organic
compounds. Orrick and Erbar reported an overall AAD of 18% for 188 organic liquids.
van Velzen29 (1972) tested their method using 314 liquids and reported AADs of 15% or
less for 272 of those. Reid et al.7 (1987) tested the Orrick-Erbar and van Velzen et al.
methods with data for 35 compounds with AADs of 14.8 and 10.8%, respectively.
Allan and Teja30 (1991) proposed to calculate the constants in equation (11) as a
function of carbon number for pure n-alkanes from C2 to C20. The regressed effective
carbon numbers (ECN) for 50 hydrocarbons based on values of liquid viscosity for one
reference substance. They reported an AAD of 2.3%. The method was extended to
mixtures using a simple mixing rule.
However, Gregory31 (1992) showed that the
method predict incorrectly the change of viscosity with temperature for ECNs above 22.
Orbey and Sandler32 (1993) proposed the following equation for liquid hydrocarbon
viscosity,
ln µ/µref = k [ -1.6866 + 1.40010 (Tb/T) + 0.2406 (Tb/T)2]
(12).
where µref and k are parameters determined from experimental data. Equation (12)
correlated the data of 50 hydrocarbons with an AAD of 1.3%. Regressed parameters
were used in the computation of the viscosity values. The authors extended their method
to correlate high-pressure viscosities by introducing a pressure dependent constant.
They also extended their method to mixtures of alkanes by using two different
approaches, a mixture equation and a one fluid model for calculation of the mixture
boiling point. Both approaches yielded similar results, giving an overall AAD of 2.4%.
Another similar approach to the Andrade equations is the ASTM33 (1981) or
Walther34 (1931) equation.
log log (µ + 0.7) = b1 + b2 log (T)
(13).
11
The use of a double log in equation (13) should caution about the possibility of big
deviations. It is well know the property of the log function to "hide" deviations.
Mehrotra35 (1991) fitted experimental data for 273 pure heavy hydrocarbons from
API Research Project 4236 (1996) to equation (13), with the constants changed from 0.7
to 0.8 to extend the range of the equation. They used regressed values of b1 and b2 to
calculate viscosity values with AADs ranging from 0.8% for n-paraffins and olefins to
1.4% for non-fused aromatics.
They used a linear correlation between the two
parameters to derive a single parameter equation,
log (µ + 0.7) = Θ log (Φ T)b
(14).
where b = b2. A regression of experimental values yielded best values of Θ and Φ of 100
and 0.01, respectively. The authors regressed optimum values of b for each chemical
compound, and the overall AADs ranged from 2.3% for branched paraffins and olefins to
10.6% for fused-ring naphtenes. Finally, b was generalized for each hydrocarbon family
as a function of molar mass and boiling point (at 10 mmHg).
Mehrotra37 (1991) correlated experimental data for 89 light and medium
hydrocarbons using regressed values of b1 and b2. The same author using equation (14)
and regressed values for b calculated viscosity values ranging from an overall AAD of
6.6% for aromatics to 12.5% for n-alkylcyclopentanes. He did not recommend his
equation for light hydrocarbons at low temperatures though.
Mehrotra38 (1994) combined the ECN approach of Allan and Teja30 (1991) with
equation (14) to provide a simple relationship between ECN and parameter b, which can
be extrapolated reliably to ECN bigger than 22. Chabra39 (1992) proposed a binary
mixing rule without adjustable parameters based on equation (14). He reported an overall
AAD of 7% for 57 different polar and nonpolar compounds. His results were correlated
with an AAD of 6%.
A different approach is given by the viscosity equation of state method (EOS). This
approach is based on the similarity between the P-V-T and P-µ-T surfaces plotted in a 3D space. The EOS method yields explicit equations as a function of T and P. Lawal40
(1986) used a cubic equation of state to propose a viscosity equation with reversed places
for T and P, and viscosity replacing the V.
temperature dependent parameters.
12
The EOS involves 4 constants and 2
Heckenberger and Stephan41, 42 (1990, 1991) also proposed a viscosity EOS based on
the fact that a residual transport property (TP) surface P-∆TP-T corresponded better than
the P-ρ-T surface. Their results however, ranged from 4.7% for alkanes up to C8 to
32.9% for some organic compounds.
The viscosity of liquid mixtures is calculated mostly using a single fluid approach,
and applying mixing rules to the parameters or correlated with mixture-viscosity
equations. The simplest mixture-viscosity equation is additive in form,
f( µ m ) = ∑ x i f( µ i)
(14).
where f( µ ) is the viscosity function normally linear, hyperbolic or logarithmic in form.
A common equation used successfully for liquid hydrocarbons is,
3
)
µ m = (∑ xi µ1/3
i
(15),
which gives reasonable results for mixtures of similar components.
Irving43 presented a review of various mixture equations and tested their accuracy
with 318 sets of non-polar and polar binary compounds data. He concluded that the most
effective equations are the parabolic type with one adjustable or interaction parameter.
The Grunberg equation is of this kind,
ln ( µ m) = ∑ x i ln ( µ i) + ∑ ∑ x i x j Gij
(16),
where Gij is an interaction parameter. Repeated coefficients are equal to zero. The
binary form of Grunberg equation is given by,
ln ( µ m) = x1 ln µ1 + x 2 ln µ 2 + x1 x 2 G12
(17).
The interaction parameters are system dependent and sometimes temperature
dependent and therefore difficult to generalize. Errors from 2.3% for non-polar/nonpolar to 8.9% for polar/polar mixtures have been reported by Irving43.
Crude Oil Fractions
Semi-theoretical Methods
Baltatu44 (1982) applied the method of Ely and Hanley10 to predict the viscosity of
petroleum fractions compiled by Amin and Maddox45 (1980). They reported an overall
AAD of 6.6% with a maximum deviation of 32.7%. Johnson et al.46 (1987) modified the
13
Ely and Hanley10 method in order to apply it to Canadian Bitumen. The authors changed
methane as reference fluid for a heavy hydrocarbon. Empirical factors were introduced
into the shape factors expressions to match density experimental data. Bitumen viscosity
data were calculated using a new reference fluid EOS within AADs of 6%. Mehrotra and
Svreck47 (1987) used this method to predict the viscosities of several Alberta bitumens
within overall AADs of 10-20%.
Pedersen et al.1 used a characterization procedure to match the viscosities of several
North Sea crude oil samples within an overall AAD of 6.5%.
18
Fredenslund
Pedersen and
modified the previous method to decrease the AAD from 6-14% to 3-8%
for 14 crude oil mixtures and from 9-13% to 6-10% for other crude oil fractions.
Aasberg-Petersen et al.24 applied their version of Teja-Rice22 method to calculate crude
oil samples with an overall AAD of 6.4%.
Empirical Methods
Amin and Maddox45 applied Andrade's equation to compiled viscosity data for 4
American crude oil fractions and 4 other crude oil samples. The authors modeled the
kinematic viscosity as a function of temperature by fitting the two parameters
empirically. Beg et al.48 (1988) applied the Amin-Maddox approach to 4 fractions of
Arabian crude oils. The authors calculated using generalized parameters viscosity values
with an overall AAD of 7.0%.
Dutt28 (1990) used equation (11) to calculate viscosities of crude oil fractions.
Parameter C was obtained using the method reported by Goletz and Tassios49 (1977) and
the parameters A and B were regressed to match viscosity data of 104 hydrocarbons.
They generalized all three parameters. The authors used the generalized parameters to
predict viscosity values with overall AADs of 6.8, 5.3 and 3.8% for the American,
Arabian and other crude oil samples, respectively.
Allan and Teja30 applied their ECN approach to calculate the viscosity of Arabian
light, Mid Continent and North Sea crude oil fractions with AADs of 10-15%, 8-11% and
5-11%, respectively.
14
Orbey and Sandler32 applied Eqn. (12) to several petroleum fractions. The authors
reported overall AADs for the American, Arabian and other crude oils of 4.6, 6.1 and
5.9%, respectively.
Mehrotra50 (1990) applied eqn. (13) to several Middle East crude oil and oil mixtures
data. Parameters b1 and b2 were regressed and it was found that b2 fell in such a narrow
range that is possible to use a constant value for this parameter.
Fang and Lei51 (1999) extended the equation used by Amin and Maddox45 and Beg et
al.48 to correlate the kinematic viscosity-temperature behavior for several liquid
petroleum fractions. They calculated the coefficients in the viscosity equation as a
function of the oil fractions characterization parameters. Their method only needs the
specific gravity at 15.6 oC and 50% boiling point as input parameters for the calculations.
Fang and Lei51 method was tested using 47 fractions coming from 15 different crude oils.
They reported an overall AAD of 4.2%.
MODIFICATION OF PEDERSEN’S MODEL
Model Development
Since most of the features from our correlation resemble Pedersen et al.1 model we
rewrite their model here,
T
µ m (P, T ) =  cm
 Tco



α1
 Pcm

 Pco



α2
 MWm

 MWo



α3
αm

 αo

 µ o (Po , To ) ,

(18),
where the coefficients α1, α2 and α3 in Pedersen's model are -1/6, 2/3 and 1/2
respectively.
α m = 1.000 + 7.378 × 10 −3 ρ1ro.847 MWm0.5173
(19)
α o = 1.000 + 0.031ρ1ro.847
(20).
Here, ρro is the reduced density of the reference fluid. Pedersen et al.1 used methane as
the reference fluid. They used a BWR-equation in the form suggested by McCarty52 to
evaluate the density of methane. This density is evaluated at a reference pressure and
temperature as indicated in equation (21),
15
 PP TT 
ρ o  co , co 
P
Tcm 
ρ ro =  cm
ρ co
(21),
the pressures and temperatures at which the reference viscosity (µo) is evaluated are given
by,
Po =
PPco αo
Pcm α m
and
To =
TTco αo
Tcm α m
(22).
The critical temperature and pressure are found using the mixing rules suggested by
Mo and Gubbins53 using the composition of the oil mixture. The method is highly
sensitive to the characterization of the heavy fraction, usually known as the C7+ fraction.
This issue is discussed in a later section.
The limitation of methane as the reference substance is that when the reduced
temperature of methane is below 0.4, it will freeze.
This is above the reduced
temperatures for most reservoir fluids. Pedersen et al.1 solved this problem by modifying
the viscosity model of Hanley et al.11, while Monnery et al.12 suggested using propane as
a reference fluid.
To use equation (18) we needed to find simplified expressions for the molecular
weight (MWm), critical temperature and pressure (Tcm, and Pcm) of the mixture, and for the
density and viscosity of the reference fluid. We initially used methane as the reference
fluid, but rather than implementing Pedersen’s modifications, which are tedious and add
additional complexity to the model, we decided to use an alternative reference fluid. We
selected n-decane for this purpose.
The viscosity and density data for n-decane were taken from various sources reported
by Geopetrole54 covering pressures from 14.7 psia to 7325 psia and temperatures from
492ºF to 762ºF. The density and viscosity of n-decane were fitted as a function of P and
T using a stepwise regression procedure and the statistical software SAS55. The density,
in lb/ft 3, is calculated by
ρ C10 = exp(- 1847.7998 × T −1 + 168.1906 × T −1 / 2 + 1.5043 × 10 −8 TP ) .
(23),
while the viscosity, in cp, is given by,
1
P
µ C10 = 50991.51 + 2321.5418 × T −1 / 3 - 8775.2881 × T −1 / 2 + 0.4775
T
T (24).
−9
3
−7
− 0.001272 × P − 6.7057 × 10 × T + 8.87 × 10 × PT
16
The correlation coefficient for equation (23) is R2 = 0.9996 with minimum and
maximum errors of –1.47 % and +1.82% respectively. Equation (24) has a correlation
coefficient of R2= 0.9998 and gives minimum and maximum errors of –3.11% and
+8.21% respectively.
Pressures and temperatures that appear in equations (23) and (24) should be given in
psia and Ranking degrees units, respectively.
Heavy Oil Fraction Characterization
The oil composition is determined experimentally by distillation (TBP Tests) and gas
chromatography. The thermodynamic properties are calculated from the experimental
information provided by the tests. A description is provided below.
True Boiling Point Tests (TBP Tests)
The tests are used to characterize the oil with respect to the boiling points of its
components. In these tests, the oil is distilled and the temperature of the condensing
vapor and the volume of liquid formed are recorded. This information is then used to
construct a distillation curve of liquid volume percent distilled versus condensing
temperature. The condensing temperature of the vapor at any point in the test will be
close to the boiling of the material condensing at that point. For a pure substance, the
boiling and condensing temperature are exactly the same. For a crude oil the distilled cut
will be a mixture of components and average properties for the cut are determined. Table
1 shows typical results of a TBP test.
In the distillation process, the hydrocarbon plus fraction is subjected to a standardized
analytical distillation, first at atmospheric pressure, and then in a vacuum at a pressure of
40 mm Hg using a fifteen theoretical plates column and a reflux ratio of five. The
equipment and procedure is described in the ASTM56 2892-84 book. It is also common
to use distillation equipment with up to ninety theoretical plates. Usually the temperature
is taken when the first droplet distills over. The different fractions are generally grouped
between the boiling points of two consecutive n-hydrocarbons, for example: Cn-1 and Cn.
The fraction receives the name of the n-hydrocarbon. The fractions are called hence,
17
single carbon number (SCN). Every fraction is a combination of hydrocarbons with
similar boiling points. . For each distillation cut, the volume, specific gravity, and
molecular weight, among other measurements, are determined. Other physical properties
such as molecular weight and specific gravity may also be measured for the entire
fraction or various cuts of it. The density is measured by picnometry or by an oscillating
tube densitometer. The average molecular weight of every fraction is determined by
measuring the freezing point depression of a solution of the fractions and a suitable
solvent, e.g., benzene.
If the distillate is accumulated in the receiver, instead of collected as isolated
fractions, the properties of each SCN group cannot be determined directly. In such cases,
material balance methods, using the density and molecular weight of the whole distillate
and the TBP distillation curve, may be used to estimate the concentration and properties
of the SCN groups57. A typical true boiling point curve is depicted in Figure 1. The
boiling point is plotted versus the collected volume.
There are several ways of
calculating each fraction boiling point.
Gas Chromatography (GC)
The composition of oil samples can be determined by gas chromatography. Whilst an
extended oil analysis by distillation takes many days and requires a relative large volume
of sample, GC analysis can identify components as heavy as C80 in a matter of hours
using only a small fluid sample58. Individual peaks in the chromatogram are identified by
comparing their retention times inside the column with those on known compounds
previously analyzed at the same GC conditions. The intermediate and heavy compounds
are eluted as a continuous stream of overlapping compounds. This is very similar to the
fractionation behavior and treated similarly. All the components detected by the GC
between two normal neighboring n-paraffins are commonly grouped together, measured
and reported as a SCN equal to that of the higher normal paraffin. A major drawback of
GC analysis is the lack of information, such as the molecular weight and density of the
different identified SCN groups. The very high boiling point constituents of an oil
sample cannot be eluted, hence, they can not be analyzed by GC methods.
18
Thermodynamic Properties Prediction
To use any of the thermodynamic property-prediction models, e.g., equation of state,
to predict the phase and volumetric behavior of complex hydrocarbon mixtures, one must
be able to provide: the critical properties, temperature (Tc), pressure (Pc), acentric (ω) and
molecular weight (Mw).
Petroleum engineers are usually interested in the behavior of hydrocarbon mixtures
rather than pure components. However, the above characteristic constants of the pure and
of the hypothetical components are used to define and predict the physical properties and
the phase behavior of mixtures at any reservoir state.
The properties more easily
measured are normal boiling points, specific gravities, and/or molecular weights.
Therefore existing correlations target these as the variables used to back up the
parameters needed for EOS simulations. (Tc, Pc, ω, MW).
Many correlations of the critical properties of each pseudo-component as a function
of experimentally determined variables such as; boiling point, specific gravity, average
molecular weight, have been published in literature. Whitson59 provides an excellent
review. For the sake of brevity only a brief list is include here.
Riazi and Daubert60 developed a simple two-parameter equation for predicting the
physical properties of pure compounds and undefined hydrocarbon mixtures.
The
proposed generalized empirical equation is based on the use of the normal boiling point
and the specific gravity (γ) as correlating parameters. The basic equation is:
ψ = aTbbγ c
(25),
where Tb is the normal boiling point temperature expressed in R and the constants a, b, c,
depend upon the physical property indicated by ψ .
Riazi and Daubert61 modified their equation while maintaining its simplicity and
significantly improving its accuracy:
ψ = aTbbγ c exp[dTb + eγ + fTbγ ]
(26)
ψ = aM wbγ c exp[dM w + eγ + fM wγ ]
(27).
The constants a to f for the two different functional forms of the correlation are
presented in Table 2, and depend upon the correlated property.
19
Cavett62 proposed correlations for estimating the critical pressure and temperature of
hydrocarbon fractions.
The correlations have received a wide acceptance in the
petroleum industry due to their reliability in extrapolating at conditions beyond those of
the data used in developing the correlations. The proposed correlations were expressed
analytically as functions of the normal boiling point (Tb) and API gravity (γ).
Lee and Kesler63 proposed a set of equations to estimate the critical temperature,
critical pressure, acentric factor, and molecular weight of petroleum fractions. The
equations use specific gravity and boiling point (oR) as input parameters. They also
proposed an equation to calculate molecular weight (Mw),
(
)
Mw = −12,272.6 + 9,486.4γ + (4.6523 − 3.3287γ )Tb + 1 − 0.77084γ − 0.02058γ 2 ×

720.79  10 −7
 ×
1.3437+ 1 − 0.80882γ + 0.02226γ 2
T
T
b
b


(
)1.8828 − 181T.98  × 10T

b

12
3
b
(28)
Lee and Kesler63 stated that their equations for Pc and Tc provide values that are
nearly identical with those from the API Data Book up to a boiling point of 1,200oF.
Edmister64 proposed a correlation for estimating the acentric factor ω, of pure fluids
and petroleum fractions. The equation, widely used in the petroleum industry, requires
boiling point, critical temperature, and critical pressure. The proposed expression is
given by the following relationship:
ω=
3 log(Pc / 14.7 )
−1
7(Tc / Tb − 1) )
(29),
with the temperatures expressed in degrees R.
Katz and Firoozabadi65 presented a generalized set of physical properties for the
petroleum fractions C6 through C45. The tabulated properties include the average boiling
point, specific gravity; and molecular weight. The authors proposed tabulated properties
are based on the analysis of the physical properties of 26 condensates and naturally
occurring liquid hydrocarbons.
Figure 2 shows the relationship between molecular
weight and the normal boiling point (Tb) or API gravity (γ) according to Katz and
Firoozabadi65.
20
Schou Pedersen et al.66 used extensive experimental data for seventeen North Sea oil
samples obtained using high temperature chromatography. They used experimental data
up to the C80+ fraction. They checked the validity of the equation,
zn = exp[A + B Cn]
(30),
proposed by Pedersen et al.67. A and B are empirical constants determined by fitting the
experimental data, zn is the total molar fraction of components belonging to the fraction
with n carbon number. The study found that the experimental data are well represented
by equation (26). Schou Pedersen et al.66 also reported that a good representation of the
heavy fraction is given by using compositional analysis up to C20+. The authors reported
that there is no significant advantage increasing the accuracy of the analysis from C20+ to
C80+.
Whitson’s Lumping Scheme
Whitson68 proposed a regrouping scheme whereby the compositional distribution of
Lumping is the reversed problem of splitting. The C7+ fraction is reduced to only a few
Multiple-Carbon-Number (MCN) groups. Whitson suggested that the number of MCN
groups necessary to describe the plus fraction is given by the following empirical rule:
N g = Int [1 + 3.3 log( N − n)]
(31), where:
Ng =
number of MCN groups
Int =
Integer
N =
number of carbon atoms of the last component in the hydrocarbon system
n =
number of carbon atoms of the first component in the plus fraction
The integer function requires that the real expression evaluated inside the brackets be
rounded to the nearest integer. The molecular weights separating each MCN group are
calculated from the following expression:
  1  MwN
ln
MwI = Mwn exp 
  N g  Mwn
 
 
 
I
(32),
21
where MwN = molecular weight of the last reported component in the extended analysis
of the plus fraction and Mwn = molecular weight of the first hydrocarbon group in the
extended analysis of the plus fraction.
I = 1, 2,..., Ng
Molecular weight of hydrocarbon groups (molecular weight of C7-group, C8-group,
etc.) falling within the boundaries of these values are included in the Ith MCN group.
A sample calculation is shown in Table 3. The molecular weight of fraction 1 is 96
while the molecular weight of fraction 45 is 539. The method predicted 6 pseudofractions with the molecular weights shown in the Table.
The components with
molecular weights between pseudo-components k-1 and k are ascribed to pseudocomponent k. Calculation results for several oil samples are presented in the Appendix.
Compositional Oil Samples
We used Whitson68 technique to characterize several oil samples collected from
literature and obtained from Bio-Engineering Inc. and other sources. A complete list
including compositional information and results is presented in the Appendix. The
procedure used involved the following steps:
1. Data corresponding to maximum and minimum carbon numbers and
molecular weights were collected. Normally we used 20 as the maximum
carbon number and 7 as the minimum. Some runs were done using 30 and 80,
but the results did not differ significantly from using 20. Schou Pedersen et
al.66 reported similar conclusions.
2. A computer program was developed to implement Whitson68 method using
equation (31) to calculate the number of pseudo-components and equation
(32) to calculate the limits between them.
3. The carbon number fractions in between the calculated limits were lumped
together.
Molecular weights, specific gravities and molar fractions were
calculated for the different pseudo-components using the set of equations
reported by Whitson59.
22
4. The general equation proposed by Riazi and Daubert61, equation (27), with the
data presented in Table 2 was used to calculate critical temperatures (Tc),
pressures (Pc) and volume (Vc). The same equation was also used to calculate
saturated boiling temperature (Tb).
5. Edminster equation, equation (29), was used to calculate the Pitzer acentric
factor.
69
6. Wong and Sandler mixing rules were used to calculate the pseudocomponents thermodynamic properties.
After these calculations we have a complete set of data to be used in validating our
viscosity model. A computer program was developed to calculate viscosity using our
modified Pedersen’s model, equations. (18) to (24).
Results
We first compared results calculated using the model presented here against
experimental data for pure liquid hydrocarbons. In this case the pure hydrocarbons are
treated as non-standards, i.e., pseudo-components. We used experimental data reported
by Baltatu44 (1982). These results are shown in Fig. 3. We calculated viscosities for 15
liquid hydrocarbons including, paraffins, naphthenes and aromatics. Two temperatures,
311 and 372 K were used. In general, equation (18) tends to slightly underpredict the
experimental data. A global AAD of 7.37% was calculated. The agreement between
predicted and experimental data was very good for aromatic compounds, AAD = 0.53%,
while the paraffins presented the highest deviation, AAD = 14.4%. This AAD value
compares well with corresponding state calculated values, Baltatu44 (1982) and Pedersen
et al.1 (1984) for example.
The predicted values show more error than the ones
calculated using empirical single compound correlations such as, Mehrotra35 (1991).
Results for oil samples are shown in Figs. (4) to (6).
Pedersen70 provided the
compositional information for most oil samples. Dr. Bryant provided the compositional
information corresponding to some heavy oil samples.
Fig. (4) shows that the value of viscosity decreases as temperature increases. This
was a typical result in all our calculations. It also agrees with literature data, Andrade25
(1934), Baltatu44 (1982), Monnery at. al.4 (1995), Mehrotra et al.5.(1996), among others.
23
Fig. 5 shows that the value of viscosity increases linearly with pressure. According to
equation (23) the linear term in the calculation of the reference fluid viscosity is the
predominant factor. It should be noticed that pressure values can also influence the
values of the parameters (αm, αo) defined in equations (19) and (20). The pressure will
modify in a non-linear way the value of the mixture relative density, equation (21). We
did not observed any non-linear effect in several calculations for different oil samples.
Fig. 6 shows a comparison between experimental and predicted values for several oil
samples. An AAD equal to 5.656% was calculated for 158 data for oil samples 3 to 9, 11
(Pedersen70) and A (Bryant). Only a handful of points showed a deviation above 12.64%.
Different values of temperature and pressure were used in this comparison.
This AAD value compares well with the values reported by most authors. Only Fang
and Lei51 reported an AAD smaller (4.2%) than the one calculated in this work. The
small deviation value is also a measurement of the accuracy of the procedure outlined
above, equations (18) to (24).
The oils compositions and results of the characterization process described in the
previous section are shown in the Appendix.
PROCEDURE TO SCREEN CRUDE OIL VISCOSITY DATA
Introduction
Crude oil viscosity correlations are usually developed for three situations: above the
bubble-point pressure, at and below the bubble-point pressure, and for dead oil71. Dead
oil is oil without gas in solution at atmospheric pressure. Above the bubble-point, the
composition of the oil mixture is constant and the viscosity changes result from
compressibility: The fluid becomes heavier and its viscosity increases. At some point
during production, the pressure drops below the bubble-point value, gas comes out of
solution, and the oil composition changes continuously. The oil becomes heavier and
more viscous, and two phases will flow in the reservoir.
Most correlations for crude oil viscosity require additional tuning to provide
acceptable predictions for a given reservoir fluid. Before recalibrating these correlations,
data must be quality controlled to ensure suitable performance of regression procedures.
24
For large data sets this data preprocessing could become tedious and laborious unless a
systematic and automated consistency check is used.
For this study, we had a database of almost 3,000 records of PVT properties and black
oil viscosity data, coming from 324 differential liberation tests performed in commercial
laboratories.
We have developed a procedure to "clean up" the data on a test basis, before
processing it with a regression routine. We individually screened each test, identified
outlying observations and removed those from the regression calculations.
The criteria used to discard data relied on the numerical evaluation of the first
derivative of selected functions of one variable. These functions should either always
increase or decrease, when the physical behavior is predicted appropriately. For example
oil viscosity (observed function) should always increase as the pressure in the differential
liberation tests is decreased. Forward and backward derivatives were used to account for
the end points. The filtered data resulting from this quality control process consisted of
2,324 observations.
The data were used to adapt two compositional viscosity models, Pedersen et al.1 and
Lohrenz, Bray and Clark72 (LBC), so that these models can be used for black oil systems
when compositional data are missing. The oil viscosity ranged from 0.18 to 78 cp, with
pressure ranging from 63 to 4,014 psia and temperature from 80 oF to 288 oF. The oil
API gravity ranged from 18.6 to 53.6.
These models were validated against an
independent data set consisting of 150 observations.
The two models had lower
statistical errors than current correlations.
Live oil viscosity is a strong function of pressure, temperature, oil gravity, gas
gravity, gas solubility, molecular sizes, and composition of the oil mixture. The variation
of viscosity with molecular structure is not well known because of the complexity of
crude oil systems. However, paraffin hydrocarbons do exhibit a regular increase in
viscosity as the size and complexity of molecules increases.
Crude oil viscosity correlations are usually developed for three situations: above the
bubble-point pressure, at and below the bubble-point pressure, and for dead oil71. Dead
oil is oil without gas in solution at atmospheric pressure. Above the bubble-point, the
composition of the oil mixture is constant and the viscosity changes result from
25
compressibility: The fluid becomes heavier and its viscosity increases. At some point
during production, the pressure drops below the bubble-point value, gas comes out of
solution, and the oil composition changes continuously. The oil becomes heavier and
more viscous, and two phases will flow in the reservoir.
Viscosity Correlations
Numerous viscosity-correlation methods have been proposed. None, however, has
been used as a standard method in the oil industry. Since the crude oil composition is
complex and often undefined, many viscosity estimation methods are geographically
dependent. Most correlation methods can be categorized either as ‘black oil’ or as
compositional.
Black oil correlations predict viscosities from available field-measured variables by
fitting of an empirical equation.
The correlating variables traditionally include a
combination of solution gas/oil ratios (Rs), bubble-point pressure, oil API gravity,
temperature, specific gas gravity, and the dead oil viscosity or the viscosity at the bubblepoint. Chew and Connally71, Beggs and Robinson73, Khan et al.74, Kartoatmodjo and
Schmidt75 and Petrosky76 correlated oil viscosity with temperature, pressure, oil gravity
and solution gas/oil ratio.
The second method derives mostly from the principle of corresponding states and its
extensions. Lohrenz et al.72, Ely and Hanley10, Pedersen and Fredenslund18, Pedersen et
al.1, and Monnery et al.12 are among the researchers following this trend. Lohrenz et al.72
and Pedersen et al.1 are probably the most common methods implemented in the majority
of the commercial compositional reservoir simulators.
Methods based upon the corresponding states theory predict the crude-oil viscosity as
a function of temperature, pressure, composition of the mixture, pseudo-critical
properties of the mixture, and the viscosity of a reference substance evaluated at a
reference pressure and temperature.
A thorough description of the viscosity prediction methods to be used in this research
has been shown in the previous section dealing with the modification to Pedersen et al.1
method.
26
Reservoir Fluid Studies for Reservoir Engineering
A black oil reservoir fluid study consists of a series of laboratory procedures designed
to provide values of the physical properties needed in the calculation method known as
material balance calculations. The experiments are performed with live oil samples at
pressures above and below the bubble-point pressure. Sampling procedures are discussed
in detail elsewhere77. In general two types of samples are obtained. For bottom-hole
samples, or subsurface samples, the well is shut in and the liquid at the bottom of the
wellbore is sampled.
In the other sampling method, production rates are carefully
monitored and the gas and liquid from the separators are recombined at the producing
volumetric gas/oil ratio. Oil reservoirs must be sampled before the reservoir pressure
drops below the bubble-point pressure of the oil, since at pressures below that no
sampling method will give a sample representative of the original reservoir mixture.
Determining the composition of all chemical species present in the black oil is
virtually impossible and impractical. In the majority of cases the composition of the light
components is determined, from methane to hexane, and all the heavier components are
grouped together in a plus fraction commonly labeled as the heptane plus fraction.
Material balance calculations are in fact volumetric calculations in which the
reservoir fluids volumes filling the pore space are determined as a function of pressure.
Corrections to account for rock compressibility effects and water encroachment are also
included.
The reservoir is considered as a tank filled with oil, gas and water. As
production takes place these volumes change as illustrated in Fig. 7.
Standard reservoir PVT fluid studies are designed to simulate processes at which oil
and gas displace from the reservoir to surface.
In a constant composition expansion test (CCE) a sample of the reservoir fluid is
placed in a variable volume PVT cell at the reservoir temperature. The pressure is
adjusted at or above the original reservoir pressure. Pressure is reduced by incrementally
increasing the cell volume, and pressure/volume pairs are recorded and plotted. The
pressure at which the slope changes is the bubble-point pressure and the volume at this
point is the bubble-point volume. All of the liberated gas remains in contact with the oil
until the two phases reach equilibrium, neither gas or liquid is removed from this cell
27
during the process; therefore, the overall composition remains constant. This test also
provides isothermal oil compressibility. Fig. 8 shows a sketch of this laboratory process.
The production path of reservoir fluids from the reservoir to surface is simulated in
the laboratory by a set of stage-wise flashings of the live oil at reservoir temperature.
These tests are labeled differential liberation tests (DL). Here the sample is placed in a
PVT cell at its bubble-point pressure. Then, pressure is reduced by incremental increases
in the cell volume. The difference in this test is that all the gas liberated is expelled from
the cell while the pressure is held constant by using a dual-cell arrangement. The gas is
collected, and its quantity and specific gravity are measured. During this process the oil
volumes and the amount of gas released are measured and used to determine oil and gas
formation volume factors (Bo, and Bg) and solution gas/oil ratios as a function of pressure
Rs.
Fig. 9 shows a schematic of the differential liberation process that ends at
atmospheric pressure. The liquid phase is called ‘dead’ oil. The temperature is then
reduced to 60oF and the volume of this oil is identified as residual oil. Table 4 shows one
out of the 324 differential liberation (DL) sets used in this study, and Table 5 shows the
corresponding viscosity data.
The oil formation volume factor Bo gives an idea of the shrinkage experienced by a
unit volume of reservoir as it goes from reservoir pressure and temperature to standard
pressure and temperature, or stock tank conditions, while the solution gas/oil ratio at a
given pressure provides the amount of dissolved gas (which will be eventually produced)
expressed as standard cubic feet per barrel of oil at standard conditions.
The oil viscosity is usually measured in a rolling-ball viscometer or a capillary
viscometer, either designed to simulate differential liberation. The composition of the oil
sample is not measured in either of the DL stages. The viscosity measured at the lowest
pressure usually has the highest uncertainty.
Data Preparation and Data Screening Routine
The viscosity correlations proposed are expressed as functions of other variables or
properties that are either measured or calculated from correlations. These variables
include oil density, molecular weight, pseudo-critical properties, pressure and
temperature, among others. The correlation will be meaningless if the quality of these
28
variables, or the quality of the data, is questionable. In that case one may be attempting
to calculate parameters by fitting errors.
During the DL process the oil becomes heavier and some physical properties should
monotonically increase as the pressure decreases. These include Vcm, Tcm, Tb, Mwm, oil
density and oil viscosity. The mixture critical properties are not known and rather
pseudo-critical properties are used, but they should follow the same trend as the true
critical properties.
These pseudo-critical properties and molecular weights are not
actually measured but correlated to measurable variables such as the oil density and the
normal boiling point. For lighter oils the critical pressure may go through a maximum
before it starts decreasing, as the oil becomes heavier78.
Most correlations for crude oil viscosity require additional tuning to provide
acceptable predictions for a given reservoir fluid. Before recalibrating these correlations,
data must be quality controlled to ensure suitable performance of regression procedures.
For large data sets this data preprocessing could become tedious and laborious unless a
systematic and automated consistency check is used.
For this study, we had a database of almost 3,000 records of PVT properties and black
oil viscosity data, coming from 324 differential liberation tests performed in commercial
laboratories.
Sometimes the data may be of good quality but the correlation may be applied beyond
its range. We verified that Mwm,, Tcm and Vcm were monotonically increasing. The
correlations used provide the correct behavior for oil specific gravities above 0.6. Since
we had oils with lower specific gravities below 0.6 we extrapolated the correlations
following a consistent trend as indicated in Fig. 10.
We have developed a procedure to "clean up" the data on a DL test basis, before
processing it with a regression package. We individually screened each test, identified
outlying observations and removed those from the regression calculations.
The criteria used to discard data relied on the numerical evaluation of the first
derivative of selected functions of one variable. These functions should either always
increase or decrease, when the physical behavior is predicted appropriately. For example
oil viscosity (observed function) should always increase as the pressure in the differential
liberation tests is decreased. Forward and backward derivatives were used to account for
29
the end points. The filtered data resulting from this quality control process consisted of
2,324 observations.
The data were classified according to test number. Each DL is characterized by
temperature and API gravity of the residual oil. The highest pressure in every set
corresponds to the bubble-point pressure at that temperature. This pressure is extracted
and written to a file for use in the correlations for solution gas/oil ratio and formation
volume factor. The viscosity data were contained in separate files and even though these
corresponded to the same DL tests, some viscosity measurements were missing or were
done at different pressures. Assembling of these two sets of files was done one a one-to
one match. The missing pair was removed from either set and stored in a separate file.
Each matched DL and viscosity set contained between 6 and 10 observations at
declining pressures. Properties were evaluated for these observations and stored.
Forward and backward derivatives were used for viscosity and oil density versus
pressure. The first derivative of these functions should always be negative. If a point
violated this monotony criterion all measured properties at that pressure were discarded.
Occasionally the oil density exhibited a consistent behavior within some acceptable
scatter and the data points passed the consistency test. However, if derived properties
(Mwm, Tcm, Vcm) magnified the inconsistency, these were included in the list of checking
variables and provided a more rigorous screening.
The number of points left in a DL set should be at least 4. Even if these appeared to
be correct, the fact that the remaining points were discarded made the test questionable.
Data Screening Results
Figures 11 to 13 indicate examples of removed data. You can find deviations from a
monotonic trend for different properties. These deviations are caused by experimental
and/or human errors. With all the cleaned data we proceeded to develop correlations for
the viscosity based upon the modified Pedersen1 and Lohrenz72 models. Additionally we
proposed new correlations for solution gas-oil ratios and formation volume factors to be
used in these models.
30
MODIFICATION OF PEDERSEN’S MODEL FOR BLACK OIL SAMPLES
Introduction
This section presents a modification of Pedersen’s corresponding states compositional
viscosity model that enables viscosity prediction for black oil systems when there are no
compositional data available. This model can be easily implemented in any reservoir
simulation software, it can be easily tuned, and it provides better estimates of oil viscosity
than the existing correlations.
Viscosity from 324 sets of differential liberation data consisting of 2343 observations
covering a wide range of pressure, temperature, and oil density were used to develop the
correlation. This correlation retains most of the functional form of Pedersen’s model.
These modifications include (1) use of n-decane as the reference fluid, (2) consider the
oil mixture as a single pseudo-component with molecular weight and critical properties
correlated to its density, and (3) addition of a functional dependence to solution gas/oil
ratio and gas-specific gravity. The average error over 2343 viscosity observations was
0.9%. The model was tested against a second data set consisting of 150 observations and
the average error was 0.7 %.
The predictions were compared with those predicted from the correlations of Khan et
74
al. and of Petrosky76 that are applicable to the experimental conditions of our data sets.
These average errors for these correlations were -28 % and 4.9 % respectively for the first
data set; and –60.8 % and –1.4 % for the second data set.
Viscosity Correlations
Numerous viscosity-correlation methods have been proposed. None, however, has
been used as a standard method in the oil industry. Most correlation methods can be
categorized either as ‘black oil’ or as compositional.
Black oil correlations predict viscosities from available field-measured variables by
fitting of an empirical equation.
The correlating variables traditionally include a
combination of solution gas/oil ratios (Rs), bubble-point pressure, oil API gravity,
temperature, specific gas gravity, and the dead oil viscosity or the viscosity at the bubblepoint.
31
The second method derives mostly from the principle of corresponding states and its
extensions. Methods based upon the corresponding states theory predict the crude-oil
viscosity as a function of temperature, pressure, composition of the mixture, pseudocritical properties of the mixture, and the viscosity of a reference substance evaluated at a
reference pressure and temperature.
Lohrenz et al.72 published the now well-known LBC correlation suitable for gases and
light oils. The LBC correlation is a fourth-degree polynomial in the pseudo-reduced
density of the mixture and this makes it very sensitive to this variable.
[(µ − µ )ξ + 10 ]
−4 1 / 4
*
5
= ∑ ai ρ r
i −1
(33)
i =1
Here µ* is the low-pressure gas mixture viscosity, and ξ is the viscosity-reducing
parameter, which is defined as,
ξ = Tcm
1/ 6
M wm
1/ 2
Pcm
2/ 3
(34).
Here and in other sections of this report we refer to information presented previously.
In order to facilitate the understanding of the subject we will repeat the necessary
information using the original equation numbers.
Ely and Hanley10 (1981) proposed the following extended corresponding states
model:
µi(ρ,T) = µo( ρ h i,o , T
1/ 2
f i,o
) (M i / M o)
-2/3
1/2
hi,o f i,o
(1)
h i ,o = h i ,o (ρc,i / ρc,o) φi,o
(2)
f i ,o = (Tc,i / Tc,o) θi,o
(3)
where θi,o, and ϕi,o are shape factors depending on the chemical components. Viscosity
calculations require correlations for a reference fluid viscosity and density along with
critical properties values, acentric factor and molar mass. Methane was selected as a
reference fluid because of the availability of highly accurate data. A problem using
methane is its high freezing point (Tr = 0.48), which is well above the reduced
temperatures of other fluids in the liquid state. In order to overcome this difficulty they
extrapolated the density correlation for methane and added an empirical correlation for
non-correspondence and extended the viscosity correlation of Henley et al.11 (1975).
32
Pedersen et al.1 introduced a third parameter (α) to correct for this deviation from the
conventional corresponding states principle. This term accounts for the molecular size
and density effects on viscosity. Their model eliminates the iterative procedure in Ely
and Hanley10 and performs a direct calculation of the viscosity.
Model Development
Here we will repeat some of the material presented above in order to improve the
understanding of the subject. Since most of the features from our correlation resemble
Pedersen et al.1 model we rewrite their model here.
T 
µ m (P, T ) =  cm 
 Tco 
α1
 Pcm 


P
 co 
α2
 MWm 


MW
o 

α3
 αm 

µ o (Po , To ) ,
α
 o
(18),
where the coefficients α1, α2 and α3 in Pedersen's model are -1/6, 2/3 and 1/2
respectively.
α m = 1.000 + 7.378 × 10 −3 ρ1ro.847 MWm0.5173
(19)
αo = 1.000 + 0.031ρ1ro.847
(20).
Here, ρro is the reduced density of the reference fluid (n-decane). This density is
evaluated at a reference pressure and temperature as indicated in equation (13)
 PP TT 
ρ o  co , co 
P
Tcm 
ρ ro =  cm
ρ co
(21),
the pressures and temperatures at which the reference viscosity (µo) is evaluated are given
by,
Po =
PPco αo
Pcm α m
and
To =
TTco αo
Tcm α m
(22).
The critical temperature and pressure are found using the mixing rules suggested by
Mo and Gubbins53 using the composition of the oil mixture. The method is highly
sensitive to the characterization of the heavy fraction, usually known as the C7+ fraction.
Our objective in this section was to extend this model to black oil mixtures for which we
do not have compositional information.
33
To use equation (18) we needed to find simplified expressions for the molecular
weight (MWm), critical temperature and pressure (Tcm, and Pcm) of the mixture, and for the
density and viscosity of the reference fluid (n-decane).
The viscosity and density data for n-decane were taken from various sources reported
by Geopetrole54 covering pressures from 14.7 psia to 7325 psia and temperatures from
492ºF to 762ºF. The density and viscosity of n-decane were fitted as a function of P and
T using a stepwise regression procedure and the statistical software SAS55. The density,
in lb/ft 3, is calculated by
ρ C10 = exp(- 1847.7998 × T −1 + 168.1906 × T −1 / 2 + 1.5043 × 10 −8 TP ) .
(23),
while the viscosity, in cp, is given by,
1
P
µ C10 = 50991.51 + 2321.5418 × T −1 / 3 - 8775.2881 × T −1 / 2 + 0.4775
T
T (24).
−9
3
−7
− 0.001272 × P − 6.7057 × 10 × T + 8.87 × 10 × PT
The correlation coefficient for equation (23) is R2 = 0.9996 with minimum and
maximum errors of –1.47 % and +1.82% respectively. Equation (24) has a correlation
coefficient R2= 0.9998 and gives minimum and maximum errors of –3.11% and +8.21%
respectively. The pressures and temperatures values that appear in equations (23) and
(24) are in psia and Ranking degrees units, respectively.
The specific gravity of the oil was evaluated from a material balance using the
reported values of formation volume factor (Bo), solution gas/oil ratio (Rs), and gas
specific gravity according to McCain78. The reported specific gravity of the gas was for
the separator at 100 psia rather than at atmospheric pressure, however; the error
introduced in the determination of specific gravity of the oil is negligible.
The oil mixture was lumped into a single pseudo-component for which the critical
temperature, the critical pressure, and the molecular weight were correlated to the oil
specific gravity.
Most correlations for the critical properties require at least two properties from the
molecular weight, the density, and the normal boiling point. We had only one of these
variables. To overcome this problem we assumed that for most oils the percentage of
paraffinic compounds dominates and in that case we correlated the normal boiling versus
specific gravity of oil at reservoir conditions (γo,R).
34
Once this was determined the
molecular weight was correlated to the normal boiling point in R. The data to develop
these correlations were reported by Ahmed79 and Whitson68.
The normal boiling point in R, and the mixture molecular weight are given by:
1
3
Tb = 3540.53 - 385.934312
- 5431.82548 × γ o,R + 4193.44761 × γ o,R
γ o ,R
(35)
MWm = 64.611 × exp(0.0022 × Tb )
(36).
Once these two properties were obtained the critical pressure Pcm was obtained using
the Riazi-Daubert61 correlation, while the Tcm was calculated using the following
relationship:
Tcm = 24.2787 × Tb
0.58848
× γ o, R
0.3596
(37).
We observed that the critical pressure, Pcm, was not always monotonic as the oil
became heavier.
Particularly for lighter oils, Pcm went through a maximum and it
decreased at the later stages of depletion. Since we wanted to generalize the equation for
heavier and lighter oils, we selected Vcm as the correlating variable since it increases
monotonically as the oil becomes heavier. The correlation used for Vcm was also from
Riazi-Daubert61.
If the hydrocarbon mixture had a larger percentage of aromatic compounds, the
correlation for the molecular weight and normal boiling points would have to be
modified. For example, the molecular weight of an aromatic component with a Tb of
640°F is approximately 179 lb/lb-mol, while the same boiling point corresponds to a
paraffinic mixture with average molecular weight of about 260 lb/lb-mol.
The database was screened for consistency following and automated scheme shown
above. The method screens for outliers in a given data set and discards the viscosity
points that do not follow a consistent pattern, i.e. viscosity should increase monotonically
as the pressure decreases.
In conclusion oil viscosity is calculated using,
35
1.0286
 T × Tcm 

µ m (P, T ) = 
2 
 TcC10 
 Vcm 


 VcC10 
0.9841
 Rs × γ o,R 


 Rsb 
−0.1362 γ o , R
 MWm 


 MWC10 
-3.9243
Rsb
−0.4471
API −2.2902
3




ρ
B
P × Vcm
o
 − 0.02359
+ 2.1388 C10 + 0.1930µ C10 (Po , To )
× exp − 0.2606 × 


ρ c C10.
TC10
 Bob 


(38),
where Bo, the formation volume factor, is dimensionless, γ o,R is the specific gravity of oil
at reservoir conditions, API is the gravity of the oil at standard conditions, Rs is the
solution gas/oil ratio in SCF/STB (standard cubic feet per stock tank barrel). Rsb and Bob
are evaluated at the bubble point pressure.
The advantage of this model is that it can be easily retuned if necessary using linear
regression. The exponent for the variable (Bo/Bob) was determined independently and it is
left as a fixed parameter. The n-decane density and viscosity were evaluated at the same
reference pressure and temperature indicated in equations (21) and (22), and the same
values for αm and α0 defined in equations (19) and (20) were used. No attempt was made
to retune these values.
Results
Our model was developed using a data set of 2,343 points (Data Set 1) and it was
validated with an independent data set from Core laboratories consisting of 150
observations (Data Set 2). Table 6 indicates the ranges of viscosity, temperature, and
pressure for the two sets.
To evaluate the performance of this model we selected two different models. These
models do not assume the knowledge of the dead-oil viscosity. Khan et al.74 proposed a
correlation for the bubble point viscosity, while Petrosky76 proposed a correlation for the
dead-oil viscosity. The experimental ranges of pressure, oil gravity, temperature, and
solution-gas/oil ratios are similar to those of our databases.
Figs. 14 and 15 show predicted versus experimental viscosities for Data Set 1
according to Khan's et al. correlation, and to Petrosky's correlation. Fig. 16 shows the
performance of the adapted untuned Pedersen model, equation (18) with the original
coefficients but using n-decane as the reference fluid, while Fig. 17 shows the predicted
36
versus the experimental viscosity for from this work.
Figures 18 to 21 depict the
predicted versus experimental viscosities for Data Set 2 according to Khan's et al.
correlation; Petrosky's correlation; the untuned Pedersen's model, equation (18), and this
work respectively.
If the parameters α1 to α3 from equation (18) are determined for every set, then the fit
can be substantially improved as indicated in Fig. 22. Current research efforts seek to
generalize the dependence of the parameters α1 to α3 with ºAPI, Rsb and other field
derived variables. Table 7 summarizes the statistics for these models.
37
CONCLUSIONS
We presented a new viscosity correlation derived from Pedersen’s corresponding
states model. The model replaces the reference compound to avoid known problems.
The procedure presented in this work can be used to calculate viscosities of
compositional and black oils. The application to black oils, in absence of compositional
data, is particularly important from the practical point of view. This model can be easily
implemented in any reservoir simulation software, it can be easily tuned, and it provides
equal or better estimates of oil viscosity than other existing correlations.
38
NOMENCLATURE
API
=
Oil gravity, (API = 145/γo,STC -135)
Bo
=
Oil formation volume factor, (RB/STB)
Bob
=
Oil formation volume factor at the bubble-point, RB/STB
f i ,o
=
Parameter defined in Eqn. (1)
Gij
=
Interaction parameter used in Eqn. (16)
h i ,o
=
Parameter defined in Eqn. (1)
MWm =
Mixture molecular weight
Pcm
=
Mixture critical pressure (psia)
P
=
Pressure (psia)
Pb
=
Bubble-point pressure, psia
Pr
=
Reduced pressure, P/Pc
Rs
=
Solution gas/oil ratio, (SCF/STB)
Rsb
=
Solution gas-oil-ratio at the bubble-point, (SCF/STB)
Rsr
=
Reduced solution gas-oil ratio, Rs/Rsb
T
=
Reservoir temperature, (oF, R)
Tb
=
Normal boiling point temperature, (oF, R)
Tcm
=
Mixture critical temperature (R)
Vcm
=
Mixture critical volume, (ft3/lbmol)
x
=
Molar fraction
Greek Letters
αm
=
Parameter defined in Eqn. (11)
αo
=
Parameter defined in Eqn. (12)
αTG
=
Tham-Gubbins17 (1971) rotational coupling coefficient
γo,R
=
Oil specific gravity at reservoir conditions
ξ
=
Viscosity-reducing parameter, which is defined as
ξTR
=
Parameter defined Eqn. (9)
ϕi,o
=
Shape factor used in Eqn. (1)
39
ρ
=
Density (lb/ft3)
ω
=
Compressibility factor
θi,o
=
Shape factor used in Eqn. (1)
µ
=
Oil viscosity, cp
Z
=
Compressibility factor
Subscripts
o
=
reference conditions, oil
c10
=
n-decane.
r
=
reduced
c
=
critical
m
=
mixture
b
=
at bubble point, or normal boiling point (Eqn. 8).
o,R
=
oil at reservoir conditions
g,100 =
gas at 100 psia.
40
REFERENCES
1. Pedersen, K. S., A. Fredenslund, P. L. Chirstensen and P. Thomassen, Chem. Eng.
Sci., 39, 1011-1016 (1984).
2 Butler, R. M. “Thermal Recovery of Oil and Bitumen,” Prentice-Hall Inc., New
Jersey, 1991.
3 Willman, B.T. "Laboratory Studies of Oil Recovery by Steam Injection," J.Pet.
Tech. (July 1961) 681-698.
4 Monnery, W. D., Svrcek, W. Y., and Mehrotra, A. K., Can. J. Chem. Eng., 73, 3,
1995.
5 A. K. Mehrotra, W. D. Monnery and W. Y. Svrcek, Fluid Phase Equilib., 117,
344-355 (1996).
6 Reid, R. C., J. M. Prausnitz and T. K. Sherwood, “The Properties of Gases and
Liquids,” McGraw-Hill (1977).
7 Reid, R. C., J. M. Prausnitz and B. E. Poling, “The Properties of Gases and
Liquids,” McGraw-Hill (1987).
8 Stephan, K. and K. D. Lucas, “Viscosity of Dense Fluids,” Plenum Press (1979).
9 Viswanath, D. S. and G. Natarajan, “Data Book on the Viscosity of Liquids,”
Hemisphere Publishing (1989).
10 Ely, J. F. and H. J. M. Henley, Ind. Eng. Fundam. 20, 323-332 (1981).
11 Henley, H. J. M., R. D. McCarty and W. M. Haynes, Cryogenics, July, 413-415
(1975).
12 Monnery, W. D., A. K. Mehrotra and W. Y. Svreck, Can. J. Chem. Eng., 69,
1213-1219 (1991).
13 Ely, J. F., “Prediction of Dense Fluid Viscosities in Hydrocarbon Mixtures,” GPA
Proc. Of 61st Ann. Conv., 9-17 (1982).
14 Ely, J. F., ”Application of the Extended Corresponding States Model to
Hydrocarbon Mixtures,” GPA Proc. Of 63rd Ann. Conv., 9-22 (1984).
15 Haile, J. M., K. C. Mo and K. E. Gubbins, Adv. Cryo. Eng., 21, 501-508 (1976).
16 Hwang, M.-J. and W. B. Whiting, Ind. Eng. Chem. Res., 26, 1758-1766 (1987).
41
17 Tham, M. J. and K. E. Gubbins, J. Chem. Phys., 55, 268-279 (1971).
18 Pedersen, K. S. and A. Fredenslund, Chem. Eng. Sci., 42, 182-186 (1986).
19 Teja, A. S., N. C. Patel and S. I. Sandler, Chem. Eng. J., 21, 21-28 (1981).
20 Lee, B. I., and M. G. Kessler, AIChE J., 21,510-527 (1975).
21 Teja, A. S. and P. A. Thurner, Chem. Eng. Commun., 49, 69-79 (1986).
22 Teja, A. S. and P. Rice, Ind. Eng. Chem. Eng. Fundam., 20, 77-81 (1981).
23 Wong, D., S. I. Sandler and A. S. Teja, Ind. Eng. Chem. Fundam., 23, 38-44
(1984).
24 Aasberg-Petersen, K., K. Knudsen and A. Fredenslund, Fluid Phase Equil., 70,
293-308 (1991).
25 Andrade, E. N. da C., Phil. Mag., 17, 497-511 (1934).
26 Guzman, J. de., J. An. Soc. Espan. Fis. Quim., 11, 353 (1913).
27 Vogel, H., Physik. Z., 22, 645-646 (1921).
28 Dutt, N. V. K., Chem. Eng. J., 45, 83-86 (1990).
29 Van Velzen, D., R. L. Cardozo and H. Langenkamp, “Liquid Viscosity and
Chemical Constitution of Organic Compounds: A New Correlation and a
Compilation of Literature Data,” Euratom report EUR 4735e, (1972).
30 Allan, J. and A. S. Teja, Can. J. Chem. Eng., 69, 986-991 (1991).
31 Gregory, G. A., Can. J. Chem. Eng., 70, 1037-1038 (1992).
32 Orbey, H. and S. I. Sandler, Can. J. Chem. Eng., 71, 437-446 (1993).
33 ASTM, Annual Book of ASTM Standards, American Society for Testing and
Materials, Philadelphia, PA, 205 (1981).
34 Walter C., Erdol Teer, 7, 382-384 (1931).
35 Mehrotra, A. K., Ind. Eng. Chem. Res., 30, 420-427 (1991).
36 API, “Properties of Hydrocarbons of High Molecular Weight,” API Research
Project 42, American Petroleum Institute, Washington DC, 1966.
37 Mehrotra, A. K., Ind. Eng. Chem. Res., 30, 1367-1372 (1991).
38 Mehrotra, A. K., Can. J. Chem. Eng., 72, 554-557 (1994).
39 Chabra, R. P., AIChE J., 38, 1657-1661 (1992).
42
40 Lawal, A. S., “Prediction of Vapor and Liquid Viscosities from the LawalSilberberg Equation of State,” SPE/DOE paper 14926 (1986).
41 Heckenberger, T. and K. Stephan, Int. J. Thermophys., 11, 1011-1017 (1990).
42 Heckenberger, T. and K. Stephan, Int. J. Thermophys., 12, 333-356 (1991).
43 Irving, J. B., “Viscosity of Binary Liquid Mixtures: The Effectiveness of Mixture
Equations,” National Engineering Lab. Report No. 631, East Kilbride, Glasgow,
Scotland (1977).
44 Baltatu, M. E., Ind. Eng. Chem. Process. Des. Dev., 21, 192-195 (1982).
45 Amin, N. B. and R. N. Maddox, Hydrocarbon Process., 59 (12), 131-135 (1980).
46 Johnson, S. E., W. Y. Svrcek and A. K. Mehrotra, “Viscosity Prediction of
Athabasca Bitumen Using the Extended Principle of Corresponding States,” Ind.
Eng. Chem. Res., 26, 2290-2298 (1987).
47 Mehrotra, A. K. and W. Y. Svrcek, “Viscosity of Compressed Athabasca
Bitumen,” Can. J. Chem. Eng., 64, 844-847 (1987).
48 Beg, S. A., M. B. Amin and I. Hussain, “Generalized Kinematic ViscosityTemperature Correlation for Undefined Petroleum Fractions,” Chem. Eng. J., 38,
123-136 (1988).
49 Goletz, E. and D. Tassios, Ind. Eng. Chem. Proc. Des. Dev., 16, 75-79 (1977).
50 Mehrotra, A. K., “Modeling the Effects of Temperature, Pressure and
Composition on the Viscosity of Crude Oil Mixtures,” Ind. Eng. Chem. Res., 29,
1574-1578 (1990).
51 Fang, W. and Q. Lei, “Generalized Correlation for Predicting the Kinematic
Viscosity of Liquid Petroleum Fractions,” Fluid Phase Equilib., 166, 125-139
(1999).
52 R. D. McCarty, Cryogenics, (1974) 276-280.
53 K. C. Mo, and K. Gubbins, Chem. Eng. Commun. 1, (1974) 281-290.
54 Geopetrole, Viscosity and Density of Light Paraffins, Nitrogen and Carbon
Dioxide, Editions Technip-Paris (1970).
55 SAS language. Version 6. Publisher Cary, NC. SAS Institute, (1990).
43
56 ASTM, "Distillation of Crude Petroleum, Designation D2892-84," Annual Book
of ASTM Standards, (1984) 821-860.
57 Hernandez, M. J. and Casrells, F. "A New Method for Petroleum Fractions and
Crude Oil Characterization," SPE Reservoir Engineering (1992) 265-270.
58 Curvers, J. and van den Engel, P. "Gas Chromatographic Method for Simulated
Distillation up to a Boiling Point of 750 oC Using Temperature-Programmed
Injection and High Temperature Fused Silica Wide-Bore Columns," J. High
Resolution Chromatography, (1989) 12, 16-22.
59 Whitson, C., "Characterizing Hydrocarbon Plus Fractions," Paper EUR 183,
presented at the European Offshore Petroleum Conference held in London,
October 21-24, 1980.
60 Riazi, M. R. and Daubert, T. E. "Simplify Property Predictions," Hydrocarbon
Proc. (1980) 59(3), 115-116.
61 Riazi, M. R. and Daubert, T. E. "Characterization Parameters for Petroleum
Fractions," Ind. Eng. Chem. Research (1987) 26, 755-759.
62 Cavett, R. H. "Physical Data for Distillation Calculations, Vapor-Liquid
Equilibria," Proc. Of 27th API Meeting, San Francisco (1962) 351-366.
63 Lee, B. I. And Kesler M. G. "Improve Vapor Pressure Prediction," Hydrocarbon
Proc., (July 1980) 163-167.
64 Edminster, W. C. "Applied Hydroccarbon Thermodynamics Part 4:
Compressibility Factors and Equations of State," Pet. Refiner, (April 1958), 37,
173-79.
65 Katz, D. L. and Firoozabadi, A. "Predicting Phase Behavior of Condensate/Crude
Oil Systems Using Methane Interaction Coefficients," JPT, (1978) 1649-55.
66 Schou Pedersen, K., Blilie, A. L. and Meisinget, K. K. "PVT Calculations on
Petroleum Reservoir Fluids Using Measured and Estimated Compositional Data
for the Plus Fraction," Ind. Eng. Res., (1992) 31, 1378-1384.
67 Pedersen, K. S., Thomassen, P., and Fredenslund, As. "Characterization of Gas
Condensates Mixtures," Adv. Thermodyn., (1989b) 1, 137-148.
44
68 Whitson, C., "Characterizing Hydrocarbon Plus Fractions,” Soc. Of Pet.
Engineers Journal, (August 1983), 37, 683-694.
69 Wong, D.S. and Sandler, S.I. Theoretically Correct New Mixing Rule for Cubic
Equations of State. AIChE J., 38, (1992) 671.
70 Pedersen, K. S., Data provided upon request (2000).
71 J. Chew, and C. A. Connally Jr., Trans. AIME (1959) Vol. 216, 23-25.
72 J. Lohrenz, B.G Bray, and C. R. Clark, J. Pet. Technol., Oct. 1964 – 1171.
73 H. D Beggs,.and J. R Robinson, J. Pet. Tech. (Sept., 1975) 1140-1141.
74 S. A. Khan, M.A. Al-Marhoun, and S.O. Duffuaa, SPE 15720, (1987).
75 R.S. Kartoatmodjo, and Z. Schmidt, Oil and Gas J. (1994), 51-55.
76 G.E. Petrosky, and F.F. Farshad, SPE 29468, (1995).
77 F. O. Reudelhuber.: “Sampling Procedures for Oil Reservoir Fluids,” J. Pet. Tech.
(Dec 1957), 9, 15-18.
78 W. D. McCain, Jr.: The Properties of Petroleum Fluids, Second Ed., Pennwell
Publishing Co., Tulsa, OK (1990).
79 T. Ahmed, Hydrocarbon Phase Behavior, Gulf Publishing Co. 1st Ed. (1990).
45
TABLES AND FIGURES
Table 1. Typical results of a TBP test.
Component
Ti
Hypo1
99
Hypo2
214
Hypo3
323
Hypo4
432
Hypo5
526
Hypo6
612
Hypo7
693
Hypo8
765
Hypo9
821
Hypo10
908
Residual
Whole Oil
Residual Volume Left
Tf
220
323
432
526
612
693
765
821
908
1010
∆V (cm3)
5.1
8.0
7.9
8.1
7.9
7.9
7.9
7.8
8.1
5.2
22.9
T mean
159.5
268.5
377.5
479
569
652.5
729
793
864.5
959
1261.1692
729
Σ(∆
∆V)
5.1
13.1
21.0
29.1
37.0
44.9
52.8
60.6
68.7
73.9
96.8
V % Off
5.3
13.5
21.7
30.1
38.2
46.4
54.5
62.6
71.0
76.3
100.0
22.9
Table 2. Parameters for Riazi and Daubert Equations (26) and (27).
Form (1)
Constant
Mw
Tc ( oR)
Pc (psia)
Vc (ft 3 / lbm )
a
581.96
10.6443
6.162x106
6.233x10-4
b
0.97476
0.81067
-0.4844
0.7506
c
6.51274
0.53691
-4
4.0846
-4
-1.2028
-3
d
5.43076x10
-5.1747x10
-4.725x10
-1.4679x10-3
e
9.53384
-0.54444
-4.8014
-0.26404
f
1.11056x10-3
3.5995x10-4
3.1939x10-3
1.095x10-3
Constant
Tc (oR)
Pc (psia)
Vc (ft 3 / lbm )
Tb ( oR)
a
544.4
4.5203x10-4
1.206x10-2
6.77857
b
0.2998
-0.8063
0.20378
0.401673
c
1.0555
1.6015
-1.3036
-1.58262
d
-1.3478x10-4
-1.8078x10-4
-2.657x10-3
3.77409x10-3
e
-0.61641
-0.3084
0.5287
2.984036
f
0.0
0.0
2.6012x10-3
-4.25288x10-3
Form (2)
46
Table 3. Grouping data for characterization of fractions with up to 45 components.
Pressure
(psig)
Group
Molecular Weight
1
127
2
170
3
227
4
303
5
404
6
539
Table 4. Differential Vaporization Test at 80 °F.
Rs
Bo [2]
Oil
Gas
Bg [3]
Density
Deviation
(SCF/
(RB/
Factor
ρo
(RCF/
STB)
STB)
Z
(gm/cc)
SCF)
[1]
1690
210
1.069
0.9022
1500
188
1.063
0.9052
1300
165
1.056
0.9083
1100
141
1.049
0.9113
900
117
1.042
0.9143
700
93
1.036
0.9174
500
68
1.029
0.9205
300
42
1.022
0.9235
100
15
1.014
0.9268
0
0
1.008
0.9305
Gravity of Residual Oil = 19.2°API @ 60°F
0.822
0.835
0.852
0.872
0.896
0.922
0.951
0.983
0.00825
0.00966
0.01162
0.01450
0.01906
0.02724
0.04593
0.13004
Gas
Gravity
γg
0.581
0.576
0.573
0.571
0.572
0.574
0.581
0.600
0.724
[1] Cubic feet of gas at 14.7 psia and 60°F per barrel of residual oil at 60°F.
[2] Barrels of oil at indicated pressure and temperature per barrel of residual oil at 60°F.
[3] Barrels of oil plus liberated gas at indicated pressure and temperature per barrel of
residual oil at 60°F.
[4] Cubic feet of gas at indicated pressure and temperature per cubic foot at 14.7 psia
and 60°F.
47
Table 5. Viscosity Data Accompanying DL Set at 80 °F and 19.2 oAPI Residual Oil.
Pressure
Oil Viscosity
Calculated
Oil/Gas
(psig)
(cp)
Gas Viscosity
Viscosity
(cp)
Ratio
1690 (Pb)
35.4
1500
40.0
0.0146
2740
1300
45.2
0.0140
3230
1100
51.2
0.0134
3820
900
58.5
0.0129
4530
700
68.4
0.0124
5520
500
82.5
0.0120
6880
300
102.1
0.0116
8800
100
127.6
0.0113
11300
0
177.0
0.0106
16700
Table 6. Range of input data. SI units and values in are indicated in parenthesis.
Variable
Minimum
Maximum
Dataset No Points.
#1
#2
#1
#2
#1
#2
#1
#2
2343
150
2343
150
2343
150
2343
150
Oil Density: lbm/ft3 (g/cm3)
Oil Density, lbm/ft3 (g/cm3)
Oil Viscosity, cp
Oil Viscosity, cp
Temperature, R (K)
Temperature, R (K)
Pressure, psia (MPa)
Pressure, psia (MPa)
35.11 (0.562)
24.31 (0.389)
0.132
0.13
540 (300)
537 (303.9)
14.7 (0.1)
102.7 (0.708)
57.31(0.92)
57.50(0.921)
78.30
68.90
766 (425.5)
762 (423.3)
5601.7 (38.62)
5434.7 (37.47)
Table 7. Summary of the black oil viscosity models performance.
Model
Number of
Maximum
Minimum
Average
Observations
Error, %
Error, %
Error, %
Khan
2343
81.6
-567
-28
Khan
150
66.1
-636
-60.8
Petrosky
2343
80.1
-214
4.9
Petrosky
150
44.8
-111
-1.4
Adapted Pedersen
2343
99.2
-384.
62
Adapted Pedersen
150
98.9
-382
54
This Work
2343
77.7
-317
0.9
This Work
150
58.7
-189
-0.7
48
o
Boiling Temperature F
TBP Distillation Curve
1400
T initial
T final
T average
1200
1000
800
600
400
200
0
0.0
20.0
40.0
60.0
80.0
100.0
Distilled Volume %
Fig. 1. True boiling point distillation curve for a standard oil.
Hydrocarbon Physical Properties (Katz & Firoozabadi)
1600
1.1
Tb(F)
SG
1200
0.9
Tb
S
G
800
0.7
400
0
0
250
500
750
1000
1250
0.5
1500
Molecular Weight
Fig. 2. True boiling point as a function of molecular weight for a standard oil.
49
Experimental Viscosity (cp)
3
2.5
2
1.5
Paraffins
1
Naphtenes
Aromatics
0.5
0
0
0.5
1
1.5
2
2.5
3
Predicted Viscosity (cp)
Fig. 3. Experimental Viscosity vs. Predicted Viscosity – Pure components.
5
Viscosity (cp)
4
3
2
1
0
300
350
400
450
500
550
600
T (K)
Fig. 4. Viscosity change with temperature.
50
650
1.6
Viscosity (cp)
1.5
1.4
1.3
1.2
1.1
1
0
50
100
150
200
250
P (ATM)
Fig. 5. Viscosity change with pressure.
Experimental Viscosity (cp)
6
OIL A
5
OIL 3
4
OIL 4
OIL 5
3
OIL 6
OIL 8
2
OIL 9
OIL 11
1
0
0
1
2
3
4
5
6
Predictive Viscosity (cp)
Fig. 6. Experimental Viscosity vs. Predicted Viscosity – Oil samples.
51
N p, G p, W p
B e g in n in g
End
G I, W I
V g = (R s i -R s )V o B g
V oi = N B oi
V o = (N -N p )B o
T im e
in te rv a l
V wi
Vw
V re
Pi
P fin a l
W a te r In flu x
Fig. 7. PVT Properties Used in Material Balance Computations.
P3
P2
P1
Gas
V1
P1
V2
V3
Oil
Oil
Hg
Hg
P5
P4
Hg
Gas
V4
Oil
Hg
Hg
V5
Hg
T = TR
P
Pb
Vb
V
Fig. 8. Crude Oil Constant Composition Expansion Test.
52
Gas off
Gas off
Gas
Oil
Hg
Stage 1
Stage 0
P = Pb
Stage 2
Stage N
P= atm
Fig. 9. Crude Oil Differential Liberation Test.
10000
Vcm
Tb (R) (ex)
Vcm (ex)
Mwm
1200
Tb (R)
Mwm (ex)
1000
800
600
3
100
400
10
Molecular Weight
Vcm /ft / lb-mol, Tb /R
1000
200
1
0
0.5
0.6
0.7
0.8
Oil Specific Gravity
0.9
1
Fig. 10. Pseudo-critical Volume (Vcm), Mixture Molecular Weight (Mwm) and Normal
Boiling Point (Tb) as a Function of Oil Specific Gravity.
53
100
540
Tcm (K)
Mwm
Tcm (K)
90
500
85
480
Mwm (g/gmol)
95
520
80
Differential Liberation Test (API = 45 T = 319 K)
460
0
50
100
150
Pressure (bar)
75
250
200
Fig. 11. Removed Data at P=200 Bar Due to Inconsistent Physical Trend.
0.7
0.76
Differential Liberation Test (API = 35.7, T = 380 K)
0.74
0.72
3
0.5
0.4
0.7
0.3
0.68
0.2
Viscosity (cp)
0.1
40
60
80
Density ( g/cm3)
100
120
140
0.66
160
180
Pressure (bar)
Fig. 12. Removed End Point Viscosity. Violation of Monotonic Behavior.
54
Density (g/cm )
Viscosity (cp)
0.6
1
0.72
Differential Liberation Test (API = 45, T = 319 K)
0.4
3
0.68
0.6
0.64
0.2
Viscosity (cp)
0
0
Density ( g/cm3)
50
100
150
Density (g/cm )
Viscosity (cp)
0.8
0.6
200
250
Pressure (bar)
Fig. 13. Removed Oil Density at P = 200 Bar Due to Inconsistent Trend.
100
Experimental Viscosity (cp)
Khan et al.
10
1
0.1
0.1
1
10
100
Predicted Viscosity (cp)
Fig. 14. Predicted Viscosity vs. Experimental Viscosity – Khan et al. model (Data Set 1)
55
100
Experimental Viscosity (cp)
Petrosky
10
1
0.1
0.1
1
10
100
Predicted Viscosity (cp)
Fig. 15. Predicted Viscosity vs. Experimental Viscosity – Petrovsky model (Data Set 1)
Experimental Viscosity (cp)
100
Untuned Adapted
Pedersen
10
1
0.1
0.1
1
10
100
Predicted Viscosity (cp)
Fig. 16. Predicted Viscosity vs. Experimental Viscosity – Untuned Adapted Pedersen’s
model (Data Set 1).
56
100
Experimental Viscosity (cp)
This Work
10
1
0.1
0.1
1
10
100
Predicted Viscosity (cp)
Fig. 17. Predicted Viscosity vs. Experimental Viscosity – This work (Data Set 1)
1000
Experimental Viscosity (cp)
Khan et al.
100
10
1
0.1
0.1
1
10
100
1000
Predicted Viscosity (cp)
Fig. 18. Predicted Viscosity vs. Experimental Viscosity – Khan model (Data Set 2 –Core
Lab).
57
1000
Experimental Viscosity (cp)
Petrosky
100
10
1
0.1
0.1
1
10
100
1000
Predicted Viscosity (cp)
Fig. 19. Predicted Viscosity vs Experimental Viscosity – Petrosky model (Data Set 2 –
Core Lab).
Experimental Viscosity (cp)
1000
Untuned Adapted
Pedersen
100
10
1
0.1
0.1
1
10
100
1000
Predicted Viscosity (cp)
Fig. 20. Predicted Viscosity vs. Experimental Viscosity – Untuned Adapted Pedersen’s
model (Data Set 2 – Core Lab).
58
1000
Experimental Viscosity (cp)
This Work
100
10
1
0.1
0.1
1
10
100
1000
Predicted Viscosity (cp)
Fig. 21. Predicted Viscosity vs. Experimental Viscosity – This work (Data Set 2 – Core
Lab).
Experimental Viscosity (cp)
100
Adapted Pedersen Model Tuned by Set
10
1
0.1
0.1
1
10
100
Predicted Viscosity (cp)
Fig. 22. Predicted Viscosity vs. Experimental Viscosity – Adapted Pedersen’s Model
tuned per set.
59
APPENDIX
The tables listed in the Appendix have been prepared in a such a way to maximize the
amount of information in the minimum possible space. A brief explanation is provided
here. The first three columns list the single carbon number fractions (SCN) along with
the corresponding molar fractions and molecular weights obtained from the TBP tests.
The first six rows also list the thermodynamic properties; Tb, Tc, Pc ,Vc and w, for the
first six single carbon number fractions. The thermodynamic properties of the first four
fractions are constant for all the oil samples and we found only small variations on the
values of the fifth and sixth fractions, therefore, we used always the same values for the
first six fractions for all the oils considered in this work. The rows from the seventh on
present the thermodynamic properties corresponding to the pseudo-components
calculated using Whitson's procedure68. A small independent table on the right bottom
corner summarizes the information corresponding to the pseudo-components including,
number, molar fractions, molecular weights and thermodynamic properties. The oil
samples that are listed by number are data taken from Schou Pederssen et al.66.
Component
C1
C2
C3
C4
C5
C6
C7 *
C8 *
C9 *
C10*
C11
C12
C13
C14
C15
C16
C17
Table A1. Oil A Composition and Properties
Molar
Molecular
Tb
Tc
Pc
Vc
ω
Fractions Weight
(K)
(K)
(bar)
(cm3/mol)
0.0047
16.0
111.7
190.6
45.4
98.0
0.008
0.0210
30.1
184.5
305.4
48.2
148.0
0.098
0.0423
43.5
231.1
369.8
41.9
203.0
0.152
0.0617
56.8
272.7
425.2
37.5
255.0
0.193
0.1058
71.1
309.2
469.6
33.3
304.0
0.231
0.1336
84.8
341.9
507.4
29.3
370.0
0.296
0.1135
88.1
372.0
540.5
26.9
427.8
0.353
0.0943
99.7
469.1
639.7
19.5
663.3
0.519
0.0785
112.6
557.2
717.3
14.0
975.7
0.709
0.0544
131.8
752.5
852.8
5.0
2796.7
1.430
0.0426
146.7
0.0363
160.1
0.0310
173.9
0.0278
186.0
0.0212
201.3
Number of Pseudo-components = 4
0.0186
212.9
MW Limits
Comp.
Molar
Average
Range
Fractions
MW
0.0162
230.4
88.10 130.02
7-9
0.3529
98.46
60
C18
C19
C20
0.0118
0.0064
0.0783
244.6
252.6
417.9
130.02
191.88
283.17
191.88
283.17
417.90
10 - 14
15 - 19
20
0.2928
0.1289
0.1350
157.50
219.66
417.9
*
Thermodynamic properties correspond to the pseudo-component fractions calculated on
the right bottom of the table.
Table A2. Oil 1 Composition and Properties
Component
Molar
Molecular
Tb
Tc
Pc
Vc
ω
Fractions Weight
(K)
(K)
(bar) (cm3/mol)
C1
0.0013
16.00
111.7
190.6
45.4
98.0
0.008
C2
0.0050
30.10
184.5
305.4
48.2
148.0
0.098
C3
0.0047
44.10
231.1
369.8
41.9
203.0
0.152
C4
0.0117
58.10
272.7
425.2
37.5
255.0
0.193
C5
0.0158
72.10
309.2
469.6
33.3
304.0
0.231
C6
0.0189
86.20
341.9
507.4
29.3
370.0
0.296
C7 *
0.0534
90.90
399.7 570.3000 24.58
487.130
0.396
*
C8
0.0854
105.00
483.45 653.1300 18.52
705.720
0.548
*
C9
0.0704
117.70
564.86 723.7000 13.56 1011.850
0.726
C10*
0.0680
132.00
662.69 806.0000 8.72
1398.140
1.022
C11*
0.0551
148.00
825.17 890.6100 3.20
3143.290
1.88
C12
0.0500
159.00
C13
0.0558
172.00
C14
0.0508
185.00
C15
0.0380
197.00
C16
0.0267
209.00
C17
0.0249
227.00
C18
0.0214
243.00
C19
0.0223
254.00
C20
0.0171
262.00
C21
0.0142
281.00
C22
0.0163
293.00
C23
0.0150
307.00
C24
0.0125
320.00
C25
0.0145
333.00
Number of Pseudo-components = 5
C26
MW Limits
Comp.
Molar
Average
0.0133
346.00
Range Fractions
MW
C27
0.0123
361.00
90.90 133.6300 7 - 10
0.2772
114.83
C28
0.0115
374.00
133.63 196.43 11 - 14
0.2117
165.80
C29
0.0109
381.00
196.43 288.76 15 - 21
0.1646
231.19
C30
0.1828
624.00
288.76 424.48 22 - 29
0.1063
335.89
424.49
624
30
0.1191
624.00
*
Thermodynamic properties correspond to the pseudo-component fractions calculated on
the right bottom of the table. Data from Schou Pedersen et al.66.
61
Table A3. Oil 3 Composition and Properties
Molar
Molecular
Tb
Tc
Pc
Vc
ω
Fractions Weight
(K)
(K)
(bar) (cm3/mol)
C1
0.0000
16.00
111.7
190.6
45.4
98.0
0.008
C2
0.0001
30.1
184.5
305.4
48.2
148.0
0.098
C3
0.0047
44.1
231.1
369.8
41.9
203.0
0.152
C4
0.0209
58.1
272.7
425.2
37.5
255.0
0.193
C5
0.1876
72.1
309.2
469.6
33.3
304.0
0.231
C6
0.0437
86.2
341.9
507.4
29.3
370.0
0.296
C7 *
0.0900
92.3
391.41 561.5100 25.3
468.780
0.383
*
C8
0.1071
105.9
487.25 656.6200 18.3
717.360
0.555
C9 *
0.0732
120.3
576.75 733.5700 12.9
1072.430
0.749
*
C10
0.0623
133.0
720.82 810.1700
7.9
1407.900
1.026
C11*
0.0550
148.0
820.08 887.9600
3.3
3037.110
1.844
C12
0.0514
163.0
C13
0.0443
177.0
C14
0.0480
190.0
C15
0.0381
204.0
C16
0.0282
217.0
C17
0.0333
235.0
C18
0.0234
248.0
C19
0.0266
260.0
C20
0.0418
269.0
C21
0.0171
283.0
C22
0.0148
298.0
C23
0.0156
310.0
C24
0.0113
322.0
C25
0.0112
332.0
Number of Pseudo-components = 5
C26
MW Limits
Comp.
Molar
Average
0.0097
351.0
Range Fractions
MW
C27
0.0110
371.0
92.30
134.74
7 - 10
0.3326
110.46
C28
0.0073
382.0
134.74 196.71 11 - 14
0.1987
168.49
C29
0.0088
394.0
196.71 287.17 15 - 21
0.2085
242.30
C30
0.0811
612.0
287.17 419.22 22 - 29
0.0897
338.29
419.22 612.00
30
0.0811
612.00
*
Thermodynamic properties correspond to the pseudo-component fractions calculated on
the right bottom of the table. Data from Schou Pedersen et al. 66.
Component
62
Table A4. Oil 4 Composition and Properties
Component
C1
C2
C3
C4
C5
C6
C7 *
C8 *
C9 *
C10*
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20
Molar
Molecular
Fractions Weight
0.0003
16.0
0.0013
30.1
0.0036
44.1
0.0074
58.1
0.0152
72.1
0.0266
86.2
0.0925
89.8
0.1714
101.4
0.1190
116.1
0.0800
134.0
0.0605
148.0
0.0526
161.0
0.0570
175.0
0.0427
189.0
0.0379
203.0
0.0286
0.0282
0.0198
0.0204
0.1350
216.0
233.0
248.0
260.0
480.0
Tb
(K)
111.7
184.5
231.1
272.7
309.2
341.9
387.56
445.95
570.84
756.00
Tc
(K)
190.6
305.4
369.8
425.2
469.6
507.4
557.38
617.42
728.66
854.78
Pc
(bar)
45.4
48.2
41.9
37.5
33.3
29.3
25.57
21.07
13.22
4.89
Vc
(cm3/mol)
98.0
148.0
203.0
255.0
304.0
370.0
460.42
599.72
1041.54
2788.09
ω
0.008
0.098
0.152
0.193
0.231
0.296
0.378
0.475
0.738
1.449
Number of Pseudo-components = 4
MW Limits
Comp.
Molar
Average
Range
Fractions
MW
89.80
136.54
7 – 10
0.4630
108.47
136.54 207.62 11 - 15
0.2510
141.31
207.62 315.68 16 - 19
0.0970
236.73
315.68 480.00
20
0.1350
480.00
*
Thermodynamic properties correspond to the pseudo-component fractions calculated on
the right bottom of the table. Data from Schou Pedersen et al. 66.
Table A5. Oil 5 Composition and Properties
Component
C1
C2
C3
C4
C5
C6
C7 *
C8 *
C9 *
C10*
Molar
Molecular
Fractions Weight
0.0005
16.0
0.0037
30.1
0.0117
44.1
0.0193
58.1
0.0236
72.1
0.0247
86.2
0.0652
88.8
0.0858
101.8
0.0486
116.1
0.0280
133.0
Tb
(K)
111.7
184.5
231.1
272.7
309.2
341.9
372.82
470.97
556.61
723.18
63
Tc
(K)
190.6
305.4
369.8
425.2
469.6
507.4
541.4
641.5
716.85
837.51
Pc
(bar)
45.4
48.2
41.9
37.5
33.3
29.3
26.83
19.34
14.04
6
Vc
(cm3/mol)
98.0
148.0
203.0
255.0
304.0
370.0
429.42
668.76
973.24
1783.69
ω
0.008
0.098
0.152
0.193
0.231
0.296
0.354
0.523
0.708
1.278
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20
0.0298
0.0308
0.0364
0.0363
0.0359
143.0
154.0
167.0
181.0
195.0
0.0304
0.0360
0.0325
0.0307
0.3881
207.0
225.0
242.0
253.0
423.0
Number of Pseudo-components = 4
MW Limits
Comp.
Molar
Average
Range Fractions
MW
88.80 131.19
7–9
0.1996
101.0356
131.19 193.81 10 - 14
0.1613
157.3323
193.81 286.32 15 - 19
0.1655
223.7184
286.32 423.00
20
0.3881
423.0000
*
Thermodynamic properties correspond to the pseudo-component fractions calculated on
the right bottom of the table. Data from Schou Pedersen et al. 66.
Table A6. Oil 6 Composition and Properties
Component
C1
C2
C3
C4
C5
C6
C7 *
C8 *
C9 *
C10*
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20
Molar
Molecular
Fractions Weight
0.0002
16.0
0.0020
30.1
0.0085
44.1
0.0160
58.1
0.0211
72.1
0.0239
86.2
0.0641
88.8
0.0884
101.8
0.0566
116.1
0.0376
133.0
0.0365
143.0
0.0366
154.0
0.0465
167.0
0.0439
181.0
0.0451
195.0
0.0386
0.0424
0.0383
0.0353
0.3181
209.0
229.0
245.0
258.0
412.0
Tb
(K)
111.7
184.5
231.1
272.7
309.2
341.9
374.13
470.45
558.76
716.27
Tc
(K)
190.6
305.4
369.8
425.2
469.6
507.4
542.84
641
718.64
833.91
Pc
(bar)
45.4
48.2
41.9
37.5
33.3
29.3
26.72
19.38
13.92
6.26
Vc
(cm3/mol)
98.0
148.0
203.0
255.0
304.0
370.0
432.13
667.24
983.07
1730.53
ω
0.008
0.098
0.152
0.193
0.231
0.296
0.356
0.522
0.713
1.245
Number of Pseudo-components = 4
MW Limits
Comp.
Molar
Average
Range
Fractions
MW
88.80
130.33
7–9
0.2091
101.69
130.33 191.27 10 - 14
0.2011
156.98
191.27 280.72 15 - 19
0.1997
225.65
280.72 412.00
20
0.3181
412.00
*
Thermodynamic properties correspond to the pseudo-component fractions calculated on
the right bottom of the table. Data from Schou Pedersen et al. 66.
64
Table A7. Oil 6 Composition and Properties
Tc
Pc
Vc
Component
Molar
Molecular
Tb
ω
(K)
(K)
(bar)
(cm3/mol)
Fractions Weight
C1
0.0000
16.0
111.7
190.6
45.4
98.0
0.008
C2
0.0011
30.1
184.5
305.4
48.2
148.0
0.098
C3
0.0121
44.1
231.1
369.8
41.9
203.0
0.152
C4
0.0474
58.1
272.7
425.2
37.5
255.0
0.193
C5
0.0524
72.1
309.2
469.6
33.3
304.0
0.231
C6
0.0549
86.2
341.9
507.4
29.3
370.0
0.296
*
C7
0.0983
92.8
391.35 561.44
25.26
468.64
0.383
C8 *
0.1065
106.3
499.5
667.73
17.49
756.16
0.581
*
C9
0.0710
120.9
580.54 736.72
12.67
1093.2
0.756
C10*
0.0606
134.0
789.18 871.87
4.06
2500.5
1.654
C11
0.0508
148.0
C12
0.0420
161.0
C13
0.0447
175.0
C14
0.0341
189.0
C15
0.0325
203.0
Number of Pseudo-components = 4
Comp.
Molar
Average
C16
0.0270
216.0
MW Limits
Range
Fractions
MW
C17
0.0283
233.0
92.80
144.40
7 – 10
0.3364
110.43
C18
0.0204
248.0
144.40 224.68 11 - 16
0.2311
177.31
C19
0.0230
260.0
224.68 349.61 17 - 19
0.0717
245.93
C20
0.1988
544.0
349.61 544.00
20
0.1988
544.00
*
Thermodynamic properties correspond to the pseudo-component fractions calculated on
the right bottom of the table. Data from Schou Pedersen et al. 66.
Table A8. Oil 9 Composition and Properties
Component
Molar
Molecular
Tb
Tc
Pc
Fractions Weight
(K)
(K)
(bar)
C1
0.0000
16.0
111.7
190.6
45.4
C2
0.0017
30.1
184.5
305.4
48.2
C3
0.0129
44.1
231.1
369.8
41.9
C4
0.0246
58.1
272.7
425.2
37.5
C5
0.0283
72.1
309.2
469.6
33.3
C6
0.0281
86.2
341.9
507.4
29.3
*
C7
0.0621
90.5
387.21
557
25.6
*
C8
0.0716
104.2
487.39 656.75
18.26
C9 *
0.0505
119.2
563.73 722.77
13.63
*
C10
0.0329
134.0
738.24 845.35
5.47
C11
0.0467
149.0
65
Vc
(cm3/mol)
98.0
148.0
203.0
255.0
304.0
370.0
459.64
7117.79
1006.43
1910.84
ω
0.008
0.098
0.152
0.193
0.231
0.296
0.376
0.556
0.723
1.353
C12
C13
C14
C15
C16
C17
C18
C19
C20
0.0345
0.0434
0.0387
0.0449
164.0
176.0
188.0
203.0
0.0281
0.0360
0.0308
0.0367
0.3436
214.0
232.0
248.0
259.0
446.0
Number of Pseudo-components = 4
Comp.
Molar
Average
MW Limits
Range
Fractions
MW
92.50
134.84
7 – 10
0.2171
108.29
134.80 200.91 11 – 14
0.1633
168.59
200.91 299.34 15 - 19
0.1765
230.16
299.34 446.00
20
0.3436
448.00
*
Thermodynamic properties correspond to the pseudo-component fractions calculated on
the right bottom of the table. Data from Schou Pedersen et al. 66.
Component
C1
C2
C3
C4
C5
C6
C7 *
C8 *
C9 *
C10*
C11
C12
C13
C14
C15
C16
C17
C18
C19
C20
Table A9. Oil 11 Composition and Properties
Molar
Molecular
Tb
Tc
Pc
Vc
ω
Fractions Weight
(K)
(K)
(bar)
(cm3/mol)
0.0000
16.0
111.7
190.6
45.4
98.0
0.008
0.0010
30.1
184.5
305.4
48.2
148.0
0.098
0.0012
44.1
231.1
369.8
41.9
203.0
0.152
0.0021
58.1
272.7
425.2
37.5
255.0
0.193
0.0021
72.1
309.2
469.6
33.3
304.0
0.231
0.0045
86.2
341.9
507.4
29.3
370.0
0.296
0.0121
90.8
406.93 577.89
24
503.59
0.408
0.0187
106.5
498.88 667.17
17.53
754.12
0.58
0.0195
122.0
571.68 729.36
13.17
1045.85
0.739
0.0556
135.0
752.50 852.80
5.01
2047.81
1.427
0.0472
149.0
0.0549
162.0
0.0640
176.0
0.0681
189.0
0.0539
202.0
Number of Pseudo-components = 4
Comp.
Molar
Average
MW Limits
0.0358
213.0
Range
Fractions
MW
0.0487
230.0
90.50
137.18
7 – 10
0.1259
118.73
0.0489
244.0
137.18 207.24 11 – 15
0.2881
176.85
0.0404
256.0
207.24 313.10 16 - 19
0.1538
237.52
0.3976
473.0
313.10 473.00
20
0.3976
473
*
Thermodynamic properties correspond to the pseudo-component fractions calculated on
the right bottom of the table. Data from Schou Pedersen et al. 66.
66