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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 Maria Barrufet Prairie View A&M State University Texas A&M University Department of Chemical Engineering Petroleum Engineering
Department Prairie View, TX 77429 College Station TX, 77204 TELE:(936) 857-2427 TELE:(979) 845-0314 FAX: (936) 857-4540 FAX:(979) 845-0325 EMAIL:[email protected] EMAIL:barrufet@spindletop. tamu.edu
1
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.
2
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.
3
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
4
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
5
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.
6
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.
7
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
8
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
evaluated at the same reduced conditions. Ely and Hanley10 (1981) proposed the
following extended corresponding states model:
µi(ρ,T) = µo(f
T ,h ρoi,
oi, ) f oi, h oi, /1/2-2/32/1)( M oM i (1)
h ,oi = h ,oi φ oi, )ρ oc,/ρ ic,( (2)
f ,oi = θ oi, )T oc,/T ic,( (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 non-
correspondence 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.
9
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. For 38 compounds, the modified
method of Hwang and Whiting16 (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 oc,/T xc, )-1/6 ( P oc,/ P xc, )2/3 α /α )Mo/Mi( 2/1oTG,xTG, µo( T ,P *
o*o ) (4),
and
T*o = )α /α( )T oc,/T xc,(
xTG,oTG, (5)
P*o = )α /α( P oc,P xc, xTG,oTG,
)/( (6),
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 .
10
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 non-
spherical fluid. In this case the subscripts r1 and r2 refer to two fluids made-up of non-
spherical 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 = )(/ )2/1(3/2 TMV cc . 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,
11
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).
12
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 3-
D 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. The EOS involves 4 constants and 2
temperature dependent parameters.
13
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( x )f( iim µµ (14).
where )f(µ is the viscosity function normally linear, hyperbolic or logarithmic in form.
A common equation used successfully for liquid hydrocarbons is,
∑= ) x( 1/3ii
3
m µµ (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,
G xx )(ln x )(ln ijjiiim ∑∑+∑= µµ (16),
where Gij is an interaction parameter. Repeated coefficients are equal to zero. The
binary form of Grunberg equation is given by,
G x x ln x ln x )(ln 12212211m ++= µµµ (17).
The interaction parameters are system dependent and sometimes temperature
dependent and therefore difficult to generalize. Errors from 2.3% for non-polar/non-
polar 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
14
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%. Pedersen and
Fredenslund18 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.
15
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,
( ) ( )oooo
m
o
m
co
cm
co
cmm TP
MWMW
PP
TT
TP ,,321
µααµ
ααα
= , (18),
where the coefficients α1, α2 and α3 in Pedersen's model are -1/6, 2/3 and 1/2
respectively. 5173.0847.1310378.7000.1 mrom MWρ×+=α − (19)
847.1031.0000.1 roo ρ+=α (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),
16
co
cm
co
cm
coo
ro
TTT
PPP
ρ
ρ
=ρ,
(21),
the pressures and temperatures at which the reference viscosity (µo) is evaluated are given
by,
mcm
ocoo P
PPPαα= and
mcm
ocoo T
TTTαα= (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
( )TPTT 82/11C10 105043.11906.1681847.7998-exp −−− ×+×+×=ρ . (23),
while the viscosity, in cp, is given by,
PT1087.8T107057.6P001272.0TP0.4775T8775.2881- T.54183212
T150991.51
739
2/13/1C10
××+××−×−
+××+=µ
−−
−−
(24).
17
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,
18
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.
19
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: cb
baT γψ = (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:
[ ]γγγψ bbcb
b fTedTaT ++= exp (26)
[ ]γγγψ wwcb
w fMedMaM ++= exp (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.
20
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),
( ) ( )( ) 3
122
7
2w
1098.1818828.102226.080882.01107972034371
02058.077084.013287.36523.44.486,96.272,12M
bbbb
b
TTTT.-.
T
×
−+−+×
×−−+−++−=−
γγ
γγγγ
(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: ( )
( ) 1)1/77.14/log3
−−
=ωbc
c
TTP
(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.
21
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:
[ ])log(3.31 nNIntNg −+= (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: I
n
N
gnI Mw
MwN
MwMw
= ln 1exp (32),
22
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 pseudo-
fractions with the molecular weights shown in the Table. The components with
molecular weights between pseudo-components k-1 and k are ascribed to pseudo-
component 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.
23
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.
6. Wong and Sandler69 mixing rules were used to calculate the pseudo-
components 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.
24
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.
25
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
26
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 bubble-
point. 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.
27
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
28
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
29
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
30
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.
31
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 al.74 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 bubble-
point.
32
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, pseudo-
critical 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.
( )[ ] 15
1
41410 −
=
− ρ=+ξµ−µ ∑ ir
ii
/* a (33)
Here *µ is the low-pressure gas mixture viscosity, and ξ is the viscosity-reducing
parameter, which is defined as, 322161 /
cm/
wm/
cm PMT=ξ (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(f
T ,h ρoi,
oi, ) f oi, h oi, /1/2-2/32/1)( M oM i (1)
h ,oi = h ,oi φ oi, )ρ oc,/ρ ic,( (2)
f ,oi = θ oi, )T oc,/T ic,( (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).
33
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.
( ) ( )oooo
m
o
m
co
cm
co
cmm TP
MWMW
PP
TTTP ,,
321
µ
αα
=µ
ααα
, (18),
where the coefficients α1, α2 and α3 in Pedersen's model are -1/6, 2/3 and 1/2
respectively. 5173.0847.1310378.7000.1 mrom MWρ×+=α − (19)
847.1031.0000.1 roo ρ+=α (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)
co
cm
co
cm
coo
ro
TTT
PPP
ρ
ρ
=ρ,
(21),
the pressures and temperatures at which the reference viscosity (µo) is evaluated are given
by,
mcm
ocoo P
PPPαα= and
mcm
ocoo T
TTTαα= (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.
34
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
( )TP105043.1T1906.168T1847.7998-exp 82/11C10
−−− ×+×+×=ρ . (23),
while the viscosity, in cp, is given by,
PT1087.8T107057.6P001272.0TP0.4775T8775.2881- T.54183212
T150991.51
739
2/13/1C10
××+××−×−
+××+=µ
−−
−−
(24).
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). Once this was determined the
35
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: 3
,,,
4193.447615431.82548-1385.934312-3540.53 RoRoRo
bT γ×+γ×γ
=
(35)
)exp(0.002264.611 bm TMW ××= (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:
24.2787 0.3596,
0.58848Robcm TT γ××= (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,
36
( )
( )
µ+
ρρ+×−
×−×
γ×
×=µ −−γ−
oo.c
cm
ob
o
sbm
sb
Roscmcmm
TPT
VPBB
APIRMWMW
RR
VV
TTTTP
Ro
,1930.01388.202359.02606.0exp
,
C10C10
C10
10C
3
2902.24471.0-3.9243
10C
1362.0,
0.9841
10cC
0286.1
210cC
,
(38),
where Bo, the formation volume factor, is dimensionless, Ro ,γ 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
37
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.
38
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.
39
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 ,oi = Parameter defined in Eqn. (1)
Gij = Interaction parameter used in Eqn. (16)
h ,oi = 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)
40
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.
41
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46
TABLES AND FIGURES
Table 1. Typical results of a TBP test.
Component Ti Tf T mean ∆∆∆∆V (cm3) ΣΣΣΣ(∆∆∆∆V) V % OffHypo1 99 220 159.5 5.1 5.1 5.3Hypo2 214 323 268.5 8.0 13.1 13.5Hypo3 323 432 377.5 7.9 21.0 21.7Hypo4 432 526 479 8.1 29.1 30.1Hypo5 526 612 569 7.9 37.0 38.2Hypo6 612 693 652.5 7.9 44.9 46.4Hypo7 693 765 729 7.9 52.8 54.5Hypo8 765 821 793 7.8 60.6 62.6Hypo9 821 908 864.5 8.1 68.7 71.0Hypo10 908 1010 959 5.2 73.9 76.3Residual 1261.1692 22.9 96.8 100.0Whole Oil 729Residual Volume Left 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 4.0846 -1.2028
d 5.43076x10-4 -5.1747x10-4 -4.725x10-3 -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
Form (2)
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
47
Table 3. Grouping data for characterization of fractions with up to 45 components.
Group Molecular Weight
1 127
2 170
3 227
4 303
5 404
6 539
Table 4. Differential Vaporization Test at 80 °F. Pressure (psig)
Rs [1] (SCF/ STB)
Bo [2] (RB/ STB)
Oil Density ρρρρo (gm/cc)
Gas Deviation Factor Z
Bg [3] (RCF/ SCF)
Gas Gravity γγγγg
1690 210 1.069 0.9022 1500 188 1.063 0.9052 0.822 0.00825 0.581 1300 165 1.056 0.9083 0.835 0.00966 0.576 1100 141 1.049 0.9113 0.852 0.01162 0.573 900 117 1.042 0.9143 0.872 0.01450 0.571 700 93 1.036 0.9174 0.896 0.01906 0.572 500 68 1.029 0.9205 0.922 0.02724 0.574 300 42 1.022 0.9235 0.951 0.04593 0.581 100 15 1.014 0.9268 0.983 0.13004 0.600 0 0 1.008 0.9305 0.724
Gravity of Residual Oil = 19.2°API @ 60°F [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.
48
Table 5. Viscosity Data Accompanying DL Set at 80 °F and 19.2 oAPI Residual Oil.
Pressure (psig)
Oil Viscosity (cp)
Calculated Gas Viscosity (cp)
Oil/Gas Viscosity 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. Dataset No Points. Variable Minimum Maximum
#1 2343 Oil Density: lbm/ft3 (g/cm3) 35.11 (0.562) 57.31(0.92) #2 150 Oil Density, lbm/ft3 (g/cm3) 24.31 (0.389) 57.50(0.921) #1 2343 Oil Viscosity, cp 0.132 78.30 #2 150 Oil Viscosity, cp 0.13 68.90 #1 2343 Temperature, R (K) 540 (300) 766 (425.5) #2 150 Temperature, R (K) 537 (303.9) 762 (423.3) #1 2343 Pressure, psia (MPa) 14.7 (0.1) 5601.7 (38.62) #2 150 Pressure, psia (MPa) 102.7 (0.708) 5434.7 (37.47)
Table 7. Summary of the black oil viscosity models performance. Model Number of
Observations Maximum Error, %
Minimum Error, %
Average 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
49
TBP Distillation Curve
0200400600
800100012001400
0.0 20.0 40.0 60.0 80.0 100.0
Distilled Volume %
Bo
ilin
g T
em
pe
ratu
re o
F T initialT finalT average
Fig. 1. True boiling point distillation curve for a standard oil.
Hydrocarbon Physical Properties (Katz & Firoozabadi)
0
400
800
1200
1600
0 250 500 750 1000 1250 1500Molecular Weight
Tb
0.5
0.7
0.9
1.1
SG
Tb(F) SG
Fig. 2. True boiling point as a function of molecular weight for a standard oil.
50
0
0.5
1
1.5
2
2.5
3
0 0.5 1 1.5 2 2.5 3
Predicted Viscosity (cp)
Expe
rimen
tal V
isco
sity
(cp)
ParaffinsNaphtenesAromatics
Fig. 3. Experimental Viscosity vs. Predicted Viscosity � Pure components.
0
1
2
3
4
5
300 350 400 450 500 550 600 650
T (K)
Visc
osity
(cp)
Fig. 4. Viscosity change with temperature.
51
1
1.1
1.2
1.3
1.4
1.5
1.6
0 50 100 150 200 250
P (ATM)
Visc
osity
(cp)
Fig. 5. Viscosity change with pressure.
0
1
2
3
4
5
6
0 1 2 3 4 5 6
Predictive Viscosity (cp)
Expe
rimen
tal V
isco
sity
(cp) OIL A
OIL 3OIL 4OIL 5OIL 6OIL 8OIL 9OIL 11
Fig. 6. Experimental Viscosity vs. Predicted Viscosity � Oil samples.
52
V w i
V o i = N B o i V o= (N -N p)B o
V g= (R si-R s)V oB g
V wV re
W ate r In flu x
G I, W I
N p, G p, W p
T im e in te rv a l
B eg in n in g E n d
P i P fin a l
V w i
V o i = N B o i V o= (N -N p)B o
V g= (R si-R s)V oB g
V wV re
W ate r In flu x
G I, W I
N p, G p, W p
T im e in te rv a l
B eg in n in g E n d
P i P fin a l
V w i
V o i = N B o i V o= (N -N p)B o
V g= (R si-R s)V oB g
V wV re
W ate r In flu x
G I, W I
N p, G p, W p
T im e in te rv a l
B eg in n in g E n d
P i P fin a l
Fig. 7. PVT Properties Used in Material Balance Computations.
V2 V3 V4 V5
P2
Hg
Gas
Oil
Hg
Gas
Oil
Hg
P3
Gas
Oil
HgHg Hg
P1
P1
Hg
P
V
Pb
Vb
P4P5
T = TR
V1V2 V3 V4 V5
P2
Hg
Gas
Oil
Hg
Gas
Oil
Hg
P3
Gas
Oil
HgHg Hg
P1
P1
Hg
P
V
Pb
Vb
P4P5
T = TR
V1
Fig. 8. Crude Oil Constant Composition Expansion Test.
53
Gas
Oil
Hg
Stage 1 Stage 2 Stage NP= atm
Stage 0P = Pb
Gas off Gas off
Gas
Oil
Hg
Stage 1 Stage 2 Stage NP= atm
Stage 0P = Pb
Gas off Gas off
Gas
Oil
Hg
Stage 1 Stage 2 Stage NP= atm
Stage 0P = Pb
Gas off Gas off
Gas
Oil
Hg
Stage 1 Stage 2 Stage NP= atm
Stage 0P = Pb
Gas off Gas off
Fig. 9. Crude Oil Differential Liberation Test.
1
10
100
1000
10000
0.5 0.6 0.7 0.8 0.9 1Oil Specific Gravity
Vcm
/ft 3 / l
b-m
ol,
Tb /R
0
200
400
600
800
1000
1200
Mol
ecul
ar W
eigh
t
Vcm Vcm (ex) Tb (R)Tb (R) (ex) Mwm Mwm (ex)
Fig. 10. Pseudo-critical Volume (Vcm), Mixture Molecular Weight (Mwm) and Normal Boiling Point (Tb) as a Function of Oil Specific Gravity.
54
Differential Liberation Test (API = 45 T = 319 K) 460
480
500
520
540
0 50 100 150 200 250Pressure (bar)
Tcm
(K)
75
80
85
90
95
100
Mw
m (g
/gm
ol)
Tcm (K) Mwm
Differential Liberation Test (API = 45 T = 319 K) 460
480
500
520
540
0 50 100 150 200 250Pressure (bar)
Tcm
(K)
75
80
85
90
95
100
Mw
m (g
/gm
ol)
Tcm (K) Mwm
Fig. 11. Removed Data at P=200 Bar Due to Inconsistent Physical Trend.
Differential Liberation Test (API = 35.7, T = 380 K)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
40 60 80 100 120 140 160 180
Pressure (bar)
Visc
osity
(cp)
0.66
0.68
0.7
0.72
0.74
0.76
Den
sity
(g/c
m3 )
Viscosity (cp) Density ( g/cm3)
Differential Liberation Test (API = 35.7, T = 380 K)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
40 60 80 100 120 140 160 180
Pressure (bar)
Visc
osity
(cp)
0.66
0.68
0.7
0.72
0.74
0.76
Den
sity
(g/c
m3 )
Viscosity (cp) Density ( g/cm3)
Fig. 12. Removed End Point Viscosity. Violation of Monotonic Behavior.
55
Differential Liberation Test (API = 45, T = 319 K)
0
0.2
0.4
0.6
0.8
1
0 50 100 150 200 250
Pressure (bar)
Visc
osity
(cp)
0.6
0.64
0.68
0.72
Den
sity
(g/c
m3 )
Viscosity (cp) Density ( g/cm3)
Fig. 13. Removed Oil Density at P = 200 Bar Due to Inconsistent Trend.
0.1
1
10
100
0.1 1 10 100Predicted Viscosity (cp)
Expe
rimen
tal V
isco
sity
(cp)
Khan et al.
Fig. 14. Predicted Viscosity vs. Experimental Viscosity � Khan et al. model (Data Set 1)
56
0.1
1
10
100
0.1 1 10 100Predicted Viscosity (cp)
Expe
rimen
tal V
isco
sity
(cp)
Petrosky
Fig. 15. Predicted Viscosity vs. Experimental Viscosity � Petrovsky model (Data Set 1)
0.1
1
10
100
0.1 1 10 100Predicted Viscosity (cp)
Expe
rimen
tal V
isco
sity
(cp)
Untuned AdaptedPedersen
Fig. 16. Predicted Viscosity vs. Experimental Viscosity � Untuned Adapted Pedersen�s model (Data Set 1).
57
0.1
1
10
100
0.1 1 10 100Predicted Viscosity (cp)
Expe
rimen
tal V
isco
sity
(cp)
This Work
Fig. 17. Predicted Viscosity vs. Experimental Viscosity � This work (Data Set 1)
0.1
1
10
100
1000
0.1 1 10 100 1000Predicted Viscosity (cp)
Expe
rimen
tal V
isco
sity
(cp)
Khan et al.
Fig. 18. Predicted Viscosity vs. Experimental Viscosity � Khan model (Data Set 2 �Core
Lab).
58
0.1
1
10
100
1000
0.1 1 10 100 1000Predicted Viscosity (cp)
Expe
rimen
tal V
isco
sity
(cp)
Petrosky
Fig. 19. Predicted Viscosity vs Experimental Viscosity � Petrosky model (Data Set 2 �
Core Lab).
0.1
1
10
100
1000
0.1 1 10 100 1000Predicted Viscosity (cp)
Expe
rimen
tal V
isco
sity
(cp)
Untuned AdaptedPedersen
Fig. 20. Predicted Viscosity vs. Experimental Viscosity � Untuned Adapted Pedersen�s
model (Data Set 2 � Core Lab).
59
0.1
1
10
100
1000
0.1 1 10 100 1000Predicted Viscosity (cp)
Expe
rimen
tal V
isco
sity
(cp)
This Work
Fig. 21. Predicted Viscosity vs. Experimental Viscosity � This work (Data Set 2 � Core
Lab).
0.1
1
10
100
0.1 1 10 100Predicted Viscosity (cp)
Expe
rimen
tal V
isco
sity
(cp)
Adapted Pedersen Model -Tuned by Set
Fig. 22. Predicted Viscosity vs. Experimental Viscosity � Adapted Pedersen�s Model
tuned per set.
60
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.
Table A1. Oil A Composition and Properties Component Molar
Fractions Molecular
Weight Tb (K)
Tc (K)
Pc (bar)
Vc (cm3/mol)
ω
C1 0.0047 16.0 111.7 190.6 45.4 98.0 0.008 C2 0.0210 30.1 184.5 305.4 48.2 148.0 0.098 C3 0.0423 43.5 231.1 369.8 41.9 203.0 0.152 C4 0.0617 56.8 272.7 425.2 37.5 255.0 0.193 C5 0.1058 71.1 309.2 469.6 33.3 304.0 0.231 C6 0.1336 84.8 341.9 507.4 29.3 370.0 0.296 C7
* 0.1135 88.1 372.0 540.5 26.9 427.8 0.353 C8
* 0.0943 99.7 469.1 639.7 19.5 663.3 0.519 C9
* 0.0785 112.6 557.2 717.3 14.0 975.7 0.709 C10
* 0.0544 131.8 752.5 852.8 5.0 2796.7 1.430 C11 0.0426 146.7 C12 0.0363 160.1 C13 0.0310 173.9 C14 0.0278 186.0 C15 0.0212 201.3 Number of Pseudo-components = 4 C16 0.0186 212.9 MW Limits Comp.
Range Molar
Fractions Average
MW C17 0.0162 230.4 88.10 130.02 7 - 9 0.3529 98.46
61
C18 0.0118 244.6 130.02 191.88 10 - 14 0.2928 157.50 C19 0.0064 252.6 191.88 283.17 15 - 19 0.1289 219.66 C20 0.0783 417.9 283.17 417.90 20 0.1350 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
Fractions Molecular
Weight Tb (K)
Tc (K)
Pc (bar)
Vc (cm3/mol)
ω
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
0.0133 346.00 MW Limits Comp.
Range Molar
Fractions Average
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.
62
Table A3. Oil 3 Composition and Properties
Component Molar Fractions
MolecularWeight
Tb (K)
Tc (K)
Pc (bar)
Vc (cm3/mol)
ω
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
0.0097 351.0 MW Limits Comp.
Range Molar
Fractions Average
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.
63
Table A4. Oil 4 Composition and Properties
Component Molar
Fractions Molecular
Weight Tb (K)
Tc (K)
Pc (bar)
Vc (cm3/mol)
ω
C1 0.0003 16.0 111.7 190.6 45.4 98.0 0.008 C2 0.0013 30.1 184.5 305.4 48.2 148.0 0.098 C3 0.0036 44.1 231.1 369.8 41.9 203.0 0.152 C4 0.0074 58.1 272.7 425.2 37.5 255.0 0.193 C5 0.0152 72.1 309.2 469.6 33.3 304.0 0.231 C6 0.0266 86.2 341.9 507.4 29.3 370.0 0.296 C7
* 0.0925 89.8 387.56 557.38 25.57 460.42 0.378 C8
* 0.1714 101.4 445.95 617.42 21.07 599.72 0.475 C9
* 0.1190 116.1 570.84 728.66 13.22 1041.54 0.738 C10
* 0.0800 134.0 756.00 854.78 4.89 2788.09 1.449 C11 0.0605 148.0 C12 0.0526 161.0 C13 0.0570 175.0 C14 0.0427 189.0 C15 0.0379 203.0 Number of Pseudo-components = 4 C16
0.0286 216.0 MW Limits Comp.
Range Molar
Fractions Average
MW C17 0.0282 233.0 89.80 136.54 7 � 10 0.4630 108.47 C18 0.0198 248.0 136.54 207.62 11 - 15 0.2510 141.31 C19 0.0204 260.0 207.62 315.68 16 - 19 0.0970 236.73 C20 0.1350 480.0 315.68 480.00 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 Molar
Fractions Molecular
Weight Tb (K)
Tc (K)
Pc (bar)
Vc (cm3/mol)
ω
C1 0.0005 16.0 111.7 190.6 45.4 98.0 0.008 C2 0.0037 30.1 184.5 305.4 48.2 148.0 0.098 C3 0.0117 44.1 231.1 369.8 41.9 203.0 0.152 C4 0.0193 58.1 272.7 425.2 37.5 255.0 0.193 C5 0.0236 72.1 309.2 469.6 33.3 304.0 0.231 C6 0.0247 86.2 341.9 507.4 29.3 370.0 0.296 C7
* 0.0652 88.8 372.82 541.4 26.83 429.42 0.354 C8
* 0.0858 101.8 470.97 641.5 19.34 668.76 0.523 C9
* 0.0486 116.1 556.61 716.85 14.04 973.24 0.708 C10
* 0.0280 133.0 723.18 837.51 6 1783.69 1.278
64
C11 0.0298 143.0 C12 0.0308 154.0 C13 0.0364 167.0 C14 0.0363 181.0 C15 0.0359 195.0 Number of Pseudo-components = 4 C16
0.0304 207.0 MW Limits Comp.
Range Molar
Fractions Average
MW C17 0.0360 225.0 88.80 131.19 7 � 9 0.1996 101.0356C18 0.0325 242.0 131.19 193.81 10 - 14 0.1613 157.3323C19 0.0307 253.0 193.81 286.32 15 - 19 0.1655 223.7184C20 0.3881 423.0 286.32 423.00 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 Molar
Fractions Molecular
Weight Tb (K)
Tc (K)
Pc (bar)
Vc (cm3/mol)
ω
C1 0.0002 16.0 111.7 190.6 45.4 98.0 0.008 C2 0.0020 30.1 184.5 305.4 48.2 148.0 0.098 C3 0.0085 44.1 231.1 369.8 41.9 203.0 0.152 C4 0.0160 58.1 272.7 425.2 37.5 255.0 0.193 C5 0.0211 72.1 309.2 469.6 33.3 304.0 0.231 C6 0.0239 86.2 341.9 507.4 29.3 370.0 0.296 C7
* 0.0641 88.8 374.13 542.84 26.72 432.13 0.356 C8
* 0.0884 101.8 470.45 641 19.38 667.24 0.522 C9
* 0.0566 116.1 558.76 718.64 13.92 983.07 0.713 C10
* 0.0376 133.0 716.27 833.91 6.26 1730.53 1.245 C11 0.0365 143.0 C12 0.0366 154.0 C13 0.0465 167.0 C14 0.0439 181.0 C15 0.0451 195.0 Number of Pseudo-components = 4 C16
0.0386 209.0 MW Limits Comp.
Range Molar
Fractions Average
MW C17 0.0424 229.0 88.80 130.33 7 � 9 0.2091 101.69 C18 0.0383 245.0 130.33 191.27 10 - 14 0.2011 156.98 C19 0.0353 258.0 191.27 280.72 15 - 19 0.1997 225.65 C20 0.3181 412.0 280.72 412.00 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.
65
Table A7. Oil 6 Composition and Properties Component Molar
Fractions Molecular
Weight Tb (K)
Tc (K)
Pc (bar)
Vc (cm3/mol)
ω
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
C16 0.0270 216.0 MW Limits Comp. Range
Molar Fractions
AverageMW
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
Fractions Molecular
Weight Tb (K)
Tc (K)
Pc (bar)
Vc (cm3/mol)
ω
C1 0.0000 16.0 111.7 190.6 45.4 98.0 0.008 C2 0.0017 30.1 184.5 305.4 48.2 148.0 0.098 C3 0.0129 44.1 231.1 369.8 41.9 203.0 0.152 C4 0.0246 58.1 272.7 425.2 37.5 255.0 0.193 C5 0.0283 72.1 309.2 469.6 33.3 304.0 0.231 C6 0.0281 86.2 341.9 507.4 29.3 370.0 0.296 C7
* 0.0621 90.5 387.21 557 25.6 459.64 0.376 C8
* 0.0716 104.2 487.39 656.75 18.26 7117.79 0.556 C9
* 0.0505 119.2 563.73 722.77 13.63 1006.43 0.723 C10
* 0.0329 134.0 738.24 845.35 5.47 1910.84 1.353 C11 0.0467 149.0
66
C12 0.0345 164.0 C13 0.0434 176.0 C14 0.0387 188.0 C15 0.0449 203.0 Number of Pseudo-components = 4
C16 0.0281 214.0 MW Limits Comp. Range
Molar Fractions
AverageMW
C17 0.0360 232.0 92.50 134.84 7 � 10 0.2171 108.29 C18 0.0308 248.0 134.80 200.91 11 � 14 0.1633 168.59 C19 0.0367 259.0 200.91 299.34 15 - 19 0.1765 230.16 C20 0.3436 446.0 299.34 446.00 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.
Table A9. Oil 11 Composition and Properties Component Molar
Fractions Molecular
Weight Tb (K)
Tc (K)
Pc (bar)
Vc (cm3/mol)
ω
C1 0.0000 16.0 111.7 190.6 45.4 98.0 0.008 C2 0.0010 30.1 184.5 305.4 48.2 148.0 0.098 C3 0.0012 44.1 231.1 369.8 41.9 203.0 0.152 C4 0.0021 58.1 272.7 425.2 37.5 255.0 0.193 C5 0.0021 72.1 309.2 469.6 33.3 304.0 0.231 C6 0.0045 86.2 341.9 507.4 29.3 370.0 0.296 C7
* 0.0121 90.8 406.93 577.89 24 503.59 0.408 C8
* 0.0187 106.5 498.88 667.17 17.53 754.12 0.58 C9
* 0.0195 122.0 571.68 729.36 13.17 1045.85 0.739 C10
* 0.0556 135.0 752.50 852.80 5.01 2047.81 1.427 C11 0.0472 149.0 C12 0.0549 162.0 C13 0.0640 176.0 C14 0.0681 189.0 C15 0.0539 202.0 Number of Pseudo-components = 4
C16 0.0358 213.0 MW Limits Comp. Range
Molar Fractions
AverageMW
C17 0.0487 230.0 90.50 137.18 7 � 10 0.1259 118.73 C18 0.0489 244.0 137.18 207.24 11 � 15 0.2881 176.85 C19 0.0404 256.0 207.24 313.10 16 - 19 0.1538 237.52 C20 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.