heavy oil viscosity

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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


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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.


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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.


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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 M ethods 7

Empir ical methods 10

Crude Oil Fractions 13

Semi-theoretical M ethods 13

Empir ical methods 14

MODIFICATION OF PEDERSENS MODEL 15

Model Development 15

Heavy Oil Fraction Characterization 17

Tr ue Boil ing Point Tests (TBP Tests) 17

Gas Chromatography (GC) 18

Thermodynamic Proper ties Prediction 19

Whi tson s Lumping Scheme 21

Compositional Oil Samples 22

Results 23

PROCEDURE TO SCREEN CRUDE OIL VISCOSITY DATA 24

Introduction 24

Viscosity Correlations 26


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Reservoir Fluid Studies for Reservoir Engineering 27

Data Preparation and Data Screening Routine 28

Data Screening Results 30

MODIFICATION OF PEDERSENS MODEL FORBLACKOIL SAMPLES 31Introduction 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


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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. Bryants 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 projects findings

to small independent producers.


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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.


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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 M ethods


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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:

i(,T) = o(f

T,hoi,

oi,) f oi,h oi,/

1/2-2/32/1

)( MoMi (1)

h ,oi = h ,oi oi,) oc,/ ic,( (2)

f ,oi = oi,)T oc,/T ic,( (3)

wherei,o, and i,oare 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.


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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:

x(P,T) = ( T oc,/T xc, )-1/6

( P oc,/P xc, )2/3

/)Mo/Mi(2/1

oTG,xTG, o( T,P*o

*o ) (4),

and

T*o = )/(

)T oc,/T xc,( xTG,oTG, (5)

P*o = )/(P oc,P xc, xTG,oTG,

)/( (6),

where TG 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 contributionto 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 (TG ) 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

TG .


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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 = Z

r1

+ [( -

r1

)/(

r2

-

r1

)] (Z

r2

- Z

r1

) (8),

where is the acentric factor factor of a single reference fluid made-up of spherical

molecules; ri , 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( TR)r= ln ( TR)

r1 + [( - r1)/(r2 - r1)] (ln ( TR)r2- ln ( TR)

r1) (9),

where TR = )(/)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.

Empir ical methods

The Andrade25

(1934) equation, first proposed by de Guzman26

(1913), is given by,

ln = 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,


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ln= 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/ref= k [ -1.6866 + 1.40010 (Tb/T) + 0.2406 (Tb/T)2] (12).

where ref 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 (+ 0.7) = b1 + b2 log (T) (13).


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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 42

36

(1996) to equation (13), with the constants changed from 0.7to 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 (+ 0.7) =log (T)b (14).

where b = b2. A regression of experimental values yielded best values ofand 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--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.


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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--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 (14).

where )f( is the viscosity function normally linear, hyperbolic or logarithmic in form.

A common equation used successfully for liquid hydrocarbons is,

= )x( 1/3

ii

3

m

(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,

Gxx)(lnx)(ln ijjiiim += (16),

where Gij is an interaction parameter. Repeated coefficients are equal to zero. Thebinary form of Grunberg equation is given by,

Gxxlnxlnx)(ln 12212211m ++= (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 M ethods

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


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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%.

Empiri cal 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.


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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.

Mehrotra

50

(1990) applied eqn. (13) to several Middle East crude oil and oil mixturesdata. 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 15.6oC 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 PEDERSENS 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

cm

m TPMW

MW

P

P

T

TTP ,,

321

= , (18),

where the coefficients 1, 2 and 3 in Pedersen's model are -1/6, 2/3 and 1/2

respectively.

5173.0847.1310378.7000.1 mrom MW+=

(19)

847.1031.0000.1 roo += (20).

Here, ro 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),


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co

cm

co

cm

coo

ro

T

TT

P

PP

=

,

(21),

the pressures and temperatures at which the reference viscosity (o) is evaluated are given

by,

mcm

ocoo

P

PPP

= and

mcm

ocoo

T

TTT

= (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, andPcm) 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 Pedersens 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

492F to 762F. The density and viscosity of n-decane were fitted as a function ofPand

Tusing 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 ++= . (23),

while the viscosity, in cp, is given by,

PT1087.8T107057.6P001272.0

T

P0.4775T8775.2881-T.54183212

T

150991.51

739

2/13/1

C10

+

++=

(24).


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The correlation coefficient for equation (23) is R2

= 0.9996 with minimum andmaximum errors of 1.47 % and +1.82% respectively. Equation (24) has a correlation

coefficient ofR2= 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.

Tr ue Boil ing 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,


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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.


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19

Thermodynamic Proper ties 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 () andmolecular 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, , 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 () as correlating parameters. The basic equation is:

cb

baT = (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 .

Riazi and Daubert61

modified their equation while maintaining its simplicity and

significantly improving its accuracy:

[ ] bbcbb fTedTaT ++= exp (26)

[ ] 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.


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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 ().

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

1098.1818828.102226.080882.01

107972034371

02058.077084.013287.36523.44.486,96.272,12M

bbbb

b

TTTT

.-.

T

++

+++=

(28)

Lee and Kesler63

stated that their equations forPc and Tcprovide 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, 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

)1/7

7.14/log3

=

bc

c

TT

P(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 naturallyoccurring liquid hydrocarbons. Figure 2 shows the relationship between molecular

weight and the normal boiling point (Tb) or API gravity () according to Katz and

Firoozabadi65

.


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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+.

Whi tson 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

g

nIMw

Mw

NMwMw

= ln

1exp (32),


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22

whereMwN= 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 andmolecular 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

.


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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 Pedersens 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.


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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 (m, o) 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.


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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 commerciallaboratories.

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 80o

F to 288o

F. 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


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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.

12are 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.


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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 atpressures 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 alsoincluded. 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


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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, andBg) 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 factorBo 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


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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, oildensity 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


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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 setcorresponds 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 forthe viscosity based upon the modified Pedersen

1and Lohrenz

72models. Additionally we

proposed new correlations for solution gas-oil ratios and formation volume factors to be

used in these models.


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31

MODIFICATION OF PEDERSENS MODEL FORBLACKOIL SAMPLES

Introduction

This section presents a modification of Pedersens 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 Pedersens 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 byfitting 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.


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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

=

=+ iri

i

/* a (33)

Here * is the low-pressure gas mixture viscosity, and is the viscosity-reducing

parameter, which is defined as,322161 /

cm

/

wm

/

cm PMT= (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:

i(,T) = o(f

T,hoi,

oi, ) f oi,h oi,/1/2-2/3

2/1

)( MoMi (1)

h ,oi = h ,oi oi,) oc,

/ ic,( (2)

f ,oi = oi,)T oc,/T ic,( (3)

where i,o, and i,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 usingmethane 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).


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33

Pedersen et al.1

introduced a third parameter ()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

MW

MW

P

P

T

TTP ,,

321

=

, (18),

where the coefficients 1, 2 and 3 in Pedersen's model are -1/6, 2/3 and 1/2

respectively.

5173.0847.1310378.7000.1 mrom MW+=

(19)

847.1031.0000.1 roo += (20).

Here, ro 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

T

TT

P

PP

=,

(21),

the pressures and temperatures at which the reference viscosity (o) is evaluated are given

by,

mcm

ocoo

P

PPP

= and

mcm

ocoo

T

TTT

= (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.


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34

To use equation (18) we needed to find simplified expressions for the molecular

weight (MWm), critical temperature and pressure (Tcm, andPcm) 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 reportedby Geopetrole

54covering pressures from 14.7 psia to 7325 psia and temperatures from

492F to 762F. The density and viscosity of n-decane were fitted as a function ofPand

Tusing a stepwise regression procedure and the statistical software SAS55

. The density,

in lb/ft3, is calculated by

( )TP105043.1T1906.168T1847.7998-exp 82/11C10 ++= . (23),

while the viscosity, in cp, is given by,

PT1087.8T107057.6P001272.0

T

P0.4775T8775.2881-T.54183212

T

150991.51

739

2/13/1

C10

+++=

(24).

The correlation coefficient for equation (23) is R2

= 0.9996 with minimum andmaximum errors of 1.47 % and +1.82% respectively. Equation (24) has a correlation

coefficientR2= 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 thereported 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 (o,R). Once this was determined the


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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 RoRo

Ro

bT +=

(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:

24.27870.3596

,

0.58848

Robcm TT = (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 forVcm 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

640F 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,


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36

( )

( )

+

+

=

oo

.c

cm

ob

o

sbm

sb

Roscmcmm

TPT

VP

B

B

APIRMW

MW

R

R

V

V

T

TTTP

Ro

,1930.01388.202359.02606.0exp

,

C10

C10

C10

10C

3

2902.24471.0

-3.9243

10C

1362.0

,

0.9841

10cC

0286.1

2

10cC

,

(38),

whereBo, the formation volume factor, is dimensionless, Ro, 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 andBob

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 isleft 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 form and 0 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


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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 1to3 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 1to3 with API, Rsb and other field

derived variables. Table 7 summarizes the statistics for these models.


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38

CONCLUSIONS

We presented a new viscosity correlation derived from Pedersens 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.


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39

NOMENCLATURE

API = Oil gravity, (API = 145/o,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

m = Parameter defined in Eqn. (11)

o = Parameter defined in Eqn. (12)

TG = Tham-Gubbins

17

(1971) rotational coupling coefficiento,R = Oil specific gravity at reservoir conditions

= Viscosity-reducing parameter, which is defined as

TR = Parameter defined Eqn. (9)

i,o = Shape factor used in Eqn. (1)


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= Density (lb/ft3)

= Compressibility factor

i,o = Shape factor used in Eqn. (1)

= 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.


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