Thermodynamics 4

Characterization of C7+ Petroleum Fractions for Compositional Simulations, Colombia, Summer 2000

Au

InAt

Characterization of C7+ Petroleum Fractions for

Compositional Simulations Generalized Phase Equilibria Models Fluid Characterization of the Hydrocarbon Plus Fraction ( C7

+ ): Gas Chromatography, True Boiling Point Tests (TBP), Viscosity, and Specific Gravity. Estimation Methods for Critical Properties. Characterization Factors. Splitting and Lumping Schemes of Petroleum Fractions. Applications to Reservoir Simulation. Group selection. Tuning of Equations of State (EOS). Selection of regression variables. Limits of Tuned Parameters. Model validation. Improvement of EOS Volumetric Predictions by Volume Translation. Building Fluid Models for Reservoir Simulation Using PVTi and Eclipse 300. S t d din : MAB

thor: Dr. Maria Barrufet - Summer, 2000 /38

structional Objectives the end of this module, the students should be able to

• Understand the need for characterizing the heavy fraction for reservoir simulation.

• Use the correlations presented in class to determine critical pressures when boiling point and specific gravity data are available

• Use the correlations presented in class to determine critical temperatures of the heavy fraction when boiling point and specific gravity data are available

• Use the correlations presented in class to determine acentric factors of the heavy fraction when boiling point and specific gravity data are available

• Compare physical properties from various correlations

• Be able to lump and characterize a hydrocarbon mixture (Gas Condensate Example in class)

• Be able to split and characterize the C7+ fraction of a mixture (Gas Condensate

Example in class).


Author: Dr. Maria Barrufet - Summer, 2000 /38

Introduction Nearly all naturally occurring hydrocarbon systems contain a quantity of heavy fractions that are not well defined and are not mixtures of discretely identified components. These heavy fractions are often lumped together and identified as the plus fraction, e.g., C7

+ fraction. This fraction is composed of any molecules of the crude oil that have boiling points higher than n-hexane.

A proper description of the physical properties of the plus fractions in hydrocarbon mixtures is essential in performing reliable phase behavior calculations and compositional modeling studies.

Frequently a distillation analysis or a chromatographic analysis is available for this undefined fraction.

True Boiling Point Tests (TBP 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 will be determined.

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. Usually the temperature is taken when the first droplet distills over. Ten fractions (cuts) are then distilled off, the first one at 50oC and each successive one with a boiling range of 25oC. For each distillation cut, the volume, specific gravity, and molecular weight, among other measurements, are determined. Cuts obtained in this manner are identified by the boiling-point ranges in which they were collected.

The following figure shows the equipment used for this type of tests.



Figure 1 - Equipment used for TBP tests.

The following table is an example of a distillation test with 11 cuts.

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 1 - Example of a distillation test with 11 cuts.

The boiling point of the residual it is unknown and it may be used as a tuning parameter.

One of the methods to estimate the boiling temperature of the residual is by using a Watson characterization factor and the specific gravity of the residual.



TBP Distillation Curve

0200400600800

100012001400

0.0 20.0 40.0 60.0 80.0 100.0

Distilled Volume %

Boi

ling

Tem

pera

ture

o F T initialT finalT average

Figure 2 - TBP distillation curve.

Other physical properties such as molecular weight and specific gravity may also be measured for the entire fraction or various cuts of it.

Average Boiling Points Used Generally, there are five different methods of defining the normal boiling point for petroleum fractions. These are:

Method 1: Volume Average Boiling Point (VABP)

This is defined by the following expression:

∑=i

biiTvVABP (1)

where



Tbi = boiling point of the distillation cut i, oR

vi = volume fraction of the distillation cut i

Method 2: Weight Average Boiling Point (WABP)

This is defined by the following expression:

∑=i

biiTwWABP (2)

where wi = weight fraction of the distillation cut i

Method 3: Molar Average Boiling Point (MABP)

This is given by the following relationship:

∑=i

biiTxMABP (3)

where xi = mole fraction of the distillation cut i

Method 4: Cubic Average Boiling Point (CABP)

This is defined as 3

3/1

= ∑

ibiiTxCABP (4)

Method 5: Mean Average Boiling Point (MeanABP):

2CABPMABPMeanABP += (5)

These five expressions for calculating normal boiling points result do not differ significantly from one another for narrow boiling petroleum fractions.



Key Physical Properties Required for Reservoir Engineering To use any of the thermodynamic property-prediction models, e.g., equation of state, to predict the phase and volumetric behavior of hydrocarbon mixtures, one must be able to provide the:

Critical temperature, Tc

Critical pressure, Pc

Critical volume, Vc

Critical compressibility factor, Zc

Acentric factor, ω

Molecular weight, Mw

for both the defined and the undefined (heavy) fractions in the mixture.

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, etc)

Importance of the C7+ Fraction in Reservoir

Engineering Most oil recovery processes are greatly influenced by the C7

+ fraction. CO2 injection for miscible displacement is highly dependent upon the C7

+ fraction but waterflooding above the bubble point is insensitive except through the oil viscosity.

The C7+ fraction may greatly influence how fluids flow in the reservoir. Fluid viscosities

are based on volumetric fractions of the phases, which are determined by the phase behavior, which is highly dependent upon the C7

+ fraction. The fractional flow of each



phase is determined by the relative permeability and viscosity of each phase. The C7+

fraction is especially important for condensate flow calculations since the amount of liquid dropout is strongly affected by the C7

+ molecular weight and fraction.

Overview Of Generalized Correlations For Estimating Physical Properties Of Hydrocarbon Fractions Most correlations use the specific gravity and the normal boiling point Tb as correlation parameters. Selecting proper values for these parameters is very important because slight changes can cause significant variations in the predicted results.

The vast number of hydrocarbon compounds making up naturally occurring crude oil have been grouped chemically into several series of compounds. Each series consists of those compounds similar in their molecular make-up and characteristics. In general, it is assumed that the heavy (undefined) hydrocarbon fractions are composed of three hydrocarbon groups, namely:

Paraffins (P)

Naphthenes (N)

Aromatics (A)

The PNA content of the plus fraction of the undefined hydrocarbon fraction can be estimated experimentally from a distillation analysis and/or a chromatographic analysis.

Riazi-Daubert Riazi and Daubert (1980) 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 γ=ψ (6)



where Tb is the normal boiling point temperature expressed in degrees R and the constants a, b, c, depend upon the physical property indicated by ψ .

Propertyψψψψ Coefficients Percent Deviation

a b c Average Maximum

MW 4.5673 x 10-5 2.1962 - 1.0164 2.6 11.8

Tc , º R 2.42787 0.58848 0.3596 1.3 10.6

Pc, psia 3.12281 - 2.3125 2.3201 3.1 9.3

Vc ft3/ lbm 7.5214 x 10-3 0.2896 - 0.7666 2.3 9.1

Table 2 - Correlation constants for Eq. (6).

The prediction accuracy is reasonable over the boiling point range of 100-850 oF.

Later, in 1987, Riazi and Daubert (1987) modified their equation while maintaining its simplicity while significantly improving its accuracy. Eq. (7) expresses the property as a function of normal boiling point (Tb) and oil specific gravity at standard conditions (Form 1 in Table 3).

[ ]γ+γ+γ=ψ bbcb

b fTedTexpaT (7)

Eq. (8) expresses the property as a function of normal boiling point (Tb) and oil molecular weight at standard conditions (Form 2 in Table 3).

[ ]γ+γ+γ=ψ wwcb

w fMedMexpaM (8)

The constants a to f for the two different functional forms of the correlation are tabulated in Table 3, and depend upon the correlated property.



Form (1) – Equation (7)

Property Coefficient

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) – Equation (8)

Property Coefficient

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

Table 3 - Coefficients from Riazi and Daubert Correlations.

The limitations for using Riazi and Daubert correlations are that the molecular weight should be in the range of 70-300 lb/lb-mol and the normal boiling point range: 80-650oF.

Lin-Chao Generalized Correlation Lin and Chao expressed the physical properties of C1 to C20 n-alkanes by the following generalized equation



( ) ( ) ( ) 15

34

2321

−++++=ξ MwaMwaMwaMwaa (9)

The constants a1 to a5 depend upon the property that is being correlated. Table 4 reports the values.

Property

C Tc ( oR) ln (Pc) (psi) ω γ Tb ( oR)

a1 490.8546 6.753444 - 28.21536 0.66405 240.8976

a2 7.055982 - 0.010182 2.209518 1.48130 x 10-3 5.604282

a3 - 0.02118708 2.51106 x 10-5 17.943264 x10-5 - 5.0702 x 10-6 - 0.0127616

a4 2.676222 x 10-5 - 3.73775 x 10-8 - 3.685356 x 10-5 6.21414 x 10-9 13.84353x 10-6

a5 - 4,100.202 3.50737 - 124.35894 - 8.45218 - 2,029.158

Table 4 - Values of the constants a1 to a5.

Cavett's Correlations Cavett (1962) 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. Cavett proposed the following expressions for estimating the critical temperature and pressure of petroleum fractions:

( )( ) ( ) ( ) 226

25

343

2210 bbbbbc TAPIaTAPIaTaTAPIaTaTaaT b ++++++= (10)

( )( ) ( ) ( )( ) 22

7

26

25

343

2210

ln

b

bbbbbc

TAPIb

TAPIbTAPIbTbTAPIbTbTbbP b +++++++= (11)



where the critical temperature (Tc) is expressed in oR, the normal boiling point (Tc) is in oF, and the critical pressure (Pc) is in psia.

Table 5 lists the coefficients used in Eqs. (10) and (11).

Index ai bi

0 768.07121 2.8290406

1 1.7133693 0.94120109 x 10-3

2 - 0.0010834003 - 0.30474749 x 10-5

3 - 0.0089212579 - 0.20876110 x 10-4

4 0.38890584 x 10-6 0.15184103 x 10-8

5 0.53094920x10-5 0.11047899 x10-7

6 0.32711600 x 10-7 - 0.48271599 x 10-7

7 0 0.13949619 x 10-9

Table 5 - Coefficients used in Cavett’s correlations (Eqs. (10) and (11)).

Kesler-Lee Correlations Kesler and Lee (1976) proposed a set of equations to estimate the critical temperature, critical pressure, acentric factor, and molecular weight of petroleum fractions. The equations, as expressed below; use specific gravity and boiling point in oR as input parameters.

3102

272

32

1010472270648346851

10118570289822424400566036348

bb

bc

T1.69770.42019 T...

T.....P

−−

−

×

γ

++×

γ

+γ

+

+×

γ

+γ

+−γ

−=

(12)



( ) ( )b

bc T..T....T

51026238346690117404244018117341−

×γ−+γ++γ+= (13)

( ) ( )( ) 3

122

7

2

1098.1818828.102226.080882.01107972034371

02058.077084.013287.36523.44.486,96.272,12

bbbb

b

TTTT.-.

TMw

×

−γ+γ−+×

×γ−γ−+γ−+γ+−=− (14)

The above equation was obtained by regression analysis using the available data on molecular weights ranging from 60 to 650.

Acentric Factor

Defining the Watson characterization factor Kw and the reduced boiling point Tbr by the following relationships.

( )γ

=3/1

bTKw (15)

and

cbbr TTT /= (16)

where Tb = boiling point, oR

Kesler and Lee proposed the following two expressions for calculating the acentric factor,

For 8.0>brT ,

( )br

br TKwTKwKw 01063.0408.1359.8007465.01352.0904.7 2 −++−+−=ω (17)

and, for 8.0<brT ,



( )6

6

43577.0)ln(4721.13/6878.152518.151693347.0)ln(28862.1/09648.692714.5696.14/ln

brbrbr

brbrbrc

TTTTTTP

+−−−++−−

=ω (18)

Kesler and Lee stated that their equations for Pc and Tc provide values are nearly identical with those from the API Data Book up to a boiling point of 1,200oF.

There are many other correlations worth mentioning, Whitson (1984) provides and excellent review.

Edmister's Correlation Edmister (1958) 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 (19)

with the temperatures expressed in degrees R.

Note that the acentric factor can be used as a tuning parameter in EOS simulations.

Katz-Firoozabadi Method Katz and Firoozabadi (1978) 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.

The following figures show the relationship among these properties according to Katz & Firoozabadi relations.



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

Table 6 - Tb vs. molecular weight according to Katz & Firoozabadi relations.

Hydrocarbon Physical Properties (Kats & Firoozabadi)

0

400

800

1200

1600

0.6 0.7 0.8 0.9 1Specific Gravity

Tb, M

w

Tb(F) Mw

Table 7 - Tb vs. specific gravity according to Katz & Firoozabadi relations.

The following figures illustrate the predicted values of of Tc, Pc, and ω as a function of boiling point and specific gravities using Katz and Firoozabadi (1978). (See excel file characterizations.xls)



Physical Properties of Normal Alkanes

0

400

800

1200

1600

0 200 400 600 800 1000 1200 1400Tb (F)

Tc ,

Pc

0

0.5

1

1.5

2

2.5

Ace

ntric

Fac

tor

Tc /FPc (psia)w

Figure 3 - Predicted values of of Tc, Pc, and ω as a function of boiling point using Katz and Firoozabadi (1978).



Physical Properties of Normal Alkanes

0

400

800

1200

1600

0.6 0.7 0.8 0.9 1Specific Gravity

Tc ,

Pc

0

0.5

1

1.5

2

2.5

Ace

ntric

Fac

tor

Tc /FPc (psia)w

Figure 4 - Predicted values of of Tc, Pc, and ω as a function of specific gravity using Katz and Firoozabadi (1978).

Willman-Teja Correlations Willman and Teja (1987) proposed correlations for determining the critical temperature and pressure of the undefined petroleum fraction. The empirical formulas are given by using a nonlinear regression model to best match the critical properties data of Berman et al. (1977) and Whitson (1980).

( )[ ]884540633.0137242.025127.11 −++= nTT bc (20)

[ ][ ] 9265669.154285.0873159.0157759.184,10416805.339 −++= nnPc (21)



Splitting and Grouping Schemes of Petroleum Fractions Accurate phase behavior data of petroleum fluids is required since this has a profound effect on reservoir simulation, particularly for gas condensates, highly volatile fluids, and gas injection recovery.

Experimental PVT data is available, but often this is insufficient for the entire range of pressures temperatures and compositions likely to occur in depletion or injection scenarios.

A thermodynamic equation of state (EOS) can overcome the lack of measured data, however the incomplete fluid description and limitations and the limitations of the cubic EOS means that some type of calibration in the EOS parameters will be needed.

The uncertainty in the properties of the hydrocarbon plus fraction, which comprises a significant portion of naturally occurring hydrocarbon fluids, provides at the same time great flexibility for calibrating the EOS.

Normal boiling points and specific gravity of the C7+ fraction may be the only measured

data available. Other types of hydrocarbon analysis provide molecular weights.

In the absence of detailed analytical data for the plus fraction in a hydrocarbon mixture, erroneous predictions and conclusions can result if the plus fraction is used directly as a single component in the mixture phase behavior calculations. Numerous authors have indicated that these errors can be substantially reduced by "splitting" or "breaking down" the plus fraction into a manageable number of fractions pseudo-components) for equation of state calculations.

The problem, then, is how to adequately split a C7 + fraction into a number of pseudo-components characterized by:

• Mole fractions

• Molecular weights

• Boiling Points

• Specific gravities



These characterization properties, when properly combined, should match the measured plus fraction properties; i.e., (Mw)7

+ and (γ)7+.

Splitting Schemes Splitting schemes refer to the procedures of dividing the C7

+ fraction into hydrocarbon groups with a single carbon number (C7, C8, C9, etc.) and are described by the same physical properties used for pure components.

Several authors have proposed different schemes for extending the molar distribution behavior of C7

+, i.e., the molecular weight and the specific gravity. In general, the proposed schemes are based on the observation that lighter systems such as condensates usually exhibit exponential molar distribution, while heavier systems often show left-skewed distributions. This behavior is shown schematically in the following figure.

Molecul ar Weight

Mol

e F

ract

ion

Di stri bution Skew ed to the Left (heavy oi l s)

Exponenti onal Distri buti on(Condensate Systems)

Figure 5 - Schematic of the distribution of lighter and heavier systems.



Splitting Constraints

Three important requirements should be satisfied when applying any splitting models.

First Constraint: The sum of the mole fractions of the individual pseudo-components is equal to the mole fraction of C7

+.

∑+

+

=

=N

iCi zz

77

(22)

Second Constraint: The sum of the products of the mole fraction and the molecular weight of the individual pseudo-components is equal to the product of the mole fraction and molecular weight of C7

+.

+

+

+∑=

=77

7C

N

iCii MwzMwz (23)

Third Constraint: The sum of the product of the mole fraction and molecular weight di-vided by the specific gravity of each individual component is equal to that of C7

+

+

+++

γ=

γ∑=

7

77

7 C

CCN

i i

iiMwzMwz (24)

Where

i = number of carbon atoms

N+= last hydrocarbon group in the C7 + with n carbon atoms, e.g., 20 +

It is assumed that the molecular weight and specific gravity of the C7+ are measured.

Several splitting schemes have been proposed recently. These schemes, as discussed below, are used to predict the compositional distribution of the heavy plus fraction.



Katz's Splitting Method Katz (1983) presented an easy-to-use graphical correlation for breaking down into pseudo-components the C7

+ fraction present in condensate systems. The method was originated by studying the compositional behavior of six condensate systems using detailed extended analyses. On semi-log scale, the mole percent of each constituent of the C7

+ fraction versus the carbon number in the fraction was plotted. The resulting relationship can be conveniently expressed mathematically by the following expression:

)25903.0exp(38205.1 7 nzzn −= + (25)

This equation is repeatedly applied until the first splitting constraint is satisfied. The molecular weight and specific gravity of the last pseudo-component can be calculated from the second and third constraint, respectively.

Pedersen's Method Pedersen et al. (1982) proposed that for naturally occurring hydrocarbon mixtures, an exponential relationship exists between the mole fraction of a component and the corresponding carbon number as follows,

−=

BAnzn exp (26)

where A and B are constants.

For condensate and volatile oils, Pedersen and coworkers suggested that A and B can be determined by a least squares fit to the molar distribution of the lighter fractions. This equation can be used to calculate the molar content of each of the heavier fractions by extrapolation. The classical three constraints are also imposed.

Whitson's Method Whitson (1980) proposed that a probability function called the Three Parameter Gamma Function can be used to model the molar distribution. Unlike the previous models, the gamma function has the flexibility to describe a wider range of distributions by adjusting



its variance, which is left as an adjustable parameter. Whitson expressed this function in the following form:

)!()exp()exp(

)(11

0

7

jYYYYz

z nj

nnj

n

jn +α

−−−αΓ

=+α

++α

+∞

=∑+ (27)

where

z7 + = mole fraction of the heptanes-plus fraction

zn = mole fraction of the pseudo-component with a number of carbon atoms of n

Using this equation the heptanes plus fraction with a mole fraction of z7 + can be divided into several pseudo-components each with composition zn and molecular weight Mwn.

αη−

η−=+7Mw

MwY nn (28)

where

α = adjustable parameter

Γ(α) = gamma function

η = the lowest molecular weight expected to occur in the pseudo-component state. A good approximation of η is given by:

614 −=η n (29)

The shape of the curve (exponential or left-skewed) representing the molar distribution depends on the value of the adjustable parameter α.

Whitson suggested that reasonable limits for α are 0.5 to 3.0. For α = 1, the distribution is exponential. Values equaling less than one give accelerated exponential distributions, while values greater than one yield left-skewed distributions.



Behrens and Sandler Splitting Scheme Behrens and Sandler (1986) used semi-continuous thermodynamic distribution theory to model the C7

+ fraction for equation of state calculations. The authors suggested that the C7

+ fraction could be fully described with as few as TWO pseudo-components.

A semi-continuous fluid mixture is defined as one in which the mole fractions of some components, such as C1 through C6, have discrete values, while the concentrations of others, (the unidentifiable components) such as C7 +, are described by the continuous distribution function F(I). This continuous distribution function F(I) describes the heavy fractions according to the index I, selected to be a property such as boiling point, carbon number, or molecular weight.

For a hydrocarbon system with k discrete components, the following relationship applies:

0.17

1=+ +∑

=

zzk

ii (30)

The mole fraction of C7+ in this equation is replaced with the selected distribution

function, to give

0.1)(1

=+ ∫∑=

B

A

k

ii dIIFz (31)

where A and B are the beginning and upper cutoff limit of integration of the continuous distribution).

The above molar distribution behavior is shown schematically in Figure 6. The parameter A can be determined from the plot, or can be defaulted to

C7, i.e., A=7.

The value of the second parameter B ranges from 50 to infinity; i.e., 50 < B < ∞; however, Behrens and Sandler pointed out that the exact choice of the cutoff is not critical.



Continuous

Molar Distribution

Discrete

Property

P

Figure 6 - Molar distribution behavior according to Behrens and Sandler.

Selecting the index I of the distribution function F(I) to be the carbon number n, Behrens and Sandler proposed the following exponential form of F(I).

nnDnF α−= exp)()( (32)

with

A < n < B

where the parameter α is given by solving the following expression

[ ]( ) 0

expexpexp1 =

−−−+−

α α−α−

α−

BA

B

nBAAC (33)

where nC is the average carbon number defined by the relationship:

[ ]14

47 += +Mw

Cn (34)

This equation can be solved for α by successive substitutions or the Newton-Raphson method, using an initial value of α as:



ACn −=α 1 (35)

Substituting the distribution function yields

0.1exp)(1

=+ ∫∑ α−

=

B

A

nk

ii dnnDz (36)

or

∫ α−=+

B

A

n dnnDz exp)(7 (37)

By a transformation of variables and changing the range of integration from A and B to 0 and c = (B-A)α the equation becomes:

∫ −=+

cr drrDz

07 exp)( (38)

The authors applied the “Gaussian quadrature numerical integration method” with a two-point integration to evaluate this equation as

∑=

=+

2

17

)(i

ii wrDz (39)

where,

ri = roots for quadrature of integrals after variable transformation

wi = weighing factor of Gaussian quadrature at point i.

The following figure provides the values of ri and wi as a function of the Upper limit of integration. For c > 10 these values remain essentially constant.



The weight factor w2 is

12 1 ww −= (40)

Quadrature Points & Weights (Behrens & Sandler Splitting

0

0.5

1

1.5

2

2.5

3

3.5

0 2 4 6 8 10Upper Integration Limit (c)

r2

0

0.2

0.4

0.6

0.8

1

r1, W

eigh

t (w

1)

r2 r1 w1

Figure 7 - Behrens and Sandler Roots and Weights for Two-point Integration

The computational sequences of the proposed method are summarized in the following example.

Example (Given by Behrens and Sandier).

A heptanes-plus fraction in a crude oil system has a mole fraction of 0.4608 with a molecular weight of 226. Using the Behrens and Sandler lumping scheme, characterize the C7 + by two pseudo-components and calculate their mole fractions.

Solution.



Step 1.

Assuming the starting and ending carbon numbers to be C7 and C50, calculate A and B from

A = starting carbon number - ½

B = ending carbon number + ½

5.65.07 =−=A

5.505.050 =+=B

Step 2.

Calculate the average carbon number nC from

[ ]14

47

+= +Mw

Cn

43.1614

4226 =+=nC

Step 3.

Solve equation for α, to give

[ ]( ) 0

expexpexp1 =

−−−+−

α α−α−

α−

BA

B

nBAAC

A good initial guess is:

−

=αACn

1

0938967.0=α



Step 4.

Evaluate the range of integration c

c = (B-A) α

13.40938967.0)5.65.50( =×−=c

Step 5.

Find the weights and the integration points for c = 4.13 from excel file (Characterizations.xls) or from the figure.

2267.0 7733.07733.0 4741.0

22

11

====

wrwr

Step 6.

Find the equivalent pseudo-component carbon numbers n1 & n2 and mole fractions zi

from the following expressions:

Arn +α

= 11 & Arn +

α= 2

2

+= 711 zwz & += 722 zwz

First pseudo-component

55.115.60938967.0

4741.01 =+=n

3563.0)4608.0)(7733.0(1 ==z

Second pseudo-component



08.335.60938967.0

4965.22 =+=n

1045.0)4608.0)(2267.0(2 ==z

The C7+ fraction is represented then by the following TWO pseudo-components:

Step 7.

Find the molecular weight of the individual pseudo-components from constraint two.

Pseudo-Component

Molecular Weight

Carbon Number Mole Fraction

1 157.6572 C11.55 0.3563

2 459.0724 C33.08 0.1045

Step 8.

Assign the physical properties of the two pseudo-components according to selected correlations. Satisfy the 3 lumping/splitting constraints.

Pseudo-Component

n Tb

(oR) γγγγ Mw

Tc

(oR)

Pc

(psia) ωωωω

1 11.55 215.9 0.7315 154 1003.7 432.41 0.3167

2 33.08 647.9 0.8636 426 1425.23 202.60 0.7906

Lumping Schemes Equation of state calculations are frequently burdened by the large number of components necessary to describe the hydrocarbon mixture for accurate phase



behavior modeling. Often, the problem is either lumping together the many experimental determined fractions, or modeling the hydrocarbon system when the only experimental data available for the C7

+ fraction are the molecular weight and specific gravity.

Generally, with a sufficiently large number of pseudo-components used in characterizing the heavy fraction of a hydrocarbon mixture, a satisfactory prediction of the PVT behavior by the equation of state can be obtained. However, in compositional models, the cost and computing time can increase significantly with the increased number of components in the system. Therefore, there are strict limitations on the maximum number of components that can be used in compositional models and the original components have to be lumped into a smaller number of new pseudo-components.

The term “lumping”or “pseudoization” then denotes the reduction in the number of components used in equation of state calculations for reservoir fluids. This reduction is accomplished by employing the concept of the pseudo-component. The pseudo-component denotes a group of pure components lumped together and represented by a single component.

There are several problems associated with “regrouping” the original components into a smaller number without losing the predicting power of the equation of state. These problems include:

• How to select the groups of pure components to be represented by one pseudo-component each.

• How to determine (Tc, Pc, and ω) for the new lumped pseudocomponents for EOS use.

The molecular weight and specific gravity of the lumped compound are determined by reversed application of the constraints #1 and #2 defined in the splitting procedures.

General rules of thumb are:

• Group components with similar critical properties

• Group all components with smaller compositions (i.e. <1%)



• Do not group light and heavy components

• Proportion of every lumped group should be similar whenever possible

There are several unique published techniques which can be used to address the above lumping problems; among those:

• Lee et al. (1979)

• Whitson (1981)

• Mehra (1982)

• Montel-Gouel (1984)

• Schlijper (1984)

• Behrens-Sandler (1986)

• Gonzalez-Colonomos-Rusinek (1986)

Whitson’s Lumping Scheme Whitson (1980) proposed a regrouping scheme whereby the compositional distribution of Lumping is the reversed problem 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 −+= (41)

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

= ln1exp (42)

where:

MwN = molecular weight of the last reported component in the extended analysis of the plus fraction.

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.

Suggested Exercises

Suggested Exercise 1 - A heptanes plus fraction with a molecular weight of 203 and a specific gravity of 0.8125 presents a naturally occurring condensate system. The reported mole fraction of the C7

+ is 0.12.

• Predict the molar distribution of the plus fraction by using Katz's correlation

• Characterize the last fraction in the predicted extended analysis in terms of its physical and critical properties.

Suggested Exercise 2 - A crude oil system has the following composition:



Component ZI

Cl 0.3100

C2 0.1042

C3 0.1187

C4 0.0732

C5 0.0441

C6 0.0255

C7 0.0571

C8 0.0472

C9 0.0246

C10 0.0233

C11 0.0212

C12 0.0169

C13+ 0.1340

The molecular weight and specific gravity of C13+ are 325 and 0.842.

• Calculate the appropriate number of pseudo-components necessary to adequately represent the above components. Use:

♦ Whitson's Lumping Method

♦ Behrens-Sandler Method

• Characterize the resulting pseudo-components.

Volume Translations to Improve Volumetric Predictions from EOS Cubic EOS such as SRK, (Soave-Redlich-Kwong) and PR (Peng-Robinson) give satisfactory VLE predictions for all reservoir engineering applications. However,



volumetric predictions, particularly for the liquid phase can be quite poor. Good predictions of both VLE and volumetric data was not possible until the volume translation concept was developed.

Using VLE information from EOS and independent estimates of liquid densities (from the many available correlations) may create inconsistencies in material balance computations, such as negative saturations, "creation" of mass, negative compressibilities, etc. The volume translation concept is a permissible transformation that leaves the VLE unchanged.

Liquid and gas volumes can then be estimated with good accuracy by correcting the volumes predicted from EOS using the proposed translating procedure from Peneloux et.al (1982) (Fluid Phase Equilibria 8, 7-23 )

Equation of State Transforms by Volume Translations Any EOS is a functional relationship relating pressure, volume, temperature and composition.

( )cNt n,...,n,n,n,V,TfP 321= (43)

with Nc being the number of different species (components) in the system.

The expression for the fugacity coefficient of component "i" using an integral in P is:

dPPRT

VˆlnP

ii ∫

−=φ

0

1 (44)

Where the partial molar volume of component "i" is defined and evaluated as:

in,P,Ti

VnnV

j

=

∂∂ (45)

The equilibrium conditions for two phases labeled as v and l (vapor and liquid),

)x,.....,x,x,P,T(ˆx)y,.....,y,y,P,T(ˆy ncliinc

vii 2121 φ=φ (i = 1, 2, 3, ....nc) (46)



The proposed volume translation consists in defining a “pseudo-volume” according to:

∑=

+=cN

iiiit

pst VncVV

1

(47)

The ci's are component characteristic constants.

Substituting this pseudovolume instead of the total volume into Eq. (43) yields a “pseudo EOS”:

( )cN

pst

ps n,...,n,n,n,V,TfP 321= (48)

we can also define “pseudo-partial molar volumes” as

iiin,P,Tin,P,Ti

pstps

i cVcnnV

nVV

jj

+=+

∂∂=

∂

∂= (49)

and “pseudo-fugacity coefficients”,

dPPRT

VˆlnP ps

ipsi ∫

−=φ

0

1 (50)

Thus the “pseudo EOS” leads to equilibrium conditions given by.

)x,.....,x,x,P,T(ˆx)y,.....,y,y,P,T(ˆy nc)l(ps

iinc)v(ps

ii 2121 φ=φ (i = 1, 2, 3, ....nc) (51)

or replacing Eq. (49) for the pseudo partial molar volume the expression in terms of the true fugacity coefficients is

φ=

φ

RTPcexp)x,.....,x,x,P,T(ˆx

RTPcexp)y,.....,y,y,P,T(ˆy i

nclii

inc

vii 2121 (52)

which is EXACTLY the same result given by the unmodified Eq. (46) .Thus, translations along the volume axis according to Eq. (47) leave the predicted VLE conditions unchanged.



Not all forms of volume correction will preserve consistency. For example Lin and Daubert (1980) (Ind. Eng. Chem. Process Des. Dev. 19, 51-59) correction which divides volume by some factor does not preserve consistency.

Corrections for Soave-Redlich-Kwong EOS The volume correction coefficient ci has been successfully correlated to the Rackett compressibility factor ZRAi appearing in Spencer and Danner's (1973) correction of the Rackett equation for saturated liquid volumes.

ZRAi is tabulated for most hydrocarbon species in the API Technical Data book Vol. II. This set of notes includes part of that table for illustration purposes.

Many other correlating functions in terms of pseudo compressibility factors from other sources have been tried for ci as well as similar corrections such as the “shift factors”

The first correction presented in Peneloux et.al (1982) paper (RSKc1) uses the following relationship between ci and ZRAi

( )RAici

cii Z.

PRT.c −

= 294410407680 (53)

The second correction RSKc2 was obtained by substituting ZRai in Eq. (53) for the critical compressibility factors Zci calculated from the experimental critical volumes.

The third correction RKSc3 was expressed in terms of the acentric factors was

( )ii

ii ..

PcRTc.c ϖ+

= 08770003850407680 (54)

Variations of this volume translation concept developed by Peneloux-Rauzy and Freze (PRF) are commonly used for others EOS as well in most reservoir engineering packages.



The following figure shows the shift parameter used in the Peng Robinson EOS as a function of molecular weight of the alkane hydrocarbon component.

Shift Factor for Normal Alkanes

0

200

400

600

0 50 100 150 200 250 300 350 400 450Molecular Weight

Pc, V

c

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Shift

Fac

tor

Pc (psia) shift

Figure 8 - Shift factor for normal alkanes used in Peng-Robinson EOS



References • Ahmed, T., Cady, G., and Story, A., "A Generalized Correlation for Characterizing

the Hydrocarbon Heavy Fractions," Paper SPE 14266,presented at the 60th Annual Technical Conference of the SPE, held in Las Vegas, September 22-25, 1985.

• Behrens, R. and Sandler, S., "The Use of Semi-continuous Description to Model the C7

+Fraction in Equation of State Calculation," Paper SPE/DOE 14925, presented at the 5th Annual Symposium on EOR, held in Tulsa, Oklahoma, April 20-23, 1986.

• Gonzalez, E., Colonomos, P., and Rusinek, I., "A New Approach for Characterizing Oil Fractions and for Selecting Pseudo-Components of Hydrocarbons," Canadian JPT, March-April 1986, pp. 78-84.

• Hong, K. C., "Lumped-Component Characterization of Crude Oils for Compositional Simulation," Paper SPE/DOE 10691, presented at the 3rd Joint Symposium on EOR, held in Tulsa, Oklahoma April 4-7, 1982.

• Hariu, 0. and Sage, R., "Crude Split Figured by Computer," Hydrocarbon Process., April 1969, pp.143-148.

• Katz, D., "Overview of Phase Behavior of Oil and Gas Production," JPT, June 1983, pp. 120-1214.

• Lee, S., et al., "Experimental and Theoretical Studies on the Fluid Properties Required for Simulation of Thermal Processes," Paper SPE 8393, presented at the 54th Annual Technical Conference of the SPE, held in Las Vegas, September 23-26, 1979.

• Maddox, H. N. and Erbar, J. H., Gas Conditioning and Processing, Vol.3-Advanced Techniques and Applications, Campbell Petroleum Series, Norman, Oklahoma, 1982.

• Maddox, R. N. and Erbar, J. H., "Improve Predictions," Hydrocarbon Processing, January, 1984, pp. 119-121.

• Mehra, H., et al., "A Statistical Approach for Combining Reservoir Fluids into Pseudo Components for Compositional Model Studies," Paper SPE 11201, presented at the 57th Annual Meeting of the SPE, New Or-leans, September 26-29, 1983.



• Montel, F. and Gouel, P., "A New Lumping Scheme of Analytical Data for Composition Studies," Paper SPE 13119, presented at the 59th Annual SPE Technical Conference held in Houston, TX, September 16-19, 1984.

• Pedersen, K., Thomassen, P., and Fredenslund, A., "Phase Equilibria and Separation Processes," Report SEP 8207, Inst. for Kemiteknik, Denmark Tekniske Hojskole (July 1982).

• Schlijper, A. G., "Simulation of Compositional Process: The Use of Pseudo-Components in Equation of State Calculations," Paper SPEI DOE 12633, presented at the SPE/DOE 4th Symposium on EOR, held in Tulsa, Oklahoma, April 15-18, 1984.

• Whitson, C., "Characterizing Hydrocarbon Plus Fractions," Paper EUR 183, presented at the European Offshore Petroleum Conference held in London, October 21-24, 1980.

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