12634 critical properties estimation from an equation of state

SPEIDOESOciotyof U.S. Depmtm.nt

Potrolwm Engirmore of Enwgy

SPEMX3E 12634

Critical Properties Estimation From an Equation of Stateby C.H. Whitson, Consultant

Member SPE-AIME

Copyright 1984 Society of Petroleum Engineers of Al ME

This paper was presented at the SPE/DOE Fourth Symposium on Enhanced Oil Recovery held in Tulsa, OK, April 15-18, 1984. The material is subject tocorrection by the author. Permission to copy is restricted to an abstract of not more than 300 words. Write SPE, 6200 North Central Expressway, Drawer6470S, Dallas, Texaa 75206 USA. Telex 730S89 SPEDAL.

AESTRACT

This paper describes a new method for calculating INTRODUCTIONcritical properties of petroleum fractions used asinput to a cubic equation of state (EOS). The Cubic equations of state are used to calculatemethod differs from existing nethods in that it volumetric and phase behavior of petroleumforces the EOS to match measured values of boiling reservoir fluids. Input data required by anpoint and molar volume (molecular weight divided equation of state (EOS) usually includes criticalby specific gravity) for each petroleum fraction. pressure, critical temperature, and acentric

factor of each component in the mixture. For purePVT predictions are made with the proposed compounds these properties are known. Critical

method using the Peng/Robinson EOS for selected properties must be estimated for the petroleumreservoir fluids reported in the literature. fractions making up heptanes-plus. Having definedSaturation pressure and saturated density are critical properties for all the components in acalculated With the EOS and compared with mixture the EOS can be used to predict PVTexperimentally determined values. Heptanee-plus

(C7+)properties.

fractions are characterized using theproposed method and results indicate that both This paper proposes a new method forvolumetric and phase behavior are L=preved= calculating critical properties of petroleumReservoir fluids used in the study repreeent a fractions. The method requires normal boilifigwide range of compositions with PVT properties point and molar volume for each petroleum

-.noinm f~Qg 38-120 ‘JC.reportea at temperzt”tires.=..5...0 fraction. The EOS chosen is forced to fit theboiling point and moiar voiume by adjusting

A new method is -suggested for matching critical pressure and critical temperature.experimental saturation pressure with an EOS. Acentric factor is estimated by an empiricalUsing the proposed method for calculating critical correlation using boiling point and molar volume;properties, the boiling point of the heaviest this correlation is general and does not depend onpetroleum fraction is adjusted until mixture the EOS.saturation pressure is matched. A near-linear

16relation exists betiiesr, hfi;limo““~--..~ r-mint of=_____ the Whitson reviews the most common empiricalheaviest fraction and saturation pressure. It is correlations used for estimating ~ri~ica~

suggested that this method has more physical properties of petroleum fractions. He studies themeaning than the common practice of adjusting effect of C7+ characterization on EOS predictionsmethane binary interaction coefficients. and concludes that none of the existing

correlations gives consistently better PVTProposed methods can be used with any cubic predictions. Observations from this earlier work

EOS. Critical properties are presented graphically led to the idea that an alternative approach couldfor the Peng/Robinson and Soave/Redlich/Kwong be used for defining critical properties ofequationa. A generalized form of the two-constant individual petroleum fractions based on the EOScubic EOS is proposed, and necessary expressions used for mixture calculations.

i“for p“nase aiid ‘val”umetrG ea~c!&~ations are given.

The critical-property method iS diagramedschematically to facilitate programming. It can beeasily incorporated into existing PVT software-already baaed on an EOS.

References and illustrations at end of paper.

.,.!JY

2 CRITICAL PROPERTIES ESTIMATION FROM AN EQUATION OF STATE SPE 12634

.. .in ~nls work LUD pZ~UILLLUIL. ~~~ ~~~~~~d k’~~h-r-m -- . . --------

experimental PVT data and results indicate thatthe proposed method gives improved VLE and phasedensities. PVT data for reservoir fluids havebeen taken from the literature (Refs. 3,8,9). Theyinclude fluid compositions and detailed C7+description, saturation pressures and phasedensities. The fluids have a wide range incompositions and measured PVT data are reported atseveral temperatures. Neither composition ortemperature variation appear to affect thereliability of the proposed method. Also, thetraditional method of adjuating binary interactioncoefficients to fit saturation pressure isreplaced by a new method of adjusting the boilingpoint of the C7+ residue (heaviest petroleumfraction).

The Peng/Robinson 10 (PR) EOS has been usedexclusively in this study, though the method isequally applicable to other two-constantj3cubic

equations such as the Soave-Rediich-Kwong Eos.The method i~l not recommended for the modifiedRedli~$-Kwong EOS propos~$ by Zudkevich andJoffe (see also Yarborough ).

Since the critical property method is based onmanipulation of a cubic EOS, the first sectionreviews the basic calculation procedures forsolving an EOS for a single component. Ageneralized form of the two-constant cubicequation is given to make the method readilyapplicable to any EOS. By specifying a numericalQQnstant: A, and the universal criticalcompressibility factor, Zc, the general EOS

.reauces to aiiYt-wo-co~si.~~t ~tibi~ ~q=~ti~=e

Equation of State Thoery

Martin shows that almost all cubic equations canbe generalized by a four-constant EOS. Thepressure-explicit form of the equation is written,

p=RT_ a............. (1)

(v;b) (v-b+c)(v-bti)

where p = pressure, T = temperature, v = molarvolume, and R = universal gas constant. Constantsa, b, c, and d define the EOS and they can bedetermined by (1) defining thermodynamicconditions or (2) fitting the equation toexperimental data. Eq. 1 states that pressureresults from the interaction of repulsive forces,RT/(v-b), and attractive forces, a/(v-b+c)(v-b-ki).

The first cubic equation of state was proposedby van der Waals in 18732and results from Eq. 1 ifb=c=d, or p=RT/(v-b)-a/v . van der Waals chose twothermodynamic criteria for defining EOS constants.They have since been named van der Waals (VDW)critical criteria which, stated mathematically,require that first and second derivativea ofpressure wi$h vo1ume equal zero:(apfav) = (a p/a~f~psc~attoTc and PC.

In Fig. 1 pressure is plotted versus volumefor a pure compound at several temperatures. Onthe critical temperature isotherm the slope equalszero at (VC,PC) and it does not change sign at theinflection point. These two graphicalobservations are equivalent to the VDW’S criteriastated mathematically above.

Expressing an EOS in terms of pressure isackward for practical purposes because volume isusually the unknown, while pressure andtemperature are measured quantities defining thestate of the system. To solve an EOS in terms ofvolume the compressibility factor, Z, is usuallyintroduced as a shorthand for pv/RT - i.e.,

Z = pv/RT

v = ZRT/p ......● ...................... (2)

~ . ~@~~

where p is densi~y expressed as mssavolume (e.g., kgfm ).

The Z-explicit form of Eq. 1 is

Z3 + Z2{-3B + (C+D-1)}

2+ 2 [2RLJ” - z~(~+~-1) + (A+CD-C-D)

+{-B3 + B2(C+D-1) - B(A+CD-C-ilj-

per unit

..,.1LUJ = G

... (3)

where a new set of EOS constants A, B, C, and Dare expressed as

A = a“p/R2T2 = a“~”Pr/Tr2 ........... (4a)

B = b“p/RT = ~Opr/Tr .**.********* (4b)

C = c*p/RT = ~Opr/Tr .***.******** (4c)

D = dOp/RT = Qd=pr/Tr ..**..**..*.. (4d)

where Tr = T/Tc and pr = p/Pc. Eqs. 4 are

definitions which ease the mathematics used todevelop an EOS.

In Eq. 1 the undefined EOS constants were a,b, c, and d. For Eqs. 4 the EOS constants are nowQa, ~, Clc,and ~d, numerical constants which willbe defined using the VDW critical criteria and theassumption that ~ is proportional to ~b, i.e.,Q= = A=%, where A is a numerical constant.

SPE 12634 CURTIS H. WHITSON 3

The conditions defining Q constants are then

ap/av = o .....................● ....... (5a)

a2pfav2 = o ........................... (5b)

Z=zc = constant ..................... (5C)

n= = A*.% where A= constant ......... (5d)

Eqa. 5a-5c apply at the critical point defined by

~; :)$ P.>T and can be written in shorthand as= o. The last constraint, Eq. 5d, is

chosen so that the generai EOS can be easilysimplified to the most common two- and three-constant equations. Results of algebraicmanipulation shows that 0 constants are definedby:

~b3 - ~b2(k2-3~+3Zc) + ~b(~-3~Z~+3Z~2)

- Zc3. . .......................... (6a)

~d =

S2a=

+

flb(j-a)-3ZC+1 ............... (6b)

~b2(k2-3~+3) + ~(3AZc-X-6Zc+3)

(3ZC2-3ZC+I) .................... (6c)

The smallest root of Eq. 6a is chosen for ~.fl&and Qa follow from Eqs. 6b and 6c, while flcisdefined by Eq. 5d. Zc is the universal criticaicompressibility factor which is assumed to be thesame for all compounds. By specifying i and Zc theEOS is defined, for example the RK and PRequations,

RK EOS : A=lor2andZc=l/3

(7)PR EOS : ~ = 2 T fiand Zc = 0.307401...

ReturnLng to Eq. 4a, the term aisa

correction to EOS constant $la to ensue accurateVLE predictions of pure compounds. a istemperature dependent and acentric factor, W, istypically used to correlate variation intemperature dependence for compounds of varyingspecies. Both the SRK and PR EOS’s use thecorrection factor for %~ resulting in thecomplete definitions of their respectiveequations:

Soave/Redlich/Kwong (SRK)

k=l(or2)

Zc = 1/3

G= l+m(l-~)

m = o.480 + 1.574uJ- 0.17(jw2 ............. (8)

Peng/Robinson (PR)

A = 2-E (or 2+ti)

~c = g*3fi7&g~,..

G= 1 +m(l-~)

m = 0.37464 + 1.54226u - 0.26992w2 ....... (9)

To complete the development of Eq. 1 it iSnecessary to express fugacity, f, for single andmulticomponent systems. Without introducing theformal definition of fugacity it can be visualizedby referring to Fig. 1. The subcritical isothermlabeled Tb has a cubic form. The Pressure whichdivides the loop into two sections with equal area(shaded) is the vapnr pressure for the temperaturedefining the isotherm. This equilibrium condition

is alao satisfied when cnmponent fugacitiea for

I vapor and liquid equal one another - i.e., fv=fL.At equilibrium conditions the vapor (which iS

represented by the volume solution at the right)is aa content being a vapor as the liquid (whichis represented by the voiume at the left? iscontent being a liquid.

For a single-component system the fugacity ofa given phaae is,

In(f/p) = -ln(Z-B) + L+ LE+EIZ-B D-C Z-B+D Z-B-I-C

+ A ~n(Z-B+D)—— .............. (lo)C-D Z-WC

Fugacity of a single component system iscalculated by first specifying the criticalpressure, critical temperature, and acentricfactor. Defining the state with pressure andtemperature, Eqs. 4 are used to calculate A, B, C,

~ ~~ ~Q~.J~d fQr z+and D, and Eq. If only onesolution exists then fugacity is not of interest.If three solutions are found then the smallest Zis defined as liquid and the largest Z ia definedaa vapor. Eq. 10 is solved for fugacity of eachphase and if fv = fL then the pressure andtemperature define an equilibrium point on thevapor pressure curve.

61


For a single component in a mixture thefugacity is given by,

Bi A Bi-Di Ci-Biln(fi/ziP) = -ln(Z-B) +_+_ e+—}

Z-B D-C Z-B+D Z-B+C

21jzjAij +A(Di-ci)l ~n(z-B+D)+

(C-D) (C-D)d Z-B-I-C

. . . (11)

where Aij=~. A, B, C, and D represent mixtureaverages calculated using the following mixingrules:

A = Ii ~j Zi Zj Aij (l-kij) •**~*** (12a)

B = Ii Zi Bi ........................ (12b)

c = Ii Zi Ci ...................*.... (12C)

D= Ii Zi Di ........................ (12d)

where Z1 iS mole fraction of the phase and kij are

binary interaction coefficients. Fugacitycalculation for mixtures are complicated and theywill not be covered here since the method fordererning critical properties based on an EOS only

requires fugacity calculation for single-componentsystems.

EOS Critical Property Estimation

The traditional method for estimating criticalprcper~ies ~f petroleum fractions is illustratedin Fig. 2A. The first step ia to determine orestimate normal boiling point, Tb, specificgravity, y, and molecular weight, M, of eachfraction. Empirical correlation are chosen toestimate critical temperature, Tc, criticalpreaaure, p=, and acentric factor, u. Usually itis f3Ufficient to specify Tb and y for Siieh

correlations. Estimated values Tc, Pc, and u forpetroleum fractions are input into the EOStogether with known values for pure compounds.Mixture phase behavior and volumetric celculationaare performed as necessary. Composition,temperature, ~il~ ----“.,-ap.csrau.= a~~ WCSlly Che datadefining the state of the system.

One might ask why the EOS is not used tocheck, for an individual petroleum fraction, ifthe EOS-calculated boiling point and specificgravity (i.e., molar volume, M/Y) match themeasured vduea. It seems reasonable that if theEOS does in fact match meaaured Tb and ii/Y datathen mixture behavior also will be predicted withreasonable accuracy. The proposed method requiresthe selected EOS to =tch Tb and H/Y by adjustingT= and PC. Acentric factor is estimated by anempirical correlation.

Fig. 3 shows acentric factor plotted versusnormai ‘boiiingpoint Eer pa~dfi~S and ~~(XMtiC&.

Approximate linear relations can

~ = -0.477 + 0.00218°Tb(K)

‘A= -0.587 + 0.00218°Tb(K)

Assuming an averaee Watson

be written

.......... (13a)

.......... (13b)

characterize={1-,

factor, ~,- of 12.7 fir paraffins and 10.0aromatics, the following approximate relationu can be derived,

u= -1.0 + 0.042 ~+ 0.00218 Tb(K) ... (

ionforfor

4)

If Tb is given in Rankine then constant 0.00218becomes 0.000983.

Since the measured data for petroleumfractions are Tb and F1/Y it would be helpful toexpress u in terms of these two data. Using theRiazi-Daubert correlation for molecular weight asa function of Tb and Y it haa been possible toderive the following relation,

Kw = 91.127 (M/Y)0”49593 Tb(K)-0.75584

(15)

If Tb is given in Rsnkine, constant 91.127 becomes142.1.

Combined, Eqa. 14 and 15 estimate acentricfactor knowing only Tb and M/Y. It is asaumedthat the estimate of u is sufficient for theproposed method since normal boiling point itselfis matched exactly by the EOS.

Having defined w it is now necessary to usethe EOS to match meaaured ~aiu~~ of Tb ~p-~u.!y,

This is done by adjusting Tc and Pc until asatisfactory match is achieved. The procedure isoutlined in Figs. 2B and 2C. The trial and errorprocedure is begun by first guessing values of Tcand pc. Reasonable estimates are given ~~ the~impie ~Helatians of Riazi and Daubert (seeWhitson ).

To calculate Tb with an EOS it ia neceaaary tospecify Tc, PCS % and pressure, which bydefinition is atmospheric (i.e., normal) pressure,

P~~* Temperature ia aasumed, EOS constants A, B,C, and D are calculated, and Eq. 3 is solved forz. If only one Z solution is found then thetemperature is not equal to Tb. If threesolutions are found then the lowest value iadesignated as liquid and the largest value asvapor. Fugacities are calculated for each phaseand if they are equal then the assumed temperatureequals Tb. If fugacities do not equal then a newtemperature ia assumed.

I


Instead of using this procedure to solve forTb it is sufficient to check if vapor and liquid

f;gacities are equal at the specified Tb (and

Pat)” If they are not equal then an errorfunction is calculated, E(Tb), defined as thedifference in fugacities of vapor and liquidphases. Holding p= constant, Tc is varied until

E(Tb) = O. A simple Newtwon or chord method ia

sufficient to determine Tc.

Having matched Tb, M/T is calculated with theEOS using Tc (from previous step), p= (heldconstant in the previous step), w (defined by Eqa.14 and 15), pressure, psc, (equal to standardpressure), and temperature, Tsc, (equal tostandard temperature). Constants A, B, C, and Dare calculated and Eq. 3 is solved for thesmallest Z factor. M/Y equala molar volume (i.e.,ZRT/p).

An error function, E(M/Y), is defined as thedifference in measured and calculated Fi/Y. Thecritical pressure, PC> is adjusted and theprocedure for matching Tb is repeated. When bothE(Tb) and E(M/Y) equal zero then the Tc and Pcvalues have been determined. The double trial-and-error procedure is relatively simple toprogram. With the generalized EOS presented inthis paper it ia eaay to determine criticalproperties of petroleum fractions for any two- orthree-constant EOS.

Tables of PR and SW critical properties forextended ranges of Tb and M/Y can be obtained fromthe author. Figs. 4a - 4d present the resultsgraphically. Instead of reporting criticaltemperature directly, it ia given as the ratio ofT= to Tb, i.e., Tc/Tb. Critical pressure isreported in MPa and the conversion to psia ispc(paia)=145.0377 *pc(MFa).

PVT Predictiorisusing EOS Critical Properties

Hoffman, Crump, and Hocott 3 presentcompositional and PVT data for a reservoir oil andits gas cap fluid. Extended analysia of heptanes-plus ia given to C35 for the oil, including malefractions, molecular weights, and specificgravities (corresponding to cuts with boilingpointa of normal paraffin). Methane content is52 mol-% and C7+ is 36.84 mol-%, with an averageC7+ molecular weight of 198.7 and specific gravityof 0.8409. Reported bubble point pressure is 3840psia (26.47 MPa) at 201 ‘F (93.89 ‘C).

Using critical properties based on the PR EOSa bubble point is predicted at 3829 psia, or 0.3%lower than the measured pressure. Interactioncoefficients have not been used. Predictedsaturated oil density is 0.6867 gfcc, comparedwith the experimental value of 0.6672, or 2.9%high.

Katz and Firoozabadi match the measured bubblepoint using the PR EOS with increasing interactioncoefficients between methane and C7+ fractions,and a methane binary of 0.17 for the heaviestfraction, C35. The methane-C35 binary waa used

to match the measured bubble point pressure.Cavett’s correlations are used for Tc and Pc whileit ia assumed that the Edmister correlation isused for u. Estimated bubble-point oil density isnot reported by Katz and Firoozabadi.

The gas-cap fluid reported by Hoffman, et al.is analyzed in SCN fractions up to C22. SCNproperties are similar, but not the same for thereservoir oil and equilibrium gaa. Methanecontent of the gas cap fluid is 91.35 mol-% andheptanes-plus constitutes 1.54 mol-% with anaverage molecular weight of 141.25 and specificgravity of 0.7867.

Using the gaa compositions reported byHoffman, et al. and critical properties calculatedwith the proposed method the PR EOS predicts a dewpoint at 201 ‘F of 4044 paia, some 5.3% high. Thisis worse than the prediction given by Katz andFiroozabadi (+0.8%) using methane binaries forC7+ fractions calculated from an empiricalcorrelation.

Table 1 illustrates how critical propertiescalculated from different correlations affectpredicted dew point pressure. It is obvious thatdew point pressure is dependent on the propertiesused to describe the petroleum fractions.

Predicted equilibrium gas composition from thebubble point VLE calculation using EOS criticalproperties resulted in methane content of 91.70mo.1-%, C7+ having 1.318 mol-% with an average

molecular weight of 128.00 and specific gravity of0.7854. C7+ specific gravity is in good agreementwith the experimental value but molecular weightshows more deviation, possibly suggesting that theuse of paraffin molecular weights by Hoffman,et al. ia not correct.

If equilibrium vapor composition resultingfrom the bubble point flash is used to representthe gas cap fluid, retaining only components toC22,” a dew point is predicted at 3706 psia. Ifthe exact composition to C35 ia used, then dewpoint is predicted exactly (equal to the bubblepoint, 3865 psia). This shows the sensitivity indew point prediction due to a round-off error incomposition of less than 0.001 mol-%.

cgIc~~Eted Iiqa&d densities and yields are

compared with experimental data in Table 2.Peng/Robinson predictions correspond to thoselisted in Table 1, where PR-2, PR-3, and PR-4 usemethane binaries suggested by Katz andFiroozabadi. Binaries are not used when EOS-basedcritical properties are used (PR-1).


Results indicate that the Cavett/Edmister(PR-3) combination performs best, though theproposed method gives excellent results as well.Interestingly, liquid yields for predictions basedon calculated gas composition from the bubble-point calculation are very poor, despite goodprediction of liquid densities.

Next we consider the data presented by Olds,Sage, and Lacey in 1945. Extensive experimentalresults are given for six recombined systems, eachmeasured at three temperatures (100, 190, and 250oF). Hexanes-plus is fractionated into seven CU’CS

with approximately equal weight fractions (-10%).Properties of the separator gas and liquid aregiven in the original work and recombinedreservoir mixtures have been calculated usingreported gasfoil ratios.

Critical properties of the petroleum fractionaare calculated using the PR EOS. EOS predictionsare presented in Table 3 for saturation pressure,vapor and liquid densities. Binary interactioncoefficients are set equal to zero in all cases.

Absolute average deviation for saturation------pre~au.= /.<77 “,.+iS Y.-.JJO .tvs ip.c~,d~~p.g ~h~ leanest—-—..—.

mixture (GOR=14440 scffatb), and 5.45% includingthe leanest mixture. Maximum error is 10%. Dewpoints for very lean condensates are difficult tomeasure experimentally and therefore it iareasonable to question measurements of the14440-GOR mixture. Considering the range oftemperatures and compositions, and lack of anyadjustment or use of binary interactioncoefficients, the predictions are very good. Thetransition from bubble- to dew-point is alsopredicted correctly.

Saturated iiquid and vapor defi~itie~Zie

overpredicted. Absolute average deviation is 5.14%with maximum error of 8.1%. Saturated liquiddensities are predicted more accurately than vapordensities. The general trend of overprediction iscwoaite of that which is usually observed withthe PR EOS (using empirical propertycorrelations).

EOS Predictions of the Olds, Sage, and Laceydata presented in 1949 are reviewed in Table 4.The PR EOS iS used to calculate criticalproperties. Binary interaction coefficients arenot used. All systems studied by Olds, et al.are oils with reported bubble point pressures andsaturated oil densities. Measurements are made atthree temperatures for each mixture of differingGOR.

Absoiute average deviation f~i biibb~t2-~CiIlt

pressure is 5.61%, with a maximum error of 13.7%.For all GOR mixtures deviation in bubble-pointpressure increasea with increasing temperature.Temperature dependence may result in part fromincorrect extrapolation of correction factor, a,(see Eqa. 4a, 8, and 9) for methane atsupercritical temperatures.

Saturated liquid densities are predicted withan absolute average deviation of 3.30% with amaximum error of 6.3%. Deviation also increaseswith temperature for all GOR mixtures.

From results presented in this section it canbe concluded that the present method of definingcritical properties based on an EOS is reasonablyaccurate for predictions of saturation pressureand mixture densities. It also appears that VLEpredictions are good. The method is not inferiorto empirical correlations and has the advantage ofbeing consistent with the EOS used. For accuratepredictions using the proposed method it isnecessary to have detailed analysis of the C7+fraction, preferably distillation data includingboiling point, specific gravity, and molecularweight for each fraction.

Matching Experimental Saturation Pressure

Traditionally, experimental saturation pressure ismatched by an EOS using binary interactioncoefficients between methane and C7+ fractiona. Ascan be seen from the mixing rules (Eqs. 12),binaries only affect EOS constant A. It can beshown that VLE calculations are dominated by A, asindicated by the choice to introduce a, a

correction factor for modifying vapor pressures ofpure compounds.

Practically, the use of binaries to matchsaturation pressures is very efficient.Interaction coefficients are, however, not easilygiven physical meaning and there is a tendency toovercompensate for bad input data (e.g., criticalproperties) by introducing large binaries. Whitsongives several examples Indicating that interaction~aefficiez~~ h2.Jes StrOSg temperature dependence.

This can be an important factor in thermalrecovery and surface process appiicatioiis.

The present method suggests that boiling pointof the heaviest petroleum fraction is the mostuncertain C7+ data and that it alone can beadjusted to match saturation pressure. A direct~eis~iar, -w{.*S=’,.”. he~ween Tb of the heaviest

fraction and saturation pressure, as indicated fora North Sea fluid in Fig. 5. An increase in Tbresults in an increase of saturation pressure.Physically, boiling point is easily defined andhas obvious limfts.

A first estimate Of Tb+ for the heavieat

fraction =n be made using molecular weight, M+,and specific gravity, Y+O me Watson

characterization factor is estimated from

and boiling point ts found directl~ from thedefinition of ~ - i.e., Tb+=(R+-Y+) /1.8 whereTb hSS UnitS Kdvin.


CONCLUSIONS

1. A new method for calculating criticalproperties and acentric factor of petroleumfractions based on an EOS appear to give promisingimprovements in both volumetric and phase behaviorpredictions. The method requires molar volume(ratio of molecular weight to specific gravity)and boiling point. The acentric factor relationis empirical, accounting for oil type (i.e.,n.raff~p.i~i~v) Using the Watson characterization~-.-.. ..factor.

2. The method for calculating critical propertieshas been tested using the Peng/Robinson EOS.Results indicate that saturation pressure andsaturated density predictions are reasonablyaccurate without using binary interaction~~eff~~i~~.csfo ~tch experimental VLE data.

3. If experimental saturation pressure is notpredicted accurately with the proposed method, aprocedure is recommended whereby the boiling pointof the heavieat C7+ fraction is adjusted to matchmeasured saturation pressure of the mixture. Thisprocedure has a more direct physical understandingtF,ar, ~e~hc& ,J~ip.g binary interactioncoefficients. It is felt that boiling point ofthe heaviest fraction is the greatest unknown ofmeasurable C7+ properties and can therefore beadjusted within reasonable limits.

NOMENCLATURE

a,b,c$d =

A,B,C,D =E=f=

Kw =.

m=M=

P=Pr =Pc =Pv =R=T=

Tb =

Tc =Tr =v=v.

z=Zc =

A=p=

w=

Q=

constants in the general form ofthe cubic EOSEOS constantserror functionfugacityWatson characterization factorTb(”R)l/3/Ycorrelating parameter for amolecular weightabsolute pressurereduced pressure, Pr = PIPCcritical pressurevapor pressureuniversal gas constantabsolute temperaturenormal boiling pointcritical temperaturereduced temperature, Tr = T/Tc

molar volumevolume

.. ...compressibility factor, Z = pv/iiTcritical compressibility factor

correction factor for EOS constant Aproportionality constant between EOSconstants SICand fib,defining the

proposed family of cubic equationsdeviation, A = (talc.-meas.)/meas.densityacentric factor, u = -log(pv/pc)-l.oat Tr=0.7numerical EOS constants

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10. Peng, D.-Y. and Robinson, D.B.: “A NewTwo-Con~tant Equation of State,” [email protected].(1976)~,1,59-64.

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12. Riazi, M.R. and Daubert, T.E.:

“On theEquation ofons,” Chem.

“Simplifyproperty Predictions,” ~dro.Proc. (Marc~l,1980).——

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14. van der Waals, J.D.: Doctoral Dissertation,Leiden,Holland, 1873.

15. Watson, K.M., Nelson, E.F. and Murphy, G.B.:Uflh.-.,,++z+FifiPiP~f PeC~Q~~IJm Fractions$”“ ,a.a&..=.. ~.-- . -. .-Ind.~.Chem.( 1935)~,1460-1464.——

16. Whitson, C.H.: “Characterizing HydrocarbonPlus Fractions,” Soc.Pet.Eng.J.(August, 1983).—683-693.

17. Whitson, C.H.: “Effect of Physical Propertiesm...--.4-- -n m“s,.++fi”-nf-~~~~~pra~ictions:ltnsLLulaLL”L,v,. uq.Ae.A”.. “.

Paper SPE 11200 presented at the 57th Annual FallTechnical Conference and Exhibition, New Orleans,Sept. 26-29, 1982.


18. Yarborough, L.: “Application of a Generalized

Equation of State to Petroleum Reservoir Fluids,”Equations of State in Engineering and Research,K.C. Chao and R.L. Robinson, Jr. (eds.)

19. Zudkevitch, D. and Joffe, J.: “Correlation.-. . . = .,----=,.-..4Aand Frealctlon or vapur LLI+UAU .Yu . . . . ..u.. . . . . .

u-..4 1 4 h-i,. m ,.r+*h

the Redlich-Kwong Equation of State,” AIChE J.(1970)16,112-119.—

TABLE 1 - EFFECT OF CRITICAi.PROPERTIES USED WLM Li-m,..-. “.7,”

PENG/ROBINSON EOS TO PREDICT THE HOFFMAN ET AL.RESERVOIR GAS DEW POINT PRESSURE.

Dew Point Pressure (psia)

Predicted With Katz/without FiroozabadiMethane Methane

Correlation Binaries Binaries

Experimental ............... 38411. EOS-Based (proposed) .... 4044 (+5.3)2

2. Riazi/Daubert + E~mister: 3322(-13.5) 3632 (-5.4)3. Cavett + Edmister ...... 3668 (-4.5) 3870 (+0.7)4. Lee/Kesler .............. 3523 (-8.3) 4049 (+5.4)

~ Same as reported by Katz and Firoozabadi.Deviation, A, %, given in parentheses.

TABLE 2 - COMPARISON OF MEASURED AND EXPERIMENTAL LIQUIDDENSITIES FOR THE HOFFMAN, ET AL. RESERVOIR GAS

Liquid Density, gleePress.

PR-1 PR-2@@% —— . —PR-3 PR-4

2915 0.6565 0.6637 0.5921 0.6364 0.6300

2515 0.6536 0.6709 0.6020 0.6475 0.6392

2015 0.6538 0.6822 0.6159 0.6619 0.6520

1515 0.6753 0.6967 0.6315 0.6779 0.6665

1015 0.7160 0.7148 0.6486 0.6957 0.6834

515 0.7209 0.7390 0.6675 0.7164 0.7029

IAI:%) 2.32 2.80 1*5O 2.48

Liquid Yieid (bbil’iilisci)Press.

PR-1 PR-2@!@l&E!&_ .— — —PR-3 PR-4

2915 9*O7 9.31 8.37 9.75 10.21

2515 12.44 11.97 12.01 12.61 12.93

2015 15.56 14.53 15.33 15.23 15.48

1515 16.98 16.09 17.30 16.74 17.00

515 15.08 14.97 16.17 15.45 15.73

IAI(%) 3.65 4.46 2.54 3.93

66

TABLE 3 - COMPARISON OF SATURATED PROPERTIES FOR OLDS-SAGE-LACEYRESERVOIR MIXTURES (1945) USING THE EOS-BASED CRITICALPROPERTY METHOD WITH THE PR EOS.

Temp- Saturation Pressure (psia) Saturated Density (lb/ft3)

GOR erature(scf/bbl) (F)

1Meas. Calc. A(%) Exp. Calc. A(%)

552 100 1112 1065 -4.2 42.55 43.34 +1.8

552 190 1410 1383 -1.9 39.68 40.52 +2.1

552 250 1675 1549 -7.5 37.52 38.50 +2.6

940 100 1870 1723 -7.9 39.65 40.88 +3.1

940 190 2210 2174 -1.6 36.13 37.78 +4.6

940 250 2380 2375 -0.2 33.66 35.44 +5.3

2205 100 3430 3102 -9.6 34.13 35.44 +3.8

2205 190 3542 3691 +4.2 29.61 31.52 +6.5

z 2205 250 3595 3880 +7.9 27.07 28.75 +6.2

..— ——— —— -——— —— ——-— ———— ———— -——— -— ———- ———

5361 100 4490 4348 -3.2 27.01 28.54 +5.7

5361 190 4590 4836 +5.4 22.90 24.40 +6.6

5361 250 4630 4869 +5.2 20.37 21.85 +7.3

7393 100 4560 4615 +1.2 24.05 25.99 +8.1

7393 190 4730 4981 +5.3 20.46 22.02 +7.6

7393 250 4780 4933 +3.2 18.35 19.65 +7.1

14440 100 3835 4556 +18.8 19.08 20.05 +5.1

14440 190 4305 4709 +9.4 16.24 16.96 +4*4

14440 250 4440 4504 +1.4 14.49 15.15 +4.6

1

2

Saturated densities are calculated at the experimental saturationpressure assuming a single phase mixture.

Lower GOR’s are bubble-point mixtures and higher GOR’s are dew-pointmixtures, as is correctly predicted by the PR EOS.

TABLE 4 - COMPARISON OF SATURATED PROPERTIES FOR OLDS-SAGE-LACEYRESERVOIR MIXTURES (1949) USING THE EOS-BASED CRITICALPROPERTY METHOD WITH THE PR EOS.

Temp- Bubble-Point Pressure (psia) Bubble-Point Density (lb/ft3)GOR erature

(scf/bbl) (F) Meas. Calc. A(%) Exp. Calc. 1 A(%).—

274 100 817 796 -3.8 49.12 49.87 +1.5274 190 1020 1067 +4.6 46.36 47.96 +3.5274 250 1170 1236 +5.6 44.66 46.54 +4.2

460 100 1540 1361 -11.6 47.96 48.34 +0.8460 190 1830 1805 -104 44.92 46.29 +3.0460 250 2020 2051 +1.5 43.08 44.78 +3*9

620 100 2085 1799 -13.7 46.62 47.19 +1.2620 190 2439 2360 -3.2 43.55 45.04 +3.4620 250 2670 2656 -0.5 41.53 43.50 +4.7

811 100 2656 2273 -14.4 44.68 45.96 +2.9811 190 3067 2944 -9.0 41.98 43.73 +4.2811 250 3384 3284 -3.0 39.70 42.21 +6.3

1. ..I...l=A-Aat the experimental saturationSaturateciciensit%esZrs ~=ab...-.v--<pressure assuming a single phase mixture.

SPE12534

*

TRADITIONAL tETHOo

Pc

Ps(

t

c-

FkIncreasingTemperature

\\~:~\T>Tc

‘-=;CL

w-i ~c

MOLAR VOLUME

Fig. l-PreaaurelV@Jme schematic showing aaveral ikothermspredic$ed by a cubic EOS.

NEW t4ZTHO0

I Yes

ICTWI c rrmAmhHdT I-., =.. ~-. . . .. .. .

i---- ---- ---- ---- ---- ---- ---- --MIxTIK?E

dF*. zkF1OW diagram OuWiningthe propoaad method for cdculaWI Pattieum

fraction critical propartii.

%

t@fAIREO

PDistillation Tby M

ESTIMATEO

Enpiricalcorrelations TC pc ~

I

SIhGLE CCWOfWT

!----- ----- ----- ----- ----- ‘-

---

MIX~U~I

Fig. 2A-FlOw diagram outlining the traditional method for calculatingpatrolaum fraction critical pmparfias.

ITERATIVE PROCESS FOR CRITICAL PRLWERTV CALCULATION

EST WATEO

w

1

ASS4Ko

Pc

Tc

J t I

+

I Yes

e.,--.,-,AL..,. - ., itli. i.. maiterative procaaa used by the WJ_rw. s-.,9. ..,..=, -..--.....:.=

method for cafculatmg petroleum Iracfii cdcicd prc9atis.

SP E12S34

h , 1 I, 1 1

I. NoRMAL PARAFFINS

+ AROMATICS

cORRELATION:

~ =. 1 +0.042 .Kw+0.002180Tb(K)

o.4959%Tb(K)-o.75584\ = 91.127 .(M/Y)

IKW=l Z.7

,Kw=lo. o

/

,,#”]enzene

1 I I

400 600 800 moo

NORMAL BOILING POINT, K

Fig.3—Relation between acentric factor and normal boilingpoint for normal parafflna,generalized correlation.

aromatics, and the

12.6I

12.4

..x’

12.2

12.0 I I t J

750 ‘ I I

II

700

650I

40 50 60

.

c:c.,--

Dew-Point Pressure, MPa

Fig. 5—Empifical relation between dewpoint pressure ofa North Sea gas condensate and titling pointand Watson characterization factor of theheaviest petroleum fracfion (Cm+).

SP E12634

I

PENG/ROBINSCIN EQUATION OF STATP

U)1 I

1“’<’’’””’WIA

t-

.~~~i!-zoo

BOILING POINT, K

PENG/ROBINSON EQUATION OF STATE

q

r “’’’’””

LQ-1

““7

c

P

I_ 1 1?_ 1 I

600 mo moo-200 400

BOILING POINT, K

SOAVE/REOLICH~tKU~G EQUATION OF STATE SOAVE/REDLICH/KUONG EQUATION OF STA.?E

q : r“~~B:

~

“y<*. .-

?

c

, -.

J___.--&-&----&-200

BOILING POINT, K

~-L-1 1 t 1 J

400 6W Wo mad!oo

BOILING POINT. K

10

FIm. 4_crItical property correkti.ns fw petroleumImctions, cak.latd us,w the PrOPOS4mathd withm.P6n@Rtinmn and SOave/RedtchKwongEOS.%.

12634 critical properties estimation from an equation of state

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