.

SPE 15082

A Continuous Mixture Computational Algorithm for ReservoirFluids Phase Behavior

by P.-C. Du and G.A. Mansoori, U. of ///inois

SPE Members

SPE

This pafw was prepared for presenlalicmat the 56th Calitomia Reg!onalMeeting of the Soc!etyof Petroleum Engineers held m Oakland, CA, April2-4,19ss,

This paper was selected for presentation by an SPE Program Committee followingreview of informationoon!ainad in an abatract submittedby theauthor(a).Contents of the papar, ae preaentad, have no! been reviewed by the Society of Petroleum Engineers and are subjad to correctionby theauthor(e),The matarial, aa presented, doea not necessarily reflect any positionof the Society of Petroleum Engineers, its officers,or members. Paperepresented al SPE maalinga are subject to publication review by Editorial Committee of the Society of Petroleum Engineers, Permissionto copy lareatricmdto an abatractof notmore than 300 words, Iilustralionamay notbe copied The abstractshouldcontainconspicuousacknowledgmentof whereand by whom the paper is presented, Write Publicaliona Manager, SPE, P.0, Box S33S3S, Richardson, TX 750S3-3S3S, Telex, 7309s9, SPEDAL.

ABSTRACT paeudocomponent method to 8 system containingvarious families of compounds, such as paraff ina,

D velopment of compositional reservoir simulators is naphthenes, and aromatics, it ia neceaaary to extandpresently hampered due to the complexity of the the pseudocomponent description of the mixtures upexisting computational ●lgorithms of reservoir fluid to C20+ or more so that the experimental data canphaae behavior calculations. Through the be matched by the prediction. Phase equilibriumapplication of the theory of polydisperse fluid calculation for the latter case is intricate, and inmixtures wa have developed a new algorithm for addition to the difficulty mentioned above, a greatreservoir fluid phase behavior calculation. In deal of computational tima will be necessary. Aa

4 for a typical systemthis algorithm a reservoir fluid is considered to nbted by Hathias ●nd Benson,consist of a continuous mixture with a defined . encountered in engineering calculations themolecular weightfcomposition distribution. The computational time depends primarily on tha termsproposed algorithm requires less computational time involving pairs of components in the equation ofthan the time required by the ●xisting state mixing rula summations.pseudocomponent VLE calculation methods. It is alao h alternative method which can significantlyahown that the new algorithm is applicable to reduce the difficulties mentioned above sterna fromvarietiea of equations of state and mixing rules. the idea of polydisperse (or continuous)Results of the proposed technique are compared with thermodynamics. That is, instead of mole fractions,results of the other ●xisting techniques, and with a density distribution function which is a functionthe simulated rnulti-pseudocomponent reservoir fluid of some measurable property such aa normal boilingmixture dsta. point, liquid density, or molecular waight is

introduced to describe composition of many-componentINTRODUCTION mixtures such as petroleum reservoir fluida.

In complax mixtures, such aa petrolaum reservoir Initially, the concept of continuous mixtures wasfluids consisting of a large number of components, ●ppliad in polymer solution theories and predi t-onit ia very difficult to identify components andtheir concentrations. Thus description of such

of liquid-liquid ~eparation in such ●olutions~-iAris and Gavalas introduced functional analysis

m:.xturas by mole fractions of components is in the thermodynamic description of polydispersepractically impossible, and a simplified technique system and kinetics of continuous reactions such aswhich ia called “paeudocomponent method” haa been polymerization or cracking. Concerning petroleumgenerally used. By this method the lighter distillation, flash calculation schemes which arecomponanta (Cl to C6) are identified ●nd theheavier components (C7+) are splitted into ●

based on simple models, such aa Raoult’s law, hfebeen reported for ontinuoua mixtutos. 9-11 For

number of fractions which are characterized by their example, Hoffmanl 6 preaentsd s numericalaverage boiling pointa, liquid denaitiva, and integration method. In this method, he ●doptedmolecular weights. Then phase equilibrium integrated form of Clauaiu8-Clapeyron equation ●ndcalculations ●re performad by ordinary thermodynamic Raoult’s law to compute vapor pressures of petroleumtechniques in which ●ach fraction ia considered as a

component. 1-3fluids. He used Gaussian distribution with normal

pure When applying the boiling temperatures considered as the independentvariable to describe the vapor and liquid phasea.

References and illustrations at end of paper, Similar procedures s Hoffman’s were presented byR~tzsch ●nd Kehlen,13 whoso treatment waa based on

991--

at ChicagoBy Ping-Chan Du and G.Ali Mansoori, University of Illinois at ChicagoSPE Members

This paper was presented at the 56th California Regional Meetingof the Society of Petroleum Engineers held in Oakland, CA

during April 2-4, 1986.

2 A Continuous Mixture Computattonal Algorithm For Reservoir Fluids Phase Behavi.or SPE ]5082

Raoult'1 law with compo1ition1 de1cribed by normal den1ity function to derive thermodrnamic propertiesof continuous mixtures, Edmister 3 developed a graphical procedure by which flash vaporization calculations of continuous mixtures were performed for petroleum distillation,

An approach using equation of state for 1olving phase equilibrium problems in continuous mixtures was developed by Gualtieri et al, 14 They used the van der Waals equation of state (vdW EOS) to 1olve the f�actionation of a polydiaperse impurity dissolved in a solvent, the 1hift of the critical point due to the presence of a polydi1per1e impurity, and calculation of the cloud-point surface and critical point of a completely polydisperse system. Similar result• for flash calculation of continuous mixtures which were baaed on the Redli�h­Kwong equation of 1tate were reported by Briano, 15 Cotterman and Prausnitz16 introdced two techniques for phase equilibrium calculation of reservoir fluids, one of which is a continuous model and the other is a semi-continuous model, In both of their technique, they used the Soave equation of state, with two interation parameters a and b (considered as linear expressions with respect to molecular weight), to perform phase equilibria c�!culation,

Due to the lack of a general algorithm for continuous mixture phase equilibrium calculation in all of the previous attempts reported above a number of crude assumptions has been made. Such approximations have hampered the effective use of continuous mixture theory in practical phaseequilibrium calculations and development of accurate pred:ction techniques. In the present report we introduce a new computational algorithm by which we can perform phase equilibrium calculation of continuous mixtures consisting of species with a wide range of molecular weight distribution such asa petroleum reservoir fluid. In part II we introduce the basic theory behind this new technique, In part III we apply this technique, together with the continuous vdW EOS, to a specific continuous mixture which resembles a gas-condensate reservoir fluid. In part IV we demostrate VLE flash calculations using the proposed technique with the continuous Peng-Robinson equation of state (PR EOS) and compare results of the calculation with simulated gas-condensate VLE flash data.

THEORY OF PHASE EQUILIBRIUM OF CONTINUOUS MIXTURES

For a mixture with a large number of components, composition may be replaced with a composition distribution function F(I,1

0,�) whose independent

variable I i1 some meaaureable property such asmolecular weight, boiling point, or the likes,15 with mean value of 1

0 and variance of �- The

composition distribution function F(I,10

,�) is normalized such that

f F(I)dl • 1I

over the entire range of I.

Several single-variable distribution functions have been proposed for the composition distribution functions of continuous mixtures by different

392

investigators as it is reported in Table 1, For example, normal distribution function was agopted by Kehlen and Ratzach12 and also by Hoffman1 for

7polymers and reservoir fluids; the Schulz1

distribution function was used by Gualtieri et a1, 14 for their fundamental studies on continuous mixtures; the gcmma distribution funyiion wasselected by Cotterman and Prauanitz for petroleum reaervoir fluids. To choose a certain continuous distribution function, some generalknowledge about composition of the many-component mixture, under consideration is necessary, For example, the exponential-decay distribution function18 can be applied for the gas condensate reservoir fluids; while the 10 called "gamma"distribution function18 may be applied for heavy petroleum fluid mixtures, asphaltenea, and resins,

In the case of the continuous mixtures an extensive thermodynamic property, such as the compressibility factor Z, may be considered as a function of temperature T, pressure P and the extensive distribution function F(I):

J•Z[T,P,F(I)] (1)

where I is an independent variable �uch asmolecular weight or boiling point.

For such a mixture from the expression of the compressibility factor one can derive expre11ion1 for other thermodynamic properties, Here, we present expression for the chemical potential of a fraction in a continuous mixtures which is neededfor the phase equilibrium calculation, that is

o • ap RT v µ(I)•µ (I)+/[-- -]dv-RTln,--- +RT (2)

v bF(I) v RTF(l)

or V

µ(I)•µ0(I)+o(v,T,Io,�:{I})-RTln---­F(I,Io,�)

(3)

where µ0(I) is the chemical potential of the continuous reference state at temperature T. With regard to the above equation we have assumed that there is only one family of continuous mixtures in the system. Extension of the above formulas to the case of a multi-family continuous mixtures is 1traightforward,

According to classical thermodynamics for phaseequilibrium calculation between several phases in equilibrium, it is necessary to equate temperatures, pressures, and chemical potentials of components in every phase. For vapor-liquid equilibrium of a continuous mixture, this means that the following conditions must be satisfied:

p • pL[vL,T,FL(I)) • pY [vy,T,Fy(I))

µL(I)• µY(I)

(41

(S)

(6)

Considering the varialbe I changes very large values, many equations,

fact that distribution function from some initial value of It to eqn.(6) is representative of very

SPE 15082 P. C. Du & G. A. Mansoori

To illustrate the difference between discrete andcontinuous vapor-liquid equilibrium concepts, Fig,(l) which represents a comparative flash VLE calculation scheme is reported, Considering thefact that distribution functions are not additive Ff (I), F1. (I), and Fy(I) can not possess the same functional forms in a specific flash calculation scheme. Considering eqn.(3), we can rewrite the phase equilibrium conditions for a one­family continuous mixture as the followings:

P • pL(T,v1.,Io1.,�1.)• pY(T,vy,Ioy,�y)

oL(T ,v1., IoL,�L: {I}) - RTln vi

{ } F(IoL,�L: I )

"' oY(T,vy,loy,�y:!I))- RTln Vu {} F(Ioy,�y: I )

(7)

(8)

In order to perform calculation, it is necessary to state to calculate pressure andin the liquid and vapor phases in

phase equilibriumuse an equation ofchemical potential equilibrium.

Generally, it is impossible to· solve eqn's.(7) and (8) simultaneously for the liquid.and vapor continuous mixtures in equilibrium. that ha: beenthe reason for the crude simplifying assumptions proposed by the previous investigators, 14-16

At this stage we assume that first n derivatives of the chemical result eqn' s. (7) and (8) can be following set of equations:

there exist the potential. As a replaced by the

L V P •P (T,v

1.,Io1.,�1.) • P (T,vy,Iov,�v) L V

µ (T,vL,IoL,�L:{I})•µ (T,vy,Ioy,��:{r})

oµL(T,vL,IoL,�L:{I}) oµ

V(T,vy,loy,�y:fI})

oI a1

• • I • I • 9 t t • e • t t t • t . • t t • • t t I t I I I I I t I I

I I I I I I I I I I I I I I I I t I I I I I I I I I I I I I 1 1 I I I

I I I I I t I I I I I I I I I I I I I I I t I I I I I I I I I I t I I

n L o µ (T,v1.,Io1.,�L:{I})

oI

n V • o µ (T,vy,Ioy,�y:{I})

nn

(9)

(IO)

(11)

02)

To perform vapor-liquid e�uilibrium calculation for a one-family continuous mixture, according to the phase rule of continuous mixtures(to be published later) we need at most 5 equations. We now assume that there exists a field distribution function

F (I) • F (I,� , I ) X X X f.X

such that the following effectively represent the n eqn's.(10)-(12).

four equationsequations given

(13)

can by

393

ft • J µV

(I) F (I)dl ( I) F (I) dlX I X

(14)

oµ L (I) V

JI F (I)dI - J

oµ (I) F (I)dl

oI X I a1 X

(15)

JI 02/(I)

F (I)dl ·f0 2 / {I)

F (I)dl01 2

X I 01 2 X

(16)

o l /cn V

JI -Jo l i. en

F (IldI F (I)dl �I' X I oi >

X

(17)

The equality of pressures, eqn.(9), together with eqn's.(14)-(17) will constitute 5 equations which are needed for VLE calculation of a one-familycontinuous �ixture. Functional form and parameters of the field distribution function fx(I,�x,Iox> will depend on the nature of the system under consideration. In some instances, it may be chosen to be identical with the distribution function of the feed,

CONTINUOUS vdW EOS FOR GAS-CONDENSATES

To illustrate utility of the proposed continuous mixture phase equilibrium algorithm we present here a simple example in which for the equation of state the vdW EOS of mixtures,

P • RT/(v-b) - a/v 2

2: 2:x · x · (a· a ·) ½ij'J lJ

a •

(18)

(19)

b • 2: X • b • (20) i 1 1

and for the distribution function the exponential­decay distribution function

F (I)• (1/�) exp [-(I-Io)/�] (21)

are used.

In order to extend the vdW EOS to continuous mixtures eqn's,(19) and (20) for a and b must be replaced with the following two expressions, respectively,

a • {J F(I) (a(I)]½dI} 1

I

b • J F(I)b(I)dI I

(22)

(23)

When we apply the above equations for ahomologous series of hydrocarbon coapounds,[a(I)]5i, and b(I) will be polynomials relating to the molecular weight of hydrocarbons compound. Forthe family pf paraffins, Fig's.(2) and (3) indicatethat [a(I)]5i and b(I) can be accurately correlated to the following 3rd order polynomials, reapectively,

(24)

3

4

A Continuous Mixttire Computational Algorithmfor Reservoir Fluids Phase Behavior SPE 15082

(25)

By introducing eqn' 1. (24), (25) and (21) into eqn'a.(22) and (23), the following expres1ion1 for a and b will be obtained

a • [ao+a1(Io+�)+a2(Io2+2Io�2)

+a3(103+310

2�+6Io�2+6�3)] 2

b • bo+b1(Io+�)+bz(Io2+2Io�2)

+b3(Io3+3102�+6Io�2+6�3)

(26)

(27)

In order to derive expression for the chemical potential in the continuous mixtures eqn,(18) will be replaced in eqn. (3) with the following result

where bo 2aoQ (I,�)

Do m-ln(v-b)+---- ---- -ln�+Io/�v-b RTv

b, 2a,Q(I .� )D1 • -1/�

V -b RTv

b2 2uQ(I, � )D2 • V -b RTv

D3 • ba 2uQ(I,� )

V -b RTv

(28)

(29)

(30)

(31)

(32)

Q(I,�)•2[a(I)]'5[a0+a1(I0+�)+a2(101+2J

0�

+2�1)+a3(10

1+3I0

1�+6I0

� 1+6�1)]

According to eqn'a,(l))-(32) Do, D,, D2 and D 3

are all functions of te�perature, volume, and the distribution variance �.

To perform VLE flash calculation we need to replace eqn.(28) in eqn'i.(14)-(17). For the present example n•J, but since, Io• IoL� Ioy• Iof• molecular weight of methane, we will only need to consider up to the first derivatives of the chemical potential in the criteria of equilibrium. A1 a result, we will need to solve the following 1et of 4 equations:

RT � RT _ _!L p - --- - -:-:--z- - ---

---::-z-VL vy-by vy

D0L-00V+(D1 L-o, V)(Io +� ) X X

+(D2L-02Y)(Io �+2Iox�x+2� �)

(33)

+(DaL-Day)(Io 3+310 2� +6Io � 2+6� 3)•0 (34) X X X X X X

+3(DaL-oaY)(Iox2+2Iox�x+2�x2> • 0 (35)

On Fig,(4) the P-T diagrams resulting from thesimultaneous solution of the above 4 equations for �f·15.0 and for various values of �x are

394

reported, Al10 reported on this figure is the P-T diagram based on a previous continuous mixture VLE com��tational scheme proposed by Gualtieri etal. According to Fig.(4) the present technique is more versatile in representing VLE data of different reservoir fluids,

Application of the proposed algorithm for the vdW EOS is a simple example of continuous mixture phaseequilibrium calculation. In what follows we introduce the proposed algorithm for a sophisticated equation of state which is extensively u1ed for reservoir fluid phase behavior calculation.

CONTINUOUS PR EOS FOR GAS-CONDENSATES

The PR EOS for mixtures, RT

p - ___ _ a(T)

v-b v(v+b)+b(v-b)

a (T) •I: 2: x · x · (a · · a · ·) � • [I: x • a · · ½J 2i j 1 J 11 JJ i 1 11

b•� xibi 1

where

a••(T)•a(T ·) (l+k,(1-T ,'5)] 2 11 Cl 1 r1

a(Tci)•0.45724R2Tci2/Pci

bi•0,0778RTci/Pci

ki•0.37464+1,54226w-0,26992w2

(36)

(37)

(38)

(39)

(40)

(41)

(42)

ha1 received a wide spread acceptance in phase

behavior calculation of reservoir fluida,1 In order to extend thi1 equation of state to continuous mixture, we introduce eqn.(37) in the following form:

(43)

where

and

a1•� Xiail' a2•� Xiai2 1 1

a· 1· (a (T · ) ] '5(1+k ·)1 lC 1

a· 2•[a(T ·)/T -]� k· 1 Cl Cl 1

(44)

(45)

(46)

On Fig'1(5)-(7) graphical representation of •il• ai2 • and bi for paraffin hydrocarbons versus molecular weight are reported, We have been able to represent a1(I), a2(I), and b(I) of paraffins, by the following third order polynomials with respect to molecular weight I:

a1(I)•a1o•a11I+•12I1+a13I 1

a2(I)•a2o•a21I+a22I•+a23I1

b(I)•b0+b1I+b2I 1+b311

(47)

(48)

(49)

Now, by using the exponential decay-distribution

P. c. Ou & G. A, Mansoori 5

function and a similar procedure a1 in the case of the vdW EOS we can derive the continuous mixture expre1sions for oarameters ••• a2, and b of the PR EOS as the following

•1·•10••11<Io+�)+a12<102+2Io�+2�2)

+a13(Io3+3 Io

2�+6Io�2+6�3)

•2••20••21<Io+l)+azz(Io2+2Io1+212)

+a23 Ciol+31021+6Io�2+6�3)

b • b0+ b1(I0+�)+ b2(I0

2+2I0�+2�2)

+b3(Iol+310

2�+6Io�2+6�3)

(50)

(51)

(52)

Knowing the continuous PR EOS chemical potential of components of the mixture can be derived in the following form:

where

D0 •-RTln(v -b) +C1 b0

and

+c2<•10+a20rl:i)-RTin,+RTI0/�

D1 •C1 b1+C2 (a11+•21t'-l)-RT/�

D2 •C1 b2+Cz (a12

+•22t'-l)

°3 •C1 b3+C2 (a13+a23t'-l)

RT '1. --­v -b

a 2.828b2

C2 • Q(�·)

1.414b

a (v +b ) b (v 2+2v b -b 2)

V -0,414b ln

V +2,414b

ln V -0.414b

V +2.414b

Q(� )•a10+•11(Io+� )+a12(lo

2+2Io� +2�2 )

+a13 Clo3

+3Io2� +6Io�2

+6�3 )-[a

20••21<Io+� )+•22 <1o2+2Io� +212 )

(53)

(54)

(55)

(56)

(57)

(58)

(59)

To perform VLE flash calculation we need to replace eqn.(36) in eqn'a,(14)-(17), In this example n•l, but since I0x• loL• Ioy• Io1• molecular weight of methane, we will only need to ccmaider up to the first derivative of the cnemical potential in the criteria of equilibrium. Aa a reault, we need to aolve the followin& aet of 4 equation,.

•RT (a1y-a2yT�)2

(61) vy-by vy(vy+by)+by(vy-by)

<00L-00

Y)+(o1L-01Y)(Io•1x)+(o2L-02V)

(Io2+2lo �x+2�x

2)+(D3L-D3V)

(I! +3I02�x+6I0 �x2+6�x3)• 0 (62)

(01L-01Y)+2(DzL-02Y)(Io•�x>

+3(D3L- D3

Y)( I02+2I

0�x+2�x2

>• 0 (63)

It should be noted that Eqn'a.(62) and (63) are similar to eqn's,(34) and (35) which were derived for the vdW EOS, By u1ing eqn'i.(61)-(63) the saturation pres1ure, P-T diagrams, of three different hypothetical 1a1-conden1ate reservoir fluid, are calculated and they are reported in Fig's.(8)-(10). Al10 reported in the1e figures are the P-T 'diagram, of the aame reservoir fluid1 assuming to contain 6, 10, and 20 pr docomponentl, respectively, According to these igurea, the propo1ed continuous mixture model can represent phaae behavior of a many-component mixture effectively.

CONCLUSIONS

We have proposed a continuous mixture phase equilibrium algorithm which can be uaed for vapor­liquid equilibrium calculations involving re1ervoir fluids. The proposed algorithm it applied to the ca1e of &••-condensate 1y1tea1 with analyticl re1ult1. Accurate prediction of reservoir fluid phase behavior by usin& a paeudocomponent Mthod require• the a11umption of a large number of p1eudocomponent1 and as a re,ult excessive computer time is needed, The proposed continuous mixture technique reducaa the required computer time significantly while it retains accuracy of the prediction,. The computer time needed for the propoaed acheme ia rouahly equivalent to th• time needed for a two-components mixture VLE calculation.

Tht proposed continuou1 mi7,ure technique i1 applicable to varietiea of reservoir fluids, equations of 1tate, and mixing rulea. In tha present report its application for some hypothetical ga1 conden1at1 systems uaing two repreaantive equation, of atate, the van der Waala and the Peng­Robinson equations, is demonatrated.

In order to apply the propoaed alaorithm for phase behavior prediction of realistic re1ervoir fluida we need to include the unlike-interaction parameters in the equation of state mixin1 rulea, In addition, knowledae about the nature of the molecular wei&ht di1tribution function of specie, of reaervoir fluids will be neceaaary. In our cmaoing re1earch activities we are addre11in1 th••• and other pertinent que1tion1 for the developaant of an accurate and 1iaple reaervoir fluid phase behavior prediction package.

6

A Continuout Mixture Computational Algortthro

For Reieryotr flui.di pb.ue Behavior SPE 15082

ACKNOWLEDGMENT

Thia research is supported by the U.S. Science Fundation, Grant CPE-8306808,

REFER.ENCE

National

1. Katz, C.H., and Firoozabadi, A.: "PredictingPhase Behavior of Condensate/Crude Oil sy1tem1Ua ing Methane Interaction Coeff icient1, 11 J.Pet, Tech. (Nov. 1978) 1649-SS; Trans. AIME,265.

2. Mannan, M,, Watanasiri, S., and Starling, K,E,:"Characterization Methods for SystemsContaining Many Components for Use in Phase

Equilibrium ani Thermodynamic PropertiesCalculations," AIChE 1984 Winter NationalMeeting, Altanta.

3. Coats, K,H.: "Si111ulation of Gas CondensateReservoir Performance," J, Pet. Tech, (Oct.1985) 1870-86.

4. Mathias, P.M. and Benson, M.S.: "ComputationalAspect. Of Equations of State. 11 AIChE 1985National Meeting, Chicago.

S. Scott, R.L.: "The Thermodynamics Of High­Polymer Solution&, 11 J. Chem, Phys. (1949) 17,268-219.

6. Komingsveld, R,: "Phase Relationships andFractionation in Multicomponent PolymerSolutions," Pure Appl. Chem. (1969) 20,271-307.

7. Flory, P,J., and Abe, A,: "StatisticalThermodynamics Of Mixtures of RodlikeParticles," Macromol ecul ea (1979) 11,1119-1126.

8, Ari1, R, and Gavalas, G,R.,: "On the Theory of Reaction in Continuous Mbturea," Phil, Trans. R.. Soc. (1966) A260, 351-93,

9. Bowman, J,R.: "Distillation of an IndefiniteNu11ber of Co11ponent1, 11 Ind. Eng, Chem. (1949)41, 2004-2007.

10. Hoffman, E.J.: "Flash Calculations forPetroleum Fractions," Chem, Eng, Sci. (1968)23, 957-64.

11. Taylor, D,L, and Edmister, W,C.: "Solutions forPetroleum Di11tillation Calculations," AIChE J,(1972) 17, 1324-29.

12, Ratzach, M.T., and Kehlen, H,: "Continuous Thermodynuica of Complex MixturH," Flu.: d Phaae Equilib. (1983) 14, 225-34.

13. Achliater, W,C.: "Improved Integral Techniquefor Petroleum Diatillation Calculation,, 11 Ind.Eng. Chem, (19S5) 47, 1685-90.

14. Gualtieri, J,A,, Kinacid, J.M., and Horri1on,G. J,: "Phase Equilibria in PolydiaperaeFluid•," Ch ... Phya. (1982) 77, 121-36.

398

15. Briano, J,G.: Classical and StatisticalThermodynamics Of �ontinuou1 Mixtures, Ph, D,The1i1, Univer1ity of Pennsylvania (1983).

16, Cotterman, R.. L, and Prausnitz., J ,M,: "F'lHh Calculation for Continuous or Seaicontinuoua Mixtures Using an Equation of State," Ind. Eng, Chem. Proce11 Dea. Dev, (1985) 24, 434-443,

17. Schulz, G.V.: "Uber die Kinetik derKettenpolymerisationen," z. Phyaik, Chem.(1939) b43, 25-46.

18. Walpole, R,E, and Myers, R.H.: Probability andStati1tic1 for Engineers and Scientists,Macmillan Inc., New York (1985) 154.

m!.! !: Several Single Variable Distribution Function Di1tributjon Mean

Normal l/(2ll'11)'2exp[-(I-Io) 1/211] Io

Gamma l/[11'T(11)](I-Io)ar-lexp[-(I-Io)/11] QI)

Schulz 1/ [Ir(11)] (I11/Io)l1exp(-I11/Io) Io

Exponen- l/11exp[-(I-Io)/11] 10+11 tial

Weibull ar11(I-Io)17-1exp[-ar(I-Io)l1] Q-l/11ro + 1/ar)

Tung 11/Io(I/Io)'1-1exp[-(I/Io)'1] I or(l + l/11)

Fy(I) t

- p1ov I

;of I FL(I)�

Continuou, Mixture

IoL I

Discrete Mixture

Fig. 1-Ftash VLE material balance comparison of continuous and discrete mixtures.

m��--------------�

;: c.

I• a;;

z '9

0•

I ., a: 0

a

i o..-.,.--,--,-........... _,_...,....... 0 •o IO 110 180 100 NO 110 110

MOLECULAR WEIGHT I

Fig. 3-lnteractlon .............. bl of the vdW E08 vs. molecular weight of parattlna.

Variance

11

Application Petroleum,Polymer

Heavy Petroleum, Polymer

Heavy Petroleum, Polymer

Caa Condensate

Q-2/.-; tro +2/11)- cro + l/17)] 1} Petroleum, polymer

Polymer,Colloids

0 40 IO 110 110 IOO NO 110 110 MOLECULAR WEIGHT I

Fig. 2-lnteractlon parameter al of the vdW EOS va. molecular weight of paraffins.

:..---------------..11,•15.0

,, , , .

I I

I I

/1 '1x•328.70

I / •.. ,. ,o ··· ., :-+-----..-----..------1

110 100 110 400 TEMPERATURE,DEG.K

Fig. 4-P-T dllgram of a contlnuoue mixture blNd on thevdW 1!08,

SPE 150827P. c. Ou & G. A, Mansoori

Fig. 5-Parameter 1111 of the PR EOS vs. molecularweight of paraffins.

m��------------�

a: I!!� u,o

2c. a: c• D,;

��s:: !i O!a: 0

io.....__ __ ---' 0 40 10 180 110 IO0 140 110 110

MOLECULAR WEIGHT I

Fig. 7-Parameter bl of the PR EOS vs. molecularweight of paraffins.

I I

c,-. fg

Ii O')O O'). ..,.

If!

I

'lf•9,0S

"x•36S,82

cc/ .......... -..... -......... -...... -.... -....­

•140•1IO •100 •IO •ID •40 •10 0 10

TEMPERATURE F

Fig. 9-P· T diagram of a continuous mixture with11, = 9.05 baaed on the PR EOS.

c��-------------

l:s :I c.a:.f

��s::

1� .... ....,-.,..-i--,..-........ -.... �0 40 10 110 180 100 140 IIO 110

MOLECULAR WEIGHT I

Fig. &-Parameter al2 of the PR EOS va. molecularweight of paraffins.

i.,-------------

cl i. ..,. ac• •• O'). Ill • o .,,,-s. 72

IE! •,·'·':'/' ..... --.... -..... --.... --.... ·---

.. ,., •140 •110 •100 •IO •IOTEMPERATURE F

Fig. 8-P-T diagram of a continuous mixture with'It= 5. 72 baaed on the PR EOS.

1-,--------------

'1f•25.0"··59.527

..... -..-.... -...--.......... -.... -......0 40 IO 110 1IO 100 140 IIO 110

TEMPERATURE F

Fig. 10-P. T diagram of • continuous mixture with11, • 25.0 baaed on the PA E08.

SPE 15082 8A Continuous Mixture Computattonal Algorithm For Reservoir Fluids Phase Behavi.or