SPESPE 19809

Predicting Retrograde Phenomena and Miscibility UsingEquation of Sate42A, Mansoori, U, of Illinois, and J.L. Savidge, Gas Research Inst.

SPE Members ,& T-

Copyright 1989, Society of Petroleum Engineers, Inc.

Thk paper was prepared for presentationat the 64h Annual Technical Conference and Exhibitionof the Society of ‘etroleum Engineers held In San Artonio, TX, October S-1 t, 1SS9.

This paper wae eelacted for presentation by an SPE Program Committee followingreview’of Informai!onccmta,,led In an abetract submitted by the author(a). Contents of the paper,ae presented, have not been reviewed by the SWiely of Petroleum Engineers and are eubjact to correction by the author(a). The material, es praasmtad,dose not necessarily reflectany positionof the Society of Petroleum Enginears, its officers,or members. Papera presented at SPE meetings are subject to publicationreview by EditorialCommittees of the Sociatyof PetroleumEngineers.Permieaion’tocopyis restrictedtoan abstractof notmorethan SW words.Illustrationsmay notbe copkf. The abetractshouldcontainccscepkuouaeckn4edgmdof where and by whom tlw nqer ie presented. Write Publication Manager, SPE, P.O. Sox SSSS36, Richardaom,TX 7S0SS-SS3S.Tetex, 7S09S9 SPEDAL

equations of state it is possible to calculate accurately theminimum miscibility conditions and aolubllity of heavy and

While the knowledge dm.tt the retrograde condensation intermediate hydrocarbons in miscible solvents (gases andand condensing-gas drive phenomena is not new their utilization gaseous mixtures) and phase behavior (dew point and bubblein production-stimulation and simulation of the natural gas andNGL reservoirs has been delayed due to the lack of aeeurate

point in VLE and VLLE cases) of complex reservoir fluids. Thebasisof these developments lie in statistical mechanical mixing

predictive computer algorithms. In this report the groundworkfor accurate computation and prediction of the behavior of

rules and the conformal solution theory of polar fluid mixtures.

‘ reservoir fluids under retrograde condensation and in RETROGRADE PHENOMENA: Retrogradecondensing-gas-drive conditions am presented It is shown that condensation has important applications in enhaneed oil and gasthrough equations of state one can predict the following recovery, industrial separation of chemical compounds, andproperties of the reservoir fluid systetnx processing of fossil fuelsl’4”13. The phenomena was frost

[i] The minimum miscibility conditions and solubilities of recognized in 1879 by Hannay attd Hogarth1*14. They

intermediate and heavy hydrocarbons in miscible discovered that solid compounds eotdd be dissolved in dens

solvents. gases having densities near that of a liquid The interest in thisprocess is beeauae of the appreciable increase in solvent power

[ii] The effect of mixed miscible solvents and entrainers on of dens fluids at temperatures and pressures above, but not farlowering the minimum miscibility pressure of !~eavy removed from, their critical point. A dens gas can be an effkctive

componentsof reservoir fluids. solvent for a condensed mmpound (solu@ which m~s in thecondensed state (liquid or solid) at the superoritieal conditions of

[iii] The role of variations in temperature, pressure, and the solvent. This requires a large molwulw weight and sizesolvent composition on the miscibility and solubilitiies. difference between the dens gas solvent and the eondenaed

solute. Thermodynamic modeling and prediction of solubilities[iv] Flash (dew point and bubble point) calculation for of heavy solutes in dens gas solvents has been hampered due to

complex reservoir fluids in VLE and VLLE cases. the lack of thermodynamic models of asymmetric mixtures.

DUCTIOI+( MINIMUM MISCIBILITY CONDITION: Thevaporizing gas drive is used in enhaneed oil and gas condensate

Retrograde ..ondensation and condensing-gas-drive recovery to achieve dynamic miscible diaplaeement or multiplephenomena have important applications in NGL production and contact miscible displacernen$. Miaeible diaplaccment relies onprocessing*4. The knowledge about retrograde condensation multiple contset of injected gas and resesvoir fluid to develop anand condensing-gas drive is not new, However, the application in-situ vaporization of intermediate moleeular weightof this knowledge haa been delayed due to the lack of accurate hydrocarbons fkom the reservoir fluid into the injeeted gas andpredictive equations of state. Recently under a ccmrraetwith tie create a miscible transition zone4. me -miaeibl~mea which areGm Research htStitutC tk UItdWOrkfor aeeurate pdiCtiOtl of

rthe behavior of reservoir uids under retrograde condensationused in such a process may include natural W, inert ~ andcarbon dioxide. For a given rekervoir fluid and a miaeble a t

and in condensing-gas-d+ -- -conditions are developed at the %?system at a given temperature, the minimum presstm at w “ch412. It is shown that by usingUniversity of Illinois w Chicago - miscibfity can be achieved through multi le eo.ltael:. is ref~

to as the &)minimum miacibiity pressure .

References and figures at end of paper.

383

2 predicting Retrograde Phenomena and MiscibilityUsing Equations of Smte SPE 19809

In order to describe and model the retrograde a~idcondensing gas drive phenomena and predict the minimummiscibility conditions accurately one has to utilize the theory ofconformal solution of asymmetric mixture: and the tfieory ofmany-body interactions.

THEORY OF CONFORMAL SOLUTIONMIXING RULES: ‘I%erehas been substantial rwcmressmade.-in recent years t“ unprove conformal solution mixing rulesis-22.Utilizing such mixing rules for calculation of mixturethermodynamic properties requires to combine them with anequation of state. There exist varieties of cubic equations ofstate available in the literature. The Peng-Robinson (PR)equation of state

z = v/(v-b) - a(T)/[v(v+b)+b(v-b) (1)

where

a(T) = a={1 + k (l-T,lBJ ]2, (2)

aC= a(TC)= 0.45724 R2 TC2/PC,

k = 0.37464+ 1.54226 O)-0.26992 C02, (3)b = 0.0778 R TC/ PC (4)

is one such equation of state which is used in the present paper,We will limit our calculations and discussions to the PR quationof state with the understanding that similar computations can beperformed with other equations of state such as SR?S,etc. Inorder to utilh,e conforrnal solution mixing rides for the PRequation of state we fmt need to separate themwdynamicvariables horn constants of this quation of state. For thispurpose we may write this equation of state in the followingformly

Z= v/(v-b)-[(A/RT+c-2(Ac/R’@~M(v+b)+(b/v)(v-b)l (5)

whereA = aC(l+k)2 (6)

C = aCk2/RTc (7)

This form of the equation of state indicates that thereexist thee independent constants (A, C, and b) in the quation.In Table I the mixing rules for these parameters are based ondifferent conformal solution theories of mixtures. Thecombining rules for the unlike interaction parameters of thisequation of state are

Aij = (I-kij) bij(aiia#biibti)12 (8)

bij =(1-lij)(bti1B+b$~)3/8 (9)

Cij = (l-~j)(ctilp + CtilB)3/8 (lo)

IrI the present report we use the van der Waals (vdW) conformalsolution mixing rules which are the simplest among thesuccessful conformal solution mixing rules. It will bedemonstrated that when the vdW mixing rules are combinedwith Equation 5 accurate retrograde condensation andcondensing gas drive calculations can be made.

THEORY OF MANY-BODY INTERACTIONS:Inafhddsystemt hetotalin temoledar potential energy of theinteracting molecules is expressed in the following form5’&m-*:

N N

U = ~ ~ U~2}(ij)+ ~ ~ ~ Uf3J(ijk)+. . . (11)i<j icj<k

In this expression u(z)(ij) is the pair intermolecular potentialenergy between molecules i and j, u(3)(ijk.) is the tripletintermolecular potential energy between molecules i, j, and k,etc. It i~ shown that23-25 the contribution of ii?e tripletintermolecular interaction energy to the total intermolemtlsrpotential energy is of the order of 5 to 10%. H6wever, higherorder terms (four-body interactions and higher) of Equation 11are negli .ble, Mureover it is shown that the leading term in tttethree-bo? y interaction energy is the dipole-dipole-dipole termwhich is known as the Axilrod-Teller triple-dipole dispersionenergy as expressed by the following expression%:

where i, j and k are t!:e three molecules formi::g a triangle withsides ril rjk i~d rk and interior angles yi, 7j ~d ‘fk~d vix h

the triple-dipole constant. Contribution of this three-body effectsto the Heim:ioltz free energy of a pum fluid is

A3b = NVd fl(@/f2 (q) (13)

where

fl (q) = 9.87749q2+11.76739 q3 -4.20030114,

f2(q) = 1-1.12789q2+0,73166113, .

q = (7r/6)(Nd3/V),

N is the number of molecules in volume V and d is a hard coremolecular diameter.

An ec aation of state, such as Equation 1 with a choice of mixingrules is based on binary interactions. The parameters Ati, bfi,and Cii of this equation of state are derived from pm

component properties while parameters Aij, bijt nd Ci. arederived from binary data of i and j. However, it has bnshown that in utilizing such equations of state, even with bettermixing rules, to predict ternary and multi-component systemphase behavior, the results deviate from experimental data~~.This is specially observed for ternary mixtures around the criticalpoint for which them is sufficient experimental data available,

The mason for the deviadon of paediclionof the equatia”of StStC around the ternary rnix~ did point CSn bS (kSdbCd

as the followiny In a ternary mixture, in addition to the kinds ofinteractions which exist between molecules in “bimuymixtuma,there also exists an interaction which we would call the unlike-three-b@y interaction which happens when there are threemolecules of three dtiferent kinds interacting in a mixture.Considering the fact that there exist no such terms in an equationof state, such as the two parameter cubic equations of state, wewould expect that the unlike three-body interaction contributionto the quation of state will not be accounted for when theequation of state is based on the pure component and binarymixtum data. Symbolically we will define the unlike-three-bodyinteraction contribution to compressibility of a ternary mixture

by XIX2X3AZ(V123,T,N) where v 123 is a parameter

corresponding m the urdike-three-imdy interaction. As a resultthe equation of state of a mixture must be written in thefolbwing form

k = q (v. T, all, a12,. ... ~, xl, ~ ,..., ~) +

~i~j~ xi XjXkAz(vix. T, N ) ; (i # j * k) (14)

,.884 ...

The second term in the right hand side of this quation includesall the unlike-three-body interaction between every threedifferent molecules in the mixture

The basic question to be answered at this stage is how toarrive at an expression for the unlike-three-body interactionterm? This problem has been addressed in the statisticalmechanical theory of man; -body interactions, Starting with thework of Axilrod and Tell:.’ in the development of algebrtdcexpressions for three-body intermolecular interactions ener

l?’and the development of an analytic expression for the unli ethree-body interaction Helmholtz free energy using theperturbation themy of statistical mechanics2c*n. Details of the

26. However, the final expressiontheory are given elsewherefor the unlike Helmho!tz free energy is as the following:

Aij~= N d-gvijkfl(~)/fz(~) (15)

Knowing the unlike-three-body interaction Helmholtz freeenergy we can now calculate other thermodynamic properties.For example as a result the correct compressibility factor of amixture willtx? .

Zm = Zm’mp+ zixjqxixjxk(?’l p/b3v)(f ~f_ - f1f2)/f22 (16)

where ZmemPis the expression for the empirical equation ofState and

~ = (W27)ZN$ qig

f ~= (dfl/d@ = 19.75498 ?l + 35.30217 ~2 -16,80120 ?13

f2 = (df2/dm)= -1.12789 +1,46332 ?l

The co-volume parameter, b, is related to q as

PHASE EQUILIBRIUM CALCULATIONS: Inthe present calculations we compare the resuits of the aboveformulations with the PR equation of state fw mixtures. In thecalculations reported here it is assumed that the equation ofstate is valid for all the phases in equilibrium. The followingequilibrium conditions between fugacities of the phases areSSSUUXd to hold

fIi (T$P${x}) = fIIi(T,P$(x)) = ... = pi (T,P,{x));

i=l,2,...,n

which can be expressed with respect to the fugacity coefficientsof the phases as the following

where I, II,..., N represent the phases in equilibrium. For amixture of three components exhibiting two iiquid-phases and avapor phase, the following algorithm is used in the phaseequilibrium CalculationSC:

1) Gverall mass balance LA+~+V=l

2) Species Mass Balance XNLA+x~i~ +y BiVi= %

4) Equilibrium Criteria fti = fBi= fi i=l,2,3

where the subscripts A and B are used to identify the two liquidphases in quilitnium. The quilibrium constants or distributioncoeffkients of m wments between phases are defined a~

As a result we obtain

X&s ~ /(LA(l-Kfi) + LB(KAi/KBi-K~) + KAi~Bi= ~K~BJ(LA(l -Kti) + ~(K~Bi-KBi) + KAJ

X = ~KA/(h(l-KJ + ‘i(KAiiBi-KAi)+KAi)

Three different objective funcions can be used

1) Liquid-liquid-vapor bubble point calculationSX&- sxBi= S%-1 = O

2) Liquid-liquid-vapor dewpoint calculationsSXti - Syi = s~Bi -1 = O

3) Liquid-liquid-vapor dewpoint calculationsSXN - Syi= sxBi -1 =0

The governing quations of the critical state of a three-component system are given by the following determinantequations$

laz~xlz a&k1ax2iu= I I =0 (18)

Iazgfaxla% a2gax27

where the partisl derivatives of the molsx Gibbs free energyg(P,T,x.i.) are obtained at constant P, T and X3. When theabove determinant equations are solved for the criticalcompositions, the tangent to the binodal curve at the critical pointtill %3dxained as the following

*.* (atthe critical point) (20)

where XICtmd ~c are the critical compositions of the light andintermediate components, respectively. Pn is the interpolatingpolynomial of the binodal cwe, and the first derivative of theinterpolating polynomkd at the critical point is approximated by acentral diffemncc fonnuiz Whh the implementation of the thme-txxly effects the mixture quation of stare will be

where P is the expression for the empirical quation of state.The fugacity mefticient will take the foilowing form:

fin fi = iln f~mp+ a(xlx2x3A3b@ni (21)

With the aid of a computational algorithm the above quations* used to generate the binodal curves of binary and ternarysystems.

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,,

4 Predicting Retrograde Phenomena and MiscibilityUsing Equations of State SPE 19809

VAPORIZING GAS DRIVE MMPPREDICTION: The subsunti~ limitations of the PR and othercubic equations of state using empiric~ mixing rules are reportedby Kuan30 and by Firoozabadi31 in predicting the phasebehavior and minimum miscibility pressures of simulatedreservoir fluids, According to these investigators by using theempirical mixing rules an over-prediction of the MMP ofhydrocarbon systems was observed, The mixture equation ofstate discussed above is applied to predict the minimummiwibllity pressure of simulated reservoir fluids.

The fmt and simplest ternary system which is analyzedhere is the mixture of methane + ethane + propane as reported onFigure 1, According to this figure while the Peng-Robinsonequation of state (dashed line) is capable of predicting thebehavior of binary VLE data of methane + propane, it fails topredict the VLE behavior (solid dots) around the critical point ofthe ternary mixture, With the consideration of three-body forces(solid line) the proposed technique is capable of predicting theVLE behavior of this iemary mixture.

The next temmy system analyz~ was carbon dioxide+n-decane + n-butane as reported on Figure 2. According to thefigure the PR equation of state (dashed lines) again fails topredict the binary data (solid dots) of these components. Withthe consideration of three-body forces ‘(solid line in ternarydiagram) the proposed technique is capable of predicting theVLE behavior of this ternary mixture quite well. Also reportedon this figure are the large errors which would occur on locatingthe tangent at the critical point which passes through thereservoir fluid composition at the MMP condition.

TENT P~,UDO~IO~~7

Let us assume the equation of state which is consideredfor phase equilibrium calculation to be in the following form

Z = Z(V,T, an, bJ (22)

where its mixing rules can be shown by the following generalexpressions:

.)” i,j = 1,......c= ~(xi. ‘j, a,, , (23)~= b~(xi, Xj,bij); iJ = 1,......c (24)

and with the following combining rules:

iiij = (1 - kij)(aii aj)l~ (25)bij = (1 - lij)(bu + bi)/2 (26)

Provided one knows the exact number of components of themixture this equation of state can be used for phase equilibriumcalculation of that mixture. In the proposed technique it isassumed that one can group the (c) components of the mixture to(s) pseudo-components (for example s=3 when one wants torepresent the data in a ternary diagram). Then the mixing rulescan be shown in the following fomw

am= %04,,<Q,a,q); VP= 1,......s (27)

bm = bm(!t. ~, AJ t,~ = l,..,...s (28)

andwith the following combining rules:

h should be pointed out that contrary to the case of Eqs 25 and

26 parameters Ktq and Atq will be in general non-zero

parameters for both cases of wp and t=(p,When t=q parametersKtq and ~q will be called “Lumping Parameters” and when tq

parameters Ktq and Itq will be called “Pseudo-binary

Interaction Parameters”, In Eqs. 27 and 28 ~t and {Q are

“Group Mole Fractions” and all, r+q, ~tt, ~qq are pseudo-

component parameters associated with each group. At thisstage we have to address three questions: (i) How to define thepseudo-component paramet :rs? (ii) How to calculate thelumping parameters? (iii) How to calc~’ate the pseudo-binaryinteraction parameters?

(i) ~ of ~ Par~. - In thepresent technique one can use any of the available techniqueswithout loss of generality. So long as the same pseudo-component calculation technique is used for defining pseudo-components the present technique will predict the same phasebehavior for the multi-component system under consideration.

(ii) ~on of - Theseparameters are calculated by assuming that a pseudo-compound‘with equation of state parameters apsttand &ll can represent

properties of a lumped group of compounds, Parameters a~tt

and ~tt are then calculated by matching properties of a pseudo-compound with the mixture properties of the group. ofcompounds which are lumped together.

(iii) ~n. P~v .of .

.~ - After the pseudo-component parameters aredefined and the lumping parameters are calculated pseudo-binaryinteraction parameters can be calculated by matching theproperties of every pseudo-binary mixture with a true multi-component mixture consisting of all the compounds appearing inthe pseudo-binary mixture.

Firat Application: A synthetic oil of 10 components32 forwhich the composition is given in Table 2 is selected to test theproposed teehnique. When the heavy-end fractions are describedby the molecular weight, the specific gravity, and the tidingpoint, empirical correlations2’33 are used to estimate theproperties (critical pressure and temperature and acentric factor)of the fiwctions. A lumping configuration is selected such thatthe synthetic mixture is reduced to 3 pseudo-components,consisting of [Cl] (methane), [~-C15] (ethane to hexanes), and

[~J (heptmes and heavier fractions).

Figure 3 shows the P-X diagram of the [C2-C6]+[C7$pseudo-binary system where the symbols represent thecalculation with the exact compositional description of thesynthetic mixture. l%c dashdotted line is the calculation with 2pseudo-components but without the pseudo-bm~ interactionparameters and lumping parameters, The dashed line is the resultwith two pseudo-components and the pseudo-binary interactionparameter, but, without the lumping parameters. The solid line isobtained with two pseudq-components, the pseudo-binaryinteraction parameter and i the lum ing ,parameters. Exact

Imulticomponent calculation is shown y (x). In F@um4 the P.X diagram of the [CJ+[~J pseudo-binary sYs*m i.$_where only the pseudo-biqary interaction pmmeter is evaluat$dto match the exact calculation represented by the symbols. Forthis particular example two lumping parameters are dy n-one lumping parameter fOr the [~-cfj] grOUPWd a ~~lumping parameter for the [~J group.

366

SPE 19809 G. Ali Mansooriand Jeff Savidge 5

Figures 5-7 are for prediction of the phase behavior of avaporizing gas drive process with C02 as the injected gas andthe synthetic oil as the reservoir oil. Prediction with thepseudoization technique is represented by the solid line. Exactrnuki- component calculation is shown by (x). The objectivehere is to test the performance of the proposed twhnique atdifferent pressures and in all ranges of composition of solventand oil, The P-X diagrams shown in F@res 5 and 6 are usedto evaluate the binary interaction parameters between C02 andthe pseudo-components of the synthetic mixture. The predictedpseudo-ternary diagram for this system is shown in Figure 7whertYf?%demonstrated that the agreement bttween the exactcalculation and the lumping technique is excellent.

Second Application: A P-T phase envelope of a gascondensste34 system (high in methane and low in heavyhydrocarbons) with the composition as in Table 3 is constructedand it is reported by Figure 8, In this calculation 4 pseudo-components are chosen which are [C!H41,[N21, [C02,C2-C5],and [Cb+]. Also, a P-X diagram of carbon dioxide-reservoir

oi134system (low in methane and high in heavier hydrocarbons)with the composition as in Table 4 is constructed using the same4 pseudo-components and it is reported by Figure 9. in both ofthese cases the results of calculations obtained with the presentself:cmsisteut approach are in very good agreement with theexact multi-component calculations.

Third Application: The final application is to establish arelationship between the cricondenbar locus of P-X diagram of amixture and the phase separation regions of the pseudo-ternarydiagram of the same mixture. This kind of relationship is ofsignificant importance in application of pseudoization techniquesin high pressure processes such as extraction of heavycompounds from mixtures by dens gases and miscible floodenhanced oil recovery,

in the present computation we categorize our mixture bythree pseudo-components [heavy (3), intermediate (2), and light(l)] and we study the P-X diagrams by wwying the ratio ofcompositions of heavy and intermediates, Rather than plottingvarious P-X diagrams the locus of the cricondenbars of suchmixtures versus C=X(2)flX(2)+X(3)] are reported by Figures10 and 11, Also reported in Figures 10 and 11 are the psuedo-ternary diagrams related to the same mixtures at differentpressures. The major difference between Figures 10 and 11 isthe difference in shapes of the crhxmndcnbar loci. These twoshapes (one a decreasing function of C and the other having aminimum point) are the only possible trends that one canproduce by choosing all possible relative compositions for thelight, intermediate, and heavy fractions of a mixture.

ORY OF Pcontinuous ~~

UM OF

Development of compositional ~reservoir simulators ispresently hampered due to the ‘complexity of the existingcomputational algorithms of reservoir fluid phase behaviorcalculations. Through the application of the theory ofpolydisperse fluid mixtures a number of algorithms for phasebehavior calculations are developedg-l 1. In this part of thepresent report we introduce one such algorithms. In thisalgorithm reservoir fluid is considered to consist of a continuousmixture with a defined molecular weight/compositiondistribution function, Computational computer time required forthe new algorithm is shown to be one order of magnitudesmaller than the existing algorithms, Comparisons .ofexperimental data with the calculated results indicate goodagreement between the two.

For a mixture with many components being continuousin cl.~ acter, the compositions can be described by a densitydistribution function, F(I,Io,q) whose independent variable I is

1

some measurable property such as molecular weight, boilingpoint or density with a mean value of ~ and a variance of q,The normalization of the density distribution function is given by

JwIF(I = 1 (31)

In case the mixture is in part continuous and contain sufficientlylarge amounts of components which should be considered asdiscrete components Equation will be valid for thecontinuous fraction of the mixture while for the whole mixturethe following normalizing condition will hold

~Xi+X~=l (32)i

where xi and XCare the mole fraction of component i and thecontinuous fraction, respectively and d is the total number oidiscrete components in the mixture, Regarding the materialbalance of component I in a tlash calculation we get

The distribution functions, F~I), FL(I), FV(I), are not additiveand as a result they cannot possess the same functional forms ina specific flash calculation scheme.

In order to extend continuous thermodynamics toengineering applications, we introduce a g~neral phaseequilibrium calculation technique which is based on THEMINIMIZATION OF THE TOTAL GIBBS FREE ENERGY

~LGORITHM1°. Using this technique, flash calculations of“complex mixtures containing discrete components and a

continuous fi’action,which could have a wide molecular weightdistribution, can be performed. For a system in equilibrium(constant T and P), for any diffemttial “virtual displacement”occurring in the system, a general criterion should be imposedon the system suchthat thetotalGibbs ffee energy is minimal.

(dG~,P = O (34)

To restrict our consideration only to the vapor-liquidequilibrium, we write the total Gibbs free energy of the systemas:

G= GL+GV (35)

For a mixture consisting of discrete components and a one-family continuous fraction, we derive an expression of theGibbs free energy for the liquid phasq

m

G~=~lp -N~RT/v~]dv-RT~ lNi n[v~,RTjv i

- NCRTIIFL(I)k?,n[vJNcRTFL(I)]dI+ PVL+ G“L (36)

where G*Lis the Gibbs free energy of the reference state. Asimilar expression will hold for the vapor phase. Since, thesystem is in phase equilibrium, thus, ac~ordlng to Equation(34), all of the first derivatives of the total Gibbs free energywith respect to the system variables must be qual to zero, i.e.

@G/~dT~..yi,ll@o@v=0,,~.@G/WT p,.~i,l’tL,I@ov=0,,~,

(37)

(38)

(aG/i)IO~ o (39)‘l’.p,XiJ’i.’lnvtItv=v=

(aG/aI ) =0‘VT?.Xi.Yi,’lLt’l~tIeL

(40)

To perfam flash calculations for a system composed of discretecomponents and a continuous fraction, Eq’s. (5) and (6),coupled with Eq’s.(37)-(40), will form a set of nonlinearequations which must be solved simultaneously. To illustratethe application of this technique let us assume the PR equation ofstate

P= RT/(v-b) - a(T)flv(v+b)+b(v-b)] (41)

a(T) = a(T~[l+k(l-Trl~)]z,a(T~ =0.45724RZTCVP.b= 0.0778 RT#C (42)

k = 0.37464+ 1.54226 co- 0.269926)2 (43)

with its empirical mixing rules

a= ~i~j Xi Xj,ijb= ~ ~biaij = (I-kij) (~aj)

(44)(45)(46)

is valid. In order to extend this equation of state to a mixturecontaining d discrete components and a continur -- .Saction witha wide molecular weight distribution, we neeu to rewrite themixing rules as the following forms

dd d

a=~i~jxix~~j +2Zi~i~j.fIF(I)a(i,I)& +(47)xCz\l~JF(I)F(J)a(I,J)dJdI

d

/b= Xi xibi + ~ F(I)b(I)dI (48)

I:

where

a(i,I)”= a,Wal~(I)(l-ku);a(I,Jj = alE(l) aV2(J)(l-ku)

The parameters alE(I) and b(I) can be accurately represented bythird-order polynomials with respect to molecular weight I for ahomologous series of paraffinic”hydrocarbons as given below9,10:

aUz(I) = cxl+ a21 + ct312+ a4P (49)

where

al = 0.4771 + 0.0157Tl~;

~= 0.1055+ 0.0017T12,

a3 = -O.4O66X1O-4+ 0.2960x 10-STl~;

U4 = O,27OOX1O-G-0, 1318x1O-8TW

and

b(I) = PI + ~21+ ~312+ ~413 (50)

where

i3~= 0.0071, ~ = 0.0013, p~ = -O,1371X1O”5and~4 = 0,9686x1W (51)

‘WW~&yfurther introduce an exponential-decay distribution

F(l) = (1/q)exp[-(1-I#q] (52)

which is proper to describe the composition of gas-condensatefluids.9-11 Knowing that methane mnstitute a large fkaction ofevery gas-condensm sys~ it is appropriate to treat the systemas a mixture of methane and a cxminuous fkaction of the rest ofhydrocarbons, As a result, by substituting Equation (41) intoEquation (35) and considering Equation (52) as the continuousfi’actiondistribution function, the total Gibbs free energy of thesystem will be

G = cPLgL+ CDvgv (53)

whexe

t3L= -RTOn(vL - “tq) + (~[2.828b~ ) ~n[(v~ - 0.414bL)/(vL +

2,414bL)] - RT(x1-xl~rixl ‘+ @n flL) + VLRT/~VL- bL] -aLv@@L + b~ + bL(vL- ~]

and gv will have a similiar expression. The expressions fo aL

and b~ areas the following:

8L = x~2all+~2~L2+2xlxc(al l)ln(~L1f%lL) (54)

with

alL = exp(-70hL)(q1+ ~qL+ ~vL2+q4qL3+~qL4);

a&= [~1+ h2&+qL) + V3@2+2101’IL+2qL2) + t14(L3+31.hL+

61.TlL2+ 61’IL3)]2- 0,02( (a,#[exp(-70/qL) - exp(-84/qL)] +

(a33)l~[exp(- 84111L)exp(98/qL)l)(sl+s2qL+s3qL2+s4qL3)

ql = -0.0579al-5,344a2- 10.257a3 -1.662X102U4;

~= 0.441xlWa1-0.0491a~10t257a3-1.662xl($tx4;

q3 = 0.8822x104a2 + 2.5308a3 - 34.398q;

q4 = 2,6466x 10dvs - 0.393a4;

q5 = 1.0586q;

s~= al+ WX102 + a3x104 + a4x106;

S2= a2+2v3x102 +3a4x104;

q = 2a3 +6a4x102; S4= 61kx4s

and

bL.= xlbl + XC[~~+ ~2(1.+~L) + ~3(Io2+210~L+2~L2) +

134(IJ+310313L+61.13L2+613L3)l

*

SPE 19809 G. Ali Mmsoori and Jeff Savidge 7

In the above quations bl is the @)parameter of the quation ofstate for methane and all, %2, and a33are the (a) parameter ofthe equation of state for methane, ethane, and propane,respectively. Similar relations will hold for parameters av andbv, To perform flash calculations, we need to substituteEquation (53) into Equations. (37)-(40). Since for gas-condensates ~= ~ = ~v, Equation (39) and Equation (40)will vanish. We only need to solve uations (37) and (38),

7simultaneously and considering the act that pressure andMnpemmm in the two phases ranain identical. In what followsthe proposed algorithm is used for calculation of properties oftwo gas-condensate systems.

COMPARISON OF CALCULATED ANDEXPERIMENTAL RESULT : In our calculations, weperform flash calculations for the a gas-condensate system35 inwhich the crude oil composition, quilihium condensate base

?equilibrium data and liquid-vapor volume ratios at dif erentpressures am given.

The gas-condensate systems is treated as a fluid with adiscrete component of methane and a continuous fkaction ofother hydrocarbons. The continuous fraction is described by anexponential-decay distribution function starting from themolecular weight of ethane. For calculation of the binaryinteraction parameter kij of gas-condensate systems, thefollowing procuiure is u*, (i) It is assumed that ~j = Ofor allbinary paics except for methane-C7+,ethane-C7+, and propane-C7+interactions. (ii) It is assumed that ~j4.01 for ethane-~+and propane-C7+ binary pairs as proposed by Katz and

Firoozabadi36. (iii) Du md Mansoorill produced the followingexpression relating the pair-interaction parameter of methane-~to molecular weight of hydrocarbons will be derived.

kl,i ~ .0.0579 + 0.441 lX104?dWi (55)

The gas condensates system is treated as a discretemixture of methane (with a mole ftaction of 0.74133) and acontinuous fraction of other hy&ocarbons. The continuousfraction is described by an exponential distribution function(with a variance of q 48.9). In Figures (12-14) the componentmole fraction quilibrium ratio,(K value), liquid volume percent(with respwt to total volume of the system), and the P-Tdiagram as calculated by the present technique are comparedwith the experimental data35. Dots are the experimental datawhile the solid lines are the results of the proposed continuousmixture phase quilibrium algorithm. As shown in Figures (12-14), the flash calculations performed by using the proposedtechnique am in good agreement with the experimental data.

This technique provides a general and convenientprocedure for performing flash calculations for a complexmixture consisting of both discrete and continuous components.This technique can also reduce the required computer time andovercome the complexity for solving a multitude of simuhanmmquations. The present technique is applicable to varieties ofcomplex reservoir fluid mixtures, quations of state, mixingrules, and combining rules.

The authors appreciate assistance of Mr. Sung-Tae Kimin preparation of this paper. This research is supported ,by theGas Research Institute.

1.

2.

3,

4.

5.

6.

7,

8.

9.

10.

11.

12.

13.

14,

15.

Katz, D. L. and Kurata, F.: “Retrograde Condensation,”Ind. Eng. Chem., 32, (1940) 817:; Katz, D. L., et al.,Handbook of Natural Gas Engineering McGraw-Hill,New York, N.Y., (1959).

Standing, M. B.: “Volumetric and Phase Behavior of GilField Hydrocarbon Systems,” Millet the printer, Inc.,Dallas, TX,(1977).

Stalkup, F. I. “Miscible Displacement,” SPEMonograph, Society of Petrol. Eng. of AIME, Dallas,Texas, (June 1983).

Park, S. J., Kwak, T. Y. and hhtsoori, G. A.:“StatisticalMechanical Description of Supercritical FluidExtraction and Retrograde Condensation.” InternationalJournal of Tlwrmophysics, 8, (1987) 449-47L

Benmekki, E. H, and Mansoori, G. A.: “MinimumMiscibility Pressure Pm.dictionwith uations of State;

7SPE Reservoir Engineering, (May 19 8) 559-564.

Benmekki, E. H. and Mansoori, G. A.: “The Role ofMixing Rules and Three-Body Forces in the PhaseBehavior of Mixtures,” Fluid Phase Equilibria, 41,(1988) 43-57.

Benmekki, E. H. and Mansoori, G. A.: “PseudoizationTechnique and Heavy Fraction Characterization withEquation of State ModelstAdvance in Thetmodynatnics,1, (1989) 57-78.

Chorn, L. G. and Mansoori, G. A.: “C,+ FractionChamcterhtion” Advances in Thermodynamics, Vol. 1,Taylor & Francis Pub. Co., New Y* N.Y., (1989).

Du, P. C. and Mansoori, G. A.: “A cOIlthlOUS MixtureComputational Algorithm for Reservoir Fluids phaseBehavior,” SPE Paper #15082, Society of PetroleumEngineers, Richardson, TX (1986).

Du, P: C. and Mansoori, G. A.: “ Phase EquilibriumComputational Algorithm of Continuous Mixtures:Proceedings of Fourth Inderantional Conference onFluid Properties & Phase Equilibria for ChemicalProcess Design, Lo-Skolen, Denmark, 8, (1986) 58-64.

Du, P. C. and Manaocai, G. A.: “Phase E@ibrium ofMukicomponent Mixtures: Continuous Mxture GibbsFree Energy Minimization and Phase Rulcv Chem. Eng.Communication, 54, (1987) 139-148.

Mansoori, G. A. and Chorn, L. G.: “Multi-componentFractions Characterization: Principles and TheoriesflAdvance in Thermodynamics, 1, (1989) 1-11.

Mansoori, G. A., Schulz, K. and Martinelli, E.:“Bioseparation Using Supercritical Fluid Extraction/R__3ro~de Condensation,” Biotechnology, 6, (1988)

-,

Ely, J. F. and Baker, J. K.: “A Review Of SupercrhicalFluid Extraction,” National Bureau of Standards,Boulder, CO, (1983).

Mansoori, G. A. and Leland, T. W.: “StatisticalThnnodynamics of Mixtures (A New Version fbr theTheory of Conformal Solutions),*’ FaradayTransactions II, 68, (1972) 320-344.

aM

8 Predicting Retrograde Phencanenaand MiscibilityUsing Equations of State SPE 19809

16.

17.

18.

19.

=0.

=1.

22.

23.

=4,

=5.

=6.

=7.

=8$

=9.

30.

Mansoori, G. A. and Ely, J. F.: “Density Expansion(DEX) Mixing Rules (Thermodynamic Modeling ofSupercriticai Extraction),” J, Chem. Phys., (Jam 1985)406-413.

Mansoori, G. A.: “Mixing Rules for Cubic Equations ofState,” Symposium on Equations of State, ACSSymposium Series No. 300, (1986) 314-330.

Chang, J. I. C., Hwu, F. S. S. and Leland, T. W.:“Effective NIolecular Diameters for Fluid Mixtures: inEquations of State in Engineering and Research,Advances in Chemistry Series No. 182, (Editors)Chao, K. C. and Robinson, R. L., ACS, (1979) 72-95.

Kwak, T.Y. and Mansoori, G. A.:’’Van der WaalsMixing Rules for Cubic Equations of State,” Chem.Eng. Sci. 41,5, (1986) 1303-1309.

Chen, L. J., Ely, J, F. and Mansoon, G. A.: “MeanDensity Approximation and Hard Sphere ExpansionTheory: A Review,” Fluid Phase Equilibria, 37, 1987(l-27).

Hanmd, E. Z. and Mansoori, G. A.:’’DenseFluid~h~ry of Mixtures,” J. Chem. Phys., 87, (1987)6046-

Kwak, T. Y., Benmekki, E. H. and Mansoori, G. A.:“Van der Waais Mixing Rules for Cubic Equations ofState (Applications for Supercritical Fluid ExtractionModeling and Phase Equilibria Calculations)” ACSSymposium Series 329,American Chemical Society,Washington, D.C., (1987) 101-114.

Lan, S. S. and Mansoori, G, A.: “Perturbation Equationof State of Pure Fluids,” International Journal ofEngineering Science, 14, (1975) 307-317.

Lan, S. S. and Mansoon, G. A.: “StatisticalThermodynamic Approach to the Pre&ction of Vapor-Liquid Equilibria Properties of Multi-ComponentMixtures,” International Journal of Engineering Science,15, (1977) 323-341.

Mansoori, G. A. “Analytic VIM Equation of State,”Fluid Phase Equilibria 13, (1983) 153-160.

Axiirod, B. M. and Teller., E.: “Interaction of the vander Waals Type Between Three Atoms,” J. Chem.Phys., 11, (1943) 299-300.

Somait, F. A. and I@nay, A.: “Liquid-Vapor Equilibriaat 270.00 K for Syixems Containing Nitrogen, Methane~0~ $~5xm Dioxide,” J. Chem, Eng. Data, 23, (1978)

-,

Zeck, S. and Knapp, H.: “Vapor-Liquid and Vapor-Liquid-Liquid Phase Equilibria for Binary aod TernarySystems of Nitrogen, Ethane and Methanol: Experimentand Data Reduction,” Fluid Phase Equilibria, 25,(1986) 303-322.

Barke, J. A,, Henderson, D. and Smith, W. R.: “ThreeBody Forces in Dense Systems,” Physical ReviewLetters, 21, (1968) 134-136.

Kuan, D. Y., Kilpatric, P. R., Sahimi, M., Striven, L.E. apd Davis, H. T.: “Multi-component CarbonDioxide/Water/ Hydrocarbon Phase Behavior ModelingA Comparative Study, SPE Paper #1 1961 presented atthe 58th Annual Technical Conference and Exhibhion,San Francisco, CA, (Oct. 5-8, 1983).

31..

32.

33.

34.

35.

36.

Firoozabadi, A.: “Reservoir Fluid Phase Behavior andVolumetric Prediction with Equations of State,” J.Petrol. Tech, (1988) 397-406,

Metcalfe, R. S. and Yarborough, L. :“The Effect ofPhase Equilibria on the C02 Displacement Mechanism.”SPE J., (Aug. 1979) 242-252,

Cavett, R. H.: API proceedings, DNision of Refining,American Petroleum Institute, Washington, D.C!.42(3), (1%2) 351.

Wu,, R. S. and Batycky, J. P.: “PseudoComponentCharacterization “for Hydrocarbon MiscibleDisplacement,” SPE Paper #15404, Proceedings of the1986 Annual Technical Conference and Exhibition inNew Orleans, LA, Society of Petroleum Engineers,Richardson, TX., (Oct. 26-29,1986).

Ng. H. J., Chen, C, J, and Robinson, D. B,: “VaporLiquid Equilibria and Condensing Curves in the Vicinityof THe critical Point for a Typical Gas Condensate:Reject 815-A-84, GPA Research Report # RR-96, GasProcessors Association, Tulsa, OK, (Nov. 1985),

Katz, D. L., and Firoozabadi, A.: “Predicting PhaseBehavior of Condensate/Crude Oil Systems UsingMethane Interaction Coefficients; J. Pet, Tech., 228,(Nov. 1978) 1649-1655.

rable 1, Conformalsolution mklng rules for the PR equation of stat{

3 1/2A = [~~jxixjAijbij]32/[XiZjxixjAijbij ]

?MA Theory b = [XiZjxixjAijbij13/ziXjxixjAijbijll;nC = [~i~jxixjAijCij 3~i~jxixjA1j ,J..c.p

A = ~zjxixjAij~dWTheory b = Xizjxixjbij

C = ZiZjXiXjCij

A = XizjxixjAij

+SE Them-y b = [XiXjxixjAij]z/XiXjxixjAij2/bij

C = [~i~jxixjAij]2/~i~jxixjAij2/Cij

~~~xi~~ij{ l“[(Aij~j)(b/A)-l]~)>EX Theory ijijij

C ‘~ZjXiXjCij

A = 2~jxixjAij

3A Theory l+AXX=iB*i/~zjxix~B*lijB*ij=xi(dij+xjAxxbij/b)C = ZiZjXiXjCij

\ = pRTtc~xx-l = -l+RT/(R’IW2/(v-b)2-2Av3/(v2+b2)2)

~fl {[A-d(ACRT)]/(2bRT~2)) Qn[(v+b-b~2)/(v+b+b~2)l+d(AcRT)/(2[4(AcRT)-A] )

SPE 19809

Table 2, Synthetic oil composition used for the first application

Components Mohu Percentage

Methane “ 35EthanePropanen-Butanen-Pentanen-Hexanen-Heptanen-Octanen-Decanen-Tetradecane

;’:355305

Table 3. Gas condensate composition used for the secondapplication

Components Molar Percentage

.. ... Methape-,”’ “’”-e

pr~panei-Butanen-Butanei-Pentanen-Pentane

Hexane+NitrogenCarbon dioxide

76.348.864.290.791.26 E0.560.584.060.942.32

Table 4. Reservoir oil composition used for the secondapplication

Components Molar Percentage

Methane 32.54EthanePropanei-Butanen-Butanei-Pentanen-Pentane

Hexane+NitrogenCarbon dioxide

9.097.731.364.281.672.30

38.411.190.63

391

T = 283.1S KMethane (100%)

P = 7S.84 BARS

I Promne (100%) Ethane (100%) IFigure1. Ternarymixtureof smthane+ edwsse+ psopane. Accontiig to sfdsfigurewhtiethe Peng-Robinsoncqoadonof state(dashedline) is cs ble of pmdkting the behaviorof

Xct@WE&hatim(wliddots)binasyVLE dataof methane+ propan~ it fails to paroundthecriticalpointof the ternarymixture.WitJIthecnnsi&radonof three-bodyforces(solidline) !hcpmpsed techniqueis capableofpmdkxingtheVLEbchaviosof thistesnarymixnue.

Carbon Dloxlde (100%)T = 344.26 K

.,,P = 103.42 BARS

n-Decane (50%) n-Butan8 (SO%

Figuse2 Tmsasy mixtwes of carbondioxid%n-dccane,usd n-~tane. Aeccmfingto shebinasyfiguressbePRequsdon of state(dashedlines) faifsso~ahsbinarydata(aofidd-m) of these components. Wkh the mmidesation of ant onsuf aofudonmixing Nfea(solidtines)we ameapsbk ofpmdkdng thebinaryndxtusesandtbe@stay odxtumswayfrom the.temaryeritksdpoins.Withtheconsidssndonof ahsadmdy fosca (solid fine in

‘z%%%%-”””” isc@feo#pscdkdngUwVLEWaavloroftids

*.,

T _ 338,70 K

.’

,’

., , 8 , a

,0 0.1I

O.t 0.$1 1 I

0.4 0.6 0.0 0.? 0.9 0.9COMPOSITION X(C2-C6)

figure 3. The P-X diagrsmfor the [~~]+[~+] pseusbbinsry system aI 338.70 K.Calculationwith two pssudoampmrcnts but withoutthehim ieg andsitepscuddinay

IIinteractionparametersis reprsscntedby the dashed.dotted ne. Calculationwith Iwo

1’scudo-componwws and the pseudo-binmy interaction parameterbut without theumpingpUamctcrais shownby thedashediina calculationwithtwopa@o-componsnrs

including*.* iumpingandpsmdo%rrmyintcrsctisinpammstas is shownby the solid line.Exact multimnponentcalculationis shownby (x).

cl t C7+860.0

Soo.o

260.6

C&

<m 200.0

100.6

60.[

0,1

T _ 33&70 K

i # 1

,0 0.1 I I

0.2 0.8,

0.4 0.89 , I

0.0 0,? O.a O.*COMPOSITION X(U)

HgU 4. ‘ilte P.x diagramf~ the[C,]+[C,+]psaudetinuy system at 338.70 K.

I

ficulation with two p~odrt-compoa~l; ssrdkluding shelumpingsnd pseudo-binaryinteractionpsrsmetcrs is shown by the sefid iii. fksct muiticompoocntcalculation Isshown by (X).

$08.

(C1-C02)/ (C2-C8)

0)a~

T = 338,70 K

.- . . -0.$ 0.4 !J.S 0.9

COMPO&ON &C0!i2:to

Figure 5. llc P-X diagram for the [C1-C02]+[~C6] systemat 338.70 K. Calculationwith pseudoizationkchniquc is shownby the solidline. Exactmulticomponcntcalculationis shownby (x) and(sl).

160.0

140,0

130s

120.(

110.[

10001

,0,1

SPE 19809

C02 / SYNTHETIC OIL

T .338,70 K

C1-C02

Figure7. P.wudo-temarydiagramforCOJ8yn!heticOil a!differentpressures,Predictionwilh the pseudoizationtcchniqucis representedby the solid tine.Exactmulti.componentcalculationis shownby (x).

240.0-

220.0.

200.0.

11O.O

180,040a# 140.0

wc 120.0

%mu 100.0g

00.0

60.0

40.(

90.(

0.1

●

✎✎✎

ico,o soo.o aio.o a40.o aio.o oio,o 400.0 420.0

TEMPERATLIRE KELVIN4io.o

Figure 8. Phase envelope of gas condensate at 369.85 K. Prediction ‘with thepseudoizationtechniqueis shownby the tolid line, Exact mukicomponentcalculationisshownby (x).

m

PHASE EQUILIBRIA OF C02 / OIL

3!

$10.1

100,1

100. m , , 1 , , ,

W* 0.1 0.2 *.8 0.4 0.s 0.0 is 0.0 0.sooMPostmoN C02

l.o

Figure 9. I%s P-xdiagswlOfctin dioxide-massvtioilayassm(lowinSocshancandhighin hcavk hydmmbmw) wish she compositionas in Tabfs4 cosaasmcscdusing shasame4 pseudo-corn-as in F@m 13.Predictionwithshapacodoimticmacchniquaisshownby shsaolii fmc. Ea3camukmoponsat calculationis ahowmby (x).

CricondenM

PaC=3W.KX(2)+X(3D

IPb

Ft

Pd

Pe

0.0 1.0

cc

(1)

//’J,$,~m Q) co o) (3 e (3)

0) Q) (2) (3)

Figura 10. Phase diagramsefationshiptctween P-X and ternarydiagramsfos sysscmstilhilhg omfyDnaphasecnvslops io thepaswbsanasy mpmacnsadon,This is I!SCfti ofshe tw~ possible s@elionshipIscswccnsriocondcnbarlocus of P-X diagramand phaseW@Jon mgmnsm pscusbtmnarydia~

G-imndcntas

Pa

Pb

R

N

R

Pf

f%

~.C=X(2)/&(2)+X(3)) I

iiCc Qlcef

(1) (1)

~~~

A( .&.&

(a @ -(3) m a (3) (2) m

FigmII. phasedi~gmmrelationshipbctwscssP-X and tematy diagmms fos systemswhichsmy exhibitswoclod phasemvclopcs in thepseudo-ternaryaeprwmmtionIlk isthe secondof Ifsctwo possible rclationshipbcswccncriocotsdasbarlocus of P-X diagsamandphasesqmatim acgionsin pacudo-tssnmydmgmm.

cl

●

4

>●

C3 ●b

●

Cb

w

B n8 wi. do 8iioPmsaum IS81A

a?io 10

Figure12 Componsmmole fmction equilibriummsio, K-values,,@ a gas condensatefluid%at MYF. Dots am the cxpmiotcnsaldatawhifethe did lines am* scads of bWP* =Iodn~s fi~m phaseU@-iiIiUMafguirhm.

09wm

.8 #o@ 1000 Woa 1009 woo sootPRESSURE PSIA

Figure 13. Liquid volume percent ofagoscondsmsatc fluid3sat lWPF. Dots a-rethcexperimentaldata while the solid lines arc the stsuhsof the proposedcontinuoussoixtusephasecquilibsiwnalgorithm

b,00

●●

●●

~

.*

f)L=70 5 ● o●

●

...-

0 .80 s 200 MO~M13Etil’’RE’%

Figure.14. ~ diagram of a gaa condcn~te.fluid3s.Dots arc the expx’imentd data whileIhe solid lines are the resuhs of the pmpoacd continuous mixture phase equilibriumalgorithm,

Ma