modeling multicomponent diffusion and … multicomponent diffusion and convection in ... rotkegel12...

Modeling Multicomponent Diffusionand Convection in Porous MediaKassem Ghorayeb, SPE, and Abbas Firoozabadi, SPE, Reservoir Engineering Research Inst.

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SummaryNumerical investigations of diffusion and convection in mulcomponent hydrocarbon mixtures in two-dimensional~2D! cross-sectional (x,z) porous media are performed using the finitvolume method. Spatial discretization is performed by a secoorder centered scheme. It is shown that methane, unlike in bihydrocarbons where it often segregates towards the bottomside of the porous media, may be at higher concentration atcold-top side in ternary mixtures and in multicomponent reservfluids. This behavior, which is consistent with oilfield data, is dto competing diffusion mechanisms which have not been propaccounted for in the past. It is also demonstrated that convecmay significantly affect the compositional variation in some hdrocarbon reservoirs. Depending on fluid mixtures, a weak cvection may drastically change compositional variation.

IntroductionField observations show a wide range of compositional variatthere is generally vertical compositional variation of the compnents in some reservoirs.1 In other reservoirs, a pronounced horzontal compositional variation is seen.2 In yet others, there is verylittle compositional variation with depth.3 One may even observa decrease of heavy components such as C71 with depth.4 It isbelieved that multicomponent diffusion and convection affectdistribution of various components in hydrocarbon reservoirs5,6

In a multicomponent fluid, the total molar flux of a given compnent consists of two parts:~1! the convective flux from the velocity of the bulk fluid, and~2! the diffusive flux as a result of thedifference between component velocity and bulk velocity. Tmolecular diffusive flux of a given component depends not oon its composition gradient~mutual diffusion!, but also on thecomposition gradient of all the other components in the mixt~cross-molecular diffusion!. The diffusive flux also depends opressure and thermal gradients—the so-called thermal diffu~Soret effect! and pressure diffusion~gravitational segregation!,respectively.7 All these effects can be modeled using thermodnamics of irreversible processes.

The compositional variation of a component in multicompnent mixtures~more than two components! in a porous mediummay radically differ from that in binary mixtures. The cross efects ~cross-molecular diffusion and thermal diffusion! are themain processes contributing to this behavior. The primary goathis work is to model the compositional variation of multicompnent mixtures in porous media, taking various cross effectsaccount, and to show that all the trends in field data~from Refs. 1through 4! can be predicted.

Methane in binary mixtures of C1 /C2, C1 /C3, and C1 /nC4,where experimental data are available, segregates to the hotof a differentially heated apparatus.8-10 On the other hand, in hy-drocarbon reservoirs, there is generally more methane on theside ~top of the reservoir!.4,5 These facts imply that one may nouse effective thermal diffusion factors to study the segregationmethane in mixtures with more than two components in a nonthermal medium. Cross-molecular diffusion has also been shto be important in some ternary and higher mixtures. There isreference in the literature to studies of the combined effec

Copyright © 2000 Society of Petroleum Engineers

This paper (SPE 62168) was revised for publication from paper SPE 51932, presented atthe 1999 SPE Reservoir Simulation Symposium held in Houston, 14–17 February. Originalmanuscript received for review 19 February 1999. Revised manuscript received 25 Octo-ber 1999. Manuscript peer approved 14 December 1999.

158 SPE Journal5 ~2!, June 2000

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thermal diffusion, molecular diffusion, and convection in ternaand higher mixtures. Larreet al.11 investigated the stability of ahorizontal layer heated from below filled with a wateisopropanol-ethanol mixture. The authors neglected the crmolecular diffusion coefficients and assumed that the thermalfusion factor of a component could be expressed as the sum othermal diffusion factors of the binaries consisting of the givcomponent and the two others, respectively. The model resultnot, however, agree with the experimental data. A similar concsion is drawn by Krupiczka and Rotkegel12 who investigated masstransfer in ternary mixtures of isopropanol-water-air aisopropanol-water-helium. Considerable discrepancies betwexperimental data and theoretical predictions were observed wcross-diffusion terms were neglected.12

There is a vast body of literature on molecular diffusion coficients in binary systems.13 A sizable amount of binary data ohydrocarbons is also available.14-16 However, molecular diffusiondata for mixtures consisting of three or more components, escially for hydrocarbon mixtures at reservoir conditions, are scaReviews of the available data in ternary mixtures with summaof the measurement methods are presented by Cussler13 and Ty-rell and Harris.17 Kooijman and Taylor18 summarized the existingmodels for predicting the molecular diffusion coefficients in muticomponent mixtures and presented a correlation based onVignes correlation19 for binary systems. Compared with expermental data for ternary systems, Kooijman and Taylocorrelation18 provides better results than do the other availacorrelations. In this work, we adopt Kooijman and Taylor’s corelation to calculate the molecular diffusion coefficients.

Surprisingly, no experimental data for ternary and higher mtures exist in the literature for thermal diffusion factors, a necsary requirement for the calculation of thermal diffusion in a mticomponent mixture. Firoozabadiet al.20 have recently developeda theoretical model to estimate thermal diffusion factors for tnary and higher mixtures. The new model accurately predictsthermal diffusion factors of binary mixtures. Furthermore, usithe thermal diffusion factors obtained by this model, we are ato successfully simulate the compositional variation in a thermgravitational porous column containing a ternary mixture reporby El Maataoui.21 The agreement between compositional dafrom experiments and the model provides indirect validationmulticomponent mixtures.

Several attempts have been made in the last 20 years to mcompositional variation in hydrocarbon reservoirs. The earstudies considered the gravitational effect on compositional vation in a one-dimensional~1D! convection-free system.22-25 Themain conclusion from those studies is that the gravitational efcauses the heavier components to segregate towards the bottthe reservoir. Thermal diffusion in a 1D convection-free systhas been accounted for in both binary and multicomponent mtures in some studies.5,26,27 From those studies one can mainobserve that thermal diffusion may have the same order of mnitude and may have an opposite effect than pressure diffusThe above studies neglect the effect of convection on comptional variation~although it may be very important! and have beenperformed in a 1D vertical system; they do not, therefore, allfor investigating areal compositional variation. Furthermore,appropriate model for thermal diffusion has been consideredthe studies which consider thermal diffusion in compositionvariation.5 For multicomponent systems, Belery and da Silva5 in-vestigated the compositional variation in the Ekofisk field usinone-dimensional model. They took into account molecular, prsure, and thermal diffusion and assumed a convection-free sys

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Their model is based on effective molecular diffusion coefficieand effective thermal diffusion factors. The results from the woof Belery and da Silva show some qualitative agreement wcompositional variation in the field; methane is more concentraat the top of the reservoir. The combined effect of convectionpressure diffusion was considered by Jacqmin6 for a two-component mixture. Jacqmin’s work shows the mixing effectconvection; the stronger convection is, the more homogeneouthe composition. However, neglecting thermal diffusion mmask an inverse role that convection may play on compositiovariation in hydrocarbon reservoirs in some range of permeabias it has been shown in recent studies.28,29

Compositional variation of methane in a C1 /nC4 single-phasebinary mixture has been studied in the same range of therphysical and geometrical conditions as in this study.28,29In a typi-cal hydrocarbon binary mixture, for low permeabilities, compotional variation is mainly affected by the ratio (2DTTz)/(D

ppz)whereDp, DT, pz , andTz are the pressure and thermal diffusiocoefficients~see Riley and Firoozabadi28! and the vertical pressurand thermal gradients, respectively. When this ratio is.1, thehorizontal compositional variation is more pronounced thanvertical variation. The horizontal compositional gradient reachemaximum value fork'1 md and then decays as 1/k.28 A smallamount of convection can cause the horizontal compositionaldient to increase. A similar phenomenon has been experimenobserved in vertical columns~separation by thermogravitationaeffect! where there exists an optimal value ofk leading to themaximal separation.21 Jametet al.30 had some success in modeing these phenomena. When (2DTTz)/(D

ppz),1, compositionalvariation is effected mostly by pressure diffusion;29 the verticalcompositional variation is more pronounced than the horizovariation.

The main objective of this paper is to provide a sound basisthe study of compositional variation in hydrocarbon reservoboth multicomponent diffusion and convection are taken intocount. The results for a ternary system are presented in detashow the effect of cross-molecular diffusion and thermal diffuson the compositional variation of methane, and to explaindiscrepancies occurring between field observations and resubinary systems. We also present the modeling of compositiovariation in the Cupiagua gas condensate field.3

The paper is organized as follows; first, we briefly presentmathematical formulation of the problem. Then, an overviewthe numerical method is presented. Results from several cotions of composition, temperature, and pressure are discussegether with a comparison with previous results obtained inbinary mixture C1 /nC4.

Mathematical FormulationWe consider a two-dimensional porous medium with widthb andheight h ~Fig. 1! saturated by a single-phase mixture ofn com-ponents. We assume that the Oberbeck-Boussinesq approximis valid for the ternary system. This assumption is relaxed fornear-critical reservoir fluid for the field example. In the OberbeBoussinesq approximation, the densityr is assumed to be con

Fig. 1–Geometry and boundary conditions.

K. Ghorayeb and A. Firoozabadi: Modeling Multicomponent Diffusion

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stant, except in the buoyancy term (rgz), where it varies linearlywith the temperatureT and the mole fractionsxi , i 51, . . . , n21:

r5r0F12bT~T2T0!2(i 51

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where

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, i 51, . . . , n21,

are the thermal expansion coefficient and the compositional cficients of componenti , respectively;r0 is the density,xi0 themole fraction of componenti andT0 is the temperature, all at thereference point. The validity of the Oberbeck-Boussinesq apprmation has been demonstrated by Ghorayeb and Firoozabadi31 forbinary systems. All the ternary mixtures considered in our stuare far enough from the critical point and the assumption oflinear variation ofr vs. T, x1 , andx2 is justified as we will seelater.

The unsteady-state conservation equations of mass and spare

]c

]t1“•~cv!50, ~2!

]~cxi !

]t1“~cxiv!1“•Ji50, i 51, . . . , n21, ~3!

wherec, v, andJi are the total molar density, the bulk velocityand the molar diffusive flux relative to molar average velocity fcomponenti ( i 51, . . . , n21), respectively. When applyingthe Oberbeck-Boussinesq approximation, Eqs. 2 and 3 read

“•v50, ~4!

cF]xi

]t1“~xiv!G1“•Ji50, i 51, . . . , n21. ~5!

The bulk velocityv is given by Darcy’s law:

v52k

fm~“p1rgz!, ~6!

wherep, g, k, m, andf are the pressure, the acceleration duegravity, the permeability, the viscosity, and the porosity, resptively. The unit vectorz points upwards. Substitution of Eq. 6 intEq. 2 and the assumption that (k/fm) is constant lead to thefollowing unsteady pressure equation:

]c

]t2

k

fm“•@c~“p1rgz!#50, ~7!

which reads with the Oberbeck-Boussinesq approximation:

“

2p5r0gS bT

]T

]z1(

i 51

n21

bxi

]xi

]z D . ~8!

We assume that the reservoir is bounded by an imperviousthat has constant temperature gradientsTx and Ty in horizontaland vertical directions, respectively. In the following, the temperature field is assumed to be a linear function ofx and z: T5Txx1Tzz1a, where a is a constant. Field data support thform of the temperature expression. If we set the temperaturx5x0 andz5z0 equal toT0 , then

T5Tx~x2x0!1Tz~z2z0!1T0 , ~9!

wherex0 and z0 are the coordinates of the reference point. Tdiffusive flux of componenti is given by~see Appendix A!:

SPE Journal, Vol. 5, No. 2, June 2000 159

160 K. Ghoray

TABLE 1– PREDICTED AND EXPERIMENTAL COMPOSITION „WEIGHT FRACTION…

IN THERMOGRAVITATIONAL COLUMN FOR A BINARY MIXTURE OF nC24 ÕnC12

k (m2)

Data of El Maataoui* Model Results

(nC24)bottom (nC24)top q (nC24)bottom (nC24)top q

2.90310211 0.200 0.095 2.38 0.202 0.094 2.44

4.90310211 0.230 0.109 2.44 0.193 0.105 2.03

6.10310211 0.205 0.116 1.96 0.187 0.111 1.83

1.18310210 0.188 0.133 1.51 0.171 0.128 1.41

1.79310210 0.169 0.140 1.25 0.164 0.135 1.26

2.69310210 0.151 0.136 1.13 0.160 0.140 1.17

4.75310210 0.165 0.148 1.13 0.156 0.144 1.09

Initial composition: 15% (of total mass) nC24; 85% nC12. T05321.5 K, DT525. Columnheight540 cm.

*Reference 21.

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T“TD , i 51, . . . , n21,

~10!

whereDi jM , Di

p , andDiT are the coefficients of molecular diffu

sion, pressure diffusion and thermal diffusion, respectively. Tdiffusive flux Ji results from the deviation in the velocity of component i from the velocity of the bulk fluid. Since, for ann-component mixture, the diffusive flux of thenth component isequal to2( i 51

n21Ji and the mole fraction of thenth component isequal to 12( i 51

n21xi , the equation expressing conservationmass for componentn will be a linear combination of Eqs. 2and 3.

Boundary Conditions. The boundary conditions for Eqs. 2, 3and 6 are based on no-fluid fluxes across the outer bound~Fig. 1!:

Ji •n50, i 51, . . . , n21, x50, b, and z50, h,~11!

v•n50, x50, b, and z50, h, ~12!

wheren is the unit normal vector. Eqs. 11 and 12 imply that

(j 51

n21

Di jM

]xj

]x1Di

p]p

]x1Di

T]T

]x50,

i 51, . . . , n21, and vx50, x50, b, ~13!

(j 51

n21

Di jM

]xj

]z1Di

p]p

]z1Di

T]T

]z50,

i 51, . . . , n21, and vz50, z50, h. ~14!

From Eq. 6, we obtain the boundary conditions that are requfor the integration of Eq. 7:

]p

]x50, x50, b, ~15!

]p

]z52g

]r

]z, z50, h. ~16!

With the above Neumann boundary conditions, mole fractionspressure must be assigned at one point in the medium. Inwork, we setxi(x0 ,z0)5xi0 , i 51, . . . , n21, and p(x0 ,z0)5p0 .

Numerical SchemeEqs. 3, 6, and 7, together with the boundary conditions givenEqs. 13–16, are integrated numerically using the finite-volu

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method~see Patankar32! with a rectangular grid system. The sptial discretization is performed using a second-order centescheme. A semi-implicit first-order scheme is used for the temral integration. We seek the steady-state solution and iterate onunsteady state until the steady state is reached. In this work,vergence to the steady state is assumed when the absolute reerror of mole fractions is less than 1027 between the two successive iterations at each grid point.

Model ValidationEl Maataoui21 measured species separation by the thermogravtional effect in nC24/nC12 and nC24/nC16/nC12 mixtures ~seeFurry et al.33 and Lorenz and Emery34 for details about the ther-mogravitational effect!. We simulated, using our numericamodel, some of those experiments. The main difference betwthe experimental apparatus and the numerical simulations isuse of different model geometry. In the experiment, a cylindricolumn was used~two concentric cylinders, the inner one heatand the outer one cooled! which we represent numerically asrectangular column~with a width equal to the difference of thradii of the two cylinders!. Since the difference of the radii issmall compared with either radius, we may proceed as if the cvection were taking place in a thin flat slab.33

Fig. 2–Predicted and measured separation factor qÄ„x nC24

Õx nC12…bottom Õ„x nC24

Õx nC12… top ; initial composition:

15% nC24 Õ85% nC12 , T0Ä321.5 K, DTÄ25, and columnheight Ä40 cm.

ffusion SPE Journal, Vol. 5, No. 2, June 2000

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Table 1 presents, for different values of permeability,nC24composition in weight fraction at the bottom and top of the cumn ~both experimental21 and numerical results! as well as theseparation factorq5(xnC24

/xnC12)bottom/(xnC24

/xnC12)top. Fig. 2

depictsq vs.k. The comparison of experimental data and numecal results shows good agreement. The maximal relative erroless than 20%. With regard to the optimal value of the permeaity, kopt , leading to a maximal segregation, experimental datanumerical results show 2.9310211,kopt,4.9310211 m2 andkopt'2.5310211 m2, respectively. There is excellent agreemebetween measured and predicted values of the Soret coeffiST5D1

T/@D11Mx10(12x10)#; El Maataoui21 reported ST

50.90 K21; from our model20 we obtainST50.85 K21.For the ternary systemnC24/nC16/nC12, which is an ideal liq-

uid mixture for the given composition and atmospheric pressthe experimental data reported by El Maaˆtaoui21 show that noqualitative change occurs between the ternary and binary sysfor the compositional variation ofnC24; nC24 segregates towardthe bottom of the column. The cross-diffusion coefficientsvery small in comparison with the mutual diffusion coefficienand thatD1

T has, in this example, the same order of magnitudeboth binary and ternary systems. Furthermore, no significant cposition variation is observed fornC16 which is mainly due to thefact that the thermal diffusion coefficientD2

T is much smaller thanD1

T ; molecular diffusion prohibits any significant segregationnC16. Table 2 shows the measured and predicted results;maximal relative error between numerical and experimentalues is less than 10%.

Results

Ternary Systems.In this study we focus on the nonideal ternamixture of C1 /C2 /nC4. Two sets of calculations are performethe composition at the center of the reservoir is fixed at~1! x10

50.25, x2050.25, andx30

50.50, and~2! x1050.35, x20

50.35,andx30

50.30. Several conditions of temperature and pressureconsidered. The mixture in all investigations is single phaTable 3 presents the relevant data used for this ternary mixtuTo study the effect of grid meshes, numerical runs were p

TABLE 2– PREDICTED AND EXPERIMENTALCOMPOSITION „WEIGHT FRACTION… IN

THERMOGRAVITATIONAL COLUMN FOR TERNARYMIXTURE OF nC24 ÕnC16 ÕnC12

(nC24)bottom (nC24)top (nC16)bottom (nC16)top

Measured data* 0.240 0.140 0.415 0.380Model results 0.265 0.135 0.408 0.389

Initial composition 20.2% nC24; 39.9% nC16; 39.9% nC12.T05321.5 K, DT525. Column height5120 cm. k56.1310211 m2.

*Reference 21.

TABLE 3– RELEVANT RESERVOIR AND FLUID PROPERTYDATA FOR THE TERNARY MIXTURE

Reservoir width 1500 mReservoir height 150 mPorosity 0.25Viscosity 0.2 cpPermeability 0–100 mdVertical thermal gradient 3.5 K/100 mHorizontal thermal gradient 1.8 K/1000 mMesh 301341


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formed with different mesh grids. We found that using finer grithan those in Table 3 do not significantly improve the accuracythe results.

Figs. 3 and 4show thep2T plots for those mixtures, as welas temperatures and pressures at which the calculations areformed. For every mixture, at each composition, three combitions of (T,p) are selected. Those combinations are represenby points 1, 2, and 3 in Figs. 3 and 4.Tables 4 and 5present, indetail, the data used in this work. The molecular diffusion coecients are obtained using Kooijman and Taylor’s correlation~seeAppendix B!. Appendix B also shows the process by which ocalculates the pressure diffusion coefficients from the molecdiffusion coefficients and the thermodynamic properties ofmixture. The thermal diffusion factors are from Firoozabaet al.20 ~see Appendix C!. The cross-molecular diffusion coefficientsD12

M andD21M may be positive or negative. As critical poin

is approached,D12M andD21

M become significant in comparison tD11

M and D22M . Thermal diffusion coefficients and pressure diff

sion coefficients show a similar trend. In this section we assuthat molecular, pressure, and thermal diffusion coefficientsconstant in the whole cross section. Their values are those areference point~the center of the reservoir!. This assumption isjustified by the fact that the mixture is far from the critical poin

Fig. 3– p -T diagram for ternary mixture C 1 ÕC2 ÕnC4 ; x 10Ä0.25,

x 20Ä0.25, x 30

Ä0.50, and d denotes the critical point.

Fig. 4– p -T diagram for ternary mixture C 1 ÕC2 ÕnC4 ; x 10Ä0.35,

x 20Ä0.35, x 30

Ä0.30, and d denotes the critical point.


162 K. Ghoray

TABLE 4– PHYSICAL AND TRANSPORT PROPERTIES FOR THE C 1 ÕC2 ÕnC4 MIXTURE

Physical and TransportProperties

p057.03106 PaT05315 K

p057.83106 PaT05345 K

p057.43106 PaT05395 K

r0 kg/m3 450.35 374.41 175.36

bT K21 20.5231022 20.8631022 21.4331022

bx1 21.25 22.17 23.98bx2 20.72 21.25 22.44D1131019 m2/s 13.14 13.52 13.96

D1231019 m2/s 20.77 21.68 23.24

D2131019 m2/s 20.32 20.83 21.87

D2231019 m2/s 14.68 16.43 18.76

D1P310116 m2

•Pa 10.32 10.87 14.54

D2P310116 m2

•Pa 10.14 10.47 12.86

D1T310112 m2/s•K 25.13 28.94 211.8

D2T310112 m2/s•K 21.35 22.71 24.36

x1050.25, x20

50.25, and x3050.50. The reference point coordinates are x05b/2, z05h/2.

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the variation of the diffusion coefficients is small within the ranof the physical and thermophysical parameters for the ternmixture.

Fig. 5 presents the variation of the density vs. temperateT,mole fraction of C1, x1 , and mole fraction of C2, x2 , using thePeng-Robinson equation of state~PR EOS!.35 The range ofT, x1 ,and x2 considered in Fig. 5 covers the variation of these paraeters in our study. Therefore, the densityr varies linearly withT,x1 , andx2 . Figs. 6 through 11depict composition contour plotfor C1 and C2 with permeabilities in the range from 0~convection-free! to 100 md. We first discuss compositional variation for tconvection-free cases before the effect of convection is conered.

From Figs. 6 through 11 it can be seen that, in most cacomponent 2~ethane! is more concentrated at the top of the reervoir. Therefore, the sign of the thermal diffusion coefficient innonideal mixture~see Tables 3 and 4! with more than two com-ponents does not imply the segregation to the hot or cold sidthe system. From Figs. 6 through 11, one can also observe thall the cases there is more compositional variation of meththan ethane. That is in agreement with the results byMaataoui21 for the ternary mixturenC24/nC16/nC12, where theobserved segregation of the component with intermediate mollar weight nC16 is negligible compared to that of the light componentnC12 and the heavy componentnC24.

eb and A. Firoozabadi: Modeling Multicomponent Di

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Let us discuss the compositional variation of methane~compo-nent 1!. Figs. 6 and 7 show that at low temperature~point denotedby ‘‘1’’ in Figs. 3 and 4!, there is more methane at the bottomthe reservoir. The trend is similar to the compositional variationmethane observed in the binary mixture of C1 /nC4.

28,30 In mostcases we studied, there is no significant vertical compositiovariation of C1 for intermediate values of temperature at highpressures~point denoted by ‘‘2’’ in Figs. 3 and 4!, see Figs. 8 and9. Figs. 10 and 11 show that at high temperatures~point denotedby ‘‘3’’ in Figs. 3 and 4!, there is more methane at the top.

Methane mole fraction contour lines have a positive slopelow temperatures~see Figs. 6 and 7!. The slope increases as thtemperature increases as shown in Figs. 8 and 9. For somebinations of (T,p) ~in the neighborhood of point ‘‘2’’!, the molefraction contour lines become almost vertical, and there isvertical compositional variation of methane in the reservoir~seeFig. 9!. As temperature increases further~point ‘‘3’’ !, the slope ofthe mole fraction contour lines becomes negative~Figs. 10 and11!. Methane in this case is more concentrated at the top ofreservoir.

Points ‘‘1,’’ ‘‘2,’’ and ‘‘3’’ differ also in terms of the effect ofpermeability on compositional variation. When we take point ‘‘1as the reference, one observes the same trend reported byand Firoozabadi28 for the binary mixture C1 /nC4. At low perme-abilities (k,1 md), methane goes towards the hot side~the

TABLE 5– PHYSICAL AND TRANSPORT PROPERTIES FOR THE C 1 ÕC2 ÕnC4 MIXTURE

Physical and TransportProperties

p058.53106 PaT05290 K

p059.33106 PaT05315 K

p058.83106 PaT05365 K

r0 kg/m3 423.20 358.66 178.58

bT K21 20.5131022 20.8631022 21.2631022

bx1 21.71 22.48 23.98bx2 20.97 21.44 22.44D1131019 m2/s 12.51 12.26 13.11

D1231019 m2/s 21.14 22.03 23.89

D2131019 m2/s 10.06 20.74 20.56

D2231019 m2/s 14.87 16.10 19.67

D1P310116 m2

•Pa 10.43 10.98 14.87

D2P310116 m2

•Pa 10.12 10.43 12.75

D1T310112 m2/s•K 26.94 210.0 213.2

D2T310112 m2/s•K 10.32 10.11 20.76

x1050.35, x20

50.35, and x3050.30. The reference point coordinates are: x05b/2, z05h/2.


Fig. 5–Density variation vs. T, x 1 , and x 2 for the ternary mixture C 1 ÕC2 ÕnC4 ; x 10Ä0.25, x 20

Ä0.25, x 30Ä0.50. a, b, and c correspond

to the density variation in the range of variation of T, x 1 , and x 2 with points 1, 2, and 3, respectively, from Fig. 3 as a reference.

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c-po-ion;OSm-ob-

nd

right-bottom corner of the reservoir!. When the permeability in-creases, due to convection, there is only horizontal compositivariation. The horizontal gradient of the mole fraction of methais almost constant. It reaches a maximum value fork51 md, thendecreases with the increase of permeability such that, ak5100 md, no significant compositional variation is observed~seeFigs. 6 and 7!. At point ‘‘2’’ as the reference, fork greater than 1md, the whole reservoir has a homogeneous compositionshown in Figs. 8 and 9. For point ‘‘3,’’ the composition in threservoir changes strongly when permeability increases fromk50 to k51 md. Fig. 12 shows the mole fraction contour lines omethane and ethane in this range of permeability. Atk50 ~that is,convection-free!, methane is more concentrated in the top-left cner of the reservoir. The slope of the mole fraction contour linenegative. At some permeability between 0.05 and 0.1 md, thcontour lines become horizontal and then, their slope becopositive. No significant effect of permeability is observed fork.1 md as shown in Figs. 10 and 11. The behavior of the ternmixture discussed above encompasses the features of comtional variation in Refs. 1 through 4.

Field Example. Recently, Lee and Chaverra3 reported data froma near-critical gas condensate reservoir with some unexpecte


nalne

t

ase

f

r-is

osees

aryposi-

be-

havior; there is very little variation of composition and dewpopressure in a gas column extending over 5,000 feet. The autprovide the vertical temperature variation. However, the horiztal temperature variation is not reported. Therefore, we assumarbitrary horizontal temperature gradient which is in the sarange of data observed in several fields from different placessingle composition shown in Table 7 is used in this example. Tcomposition differs slightly from that reported in Ref. 3. Thcomposition was provided by the first author of Ref. 3.

Tables 6 and 7list the data used in this field example and tcomposition and molecular weights at the sample point~corre-sponding tox055000 m andz0520 m!, respectively. Note thatthe sample contains a small amount of N2; it is added to C1 in ourcalculations. The pressure at (x0 ,z0) is set top5466.5 bar. Thedewpoint pressure at the sample point ispdew5380.5 bar. Thenear criticality of the fluid exhibits, as we will show in this setion, a nonlinear variation of density vs. temperature and comsition. Therefore, we do not adopt the Boussinesq approximatinstead, the density is calculated at every point using the PR Eand we assume all the diffusion coefficients as functions of teperature, pressure, and composition. We first present resultstained taking into account both cross-molecular diffusion a


Fig. 7–Composition contour plots for the C 1 ÕC2 ÕnC4 mixture;p 0Ä8.5Ã106 Pa, T0Ä290 K, x 10

Ä0.35, x 20Ä0.35, and x 30

Ä0.30„reference state set to point 1 in Fig. 4 ….


Ä0.25, x 20Ä0.25, and x 30


164 K. Ghorayeb and A. Firoozabadi: Modeling Multicomponent Di


Ä0.25, x 20Ä0.25, and x 30



Ä0.35, x 20Ä0.35, and x 30




Ä0.35, x 20Ä0.35, and x 30



Ä0.25, x 20Ä0.25, and x 30




Ä0.25, x 20Ä0.25, and x 30


TABLE 6– RELEVANT RESERVOIR AND FLUID PROPERTYDATA FOR THE FIELD EXAMPLE

Reservoir width 10 kmReservoir height 1.5 kmPorosity 0.20Viscosity 0.075 cpPermeability 0–10 mdVertical thermal gradient 2.75 K/100 mHorizontal thermal gradient 1.5 K/1000 mp0(x055000m, z0520m) 4.663107 PaT0(x055000m, z0520m) 422 KMesh 301341

TABLE 7– COMPOSITION AT THE REFERENCE POINTFOR THE FIELD EXAMPLE

Component mol% Molecular Weight

CO2 3.2290 44.010C1 62.253 16.043C2 9.6440 30.070C3–C4 9.0800 49.702C5–C6 3.4360 77.242C7–C10 5.9790 110.89C11–C14 2.4500 167.60C15–C20 2.0850 237.67C21–C29 1.2880 334.25C301 0.5560 550.00


Cl

re is

en-n-

eny

tionher ishe-ay

14,e ofost

-the

ant

-voirsidee

osi-

on

thermal diffusion. We then investigate the effect of increased thmal diffusion. We also set the thermal diffusion flux to zerodemonstrate that thermal diffusion can be an important mecnism in compositional variation studies.

Figs. 13 and 14show the mole fraction contour lines of CO2,C1, C2, C32C4, C52C6, and C71 and the fluid density forconvection-free andk510 md, respectively. One observes that1segregates toward the top of the reservoir whereas C71 segregatestowards the bottom. One also observes that the maximum segation occurs for C1, and C71 . No significant compositionalvariation is observed for the components and pseudocompon

Fig. 13–Composition and density contour plots for the field ex-ample; kÄ0 md „convection-free ….

Fig. 14–Composition and density contour plots for the field ex-ample; kÄ10 md.


er-toha-

gre-

ents

in between: C2, C32C4, and C52C6. Figs. 13 and 14 also reveathat there is no significant compositional variation for CO2.

When the permeability changes from convection-free tok510 md, one observes by comparing Figs. 13 and 14 that theless compositional variation for all the components atk510 thanfor k50 md. Furthermore, methane, which was more conctrated in the right-top corner of the reservoir for the convectiofree case is more concentrated in the left-top corner fork510 md. The compositional variation changes betweconvection-free andk510 md are similar to that for the ternarmixture C1 /C2 /nC4 depicted in Figs. 10 and 11.

From Figs. 13 and 14 one concludes that there is more variaof composition and density in the bottom than in the top of treservoir. This is due to the fact that the bottom of the reservoicloser in this field example to the critical point which causes tdiffusion coefficients~and especially the cross-molecular diffusion coefficients! to change sharply in that region resulting insignificant nonlinear variation of composition and density. Bcomparing the density contour plots depicted in Figs. 13 andone concludes that convection results in a significant decreasvertical density variation and causes density to become alm~but not exactly! constant in the horizontal direction.

Fig. 15 shows the variation of C1, C71 , pressure, and saturation pressure vs. height at three different cross sections ofreservoir: atx55000 m (b/2), x52500 m ~the cold side!, and x57500 m ~the hot side! for convection-free and fork510 md.Fig. 15 reveals that, due to convection, there is no significhorizontal variation of composition over the whole reservoir~seeFigs. 15b and 15d!. The horizontal variation of the dewpoint pressure is also small. However, because the left side of the reseris the colder region and methane is less concentrated in thatfor convection-free conditions, it exhibits a bubblepoint in thleft-bottom corner as depicted in Fig. 15e. Because the comption varies very little in the areal direction atk510 md, the dew-point pressure does not significantly vary in that directi

Fig. 15–Mole fraction of C 1 and C7¿ , pressure and saturationpressure for convection-free „left … and kÄ10 md „right ….


rly.nal

mall

edif-c-hot

esig-

ther-esiesec-ea-th-at

be-

y-res,ther

either, as shown in Fig. 15f. The results presented so far givmore pronounced variation of the dewpoint pressure and comsition than that of the field data.3

Figs. 16 and 17depict the mole fraction contour lines of CO2,C1, C2, C32C4, C52C6, C71, and density for convection-freeandk510 md, respectively, when the effect of thermal diffusiois increased. This increase is achieved by decreasingt i of theheavy pseudocomponents (C7– C10 and above! in Eq. C–8. Forthe components and pseudocomponents from C1 to C5– C6, t i54 as is the case in all the examples presented above. Fo

Fig. 16–Composition and density contour plots for the field ex-ample; kÄ0 md „convection-free … with increased thermal diffu-sion.

Fig. 17–Composition and density contour plots for the field ex-ample; kÄ10 md with increased thermal diffusion.


e apo-

n

the

pseudocomponents C7– C10, C11– C14, C15– C20, C21– C29, andC301, t i is set equal to 3.0, 2,9, 2.8, 2.7, and 2.6, respectiveFigs. 16 and 17 show a significant decrease of compositiovariation consistent with fluid data.Fig. 18 portrays variations ofcomposition, pressure, and dewpoint pressure. There is a svariation of dewpoint pressure in the entire reservoir~see Figs.18e and 18f!. The increase of the thermal diffusion provides thcondition that it has almost the same magnitude as pressurefusion but affecting compositional variation in an opposite diretion. Thermal diffusion pushes the lighter components to thebottom of the reservoir.

Fig. 19 shows the vertical variation of the mole fraction of C1and C71 , and pressure and dewpoint pressure atx55000 m withand without thermal diffusion~left! and with and without in-creased thermal diffusion~right! for the convection-free case. Onobserves that thermal diffusion reduces composition variationnificantly.

Remarks and ConclusionsThe use of general expressions for molecular, pressure, andmal diffusion in multicomponent nonideal hydrocarbon mixturcombined with natural convection allow the calculation of specdistribution in hydrocarbon reservoirs. Since steady-state convtion is governed by the horizontal temperature gradient, the msurement of horizontal temperature variation is a key factor. Wiout horizontal temperature variation, there is no convectionsteady state. Specific conclusions from this study are drawnlow.

1. There are various features of compositional variation in hdrocarbon reservoirs depending on composition. Those featusuch as uniform composition across a long column, can be eidue to thermal diffusion or convection~that is, the permeabilityeffect!.

Fig. 18–Mole fraction of C 1 and C7¿ , pressure and saturationpressure for convection-free „left … and kÄ10 md „right … with in-creased thermal diffusion.


m

,

2-on

Re-

a-

n244on-

n-PEand

i-70hibi-

npo-

re-

er-

-

e

er-

-

2. Convection and thermal diffusion can have a substantialfect on species distribution in hydrocarbon reservoirs. The therdiffusion effect increases as the critical point approaches, whicin accord with the general theory of thermal diffusion flux.

3. The sign of the thermal diffusion coefficient for a givecomponent in multicomponent mixtures~more than two compo-nents! cannot be related to the segregation of that componenthe cold top end or the hot bottom end. Such a trend is, therefdifferent from the behavior in binary mixtures.

Nomenclature

b 5 reservoir width, L, mc 5 total molar density, mol/m3

Di jM

5 molecular diffusion coefficient, L2/t, m2/sD i j 5 Maxwell-Stefan diffusion coefficient, L2/t, m2/sDi j

05 infinite dilution diffusion coefficient, L2/t, m2/s

Dip

5 pressure diffusion coefficient, L3t/m, m2/s•PaDi

T5 thermal diffusion coefficient, L2/tK, m2/s•K

f i 5 fugacity of componenti , m/Lt2, Pag 5 acceleration of gravity, L/t2, m/s2

h 5 reservoir height, L, mJi 5 molar diffusive flux of componenti , mol/m2

•sk 5 permeability, L2, md

kTi 5 thermal diffusion ratio of componentiM 5 total molecular weight, kg/molMi 5 molecular weight of componenti , kg/mol

n 5 number of componentsn 5 normal vectorp 5 pressure, m/Lt2, Pa~bar!

Fig. 19–Mole fraction of C 1 and C7¿ , pressure and saturationpressure at xÄ5000 m; „right … with and without thermal diffu-sion and „left … with increased and unadjusted thermal diffusion.


ef-al

h is

n

t toore,

p0 5 pressure at the reference point, m/Lt2, Pa~bar!pbub 5 bubblepoint pressure, m/Lt2, barpdew 5 dewpoint pressure, m/Lt2, bar

T 5 temperature, T, KT0 5 temperature at the reference point, T, KTx 5 horizontal thermal gradient, T/L, K/mTz 5 vertical thermal gradient, T/L, K/m

t 5 time, t, secondvx 5 horizontal velocity, L/t, m/svz 5 vertical velocity, L/t, m/sv 5 velocity vector, L/t, m/sxi 5 mole fraction of componenti , dimensionless

xi0 5 mole fraction of componenti at the reference pointdimensionless

x0 ,z0 5 coordinates of the reference pointz 5 upward vertical unit vector

bxi 5 compositional coefficient of componenti , dimen-sionless

bT 5 thermal expansion coefficient, T21, K21

f 5 porosity, dimensionlessm 5 viscosity, m/Lt, kg/m•sr 5 density, m/L3, kg/m3

r0 5 density at the reference point, m/L3, kg/m3

AcknowledgmentsThis work was supported by the U.S. DOE Grant No. DE-FG296BC14850 and the members of the Research ConsortiumFractured/Layered Reservoirs of the Reservoir Engineeringsearch Institute~RERI!. Their support is greatly appreciated.

References1. Metcalfe, R.S., Vogel, J.L., and Morris, R.W.: ‘‘Compositional Gr

dients in the Anschutz Ranch East Field,’’SPERE~August 1988!1025;Trans.,AIME, 285.

2. Hamoodi, A.N., Abed, A.F., and Firoozabadi, A.: ‘‘CompositioModeling of Two-Phase Hydrocarbon Reservoirs,’’ paper SPE 36presented at the 1996 Abu Dhabi Intl. Petroleum Exhibition and Cference, Abu Dhabi, UAE, 13–16 October.

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4. Temeng, K.O., Al-Sadeg, M.J., and Al-Mulhim, W.A.: ‘‘Compostional Grading in the Ghawar Khuff Reservoirs,’’ paper SPE 492presented at the 1998 SPE Annual Technical Conference and Extion, New Orleans, 27–30 September.

5. Belery, P. and da Silva, F.V.: ‘‘Gravity and Thermal Diffusion iHydrocarbon Reservoirs,’’ Third North Sea Chalk Research Symsium, Copenhagen~1990!.

6. Jacqmin, D.: ‘‘Interaction of Natural Convection and Gravity Seggation in Oil/Gas Reservoirs,’’SPERE ~May 1990! 233; Trans.,AIME, 289.

7. Ghorayeb, K. and Firoozabadi, A.: ‘‘Molecular, Pressure, and Thmal Diffusion in Nonideal Multicomponent Mixtures,’’AIChE J.~May 2000! 46, No. 4, 883.

8. Sakonidou, E.P.et al.: ‘‘The Thermal Conductivity of an EquimolarMethane-Ethane Mixture in the Critical Region,’’J. Chem. Phys.~1998! 109, 717.

9. Haase, R.et al.: ‘‘Thermodiffusion im Kritischen Vardampfungsgebiet Ninarer Systeme,’’Z. Naturforsch. A~1971! 26A, 1224.

10. Rutherford, W.M. and Roof, J.G.: ‘‘Thermal Diffusion in Methann-Butane Mixtures in the Critical Region,’’J. Phys. Chem.~1959! 63,1506.

11. Larre, J.P., Platten, J.K., and Chavepeyer, G.: ‘‘Soret Effects in Tnary Systems Heated From Below,’’Int. J. Heat Mass Transf.~1997!40, 545.

12. Krupiczka, R. and Rotkegel, A.: ‘‘An Experimental Study of Diffusional Cross Effects in Multicomponent Mass Transfer,’’Chem. Eng.Sci. ~1997! 52, 1007.

13. Cussler, E.L.:Multicomponent Diffusion, Elsevier Science PublishingCo. Inc., New York City~1976!.


n

l

d

-.

u

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9

N

f

s

ru

r

o

o

o

l

o

m

cs;he

r

-

14. Sigmund, P.M.: ‘‘Prediction of Molecular Diffusion at ReservoConditions. Part I—Measurement and Prediction of Binary DeGas Diffusion Coefficients,’’J. Cdn. Pet. Tech.~April–June 1976!48.

15. Hayduk, W. and Loakimidos, S.: ‘‘Liquid Diffusivities in NormaParaffin Solutions,’’J. Chem. Eng. Data~1976! 21, 255.

16. Dysthe, D.K. and Hafskjold, B.: ‘‘Inter- and Intradiffusion in LiquiMixtures of Methane andn-Decane,’’Int. J. Thermophys.~1995! 16,1213.

17. Tyrell, H.J.V. and Harris, K.R.:Diffusion in Liquids, a Theoreticaland Experimental Study, Butterworths, London~1984!.

18. Kooijman, H.A. and Taylor, R.: ‘‘Estimation of Diffusion Coefficients in a Multicomponent Liquid System,’’Ind. Eng. Chem. Res~1991! 30, 1217.

19. Vignes, A.: ‘‘Diffusion in Binary Solutions,’’Ind. Eng. Chem. Fun-dam.~1966! 5, 189.

20. Firoozabadi, A., Ghorayeb, K., and Shukla, K.: ‘‘Theoretical Modof Thermal Diffusion Factors in Multicomponent Mixtures,’’AIChEJ. ~May 2000! 46, No. 4, 892.

21. El Maataoui, M.: ‘‘Consequences de la Thermodiffusion en MiliePoreux sur l’Hydrolyse des Solutions de Chlorures Ferriques etles Migrations d’Hydrocarbures dans les Me´langes den-Alcanes etdans Un Pe´trole Brut: Implications Ge´ochimiques,’’ PhD dissertationU. Paul Sabatier, Toulouse, France~1986!.

22. Schulte, A.M.: ‘‘Compositional Variations Within a HydrocarboColumn Due To Gravity,’’ paper SPE 9235 presented at the 1SPE Annual Technical Conference and Exhibition, Dallas, 21–September.

23. Montel, F. and Gouel, P.L.: ‘‘Prediction of Compositional Gradinga Reservoir Fluid Column,’’ paper SPE 14410 presented at the 1SPE Annual Technical Conference and Exhibition, Las Vegas,vada, 22–25 September.

24. Hirschberg, A.: ‘‘Role of Asphaltenes in Compositional Grading oReservoir’s Fluid Column,’’JPT ~January 1988! 89; Trans.,AIME,285.

25. Wheaton, R.J.: ‘‘Treatment of Variations of Compostion With Depin Gas-Condensate Reservoirs,’’SPERE~May 1991! 239.

26. Holt, T., Lindeberg, E., and Ratkje, S.K.: ‘‘The Effect of Gravity anTemperature Gradients on Methane Distribution in Oil Reservoirpaper SPE 11761, available from SPE, Richardson, Texas~1983!.

27. Whitson, C.H. and Belery, P.: ‘‘Compositional Gradients in Petleum Reservoirs,’’ paper SPE 28000 presented at The U. of TCentennial Pet. Eng. Symposium, Tulsa, Oklahoma, 29–31 Augu

28. Riley, M.F. and Firoozabadi, A.: ‘‘Compositional Variation in Hydrocarbon Reservoirs With Natural Convection and Diffusion,’’AIChEJ. ~1998! 44, 452.

29. Ghorayeb, K. and Firoozabadi, A.: ‘‘Features of Convection and Dfusion in Porous Media for Binary Systems,’’J. Cdn. Pet. Tech.~No-vember 2000!.

30. Jamet, Ph.et al.: ‘‘The Thermogravitational Effect in Porous MediaA Modeling Approach,’’Transp. Porous Media~1992! 9, 223.

31. Ghorayeb, K. and Firoozabadi, A.: ‘‘Numerical Study of NatuConvection and Diffusion in Fractured Porous Media,’’SPEJ~March2000! 5, 12.

32. Patankar, S.V.:Numerical Heat Transfer and Fluid Flow, Hemi-sphere, Washington~1980!.

33. Furry, W.H., Jones, R.C., and Onsager, L.: ‘‘On the Theory of IsotSeparation by Thermal Diffusion,’’Phys. Rev.~1939! 55, 1083.

34. Lorenz, M. and Emery, A.H.J.R.: ‘‘The Packed Thermal DiffusiColumn,’’ Chem. Eng. Sci.~1959! 11, 16.

35. Peng, D.Y. and Robinson, D.B.: ‘‘A New Two-Constant EquationState,’’ Ind. Eng. Chem. Fundam.~1976! 15, 59.

36. de Groot, S.R. and Mazur, P.:Non-Equilibrium Thermodynamics,North-Holland Publishing Co., Amsterdam~1962!.

37. Firoozabadi, A.: Thermodynamics of Hydrocarbon Reservoir,McGraw-Hill, New York City ~1999!.

38. Hayduk, W. and Minhas, B.S.: ‘‘Correlations for Predictions of Mlecular Diffusivities in Liquids,’’Can. J. Chem. Eng.~1982! 60, 295.

39. Younglove, B.A. and Ely, J.F.: ‘‘Thermophysical Properties of Fids. II—Methane, Ethane, Propane, Iso-Butane and Normal-ButanJ. Phys. Chem. Ref. Data~1987! 16, 577.

40. Assael, M.J.et al.: ‘‘Measurements of the Viscosity ofn-Heptane,n-Nonane, andn-Undecane at Pressure up to 70 MPa,’’Int. J. Ther-mophys.~1991! 12, 801.

41. Fenghour, A., Wakeham, W.A., and Vesovic, V.: ‘‘The ViscosityCarbon Dioxide,’’J. Phys. Chem. Ref. Data~1998! 27, 31.

42. Younglove, B.A.: ‘‘Thermophysical Properties of Fluids. 1. ArgoEthylene, para-Hydrogen, Nitrogen, Nitrogen Triluoride, and Oxgen,’’ J. Phys. Chem. Ref. Data~1982! 11, 1.


irse

el

sur

n8024

in985e-

a

th

d,’’

o-lsast.-

if-

:

al

pe

n

of

s

-

u-e,’’

f

n,y-

43. Starling, K.E.:Fluid Thermodynamics Properties for Light PetroleuSystems, Gulf Publishing Co., Houston~1973!.

44. Glasstone, S., Laidler, K.J., and Eyring, H.:The Theory of Rate Pro-cesses, McGraw-Hill Book Co., New York City~1941!.

Appendix A–Diffusion FluxGhorayeb and Firoozabadi7 used: ~1! the entropy productionexpression;36 ~2! phenomenological laws of the thermodynamiof the irreversible processes;~3! Onsager’s reciprocal relationsand~4! equilibrium thermodynamics at a local level to derive tmolar diffusion flux based on the molar average velocityJ5(J1 , . . . , Jn21) in a nonideal fluid mixture ofn components.The flux reads

J52c~DM•“x1DT

•“T1Dp•“p!. ~A-1!

In the above equation

DM[@Di jM#, i , j 51, . . . ,n21,

DT[@DiT#, i 51, . . . ,n21,

Dp[@Dip#, i 51, . . . ,n21,

“x[@“xi #, i 51, . . . ,n21.

The molecular diffusion coefficientsDi jM , the thermal diffusion

coefficientsDiT , and the pressure diffusion coefficientsDi

p , areexpressed by

Di jM5ainDin

M ixi

Lii(k51

n21

Lik (l 51

n21Mlxl1Mnxnd lk

M l

] ln f l

]xjU

xj ,T,P

,

i , j 51, . . . ,n21, ~A-2!

DiT5ainDinM

kTi

T, i 51, . . . ,n21, ~A-3!

Dip5ainDin

M ixi

RTLii(k51

n21

LikF(j 51

n21

xj Vj1Mnxn

MkVk2

1

cGi 51, . . . ,n21, ~A-4!

respectively. In the above equations,Li j , Mi , M , Vi , f i , andRare the phenomenological coefficients (Li j 5L ji ),

36 the molecularweight of componenti , the total molar weight, the partial molavolume of componenti , the fugacity of componenti , and the gasconstant, respectively. The subscriptxj denotes xj

[(x1 , . . . ,xj 21 ,xj 11 , . . . ,xn21). Vi and f i can be obtained using an equation of state.37 The coefficientsain , Din , and kTi~thermal diffusion ratio of componenti ! are given by7

ain5MiMn

M2 , i 51, . . . ,n21, ~A-5!

Din5M2RLii

cMi2Mn

2xixn, i 51, . . . ,n21, ~A-6!

kTi5MixiMnxnLiq8

MRTLii, i 51, . . . ,n21. ~A-7!

Let

c15M1x11M3x3

M1

] ln f 1

]x2U

P,T,x1

1x2

] ln f 2

]x2U

P,T,x1

,

c25x1

] ln f 1

]x2U

P,T,x1

1M2x21M3x3

M2

] ln f 2

]x2U

P,T,x1

,

c35M1x11M3x3

M1

] ln f 1

]x1U

P,T,x2

1x2

] ln f 2

]x1U

P,T,x2

,

c45x1

] ln f 1

]x1U

P,T,x2

1M2x21M3x3

M2

] ln f 2

]x1U

P,T,x2

.


l

o

l

ef

the

e

nf-en-tedgan

t bi-

reliffu-

For a ternary mixture, Eqs. A-2 to A-4 then read

D11M5a13D13M1x1S c31c4

L12

L11D , ~A-8!


L12

L11D , ~A-9!


L21

L22D , ~A-10!


L21

L22D , ~A-11!

D1T5a13D13M

kT1

T, ~A-12!

D2T5a23D23M

kT2

T, ~A-13!

D1p5a13D13

M1x1

RT FM1x11M3x3

M1V11x2V22

1

c

1S M2x21M3x3

M2V21x1V12

1

cD L12

L11G , ~A-14!

D2p5a23D23

M2x2

RT FM2x21M3x3

M2V21x1V12

1

c

1S M1x11M3x3

M1V11x2V22

1

cD L21

L22G . ~A-15!

From Eqs. A-8 to A-11 the phenomenological coefficients re~provided thatc2c32c1c4Þ0!

L115cM3x3

R~c2c32c1c4!~c2D11

M2c4D12M !, ~A-16!

L1252cM3x3

R~c2c32c1c4!~c1D11

M2c3D12M !, ~A-17!

L215cM3x3

R~c2c32c1c4!~c2D21

M2c4D22M !, ~A-18!

L2252cM3x3

R~c2c32c1c4!~c1D21

M2c3D22M !. ~A-19!

Suppose the molecular diffusion coefficients are known, sayexperimental measurements or from a correlation~see AppendixB for the correlation used in this work!; the phenomenologicacoefficients can be obtained using Eqs. A-16 to A-19. Note tc2c32c1c4 ~appearing in Eqs. A-16 to A-19! vanishes at the criti-cal point because of the criticality condition.37 Once the phenom-enological coefficients are calculated, the pressure diffusion cficients are readily obtained from Eqs. A-14 and A-15.

Appendix B–Molecular Diffusion CoefficientsKooijman and Taylor18 proposed a correlation for the MaxwelStefan diffusion coefficientsD i j for multicomponent mixturesbased on a generalization of the Vignes19 equation for binary sys-tems. For a mixture ofn components, this correlation reads18

D i j 5~D i j0 !xj~D j i

0 !xi )k51kÞ i , j

n

~D ik0 D jk

0 !xk/2, i , j 51, . . . ,n21,

~B-1!

where D i j0 are the infinite dilution diffusion coefficients. Th

Maxwell-Stefan and the Fickian expressions for the molar difsive fluxes for nonideal mixtures are

J52cB21•G•“x ~B-2!

and


ad

by

hat

ef-

-

u-

J52cDM•“x, ~B-3!

respectively.18 In the above equationsB is a square matrix~ordern21! with elements given by

Bii 5xi

D in

1(k51iÞk

nxk

D ik

, ~B-4!

Bi j 52xiS 1

D i j

21

D inD , ~B-5!

andG is the matrix of thermodynamic factors with elements

G i j 5xi

] ln f i

]xjU

xj ,T,P

. ~B-6!

Thus, the Maxwell-Stefan diffusion coefficients are related toFickian diffusion coefficients by the following relationship:

DM5B21•G. ~B-7!

In this work we use the correlation of Hayduk and Minhas38 toestimate the infinite dilution diffusion coefficientsD i j ; this cor-relation reads

D i j0 513.3~1028!T1.47h j

(10.2/Vi20.791)Vi20.71,

i , j 51, . . . ,n, iÞ j ~B-8!

whereh j and Vi are the viscosity and the molar volume at thnormal boiling point of componenti , respectively.

Eqs. B-1 and B-8 allow the prediction of the Maxwell-Stefadiffusion coefficients. Following this calculation, the Fickian difusion coefficients can be obtained using Eq. B-7. The molar dsity, the partial molar volumes, and the fugacities are calculausing the PR EOS.35 We use the viscosity and the normal boilinpoint data reported in Refs. 39 through 43. We also performedindependent evaluation of the above correlation for the recennary diffusion coefficients data for the C1 /nC10 mixture at highpressures and temperatures16 with satisfactory results.

Appendix C–Thermal Diffusion CoefficientsThermal diffusion coefficients in multicomponent mixtures westudied by Firoozabadiet al.20 Here, we summarize the modereported by the authors for the sake of completeness. The dsion flux Ji of the i th component in a mixture ofn componentscan be written in the two following forms:

Ji52Liq8“T

T2 21

T (k51

n21

Lik“TS mk

Mk2

mn

MnD ,

i 51, . . . ,n21, ~C-1!

Ji521

T (k51

n21

LikF S Qk*

Mk2

Qn*

MnD “T

T1“TS mk

Mk2

mn

MnD G ,

i 51, . . . ,n21, ~C-2!

whereQk* is the net heat of transport of componentk, mk is thechemical potential of componentk, andLiq8 andLik are the phe-nomenological coefficients. Eqs. C-1 and C-2 imply

Liq8 5(k51

n21

LikS Qk*

Mk2

Qn*

MnD , i 51, . . . ,n21. ~C-3!

Let

L[@Li j #[Li j , i , j 51, . . . ,n21,

Lq[@Liq8 # i 51, . . . ,n21,

Q[FQk*

Mk2

Qn*

MnG , i 51, . . . ,n21.

Eq. C-3 can be then written in a compact form as


m

.

Lq5L "Q. ~C-4!

Let

D[@ainDind i j # i , j 51, . . . ,n21,

F[F ] ln f i

]xjU

xj ,T,PG i , j 51, . . . ,n21,

M[FMixi

Liid i j G i , j 51, . . . ,n21,

W[FM jxj1Mnxnd i j

M jG i , j 51, . . . ,n21,

KT[@kTi# i 51, . . . ,n21.

In the above equations,d i j denotes the Kronecker delta. UsinEqs. A-1 to A-4,DM andLq read

DM5D"M "L "W"F ~C-5!

and

Lq5MRT

MnxnM21

•KT , ~C-6!

respectively. Eq. C-6 can be written as

kTi5

MixiMnxn

MRTLii(j 51

n21

Li j S Qj*

M j2

Qn*

MnD , i 51, . . . ,n21. ~C-7!

The net heat of transport for thei th component in ann-componentmixture has been estimated using

Qi* 52DU i

t i

1F(j 51

nxjDU j

t jG Vi

( j 51n xj Vj

, i 51, . . . ,n, ~C-8!

whereDU i is the partial molar internal energy departure of coponenti , t i5DUi

vap/DUiv is ; DUi

vap andDUiv is are the energy of

vaporization and the energy of viscous flow, respectively.44

From Eq. C-5, one obtains the expression relating the phenenological coefficients matrixL to the molecular diffusion coef-ficients matrixDM:7


g

-

om-

L5~D"M !21•DM

•~W"F!21; ~C-9!

Eq. C-9 can be written as7

(l 51

n21

(k51

n21Mkxk1Mnxnd lk

Mk

] ln f k

]xjU

xj ,T,P

Lli 5rMnxn

RDi j

M ,

i , j 51, . . . ,n21. ~C-10!

Thus, havingDM and Q, DT can be calculated readily from EqA-3.

SI Metric Conversion Factorsbar 3 1.0* E105 5 Pa

*Conversion factor is exact. SPEJ

Kassem Ghorayeb is a scientist at the Reservoir EngineeringResearch Inst. (RERI) in Palo Alto, California. e-mail:[email protected]. His research interests include numericalsimulation and mathematical modeling of compositionalvariation and fluid flow in homogeneous and fractured porousmedia. Ghorayeb holds BS, MS, and PhD degrees in fluid me-chanics from the U. of Toulouse, France. Abbas Firoozabadiis a senior scientist and director at RERI in Palo Alto andteaches at Imperial College in London. e-mail: [email protected] previously taught at the U. of Texas, Austin, and at StanfordU. His research interests include equilibrium, nonequilibrium,and irreversible thermodynamics; multiphase flow in fracturedand layered permeable media; and tidal-force-influencedpressure transients for estimation of key parameters of frac-tured reservoirs. Firoozabadi holds a BS degree from AbadanInst. of Technology, Iran, and MS and PhD degrees from IllinoisInst. of Technology, all in gas engineering. He has served onthe Editorial Review Committee since 1986 and was a 1988–89member and 1992–93 chairman of the Forum Series in NorthAmerica Steering Committee, a 1991–92 member of the West-ern Regional Meeting Program Committee, a 1987–89 mem-ber of the Reservoir Simulation Symposium Program Commit-tee, and a 1992–96 Short Course Instructor.


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