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New Three-Phase Multicomponent Compositional Model forAsphaltene Precipitation during CO2 Injection Using CPA-EOSHadi Nasrabadi,*,† Joachim Moortgat,‡ and Abbas Firoozabadi§,∥

†Petroleum Engineering, Texas A&M University, College Station, Texas 77843, United States‡School of Earth Sciences, The Ohio State University, Columbus, Ohio 43210, United States§Reservoir Engineering Research Institute (RERI), Palo Alto, California 94301, United States∥Chemical and Environmental Engineering, Yale University, New Haven, Connecticut 06511, United States

ABSTRACT: Numerical modeling of asphaltene precipitation in petroleum reservoirs is important in relation to possibleprecipitation around the wellbore in the producing well. Production from some reservoirs results in asphaltene precipitation inthe wellbore region, leading to productivity loss and need for cleanup. Fluid injection even when there is asphaltene precipitationmay not lead to injectivity loss. There are desirable processes in which precipitation of asphaltenes can lead to “in situ” upgradingof heavy oil recovery. Reservoir compositional models that are currently in use rely on cubic equations of state for asphalteneprecipitation. The cubic equations, despite their relative reliability in describing reservoir fluids’ phase behavior, becomeunreliable in asphaltene-rich phase description. A number of noncubic equations of state have been introduced to overcome theshortcomings of cubic equations. The cubic-plus-association equation of state (CPA-EOS) is perhaps the method of choicein representing asphaltenes in compositional modeling. When the hydrocarbon fluids do not contain asphaltenes, CPA-EOSreduces to the standard cubic equation. In this work, we implement CPA-EOS in compositional modeling and introduce a simpletechnique to speed up considerably the root finding of the CPA-EOS. Our efficient algorithm reduces significantly the additionalcomputational cost from the incorporation of the CPA-EOS. We also derive the basic equations for the total compressibility andtotal partial molar volume in our implementation of the CPA-EOS compositional modeling. We present three numericalexamples for CO2 injection in 2D and 3D domains saturated with Weyburn oil and show results of asphaltene-rich phasesaturation among other predictions. This work introduces a general framework for widespread use of CPA-EOS in compositionalmodeling in three-phase flows of gas, light liquid, and asphaltene-rich phases.

■ INTRODUCTION

Asphaltenes are the most polar fraction of crude oil. Their exactchemical structure remains unknown. It is generally accepted thatasphaltenes consist of polar polyaromatic material which maycontain various metals, oxygen, sulfur, and nitrogen. Asphaltenesare operationally defined as the fraction of crude oil insoluble innormal alkanes such as n-pentane but soluble in aromatics such asbenzene and toluene at room temperature.1

Changes in reservoir pressure (e.g., during primary depletionof highly undersaturated reservoirs) and oil composition (e.g.,due to CO2 injection) can lead to precipitation of an asphaltenephase. Asphaltene precipitation can affect flow in the wellbore inthe production well and in the reservoir near the well due topermeability reduction and alteration of rock wettability fromwater-wet to oil-wet.2−4

Despite numerous publications and advances, we are notaware of a three-phase compositional simulator that can treat theasphaltene phase as an asphaltene-rich liquid phase. On the phasebehavior side, examples of models for asphaltene precipitationare solubility theory,5−13 cubic equations of state (EOS),14−18

colloidal theory,19 micellization theory,20−22 McMillan−Mayer−SAFT,23−25 and perturbed-chain-SAFT.26−28

Compositional reservoir simulators are primarily based oncubic EOS to determine the phase separation of hydrocarbonfluids. The EOSmethod has advantages over solubility theory formodeling asphaltene precipitation. Current compositional simu-lators mainly use the cubic EOS (e.g., Nghiem et al.16). This is

motivated by its compatibility with the cubic EOS frameworkthat is widely used for nonasphaltene-related modeling.However, this class of models cannot describe realistic polar−polar interactions relevant to asphaltenes and other polarcomponents, which may obscure the nature of asphalteneprecipitation and stabilization.29 Some of the models take intoaccount the interactions using multiple complex equations oradjustable parameters.30 These models have high computationalcost, which may make their implementation in current com-positional simulators unattractive.Li and Firoozabadi31 proposed a cubic-plus-association

equation of state (CPA-EOS) to study asphaltene precipitationin model solutions (asphaltene + toluene) and heavy oils fromthe addition of n-alkanes. The physical interactions are describedby the Peng−Robinson equation of state.32 The polar−polarinteractions between asphaltene molecules and betweenasphaltene and aromatics/resins (or toluene) molecules aredescribed by the thermodynamic perturbation theory. Withthis model, they reproduced the experiments for the amountof asphaltene precipitation by different n-alkanes from modelsolutions and seven heavy oils and bitumens over a wide range oftemperatures, pressures, and compositions through adjustingonly a single parameter. This parameter is the cross-association

Received: December 21, 2015Revised: February 17, 2016Published: February 19, 2016

Article

pubs.acs.org/EF

© 2016 American Chemical Society 3306 DOI: 10.1021/acs.energyfuels.5b02944Energy Fuels 2016, 30, 3306−3319

energy between asphaltene and aromatics/resins (or toluene)molecules, which depends on the types of asphaltene and n-alkane, and temperature but is independent of pressure andconcentration. In a recent investigation Jindrova et al.33 haveshown that while the PR-EOS performs well for solubility of CO2and light alkanes in bitumens, when a second liquid phasewith high asphaltene content forms, then the PR-EOS becomesunreliable and the CPA-EOS describes the phase behavior invapor−liquid−liquid equilibria. Zhang et al.34 compared theperformance of the CPA-EOS31 with the PC-SAFT for asphaltenemodeling in six live oils. Their results show that CPA-EOS ismore accurate compared to PC-SAFT for modeling asphalteneprecipitation over a wide pressure and temperature range.On reservoir simulation, Mohebbinia et al.30 implemented

PC-SAFT equation of state in a compositional simulation modelfor predicting asphaltene precipitation during gas injection. As awhole it is desirable to keep the cubic equations of state incompostional modeling due to its simplicity and superiority overthe noncubic equations of state and add additional terms whennecessary to describe the physics that is not included in the cubicequations of state. This is the basis of our desire to include theCPA-EOS in our work.In this work, we present the mathematical formulation and

an efficient numerical implementation of a three-phase multi-component compositional model based on the CPA-EOS. Thework is organized as follows: we first present the mathematicalmodel governing three-phase compositional flow in porousmedia and subsequently discuss our numerical scheme basedon the mixed finite element and higher-order discontinuousGalerkin methods. We present three examples on the use of ourmodel for simulation of asphaltene precipitation due to CO2injection in an oil reservoir. We end the work with concludingremarks.

■ MATHEMATICAL MODELAssumptions. In this work, we neglect the capillary pressure due to

low interfacial tension at the conditions of our study. We also ignoreFickian diffusion and mechanical dispersion. Diffusion effect may benegligible in gas injection in unfractured porous media,35 and diffusioncan be very slow in an asphaltene-rich phase. In addition, to the best ofour knowledge, diffusion coefficients have not been measured for CO2−asphaltene mixtures. Mechanical dispersion is generally dominant athigh flow rates so we believe, at the conditions of our study, it can beneglected. A recent study36 shows that for two- and three-phase flow attypical reservoir injection rates, even in the context of viscous and gravi-tational flow instabilities, the effects of Fickian diffusion and mechanicaldispersion are low (because Peclet numbers are generally high).We assume three phases in our model: (1) asphaltene-rich liquid

phase, (2) hydrocarbon-rich liquid phase, and (3) gas phase. We alsoassume that the reservoir fluid consists of a number of pure hydrocarbonand nonhydrocarbon components (e.g., CO2, N2, C1, C2, C3, iC4, nC4,iC5, and nC5), pseudohydrocarbon components (defined by lumping anumber of hydrocarbon components within a certain range in normalboiling points), and a hydrocarbon residue (Cn+). In this work, thehydrocarbon residue is split into two components: (1) the “heavy”component that contains the heavy alkanes, the heavy aromatics, and theresins and (2) asphaltene.29

The governing equations for isothermal compositional three-phaseflow include species-balance equations, Darcy’s law, and thermodynamicequilibrium between the phases.Species-Balance Equations. The species balance for component i

is given by

∑ϕ ∂∂

+ ∇· = =α

α α α=

czt

c x F i nu( ) , 1, ...,ii i c

a o g, ,,

(1)

Fi may represent the injection and production wells.The compositions zi and xi,α are constrained by

∑ ∑= =α= =

z x 1i

n

ii

n

i1 1

,

c c

(2)

Darcy’s Law. The velocity for each phase (uα, α = a, o, g) is given byDarcy’s law:

μρ= ∇ −α

α

αα

kpu

kg( )r

(3)

The viscosities of the asphaltene-rich liquid phase, hydrocarbon-richliquid phase, and gas phase are a function of temperature, pressure, andphase composition.We use the corresponding-state model37 to calculateviscosity. Relative permeabilities are functions of phase saturationsand are computed using Stone’s38 formulation (see further details inNumerical Results).

Pressure Equation. Based on the concept of volume balance,39,40

we use the following pressure equation:

∑ ∑ϕ ∂∂

+ ∇· − =α

α α α= =

cpt

v c x Fu[ ( ) ] 0fi

n

i i ia o g1 , ,

,

c

(4)

Thermodynamic Equilibrium. Thermodynamic equilibrium re-quires the fugacities in the three phases to be equal. We also use phasestability analysis to guarantee the global minimum of Gibbs free energy.In this work, we model asphaltene precipitation as a traditional liquid−liquid or gas−liquid−liquid phase separation. We define equilibriumratios as Ki,g = xi,g/xi,o and Ki,a = xi,a/xi,o. From the equality of fugacities,we have

φ φ α= − = =α αK i n g aln ln ln , 1, ..., ; ,i i o i c, , , (5)

We also have the following constraint for the molar fractions of eachphase (βα):

∑ β =α

α=

1g o a, , (6)

where βα is related to the overall mole fraction of component i (zi) by

∑ β= =α

α α=

z x i n, 1, ...,ig o a

i c, ,

,(7)

The mass density and molar density of all three phases are obtainedfrom

∑ρ α= =α α α=

c x M g o a, , ,i

n

i i1

,

c

(8)

α= =αα

cp

Z RTg o a, , ,

(9)

In this work,Zα for all the three phases is obtained from the CPA-EOSto take into account the polar−polar interaction (self-association andcross-association) between asphaltene and “heavy” components. Alsothe calculation of the phase fugacities (in eq 5), the total fluidcompressibility, and the total partial molar volume (in eq 4) are based onCPA-EOS. Calculation of the compressibility factor in the CPA-EOS isan important aspect of this work. Our experience shows that efficientand robust calculation of this parameter has direct effect on overallperformance of the model. We discuss the calculation steps of thisparameter later.

Boundary/Initial Conditions. Appropriate boundary and initialconditions are required to complete the multicomponent three-phaseflow formulation. We apply impermeable boundaries at all sides of thedomain. We consider a constant oil pressure in the production well(Dirichlet boundary condition) and constant rate in the injection well(Neumann boundary condition).

Energy & Fuels Article

DOI: 10.1021/acs.energyfuels.5b02944Energy Fuels 2016, 30, 3306−3319

3307

■ NUMERICAL MODEL

We solve the species-balance and flow equations using animplicit-pressure−explicit-composition (IMPEC) scheme. Wefirst calculate the pressure and overall composition and thenperform phase equilibrium calculations to obtain phase satu-rations and the species distribution in each phase. We use the mixedfinite element (MFE)method to solve the pressure equation (eq 4)and the discontinuous Galerkin (DG) method to solve the species-balance equation (eq 1). Hoteit and Firoozabadi41−43 providedetailed description of the combined MFE-DG numerical model.This numerical model has been successfully applied and tested forthree-phase compositional flow.44−47 These studies demonstratepowerful features of this combination such as reduction in numericaldispersion and grid orientation effects. In this work, we use theefficient and robust three-phase stability analysis and phase-splitroutines presented by Li and Firoozabadi.48

The main new features of our compositional model arediscussed in the following:Asphaltene-Rich Liquid Phase. While the current

compositional simulation models for asphaltene precipitationassume that the precipitated phase is composed of pureasphaltene, in this work, we assume the precipitated phase toconsist of various species including the asphaltenes anddetermine the composition based on three-phase equilibriumcomputations. Figure 1 provides the composition of an

asphaltene-rich phase revealing the presence of variousspecies. The density, viscosity, and other properties of thisphase for use in the compositional simulation are partof the computations. The fact that the asphaltene phase is notpure is supported by theoretical analysis and experimentaldata.49−51

Compressibility Factor. Traditionally in compositionalsimulation the compressibility factor of each phase (Zα) iscalculated using a cubic equation of state. For example, toobtain Zα from PR-EOS, one needs to solve the followingequation:

α=−

−+ −

=αα

α α

α α

α α α αZ

ZZ B

A ZZ B Z B

g o a2

, ,2 2

(10)

where Aα and Bα are functions of pressure, temperature,respective phase compositions, binary interaction parameters,the critical properties, and acentric factors of pure species.1

In this work, to take into account the polar−polar interac-tion between asphaltene and resin molecules and betweenasphaltene molecules themselves, we use CPA-EOS toobtain phase and volumetric properties. In CPA-EOS, eq 10changes to

η ηη η

χ χ

α

=−

−+ −

++ −− +

− + −

=

αα

α α

α α

α α α α

α α

α αα α α α

ZZ

Z BA Z

Z B Z B

x x

g o a

2

4 4 2

2 3[ ( 1) ( 1)]

, ,

a a r r

2 2

2

2 , , , ,

(11)

where ηα = Bα/4Zα. The subscripts “a” and “r” represent theasphaltenes and resins/aromatics, respectively. χa,α and χr,α arethe mole fractions of asphaltenes and resins/aromatics in phaseα not bonded at one of the association sites, respectively. In thiswork, we assume association bonding always occurs with anasphaltene molecule on one side and the other side can beeither an asphaltene or aromatics/resins molecule.29 Withoutthe asphaltenes CPA-EOS (eq 11) reduces to the originalPR-EOS (eq 10). χa,α and χr,α in eq 11 are given by29

χχ χ

α=+ Δ + Δ

=αα

α α α α α

ZZ Nx N x

g o a, ,aa a a

aar r r

ar,, , , ,

(12)

χχ

α=+ Δ

=αα

α α α

ZZ Nx

g o a, ,ra a a

ar,, , (13)

where Δij = [1 − 0.5ηα/(1 − ηα)3]kijbij[exp(εij/kbT) − 1] (i = a,

j = a or r) represents the “association strength”. bij = bi + bj/2,,where bk = 0.0778 RTck/pck.

1 kb is the Boltzmann constant. Na isthe number of identical association sites that each asphaltenemolecule has. Nr represents the same parameter for resins/aromatics molecule.A robust algorithm to find the compressibility factor in the

CPA-EOS is through the bisection method which guaranteesfinding the solution. It is, however, computationally expensive,especially for application in compositional simulation wherethe compressibility factor needs to be calculated billions oftimes. Moortgat et al.45 present an efficient modification of thisalgorithm based on the Newton method and using the Z-factorfrom the previous time step as the initial guess. However, theyconsider the CPA-EOS only for the aqueous phase, which mayhave only a small fraction of CO2. This is why, in their work, theinitial guess from a previous time step or water density is goodenough.In this work, we used an efficient algorithm based on the

classical Newton method where the initial guess is obtained fromPR-EOS (eq 10).52 In our algorithm, we first find the “correct”root of the PR-EOS based on the minimization of the Gibbs freeenergy.53 Then, this root will be used as an initial guess for theNewton method. In the CPA-EOS, the compressibility factorcomprises physical and association contributions:53

= +Z Z Zphysical association (14)

The “correct” root of the cubic EOS (based on the Gibbs freeenergy minimization) calculates the physical contribution.Therefore, it is relatively close to the final solution that takesinto account both the physical and association contributions. Inthe gas phase, the association part is zero since no asphaltene ispresent. In the hydrocarbon-rich phase (with very lowcomposition of the asphaltene and resins), the association partis small compared to the physical part and the Newton methodconverges in a few iterations. In the asphaltene-rich phase, theassociation part of the compressibility factor can be signifi-cant (due to relatively high asphaltene concentration). In ourpresented examples, even for the asphaltene-rich phase the

Figure 1. Composition of asphaltene-rich liquid phase50



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“correct” root of the PR-EOS provides a good initial guess in thevast majority of the conditions. In the rare occasions in which theNewtonmethod diverges, we switch to the bisectionmethod thatalways converges but at a relatively high computational cost. Inthe examples presented in this work, when the bisection methodis used for the asphaltene-rich phase, we find only one root forthe compressibility factor in the physical range (Z ≥ B). Oursolution for the case in which the bisection method finds morethan one root for the CPA-EOS is to calculate the Gibbs freeenergy for each root and select the one that has a minimumGibbsfree energy. Using the combined Newton−bisection method, we

improve the efficiency for the calculation of the compressibilityfactor by 2 orders of magnitude.The above modifications in the CPA-EOS are performed at a

comparable efficiency with similar calculations using a cubicEOS. The CPU time for one time step using the CPA-EOS(in the examples presented below) is about 3.7 times that of theCPU time for one time step in the PR-EOS when asphaltenesprecipitate throughout most of the domain.

Three-Phase Total Compressibility and Partial MolarVolume. Two important parameters in the pressure equation(eq 4) are three-phase total compressibility (cf) and partial molarvolume ( vi). In this work, we present the calculation of theseterms using the CPA-EOS. The details of the derivations arepresented in Appendices A−D for completeness.

Formation Damage due to Asphaltene Precipitation.Despite several experimental54−57 and modeling58−60 studies on

Table 1. Fluid Composition, Critical Properties, and Other Relevant Data for the Weyburn Oil29,61,62

components composition (mole fraction) critical temp (K) critical pressure (bar) Mw (g/mol) acentric factor

CO2 0.0058 304.14 73.75 44.01 0.2390N2/H2S/C1 0.0575 191.83 47.08 20.66 0.0245C2−C3 0.0774 350.44 44.36 39.89 0.1368C4−C5 0.0619 448.30 35.16 66.72 0.2236heavy fraction 0.7924 746.22 15.91 220.1 1.0038asphaltene 0.0050 1474.0 6.342 1800.0 2.0000

Table 2. Non-zero Binary Interaction Coefficients for theWeyburn Oil29,63 (Mw, Molecular Weight, g/mol)

CO2−HC C1−HC

0.071 0.0289 + 1.633 × 10−4Mw

Figure 2. (a) Bubble-point pressure at 332 K and (b) fraction of precipitated asphaltenes at 332 K and 160 bar as a function of CO2 mole fraction for theWeyburn oil. (Reprinted with permission from ref 29. Copyright 2010 American Chemical Society). Symbols are experimental values,64 and linesrepresent calculations by the CPA-EOS.

Figure 3. Relative permeability curves for (a) asphaltene-rich liquid/hydrocarbon-rich liquid two-phase system and (b) gas/hydrocarbon-rich liquidtwo-phase system.



3309

the formation damage caused by asphaltene precipitation, thereis significant uncertainty on its underlying mechanisms andquantification. The developed models primarily depend onphenomenological parameters that often have no clear relation tounderlying physics. These parameters are therefore difficult toestimate theoretically and are obtained by fitting experimental

data. In this work, we use a simple model to take into account theformation damage due to asphaltene precipitation. We alter therelative permeabilities of the hydrocarbon-rich liquid phase andthe gas phase based on the saturation of the asphaltene-richliquid phase. More details of this model are discussed in thefollowing section.

Figure 4. Overall CO2 composition (zCO2), asphaltene-rich liquid saturation (Sa), gas saturation (Sg), asphaltene composition in the asphaltene-rich

phase (xasp.,a), and oil viscosity (μo) at different PVI: 2D horizontal domain.



3310

■ NUMERICAL RESULTS

In this section, we present the following three examples using thenew compositional model based on the CPA-EOS for asphalteneprecipitation from CO2 injection in an oil reservoir:Example 1: 2D Horizontal Domain. In the first example,

we consider a 2D horizontal 100 m × 100 m domain. Thereservoir temperature is 332 K, and the initial pressure is 160 bar.CO2 is injected at a constant rate of 0.1 PV/year at the lowerleft corner. The producer is located at the opposite corner.The pressure at the producing well is set equal to 160 bar. Theporosity is 20%, and the permeability is 100 md. We use 32 × 32grids of dimensions 3.125 m × 3.125 m.We select the Weyburn oil as the reservoir fluid in this

example. Weyburn oil contains a very small amount of lightcomponents. Experimental data by Srivastava et al.64 showthat CO2 injection in this oil leads to asphaltene precipitation.Table 1 lists the fluid composition and the EOS parameters usedin the phase behavior calculations. The binary interactioncoefficients (BICs) between CO2 and hydrocarbons andbetween C1 and other hydrocarbons are listed in Table 2.Other BICs are set to zero. In Figure 2a, the bubble-pointpressure at 332 K is presented for different CO2/oil mixtures.Figure 2b shows the fraction of precipitated asphaltenes at 332 Kand 160 bar as a function of overall concentration of CO2 in themixture. As expected, as the CO2 overall concentration increases,asphaltene precipitation becomes more pronounced. However,at very high CO2 concentration, the asphaltene phase volume candecrease because of the appearance of a gas phase. This isreflected by the slope decrease in Figure 2b. The CPA-EOScalculations are in good agreement with the measurements.29

The relevant parameters for the CPA-EOS (eqs 12 and 13)are Na = Nr = 4, kaa = kar = 0.01, εaa/kb = 2000 K, and εar/kb =500 K.In this example, we have three phases: (1) gas phase, (2)

hydrocarbon-rich liquid phase, and (3) asphaltene-rich liquidphase. We use Stone’s38 correlations to model relative per-meability of these phases. All two-phase relative permeabilitycurves have a power of 2.0 except for the asphaltene-rich liquidphase with a power of 3.0 (Figure 3). The end-point relativepermeability for hydrocarbon-rich liquid with the asphaltene-richliquid is 0.69. The end-point relative permeability for hydro-carbon-rich liquid with gas is 0.95. The asphaltene-rich liquidhas an end-point relative permeability of 0.4, and the end-pointrelative permeability of gas is 0.63. We assume a critical gassaturation of 2%. The residual saturation for the asphaltene-richliquid is 40%. The residual saturation of the hydrocarbon-richliquid to both gas and asphaltene-rich liquid is assumed to be10%.Figure 4 presents the profiles for overall CO2 composition,

asphaltene-rich liquid saturation, gas saturation, asphaltene com-position (in asphaltene-rich liquid phase), and hydrocarbon-richliquid viscosity at 0.1, 0.3, and 0.7 pore volumes injected (PVI).This figure shows the CO2 composition path in the injectionprocess. CO2 causes various phase behavior effects upon mixingwith the crude oil. While the original reservoir fluid mixed with ahigh fraction of CO2 stays above the bubble-point pressure(Figure 1a), a CO2 composition above a mole fraction of 0.45(Figure 1b) leads to the formation of an asphaltene-rich liquidphase. Continuous increase in CO2 composition due to furthermixing can lead to formation of a gas phase. In this example,the gas phase forms at overall CO2 molar fractions aboveapproximately 0.73. The gas phase has a CO2 mole fraction of

more than 0.94. One expects to see high gas saturation (due toevaporation of components up to C30 in the gas phase) near theinjection well and a three-phase gas/hydrocarbon-rich liquid/asphaltene-rich liquid mixture at the mixing zone of CO2 andcrude oil. Next, we further clarify the evolution of the three-phases as CO2 injection progresses. This figure also shows that,during CO2 injection, the asphaltene-rich liquid and gas phasesform at the mixing zone of CO2 and crude oil. The asphaltene-rich liquid phase develops across a relatively large fraction of thereservoir. During CO2 injection, the asphaltene-rich liquid phasefirst forms near the injection well as CO2 concentration increasesquickly to above the 0.45 mole fraction threshold. However, asthe CO2 concentration increases near the injection well, thesaturation of the asphaltene-rich liquid phase becomes relativelyconstant due to the formation of the gas phase and disappearanceof the hydrocarbon-rich liquid phase. As CO2 injection continuesthe saturation of gas phase increases. There is also evaporation oflighter components in the asphaltene-rich liquid phase that leadsto higher concentration of asphaltene in this phase near theinjector (Figure 4).In CO2 injection, an area with high saturation of asphaltene-

rich liquid phase develops near the producer (Figure 4) from thecontinuous flow of the asphaltene-rich phase. The followingexamples show this phenomenon more clearly. Figure 4 showsthat the asphaltene-rich liquid saturation stays below the residualsaturation (40%) and therefore is immobile. The appearance ofthis phase is solely due to phase behavior effects. The molarconcentration of asphaltene in the asphaltene-rich liquid phase(xasp.,a) is shown in Figure 4. The highest mole fraction ofasphaltene in this phase is 0.38. The initial oil viscosity is 3.0 cP.Dissolution of CO2 in the crude and reduction in its asphaltenecontent result in significant reduction in the oil viscosity.The oil recovery versus PVI and bottomhole pressure at the

injector are shown in Figure 5. The oil recovery at breakthrough(0.6 PVI) is 53%. The recovery increases to 60% at 2.0 PVI. Thebottomhole pressure at the injector drops after CO2 break-through.

Example 2: 2D Vertical Domain. In this example, weconsider a 2D 100 m × 100 m vertical cross-section of thereservoir. The fluid and reservoir properties are set the same asthe previous example. CO2 is injected at the top-right corner ofthe domain and oil is produced at the bottom-left corner. We usea uniform initial composition (Table 1) and hydrostatic pressurevariation (with pressure of 160 bar at the bottom of theformation) for the initial conditions.Unlike the previous example, gravity affects the CO2 flow path.

Figure 6 shows that most of the injected CO2 stays at the top ofthe formation at early times. Similar to the previous example, theasphaltene precipitation occurs throughout a significant portion

Figure 5. Oil recovery (dashed) and injector bottomhole pressure(solid) vs PVI: 2D horizontal domain.



3311

of the reservoir due to CO2 dissolution. However, the saturationof the asphaltene-rich liquid phase stays low except near theproduction well (Figure 6). At 0.7 PVI, the asphaltene-rich liquidsaturation reaches values as high as 20% near the productionwell. Similar to the previous example, the asphaltene-richliquid saturation stays low (less than 4%) near the injection

well. Figure 6 also presents the gas saturation, composition ofasphaltene in asphaltene-rich liquid phase, and oil viscosity atdifferent PVI. Similar trends as in the previous example areobserved for these parameters.Figure 7 presents the oil recovery and bottomhole pressure at

the injection well versus PVI. The CO2 breakthrough occurs at

Figure 6. Overall CO2 composition (zCO2), asphaltene-rich liquid saturation (Sa), gas saturation (Sg), asphaltene composition in the asphaltene-rich

phase (xasp.,a), and oil viscosity (μo) at different PVI: 2D vertical domain.



3312

0.46 PVI with approximately 40% oil recovery. The recoveryfactor increases to roughly 60% after 2.0 PVI.Example 3: 3D Domain. In this example, we consider a

100 m × 100 m × 40 m domain. We use similar initial andboundary conditions as the 2D examples. The injector is locatedat the top-left corner and the producer is on the bottom-rightcorner. We use 32 × 32 × 20 grids of dimensions 3.125 m ×3.125 m × 2.5 m in x, y, and z directions, respectively. Similarto the 2D examples, the permeability is 100 md and the porosityis 20%. The 3D example allows features of asphaltene depositionin the production well due to asphaltene-rich phase buildupto high saturations and subsequent oil rate reduction. In ourexample, because of the fixed injection rate and constant pressureat the production well, the injection pressure may increasesignificantly.Figure 8 displays the variation in overall CO2 composition and

asphaltene-rich liquid saturation at PVI = 0.3. This figure showsthat CO2 flows, due to gravity, mainly at the top of the reservoir.Due to CO2 dissolution, an asphaltene-rich liquid saturationforms. The asphaltene-rich phase saturation stays low near theinjector similar to our 2D results. The asphaltene-rich saturationnear the producer is much higher (about 40%) than in the 2Dexamples. This high saturation of precipitated asphaltene phasecauses a significant increase in the injection pressure as shown inFigure 9. The injection pressure increases more than 80 bar(1160 psi) after 0.3 PVI. We stopped the simulation at 0.3 PVIdue to this high pressure.Figure 9 shows an early breakthrough after approximately

0.2 PVI. The recovery factor at breakthrough is 0.18. There is aslight increase in injection pressure at the beginning due toformation of asphaltene-rich liquid phase near the injector. Thepressure does not increase further at that point due to smallsaturation of the asphaltene-rich liquid phase (about 4%).However, due to the high saturation of the asphaltene-rich phase

at the producer the bottomhole pressure increases significantly.At this stage there is need to clean up the well.

■ CONCLUSIONSThe central theme of this work is the presentation of a new three-phase multicomponent compositional model for asphalteneprecipitation using CPA-EOS. In this work, we discuss for thefirst time a unified compositional framework to model all threephases: gas, hydrocarbon-rich liquid, and asphaltene-rich liquid.We allow all components to be present in the asphaltene-rich(and other) phases, unlike simplified models in which theasphaltene phase consists of only the asphaltene component.Therefore, the phase behavior of all phases changes whenasphaltenes precipitate, in agreement with experimentalobservations. An efficient algorithm for the calculation of thecompressibility factor in the CPA-EOS reduces the computa-tional cost for phase behavior calculations significantly. Theoutcome is an efficient CPA-EOS compositional model withcomparable performance to conventional compositional modelsbased on the cubic EOS.We use our model to evaluate the extent of asphaltene

precipitation fromCO2 injection in an oil reservoir. The reservoiris saturated with Weyburn oil, prone to asphaltene precipitationwhen mixed with CO2. Our results show that the asphaltene-richliquid phase will form in a relatively large volume of the reservoirdue to CO2/oil mixing. The asphaltene-rich liquid saturationclose to the injection well reaches a maximum and stops toincrease as more CO2 is injected. The most significant accu-mulation of the asphaltene-rich liquid phase is around theproduction well. This accumulation, in our 3D example, led tosignificant increase in injection pressure. When the injectionpressure is not allowed to increase there will be a decrease inproduction rate due to asphaltene-rich phase accumulation in thewellbore region of the producer.

Figure 7. Oil recovery (dashed) and injector bottomhole pressure(solid) vs PVI: 2D vertical domain.

Figure 8. (a) overall CO2 composition and (b) asphaltene-rich liquid saturation at PVI = 0.3: 3D domain.

Figure 9. Oil recovery (dashed) and injector bottomhole pressure(solid) vs PVI: 3D domain.



3313

■ APPENDIX A: THREE-PHASE TOTAL FLUIDCOMPRESSIBILITY

The equations used to calculate three-phase total fluidcompressibility from the CPA-EOS are presented in thisAppendix:1

= − ∂∂

⎛⎝⎜

⎞⎠⎟c

VVp

1f

T

T

T n, (A-1)

where

∑ ∑= =α

αα

α α=

V VRTp

Z nTg o a, , (A-2)

From (A-2), we have

∑

∑

∂∂

= −

+ ∂∂

+ ∂∂

αα α

αα

αα

α

=

=

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎤

⎦⎥⎥

Vp

RTp

Z n

RTp

nZp

Znp

T

T g o a

g o a T T

n

n n

,2

, ,

, , , ,

(A-3)

Therefore, to calculate the total compressibility, one needs tocalculate (∂Z/∂p)T,n and (∂n/∂p)T,n. We first present the steps tocalculate (∂nα/∂p)T,n and then (∂Zα/∂p)T,n. We have

∑ α∂∂

=∂∂

=α α

=

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

np

n

pg o a, ,

T i

ni

Tn n, 1

,

,

c

(A-4)

where ni,α is the number of moles of component i in phase α. Tocalculate (∂n/∂p)T,n, we have 3nc unknowns in eq A-4. We need3nc equations to find these unknowns. Material balance yields

∑= =α

α=

n n i n1, ...,ig o a

i c, ,

,(A-5)

where ni is the total number of moles of component i in the three-phase system that is constant. Therefore, from eq A-5 we havethe following nc equations:

∑ ∂∂

= =α

α

=

⎛⎝⎜

⎞⎠⎟

n

pi n0 1, ...,

g o a

i

Tc

n, ,

,

, (A-6)

The remaining 2nc equations come from the followingequilibrium conditions:

∂∂

−∂∂

=∂∂

−∂∂

= =⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

f

p

f

p

f

p

f

pi n0 1, ...,

i g i o

T

i o i a

T

c

n n

, ,

,

, ,

,

(A-7)

where f i,α(α = g, o, a) is the fugacity of component i in phase α.The calculation of this parameter using CPA-EOS is explained inAppendix C. One can write eq A-7 as

∑ ∑∂∂

∂∂

−∂∂

∂∂

+∂∂

−∂∂

= =

= =

⎡

⎣⎢⎢⎢

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

⎤

⎦⎥⎥⎥

⎡

⎣⎢⎢⎢⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎤

⎦⎥⎥⎥

f

n

n

p

f

n

n

p

f

p

f

pi n0 1, ...,

k

ni g

k g T

k g

T k

ni o

k o T

k o

T

i g i oc

n n n n

n n

1

,

,,

,

, 1

,

, ,

,

,

, ,

c

g k

c

o k

g o

, ,

(A-8)

∑ ∑∂∂

∂∂

−∂∂

∂∂

+∂∂

−∂∂

= =

= =

⎡

⎣⎢⎢⎢

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

⎤

⎦⎥⎥⎥

⎡

⎣⎢⎢⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎤

⎦⎥⎥

f

n

n

p

f

n

n

p

f

p

f

pi n0 1, ...,

k

ni o

k o T

k o

T k

ni a

k a T

k a

T

i o i ac

n n n n

n n

1

,

, ,

,

, 1

,

, ,

,

,

, ,

c

o k

c

a k

o a

, ,

(A-9)

The calculation steps for (∂f i/∂nk)T,nk and (∂f i/∂p)n using theCPA-EOS are reported in Appendix C.To calculate (∂Zα/∂p)T,n (α = g, o, a), we use the following

equation:

∑

α

∂∂

= ∂∂

+ ∂∂

∂∂

=

α α α

α

α

=α α α

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

Zp

Zp

Zn

n

p

g o a, ,

T T k

n

k T

i

Tn n n n, , 1 , ,

,

,

c

k,

(A-10)

The calculation steps for (∂Zα/∂p)T,nα and (∂Zα/∂nα,k)T,nα,k usingthe CPA-EOS is explained in Appendix D.

■ APPENDIX B: THREE-PHASE PARTIAL MOLARVOLUME

The calculation steps for three-phase partial molar volume arepresented in this section. Partial molar volume of component i( vi) is defined as

∑ ∑

=∂∂

= + ∂∂

∂∂

=

α

α α α

α

α

= =α α

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢⎢⎢

⎡

⎣⎢⎢⎢

⎛⎝⎜⎜

⎞⎠⎟⎟

⎤

⎦⎥⎥⎥⎛⎝⎜

⎞⎠⎟

⎤

⎦⎥⎥⎥

vVn

Z RTp

n RTp

Zn

n

n

i n1, ...,

iT

i T p

g o a k

n

k T p

k

i T p

c

n

n n

, ,

, , 1 , , ,

,

, ,

i

c

k i, ,

(B-1)

To calculate partial molar volume from the above equation, oneneeds to find 3nc unknown values of (∂nk,α/∂ni)T,p,nα,i (α = g, o, a).Similar to calculation of total fluid compressibility, the followingmaterial balance and equality of fugacities (due to equilibrium)provide the required 3nc equations:

∑ ∂∂

= =α

α

= α

⎛⎝⎜

⎞⎠⎟

n

ni n0 1, ...,

g o a

i

i T pc

n, ,

,

, , i, (B-2)

∂∂

−∂∂

=∂∂

−∂∂

=

=α α

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

f

n

f

n

f

n

f

n

i n

0

1, ...,

i g

i

i o

i T p

i o

i

i a

i T p

c

n n

, ,

, ,

, ,

, ,i i, ,

(B-3)

■ APPENDIX C: EXPRESSIONS OF (∂f i/∂nk)T,nkAND

(∂f i/∂p)n FROM CPA-EOS

To calculate three-phase total compressibility and three-phasepartial molar volume, we need to calculate f i, (∂f i/∂nk)T,nk and(∂f i/∂p)n for each phase using the CPA-EOS. The expressions forthese parameters are presented in this Appendix. To calculate f i,we have



3314

φ

ηη η

χ χ χ

= − − +−

−+ −

−∑

− + +− +

+ −− +

− + − +

=

=

⎜ ⎟⎛⎝

⎞⎠

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

Z BBB

BZ B

AZZ BZ B

AB

xA

ABB

Z BZ B

BZ

x x

i a r

ln ln( )2

2 2

2ln

(1 2 )(1 2 )

2.52 3

[ ( 1) ( 1)] 4 ln

or

ii

jn

j ij i

ia a r r i

2 2

1

2

c

(C-1)

φ = − − +−

−+ −

−∑

− + +− +

≠

=

⎜ ⎟⎛⎝

⎞⎠

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

Z BBB

BZ B

AZZ BZ B

AB

xA

ABB

Z BZ B

i a r

ln ln( )2

2 2

2ln

(1 2 )(1 2 )

,

and

ii

jn

j ij i

2 2

1c

(C-2)

where φi = f i/(xip). To calculate the derivatives of φi with respectto composition, we consider the following chain rule:

φ φ φ φ φ

φχ

χ φχ

χ φη

η

∂∂

=∂

∂∂∂

+∂

∂∂∂

+∂

∂∂∂

+∂

∂

+∂

∂∂∂

+∂

∂∂∂

+∂

∂∂∂

x

F

ZZx

F

AAx

F

BBx

F

x

F

x

F

x

F

x

ln (ln ) (ln ) (ln ) (ln )

(ln ) (ln ) (ln )

i

j

i

j

i

j

i

j

i

j

i

a

a

j

i

r

r

j

i

j (C-3)

The terms in the above equation are obtained from

φ

ηη η

χ χ

∂∂

= −−

+ −−

+ −+ −

+∑

+ −− −

− +− + −

=

=

⎛⎝⎜

⎞⎠⎟

F

Z Z BBB

BZ B

AB B ZZ BZ B

xA

Z BZ BBZ

x x

i a r

(ln ) 1( ) ( )

2 ( )( 2 )

2

22.5

2 3[ ( 1) ( 1)]

or

i i

jn

j ij ia a r r

2 2 2 2

12 2 2 2

c

(C-4)

φ∂∂

= −−

+ −−

+ −+ −

+∑

+ −≠=

⎛⎝⎜

⎞⎠⎟

F

Z Z BBB

BZ B

AB B ZZ BZ B

xA

Z BZ Bi a r

(ln ) 1( ) ( )

2 ( )( 2 )

2

2and

i i

jn

j ij

2 2 2 2

12 2

c

(C-5)

φ∂∂

=+ −

+ + +− +

⎜ ⎟⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

F

ABB

AZZ BZ B

ABB

Z BZ B

(ln )

2 2 2ln

(1 2 )(1 2 )

i i i2 2 2

(C-6)

φ∂∂

=−

−−

−+ −

+−

+ −+ −

+∑

− + +− +

−∑

+ −

= = −

= =

⎜ ⎟⎛⎝

⎞⎠

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

F

B Z BBB

BZ B

AZZ BZ B

BB

ZZ B

AZ Z BZ BZ B

xA

BABB

Z BZ B

ZB

xA

Z BZ B

i a r j n

(ln ) 1( )

22

( )2 ( )

( 2 )

2 2ln

(1 2 )(1 2 )

2

2

or , 1, ..., 1

i i

i

jn

j ij i jn

j ij

c

2 2 2

2 2 2 2

12 2

12 2

c c

(C-7)

φ ηη η

χ∂

∂=

− + +− +

+ −− +

= = −

⎛⎝⎜

⎞⎠⎟

F

x

A A

BZ BZ B

BZ

i a r j n

(ln )

2ln

(1 2 )(1 2 )

2.52 3 2

or , 1, ..., 1

i

j

in ij ii

c

2c

(C-8)

φ∂∂

=− + +

− +

≠ = −

⎛⎝⎜

⎞⎠⎟

F

x

A A

BZ BZ B

i a r j n

(ln )

2ln

(1 2 )(1 2 )

and , 1, ..., 1

i

j

in ij

c

c

(C-9)

φχ

ηη η χ

∂∂

= −− +

+

= =

F BZ

x

i a r k a r

(ln ) 2.52 3

4

or , or

i

k

ik

k2

(C-10)

φχ

∂∂

= ≠ =F

i a r k a r(ln )

0 and , ori

k (C-11)

φη

η ηη η

χ χ∂

∂= − +

− +− + −

=

F BZ

x x

i a r

(ln ) 5.5 5(2 3 )

[ ( 1) ( 1)]

or

i ia a r r

2

2 2

(C-12)

φη

∂∂

= ≠F

i a r(ln )

0 andi

(C-13)

∑∂∂

= −=

Ax

x A A2 ( )j i

n

i ij in1

c

c(C-14)

∂∂

= −Bx

B Bj

j nc(C-15)

The terms ∂Z/∂xj, ∂χa/∂xj, and ∂χr/∂xj can be obtained fromAppendix D. To calculate (∂f i,α/∂p)nα, we use the followingequation:

∂∂

= + ∂∂

α α α α α

αα α

⎛⎝⎜⎜

⎞⎠⎟⎟

⎡

⎣⎢⎢⎢

⎛⎝⎜⎜

⎞⎠⎟⎟

⎤

⎦⎥⎥⎥

f

p

f

RTZ RTp

n RTp

Zn

i i

k T pn n

, ,

, , , k, (C-16)

■ APPENDIX D: EXPRESSIONS OF (∂Z/∂nk)T,nkAND

(∂Z/∂p)T,nj FROM CPA-EOS

In this Appendix, the expressions to calculate (∂Z/∂nk)T,nk and(∂Z/∂p)T,nj using the CPA-EOS are presented. From the CPA-EOS, we have

η ηη η

χ χ α

=−

−+ −

+ + −− +

− + − =

Z ZZ B

AZZ BZ B

x x g o a

24 4 22 3

[ ( 1) ( 1)] , ,a a r r

2 2

2

2

(D-1)

To calculate the derivative of Z with respect to composition, wehave

χχ

χχ

ηη

∂∂

= ∂∂

∂∂

+ ∂∂

∂∂

+ ∂∂

∂∂

+ ∂∂

+ ∂∂

∂∂

+ ∂∂

∂∂

+ ∂∂

∂∂

Zx

F ZZ

Zx

F ZA

Ax

F ZB

Bx

F Zx

F Zx

F Zx

F Zx

( ) ( ) ( ) ( )

( ) ( ) ( )k k k k k

a

a

k r

r

k k (D-2)

Using the chain rule, we have the following equations for ∂χa/∂xk,∂χr/∂xk, and ∂η/∂xk:

χ χ χ χχ

χ χχ

χ

χη

η χη

η

∂∂∂

=∂

∂ ∂+

∂∂

+∂

∂∂∂

+∂

∂∂∂

+∂∂Δ

∂Δ∂

∂∂

+∂∂Δ

∂Δ∂

∂∂

x

F

ZZx

F

x

F

x

F

x

F

x

F

x

( ) ( ) ( ) ( )

( ) ( )

a

k

a

k

a

k

a

a

a

k

a

r

r

k

aaa

aa

k

aar

ar

k (D-3)

χ χ χ χχ

χ χη

η∂∂

=∂

∂∂∂

+∂

∂+

∂∂

∂∂

+∂∂Δ

∂Δ∂

∂∂x

F

ZZx

F

x

F

x

F

x

( ) ( ) ( ) ( )r

k

r

k

r

k

r

a

a

k

aar

ar

k

(D-4)



3315

η η η∂∂

= ∂∂

∂∂

+ ∂∂

∂∂x

FZ

Zx

FB

Bx

( ) ( )

k k k (D-5)

We define f1, f 2, and f 3 as

χ χχ

χ χχ

χη

η

χχ

χη

η χη

η

χη

η

=∂

∂+

∂∂

∂∂

+∂

∂∂∂Δ

∂Δ∂

∂∂

+∂

∂∂∂Δ

∂Δ∂

∂∂

+∂∂Δ

∂Δ∂

∂∂

+∂∂Δ

∂Δ∂

∂∂

fF

Z

F

Z

F F FZ

F F FZ

F FZ

F FZ

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( )

a a

r

r a

r

raa

aa

a

r

rar

araaa

aa

aar

ar

1

(D-6)

χχ

χη

η χχ

χη

η

χη

η χη

η χ

χχ

χ

= ∂∂

∂∂

∂∂Δ

∂Δ∂

∂∂

+∂

∂∂∂Δ

∂Δ∂

∂∂

+∂∂Δ

∂Δ∂

∂∂

+∂∂Δ

∂Δ∂

∂∂

+∂

∂

+∂

∂∂∂

⎛⎝⎜⎜

⎞⎠⎟⎟

f Bx

F F FZ

F F FZ

F FZ

F FZ

F

x

F

x

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( )

k

a

r

raa

aaa

r

rar

ar

aaa

aarar

ara

k

a

r

r

k

2

(D-7)

χχ

χχ

χχ

= −∂

∂−

∂∂

∂∂

fF F F

1( ) ( ) ( )a

a

a

r

r

a3

(D-8)

If we substitute eqs D-4 and D-5 into eq D-3, using the abovedefinitions, we obtain

χ∂∂

= ∂∂

+x

f

fZx

f

fa

k k

1

3

2

3 (D-9)

We define f4 and f5 as

χ χχ

χη

η

χη

η

=∂

∂+

∂∂

+∂∂Δ

∂Δ∂

∂∂

+∂∂Δ

∂Δ∂

∂∂

fF

Z

f

f

F F FZ

F FZ

( ) ( ) ( ) ( )

( ) ( )

r r

a

raa

aa

rar

ar

41

3

(D-10)

χ χχ

χη

η=∂

∂+

∂∂

+∂∂Δ

∂ Δ∂

∂∂

∂∂

fF

x

f

f

F F F FB

Bx

( ) ( ) ( ) ( ) ( )r

k

r

a

rar

ar

k5

2

3(D-11)

Then, substituting eqs D-5 and ) into eq D-4, we have

χ∂∂

= ∂∂

+x

fZx

fr

k k4 5 (D-12)

We define f6 and f 7 as

ηη

χ χ

= ∂∂

∂∂

+ ∂∂

+ ∂∂

∂∂

∂∂

+ ∂∂

+ ∂∂

+ ∂∂

⎛⎝⎜

⎞⎠⎟f

F ZA

Ax

F ZB

F ZB

Bx

F Zx

f

fF Z

fF Z

( ) ( ) ( ) ( )

( ) ( )k k k

a r

6

2

35

(D-13)

χ χ ηη= − ∂

∂− ∂

∂− ∂

∂∂

∂∂

∂f

F ZZ

f

fF Z

fF Z F Z F

Z1

( ) ( ) ( ) ( ) ( )

a r7

1

34

(D-14)

If we substitute eqs D-5, D-9, and D-12 into eq D-2, using theabove definitions, we obtain

∂∂

=Zx

f

fk

6

7 (D-15)

η η η∂∂

= ∂∂

+ ∂∂

∂∂x

f

ffZ

fB

Bx

( ) ( )

k k

6

7 (D-16)

The partial derivatives in the above equations are given by

∂∂

=−

+ ++ −

F ZZ

BZ B

A Z BZ BZ B

( )( )

( )( 2 )2

2 2

2 2 2 (D-17)

∂∂

= −+ −

F ZA

ZZ BZ B

( )22 2 (D-18)

∂∂

=−

+ −+ −

F ZB

ZZ B

AZ Z BZ BZ B

( )( )

2 ( )( 2 )2 2 2 2

(D-19)

η ηη η

χ∂∂

= + −− +

− =F Zx

k a r( ) 4 4 2

2 3( 1) or

kk

2

2

(D-20)

∂∂

= ≠F Zx

k a r( )

0 andk (D-21)

χη ηη η

∂∂

= + −− +

=F Zx k a r

( ) 4 4 22 3

ork

k

2

2(D-22)

ηη η

η ηχ χ∂

∂= − +

− +− + −F Z

x x( ) 20 16 2

(2 3 )[ ( 1) ( 1)]a a r r

2

2 2

(D-23)

∑∂∂

= −=

Ax

x A A2 ( )k i

n

i ik in1

c

c(D-24)

∂∂

= −Bx

B Bk

k nc (D-25)

χ χ χχ χ

∂∂

=Δ + Δ

+ Δ + Δα α α αF

Z

N x N x

Z N x N x

( )

( )a a a a

aar r r

ar

a a aaa

r r rar

, , , ,2

(D-26)

χ χχ χ

∂∂

= −Δ

+ Δ + Δ=

F

x

N Z

Z Nx N xk a r

( )

( )ora

k

k kak

a a aaa

r r rar 2

(D-27)

χ∂∂

= ≠F

xk a r

( )0 anda

k (D-28)

χχ χ χ

∂∂

= − Δ+ Δ + Δ

F N x ZZ N x N x

( )

( )a

a

a aaa

a a aaa

r r rar 2

(D-29)

χχ χ χ

∂∂

= − Δ+ Δ + Δ

F N x ZZ N x N x

( )

( )a

r

r rar

a a aaa

r r rar 2

(D-30)

χ χχ χ

∂∂Δ

= −+ Δ + Δ

F N x Z

Z N x N x

( )

( )aaa

a a a

a a aaa

r r rar 2

(D-31)

χ χχ χ

∂∂Δ

= −+ Δ + Δ

F N x Z

Z N x N x

( )

( )aar

r r r

a a aaa

r r rar 2

(D-32)

χ χχ

∂∂

= −Δ

+ ΔF

Z

N x

Z N x

( )

( )r a a a

ar

a a aar 2

(D-33)



3316

χ χχ

∂∂

= −Δ

+ Δ=

F

x

N Z

Z Nxk a

( )

( )r

k

a aar

a a aar 2

(D-34)

χ∂∂

= ≠F

xk a

( )0r

k (D-35)

χχ χ

∂∂

= − Δ+ Δ

F N x ZZ N x

( )

( )r

a

a aar

a a aar 2

(D-36)

χ χχ

∂∂Δ

= −+ Δ

F N x Z

Z N x

( )

( )rar

a a a

a a aar 2

(D-37)

η∂∂

= −FZ

BZ

( )4 2 (D-38)

η∂∂

=FB Z( ) 1

4 (D-39)

ηη

ηκ

ε∂Δ∂

= −−

− = =⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥b

k Ti a j a r

2.5(1 )

exp 1 , orij

ij ijij

b4

(D-40)

To calculate the derivative of Z with respect to pressure, we have

χχ

χχ

ηη

∂∂

= ∂∂

∂∂

+ ∂∂

∂∂

+ ∂∂

∂∂

+ ∂∂

∂∂

+ ∂∂

∂∂

+ ∂∂

∂∂

Zp

F ZZ

Zp

F ZA

Ap

F ZB

Bp

F Zp

F Zp

F Zp

( ) ( ) ( )

( ) ( ) ( )

a

a

r

r

(D-41)

Using the chain rule, we have the following equations for ∂χa/∂p,∂χr/∂p, and ∂η/∂p:

χ χ χχ

χ χχ

χ

χη

η χη

η

∂∂

=∂

∂∂∂

+∂

∂∂∂

+∂

∂∂∂

+∂∂Δ

∂Δ∂

∂∂

+∂∂Δ

∂Δ∂

∂∂

p

F

ZZp

F

p

F

p

F

p

F

p

( ) ( ) ( )

( ) ( )

a a a

a

a a

r

r

aaa

aaaar

ar

(D-42)

χ χ χχ

χ χη

η∂∂

=∂

∂∂∂

+∂

∂∂∂

+∂∂Δ

∂Δ∂

∂∂p

F

ZZp

F

p

F

p

( ) ( ) ( )r r r

a

a aar

ar

(D-43)

η η η∂∂

= ∂∂

∂∂

+ ∂∂

∂∂p

FZ

Zp

FB

Bp

( ) ( )(D-44)

ηη∂Δ

∂= ∂ Δ

∂∂∂

= =p

Fp

i a j a r( )

, orij ij

(D-45)

We define g1, g2, and g3 as

=g f1 1 (D-46)

χχ

χη

η χχ

χη

η

χη

η χη

η

= ∂∂

∂∂

∂∂Δ

∂Δ∂

∂∂

+∂

∂∂∂Δ

∂Δ∂

∂∂

+∂∂Δ

∂Δ∂

∂∂

+∂∂Δ

∂Δ∂

∂∂

⎛⎝⎜⎜

⎞⎠⎟⎟

g Bp

F F FZ

F F FZ

F FZ

F FZ

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

a

r

raa

aaa

r

rar

ar

aaa

aarar

ar

2

(D-47)

=g f3 3 (D-48)

If we substitute eqs D-43, D-44, and D-45 into eq D-42, using theabove definitions, we can obtain

χ∂∂

= ∂∂

+p

g

gZp

g

ga 1

3

2

3 (D-49)

Substituting eqs D-44, D-45, and D-49 into eq D-43 and definingg4 and g5 as

=g f4 4 (D-50)

χχ

χη

η=∂

∂+

∂∂Δ

∂ Δ∂

∂∂

∂∂

gg

g

F F F FB

Bp

( ) ( ) ( ) ( )r

a

rar

ar

52

3 (D-51)

we have

χ∂∂

= ∂∂

+p

gZp

gr4 5 (D-52)

We define g6 and g7 as

ηη

χ χ

= ∂∂

∂∂

+ ∂∂

+ ∂∂

∂∂

∂∂

+ ∂∂

+ ∂∂

⎛⎝⎜

⎞⎠⎟g

F ZA

Ap

F ZB

F ZB

Bp

g

gF Z

gF Z

( ) ( ) ( )

( ) ( )

a r

6

2

35

(D-53)

=g f7 7 (D-54)

If we substitute eqs D-44, D-45, and D-49 into eq D-41, using theabove definitions, we obtain

∂∂

=Zp

g

g6

7 (D-55)

η η η∂∂

= ∂∂

+ ∂∂

∂∂p

g

gfZ

fB

Bp

( ) ( )6

7 (D-56)

The partial derivatives in the above equations are definedpreviously except for ∂A/∂p and ∂B/∂p given by

∂∂

=Ap

Ap (D-57)

∂∂

=Bp

Bp (D-58)

■ AUTHOR INFORMATIONCorresponding Author*Phone: (979)8626483. Fax: (979)8451307. E-mail: [email protected] authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis research is supported by the Qatar National Research Fundunder Grant NPRP 09-1050-2-405. This support is greatlyappreciated. We also thank Dr. Zhidong Li, currently withExxonMobil, for helpful technical discussions.

■ NOMENCLATUREA = Peng−Robinson equation of state parameter, dimension-lessB = Peng−Robinson equation of state parameter, dimension-lessc = overall molar density, mol/m3

cf = total fluid compressibility, 1/Pa



3317

Fi = the sink or source term for component i, mol/(s·m3)f i,α = fugacity of component i in phase α, Pag = acceleration of gravity, m2/sk = absolute permeability, m2

kb = Boltzmann constant, J/Kkrα = relative permeability of phase α, dimensionlessMi = molecular weight of component i, g/molNi = number of association sites of molecule in = number of molesnc = number of componentsp = pressure, Papc = critical pressure, PaR = universal gas constant, J/(K·mol)T = temperature, KTc = critical temperature, Kuα = phase velocity, m/sV = volume, m3

vi = total partial molar volume of component i, m3/molxi,α = mole fraction of component i in phase α, dimensionlessZα = compressibility factor of phase αzi = overall composition of component i, dimensionless

Greek Lettersβα = molar fractions of phase α, dimensionlessΔij = association strength between site i and site j,dimensionlessεij = energy parameter, Jη = cubic-plus-association equation of state parameter,dimensionlessκij = association volume, dimensionlessμα = viscosity of phase α, kg/(m·s)ρα = mass density of phase α, kg/m3

ϕ = porosity, dimensionlessφi,α = fugacity coefficient of component i in phase α,dimensionlessχi = mole fractions of component i not bonded at one of theassociation sites, dimensionless

Subscriptsa = asphaltene-rich liquid phaseg = gas phasei = componento = hydrocarbon-rich liquid phaser = resins/aromaticsα = phase

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