Thermodynamic Model of Petroleum Fluids Containing Polydisperse

Javier Dufour , José A. Calles , Javier Marugán , Raúl Giménez-Aguirre , José ... Maria Magdalena Ramirez-Corredores ... Jinsheng Wang , Edward J...
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Ind. Eng. Chem. Res. 1998, 37, 3242-3251

Thermodynamic Model of Petroleum Fluids Containing Polydisperse Asphaltene Aggregates Alexey I. Victorov and Natalia A. Smirnova* Department of Chemistry, St. Petersburg State University, Universitetsky pr. 2, 198904 St. Petersburg, Russia

New version of thermodynamic aggregation model of asphaltene-containing crude is formulated and analyzed with regard to characteristics of resin molecule shape and deformation of the resin shell of the asphaltene aggregates. The relation between the shape factors and elastic constants is discussed, and their effect on the size distribution of asphaltenic aggregates is considered. The model is applied to describe asphaltene precipitation from petroleum fluid upon its dilution with a liquid alkane. The effect of dilution on the asphaltene particle size distribution is studied. The model calculation has shown that the behavior of asphaltenic crude varies dramatically with the change of molecular characteristics of resins. 1. Introduction Asphaltenes are heavy polyaromatic materials in the high-boiling fractions of petroleum fluids. Under certain conditions asphaltenes can flocculate in the crude causing the formation of heavy deposits. Various reasons, such as CO2 injection, mixing of different oils in oil storage, pressure change, etc., can cause the asphaltene dropout.1 The consequences of asphaltene deposition during the production and processing of asphaltenic crudes are rather undesirable and cause many problems, e.g., reduction of reservoir permeability, equipment plugging, wettability reversal, etc. Whereas there is indisputable progress in understanding the phase behavior of petroleum fluid phases, especially owing to the advanced equations of state of bulk fluids,2-4 much less can be said as yet about the asphaltene-related phase behavior of the crudes. Modeling of the stability of crudes with respect to asphaltene dropout, prediction of the conditions of the onset of precipitation, and the amount of precipitate still remains among the challenging tasks of chemical engineering thermodynamics. Vast literature exists on the chemistry of asphaltene materials with rather detailed consideration of various versions of their molecular structure and chemical properties.5-8 Nevertheless there are still discussions on the nature and possible mechanisms of interactions between asphaltenic molecules in the petroleum fluid, and different arguments are used to explain the extreme polarity and association of asphaltenes.5 Despite all the disagreement, the polyaromaticity of asphaltenes and their association into large aggregates, which are stabilized by highly aromatic bipolar resins, are wellrecognized and are part of a conventional point of view on the behavior of asphaltenes.9 Thermodynamic data on asphaltenes include the results of titration experiments (using liquids or gases as titrants10-12), studies of asphaltene precipitation with changing pressure,10,11,13,14 electric conductivity,15 surface tension, and critical micelle concentration (cmc) measurements.16,17 In addition there are spectroscopic data on the structure of asphaltenic aggregates in various media16,18-20 and * Phone: 7 (812) 428 6790. Fax: 7 (812) 428 6939. E-mail: [email protected].

many controversial data on the molecular weights of asphaltenic species9 with very few reliable estimates of monomeric asphaltene molecular weight. The success in modeling the phase behavior of asphaltenic crudes is still rather limited, and there are two main directions in this work, as it is discussed below. In principle, quite different mechanisms can be responsible for a bulk-phase formation in a fluid. A regular van der Waals loop, as in the common liquidliquid or liquid-gas phase split, or the osmotic effects, as for the case of protein flocculation21,22 are among the most familiar signs of instability resulting in the formation of a new phase. In this very line were the first models of asphaltene dropout,10 explaining the formation of the asphaltene precipitate by just its low solubility in the bulk of a petroleum fluid. Various versions of the bulk-phase equations of state or GE models (like lattice-based Flory theory) were used to describe this solubility and petroleum fluid-asphaltene phase equilibrium.10,11,23 However, the observed macroscopic behavior of asphaltene-containing crudes, in particular, (1) precipitation upon adding a precipitant (nonpolar chainlike substance of medium molecular mass, normally an alkane), (2) precipitation upon depressurizing the crude, and (3) precipitation from rather dilute (with respect to asphaltenes) crudes and the stability of crudes with higher asphaltene content, shows that the precipitation occurs upon “expanding” of the crude, when the average asphaltene interparticle distances tend to increase. These observations suggest that the mechanism of asphaltene precipitation can be different from that for bulk fluid phase splitting, or protein flocculation, and it is also unlikely that the osmotic effects are responsible for the asphaltene dropout. Many experimental data (measurements of surface tension, electrical conductivity, cmc, and spectroscopy) indicate that it is the aggregation of asphaltenes and resins that is responsible for asphaltene precipitation phenomena. Thermodynamic behavior of systems whose components can self-assemble into aggregated structures (micelles, vesicles, bilayers, swollen micelles, and microemulsion droplets) has received much attention, mainly in connection with the studies of aqueous surfactant solutions, biologically active membranes, and microemulsions (see, e.g., refs 24-29). However, very few attempts were made so far to apply these ap-

S0888-5885(97)00869-5 CCC: $15.00 © 1998 American Chemical Society Published on Web 07/17/1998

Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3243

proaches for asphaltene-containing mixtures. In the present paper we discuss recent advances and problems in this area of research.

n2 monomers:

2. Thermodynamic Model of Asphaltene Aggregation

For a diluted solution, i.e., at low concentrations of aggregates and monomers (though in the case of large aggregates this may correspond to some considerable gross concentrations; see eq 1), this equation can be rewritten as

The first thermodynamic colloid model of asphaltenes is due to Leontaritis and Mansoori.30 They recognized the importance of resins stabilizing the asphaltene dispersion and derived the criterion of the asphaltene dropout on the basis of the application of the Langmuir adsorption isotherm. It is understandable however that aiming at the derivation of a most simple and practical version of this criterion, they did not include in their model any detailed picture of the aggregates with regard to their structure, size, and shape. A model of asphaltenic crudes, in which these factors, crucial for the stability of a dispersion of aggregates, were incorporated explicitly, was proposed somewhat later.31 The model exploits basic ideas of thermodynamics of self-assembling systems, and its essence can be outlined as follows. The stability of asphaltene-containing crudes is due to solubilization of asphaltene material inside aggregates having the resin shell, asphaltenes forming the core of aggregates. If the aggregates are destroyed (e.g., by washing the resins away from the shell), then the asphaltenes are to be dispersed in the bulk of the crude, but they are immediately expelled from it (to form the precipitate) due to their low-bulk solubility. This is in line with experimental observations of asphaltene dropout upon “expanding” the crude. The ensemble of aggregated particles in this model thus has the role of some “reservoir” of asphaltenic material within the crude, so that this material, although having rather low bulk solubility, still can be accommodated in a macroscopically homogeneous petroleum fluid. The material balance equations for this fluid (prior to the formation of the asphaltene phase) can be written in the following form: ∞

Xa ) Xa1 +

∑ n1XM(n1,n2)

n1,n2 ∞

Xr ) Xr1 +

∑ n2XM(n1,n2) n ,n 1

(1)

2

expressing the fact that the total amount of asphaltenes and resins (gross mole fractions Xa, Xr) is the sum of the amounts of monomeric (mole fractions Xa1, Xr1) and aggregated moieties, the latter built of n1 and n2 monomeric units (mole fraction XM(n1,n2)). The focus of any theory of self-assembly is, indeed, to build an expression for XM(n1,n2), and the central question that arises is whether the system is at thermodynamic equilibrium. This is a rather difficult question to answer, particularly for systems containing heavy viscous polyaromatic fractions, such as resins and asphaltenes, and the positive answer is to some extent an assumption. There are experimental data supporting this assumption, the results on the static and dynamic surface tensions obtained recently by Sheu16 being among the strongest arguments. The condition of aggregation equilibrium32 relates the chemical potentials of monomeric asphaltenes µβa1 and resins µβr1 to the chemical potential of an aggregate built up of n1 and

µβM ) n1µβa1 + n2µβr1

XM ) Xan11 Xrn12 exp{∆G00 M (n1,n2)/kBT}

(2)

(3)

Thus, the concentration of aggregates is related to ∆G00 M (n1,n2), the standard Gibbs energy of aggregation with the minus sign (the opposite sign convention is also used below for separate contributions into the total value). Equation 3 separates the entropic contribution (preexponential factor, which falls rapidly with growing n1 and n2) from other driving forces involved in the aggregation process and clearly shows that the entropic factor always opposes the aggregate growth and favors dispersion. The main driving force of aggregation in the present model is, after Tanford,32 the solvophobic effect, which reflects the trend of asphaltene monomers to avoid petroleum fluid, and, on the other hand, is akin to its low-bulk solubility. If the average state of asphaltene molecules in the aggregate core is assumed to be similar to that in the bulk of asphaltene precipitate (with the corresponding corrections for pressure difference), the solvophobic contribution can be related to the concentration of monomeric asphaltenes, Xons a1 , in the crude at equilibrium with the asphaltene precipitate:31 ons ∆G00 solv/kBT ) -n1 ln Xa1

(4)

Note that the above approximation implies that the aggregates are not very small and poses some limitation of the lower n1 values for which the model can still be valid. The solvophobic effect always promotes aggregation, giving, however, a constant contribution per asphaltene particle in the aggregate under the assumptions leading to eq 4. Thus, in the frame of the model this effect is independent of the aggregate size and does not favor aggregate growth, nor their dispersion. Association of the resin molecule polar heads with asphaltenes is the major driving force for the resins to be adsorbed and form a shell around the asphaltenic core of the aggregate. The corresponding term is

∆G00 ads/kBT ) n2∆Ur/kBT

(5)

where ∆Ur is the energy of desorption per resin molecule. As a rough approximation, this quantity is assumed to be independent of the specific composition of the surrounding nonpolar petroleum medium and of the configuration of the asphaltene core of aggregates. Its estimated value of 18 kJ/mol31 falls within a typical range of hydrogen bond energies (and seems reasonable in view of the nature of resin heads, where hydroxyl groups are present). A term, which always opposes the adsorption of resins on asphaltene aggregate cores, is due to the steric repulsion between the resin heads. This term accounts for the translational restrictions of resins adsorbed on the asphaltene cores and can be expressed as26,27

3244 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998

∆G00 rep/kBT ) n2 ln(1 - Θ)

(6)

where Θ ) Ar(n2)/A∑ is the average fraction of the aggregate overall surface (A∑) that is occupied by resins Ar(n2). The excess Gibbs energy of creation of asphaltene/ crude interfaces disfavors aggregation. There are two factors that can diminish the contribution of this surface term. The first is the aggregate growth reducing its surface per volume ratio (the surface area per molecule and the curvature). Hence, this term, though giving negative contribution to aggregation with respect to the monomeric state, promotes aggregate growth, at least starting from some minimum size, when σ0 can be approximately considered as size-independent. The excess energy can also be lowered considerably, when amphiphilic resin molecules adsorb on the asphaltene core somewhat decreasing the interfacial tension σ0 at the asphaltene core/petroleum fluid boundary (screening effect). The latter is expressed by the following surface term:

∆G00 surf/kBT ) -

n2σ0a0(1 - Θ) σ0(AΣ - Ar(n2)) )kBT kBTΘ (7)

where a0 is the resin molecule polar head area (the area of asphaltene core/petroleum interface that becomes screened from contact, when a single resin molecule is adsorbed on the core). An important feature of the ∆G00 surf term is its implicit dependence on the degree of the dispersion of asphaltenic material. With the different contributions described above the model for the standard Gibbs energy of aggregation can be written as

∆G00 M (n1,n2)

)

∆G00 solv

+

∆G00 ads

+

∆G00 surf

+

∆G00 rep

(8)

Unlike the other contributions to the standard Gibbs energy of aggregation, the latter two depend upon the aggregate composition, Θ. Among the aggregates of a given size of core, n1, the maximum concentration have those with a value Θ ) Θ* satisfying33

∆Ur σ0a0 Θ ln(1 - Θ) ) - ln Xr1 (9) 1-Θ kBT kBT The solution of this equation

jR Θ* ) a0/a

(10)

determines some optimum (most probable to occur) area a j R per resin molecule head for an aggregate of any arbitrary size (note that eq 9 contains neither n1 nor n2 explicitly). 3. Distribution of Aggregates over Their Size Formally the optimal value of Θ obtained by means of eq 9 can correspond to an aggregate of any size. The entropic trend for dispersion would then always drive the system toward the state with smaller aggregates, and thus the distribution of the aggregates would be monotonously falling with size. In fact the interplay between this entropic trend and the trend of the system to diminish the surface excess energy by minimizing the surface area per molecule (when assembling into bigger

aggregates) results in some optimum size of the aggregate for every allowed aggregate shape. As it was shown by Israelachvili et al.,24 the allowed shape of the aggregate depends on the specific molecular architecture of the monomers and is obtained by taking into account packing constraints. These would restrict physically reasonable values of the surface areas for the aggregates of particular shape and affect the expressions for the standard Gibbs energy of aggregation. In our previous work33 on the distribution of asphaltenic platelike particles, a rapidly decaying distribution-over-size has been obtained; however, the packing constraint was implicitly taken into account by the notion that at some small enough aggregation numbers the spherical aggregates, instead of cylindrical ones, are likely to form. The main difficulty in accounting for packing in asphaltenic systems is that no certain estimates can be given regarding the geometrical characteristics of monomeric asphaltenes and resins. If there were some, we could immediately arrive at the estimates of the optimal aggregate size (as illustrated below for spherical aggregates) in complete analogy with these estimates given by Nagarajan34 for microemulsion droplets or by Israelachvili et al.24 for vesicles. Consider spherical aggregates with an asphaltene core of a radius R and the resin shell of a thickness lR. Equation 10 gives the relation between the aggregation numbers n1 and n2, since

4 n1va ) πR3 3 n2 )

AΣ 4πR2 θ*(4πR2) ) ) a jR a jR a0

(11)

(12)

where va is the molecular volume per asphaltene monomer in the core, and thus

n2 )

(4π)1/3(3va)2/3 2/3 θ* (4π)1/3(3va)2/3n12/3 ) n1 a0 a jR

(13)

The volume of the resin shell is given by

4 n2vR ) π[(R + lR)3 - R3] 3

(14)

where vR is the volume per resin molecule. From eqs 11, 12, and 14 we get 1/2 R [3 + [3(4Sh - 1)] ] ) lR 6(Sh - 1)

(15)

where Sh is the resin molecule shape factor defined by

Sh )

θ*vR vR ) a j RlR a0lR

(16)

Formula 15 is equivalent to eq 7.11 of the original work;24 the difference is only in notations. If we assume that the resin shell does not swell, i.e., there is no penetration of surrounding fluid molecules into the shell, then vR is equal to the resin molecular volume. The shell thickness lR is not greater than the length of the fully extended tail of a resin molecule, and if we take the latter as an estimate for lR, then eq 15 determines the aggregate core radius in terms of characteristics of resin molecules (shape factor Sh). This

Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3245

makes it possible to choose a single pair of n1 and n2 values from the variety of their values related by eq 13 for given Θ*. As it can be obtained from eq 15, the core radius decreases with decreasing lR, so the entropic trend for dispersion diminishing the core radius should also tend to decrease the shell thickness. However, this tendency must be opposed by the stresses in the shell due to its deformation, which results in some lower limiting value of lR. The search of models for factors preventing the aggregates from dispersion to monomeric units is both an important and difficult part in understanding the aggregation phenomena. The discussion in the literature relates mainly to the consideration of deformational terms,26 whose basic effect in a revised version of asphaltene model35 is to prevent the aggregates from infinite growth. Modeling of very small aggregates is still a great problem: surface tension can no longer be considered as curvature independent, the state of asphaltene molecules in a small aggregate deviates dramatically from that in the bulk asphaltene phase, and approximation 4 becomes invalid. The free energy of a very small aggregate is essentially a functional of its shape, and simplified phenomenological models can no longer be applied. The only approaches so far, which seem to be eligible in this area, are those based on the lattice model.36 However, as it was first shown in an excellent paper by Israelachvili et al.,24 some simple geometrical considerations interfere into thermodynamics serving as conditions, which restrict the possible shapes of the aggregates. Not only spherical aggregates were considered, and it was shown that their shape is related to the values of the molecular shape factor. For large aggregates with a spherical core inside, the shape factor must have values somewhere between 1 and 2, with Sh ) 1 corresponding to a planar layer and Sh ) 2 to cylindrical particles (values close to 3 and greater are likely to result in the formation of very small spherical aggregates). Below we present some estimates, taking into account the packing constraints for spherical asphaltene aggregates. Although there are no direct data on the geometry of resin molecules, one can give some estimates of likely molecular characteristics using available experimental data and the above formulas. Having in mind that the aggregates with large asphaltenic cores are observed experimentally,18 we take Sh = 1.7 as a very rough estimate. Of course this estimate cannot be precise, also because the oil molecules penetrate into the resin shell, which increases vR and hence shifts Sh to somewhat higher values, promoting the formation of small spherical particles; however, such a shift cannot occur without bounds and must be controlled by the deformation of resin tails. With this value of shape factor we get R/lR = 1.7 for the core radius to shell thickness ratio. For the asphaltene monomer molecular volume we take va ) 1320 Å3 (based on asphaltene density and the monomer molecular weight experimental data by Sheu:16 F ) 1.1 g/cm3; MWa ) 857). The series of R and lR values satisfying eqs 11 and 15 for different values assigned to n1 is presented in Table 1. According to the model Θ* is uniquely determined by the nature of the system and the value of Xr1. The results below relate to Θ* ) 0.8, the typical value obtained in our calculations for asphaltenic crudes;31,33 i.e., the calculated optimal area tended to be 20% greater than a0. Choosing a0 ) 40 Å2, as in the previous

Table 1. Hypothetical Geometric Parameters of Asphaltenic Agglomerates Matching the Packing Constraints

works, we obtain n2 from eq 12 (Table 1). vR/a0 is calculated by eq 16. From vR/a0 one can find vR values satisfying the packing constraint (at given a0) for every radius R. Obtained vR values change with R, and this may be attributed to our assumption of constant Θ* ) j R (and hence a j R). R values in the range from 20 to a0/a 32 Å correspond to the aggregate radii of 32-51 Å (Table 1) and are in line with spectroscopic observations for asphaltenic particles.18,19 With these values and a0 ) 40 Å2 we conclude that vR must lie in the interval 10001600 Å3. To the best of our knowledge, the only estimates of this latter quantity that can be made using the literature data are vR ) 1167 Å3 (F ) 1.21 g/cm3, MWr ) 850)35 and vR ) 1093 Å3 (F ) 1.013 g/cm3; MWr ) 665),37 both of them lying within the estimated interval. The center of this interval (i.e., vR = va = 1300 Å3) can be taken as a rough estimate for resin molecular volume. We now fix the vR value and for vR/a0 from Table 1 find a0 values that satisfy the packing constraint. We get a series of a0 values (the last line in Table 1), starting from unlikely high values corresponding to small aggregates and ending with unlikely low ones for big aggregates. To summarize our speculation, the approximate intervals for the most likely parameter values for a representative asphaltenic aggregate are within the window indicated in Table 1. Even though the estimates presented above are very approximate, and cannot be regarded as reliable, sometimes they can be helpful. It seems, for instance, that, with the geometric characteristics of resin molecules used by Pan and Firoozabadi,35 it is difficult to conform the obtained large aggregation numbers (n1 g 100) to the requirement of packing. Taking from this reference vR ) 1167 Å3 and cross-sectional areas of 21 and 100 Å2 for the resin molecule head and tail, respectively, one can obtain lR = 21 Å as an estimate of the resin molecule length. Then for a close packed resin shell Sh = 2.6, which leads to R = 17 Å and n1 = 15. For getting larger n1 values the packing constraints suggest a somewhat smaller difference in head and tail cross-sectional areas for resin molecules. Possible penetration of petroleum molecules into the resin shell can only increase Sh, and thus the particles with even smaller radii will tend to form. Nevertheless the application of packing constraints to asphaltenic crudes is an obvious oversimplification and remains problematic (even if we had reliable estimates of molecular characteristics). There is specificity in packing effects for asphaltenic systems, especially in view of the structural diversity of species, as discussed by Sheu,16 and these effects are mainly responsible for the differences in the nature of phenomena observed in asphaltenic crudes and in surfactant systems. But both types of systems have much in common, aggregation being of primary importance for them. The explicit account of the influence of packing constraint on the standard Gibbs energy of aggregation (and hence on the aggregate distribution) was given by Israelachvili et al.24 for the case of formation of a

3246 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998

bilamellar vesicle. It was shown that the leading role is played by the outer monolayer packing constraint and that as this constraint comes into play for some small enough radii, it leads to an increase in the standard energy, which can be expressed as

G - Gplan. ) 4πγlc2[1 - Rout/Rcout]2

(17)

where Gplan. is the energy of a planar bilayer, γ is the interfacial tension of the aggregate, lc is the length of a fully extended tail of an amphiphilic molecule, Rcout is the outer radius of the aggregate obeying the packing constraint (equivalent to eq 15), and Rout is some actual outer radius, such that Rout < Rcout. Although eq 17 was derived by consideration of the electrostatic repulsion between the amphiphile molecule heads residing in the outer surface of the vesicle, any repulsion potentials proportional to constant/a between the parts of the amphiphiles in the outer shell will lead to the same result, as pointed out in the paper cited above.24 Utilizing this result for our case, which is similar to a unilamellar reverse vesicle, we have to substitute lR for lc, σ0(1 - Θ) for γ, and (R + lR) for Rout, respectively. Gplan. represents the energy of a planar resin monolayer, and Rcout is obtained by applying eq 15. Equation 17 leads to a nearly Gaussian size distribution with a standard deviation, given approximately by

s ) (1 + R/lR)[kBT/8πγ]1/2

(18)

where R/lR can be calculated from eq 15. Using Sh ) 1.7, Θ* ) 0.8 (as above), and σ0 ) 0.04 N/m,31 one obtains s = 4 Å. The distribution peaks at a radius close to, but slightly smaller than, Rcout. In order to obtain the size of the aggregates (Rcout) from eq 15, one has to assume, apart from Sh, some value for lR. In the described model the effects of deformation of the resin molecule tails on the distribution of the aggregates over the size and shape are expressed in terms of packing constraints. Another possible, though quite phenomenological, approach is to consider curvature corrections through the bending energy contributions, as proposed originally by Helfrich,38 Miller and Neogi,39 and Mukerjee et al.,40 among the first who attempted to quantify the bending effects for a dispersed system, and the approach was further developed for various aggregated systems by Turkevich et al.25 The approach focuses on bending of the aggregate shells from some natural state (with a curvature, unperturbed by the entropy effects of aggregation) to the state, in which the film is deformed to its actual configuration in equilibrium solution. The corresponding contribution to the standard aggregate free energy of aggregates (the difference between free energies of the spherical film having natural radius of curvature, R0, and the one having the curvature 1/Rout) can be written as

h )[1 - Rout/R0]2 ∆F00 bend ) 4π(2k + k

(19)

where k and k h are, respectively, the bending (splay) and saddle-splay (Gaussian) elastic constants of the resin film. As suggested by Israelachvili et al.,24 the corrections for curvature can be included into the Gplan. term of eq 17, leaving the right hand side of the equation unchanged. If the corrections to Gplan. are included for the natural curvature, then ∆F00 bend ≈ G - Gplan.. This indicates the parallelism of eqs 17 and 19 and allows

the estimation of the elastic constants as

(2k + k h ) ) σ0(1 - Θ*)lR2

(20)

h ) values (with This gives several units of kBT for (2k + k the estimates of lR from Table 1 we obtain ∼2-7 kBT), showing that there is a substantial energy cost to deform the resin film from its natural curvature, which is likely to lead to a pronounced maximum in the distribution of aggregates over the size. A number of molecular approaches exists relating elastic constants to molecular characteristics of amphiphile molecules at the surface. Making use of a simplified treatment proposed by Turkevich et al. (eq 5 of the original ref 25 with different notation for the elastic constants) and assuming that the resin molecule polar head is much shorter than its tail, we obtain

k h /(2k + k h ) ≈ -lR/R0

(21)

χ ) -k h /(2k + k h ) ) 1/(R/lR + 1)

(22)

This gives

and demonstrates the relation between the elastic constants and the shape factor, connected to R/lR by eq 15. For Sh ) 1.7 we get χ = 0.37. Taking as before, Θ* ) 0.8, σ0 ) 0.040 N/m, and lR from within the interval framed in Table 1, we find 1.4 ÷ 5 kBT and -0.7 h , respectively. (The values found ÷ -2.6 kBT for k and k obey the stability condition requirement, k h /k > -2). From the studies of the influence of χ on the allowed shapes of particles,25 it follows that for the value given above the only allowed shapes are spheres and lamellae (for cylindrical particles to be allowed χ must be smaller than ∼0.33). This conclusion is in agreement with spectroscopic observations of spherical and platelike aggregates in asphaltenic systems.18,19 Obviously, eq 19 will also lead to a near Gaussian size distribution with the standard deviation, whose relation to the elastic constants is readily obtained by combining eqs 18, 20, and 22 to give

s)-

[(2k + k h )]1/2 lR[kBT/8π]1/2 k h

(23)

Because R is an increasing function of lR, s also increases with the most probable particle size. The characteristics of the system polydispersity can readily be connected to the spectroscopic experimental information.16,29,41,42 A detailed molecular statistical derivation of elastic properties of chain-molecule films based on the consideration of conformational statistics of chains was given by Szleifer et al.,43 who expressed elastic constants in terms of the lateral pressure and its derivative with respect to curvature for planar mono- and bilayers and demonstrated the crucial role of chain repulsion in determining the shape of the film. To use this result, one, however, has to obtain the local profiles of properties inside the film. Another, but similar, way of using the packing constraint to express size distribution of aggregated particles is due to Bergstrom,44 who was the first to take explicitly into account the adjacent solvent molecules in the micellar shell and considered chain packing density and shape fluctuations. The Gaussian size distribution is obtained for spherical aggregates as a

Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3247

consequence of the quadratic approximation for the fluctuations of the aggregate free energy around its equilibrium value due to a fluctuating number of chains in the micellar core (for nonspherical particles similar approximation is employed with respect to the optimal surface area), which, in essence, is another way of writing the Helfrich harmonic approximation for the bending energy. But again, without the knowledge of the optimal free energy (and size), one is merely able to analyze the distribution around this value, leaving unsolved the question of how to get this size. The discussion given above shows the relation of different approaches for studying aggregate distribution and illustrates how phenomenological elastic constants are related with molecular packing parameters. Both approaches lead to the same principal features of distribution, and it is possible to estimate the parameters of one approach, having these for the other. Though all these studies are convincing that the essence of the effect lies in the deformation of resins in the shell (the asymmetry in the interaction between the resin molecule heads and their tails giving rise to the bending stresses in the shell), the same problem persists in applying the approaches: not a good enough knowledge of resins. 4. Incorporation of the Equation of State and Asphaltene Dropout Calculations If there were a need to estimate the solvophobic parameter Xons a1 at every state of the crude, the model described above could not have been used for any predictions at all. In fact this parameter can be estimated using conventional bulk-phase equations of ons ) at some specified temperastate, given its value (Xa10 ture, pressure, and composition of the crude. If we consider the bulk of the petroleum crude discarding for the moment all the aggregates present, then for the equilibrium between the crude and the asphaltene precipitate (phase γ) one can write

dµγa(T,Pβ) ) d[kBT ln fa1(T,Pβ,Xons a1 )]

(24)

where fa1(T,Pβ,Xons a1 ) is the fugacity of monomeric asphaltenes in the petroleum fluid. Since, according to our model, most of the asphaltene material resides inside the aggregates, not in its monomeric form, one can use common bulk-phase equations of state (like density cubic ones) to estimate the fugacity of asphaltenes. The extreme polarity of asphaltenic material does not prevent one from applying such equations of state, because the bulk crude remains strongly diluted with respect to asphaltenes up to their considerable gross concentrations, and many crudes (except heavy bitumen) will meet this requirement. For a titration of asphaltenic material at some specified T, P0 from eq 24 we get

Xons a1 (T,P0,t) ) ons (T,P0,t)0) Xal0

φβa1(T,P0,t)0)/φβa1(T,P0,t) (25)

where t is the dilution ratio (the ratio of the volume of titrant added to the amount of the crude taken initially for the titration) and φβa1 is the fugacity coefficient of monomeric asphaltenes in the bulk of the crude. The latter was calculated with the aid of the Peng-Robinson equation of state. Equation 25 predicts how Xons a1

changes when various titrants (e.g., n-alkanes C5, C6, etc.) are added to the crude. The influence of pressure is estimated by virtue of eq 24 under the assumption of the asphaltene-phase incompressibility:

φβa1(P0)P0

ons Xons a1 (P) ) Xa10(P0)

φβa1(P)P

(

exp

)

va(P - P0) RT

(26)

Thus, the equation of state is included in the standard Gibbs energy of aggregation via the solvophobic parts the main driving force of the aggregation process (an approximation, since other parameters of the aggregation model are certainly also equation of state dependent). The first version of the model has been formulated for the case of monodisperse platelike aggregates and applied to describe asphaltene deposition from a number of real crudes.31 Liquid titration with n-alkanes of various chain lengths, gas titration, and the effect of pressure on the asphaltene deposition have been considered and promising results obtained. The predicted pressure effect is in line with experimental observations: the model predicts asphaltene dropout upon depressurizing the crudes and gives reasonable estimates of the amount of precipitate. The monodisperse version of the model reflects the basic features observed for asphaltene titration. It gives proper amounts of asphaltene precipitate and predicts correctly the change in the precipitation power of different alkane precipitants with their molecular mass. However, there was essential arbitrariness in the choice of the aggregation number, the results of modeling being rather sensitive to its value. Consideration of a polydisperse version of the model for platelike aggregates33 has shown that spherical aggregates are likely to play an important role (at least in the frame of the approach) and have to be taken into account. The present version of the model describes liquid titration and the influence of the shape of resin molecules on the asphaltene dropout in the crudes containing polydisperse spherical aggregates. 5. Results of Calculation The following calculation scheme (see Appendix) was adopted. With the values assigned to the model paons , ∆Ur, σ0, a0, and va, the optimum coverrameters Xa10 age Θ* is estimated by eq 9, given some initial estimates for monomer mole fractions Xa1 and Xr1. For every n1, and corresponding n2 value obtained by eq 13, the standard Gibbs energy of aggregation is calculated (with ons Xons a1 found from Xa10 by means of eq 25 and the PengRobinson equation of state). For given n1 the concentration of aggregates having optimal composition is obtained as n1 n2 XM(n1,n2) ) (Xa1/Xons a1 ) (Xr1 exp[f(Θ*)]) exp(-Gcor) (27)

with

f(Θ*) ) ln(1 - Θ*) +

∆Ur σ0a0(1 - Θ*) (28) kBT kBTΘ*

where Gcor is the correction for deformation of the resin shell. Its value can be computed according to eq 17 or 19, as discussed above, given the resin molecule volume

3248 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998

Figure 1. Liquid titration curves for tank oil with n-heptane (295 K) calculated by different versions of the model: monodisperse spherical aggregates with n1 ) 40 (solid curve 1); polydisperse spherical aggregates with Sh ) 1.3 (solid curve 2); polydisperse platelike particles with the average aggregation number n1 = 42 (dotted curve);33 monodisperse platelike aggregates (dashed curve).31 Points are the experimental data.10

and shape factor, or the elasticity constants. Note that, for the monodisperse case, where all the aggregates are supposed to have some optimal size, the last factor in the right hand side of eq 27 can be dropped. Equation 1, in which the summation over all n1 values is performed numerically, is solved iteratively, together with eq 9, for monomer concentrations (Xa1, Xr1) and Θ*. Having obtained the average aggregation numbers as ∞

〈n1〉 ) (Xa - Xa1)/

∑ XM(n1,n2)

n1,n2 ∞

〈n2〉 ) (Xr - Xr1)/

∑ XM(n1,n2)

(29)

n1,n2

where the summations are performed as discussed above, one can estimate the mean radius of the aggregate population from eq 11. Upon addition of a little amount of titrant, a new composition of the crude is obtained, and the whole procedure is repeated. Inequality Xa1 > Xons a1 signals the asphaltene precipitation, and the material balance equations are corrected for the presence of pure asphaltene precipitate. The above procedure has been applied to model titration of asphaltenes from tank oil by liquid nalkanes.10 The composition of the tank oil was modeled in the same way as before31 using the Whitson characterization scheme45 and the Cotterman, Bender, and Prausnitz quadrature technique4 to perform lumping. The same parameter values of the equation of state were

Figure 2. Influence of the assumed resin molecule shape on the predicted liquid titration curve for tank oil with n-heptane (295 K): Sh ) 1.7 (dashed curve); Sh ) 1.2 (solid curve). Points are the experimental data.10

used for monomer fugacity calculations. The other parameters retained their values from the monodisperse ons ) 2.4 × 10-3, ∆Ur ) version of the model,31 viz., Xa10 2 0.073 J/(mol m ), σ0 ) 0.040 N/m, and a0 ) 40 Å2, with the only difference being that available estimates for asphaltene monomer volume and its molecular mass (va ) 1320 Å3 and MWa ) 85716) have been used in the present work. We took vR )1100 Å3 for the resins.35,37 The results of modeling of the titration of tank oil by liquid n-heptane are given in Figure 1. It can be seen that different versions of the model can lead to similar precipitation curves provided that corresponding values of the parameters are chosen. However, in the frame of the model of polydisperse spherical aggregates the predicted behavior of the asphaltenic mixture can be quite different, depending on the assumed value of the shape factor for resins. This is illustrated in Figures 2-4. The higher the shape factor is, the smaller is the amount of precipitate, and the more the onset of precipitation is shifted toward lower dilution (so that the crude becomes originally unstable with respect to asphaltene dropout at Sh ) 1.68). Low shape factors retard the onset and result in more asphaltene material to dropout (Figure 2). The trends in the dependence of the average aggregation numbers upon dilution are quite different for different shape factors (Figure 3ac). In the case of high shape factors, small aggregates are initially present in the crude and they slowly tend to be more dispersed (Figure 3a), coexisting with the precipitate, as the crude is diluted; cmc is observed at high dilution. At small shape factor values the ag-

Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3249

a

b

c

Figure 3. Influence of the resin molecule shape on the calculated average asphaltene aggregation numbers for titration of tank oil10 with n-heptane: Sh ) 1.7 (a), Sh ) 1.5 (b), Sh ) 1.2 (c).

gregates reveal rapid growth (Figure 3c) until a large amount of precipitate forms accompanied with abrupt destruction of large aggregates. A delicate balance between all different factors of aggregation and asphaltene dropout results in a maximum of the average aggregation number versus dilution curve at some intermediate shape factors (Figure 3b). As seen from the figure, the asphaltene aggregates first grow, and as the asphaltene precipitate forms, the aggregates gradually diminish their size until the cmc. Figure 4a shows the distribution of the aggregates over the size corresponding to the situation just before the onset of precipitation in the system of large aggregates (Figure 1, Sh ) 1.3). The size distribution for the case of small aggregates (Sh ) 1.5) is depicted in Figure 4b. As might be expected, the distribution is much narrower for small aggregates, while the ensemble of big aggregates reveals essential polydispersity. In the latter case the skewness of the distribution (making the average aggregation number to exceed the most probable value) is noticeable. The resin aggregation numbers are calculated within the approximation of optimum aggregate composition, eq 13, and are monotonous functions of n1 at every solution composition. Distributions have similar shape for asphaltenes and resins. 6. Conclusions A new version of the thermodynamic aggregation model of asphaltene-containing crude is formulated and analyzed in connection with the characteristics of resin

molecule shape and deformation of the resin shell of asphaltene aggregates. The relation between the shape factors and elastic constants is discussed, and their effect on the size distribution of asphaltene particles is studied. It is shown that both approaches lead to essentially the same general features of asphaltene particle distribution. While the practical determination of resin molecule shape factors and use of the packing constraints are rather problematic, owing to the diversity of resin species in petroleum fluids, the phenomenological approach based on resin shell elasticity includes a variety of molecular mechanisms and thus can be applied with more confidence. This seems promising and to a certain extent can be regarded as a validation of the packing constraint scheme and shape factors usage. The model has been applied to describe asphaltene dropout upon diluting the petroleum crude with a liquid alkane. The effect of the dilution on the asphaltene aggregate size distribution is studied. It is shown that dramatic difference in the effect of the dilution and in the crude stability with respect to dropout is predicted by the model, depending on the molecular geometry of resins. This may be helpful in the search of stabilizing agents for asphaltene dispersions. According to the model the bipolar agents with rather wedge-shaped molecules will promote dispersion into smaller aggregates (but will cause earlier precipitation upon dilution and diminish the amount of the precipitate formed), whereas the molecules closer to cylindrical shape will promote formation of larger aggregates, their growth upon dilution, and a sudden dropout of a greater

3250 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998

a

Scheme 1

b

Figure 4. Distribution of the asphaltene aggregates over the size: at the onset of precipitation, Sh ) 1.3 (a); before titration Sh ) 1.5 (b).

amount of precipitate. Our results once more illustrate the crucial role of resins in the dispersion and stability of asphaltenic crudes. Certainly the model relies on a number of assumptions and approximations, as discussed above. Several assumptions of the present model could be tested more systematically or removed, if there were more experimental data on aspahltenic systems. Better knowledge of the molecular characteristics of resins (masses and average geometrical characteristics), data on heats of adsorption of resins on asphaltenes, and information on petroleum fluid/asphaltene interfacial tensions would be of primary importance for further verification and improvement of the approach. Another problem is connected with the assumption that the asphaltene precipitate is a pure substance. This assumption, useful for estimation of Xons a1 from eq 24, implies the constancy of the asphaltene chemical potential in titration experiments (at specified temperature and pressure), a strong approximation, whose validity is still needed to be verified. Studies of the effects of pressure on the asphaltene polydispersity are among the prospects for future work. Acknowledgment The authors are thankful to the Russian Foundation of Basic Research for financial support under Grants 96-03-33992a and 96-15-97399. Appendix The calculation scheme is given in Scheme 1.

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Received for review December 1, 1997 Revised manuscript received May 11, 1998 Accepted May 14, 1998 IE970869Z