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Fluid Phase Equilibria 286 (2009) 113–119

Contents lists available at ScienceDirect

Fluid Phase Equilibria

journa l homepage: www.e lsev ier .com/ locate / f lu id

redicting adsorption isotherms of asphaltenes in porous materials

artín Castroa, José L. Mendoza de la Cruza, Eduardo Buenrostro-Gonzaleza,imón López-Ramíreza, Alejandro Gil-Villegasb,∗

Instituto Mexicano del Petróleo, Eje Central Lázaro Cárdenas Norte 152, Col. San Bartolo Atepehuacán, C.P. 07730 México, D.F., MexicoDepartamento de Ingeniería Física, DCI Campus León de la Universidad de Guanajuato, Lomas del Bosque 103, Colonia Lomas del Campestre, León, Guanajuato,.P. 37150 México, Mexico

r t i c l e i n f o

rticle history:eceived 26 February 2009eceived in revised form 17 July 2009ccepted 13 August 2009vailable online 22 August 2009

eywords:AFT-VRdsorption isothermsorous materialssphaltenes

a b s t r a c t

In this paper we present a molecular thermodynamics approach for the modeling of adsorption isothermsof asphaltenes adsorbed on Berea sandstone, Bedford limestone and dolomite rock, using a model forbulk asphaltenes precipitation and a quasi-two-dimensional approach for confined fluids [E. Buenrostro-González, C. Lira-Galeana, A. Gil-Villegas, J. Wu, AIChE J., 50 (2004) 2552–2570; A. Martínez, M. Castro,C. McCabe A. Gil-Villegas, J. Chem. Phys. 126 (2007) 074707, respectively], both based on the StatisticalAssociating Fluid Theory for Potentials of Variable Range [A. Gil-Villegas, A. Galindo, P.J. Whitehead, S.J.Mills, G. Jackson, A.N. Burgess, J. Chem. Phys. 106 (1997) 4168–4186]. The theory is applied to modeladsorption isotherms from experimental data of asphaltenes extracted from a dead sample of heavycrude oil from a Mexican reservoir. The theoretical results give the right Langmuir Type II adsorptionisotherms observed experimentally. The model requires the determination of ten molecular parametersrelated to the size of the particles and the square-well potentials used to describe the particle–surface andparticle–particle interactions at the bulk and adsorbed phases. Nine parameters are taken from previous

published results about the behavior of asphaltenes in bulk phases and the adsorption of several molecularfluids onto activated carbon and graphite surfaces. The remaining parameter, the energy strength of theparticle–surface interaction, is adjusted to reproduce the experimental data, obtaining values that areconsistent with Molecular Mechanics calculations for asphaltenes adsorbed on different surfaces andsolutions. Although the agreement between theory and experiments shows some deviations at low bulkconcentrations, the model reproduces adsorption data at high concentrations where other semi-empirical approaches fail.

. Introduction

Asphaltenes adsorption on reservoir rocks modifies the relativeroportions of the total pores surface area which are in contact withhe formation fluids [1,2], and reservoir wettability is a major fac-or controlling the location, fluid distribution, and flow propertiesf the system [3,4]. The subject of surfactant adsorption on polarurfaces from non-polar and weakly polar non-aqueous solutions isargely unexplored [5], and in the case of adsorption of asphaltenes,

he nature of those processes is still not understood [1,6–9].

The liquid phase adsorption is a very complex phenomenonue to the presence of solvent molecules, formation of molecu-

ar aggregates, irregular packing and multilayer coverage [8,10]. In

∗ Corresponding author. Tel.: +52 4777885100x8415;ax: +52 4777885100x8410.

E-mail addresses: mgcastro13@gmail.com (M. Castro), jlmendoz@imp.mx (J.L.M.e la Cruz), ebuenro@imp.mx (E. Buenrostro-Gonzalez), slopezr@imp.mx (S. López-amírez), gil@fisica.ugto.mx (A. Gil-Villegas).

378-3812/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.fluid.2009.08.009

© 2009 Elsevier B.V. All rights reserved.

the case of asphaltenes, the extent of their adsorption on mineralsurfaces relies on an important way on their tendency to aggre-gate and separate from the crude oil in response to changes in oilsolvency.

Asphaltenes are the most heavy and polar fraction in the crudeoil. The asphaltene fraction is formed by many series of relativelylarge molecules containing aromatic rings, several heteroaromaticand napthenic ring plus relatively short paraffinic branches [11,12].The adsorption of asphaltenes on solids is the result of favorableinteractions of the asphaltene species or its aggregates with chem-ical species on or near the mineral surface. Different interparticleforces are responsible of this effect, individually or due to the inter-play between them. The major forces that can contribute to theadsorption process are electrostatic, charge transfer, van der Waals,hydrogen-bonding and steric interactions [4,13,14].

Adsorption of asphaltenes on various dry mineral surfaceshas been the subject of many investigations [15]. As a result ofthese studies, both monolayer [1–4,16–18] and multilayer [4,5,7,8]adsorption behavior have been reported, depending on the sol-vent and source of asphaltenes used in experiments. For studies

114 M. Castro et al. / Fluid Phase Equilibria 286 (2009) 113–119

Table 1Properties of mineral samples used to study adsorption of asphaltenes in porous materials.

Composition (wt.%) Particle sizea (�m) Surface area, F (m2/g) Average pore diameter (Å)

osattmnftohotd

(oWbasttaftt

oVdidpdabv[

2

2

MtAwfuac[am

Bedford limestone Calcite, 97.3 SiO2, 1.7 Al2O3, 0.5 MgCO3, 0.4Berea sandstone SiO2, 93.1 Al2O3, 3.86 FeO, 0.54 MgO, 0.25 Fe2O3, 0.11Cantarell field Sample Dolomite 99.0

a Particles retained between sieve 50 and 60.

n the influence of aggregate formation on adsorption, higherolution concentrations are required. Higher concentrations ofsphaltene complicate the problem due to multilayer formation,ime dependence, aggregate formation, and precipitation [1]. Dueo the complexity of the asphaltene fraction composition and the

olecular structure of the species involved, it is well known that isot an easy task to generate molecular-based equations of state

or this substance. Usually, a whole series of aspects must beaken into account in order to satisfactorily obtain an equationf state for its use in industrial applications, and a compromiseas to be made among of all them: the primitive model devel-ped, the accuracy of the equation of state obtained, and finally,he capability of this equation of state to predict the right phaseiagrams.

Over the last decade, the Statistical Associating Fluid TheorySAFT) [19,20], based on the thermodynamic perturbation the-ry for fluids with highly anisotropic interactions developed byertheim [21–26], has been extensively used to calculate the phase

ehavior of associating and non-associating fluid systems. The SAFTpproach has proven to be a powerful molecular-based equation oftate and has been applied to a wide variety of complex and indus-rially relevant fluid systems. Over the years, several variations ofhe theory have been proposed and have improved the scope ofpplications and predictive power of the method, such as densityunctional theories to study the liquid–vapor interface and proper-ies of associating fluids [27–32]. See Refs. [33–36] for reviews onhe SAFT approach.

One of the more versatile versions of SAFT introduces the usef a variable-range potential to describe real substances (SAFT-R, see Refs. [37]). This approach has been recently adapted toescribe the adsorption of associating and non-associating flu-

ds [38,39], based on a quasi-two-dimensional approximation toescribe adsorbed fluids. Using this approach, in this paper weresent predictions for asphaltene adsorption isotherms on threeifferent rock minerals (Berea sandstone, Bedford limestone andreservoir rock) at a defined particle size of 297 �m (mesh + 50),ased on a primitive model for asphaltenes that has been pre-iously used to model asphaltene precipitation in bulk phases40].

. Experimental

.1. Asphaltenes

Asphaltenes were extracted from a dead crude oil sample of aexican reservoir fluid, heavy crude oil from a marine region of

he Mexican coast, by precipitation with n-heptane following theSTM D3279-97 method. The asphaltene content of the crude oilas 13.6 wt.%. The average number molecular weight of asphaltene

raction, Mn, was 779.7 g/mol and it was obtained experimentallysing the vapor pressure osmometry technique (VPO) with toluenet 50 ◦C by linear extrapolation from five Mn data measured in theoncentration range from 4 to 10.4 g/kg as described elsewhere41]. The content of Ni and V in the asphaltene fraction was 413.4nd 1731.0 mg/kg, respectively, determined by atomic absorptionethod (EPA-6010C).

250–300 1.15 102250–300 0.86 143250–300 1.19 164

2.2. Mineral samples

The rock types used in this work as adsorbent materials wereBedford limestone, Berea sandstone, and a dolomite rock samplecollected from the Cantarell field at the Gulf of Mexico. The samplesof powdered minerals (silica, calcite and dolomite) were preparedby crushing rock samples in laboratory followed by grinding in anagate mortar. The powder then was divided into fractions with a setof sieves (mesh sieve numbers 50, 60, and 70). The cleaning of all thepowdered minerals was performed by means of Soxhlet extractionwith toluene for 24 h; afterwards, they were dried in an oven withvacuum system at 373 K during 24 h and cooled in a dessicator. Thespecific BET surface area (F) and average pore diameter of adsorbentmaterials was determined by analysis of nitrogen adsorption. Thecharacteristics of each mineral sample are shown in Table 1.

2.3. Adsorption experiments

Static adsorption experiments were carried out at 291 K in aseries of vials containing equal volumes (V) of asphaltene-toluenesolutions at varying initial concentrations (C0

b) within the range

10–30,000 ppm (mg/L) and equal mass (q) of mineral sample,following the procedure described in a previous work [42]. Theamount of adsorbed asphaltene, � ads, was calculated from a massbalance relation, �ads = V(C0

b− Cb)/q, where Cb is the equilibrium

concentration which is calculated from the measuring of UV–visabsorbance of the solution at 400 nm wavelength [2,5,43].

3. Theory

Recently, a theoretical framework has been presented to studythe behavior of adsorption of fluids [38,39] using the SAFT-VRapproach [37] and a quasi-two-dimensional modeling of adsorbedphases. We summarize here the main expressions of this theory,for more details see Refs. [38,39].

3.1. Adsorption of monomeric fluids

The system model is a simple-component fluid in the presence ofa uniform wall. The fluid consists of N spherical particles of diam-eter �. Due to the wall, the behavior of the particles is differentdepending upon their distance to the wall. The interaction exertedby the wall on a particle is given by a square-well potential

upw(z; �, �w, εw) ={ ∞ if z ≤ 0

−εw if 0 < z ≤ �w�0 if z > �w�

(1)

where z is the perpendicular distance of the particles from the wall,εw and �w� are the energy depth and the range of the attractivepotential, respectively. Hence we can describe the system as beingcomposed of two subsystems: (i) a fluid whose particles are near tothe wall, i.e., when z ≤ �w�, which we shall refer to as the “adsorbed

fluid”, and (ii) a fluid whose particles are far from the wall, i.e.,the “fluid bulk”. In this way, the length scale that characterizes theadsorbed fluid is given by �w�. This approach is formally valid if wedo not take into account the interface between the adsorbed andbulk fluid.

e Equi

tiippwtawuS(p

bc

�

w�sa

�

ws

�

Ew2

tto

�

Tdf�

f

Z

wBcu

ab

Q

wt

Q

M. Castro et al. / Fluid Phas

The adsorbed and bulk fluids have different properties due tohe wall; it is well known that the interaction between moleculess modified by the presence of a surface and therefore the pairnteraction between particles is different for the adsorbed and bulkhases [46]. We denote upp(r;εb,�b) and uads

pp (r; εads, �ads) as the pairotential for particles in the bulk and adsorbed phases, respectively,here εb and �b (or εads and �ads) are parameters that describe

he energy depth and the range of the potential for the bulk anddsorbed particles, respectively. We will discuss the general frame-ork of the theory that allows to use different potentials models for

pp(r;εb,�b) and uadspp (r; εads, �ads). For the applications presented in

ection 4 we will restrict the discussion to square-well potentialsSW) for the particle–particle interactions at the bulk and adsorbedhases.

Due to the presence of the wall, the fluid particle density, �, wille a function of z. To describe the amount of adsorbed particles theoverage � is introduced and defined as

=∫ ∞

0

dz[�(z) − �b] (2)

here �b is the density of bulk particles, i.e., �(z → ∞),b = Cb(NAV/Mn), where NAV is Avogadro’s number. Since the lengthcale of the adsorbed fluid is defined by �w�, we can rewrite Eq. (2)s

=∫ �w�

0

dz�(z) − �b�w� (3)

here the integral of �(z) in the right-hand term of Eq. (3) is theurface of adsorbed density:

ads =∫ �w�

0

dz�(z) (4)

q. (4) can be identified with a two-dimensional density, in such aay that the properties of the adsorbed fluid are represented as a

D fluid.Since in thermodynamic equilibrium, the chemical potentials of

he adsorbed (�ads) and bulk (�b) phases must be equal for a givenemperature T and bulk pressure P, the adsorbed density can bebtained by solving the following equation:

ads = �b (5)

he formal identification of the adsorbed fluid as a 2D system can beescribed as a decoupling of x, y coordinates from the coordinate zor each adsorbed particles. In this way we can define a 2D potential(x,y;ε2D,�2D) that depends on the coordinates parallel to the wall.

According to the definitions introduced previously, the partitionunction of the adsorbed fluid is given as

ads = VNads

N!3NQads (6)

here Vads is the volume containing the adsorbed fluid, is the deroglie thermal wavelength = h/(2mpkT)1/2 in terms of Planck’sonstant h and the mass mp of the particles, and Qads is the config-rational partition function.

By introducing the adsorption area S, we have that Vads = �w�,nd after several algebraic manipulations (see Ref. [39]), Qads cane rewritten as

ads = Q1DQ2D (7)

here Q1D and Q2D are the one- and two-dimensional configura-ional partition functions, respectively, that are given by

1D = e−Nˇupw(z∗) (8)

libria 286 (2009) 113–119 115

where z* is the value of the coordinate z that guarantees the meanvalue of the Boltzmann factor, and

Q2D = 1SN

∫dxNdyNe−ˇ/2N(N−1)�(x,y) (9)

Applying the standard relation A = −kT ln Z, the Helmholtz freeenergy of the adsorbed fluid is given by

Aads

NkT= A2D

NkT− ln

(�w�

)+ ˇupw(z∗) (10)

where A2D is the Helmholtz free energy of a two-dimensional fluidinteracting via the potential �(x,y), which can be described by asecond-order perturbation theory:

A2D

NkT= ln(�ads

2) + AHD

NkT+ ˇa2D

1 + ˇ2a2D2 (11)

Here AHD is the excess Helmholtz free energy for a fluid of harddisks, ˇ = 1/kT, and a2D

1 and a2D2 are the first two terms of the per-

turbation expansion in 2D. The expression given for Q1D is exact,and this information appears in Eq. (14) as an external field thatessentially can be used to define an effective energy parameter ofthe wall. For the case where the wall–particle interaction is givenby a square-well interaction of range �w� and energy depth εw , wehave that upw(z∗) = εw .

For the bulk fluid, we consider an analogous second-order per-turbation expression:

A3D

NkT= ln(�b3) + AHS

NkT+ ˇa3D

1 + ˇ2a3D2 (12)

The chemical potentials �ads and �b can be obtained from Eqs.(10) and (12), respectively. It is important to note that the methodpresented here can be extended to describe the adsorption phe-nomena when the particle–particle and wall–particle interactionsare described by discontinuous potentials of more general formthan the square-well (SW) potential. A general theory for 3Ddiscrete-potential fluids has been presented by Benavides andGil-Villegas [44,45] that can accurately describe the properties ofLennard–Jonessium systems, and it has been also developed for 2Dfluid systems [39].

3.2. Extension to associating molecules

The case of adsorption of associating chain molecules isdescribed by Eq. (10) where the 2D Helmholtz free energy is givennow in terms of different contributions, according to the SAFTapproach:

A2D

NkT= Aideal

2DNkT

+ Amonomer2DNkT

+ Achain2D

NkT+ Aassoc

2DNkT

(13)

Expressions for the ideal, monomeric, chain and associating con-tributions have been obtained previously for 2D fluids (see Ref.[39]). We summarize here this information:

1. Ideal contribution:

Aideal2D

NkT= ln(�ads

2) (14)

2. Monomer contribution:

Amonomer A2D

2DNkT

= mNskT

(15)

where A2D is the Helmholtz free energy for 2D monomeric flu-ids, i.e., systems composed by Ns spheres interacting via a SWpotential, and m is the number of segments forming each chain

116 M. Castro et al. / Fluid Phase Equilibria 286 (2009) 113–119

Fstb

3

4

X

wd

�

ImMdb

4

iavmdicsat

Table 2Values of interaction parameters for asphaltenes in bulk phases.

ig. 1. Experimental data of asphaltene adsorption isotherms on Bedford lime-tone, Dolomite (Cantarell) and Berea at 291 K. Results are presented in terms ofhe mass adsorbed per area unit, mg/m2, as a function of the corresponding initialulk concentration value, mg/L.

molecule. The free energy A2D is obtained by a second-orderperturbation theory (see Ref. [38]).

. Chain contribution:

Achain2D

NkT= −(m − 1)ln y2DSW(�) (16)

where y2DSW is the 2D background correlation function thatis obtained from the radial distribution function using thewell known relationship y(r) = g(r)eˇu(r). Following the SAFT-VRprocedure, g(r) is obtained also from a high temperature pertur-bation expansion approach [38].

. Associative contribution:

Aassoc2D

NkT=

∑A=1

(ln XA − XA

2

)+ ˝

2(17)

where the sum is over all ˝ sites A on a molecule, and XA isthe fraction of molecules not bonded at site A. This fraction isobtained by the solution of the following mass-action equation:

A = 1

1 +∑˝

B=1�XB�AB

(19)

here �AB characterizes the association between sites A and B onifferent molecules, and is given by

AB = 2g2D(�)�fAB (20)

n the last equation, g2D is the contact value of the 2Donomer–monomer radial distribution function, and fAB is theayer function of the A–B site–site bonding interaction, that is

escribed in terms of a SW potential with energy depth εassoc andonding volume �.

. Results

The experimental data of asphaltene adsorption over calcite, sil-ca and dolomite are shown in Fig. 1, and reported as amount ofsphaltene adsorbed per area unit (m2) of substrate mineral sampleersus initial asphaltene concentration in toluene solution (ppm,g/L). Data shown represent the average of at least two indepen-

ent mineral-asphaltene solutions and adsorption measurements

n equilibrium, as previously reported in Ref. [42]. From statisti-al analysis of the experimental data, we found that the maximumtandard deviations for asphaltene adsorptions on calcite, silica,nd dolomite were 0.68, 0.78, and 0.81 mg/g, respectively, whilehe maximum standard errors of the mean (defined as sx = s/

√N

Mn (g/mol) m � (nm) �b εb (kJ/mol) εassoc (kJ/mol) �

779.7 1.0 1.7 1.6 1.318 4.284 0.05

Except Mn , the parameters values were taken from Ref. [40].

where s is the standard deviation) were 0.48, 0.55 and 0.57 mg/g forcalcite, silica and dolomite, respectively. The errors bars are pre-sented in each series for the points where they are larger than asymbol size (deviations >5%).

In order to predict adsorption isotherms by the molecularapproach summarized in the previous section, the bulk phaseasphaltene molecular parameters were taken from a previous study[40]. These parameters are (see Table 2): the number of segmentsused in the chain molecule model, m; the segments diameter, �;the range of the attractive well and energy depth of the SW poten-tial modeling the particle–particle interaction at the bulk phase,�b, and εb, respectively; the energy depth and bonding volume ofthe site-site interaction, εassoc and �, respectively. As we can seefrom Table 2, asphaltenes, adsorbed or at bulk, are represented by amonomeric spherical fluid (m = 1). The diameter of the asphaltenemonomer, �, is based on experimental data, and is the same forthe bulk and the adsorbed molecules too. The associating and SWpotential parameters for bulk phase (εassoc,�b,εb) where calculatedas well in Ref. [40] from asphaltene precipitation experimental data.

In the case of the SW potentials parameters describing theadsorbed particle–particle and particle–wall interactions, their val-ues were obtained following the same approach used for adsorptionof molecular fluids [38,39]. The energy parameter εads was chosenaccording to the quantum-mechanical results derived by Sinanogluand Pitzer [46] for a Lennard–Jones fluid, who demonstrated thatthe energy-well depth of the particle–particle potential in anadsorbed monolayer is reduced by 20–40% from its bulk phase valuedue to the influence of the substrate. In the case of molecular fluidsadsorbed on activated carbon and graphite described by SW poten-tials, we found an optimum value for the reduction factor of 20%,which corresponds to a εads/εb ratio equal to 0.8. We selected thesame value for the adsorption of asphaltenes for the three differentsubstrates. The range �ads is determined from the ratio between thecritical temperatures of the bulk (Tbulk

c ) and monolayer adsorbedphases (Tads

c ), Rc = Tadsc /Tbulk

c as explained in Ref. [39]. This ratio is afunction of the size of the chain molecule and the SW parameters ofthe particle–particle interactions at the bulk and adsorbed phases,Rc = Rc(m,�,εb,�b,εassoc,�,εads,�ads). Since the size parameters, m and�, are the same for the bulk and adsorbed phases, and all the bulkparameters were selected from an independent source [40], theratio Rc depends only on �ads, once the adsorption energy εads hasbeen selected. By knowing Rc then �ads can be determined. Sincethe experimental critical temperature for the asphaltenes adsorbedphase is not known, we approximated this value by considering thecases of adsorption of several molecular fluids on graphite and acti-vated carbon, i.e., Rc = 0.4 [39]. We also fixed �w as 0.2453, accordingto the same results of adsorption for molecular fluids [38], and thatit is within the interval 0.1305 ≤ �w ≤ 0.8165 required by the the-ory. The theoretical adsorption isotherms were calculated solvingEq. (5) for the experimental temperature (291 K), with the valuesdescribed previously for εads, �ads and �w . In order to obtain thebest agreement between theoretical and experimental results, thewall–particle energy parameter, εw , was adjusted for each porousmaterial considered in this study, see Table 3.

The reduced energy parameter (ε∗ = εw/εb) obtained forasphaltene adsorption is smaller than the corresponding molec-ular fluids adsorption case (typically ε* = 7, as reported in Ref.[39]). However, in spite of the apparent arbitrariness in the selec-tion of the values for εads/εb, �ads and �w , the obtained optimized

M. Castro et al. / Fluid Phase Equilibria 286 (2009) 113–119 117

Table 3Values of interaction parameters for adsorbed asphaltenes on porous materials.

Rock type �ads εads/εb �w εw/εba

Berea sandstone 1.45 0.8 0.2453 1.6Bedford limestone 1.45 0.8 0.2453 1.6Dolomite 1.45 0.8 0.2453 1.4

a Fitted from experimental adsorption data.

Ftrm

vtt0polvei

dRmtdpae

Frta

Fig. 4. Asphaltene adsorption isotherm on Cantarell dolomite mineral at 291 K.

ig. 2. Asphaltene adsorption isotherms on Bedford limestone at 291 K. Experimen-al results and theoretical modeling are given by solid dots and continuous line,espectively. Results are presented in terms of the mass adsorbed per area unit,g/m2, as a function of the corresponding initial bulk concentration value, mg/L.

alue for ε* is consistent with Molecular Mechanics calculations forhe particle–particle and particle–surface interactions of asphal-ene models, giving values of the reduced energy within the range.5 < ε* < 7 [47–50], where the lowest value corresponds to tolueneresent as a solvent, and the highest corresponds to interactionsbtained in vacuum. The values obtained are close to the Molecu-ar Mechanics estimate for asphaltenes adsorbed on aluminia inacuum [50]. It is important to note here that ε* could also bestimated from experimental values of adsorbate energies or thesosteric heath of adsorption.

The adsorption isotherms compared with the experimentalata for each of the rocks samples are presented in Figs. 2–4 .esults are presented in terms of the mass adsorbed per area unit,g/m2, as a function of the corresponding initial bulk concentra-

ion value, mg/L. The theory is able to reproduce the experimental

ata that corresponds to a Langmuir Type II isotherm, whichresents a transition from monolayer to multilayer asphaltenedsorption behavior between 5000 and 15000 mg/L. From thexperimental data we can observe the presence local maximum

ig. 3. Asphaltene adsorption isotherms on Berea sandstone at 291 K. Experimentalesults and theoretical modeling are given by solid dots and continuous line, respec-ively. Results are presented in terms of the mass adsorbed per area unit, mg/m2, asfunction of the corresponding initial bulk concentration value, mg/L.

Experimental results and theoretical modeling are given by solid dots and continu-ous line, respectively. Results are presented in terms of the mass adsorbed per areaunit, mg/m2, as a function of the corresponding initial bulk concentration value,mg/L.

around 2500–5000 mg/L (dolomite) and around 3500–8000 mg/L(Bedford limestone and Berea), that could correspond to the asso-ciation/clustering process of asphaltenes, lateral interactions andthe presence, of highly surface-active components like hydrocar-bons containing carboxyl groups [5]. The theory can predict thepresence of this local maximum (see Fig. 4 in Ref. [39]), althoughfor the cases studied here this prediction is not properly obtainedwith the set of parameter values obtained.

The absolute average percent deviation for the threecases shown in Figs. 2–4, AAD = (100/N)

∑Ni=1|adsortionexp

i−

adsortiontheoi

/adsortionexpi

|, used to compare the theoreticaland experimental data, and are reported in Table 4. Significanthigh deviations are obtained as a consequence of the failure todescribe the local maximum. However, considering that onlyone parameter, εw/εb, is adjusted to reproduce the experimen-tal data, the performance of this approach is very good for thethree isotherms, specially at the high initial bulk concentrationregion (>8000 mg/L), which is reflected in low values for thestandard deviation of theoretical results versus experimental

data, ˙ =√∑N

i=1(ei − e)2/N − 1, where e =∑N

i=1|ei|/N and

ei = (adsortionexpi

− adsortiontheoi

), as also reported in Table 4. Incontrast, semi-empirical models [5,42] fail to fit the adsorptiondata at high concentrations [42] and require at least four adjustableparameters that do not have a physical meaning. In the SAFT-VRapproach all the parameters have a molecular basis, modelingforces acting in the adsorption process.

The strong tendency of asphaltenes to associate among them-selves to form aggregates, plays a very important role in theadsorption behavior, even more significant than the specific inter-action of asphaltenes with the different mineral surfaces. As the

bulk concentration increases the size of asphaltenes aggregatesincreases as well [41], conditioning the adsorption process. In Fig. 1it is possible to distinguish at least four different stages in the threeisotherms, one below 2500 mg/L that could be the monolayer for-mation, a second with a maximum between 2500 and 5000 or

Table 4Absolute average percent deviation (AAD) and standard deviation (˙) of the theo-retical results versus experimental data for the three systems studied.

Rock type AAD (%) ˙ (mg/g)

Bedford limestone 54.1 12.8Berea sandstone 50.0 6.8Dolomite 62.5 13.1

1 e Equi

8ao2taataiol

5

iatufitfttmitaobdstspooidvmbsri

LkNMCCq��F���VVSmZ

18 M. Castro et al. / Fluid Phas

000 mg/L in the of Berea and Limestone, that could be assumeds a bi-layer formation, a third transition to multilayer from 5000r 8000 to 15,000 and the clear multilayer region from here to0000 mg/L. All these adsorption stages could reflect the aggrega-ion events occurring in solution as the concentration is increased,s was postulated by Acevedo et al. [6]. The first stage suggest thedsorption of small aggregates formed in this diluted concentra-ion range, at higher concentrations larger aggregates are formednd their adsorption on the multilayer would lead to steps in thesotherm form. After bi-layer–multilayer transition, the formationf bigger aggregates at higher bulk concentrations, >15,000 mg/L,ead to the formation of thick multilayers.

. Conclusions

In this article we have presented a study about adsorptionsotherms of asphaltenes contained within porous materials suchs silica, calcite and dolomite, based on experimental informa-ion obtained for these systems. The isotherms have been modeledsing the SAFT-VR approach [37] and its extension to model con-ned fluids [38,39]. Although the theory is limited by neglectinghe adsorbed–bulk interface, the heterogeneity of confining sur-aces as well as the size of the pores, our results indicate that theheory can give useful insights on the behavior of associating sys-ems under confinement, and it is encouraging that the primitive

odel for bulk asphaltenes proposed previously [40], also can bemplemented, within a quasi-two-dimensional approach, to modelhe behavior of real asphaltenes confined in porous materials. Thepplied thermodynamic model is based on a theoretical equationf state whose parameters have values justified on a molecularasis, in contrast with other approaches [42] where the systems areescribed empirically as a mixture of gases based on the use of fouremi-empirical parameters. Although the present theory requireshe specification of ten molecular parameters, we have shown thatix of them can be taken from the behavior of asphaltenes at bulkhases [40] and another three from the modeling of adsorptionf molecular fluids [38,39]. The theory only requires at the endne adjustable parameter, the energy strength of the particle–wallnteraction. However, the obtained values are consistent with thoseetermined by Molecular Mechanics. The whole consistency of thealues of the parameters indicates that, as a consequence of theolecular approach, the theoretical prediction can be improved

y a proper experimental determination of asphaltene quantitiesuch as Rc or the isosteric heath of adsorption. Future work is alsoequired to study the presence of resins, polymers and surfactantsn solution on mineral surfaces.

ist of symbolsBoltzmann’s constant

AV Avogadro’s numbern average number molecular weight of asphaltene fraction

0b

asphaltene initial bulk concentrationb asphaltene equilibrium concentration

mass of mineral in adsorption experimentamount of adsorbed particles

ads amount of adsorbed asphaltenes (mass)specific BET surface area

b density of particles at the bulkdensity of particles

ads density of adsorbed particlesads volume of adsorbed phase

volume of asphaltene–toluene solutionadsorption areade Broglie’s thermal wavelength

p particles massads partition function of the adsorbed system

libria 286 (2009) 113–119

Qads configurational partition function of the adsorbed systemQ1D one-dimensional configurational partition function of the

adsorbed systemQ2D two-dimensional configurational partition function of the

adsorbed systemAads Helmholtz free energy of the adsorbed phaseA2D two-dimensional Helmholtz free energyA3D three-dimensional Helmholtz free energy (bulk phase)�ads chemical potential of the adsorbed phase�b chemical potential of the bulk phaseAHD hard disks Helmholtz free energyAHS hard-spheres Helmholtz free energya2D

1 first-order perturbation term for the 2D Helmholtz freeenergy

a2D2 second-order perturbation term for the 2D Helmholtz free

energya3D

1 first-order perturbation term for the 3D Helmholtz freeenergy

a3D2 second-order perturbation term for the 3D Helmholtz free

energyAideal

2D 2D ideal contribution to the free energy of a chainmolecules fluid

Amonomer2D 2D monomer contribution to the free energy of a chain

molecules fluidAchain

2D 2D chain contribution to the free energy of a chainmolecules fluid

Aassoc2D 2D association contribution to the free energy of a chain

molecules fluidm number of segments of a chain molecule� diameter of segments forming a chain molecule�b range of the particle–particle SW interaction (bulk phase)εb energy depth of the particle–particle SW interaction (bulk

phase)εassoc energy depth of an associating site–site interaction� bonding volume of an attractive associating site–site

interaction�ads range of the particle–particle SW interaction (adsorbed

phase)εads energy depth of the particle–particle SW interaction

(adsorbed phase)�w range of the particle–wall SW interactionεw energy depth of the particle–wall SW interaction˝ number of associating sites per chain moleculeXA fraction of molecules not bonded at site Ag2D contact value of the 2D monomer–monomer radial distri-

bution functionfAB Mayer function of the A–B site–site bonding interaction.˙ standard deviationAAD absolute average percent deviation

Acknowledgments

The authors gratefully acknowledge Alfredo Ríos Reyes andAntonio Camacho Chávez (Laboratorio de Productividad de Pozos)for their technical support for the experimental study. We alsoacknowldge support from the Laboratorio de Productividad dePozos, Área de Termodinámica de Altas Presiones. This research wassupported by the Instituto Mexicano del Petróleo under ProjectsD31519, D.31515 and D.00476. AGV acknowledges support fromCONACYT grants 81215 and 61418.

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