Application of Continuous Polydisperse Molecular Thermodynamics

2678

Energy & Fuels 2008, 22, 2678–2686

Application of Continuous Polydisperse Molecular Thermodynamics for Modeling Asphaltene Precipitation in Crude Oil Systems Abbas Khaksar Manshad† and Mohsen Edalat*,† Oil and Gas Center of Excellence, Department of Chemical Engineering, UniVersity of Tehran, Tehran, Iran ReceiVed NoVember 2, 2007. ReVised Manuscript ReceiVed March 6, 2008

In this work, a new algorithm for polydisperse asphaltene modeling through application of continuous molecular thermodynamics is introduced. The Scott-Magat theory of polydisperse polymer solutions is applied at equilibrium condition and a necessary and sufficient condition is defined to minimize the Gibbs free energy relation. Three commonly used distribution functions were examined for characterization of asphaltenes, and experimental data reported in the literature is used to adjust parameters of the distribution functions. Among the three distribution functions, the fractal molecular weight distribution function which gives accurate results is selected. A new exponential binary interaction coefficient between asphaltene and crude oil is introduced. The solubility parameter, volume fraction of asphaltene, and the amount of asphaltenes precipitated are calculated through minimization of Gibbs free energy and phase equilibrium condition. It has been shown that the calculated results are in good agreement with experimental data.

1. Introduction Petroleum crude generally consists of a mixture of aromatic and other hydrocarbons such as wax, resin, and asphaltenes. Asphaltenes are the heaviest components of crude oil and are defined as molecular aggregates from petroleum that is not soluble in light hydrocarbon solvents such as pentane, heptane, and decane but can be dissolved in toluene, benzene, and xylene. This fraction contains aliphatic and aromatic structures with high molecular weight ranges from the order of 103 to 105.1–11 Asphaltenes are generally incompatible with light petroleum * To whom correspondence should be addressed. Telephone: 98-21-66957789. Email: [email protected]. Fax: 98-21-2222-8972. † Address: Chemical Engineering Department, P.O. Box 11365/4563, Technical Faculty, University of Tehran, Enghelab Ave., Tehran, Iran. (1) Wiehe, I. A.; Liang, K. S. Asphaltenes, Resins, and Other Petroleum Macromolecules. Fluid Phase Equilib. 1996, 117, 201–210. (2) Yen, T. F.; Erdman, J. G.; Pollack, S. S. Investigation of the structure of Petroleum Asphaltenes by X-ray Diffraction. Anal. Chem. 1961, 33, 1587. (3) Moschopedis, S. E.; Fryer, J. F.; Speight, J. G. Investigation of asphaltene molecular weights. Fuel 1976, 55, 227. (4) Tojima, M.; Suhara, S.; Imamura, M.; Furuta, A. Effect of heavy asphaltene on stability of residual oil. Catal. Today 1998, 43, 347–351. (5) Spiecker, P. M.; Gawrys, L.; Kilpatrick, P. K. Aggregation and solubility behavior of asphaltenes and their subfractions. J. Colloid Interface Sci. 2003, 267, 178–193. (6) Wargadalam, V. J.; Norinaga, K.; Iino, M. Size and shape of a coal asphaltene studied by viscosity and diffusion coefficient measurements. Fuel 2002, 81, 1403. (7) Guerra, R.; Ladavac, K.; Andrews, A. B.; Sen, P. N.; Mullins, O. C. Diffusivity of Coal and Petroleum Asphaltenes Monomers by Fluorescence Correlation Spectroscopy. Fuel 2007, 86, 2016–2020. (8) Acevedo, S.; Escobar, G.; Antonieta M.; Rizzo, A. Molecular weight properties of asphaltenes calculated from GPC data for octylated asphaltenes. Fuel 1998, 77 (8), 853–858. (9) Groenzin, H.; Mullins, O. C. Molecular sizes of asphaltenes from different origin. Energy Fuels 2000, 14, 677. (10) Hortal, A. R.; Martı´nez-Haya, B.; Lobato, M. D.; Pedrosa, J. M.; Lago, S. On the determination of molecular weight distributions of asphaltenes and their aggregates in laser desorption ionization experiments. J. Mass Spectrom. 2006, 41, 960. (11) Klein, G. C.; Kim, S.; Yen, A.; Asomaning, S.; Rodgers, R. P.; Marshall, A. G. Energy Fuels 2006, 20, 1965–1972.

fractions leading to undesirable effects in many stages of the petroleum industry. Asphaltene precipitation can cause serious problems associated with well production, transferring in pipeline, transportation, and ultimately oil refining. Asphaltene deposition within reservoir rocks has been blamed for pronounced reductions in well productivity.12 Treatment to remove or prevent the formation and deposition of solids increases operating costs and hence it is desirable to prevent or mitigate asphaltene deposition. Understanding asphaltene precipitation mechanisms and finding a solution to the problems associated with the asphaltene deposition have been the driving force behind many investigations in the past half century. The precipitation of asphaltenes can be attributed to changes in crude oil composition as well as changes in temperature and pressure. The most important questions of interest in the oil industry are under what conditions asphaltenes will precipitate (12) Cimino, R.; Correra, S.; Bianco, A. D.; Lockhart, T. P. In Asphaltenes: Fundamentals and Applications; Mullins, O. C., Ed.; Plenum: New York, 1995; p 97. (13) Leontaritis, K. J. Asphaltene flocculation during oil recovery and processing: a thermodynamic-colloidal model. Proceedings of the SPE Production Operations Symposium, Oklahoma City, OK, March 1989; Vol. 13/14, pp 599-609. (14) Kawanaka, S.; Park, S. J.; Mansoori, G. A. Organic deposition from reservoir fluids: A thermodynamic predictive technique. Proceedings of the SPE/DOE Enhanced Oil RecoVery Symposium, Tulsa, OK, April 1988; Vol. 17-20, pp 617-627. (15) Mannistu, K. D.; Yarranton, H. W.; Masliyah, J. H. Solubility Modeling of Asphaltenes in Organic Solvents. Energy Fuels 1997, 11, 615– 622. (16) Lindeloff, N.; Heidemann, R. A.; Andersen, S. I.; Stenby, E. H. A Thermodynamic Mixed-Solid Asphaltene Precipitation Model. Pet. Sci. Technol. 1998, 16 (3/4), 307–321. (17) Yudin, I. K.; Anisimov, M. A. Asphaltene aggregation in hydrocarbon solutions studied by photon correlation spectroscopy. J. Phys. Chem. 1995, 99, 9576–9580. (18) Sheu, A.; Hammami, A. G. Asphaltenes, HeaVy Oils and Petroleomics; Springer: New York, 2007. (19) Buckley, J. S.; Hirasaki, G. J.; Liu, Y.; Von Drasek, S.; Wang, J. X.; Gill, B. S. Asphaltene Precipitation and Solvent Properties of Crude Oils. Pet. Sci. Technol. 1998, 16 (3/4), 251–285.

10.1021/ef7006529 CCC: $40.75  2008 American Chemical Society Published on Web 05/22/2008

Asphaltene Precipitation in Crude Oil Systems

Energy & Fuels, Vol. 22, No. 4, 2008 2679 Table 1. Properties of Crude Oils Used for Models34

Figure 1. Binary interaction coefficient of asphaltene and solvent as function of molecular weight of solvent.

component

mol %

critical temp (K)

critical press. (atm)

nitrogen H 2S CO2 methane ethane propane i-butane n-butane i-pentane n-pentane hexanes heptanes+

0.10 0.48 2.05 0.88 3.16 1.93 2.58 4.32 84.50

190.6 305.4 369.8 408.1 425.2 460.4 469.6 506.6 771.2

45.4 48.2 41.9 36 37.5 33.4 33.3 32.3 16.7

acentric factor

mol wt (g/gmol)

0.008 0.098 0.152 0.176 0.193 0.227 0.251 0.281 0.639

16.043 30.07 44.097 58.124 58.124 72.151 72.151 84.00 249.9

Table 2. Properties of Crude Oil Used in This Investigation29–31,34 av MWtan koil (g/gmol)

SGtan koil

av Fasphaltene (g/cm3)

min MWasphaltene (g/gmol)

av MWasphaltene (g/gmol)

WaT (%)

221.5

0.873

1.2

500

4800

4

Table 3. Experimental Data of Amount of Asphaltene Precipitation for Different Solvent and Solvent Ratio at Atmospheric Conditions34 solvent ratio (cm3 diluent/g tank oil) 1.35 1.40 1.90 2.22 5 10 20 50

n-C5

n-C7

n-C10

3.61 3.79 3.87

onset 1.53 1.82 1.89 1.87

onset 1.34 1.45 1.50 -

Table 4. Transformation of Distribution Functions of Asphaltene Properties distribn function distribn parameters obtained functions from exptl data

Figure 2. Algorithm for polydisperse modeling of asphaltene precipitation.

and how much. To solve the problem, it may be useful to develop a new molecular-thermodynamic model that can describe the phase behavior of asphaltene-containing petroleum fluids. Several models have been proposed in recent years.12–22 Most of the existing thermodynamic models for describing asphaltene precipitation process fall into reversible processes consisting of molecular thermodynamics models. 1.1. Asphaltene Precipitation as Reversible Process. Since asphaltene precipitation is considered as a reversible process, molecular thermodynamics is being applied for modeling asphaltene precipitation. Most of the models in this category are based on the classical Flory-Huggins polymer-solution (20) Buckley, J. S. Predicting the Onset of Asphaltene Precipitation from Refractive Index Measurements. Energy Fuels 1999, 13, 328–332. (21) Wang, J. X.; Brower, K. R.; Buckley, J. S. Advances in Observation of Asphaltene Destabilization. Paper presented at the 1999 International Oilfield Chemistry Symposium, Houston, 16-19 Feb; No. SPE 50745. (22) Wang, J. X.; Buckley, J. S. Improved Modeling of the Onset of Asphaltene Flocculation. 2nd International Conference on Petroleum and Gas Phase Behavior and Fouling, Copenhagen, 27-31 August 2000.

Gamma Schultz

η -

fractal

a,c,ω

ξ

Ψ(ξ)

(MWai - MWa0)/β 1/Γ(R)ξ 2[(MWai - MWa0)/ MWa0 (〈MWai〉 - MWa0)] c[ξR/aR(ξ + aMWa0] a(MWai - MWa0)

theory coupled with Hildebrand regular-solution theory to describe the phase behavior of asphaltene-containing fluids.23–29 The asphaltene components are characterized by one pseudocomponent, monodisperse, or multipseudocomponent, polydisperse, with average molar volume or with the molecular weight of (23) Mofidi, M.; Edalat, M. A simplified thermodynamic modeling procedure for predicting asphaltene precipitation. Fuel 2006, 85, 2616– 2621. (24) Khaksar Manshad, A.; Dabir, B. Investigation of Thermodynamic Modeling of Asphaltene Precipitation. M.Sc. Dissertation; Amirkabir University of Technology: Tehran, Iran, 2004. (25) MacMillan, D. J.; Tackett, J. E., Jesee, M. A.; Monger, T. G. A Unified Approach to Asphaltene Precipitation. Lab. Meas. Model. 1995, 788–793. (26) Ghanei, M.; Edalat, M. The Non-Regular Solubility Parameter Term for Predicting Onset and Amount of Asphaltene Precipitation; 2001, SPE 67329. (27) Hirschberg, A.; DeJong, L. N. J.; Schipper, B. A.; Meijer, J. G. Influence of Temperature and Pressure on Asphaltene Flocculation. SPE J. 1984, 24, 283–293. (28) Joshi, N. B.; Mullins, O. C.; Jamaluddin, A.; Creek, J.; McFadden, J. Asphaltene Precipitation from Live Crude Oils. Energy Fuels 2001, 15, 979. (29) Hammami, A.; Phelps, C. P.; Monger-McClure, T.; Little, T. M. Asphaltene precipitation from live oils: An experimental investigation of onset conditions and reversibility. Energy Fuels 2000, 14, 14–18.

2680 Energy & Fuels, Vol. 22, No. 4, 2008

Manshad and Edalat

Table 5. Average Molecular Weight in Liquid and Solid Phase and the Amount of Asphaltene Deposition at Different Temperature, Pressure, and Solvent Ratio Using Gamma Distribution Function

solvent

solvent ratio (cm3/g)

temp (°F)

press. (Pisa)

wt of asphaltene deposition (wt %)

wt of asphaltene deposition in liquid phase (wt %)

NC5 NC5 NC5 NC5 NC5 NC7 NC7 NC7 NC10 NC10 NC10 NC10 NC5 NC5 NC5 NC5 NC5 NC5 NC5 NC5

1 5 10 20 50 5 10 50 5 10 20 50 10 10 10 10 10 10 10 10

60.00 60.00 60.00 60.00 60.00 60.00 60.00 60.00 60.00 60.00 60.00 60.00 70.00 80.00 90.00 90.00 90.00 90.00 90.00 90.00

14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 50.70 100.70 150.70 200.70 300.70

0.00 3.46 3.76 3.86 3.86 0.01 1.63 2.53 0.01 1.18 1.73 1.98 3.81 3.85 3.90 3.89 3.88 3.88 3.87 3.86

4.00 0.54 0.24 0.14 0.14 3.99 2.37 1.47 3.99 2.83 2.27 2.07 0.19 0.15 0.09 0.11 0.12 0.12 0.13 0.14

av mol wt of asphaltene in liquid phase

solid phase

4870.63 2436.86 2366.54 2364.41 2364.10 4868.00 3847.87 3295.22 4865.42 4187.30 3777.37 3256.37 2365.57 2364.94 2362.56 2364.58 2364.64 2364.70 2364.77 2364.92

2764.95 5779.93 5222.71 5070.53 5072.04 12351.24 7977.48 6750.08 13041.23 11216.24 7870.00 6452.25 5143.96 5074.17 4915.13 5022.37 5031.50 5040.99 5050.82 5071.54

Table 6. Average Molecular Weight in Liquid and Solid Phase and the Amount of Asphaltene Deposition at Different Temperature, Pressure, and Solvent Ratio Using Schultz Distribution Function

solvent


temp (°F)

NC5 NC5 NC5 NC5 NC5 NC7 NC7 NC7 NC10 NC10 NC10 NC5 NC5 NC5 NC5 NC5 NC5 NC5

1 5 10 20 50 5 20 50 5 20 50 10 10 10 10 10 10 10

60.00 60.00 60.00 60.00 60.00 60.00 60.00 60.00 60.00 60.00 60.00 70.00 80.00 90.00 90.00 90.00 90.00 90.00

press. (Pisa)



14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 100.70 200.70 300.70 400.70

0.00 3.67 3.80 3.86 3.88 0.00 2.13 2.24 0.01 0.54 1.64 3.81 3.83 3.85 3.85 3.84 3.83 3.82

4.00 0.33 0.20 0.14 0.12 4.00 1.87 1.76 3.99 3.46 2.36 0.19 0.17 0.15 0.15 0.16 0.17 0.18

asphaltene distribution function and solubility parameter. In both monodisperse and polydisperse models, the assumption is made that asphaltene is in equilibrium with insoluble asphaltene and thus the system is considered a saturated system. Also in the modeling, similar fractions of multicomponent systems are lumped into pseudocomponents in order to simplify the phase equilibrium calculation. 1.1.1. SH Monodisperse Model. By applying the regular solution theory to the crude oil system, activity of asphaltene, aa obtained from Scatchard-Hildebrand (SH) equation is ln aa ) ln xa +

MWa 2 [φ (δ - δa)2] RTFa s s

(1)

where MWa, Fa, δa, and xa are molecular weight, density, solubility parameter, and mole fraction (solubility) of asphaltene, respectively. φs ) Vs/Vm and δs are the volume fraction and solubility of deasphalted crude oil, respectively. (30) Funk, E. W.; Prauznitz, J. M. First Born Approximation Cross Sections for Electron Loss from Fast Hydrogen Atoms Passing Through Atomic Helium and Fast Helium Atoms Passing Through Atomic Hydrogen and Helium Ind. Eng. Chem. 1970, 62.

av mol wt of asphaltene in liquid phase

solid phase

4800.00 1693.32 1635.24 1634.55 1632.22 4800.00 3282.99 3217.58 4865.42 4401.09 3587.36 1634.99 1634.81 1634.69 1634.72 1634.77 1634.83 1634.89

2063.04 5761.45 5354.57 5167.53 5102.55 10557.51 8098.28 7821.46 13041.23 11370.11 9303.56 5301.07 5242.50 5180.78 5200.95 5224.40 5247.79 5271.02

By assuming that precipitated asphaltene is at equilibrium with asphaltene in crude oil, aa ) 1. The amount of asphaltene is predicted by (eq 2).

{

xa ) exp -

}

MWa 2 [φ (δ - δa)2] RTFa s s

(2)

The input parameters in the above equation are molecular weight of asphaltene, MWa, density Fa, solubility parameter of the asphaltene, δa, and solubility parameter of the solvent (asphaltene-free crude oil), δs. The solubility parameter of solvent and the molar volume, Vs, of the solvent can be calculated using an equation of state. 1.1.2. FH Monodisperse Model. The Flory-Huggins (FH)27 model of a polymer’s solubility has been used for predicting asphaltene precipitation. In the FH model, it is also assumed that the asphaltenic phase can be modeled as a pseudoliquid and that the asphaltene precipitation will not affect the liquid-vapor equilibrium. This last assumption justifies the use of sequential phase equilibria calculations (vapor-liquid followed by a liquid-liquid calculation) instead of a multiphase equilibria calculation.


Energy & Fuels, Vol. 22, No. 4, 2008 2681

Table 7. Average Molecular Weight in Liquid and Solid Phase and the Amount of Asphaltene Deposition at Different Temperature, Pressure, and Solvent Ratio Using Fractal Distribution Function

solvent


temp (°F)

press. (Pisa)



NC5 NC5 NC5 NC5 NC5 NC7 NC7 NC7 NC7 NC10 NC10 NC10 NC5 NC5 NC5 NC5 NC5 NC5 NC5 NC5

1 5 10 20 50 5 10 20 50 5 10 50 10 10 10 10 10 10 10 10

60.00 60.00 60.00 60.00 60.00 60.00 60.00 60.00 60.00 60.00 60.00 60.00 70.00 80.00 90.00 90.00 90.00 90.00 90.00 90.00

14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 14.70 50.70 100.70 150.70 200.70 300.70

0.00 3.10 3.63 3.80 3.81 0.00 1.75 2.62 2.72 0.00 0.99 1.79 3.71 3.77 3.83 3.87 3.88 3.86 3.85 3.83

4.00 0.90 0.37 0.20 0.19 4.00 2.25 1.38 1.28 4.00 3.01 2.21 0.29 0.23 0.17 0.13 0.12 0.14 0.15 0.17

liquid phase

solid phase

4799.99 2364.46 2363.35 2363.35 2363.35 4799.94 3449.60 2608.45 2476.94 4800.00 3974.48 3408.01 2363.35 2363.35 2363.35 2364.58 2364.64 2363.35 2363.35 2363.35

2419.54 6856.31 5357.49 5068.89 5056.06 33078.39 9662.15 8617.76 8603.21 31555.23 13085.47 9719.57 5228.63 5117.89 5027.58 5022.37 5031.50 4988.52 5001.08 5028.08

Table 8. Parameters of Model for Different Distribution Functions

The activity coefficient of FH model is

( )

ln γa ) ln φa + 1 -

av mol wt of asphaltene in

Va Va φs + φs2(δa - δs)2 Vs RT

(3)

distribn functions

av absolute dev (%)

as

bs

ω

η

δasphaltene (atm0.5)

where φa ) Va/Vmix and Va are the volume fraction and the molar volume of asphaltene, respectively. The maximum volume fraction of the asphaltenes φmax that is soluble can be calculated a by eq 4 by assuming that insoluble asphaltenes are in equilibrium with asphaltene in crude oil, γa ) 1.

Gamma Schultz fractal

1.86 2.38 0.15

0.25 0.53 0.28

-0.03 -0.04 -0.03

-0.40

6844593.10 -

55.64 55.60 55.30

[( )

) exp φmax a

]

Va Va - 1 φs - φs2(δa - δs)2 Vs RT

(4)

Two parameters, the molar volume and solubility parameter of asphaltenes, are essential in this model. The molar volume is dependent on the molecular weight and density. Density is a measurable parameter whereas the molecular weight is much more difficult to measure with any degree of accuracy. 1.1.3. Modified FH Monodisperse Model. In the SH and FH models it was assumed that precipitated asphaltene is free solvent. Contrary to this assumption, FH model is modified by considering that the precipitated asphaltenes forming a nonpure phase and containing part of the solvent in it.12 Then the equilibrium condition for precipitated asphaltene containing solvent is: ) µsa µsolid a

(5)

The FH relation for precipitated asphaltene and solvent is used in eq 5

( )

ln φa + 1 -

Va Va φ + φ 2(δ - δs)2 ) ln(φs) + Vs s RT s a Va Va 1φ + φ 2(δ - δs)2 (6) Vs a RT a a

( )

By neglecting the asphaltene volumetric fraction in the liquid phase

( )

ln(1 - φsa) + 1 -

Va Va φa + φa2(δa - δs)2 ) 0 Vs RT

(7)

where φsa is the volumetric fraction of asphaltenes in the solid phase, with φsa < 1.

In eq 7, knowledge of molar volume or molecular weight of asphaltene and solubility parameters are required to predict the amount of asphaltene in crude oil. The main drawback of the monodisperse models is that they do not take into account the aggregation phenomena of asphaltenes. This can explain why these models can be correct in some cases and not in others, depending on asphaltene dispersion, which in turn depends on the asphaltene and solvent nature. Another shortcoming with the SH and FH models is that extensive manipulation is necessary. If the molecular weight is kept at an average value, the solubility increases rapidly with increasing solvent solubility parameter and there is not a reliable relation for calculating solubility parameter of asphaltene. 1.1.4. Scott-Magat (SM) Polydisperse Model. In the SH and FH models, it was assumed a homogeneous polymer of uniform molecular weight exists in a single uniform solvent. In practice, in a polymer solution, all the molecules do not have the same molecular weight but rather have a mixture of different molecular weights, forming a continuous distribution.14 In the SM model, a molecular weight distribution was used and an interaction parameter l12 as function of average molecular weight30 is introduced. Then the FH equation and eq 5 may be written as ln φa )

Va Va - 1 - [(δa - δs)2 + 2l12δaδs] Vmix RT

(8)

where l12 ) a + b〈MWai 〉

(9)

In the above equations, the constants a and b are adjustable parameters; the molecular weight distribution, smallest molecular weight of asphaltenes, molar volume of asphaltene, and the solubility parameter of the asphaltenes are obtained from knowledge of experimental data.


Figure 3. Comparison between amount of asphaltene precipitation from experimental data and fractal distribution function model output for different dilution ratio and atmospheric conditions.

Manshad and Edalat

Figure 7. Gamma molar distribution function of asphaltene in different phases with the solvent ratio 20 cm3 of n-pentane to 1 g of tank oil.

Figure 8. Schultz molar distribution function of asphaltene in different phases with the solvent ratio 20 cm3 of n-pentane to 1 g of tank oil.

Figure 4. Fractal molar distribution function of asphaltene in different phases with the solvent ratio 20 cm3 of n-decane to 1 g of tank oil.

Figure 9. Fractal molar distribution function of asphaltene in different phases with the solvent ratio 20 cm3 of n-pentane to 1 g of tank oil. Figure 5. Comparison of fractal molar distribution function of asphaltene in the liquid phase for 10 cm3 of different solvents added.

in incorrect prediction of Vs. Because all of the above models are dependent on the square of the difference between the asphaltenes solubility parameter and that of the solvent (δa δs)2, any poor estimation of the solubility will affect the predictability of asphaltene precipitation. The solubility parameter of the solvent is calculated through the internal energy change of vaporization as δs )

Figure 6. Comparison of fractal molar distribution function of asphaltene in the solid phase for 10 cm3 of different solvents added.

In all of the above models, the equation of state is employed to calculate molar volume, Vs, and the solubility parameter of solvent, δs. Selecting an inaccurate equation of state will result

( ) ∆Uvap Vs

1/2

(10)

There are three more unknowns in the above models such as the molar volume, solubility parameter, and molecular weight of asphaltene. Since the ultimate purpose of the above models is to be able to predict the onset of precipitation and the amount of asphaltene in crude oil, the three unknowns are estimated by fitting the experimental data. While these models can partially explain a few of the experimental results and have some use in describing selected asphaltene precipitation phenomena in petroleum fluids, each of them suffers certain inadequacies and none of them is capable of predicting the precipitation of asphaltene in crude oil



satisfactorily. Attempts at relieving these shortcomings by modifying existing models or deriving a new model are being performed, and the results of research have been published and promising new ones are appearing. In this work, the continuous molecular thermodynamics polydisperse model is used by considering asphaltene precipitation as a reversible process.28,29,31 Thermodynamic equilibrium conditions for a stable system are established by minimizing Gibbs free energy. A molecular weight distribution function is selected among the three commonly used ones and the volume fraction, solubility of asphaltene, and volume fraction of solvent are calculated. Consequently, the onset of precipitation and amount of asphaltenes in crude oil can be predicted without extensive experimental data. It is worth mentioning that the analysis below presumes that large molecular weights apply. 1.2. Continuous Molecular Thermodynamics Polydisperse Model. In the polydisperse models, molecular weight and size are characterized and the asphaltene fraction is considered to be uniform in chemical composition. In this model, the asphaltenes are considered to be a mixture of polymers which only vary in molecular weight so that a continuous molecular distribution function can be used. A continuous molecular distribution function allows properties of asphaltene be related to molecular weight of asphaltene fraction.31–34 2. Development of a New Model for Predicting Asphaltene Precipitation Using Polydisperse Phenomena theory35

According to the Scott and Magat for heterogeneous polymer solutions, the chemical potential of each asphaltene segments in asphaltene and crude oil solutions is defined by following: µai - µ a°i RT

) ln φai + 1 -

( ) mai

〈ma 〉

(1 - φs) - maiφs + f maiφs L

2

Famass ) 〈Fmass 〉 a i

Each submolecule is considered to be the same size as the molecule of solvent so that mai in reality is the ratio of molecular volume of asphaltene molecule Vai to the molar volume of solvent, Vs. mai )

MWai

i

The average segment number of asphaltene is expressed as

〈mA 〉 ) ∑ xAimAi )

RT

) ln γai ) ln φai + 1 -

( ) mai

〈ma 〉

∑ (φ ) i)1

ai max )

[ ( )

∑ exp 1 i)1

mai

〈mai 〉

Vs , φs ) and φai + φs ) 1 (13) Vmix Vmix mai is the segment number of the ith fraction of asphaltene (molecular weight of ni submolecules) and is defined as follows:

(∑ )

mai )

Vai Vs

)

MWai Famax Vs i

(14)

(31) Scott, R. L.; Magat, M. J. The thermodynamics of high-polymer solutions. I. The free energy of mixing of solvents and polymers of heterogeneous distribution. Chem. Phys. 1945, 13 (5), 172–177. (32) Dabir, B.; Nematy, M.; Mehrabi, A.; Rassamdana, H.; Sahimi, M. Asphalt flocculation and deposition, III. The molecular weight distribution. Fuel 1996, 75 (No. 14), 1633–1645. (33) Vazquez, D.; Mansoori, G. A. Identification and measurement of petroleum precipitates. J. Pet. Sci. Eng. 2000, 26, 49–55. (34) Acevedo, S.; Escobar, G.; Ranaudo; M, A.; Rizzo, A. Molecular weight properties of asphaltenes calculated from GPC data for octylated asphaltenes. Fuel 1998, 77, 853–858, No. 8, pp. (35) Scott, R. L. The thermodynamics of high-polymer solutions. II. The solubility and fractionation of a polymer of heterogeneous distribution. J. Chem. Phys. 1945, 13 (No. 5), 178–187.

∑nm i

)

Ai

(17)

mAi

∑φ

]

2 L (1 - φs) - maiφs + f maiφs (18)

where (φAi)max is the maximum soluble of ith fraction of asphaltene volume percent in the liquid phase. The total amount of asphaltene deposited, W dep a , in the crude oil will be

∑φ

(19)

ai

where W Ta is the total amount of asphaltene.

(1 - φLa ) (1 - φLa )

Subscripts ai and s are i segment of asphaltene and crude oil solvent, respectively. φ is the volume fraction of component and is defined as

Ai

xai is the mole fraction of the ith fraction with respect to total asphaltene. Vai is the molar volume of ith fraction of asphaltene, is mass density of ith asphaltene fraction, and 〈Fmass Famass 〉 is a i the mass average density of asphaltene. By assuming that the asphaltene precipitation does not alter the vapor-liquid equilibrium, φai < φs and γai ) 1 in eqs 11 and 12, the maximum volume fraction of asphaltene soluble in the oil phase can be calculated from the following relation:

W Ta ) FaφLa V L )

(1 - φs) - maiφs + f Lmaiφs2 (12)

Vai

i

Ai

i

φ ai )

∑nm ∑n i

and the activity coefficient of asphaltene, γai can be calculated from µai - µ °ai

(16)

〈Fmass 〉 Vs a

T W dep a ) W a - FaVa

(11)

(15)

MWs Vs

〈MWa 〉 Va

+ φLa

MWs Vs

(20)

In eq 18, the average segment of asphaltene, the volume fraction or the molecular weight distribution function, and f L need to be known. The parameter f L in eqs 11 and 18 is defined and consists of the entropy and the heat of mixture as in the following correlation.14,31 ∆hSE 1 f L) + r RTφ 2

(21)

a

where the heat of mixture takes the form from van der Waals EOS as ∆hsE 2

RTφa

)

Vs{(δa - δs)2 + 2Kasδaδs} RT

(22)

By inserting eq 22 into eq 21 1 Vs{(δa - δs)2 + 2Kasδaδs(δa - δs)2 + 2Kasδaδs} f L) + r RT

(23)

where Kas is interaction coefficient between asphaltene and free asphaltene crude oil solvent. The important role of the oil-rich phase solubility parameter and sensitivity of the results to this property has been emphasized in all models. The solubility parameter of a liquid is the


Manshad and Edalat

square root of the cohesive energy density, where the cohesive energy density of a liquid is the energy requirement to separate the molecules present in 1 cm3 and hence to vaporize it, i.e. δs )

(

∆H - RT Vs

)

(24)

Kas ) a exp b〈Mas 〉

(25)

To calculate δa and φs, the minimization conditions of total Gibbs free energy are used. 2.1. Phase Equilibrium and Gibbs Free Energy Minimization. By assuming that the chemical potentials of asphaltene in liquid and solid phase at constant T and P and at precipitation point (onset) are equal, the chemical potential of fraction i of asphaltene in the liquid phase and solid phase will be as µaSi ) µaLi

(26)

Substituting eq 11 into eq 26 and assuming that the reference state is the same for both phases and the definition of Gibbs free energy change and activity coefficient at constant T and P 0 ai - Gai

) ) (µ

ai - µ ° ai

RT Inserting eq 12 into eq 27 ln γai ) ln φai + 1 -

( ) mai

〈ma 〉

RT

)

) ln γai

(27)

2 L (1 - φs) - maiφs + f maiφs (28)

i

The condition for a system being at thermodynamic equilibrium is that the total Gibbs free energy of the system should have a minimum value. Once the total Gibbs free energy of a system is zero, the system will be in a stable condition. The minimization conditions are 2

∂ Gai ∂x2

e0

and

∂x3

g0

(29)

or ∂ ln γai ∂xai

e0

and

∂2 ln γai 2

∂x

L L ai + φs ) 1

g0

mai 1 + mai - 2f LmaiφsL ) 0 L φai 〈mai 〉

〈ma 〉 ) ∫0

∞

i

∫

∞

0

(34)

F(MWai) d(MWai) ) 1

(35)

The average molecular weight of asphaltene 〈MWai〉 is calculated from

∫

∞

0

MwaiF(Mwai)d(Mwai) ) 〈Mwai 〉

(36)

2.2. Selection of Distribution Function for Continuous Model of Asphaltene. The asphaltene volume fraction, the asphaltene-free crude oil volume fraction, the solubility parameter of asphaltene, and finally the total amount of asphaltene at onset of precipitation can be calculated by simultaneous solution of eqs 31–33. In these equations, an accurate molecular weight distribution function is required. Since asphaltene is characterized as a continuous mixture that follows a distribution function, we examine different types of distribution for selecting the most accurate molecular weight distribution function. The Gamma distribution function has been used in the literature14,30 for asphaltene molecular weight distribution. However, we examined the three most commonly used distribution functions. The parameters of each one are adjusted using experimental data of asphaltene32–34,37 and then each one is applied to the set of eqs 31 and 32 to calculate the total asphaltene content of crude oil. The following forms of distribution have been studied for characterization of asphaltene molecular weight and properties. 2.2.1. Gamma Distribution Functions for Asphaltene Properties. The Gamma distribution function form is as follows:14,38

[

R

(MWa - MWa )R-1β i

exp -

0

Γ(R)

(MWa - MWa ) i

0

β

]

(37)

where the parameters of Gamma distribution function are obtained as

β)

(36) Anderson, S. I.; Speight, J. G. Thermodynamic models for asphaltene solubility and precipitation. J. Pet. Sci. Eng. 1999, 22, 53–66.

maiF(MWai) d(MWai)

where

(30)

(31)

(33)

Equations 31–33 can be solved simultaneously to calculate φsL, φaLi and δa. There are obviously i simultaneous equations of types (31) and (32). In solving the above equations, one needs to have an accurate molecular weight distribution function for MWai to obtain the amount of average segment number of asphaltene 〈mai〉 as defined in the following equation.

R)

Then, the first and second derivatives of eq 30 will be

( )

(32)

∑φ

F(MWai) )

3

∂ Gai

1 + 2f Lmai ) 0 L 2 φ ( ai )

and the following equation is valid.

0.5

where ∆H and Vs are the enthalpy of vaporization and the molar volume of liquid, respectively. In this work, the solubility parameter of asphaltene-free crude liquid oil is calculated using the Peng-Robinson equation of state. Many attempts have been made to derive a relation for calculating the asphaltene solubility δa.36 However, we have considered it as an unknown parameter in this algorithm, not as an input parameter. The interaction coefficient Kas was assumed as a linear function of asphaltene molecular weight in the literature.14,36 After carefully studying the variation of the binary interaction coefficient, Kas, between asphaltene and free asphaltene crude oil solvent, it is found that Kas is varying with Mas exponentially (Figure 1). We then propose the exponential relation in the form of

(G

-

(〈MWa 〉 - MWa )2 i

0

η η 〈MW 〉 - MWa0) a ( i

(38) (39)

(37) Burke, N. E.; Hobbs, R. E.; Kashou, S. F. Measurement and Modeling of Asphaltene Precipitation. J. Pet. Technol. 1990, 42, 1440– 1446. (38) Rassamdana, H.; Sahimi, M. Asphalt Flocculation and Deposition: II. Formation and growth of fractal aggregates. AIChE J. 1996, 42 (12), 3318–3332.


∫

∞ R-1

Γ(R) )

t

0


exp(-t) dt

(40)

2.2.2. Schultz-Zimm distribution functions for asphaltene properties. The Schultz-Zimm distribution function form is:39 f(MWai) )

4(MWai - 1)

(〈MWa 〉 - 1)2

[

exp -

i

2(MWai - 1)

(〈MWa 〉 - 1) i

]

(41)

and the average association parameter for each segment is obtained in the following form:

〈MWai 〉 )

∫

MWa(i+1)

MWa

∫

MWaif(MWai) d(MWai)

MWa(i+1)

MWai

f(MWai) d(MWai)

2.2.3. Fractal Distribution Functions for Properties. The fractal distribution function is38 fw(MWai) )

c(MWai - MWa0)-2w MWai

(42)

Asphaltene

MWai > MWa0(43)

and normalized form as

∫

∞ f (MWai) MWa0 w

dMWai ) c

∫

∞

×

MWa0

(MWa - MWa )-2w exp[-a(MWa - MWa )] i

0

i

0

MWai

dMWai ) 1 (44)

By defining ξ ) a(MWai - MWa0) eq 44 takes the form c aR

∫

∞

0

ξRe-ξ dξ ) 1 ξ + aMWa0

The calculation procedure for the new algorithm is shown in Figure 1. 3. Results and Discussion

exp[ - a(MWai MWa0)],

Figure 10. Sensibility analysis of the amount of asphaltene precipitation by solubility parameter and binary interaction coefficient between asphaltene and crude oil.

(45)

In this study, the experimental data at atmospheric conditions (Tables 1–3)), reported by Burke et al.,37 is used for obtaining the parameters of molecular weight distribution functions. By knowing the parameters of the distribution functions, eqs 31–33 are solved simultaneously in order to find solubility parameter of asphaltene and the amount of precipitation. The results of these calculations are shown in (Tables 5–8) for three selected distribution functions. The molar distribution in tank oil, liquid, and solid phases for different solvent ratios are shown in (Figures 3–9). In the present model, unlike all the other models where the solubility of asphaltene is kept at a fixed value or is tuned, the solubility parameter of asphaltene is obtained from continuous molecular thermodynamics relationships derived from minimization of Gibbs free energy function of the Scott-Magat theory.

and

∫

∞

c aR+1

0

4. Conclusions

ξRe-ξ dξ ) 〈MWai 〉

(46)

A relation between a, c, and mean molecular weight is established as40 R+1

c)

〈MWa 〉 a i

(47) Γ(R + 1) Substituting eq 47 into eq 45 and applying Newton-Raphson method, one can find the parameter a of eq 43 and the numerical Gauss-Laguerre quadrature method to evaluate the integral. The average molecular weight 〈MWai〉 and the minimum molecular weight MWa0 for three distribution functions are obtained from experimental data.32–34,37 A generic algorithm is used as a global optimization tool for calculating the adjusted parameters of asphaltene model and distribution function. In this optimization method, the objective function is defined as the following form: F)

∑ j

[

- W(dep)cal (W(dep)exp ) a a W(dep)exp a

]

(48) j

(39) Alboudwarej, H.; Akbarzadeh, K. A generalized regular Solution Model for Asphaltene Precipitation from n-alkane diluted heavy oils and Bitumens. Fluid Phase Equilib. 2005, 232, 159–170. (40) Jorge, E. P.; Monteagudo, A.; Paulo, L. C.; Lage, A.; Rajagopal, B. Towards a polydisperse molecular thermodynamic model for asphaltene precipitation in live-oil. Fluid Phase Equilib. 2001, 187-188, 443–471.

1. A new polydisperse model algorithm for calculation of amount of asphaltene precipitation shows that the new algorithm is very practical and efficient for polydisperse asphaltene modeling. 2. Fractal distribution function among the other two different distribution functions has the lowest average absolute deviation (%) and the best rate of convergence for prediction of amount of asphaltene deposition in crude oil systems (Table 8). 3. The accuracy of calculations of continuous thermodynamic property integral is a function of the number of asphaltene segments. Increasing the number of segments the accuracy of calculations increases and reduces the convergence (the optimum value is four segments in this work). 4. The onset of asphaltene precipitation occurs when the average molecular weight of asphaltene in liquid and solid phase is maximum. 5. Based on sensibility analysis (Figure 10), the most sensible parameter is the solubility parameter of asphaltene and oil mixture. To a lesser extent, aS, the first parameter of exponential correlation of binary interaction coefficient of asphaltene and oil mixture, is also sensible. 6. Application of the new binary interaction coefficient correlation (Figure 2) between asphaltene and solvent Kas ) aS exp(bS〈MWas〉) with optimized parameter as and bs reveals that the calculated results are in good agreement with experimental data.


Manshad and Edalat

7. The main advantage of this model compared with other models is the requirement of minimum experimental data to find unknown parameters. It also has the advantage of obtaining solubility parameters through thermodynamic relationships rather than using a fixed value or tuning one. Acknowledgment. The authors are grateful to Oil and Gas Center of Excellence for supporting this study.

Appendix A An efficient method for solution of the integral of continuous thermodynamic properties is required. Thus, the distribution functions of asphaltene properties are transformed to the soluble standard form of integral

∫

∞ -ξ -R e ξ Ψ(ξ) 0

NQP

dξ )

∑ W Ψ(ξ ) i

i

(A1)

i)1

The parameters of the standard form of integral properties (ξ,Ψ(ξ)) are shown in the (Table 4).

Nomenclature a ) activity T ) temperature P ) pressure R ) universal gas constant

µ ) chemical potential φ ) volume fraction V ) molar volume δ ) solubility parameter x ) molar percent W ) weight percent f ) fugacity K ) equilibrium constant γ ) activity coefficient φ ) fugacity coefficient ∆U ) internal energy alteration ∆H ) enthalpy energy alteration ma ) asphaltene segmentation number F ) density r ) coordination number Kas ) binary interaction of asphaltene and oil na ) mole percent of asphaltene in each phase MW ) molecular weight MWa0 ) minimum molecular weight of asphaltene 〈MWai〉 ) average molecular weight of asphaltene 〈MWas〉 ) average molecular weight of solvent η ) variance F ) distribution function EF7006529

Application of Continuous Polydisperse Molecular Thermodynamics

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