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Difference between revisions of "Thermodynamic models for asphaltene precipitation"

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[[Thermodynamics and phase behavior|Thermodynamic models]] for predicting [[asphaltene precipitation]] behavior fall into two general categories: activity models and [[Equations of state|equation-of-state]] (EOS) models. This page provides the mathematics underlying the most commonly used models of each type.
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[[Thermodynamics_and_phase_behavior|Thermodynamic models]] for predicting [[Asphaltene_precipitation|asphaltene precipitation]] behavior fall into two general categories: activity models and [[Equations_of_state|equation-of-state]] (EOS) models. This page provides the mathematics underlying the most commonly used models of each type.
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== Thermodynamic equilibrium ==
  
==Thermodynamic equilibrium==
 
 
With the precipitated asphaltene treated as a single-component or multicomponent solid, the condition for thermodynamic equilibrium between the oil (liquid) and solid phase is the equality of component chemical potentials in the oil and solid phases. That is,
 
With the precipitated asphaltene treated as a single-component or multicomponent solid, the condition for thermodynamic equilibrium between the oil (liquid) and solid phase is the equality of component chemical potentials in the oil and solid phases. That is,
  
[[File:Vol1 page 0412 eq 001.png]]....................(1)
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[[File:Vol1 page 0412 eq 001.png|RTENOTITLE]]....................(1)
  
where ''μ''<sub>''io''</sub> and ''μ'' is are the chemical potential of component ''i'' in the oil and solid phases, respectively, and ''n''<sub>''c''</sub> is the number of components. The application of activity coefficient models or EOS models gives different expressions for the chemical potential. In addition, not all components in the oil phase undergo precipitation; therefore, '''Eq. 1''' applies only to those components that precipitate.  
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where ''μ''<sub>''io''</sub> and ''μ'' is are the chemical potential of component ''i'' in the oil and solid phases, respectively, and ''n''<sub>''c''</sub> is the number of components. The application of activity coefficient models or EOS models gives different expressions for the chemical potential. In addition, not all components in the oil phase undergo precipitation; therefore, '''Eq. 1''' applies only to those components that precipitate.
  
 
== Activity models ==
 
== Activity models ==
===Activity coefficients===
 
Because [[Asphaltenes and waxes|asphaltenes]] are a solubility class that can be precipitated from petroleum by the addition of solvent, activity coefficient models have been applied to model the phase equilibrium phenomena. The introduction of activity coefficients in '''Eq. 1''' yields
 
  
[[File:Vol1 page 0413 eq 001.png]]....................(2)
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=== Activity coefficients ===
 +
 
 +
Because [[Asphaltenes_and_waxes|asphaltenes]] are a solubility class that can be precipitated from petroleum by the addition of solvent, activity coefficient models have been applied to model the phase equilibrium phenomena. The introduction of activity coefficients in '''Eq. 1''' yields
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 +
[[File:Vol1 page 0413 eq 001.png|RTENOTITLE]]....................(2)
  
 
where:
 
where:
* [[File:Vol1 page 0413 inline 001.png]] = standard state fugacity of component ''i'' in phase ''k'' (''k'' = ''o'', ''s'')
 
* ''v''<sub>''ik''</sub> = partial molar volume of component ''i'' in phase ''k'' (''k'' = ''o'', ''s'')
 
* ''y''<sub>''ik''</sub> = mole fraction of component ''i'' in phase ''k'' (''k'' = ''o'', ''s'')
 
* ''γ''<sub>''ik''</sub> = activity coefficient of component ''i'' in phase ''k'' (''k'' = ''o'', ''s'')
 
  
Several approaches that use the activity coefficient model assume the oil and asphaltene as two pseudocomponents: one component representing the deasphalted oil and the other the asphaltenes. Andersen and Speight<ref name="r1" /> provided a review of activity models in this category. Other approaches represent the precipitate as a multicomponent solid. Chung,<ref name="r2" /> Yarranton and Masliyah,<ref name="r3" /> and Zhou ''et al.''<ref name="r4" /> gave detailed descriptions of these models.
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*[[File:Vol1 page 0413 inline 001.png|RTENOTITLE]] = standard state fugacity of component ''i'' in phase ''k'' (''k'' = ''o'', ''s'')
 +
*''v''<sub>''ik''</sub> = partial molar volume of component ''i'' in phase ''k'' (''k'' = ''o'', ''s'')
 +
*''y''<sub>''ik''</sub> = mole fraction of component ''i'' in phase ''k'' (''k'' = ''o'', ''s'')
 +
*''γ''<sub>''ik''</sub> = activity coefficient of component ''i'' in phase ''k'' (''k'' = ''o'', ''s'')
  
===Flory-Huggins model===  
+
Several approaches that use the activity coefficient model assume the oil and asphaltene as two pseudocomponents: one component representing the deasphalted oil and the other the asphaltenes. Andersen and Speight<ref name="r1">Andersen, S.I. and Speight, J.G. 1999. Thermodynamic models for asphaltene solubility and precipitation. J. Pet. Sci. Eng. 22 (1–3): 53-66. http://dx.doi.org/10.1016/S0920-4105(98)00057-6</ref> provided a review of activity models in this category. Other approaches represent the precipitate as a multicomponent solid. Chung,<ref name="r2">Chung, T.-H. 1992. Thermodynamic Modeling for Organic Solid Precipitation. Presented at the SPE Annual Technical Conference and Exhibition, Washington, DC, 4-7 October. SPE-24851-MS. http://dx.doi.org/10.2118/24851-MS</ref> Yarranton and Masliyah,<ref name="r3">Yarranton, H.W. and Masliyah, J.H. 1996. Molar mass distribution and solubility modeling of asphaltenes. AIChE J. 42 (12): 3533-3543. http://dx.doi.org/10.1002/aic.690421222</ref> and Zhou ''et al.''<ref name="r4">Zhou, X., Thomas, F.B., and Moore, R.G. 1996. Modelling of Solid Precipitation From Reservoir Fluid. J Can Pet Technol 35 (10). PETSOC-96-10-03. http://dx.doi.org/10.2118/96-10-03</ref> gave detailed descriptions of these models.
The solubility model used most in the literature is the Flory-Huggins solubility model introduced by Hirschberg ''et al.''<ref name="r5" /> Vapor/liquid equilibrium calculations with the [[Equations of state|Soave-Redlich-Kwong EOS]]<ref name="r6" /> are performed to split the petroleum mixture into a liquid phase and a vapor phase. The liquid phase then is divided into two components: a component that corresponds to the asphaltene and a component that represents the remaining oil (deasphalted oil). When solvent is added into the oil, the second component represents the mixture of deasphalted oil and solvent. These two components are for modeling purposes and do not correspond to any EOS components used in the vapor/liquid calculations. It also is assumed that asphaltene precipitation does not affect vapor/liquid equilibrium.  
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 +
=== Flory-Huggins model ===
 +
 
 +
The solubility model used most in the literature is the Flory-Huggins solubility model introduced by Hirschberg ''et al.''<ref name="r5">Hirschberg, A., deJong, L.N.J., Schipper, B.A. et al. 1984. Influence of Temperature and Pressure on Asphaltene Flocculation. SPE J. 24 (3): 283-293. SPE-11202-PA. http://dx.doi.org/10.2118/11202-PA</ref> Vapor/liquid equilibrium calculations with the [[Equations_of_state|Soave-Redlich-Kwong EOS]]<ref name="r6">Soave, G. 1972. Equilibrium constants from a modified Redlich-Kwong equation of state. Chem. Eng. Sci. 27 (6): 1197–1203. http://dx.doi.org/10.1016/0009-2509(72)80096-4</ref> are performed to split the petroleum mixture into a liquid phase and a vapor phase. The liquid phase then is divided into two components: a component that corresponds to the asphaltene and a component that represents the remaining oil (deasphalted oil). When solvent is added into the oil, the second component represents the mixture of deasphalted oil and solvent. These two components are for modeling purposes and do not correspond to any EOS components used in the vapor/liquid calculations. It also is assumed that asphaltene precipitation does not affect vapor/liquid equilibrium.
  
 
Application of the Flory-Huggins solution theory gives the following expression for the chemical potential of the asphaltene component in the oil phase.
 
Application of the Flory-Huggins solution theory gives the following expression for the chemical potential of the asphaltene component in the oil phase.
  
[[File:Vol1 page 0413 eq 002.png]]....................(3)
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[[File:Vol1 page 0413 eq 002.png|RTENOTITLE]]....................(3)
  
with [[File:Vol1 page 0413 eq 003.png]]....................(4)
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with [[File:Vol1 page 0413 eq 003.png|RTENOTITLE]]....................(4)
  
where:  
+
where:
  
* subscripts ''a'', ''o'', and ''m'' are used to denote the asphaltene component, the deasphalted oil, and the oil phase mixture, respectively
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*subscripts ''a'', ''o'', and ''m'' are used to denote the asphaltene component, the deasphalted oil, and the oil phase mixture, respectively
* ''v''<sub>''a''</sub> = molar volume of pure asphaltene,  
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*''v''<sub>''a''</sub> = molar volume of pure asphaltene,
* ''v''<sub>''m''</sub> = molar volume of mixture,  
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*''v''<sub>''m''</sub> = molar volume of mixture,
* ''δ''<sub>''i''</sub> = solubility parameter of component ''i'',  
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*''δ''<sub>''i''</sub> = solubility parameter of component ''i'',
* ''δ''<sub>''m''</sub> = solubility parameter of mixture,  
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*''δ''<sub>''m''</sub> = solubility parameter of mixture,
* ''Φ''<sub>''a''</sub> = volume fraction of asphaltene in the mixture, ''μ''<sub>''am''</sub> = chemical potential of asphaltene in the mixture
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*''Φ''<sub>''a''</sub> = volume fraction of asphaltene in the mixture, ''μ''<sub>''am''</sub> = chemical potential of asphaltene in the mixture
* [[File:Vol1 page 0413 inline 002.png]] = reference chemical potential of asphaltene component
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*[[File:Vol1 page 0413 inline 002.png|RTENOTITLE]] = reference chemical potential of asphaltene component
  
Because the precipitated asphaltene is pure asphaltene, ''μ''<sub>''s''</sub> = [[File:Vol1 page 0413 inline 002.png]]. From the equality of chemical potential ''μ''<sub>''am''</sub> = ''μ''<sub>''s''</sub>, '''Eq. 3''' gives
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Because the precipitated asphaltene is pure asphaltene, ''μ''<sub>''s''</sub> = [[File:Vol1 page 0413 inline 002.png|RTENOTITLE]]. From the equality of chemical potential ''μ''<sub>''am''</sub> = ''μ''<sub>''s''</sub>, '''Eq. 3''' gives
  
[[File:Vol1 page 0413 eq 004.png]]....................(5)
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[[File:Vol1 page 0413 eq 004.png|RTENOTITLE]]....................(5)
  
 
The molar volume, ''v''<sub>''m''</sub>, of the oil mixture is calculated from the composition of the liquid phase obtained from vapor/liquid calculations that use the Soave-Redlich-Kwong EOS. The solubility parameter, ''δ''<sub>''m''</sub>, is calculated from
 
The molar volume, ''v''<sub>''m''</sub>, of the oil mixture is calculated from the composition of the liquid phase obtained from vapor/liquid calculations that use the Soave-Redlich-Kwong EOS. The solubility parameter, ''δ''<sub>''m''</sub>, is calculated from
  
[[File:Vol1 page 0413 eq 005.png]]....................(6)
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[[File:Vol1 page 0413 eq 005.png|RTENOTITLE]]....................(6)
  
where Δ''U''<sub>''v''</sub> is the molar internal energy of vaporization at the system temperature, which also can be calculated from the Soave-Redlich-Kwong EOS. The remaining parameters are the molar volume of asphaltene, ''v''<sub>''a''</sub>, and the solubility parameter of asphaltene, ''δ''<sub>''a''</sub>, which are essential to the performance of this model. The molar volume of asphaltene can only be speculated on. Hirschberg ''et al.''<ref name="r5" /> used values of v a in the range of 1 to 4 m<sup>3</sup>/kmol. The solubility parameter of asphaltene can be estimated by measuring the solubility of asphaltene in different solvents of increasing solubility parameters. The asphaltene is assumed to have the solubility parameter of the best solvent. Hirschberg ''et al.''<ref name="r5" /> suggests that the solubility parameter of asphaltene is close to that of naphthalene. '''Eq. 5''' gives the amount (volume fraction) of asphaltene soluble in the oil mixture. The amount of precipitation is determined by the difference between the total amount of asphaltenes present in the initial oil and the solubility of asphaltene under given conditions.  
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where Δ''U''<sub>''v''</sub> is the molar internal energy of vaporization at the system temperature, which also can be calculated from the Soave-Redlich-Kwong EOS. The remaining parameters are the molar volume of asphaltene, ''v''<sub>''a''</sub>, and the solubility parameter of asphaltene, ''δ''<sub>''a''</sub>, which are essential to the performance of this model. The molar volume of asphaltene can only be speculated on. Hirschberg ''et al.''<ref name="r5">Hirschberg, A., deJong, L.N.J., Schipper, B.A. et al. 1984. Influence of Temperature and Pressure on Asphaltene Flocculation. SPE J. 24 (3): 283-293. SPE-11202-PA. http://dx.doi.org/10.2118/11202-PA</ref> used values of v a in the range of 1 to 4 m<sup>3</sup>/kmol. The solubility parameter of asphaltene can be estimated by measuring the solubility of asphaltene in different solvents of increasing solubility parameters. The asphaltene is assumed to have the solubility parameter of the best solvent. Hirschberg ''et al.''<ref name="r5">Hirschberg, A., deJong, L.N.J., Schipper, B.A. et al. 1984. Influence of Temperature and Pressure on Asphaltene Flocculation. SPE J. 24 (3): 283-293. SPE-11202-PA. http://dx.doi.org/10.2118/11202-PA</ref> suggests that the solubility parameter of asphaltene is close to that of naphthalene. '''Eq. 5''' gives the amount (volume fraction) of asphaltene soluble in the oil mixture. The amount of precipitation is determined by the difference between the total amount of asphaltenes present in the initial oil and the solubility of asphaltene under given conditions.
  
 
The solubility parameter can be correlated as a linear equation with respect to temperature as
 
The solubility parameter can be correlated as a linear equation with respect to temperature as
  
[[File:Vol1 page 0414 eq 001.png]]....................(7)
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[[File:Vol1 page 0414 eq 001.png|RTENOTITLE]]....................(7)
  
where ''a'' and ''b'' are constants. parameter ''b'' is negative as the solubility parameter decreases with increasing temperature. Buckley ''et al.''<ref name="r7" /> and Wang and Buckley<ref name="r8" /> showed that the measurements of the refractive index of crude oils can be used to determine the solubility parameters required for the Flory-Huggins model.  
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where ''a'' and ''b'' are constants. parameter ''b'' is negative as the solubility parameter decreases with increasing temperature. Buckley ''et al.''<ref name="r7">Buckley, J.S., Hirasaki, G.J., Liu, Y. et al. 1998. Asphaltene Precipitation and Solvent Properties of Crude Oils. Petroleum Science and Technology 16 (3-4): 251-285. http://dx.doi.org/10.1080/10916469808949783</ref> and Wang and Buckley<ref name="r8">Wang, J.X. and Buckley, J.S. 2001. An Experimental Approach to Prediction of Asphaltene Flocculation. Presented at the SPE International Symposium on Oilfield Chemistry, Houston, 13-16 February. SPE-64994-MS. http://dx.doi.org/10.2118/64994-MS</ref> showed that the measurements of the refractive index of crude oils can be used to determine the solubility parameters required for the Flory-Huggins model.
  
The Hirschberg ''et al.''<ref name="r5" /> approach also has been used with some degree of success by:  
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The Hirschberg ''et al.''<ref name="r5">Hirschberg, A., deJong, L.N.J., Schipper, B.A. et al. 1984. Influence of Temperature and Pressure on Asphaltene Flocculation. SPE J. 24 (3): 283-293. SPE-11202-PA. http://dx.doi.org/10.2118/11202-PA</ref> approach also has been used with some degree of success by:
  
* Burke ''et al.'',<ref name="r9" />  
+
*Burke ''et al.'',<ref name="r9">Burke, N.E., Hobbs, R.E., and Kashou, S.F. 1990. Measurement and Modeling of Asphaltene Precipitation. J Pet Technol 42 (11): 1440-1446. SPE-18273-PA. http://dx.doi.org/10.2118/18273-PA</ref>
* Kokal and Sayegh,<ref name="r10" />  
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*Kokal and Sayegh,<ref name="r10">Kokal, S.L. and Sayegh, S.G. 1995. Asphaltenes: The Cholesterol of Petroleum. Presented at the Middle East Oil Show, Bahrain, 11-14 March. SPE-29787-MS. http://dx.doi.org/10.2118/29787-MS</ref>
* Novosad and Costain,<ref name="r11" />  
+
*Novosad and Costain,<ref name="r11">Novosad, Z. and Costain, T.G. 1990. Experimental and Modeling Studies of Asphaltene Equilibria for a Reservoir Under CO2 Injection. Presented at the SPE Annual Technical Conference and Exhibition, New Orleans, 23-26 September. SPE-20530-MS. http://dx.doi.org/10.2118/20530-MS</ref>
* Nor-Azian and Adewumi,<ref name="r12" />  
+
*Nor-Azian and Adewumi,<ref name="r12">Nor-Azlan, N. and Adewumi, M.A. 1993. Development of Asphaltene Phase Equilibria Predictive Model. Presented at the SPE Eastern Regional Meeting, Pittsburgh, Pennsylvania, USA, 2-4 November. SPE-26905-MS. http://dx.doi.org/10.2118/26905-MS</ref>
* Rassamdana ''et al.''<ref name="r13" />
+
*Rassamdana ''et al.''<ref name="r13">Rassamdana, H., Dabir, B., Nematy, M. et al. 1996. Asphalt flocculation and deposition: I. The onset of precipitation. AIChE J. 42 (1): 10-22. http://dx.doi.org/10.1002/aic.690420104</ref>
  
de Boer ''et al.'' used this model to screen crude oils for their tendency to precipitate asphaltene. They compared properties of some crudes in which asphaltene problems were encountered and properties of crudes that operated trouble free. They found that asphaltene problems were encountered with light crudes with high C<sub>1</sub> to C<sub>3</sub> contents, high bubblepoint pressures, large differences between reservoir pressure and bubblepoint pressure, and high compressibility. With an asphaltene molar volume of 1 m<sup>3</sup>/kmol, de Boer ''et al.''<ref name="r14" /> showed that the solubility of asphaltene in a light crude oil with '''Eq. 5''' follows the curve shown in '''Fig. 1'''. Above the bubblepoint, the decrease in asphaltene solubility is caused by pressure effects. Below the bubblepoint, the increase in asphaltene solubility is caused by the variation in the oil composition. Clearly, a minimum asphaltene solubility occurs around the bubblepoint.  
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de Boer ''et al.'' used this model to screen crude oils for their tendency to precipitate asphaltene. They compared properties of some crudes in which asphaltene problems were encountered and properties of crudes that operated trouble free. They found that asphaltene problems were encountered with light crudes with high C<sub>1</sub> to C<sub>3</sub> contents, high bubblepoint pressures, large differences between reservoir pressure and bubblepoint pressure, and high compressibility. With an asphaltene molar volume of 1 m<sup>3</sup>/kmol, de Boer ''et al.''<ref name="r14">de Boer, R.B., Leeriooyer, K., Eigner, M.R.P. et al. 1995. Screening of Crude Oils for Asphalt Precipitation: Theory, Practice, and the Selection of Inhibitors. SPE Prod & Fac 10 (1): 55–61. SPE-24987-PA. http://dx.doi.org/10.2118/24987-PA</ref> showed that the solubility of asphaltene in a light crude oil with '''Eq. 5''' follows the curve shown in '''Fig. 1'''. Above the bubblepoint, the decrease in asphaltene solubility is caused by pressure effects. Below the bubblepoint, the increase in asphaltene solubility is caused by the variation in the oil composition. Clearly, a minimum asphaltene solubility occurs around the bubblepoint.
  
 
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</gallery>
 
</gallery>
  
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de Boer ''et al.''<ref name="r14">de Boer, R.B., Leeriooyer, K., Eigner, M.R.P. et al. 1995. Screening of Crude Oils for Asphalt Precipitation: Theory, Practice, and the Selection of Inhibitors. SPE Prod & Fac 10 (1): 55–61. SPE-24987-PA. http://dx.doi.org/10.2118/24987-PA</ref> calculated the solubility of asphaltene with '''Eq. 5''' for different values of in-situ crude oil densities and asphaltene-solubility parameters. They also introduced a maximum supersaturation at bubblepoint defined as
  
de Boer ''et al.''<ref name="r14" /> calculated the solubility of asphaltene with '''Eq. 5''' for different values of in-situ crude oil densities and asphaltene-solubility parameters. They also introduced a maximum supersaturation at bubblepoint defined as
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[[File:Vol1 page 0415 eq 001.png|RTENOTITLE]]....................(8)
 
 
[[File:Vol1 page 0415 eq 001.png]]....................(8)
 
  
where ''p''<sub>''r''</sub> and ''p''<sub>''b''</sub> are, respectively, the reservoir pressure and the bubblepoint pressure at the reservoir temperature. '''Fig. 2''' shows the maximum supersaturation at the bubblepoint as a function of the difference between reservoir and bubblepoint pressure, the in-situ oil density, and the asphaltene-solubility parameter. The influence of the asphaltene-solubility parameter is very small. Supersaturations are larger for lighter crudes. The boundary between problem and nonproblem areas lies at a maximum supersaturation of approximately 1. Although these results were derived with North Sea and Kuwait crudes, Hammami ''et al.''<ref name="r15" /> showed that they also are applicable to crudes from the Gulf of Mexico.  
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where ''p''<sub>''r''</sub> and ''p''<sub>''b''</sub> are, respectively, the reservoir pressure and the bubblepoint pressure at the reservoir temperature. '''Fig. 2''' shows the maximum supersaturation at the bubblepoint as a function of the difference between reservoir and bubblepoint pressure, the in-situ oil density, and the asphaltene-solubility parameter. The influence of the asphaltene-solubility parameter is very small. Supersaturations are larger for lighter crudes. The boundary between problem and nonproblem areas lies at a maximum supersaturation of approximately 1. Although these results were derived with North Sea and Kuwait crudes, Hammami ''et al.''<ref name="r15">Hammami, A., Phelps, C.H., Monger-McClure, T. et al. 1999. Asphaltene Precipitation from Live Oils:  An Experimental Investigation of Onset Conditions and Reversibility. Energy Fuels 14 (1): 14-18. http://dx.doi.org/10.1021/ef990104z</ref> showed that they also are applicable to crudes from the Gulf of Mexico.
  
 
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===Extension of Flory-Huggins model===
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=== Extension of Flory-Huggins model ===
The Flory-Huggins model initially was developed for [[Polymers|polymer]] solutions. The Hirschberg ''et al.''<ref name="r5" /> approach is based on the representation of asphaltene as a homogeneous polymer. Novosad and Constain<ref name="r11" /> used an extension of the model that includes asphaltene polymerization and asphaltene-resin association in the solid phase. Kawanaka ''et al.''<ref name="r16" /> proposed an improvement whereby the precipitated asphaltene is treated as a heterogeneous polymer (i.e., a mixture of polymers of different molecular weights). The Scott-Magat theory was used to obtain a solubility model for a given molecular-weight distribution for asphaltene. Cimino ''et al.''<ref name="r17" /> also used the Flory-Huggins model with two components (solvent and asphaltene) but considered the solid phase to be a mixture of solvent and asphaltene instead of pure asphaltene as in Hirschberg ''et al.'' ‘s approach. Yang ''et al.''<ref name="r18" /> proposed a multicomponent Flory-Huggins model in which components are the same as the EOS components used in the oil/gas flash calculations.  
+
 
 +
The Flory-Huggins model initially was developed for [[Polymers|polymer]] solutions. The Hirschberg ''et al.''<ref name="r5">Hirschberg, A., deJong, L.N.J., Schipper, B.A. et al. 1984. Influence of Temperature and Pressure on Asphaltene Flocculation. SPE J. 24 (3): 283-293. SPE-11202-PA. http://dx.doi.org/10.2118/11202-PA</ref> approach is based on the representation of asphaltene as a homogeneous polymer. Novosad and Constain<ref name="r11">Novosad, Z. and Costain, T.G. 1990. Experimental and Modeling Studies of Asphaltene Equilibria for a Reservoir Under CO2 Injection. Presented at the SPE Annual Technical Conference and Exhibition, New Orleans, 23-26 September. SPE-20530-MS. http://dx.doi.org/10.2118/20530-MS</ref> used an extension of the model that includes asphaltene polymerization and asphaltene-resin association in the solid phase. Kawanaka ''et al.''<ref name="r16">Kawanaka, S., Park, S.J., and Mansoori, G.A. 1991. Organic Deposition From Reservoir Fluids: A Thermodynamic Predictive Technique. SPE Res Eng 6 (2): 185-192. SPE-17376-PA. http://dx.doi.org/10.2118/17376-PA</ref> proposed an improvement whereby the precipitated asphaltene is treated as a heterogeneous polymer (i.e., a mixture of polymers of different molecular weights). The Scott-Magat theory was used to obtain a solubility model for a given molecular-weight distribution for asphaltene. Cimino ''et al.''<ref name="r17">Cimino, R., Correra, S., Sacomani, P.A. et al. 1995. Thermodynamic Modelling for Prediction of Asphaltene Deposition in Live Oils. Presented at the SPE International Symposium on Oilfield Chemistry, San Antonio, Texas, USA, 14-17 February. SPE-28993-MS. http://dx.doi.org/10.2118/28993-MS</ref> also used the Flory-Huggins model with two components (solvent and asphaltene) but considered the solid phase to be a mixture of solvent and asphaltene instead of pure asphaltene as in Hirschberg ''et al.'' ‘s approach. Yang ''et al.''<ref name="r18">Yang, Z., Ma, C.F., Lin, X.S. et al. 1999. Experimental and modeling studies on the asphaltene precipitation in degassed and gas-injected reservoir oils. Fluid Phase Equilib. 157 (1): 143-158. http://dx.doi.org/10.1016/S0378-3812(99)00004-7</ref> proposed a multicomponent Flory-Huggins model in which components are the same as the EOS components used in the oil/gas flash calculations.
  
===Multicomponent activity coefficient models===
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=== Multicomponent activity coefficient models ===
These models are derived from methods for modeling wax precipitation.<ref name="r19" /><ref name="r20" /><ref name="r21" /> Multicomponent solid/liquid K values are derived from '''Eq. 2''' and then used with an EOS in a three-phase oil/gas/solid flash calculation. The solid/liquid ''K'' values are defined as
 
  
[[File:Vol1 page 0416 eq 001.png]]....................(9)
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These models are derived from methods for modeling wax precipitation.<ref name="r19">Cimimo, R., Correra, S., Del Bianco, A. et al. 1995. Solubility and phase behavior of asphaltenes in hydrocarbon media. In Asphaltenes: Fundamentals and Applications, ed. E.Y. Sheu and O.C. Mullins, 97–130. New York: Plenum Press.</ref><ref name="r20">Won, K.W. 1986. Thermodynamics for solid solution-liquid-vapor equilibria: wax phase formation from heavy hydrocarbon mixtures. Fluid Phase Equilib. 30 (0): 265-279. http://dx.doi.org/10.1016/0378-3812(86)80061-9</ref><ref name="r21">Schou Pedersen, K., Skovborg, P., and Roenningsen, H.P. 1991. Wax precipitation from North Sea crude oils. 4. Thermodynamic modeling. Energy Fuels 5 (6): 924-932. http://dx.doi.org/10.1021/ef00030a022</ref> Multicomponent solid/liquid K values are derived from '''Eq. 2''' and then used with an EOS in a three-phase oil/gas/solid flash calculation. The solid/liquid ''K'' values are defined as
 +
 
 +
[[File:Vol1 page 0416 eq 001.png|RTENOTITLE]]....................(9)
  
 
Eq. 2 gives:
 
Eq. 2 gives:
  
[[File:Vol1 page 0416 eq 002.png]]....................(10)
+
[[File:Vol1 page 0416 eq 002.png|RTENOTITLE]]....................(10)
  
with [[File:Vol1 page 0416 eq 003.png]]....................(11)
+
with [[File:Vol1 page 0416 eq 003.png|RTENOTITLE]]....................(11)
  
Eq. 11 is equivalent to<ref name="r20" /><ref name="r21" />:
+
Eq. 11 is equivalent to<ref name="r20">_</ref><ref name="r21">_</ref>:
  
[[File:Vol1 page 0416 eq 004.png]]....................(12)
+
[[File:Vol1 page 0416 eq 004.png|RTENOTITLE]]....................(12)
  
where:  
+
where:
 +
 
 +
*''T''<sub>''if''</sub> = fusion temperature of component ''i''
 +
*Δ ''C''<sub>''pi''</sub> = ''C''<sub>''po,i''</sub> – ''C''<sub>''ps,i''</sub>, heat capacity change of fusion
 +
*Δ''H''<sub>''if''</sub> = heat of fusion of component ''i''
  
* ''T''<sub>''if''</sub> = fusion temperature of component ''i''
+
Δ''C''<sub>''pi''</sub> is assumed to be independent of temperature in '''Eq. 12'''.
* Δ ''C''<sub>''pi''</sub> = ''C''<sub>''po,i''</sub> – ''C''<sub>''ps,i''</sub>, heat capacity change of fusion
 
* Δ''H''<sub>''if''</sub> = heat of fusion of component ''i''  
 
  
Δ''C''<sub>''pi''</sub> is assumed to be independent of temperature in '''Eq. 12'''.  
+
Starting with '''Eq. 12''', methods were derived through the use of different models for activity coefficients. The earliest approach is from Won<ref name="r20">Won, K.W. 1986. Thermodynamics for solid solution-liquid-vapor equilibria: wax phase formation from heavy hydrocarbon mixtures. Fluid Phase Equilib. 30 (0): 265-279. http://dx.doi.org/10.1016/0378-3812(86)80061-9</ref> in the modeling of wax precipitation. Won<ref name="r20">Won, K.W. 1986. Thermodynamics for solid solution-liquid-vapor equilibria: wax phase formation from heavy hydrocarbon mixtures. Fluid Phase Equilib. 30 (0): 265-279. http://dx.doi.org/10.1016/0378-3812(86)80061-9</ref> suggested that the term involving Δ''C''<sub>''pi''</sub> and the integral involving Δ''v''<sub>''i''</sub> are negligible and used regular solution theory to calculate the activity coefficients in '''Eq. 12''' as follows.
  
Starting with '''Eq. 12''', methods were derived through the use of different models for activity coefficients. The earliest approach is from Won<ref name="r20" /> in the modeling of wax precipitation. Won<ref name="r20" /> suggested that the term involving Δ''C''<sub>''pi''</sub> and the integral involving Δ''v''<sub>''i''</sub> are negligible and used regular solution theory to calculate the activity coefficients in '''Eq. 12''' as follows.
+
[[File:Vol1 page 0416 eq 005.png|RTENOTITLE]]....................(13)
  
[[File:Vol1 page 0416 eq 005.png]]....................(13)
+
[[File:Vol1 page 0416 eq 006.png|RTENOTITLE]]....................(14)
  
[[File:Vol1 page 0416 eq 006.png]]....................(14)
+
[[File:Vol1 page 0416 eq 007.png|RTENOTITLE]]....................(15)
  
[[File:Vol1 page 0416 eq 007.png]]....................(15)
+
where:
  
where:
+
*''δ''<sub>''ik''</sub> is the solubility parameter of component ''i'' in phase ''k'' (''k'' = ''o'', ''s'')
* ''δ''<sub>''ik''</sub> is the solubility parameter of component ''i'' in phase ''k'' (''k'' = ''o'', ''s'')
+
*''v''<sub>''ik''</sub> is the molar volume of component ''i'' in phase ''k''
* ''v''<sub>''ik''</sub> is the molar volume of component ''i'' in phase ''k''
+
*''Φ''<sub>''ik''</sub> is the volume fraction of component ''i'' in phase ''k''
* ''Φ''<sub>''ik''</sub> is the volume fraction of component ''i'' in phase ''k''
 
  
Won gave solubility parameter values, ''δ''<sub>''io''</sub> and ''δ''<sub>''is''</sub>, for normal paraffins up to C<sub>40</sub>. Correlations also are provided to calculate Δ''H''<sub>''if''</sub>, ''v''<sub>''io''</sub>, and ''v''<sub>''is''</sub>. Although Won’s model was developed for wax precipitation, Thomas ''et al.''<ref name="r22" /> have applied it with some success in predicting asphaltene precipitation. However, they have developed their own correlations for solubility parameters. MacMillan ''et al.''<ref name="r23" /> also used Won’s model but kept all the terms in '''Eq. 12''' instead of neglecting the terms involving Δ''C''<sub>''pi''</sub> and Δ''v''<sub>''i''</sub> as Won did. They also included additional multiplication factors to the different terms in '''Eq. 12''' to facilitate phase-behavior matching.  
+
Won gave solubility parameter values, ''δ''<sub>''io''</sub> and ''δ''<sub>''is''</sub>, for normal paraffins up to C<sub>40</sub>. Correlations also are provided to calculate Δ''H''<sub>''if''</sub>, ''v''<sub>''io''</sub>, and ''v''<sub>''is''</sub>. Although Won’s model was developed for wax precipitation, Thomas ''et al.''<ref name="r22">Thomas, F.B., Bennion, D.B., Bennion, D.W. et al. 1992. Experimental And Theoretical Studies Of Solids Precipitation From Reservoir Fluid. J Can Pet Technol 31 (1): 22. PETSOC-92-01-02. http://dx.doi.org/10.2118/92-01-02</ref> have applied it with some success in predicting asphaltene precipitation. However, they have developed their own correlations for solubility parameters. MacMillan ''et al.''<ref name="r23">MacMillan, D.J., Tackett, J.E. Jr., Jessee, M.A. et al. 1995. A Unified Approach to Asphaltene Precipitation: Laboratory Measurement and Modeling. J Pet Technol 47 (9): 788-793. SPE-28990-PA. http://dx.doi.org/10.2118/28990-PA</ref> also used Won’s model but kept all the terms in '''Eq. 12''' instead of neglecting the terms involving Δ''C''<sub>''pi''</sub> and Δ''v''<sub>''i''</sub> as Won did. They also included additional multiplication factors to the different terms in '''Eq. 12''' to facilitate phase-behavior matching.
  
Hansen ''et al.''<ref name="r24" /> and Yarranton and Masliyah<ref name="r3" /> used the Flory-Huggins model to calculate the activity coefficients in '''Eq. 12'''. Hansen ''et al.''<ref name="r24" /> applied their method to the modeling of wax precipitation, while Yarranton and Masliyah<ref name="r3" /> modeled precipitation of Athabasca asphaltenes. Yarranton and Masliyah<ref name="r3" /> proposed an approach for calculating the molar volumes and solubility parameters from experimental measurements of molar mass and density. Asphaltene density, molar volume, and solubility parameter are correlated with molar mass. Zhou ''et al.''<ref name="r4" /> used the Flory-Huggins polymer-solution theory with a modification to account for the colloidal suspension effect of [[Asphaltenes and waxes|asphaltenes and resins]].
+
Hansen ''et al.''<ref name="r24">Musser, B.J. and Kilpatrick, P.K. 1998. Molecular Characterization of Wax Isolated from a Variety of Crude Oils. Energy Fuels 12 (4): 715-725. http://dx.doi.org/10.1021/ef970206u</ref> and Yarranton and Masliyah<ref name="r3">Yarranton, H.W. and Masliyah, J.H. 1996. Molar mass distribution and solubility modeling of asphaltenes. AIChE J. 42 (12): 3533-3543. http://dx.doi.org/10.1002/aic.690421222</ref> used the Flory-Huggins model to calculate the activity coefficients in '''Eq. 12'''. Hansen ''et al.''<ref name="r24">Musser, B.J. and Kilpatrick, P.K. 1998. Molecular Characterization of Wax Isolated from a Variety of Crude Oils. Energy Fuels 12 (4): 715-725. http://dx.doi.org/10.1021/ef970206u</ref> applied their method to the modeling of wax precipitation, while Yarranton and Masliyah<ref name="r3">Yarranton, H.W. and Masliyah, J.H. 1996. Molar mass distribution and solubility modeling of asphaltenes. AIChE J. 42 (12): 3533-3543. http://dx.doi.org/10.1002/aic.690421222</ref> modeled precipitation of Athabasca asphaltenes. Yarranton and Masliyah<ref name="r3">Yarranton, H.W. and Masliyah, J.H. 1996. Molar mass distribution and solubility modeling of asphaltenes. AIChE J. 42 (12): 3533-3543. http://dx.doi.org/10.1002/aic.690421222</ref> proposed an approach for calculating the molar volumes and solubility parameters from experimental measurements of molar mass and density. Asphaltene density, molar volume, and solubility parameter are correlated with molar mass. Zhou ''et al.''<ref name="r4">Zhou, X., Thomas, F.B., and Moore, R.G. 1996. Modelling of Solid Precipitation From Reservoir Fluid. J Can Pet Technol 35 (10). PETSOC-96-10-03. http://dx.doi.org/10.2118/96-10-03</ref> used the Flory-Huggins polymer-solution theory with a modification to account for the colloidal suspension effect of [[Asphaltenes_and_waxes|asphaltenes and resins]].
  
 
== Equation of state models ==
 
== Equation of state models ==
These approaches model the oil, gas, and precipitate by an EOS, which is used to calculate the component fugacities in different phases. Cubic EOSs have been used to model petroleum reservoir fluids that exhibit vapor/liquid 1/liquid 2 behavior (see Fussell,<ref name="r25" /> Nghiem and Li,<ref name="r26" /> or Godbole ''et al.''<ref name="r27" />). Godbole ''et al.''<ref name="r27" /> observed that the apparent second liquid phase could be approximated as a mixture of aggregated asphaltenes (solid phase) entrained in a portion of the other liquid phase in the modeling of mixtures of crude oil from the North Slope of Alaska and enriched gas. Under certain conditions, a phase-behavior program that includes a three-phase calculation with an EOS could be used to model some aspects of asphaltene precipitation; however, the prevailing approach consists of the use of a cubic EOS (e.g., Soave-Redlich-Kwong EOS<ref name="r6" /> or Peng-Robinson EOS<ref name="r28" />) for the oil and gas phases and a solid model for the precipitate.  
+
 
 +
These approaches model the oil, gas, and precipitate by an EOS, which is used to calculate the component fugacities in different phases. Cubic EOSs have been used to model petroleum reservoir fluids that exhibit vapor/liquid 1/liquid 2 behavior (see Fussell,<ref name="r25">Fussell, L.T. 1979. A Technique for Calculating Multiphase Equilibria. Society of Petroleum Engineers Journal 19 (4): 203-210. SPE-6722-PA. http://dx.doi.org/10.2118/6722-PA</ref> Nghiem and Li,<ref name="r26">Nghiem, L.X. and Li, Y.-K. 1984. Computation of multiphase equilibrium phenomena with an equation of state. Fluid Phase Equilib. 17 (1): 77-95. http://dx.doi.org/10.1016/0378-3812(84)80013-8</ref> or Godbole ''et al.''<ref name="r27">Godbole, S.P., Thele, K.J., and Reinbold, E.W. 1995. EOS Modeling and Experimental Observations of Three-Hydrocarbon-Phase Equilibria. SPE Res Eng 10 (2): 101-108. SPE-24936-PA. http://dx.doi.org/10.2118/24936-PA</ref>). Godbole ''et al.''<ref name="r27">Godbole, S.P., Thele, K.J., and Reinbold, E.W. 1995. EOS Modeling and Experimental Observations of Three-Hydrocarbon-Phase Equilibria. SPE Res Eng 10 (2): 101-108. SPE-24936-PA. http://dx.doi.org/10.2118/24936-PA</ref> observed that the apparent second liquid phase could be approximated as a mixture of aggregated asphaltenes (solid phase) entrained in a portion of the other liquid phase in the modeling of mixtures of crude oil from the North Slope of Alaska and enriched gas. Under certain conditions, a phase-behavior program that includes a three-phase calculation with an EOS could be used to model some aspects of asphaltene precipitation; however, the prevailing approach consists of the use of a cubic EOS (e.g., Soave-Redlich-Kwong EOS<ref name="r6">Soave, G. 1972. Equilibrium constants from a modified Redlich-Kwong equation of state. Chem. Eng. Sci. 27 (6): 1197–1203. http://dx.doi.org/10.1016/0009-2509(72)80096-4</ref> or Peng-Robinson EOS<ref name="r28">Peng, D.-Y. and Robinson, D.B. 1976. A New Two-Constant Equation of State. Industrial & Engineering Chemistry Fundamentals 15 (1): 59–64. http://dx.doi.org/10.1021/i160057a011</ref>) for the oil and gas phases and a solid model for the precipitate.
  
 
The simplest model for precipitated asphaltene is the single-component solid model. The precipitated asphaltene is represented as a pure solid, while the oil and gas phases are modeled with a cubic EOS. The fugacity of the pure solid is given by
 
The simplest model for precipitated asphaltene is the single-component solid model. The precipitated asphaltene is represented as a pure solid, while the oil and gas phases are modeled with a cubic EOS. The fugacity of the pure solid is given by
  
[[File:Vol1 page 0417 eq 001.png]]....................(16)
+
[[File:Vol1 page 0417 eq 001.png|RTENOTITLE]]....................(16)
 +
 
 +
where:
  
where:
 
 
*''f''<sub>''s''</sub> = solid fugacity
 
*''f''<sub>''s''</sub> = solid fugacity
*[[File:Vol1 page 0417 inline 001.png]] = reference solid fugacity,  
+
*[[File:Vol1 page 0417 inline 001.png|RTENOTITLE]] = reference solid fugacity,
 
*''p'' = pressure
 
*''p'' = pressure
 
*''p''* = reference pressure
 
*''p''* = reference pressure
Line 143: Line 152:
 
The following fugacity equality equations are solved to obtain oil/gas/solid equilibrium.
 
The following fugacity equality equations are solved to obtain oil/gas/solid equilibrium.
  
[[File:Vol1 page 0417 eq 002.png]]....................(17a)
+
[[File:Vol1 page 0417 eq 002.png|RTENOTITLE]]....................(17a)
  
and [[File:Vol1 page 0417 eq 003.png]]....................(17b)
+
and [[File:Vol1 page 0417 eq 003.png|RTENOTITLE]]....................(17b)
  
The oil and gas fugacities, ''f''<sub>''io''</sub> and ''f''<sub>''ig''</sub>, for component i are calculated from an EOS. In '''Eq. 17b''', subscript a denotes the asphaltene component in solution. Normally, this asphaltene component is the heaviest and last component of the oil (i.e., ''a'' = ''n''<sub>''c''</sub>). The following simple stability test can be used to determine whether there is asphaltene precipitation: if ''f''<sub>''ao''</sub> ≥ ''f''<sub>''s''</sub>, asphaltene precipitation occurs, and if ''f''<sub>''ao''</sub> < ''f''<sub>''s''</sub>, there is no precipitation.  
+
The oil and gas fugacities, ''f''<sub>''io''</sub> and ''f''<sub>''ig''</sub>, for component i are calculated from an EOS. In '''Eq. 17b''', subscript a denotes the asphaltene component in solution. Normally, this asphaltene component is the heaviest and last component of the oil (i.e., ''a'' = ''n''<sub>''c''</sub>). The following simple stability test can be used to determine whether there is asphaltene precipitation: if ''f''<sub>''ao''</sub> ≥ ''f''<sub>''s''</sub>, asphaltene precipitation occurs, and if ''f''<sub>''ao''</sub> < ''f''<sub>''s''</sub>, there is no precipitation.
  
Earlier applications of the single-component solid model for asphaltene precipitation were not successful.<ref name="r22" /> Nghiem ''et al.''<ref name="r29" /> introduced a method for representing the asphaltene component in the oil that improves the capabilities of the single-component solid model to predict asphaltene precipitation. The method was subsequently refined by Nghiem ''et al.''<ref name="r30" /><ref name="r31" /><ref name="r32" /><ref name="r33" /><ref name="r34" /> The key to the approach is the split of the heaviest fraction of the oil into two pseudocomponents:  
+
Earlier applications of the single-component solid model for asphaltene precipitation were not successful.<ref name="r22">Thomas, F.B., Bennion, D.B., Bennion, D.W. et al. 1992. Experimental And Theoretical Studies Of Solids Precipitation From Reservoir Fluid. J Can Pet Technol 31 (1): 22. PETSOC-92-01-02. http://dx.doi.org/10.2118/92-01-02</ref> Nghiem ''et al.''<ref name="r29">Nghiem, L.X., Hassam, M.S., Nutakki, R. et al. 1993. Efficient Modelling of Asphaltene Precipitation. Presented at the SPE Annual Technical Conference and Exhibition, Houston, Texas, 3-6 October. SPE-26642-MS. http://dx.doi.org/10.2118/26642-MS</ref> introduced a method for representing the asphaltene component in the oil that improves the capabilities of the single-component solid model to predict asphaltene precipitation. The method was subsequently refined by Nghiem ''et al.''<ref name="r30">Nghiem, L.X. and Coombe, D.A. 1997. Modeling Asphaltene Precipitation During Primary Depletion. SPE J. 2 (2): 170-176. SPE-36106-PA. http://dx.doi.org/10.2118/36106-PA</ref><ref name="r31">Nghiem, L.X., Coombe, D.A., and Farouq Ali, S.M. 1998. Compositional Simulation of Asphaltene Deposition and Plugging. Presented at the SPE Annual Technical Conference and Exhibition, New Orleans, 27-30 September. SPE-48996-MS. http://dx.doi.org/10.2118/48996-MS</ref><ref name="r32">Nghiem, L.X., Kohse, B.F., Ali, S.M.F. et al. 2000. Asphaltene Precipitation: Phase Behaviour Modelling and Compositional Simulation. Presented at the SPE Asia Pacific Conference on Integrated Modelling for Asset Management, Yokohama, Japan, 25-26 April. SPE-59432-MS. http://dx.doi.org/10.2118/59432-MS</ref><ref name="r33">Nghiem, L.X., Sammon, P.H., and Kohse, B.F. 2001. Modeling Asphaltene Precipitation and Dispersive Mixing in the Vapex Process. Presented at the SPE Reservoir Simulation Symposium, Houston, 11-14 February. SPE-66361-MS. http://dx.doi.org/10.2118/66361-MS</ref><ref name="r34">Nghiem, L.X., Kohse, B.F., and Sammon, P.H. 2001. Compositional Simulation of VAPEX Process. J Can Pet Technol 40 (8): 54–61. JCPT Paper No. 01-08-05. http://dx.doi.org/10.2118/01-08-05</ref> The key to the approach is the split of the heaviest fraction of the oil into two pseudocomponents:
  
* One that does not precipitate (nonprecipitating component)
+
*One that does not precipitate (nonprecipitating component)
* One that can precipitate (precipitating component)
+
*One that can precipitate (precipitating component)
  
These two pseudocomponents have identical critical temperatures, critical pressures, acentric factors, and molecular weights. The differences are in the interaction coefficients. The interaction coefficients between the precipitating components and the light components are larger than those between the nonprecipitating component and the light components. The parameters of the model are the reference fugacity and the solid molar volume. The reference fugacity could be estimated from a data point on the asphatene precipitation envelope (APE), and a value for solid molar volume slightly larger than the EOS value for the pure component ''a'' is adequate.<ref name="r29" />  
+
These two pseudocomponents have identical critical temperatures, critical pressures, acentric factors, and molecular weights. The differences are in the interaction coefficients. The interaction coefficients between the precipitating components and the light components are larger than those between the nonprecipitating component and the light components. The parameters of the model are the reference fugacity and the solid molar volume. The reference fugacity could be estimated from a data point on the asphatene precipitation envelope (APE), and a value for solid molar volume slightly larger than the EOS value for the pure component ''a'' is adequate.<ref name="r29">Nghiem, L.X., Hassam, M.S., Nutakki, R. et al. 1993. Efficient Modelling of Asphaltene Precipitation. Presented at the SPE Annual Technical Conference and Exhibition, Houston, Texas, 3-6 October. SPE-26642-MS. http://dx.doi.org/10.2118/26642-MS</ref>
  
The following application of the model to a North Sea fluid from Nghiem ''et al.''<ref name="r31" /> illustrates the procedure. '''Table 1''' shows the pseudocomponent representation of the reservoir fluid with the separator gas and separator oil compositions. The reservoir oil corresponds to a combination of 65.3 mol% separator oil and 34.7 mol% separator gas. The crucial step in the modeling of asphaltene is the split of the heaviest component in the oil (e.g., C<sub>32+</sub>) into:  
+
The following application of the model to a North Sea fluid from Nghiem ''et al.''<ref name="r31">Nghiem, L.X., Coombe, D.A., and Farouq Ali, S.M. 1998. Compositional Simulation of Asphaltene Deposition and Plugging. Presented at the SPE Annual Technical Conference and Exhibition, New Orleans, 27-30 September. SPE-48996-MS. http://dx.doi.org/10.2118/48996-MS</ref> illustrates the procedure. '''Table 1''' shows the pseudocomponent representation of the reservoir fluid with the separator gas and separator oil compositions. The reservoir oil corresponds to a combination of 65.3 mol% separator oil and 34.7 mol% separator gas. The crucial step in the modeling of asphaltene is the split of the heaviest component in the oil (e.g., C<sub>32+</sub>) into:
  
* A nonprecipitating component (C<sub>32A+</sub>)  
+
*A nonprecipitating component (C<sub>32A+</sub>)
* A precipitating component (C<sub>32B+</sub>)
+
*A precipitating component (C<sub>32B+</sub>)
  
These two components have identical critical properties and acentric factors but different interaction coefficients with the light components. The precipitating component has larger interaction coefficients with the light components. With larger interaction coefficients, the precipitating component becomes more "incompatible" with the light components and tends to precipitate as the amount of light component in solution increases. Although C<sub>32B+</sub> is called the precipitating component, the amount that precipitates is governed by '''Eq. 16'''. Normally, only a portion of the total amount of C<sub>32B+</sub> will precipitate during a calculation. Hirschberg ''et al.''<ref name="r5" /> reports that the asphalt precipitate from a tank oil consists mainly (90%) of C<sub>30</sub> to C<sub>60</sub> compounds. For the purpose of modeling asphaltene precipitation, a heaviest component in the vicinity of C<sub>30+</sub> is adequate. For this example, C<sub>32+</sub> is used.
+
These two components have identical critical properties and acentric factors but different interaction coefficients with the light components. The precipitating component has larger interaction coefficients with the light components. With larger interaction coefficients, the precipitating component becomes more "incompatible" with the light components and tends to precipitate as the amount of light component in solution increases. Although C<sub>32B+</sub> is called the precipitating component, the amount that precipitates is governed by '''Eq. 16'''. Normally, only a portion of the total amount of C<sub>32B+</sub> will precipitate during a calculation. Hirschberg ''et al.''<ref name="r5">Hirschberg, A., deJong, L.N.J., Schipper, B.A. et al. 1984. Influence of Temperature and Pressure on Asphaltene Flocculation. SPE J. 24 (3): 283-293. SPE-11202-PA. http://dx.doi.org/10.2118/11202-PA</ref> reports that the asphalt precipitate from a tank oil consists mainly (90%) of C<sub>30</sub> to C<sub>60</sub> compounds. For the purpose of modeling asphaltene precipitation, a heaviest component in the vicinity of C<sub>30+</sub> is adequate. For this example, C<sub>32+</sub> is used.
  
The Peng-Robinson EOS was used to model the oil and gas phases. The critical properties and acentric factors of the pseudocomponents in '''Table 1''' are calculated as described in Li ''et al.''<ref name="r35" /> The interaction coefficients are calculated from
+
The Peng-Robinson EOS was used to model the oil and gas phases. The critical properties and acentric factors of the pseudocomponents in '''Table 1''' are calculated as described in Li ''et al.''<ref name="r35">Li, Y.-K., Nghiem, L.X., and Siu, A. 1985. Phase Behaviour Computations For Reservoir Fluids: Effect Of Pseudo-Components On Phase Diagrams And Simulation Results. J Can Pet Technol 24 (6): 29. PETSOC-85-06-02. http://dx.doi.org/10.2118/85-06-02</ref> The interaction coefficients are calculated from
  
[[File:Vol1 page 0418 eq 001.png]]....................(18)
+
[[File:Vol1 page 0418 eq 001.png|RTENOTITLE]]....................(18)
  
where:  
+
where:
  
* ''d''<sub>''ij''</sub> = the interaction coefficient between component ''i'' and ''j''
+
*''d''<sub>''ij''</sub> = the interaction coefficient between component ''i'' and ''j''
* ''v''<sub>''ci''</sub> = the critical volume of component ''i'',  
+
*''v''<sub>''ci''</sub> = the critical volume of component ''i'',
* ''e'' = an adjustable parameter
+
*''e'' = an adjustable parameter
  
A value of ''e''(C<sub>32A+</sub>) = 0.84 and a value of ''e''(C<sub>32B+</sub>) = 1.57 were found to provide a good match of the saturation and onset pressure. The reference solid fugacity was obtained by calculating the fugacity of oil at one point on the APE (recombined oil with 69.9 mol% separator gas and 30 000 kPa) with the Peng-Robinson EOS and equating it to [[File:Vol1 page 0417 inline 001.png]]. The molar volume of the asphaltene precipitate was assumed equal to 0.9 m<sup>3</sup>/kmol.  
+
A value of ''e''(C<sub>32A+</sub>) = 0.84 and a value of ''e''(C<sub>32B+</sub>) = 1.57 were found to provide a good match of the saturation and onset pressure. The reference solid fugacity was obtained by calculating the fugacity of oil at one point on the APE (recombined oil with 69.9 mol% separator gas and 30 000 kPa) with the Peng-Robinson EOS and equating it to [[File:Vol1 page 0417 inline 001.png|RTENOTITLE]]. The molar volume of the asphaltene precipitate was assumed equal to 0.9 m<sup>3</sup>/kmol.
  
<gallery widths=300px heights=200px>
+
<gallery widths="300px" heights="200px">
 
File:Vol1 Page 418 Image 0001.png|'''Table 1'''
 
File:Vol1 Page 418 Image 0001.png|'''Table 1'''
 
</gallery>
 
</gallery>
  
 
+
'''Fig. 3''' shows a good match of the experimental and calculated APE and saturation pressure curves at the reservoir temperature of 90°C. The model was able to predict precipitation conditions that are far from the reference conditions used to determine [[File:Vol1 page 0417 inline 001.png|RTENOTITLE]]. '''Fig. 3''' shows the amounts of precipitation calculated as constant weight percent of precipitate (similar to "quality lines" in oil/gas phase diagrams). As pressure decreases below the APE, the amount of precipitation increases and reaches a maximum at the saturation pressure. Below the saturation pressure, the amount of precipitation decreases with decreasing pressure. The results are consistent with the laboratory observations described [[Experimental_measurements_of_asphaltene_precipitation|here]].
'''Fig. 3''' shows a good match of the experimental and calculated APE and saturation pressure curves at the reservoir temperature of 90°C. The model was able to predict precipitation conditions that are far from the reference conditions used to determine [[File:Vol1 page 0417 inline 001.png]]. '''Fig. 3''' shows the amounts of precipitation calculated as constant weight percent of precipitate (similar to "quality lines" in oil/gas phase diagrams). As pressure decreases below the APE, the amount of precipitation increases and reaches a maximum at the saturation pressure. Below the saturation pressure, the amount of precipitation decreases with decreasing pressure. The results are consistent with the laboratory observations described [[Experimental measurements of asphaltene precipitation|here]].  
 
  
 
<gallery widths="300px" heights="200px">
 
<gallery widths="300px" heights="200px">
Line 186: Line 194:
 
</gallery>
 
</gallery>
  
 +
For nonisothermal conditions, '''Eq. 19''' can be used to calculate the solid fugacity at (''p'', ''T'') from the solid fugacity at a reference condition (''p''*, ''T''*).<ref name="r32">_</ref><ref name="r36">_</ref>
  
For nonisothermal conditions, '''Eq. 19''' can be used to calculate the solid fugacity at (''p'', ''T'') from the solid fugacity at a reference condition (''p''*, ''T''*).<ref name="r32" /><ref name="r36" />
+
[[File:Vol1 page 0419 eq 001.png|RTENOTITLE]]....................(19)
  
[[File:Vol1 page 0419 eq 001.png]]....................(19)
+
where:
  
where:
+
*''f''<sub>''ℓ''</sub> = fugacity of the asphaltene component in the pure liquid state
* ''f''<sub>''ℓ''</sub> = fugacity of the asphaltene component in the pure liquid state
+
*''T''<sub>''f''</sub> = melting point temperature
* ''T''<sub>''f''</sub> = melting point temperature
+
*''v''<sub>''ℓ''</sub> = molar volume of liquid
* ''v''<sub>''ℓ''</sub> = molar volume of liquid
+
*Δ''C''<sub>''p''</sub> = heat capacity of fusion
* Δ''C''<sub>''p''</sub> = heat capacity of fusion
+
*Δ''H''<sub>''f''</sub> = enthalpy of fusion
* Δ''H''<sub>''f''</sub> = enthalpy of fusion
 
  
Kohse ''et al.''<ref name="r36" /> used '''Eq. 19''' to model the precipitation behavior of a crude oil with changes in pressure and temperature. '''Fig. 4''' shows good agreements between the experimental and calculated APE and saturation-pressure curves. The measured data point of 1.6 wt% of precipitate also is close to the predictions.  
+
Kohse ''et al.''<ref name="r36">Kohse, B.F., Nghiem, L.X., Maeda, H. et al. 2000. Modelling Phase Behaviour Including the Effect of Pressure and Temperature on Asphaltene Precipitation. Presented at the SPE Asia Pacific Oil and Gas Conference and Exhibition, Brisbane, Australia, 16-18 October. SPE-64465-MS. http://dx.doi.org/10.2118/64465-MS</ref> used '''Eq. 19''' to model the precipitation behavior of a crude oil with changes in pressure and temperature. '''Fig. 4''' shows good agreements between the experimental and calculated APE and saturation-pressure curves. The measured data point of 1.6 wt% of precipitate also is close to the predictions.
  
 
<gallery widths="300px" heights="200px">
 
<gallery widths="300px" heights="200px">
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</gallery>
 
</gallery>
  
 +
The previous two examples illustrate the application of the single-component solid model to the modeling of precipitation behavior of crudes with changes in:
  
The previous two examples illustrate the application of the single-component solid model to the modeling of precipitation behavior of crudes with changes in:
+
*Pressure
 +
*Temperature
 +
*Composition
  
* Pressure
+
From a mechanistic point of view, the nonprecipitating component can be related to:
* Temperature
 
* Composition
 
  
From a mechanistic point of view, the nonprecipitating component can be related to:
+
*Resins
 +
*Asphaltene/resin micelles that do not dissociate
 +
*Heavy paraffins
  
* Resins
+
The precipitating component corresponds to both the [[Asphaltenes_and_waxes|asphaltenes]] that dissociate and the asphaltene/resin micelles that precipitate unaltered. Because of identical critical properties and acentric factors, the nonprecipitating and precipitating components behave as a single component in solution. The larger interaction coefficients between the precipitating and the solvent components cause the precipitation of the former with the addition of solvent. The amount of precipitation depends on the solution of '''Eqs. 17a''' and '''17b'''. Normally, only a portion of the precipitating component actually precipitates.
* Asphaltene/resin micelles that do not dissociate
 
* Heavy paraffins
 
  
The precipitating component corresponds to both the [[Asphaltenes and waxes|asphaltenes]] that dissociate and the asphaltene/resin micelles that precipitate unaltered. Because of identical critical properties and acentric factors, the nonprecipitating and precipitating components behave as a single component in solution. The larger interaction coefficients between the precipitating and the solvent components cause the precipitation of the former with the addition of solvent. The amount of precipitation depends on the solution of '''Eqs. 17a''' and '''17b'''. Normally, only a portion of the precipitating component actually precipitates.  
+
Solid precipitation with the previous model is reversible. Nghiem ''et al.''<ref name="r34">Nghiem, L.X., Kohse, B.F., and Sammon, P.H. 2001. Compositional Simulation of VAPEX Process. J Can Pet Technol 40 (8): 54–61. JCPT Paper No. 01-08-05. http://dx.doi.org/10.2118/01-08-05</ref> proposed an enhancement to the approach to obtain partial irreversibility. A second solid (Solid 2) is introduced that is obtained from the reversible solid (Solid 1) through a chemical reaction:
  
Solid precipitation with the previous model is reversible. Nghiem ''et al.''<ref name="r34" /> proposed an enhancement to the approach to obtain partial irreversibility. A second solid (Solid 2) is introduced that is obtained from the reversible solid (Solid 1) through a chemical reaction:
+
[[File:Vol1 page 0420 eq 001.png|RTENOTITLE]]
  
[[File:Vol1 page 0420 eq 001.png]]
+
If the forward reaction rate k<sub>12</sub> is much larger than the backward reaction rate k<sub>21</sub>, Solid 2 behaves as a partially irreversible solid.
  
If the forward reaction rate k<sub>12</sub> is much larger than the backward reaction rate k<sub>21</sub>, Solid 2 behaves as a partially irreversible solid.
+
== Thermodynamic models ==
  
==Thermodynamic models==
 
 
=== Thermodynamic-colloidal model ===
 
=== Thermodynamic-colloidal model ===
  
Leontaritis and Mansoori<ref name="r37" /> proposed a more mechanistic approach based on the assumption that asphaltenes exist in the oil as solid particles in colloidal suspension stabilized by resins adsorbed on their surface. This thermodynamic-colloidal model assumes thermodynamic equilibrium between the resins in the oil phase and the resins adsorbed on the surface of colloidal asphaltene (asphaltene micelle). The corresponding equilibrium equation is
+
Leontaritis and Mansoori<ref name="r37">Leontaritis, K.J. and Mansoori, G.A. 1987. Asphaltene Flocculation During Oil Production and Processing: A Thermodynamic Colloidal Model. Presented at the SPE International Symposium on Oilfield Chemistry, San Antonio, Texas, USA, 4–6 February. SPE-16258-MS. http://dx.doi.org/10.2118/16258-MS</ref> proposed a more mechanistic approach based on the assumption that asphaltenes exist in the oil as solid particles in colloidal suspension stabilized by resins adsorbed on their surface. This thermodynamic-colloidal model assumes thermodynamic equilibrium between the resins in the oil phase and the resins adsorbed on the surface of colloidal asphaltene (asphaltene micelle). The corresponding equilibrium equation is
  
[[File:Vol1 page 0421 eq 001.png]]....................(20)
+
[[File:Vol1 page 0421 eq 001.png|RTENOTITLE]]....................(20)
  
 
Assuming that resins behave as monodisperse polymers and applying the Flory-Huggins polymer-solution theory gives the volume fraction of dissolved resins as
 
Assuming that resins behave as monodisperse polymers and applying the Flory-Huggins polymer-solution theory gives the volume fraction of dissolved resins as
  
[[File:Vol1 page 0421 eq 002.png]]....................(21)
+
[[File:Vol1 page 0421 eq 002.png|RTENOTITLE]]....................(21)
  
which is analogous to '''Eq. 5''' for the asphaltene component in Hirschberg ''et al.''‘s approach. In Hirschberg ''et al.''‘s approach, the asphaltene component contains both resins and asphaltene, whereas '''Eq. 21''' applies to the resins only. As in Hirschberg ''et al.''‘s approach, EOS flash calculations with a multicomponent system are performed to obtain an oil/gas split and oil properties from which ''Φ''<sub>''r''</sub> is calculated. This value of ''Φ''<sub>''r''</sub> is compared with a critical resin concentration, ''Φ''<sub>''cr''</sub>, which is given as a function of pressure, temperature, molar volume, and solubility parameters.  
+
which is analogous to '''Eq. 5''' for the asphaltene component in Hirschberg ''et al.''‘s approach. In Hirschberg ''et al.''‘s approach, the asphaltene component contains both resins and asphaltene, whereas '''Eq. 21''' applies to the resins only. As in Hirschberg ''et al.''‘s approach, EOS flash calculations with a multicomponent system are performed to obtain an oil/gas split and oil properties from which ''Φ''<sub>''r''</sub> is calculated. This value of ''Φ''<sub>''r''</sub> is compared with a critical resin concentration, ''Φ''<sub>''cr''</sub>, which is given as a function of pressure, temperature, molar volume, and solubility parameters.
  
''Φ''<sub>''cr''</sub> is the key parameter of the model. If ''Φ''<sub>''r''</sub> > ''Φ''<sub>''cr''</sub>, the system is stable and no precipitation occurs. If ''Φ''<sub>''r''</sub> ≤ ''Φ''<sub>''cr''</sub>, asphaltene precipitation occurs. The amount of precipitated asphaltene can be made a function of the asphaltene particle sizes.  
+
''Φ''<sub>''cr''</sub> is the key parameter of the model. If ''Φ''<sub>''r''</sub> > ''Φ''<sub>''cr''</sub>, the system is stable and no precipitation occurs. If ''Φ''<sub>''r''</sub> ≤ ''Φ''<sub>''cr''</sub>, asphaltene precipitation occurs. The amount of precipitated asphaltene can be made a function of the asphaltene particle sizes.
  
 
=== Thermodynamic-micellization model ===
 
=== Thermodynamic-micellization model ===
Pan and Firoozabadi<ref name="r38" /><ref name="r39" /> proposed the most mechanistic approach to model asphaltene precipitation by calculating the Gibbs free energy of formation of the asphaltene micelle and including it in the phase-equilibrium calculations. Details of the approach can be found in Firoozabadi.<ref name="r40" /> '''Fig. 5''' portrays schematically the system to be modeled. The species in the liquid phase (L<sub>1</sub>) are monomeric asphaltenes, monomeric resins, micelles, and asphalt-free oil species. The micelle consists of a core of ''n''<sub>1</sub> asphaltene molecules surrounded by a shell containing ''n''<sub>2</sub> resins molecules. The precipitate phase is considered as a liquid mixture (L<sub>2</sub>) of asphaltene and resin molecules. An expression for Gibbs free energy of formation of the micelle, [[File:Vol1 page 0421 inline 001.png]], which includes ''n''<sub>1</sub>, ''n''<sub>2</sub>, and the shell thickness, ''D'', was proposed. The Gibbs free energy of the liquid phase, L<sub>1</sub>, then is derived with:
 
  
* An EOS for the asphalt-free oil species
+
Pan and Firoozabadi<ref name="r38">Pan, H. and Firoozabadi, A. 1997. Thermodynamic Micellization Model for Asphaltene Precipitation from Reservoir Crudes at High Pressures and Temperatures. Presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, USA, 5-8 October. SPE-38857-MS. http://dx.doi.org/10.2118/38857-MS</ref><ref name="r39">Pan, H. and Firoozabadi, A. 1998. A Thermodynamic Micellization Model for Asphaltene Precipitation: Part I: Micellar Size and Growth. SPE Prod & Oper 13 (2): 118-127. SPE-36741-PA. http://dx.doi.org/10.2118/36741-PA</ref> proposed the most mechanistic approach to model asphaltene precipitation by calculating the Gibbs free energy of formation of the asphaltene micelle and including it in the phase-equilibrium calculations. Details of the approach can be found in Firoozabadi.<ref name="r40">Firoozabadi, A. 1999. Thermodynamics of Hydrocarbon Reservoirs. New York: McGraw-Hill.</ref> '''Fig. 5''' portrays schematically the system to be modeled. The species in the liquid phase (L<sub>1</sub>) are monomeric asphaltenes, monomeric resins, micelles, and asphalt-free oil species. The micelle consists of a core of ''n''<sub>1</sub> asphaltene molecules surrounded by a shell containing ''n''<sub>2</sub> resins molecules. The precipitate phase is considered as a liquid mixture (L<sub>2</sub>) of asphaltene and resin molecules. An expression for Gibbs free energy of formation of the micelle, [[File:Vol1 page 0421 inline 001.png|RTENOTITLE]], which includes ''n''<sub>1</sub>, ''n''<sub>2</sub>, and the shell thickness, ''D'', was proposed. The Gibbs free energy of the liquid phase, L<sub>1</sub>, then is derived with:
* Activity models for the monomeric asphaltenes and resins
+
 
* Gibbs free energy of formation of the micelle, [[File:Vol1 page 0421 inline 001.png]]
+
*An EOS for the asphalt-free oil species
 +
*Activity models for the monomeric asphaltenes and resins
 +
*Gibbs free energy of formation of the micelle, [[File:Vol1 page 0421 inline 001.png|RTENOTITLE]]
  
 
Similarly, the Gibbs free energy of the precipitated phase, L<sub>2</sub>, which is a binary mixture of monomeric asphaltenes and resins, also is derived with the use of an EOS. The total Gibbs free energy of the system,
 
Similarly, the Gibbs free energy of the precipitated phase, L<sub>2</sub>, which is a binary mixture of monomeric asphaltenes and resins, also is derived with the use of an EOS. The total Gibbs free energy of the system,
  
[[File:Vol1 page 0421 eq 003.png]]....................(22)
+
[[File:Vol1 page 0421 eq 003.png|RTENOTITLE]]....................(22)
 +
 
 +
then is minimized with respect to:
  
then is minimized with respect to:
+
*''n''<sub>1</sub> = number of asphaltene molecules in the micellar core
* ''n''<sub>1</sub> = number of asphaltene molecules in the micellar core
+
*''n''<sub>2</sub> = number of resin molecules in the micellar cell,
* ''n''<sub>2</sub> = number of resin molecules in the micellar cell,  
+
*''D'' = shell thickness of the micelle
* ''D'' = shell thickness of the micelle
+
*[[File:Vol1 page 0421 inline 002.png|RTENOTITLE]] = number of asphaltene monomers in liquid phase (''L''<sub>1</sub>)
* [[File:Vol1 page 0421 inline 002.png]] = number of asphaltene monomers in liquid phase (''L''<sub>1</sub>)
+
*[[File:Vol1 page 0421 inline 003.png|RTENOTITLE]] = number of resin monomers in phase ''L''<sub>1</sub>
* [[File:Vol1 page 0421 inline 003.png]] = number of resin monomers in phase ''L''<sub>1</sub>
+
*[[File:Vol1 page 0421 inline 004.png|RTENOTITLE]] = number of micelles in phase ''L''<sub>1</sub>
* [[File:Vol1 page 0421 inline 004.png]] = number of micelles in phase ''L''<sub>1</sub>
+
*[[File:Vol1 page 0421 inline 005.png|RTENOTITLE]] = number of asphaltene monomers in precipitated phase L<sub>2</sub>
* [[File:Vol1 page 0421 inline 005.png]] = number of asphaltene monomers in precipitated phase L<sub>2</sub>
+
*[[File:Vol1 page 0421 inline 006.png|RTENOTITLE]] = number of resin monomers in phase L<sub>2</sub>
* [[File:Vol1 page 0421 inline 006.png]] = number of resin monomers in phase L<sub>2</sub>
 
  
The minimization requires a robust numerical procedure.  
+
The minimization requires a robust numerical procedure.
  
 
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</gallery>
 
</gallery>
  
The model was applied to predict precipitation from a tank oil with propane,<ref name="r5" /> Weyburn oil with CO<sub>2</sub>,<ref name="r41" /> and a North Sea oil with separator gas. '''Fig. 6''' shows the predictions of Weyburn oil with CO<sub>2</sub> obtained with the thermodynamic-micellization model. For comparison, the match obtained with the pure solid model<ref name="r31" /> also is shown.  
+
The model was applied to predict precipitation from a tank oil with propane,<ref name="r5">Hirschberg, A., deJong, L.N.J., Schipper, B.A. et al. 1984. Influence of Temperature and Pressure on Asphaltene Flocculation. SPE J. 24 (3): 283-293. SPE-11202-PA. http://dx.doi.org/10.2118/11202-PA</ref> Weyburn oil with CO<sub>2</sub>,<ref name="r41">Srivastava, R.K., Huang, S.S., Dyer, S.B. et al. 1995. Quantification of Asphaltene Flocculation During Miscible CO2 Flooding In the Weyburn Reservoir. J Can Pet Technol 34 (8): 31. PETSOC-95-08-03. http://dx.doi.org/10.2118/95-08-03</ref> and a North Sea oil with separator gas. '''Fig. 6''' shows the predictions of Weyburn oil with CO<sub>2</sub> obtained with the thermodynamic-micellization model. For comparison, the match obtained with the pure solid model<ref name="r31">Nghiem, L.X., Coombe, D.A., and Farouq Ali, S.M. 1998. Compositional Simulation of Asphaltene Deposition and Plugging. Presented at the SPE Annual Technical Conference and Exhibition, New Orleans, 27-30 September. SPE-48996-MS. http://dx.doi.org/10.2118/48996-MS</ref> also is shown.
  
 
<gallery widths="300px" heights="200px">
 
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</gallery>
 
</gallery>
  
==Nomenclature==
+
== Nomenclature ==
 +
 
 
{|
 
{|
|''a''
 
|=
 
|constant
 
 
|-
 
|-
|''b''  
+
| ''a''
|=  
+
| =
|constant  
+
| constant
 +
|-
 +
| ''b''
 +
| =
 +
| constant
 +
|-
 +
| ''d''<sub>''ij''</sub>
 +
| =
 +
| interaction coefficient between component ''i'' and ''j''
 +
|-
 +
| ''D''
 +
| =
 +
| shell thickness of the micelle, L
 
|-
 
|-
|''d''<sub>''ij''</sub>  
+
| ''f''<sub>''ao''</sub>
|=  
+
| =
|interaction coefficient between component ''i'' and ''j''
+
| fugacity of asphaltene component in oil phase, m/Lt<sup>2</sup>
 
|-
 
|-
|''D''
+
| ''f''<sub>''i''</sub>
|=
+
| =
|shell thickness of the micelle, L
+
| fugacity of component ''i'', m/Lt<sup>2</sup>
|-
 
|''f''<sub>''ao''</sub>  
 
|=  
 
|fugacity of asphaltene component in oil phase, m/Lt<sup>2</sup>  
 
 
|-
 
|-
|''f''<sub>''i''</sub>
+
| [[File:Vol1 page 0434 inline 001.png|RTENOTITLE]]
|=  
+
| =
|fugacity of component ''i'', m/Lt<sup>2</sup>  
+
| standard state fugacity of component ''i'', m/Lt<sup>2</sup>
 
|-
 
|-
|[[File:Vol1 page 0434 inline 001.png]]
+
| ''f''<sub>''ig''</sub>
|=  
+
| =
|standard state fugacity of component ''i'', m/Lt<sup>2</sup>  
+
| fugacity of component ''i'' in the oil phase, m/Lt<sup>2</sup>
 
|-
 
|-
|''f''<sub>''ig''</sub>  
+
| ''f''<sub>''ik''</sub>
|=  
+
| =
|fugacity of component ''i'' in the oil phase, m/Lt<sup>2</sup>  
+
| fugacity of component ''i'' in phase k (''k'' = ''o'', ''s''), m/Lt<sup>2</sup>
 
|-
 
|-
|''f''<sub>''ik''</sub>
+
| [[File:Vol1 page 0433 inline 001.png|RTENOTITLE]]
|=  
+
| =
|fugacity of component ''i'' in phase k (''k'' = ''o'', ''s''), m/Lt<sup>2</sup>  
+
| fugacity of pure component ''i'' in phase state ''k'' (''k'' = ''o'', ''s''), m/Lt<sup>2</sup>
 
|-
 
|-
|[[File:Vol1 page 0433 inline 001.png]]  
+
| [[File:Vol1 page 0454 inline 001.png|RTENOTITLE]]
|=  
+
| =
|fugacity of pure component ''i'' in phase state ''k'' (''k'' = ''o'', ''s''), m/Lt<sup>2</sup>  
+
| standard state fugacity of component ''i'' in phase ''k'' (''k'' = ''o'', ''s''), m/Lt<sup>2</sup>
 
|-
 
|-
|[[File:Vol1 page 0454 inline 001.png]]
+
| ''f''<sub>''io''</sub>
|=  
+
| =
|standard state fugacity of component ''i'' in phase ''k'' (''k'' = ''o'', ''s''), m/Lt<sup>2</sup>  
+
| fugacity of component ''i'' in the oil phase, m/Lt<sup>2</sup>
 
|-
 
|-
|''f''<sub>''io''</sub>
+
| [[File:Vol1 page 0454 inline 002.png|RTENOTITLE]]
|=  
+
| =
|fugacity of component ''i'' in the oil phase, m/Lt<sup>2</sup>  
+
| fugacity of pure component ''i'' in oil phase, m/Lt<sup>2</sup>
 
|-
 
|-
|[[File:Vol1 page 0454 inline 002.png]]  
+
| [[File:Vol1 page 0454 inline 003.png|RTENOTITLE]]
|=  
+
| =
|fugacity of pure component ''i'' in oil phase, m/Lt<sup>2</sup>  
+
| standard state fugacity of component ''i'' in oil phase, m/Lt<sup>2</sup>
 
|-
 
|-
|[[File:Vol1 page 0454 inline 003.png]]
+
| ''f''<sub>''is''</sub>
|=  
+
| =
|standard state fugacity of component ''i'' in oil phase, m/Lt<sup>2</sup>  
+
| fugacity of component ''i'' in the solid phase, m/Lt<sup>2</sup>
 
|-
 
|-
|''f''<sub>''is''</sub>
+
| [[File:Vol1 page 0454 inline 004.png|RTENOTITLE]]
|=  
+
| =
|fugacity of component ''i'' in the solid phase, m/Lt<sup>2</sup>  
+
| fugacity of pure component ''i'' in solid phase, m/Lt<sup>2</sup>
 
|-
 
|-
|[[File:Vol1 page 0454 inline 004.png]]  
+
| [[File:Vol1 page 0454 inline 005.png|RTENOTITLE]]
|=  
+
| =
|fugacity of pure component ''i'' in solid phase, m/Lt<sup>2</sup>  
+
| standard state fugacity of component ''i'' in solid phase, m/Lt<sup>2</sup>
 
|-
 
|-
|[[File:Vol1 page 0454 inline 005.png]]
+
| ''f''<sub>''ℓ''</sub>
|=  
+
| =
|standard state fugacity of component ''i'' in solid phase, m/Lt<sup>2</sup>  
+
| fugacity of the asphaltene component in the pure liquid state, m/Lt<sup>2</sup>
 
|-
 
|-
|''f''<sub>''''</sub>  
+
| ''f''<sub>''p''</sub>
|=  
+
| =
|fugacity of the asphaltene component in the pure liquid state, m/Lt<sup>2</sup>
+
| porous medium particle transport efficiency factor
 
|-
 
|-
|''f''<sub>''p''</sub>  
+
| ''f''<sub>''s''</sub>
|=  
+
| =
|porous medium particle transport efficiency factor
+
| solid fugacity, m/Lt<sup>2</sup>
 
|-
 
|-
|''f''<sub>''s''</sub>
+
| [[File:Vol1 page 0454 inline 006.png|RTENOTITLE]]
|=  
+
| =
|solid fugacity, m/Lt<sup>2</sup>  
+
| reference solid fugacity, m/Lt<sup>2</sup>
 
|-
 
|-
|[[File:Vol1 page 0454 inline 006.png]]
+
| ''G''
|=  
+
| =
|reference solid fugacity, m/Lt<sup>2</sup>  
+
| total Gibbs free energy of the system, m/L<sup>2</sup>t<sup>2</sup>
 
|-
 
|-
|''G''  
+
| ''G''<sup>''E''</sup>
|=  
+
| =
|total Gibbs free energy of the system, m/L<sup>2</sup>t<sup>2</sup>  
+
| total excess Gibbs energy for a phase, m/L<sup>2</sup>t<sup>2</sup>
 
|-
 
|-
|''G''<sup>''E''</sup>  
+
| ''K''<sub>''is''</sub>
|=  
+
| =
|total excess Gibbs energy for a phase, m/L<sup>2</sup>t<sup>2</sup>
+
| solid/liquid ''K'' value for component ''i''
 
|-
 
|-
|''K''<sub>''is''</sub>  
+
| ''n''<sub>''c''</sub>
|=  
+
| =
|solid/liquid ''K'' value for component ''i''
+
| number of components
 
|-
 
|-
|''n''<sub>''c''</sub>  
+
| [[File:Vol1 page 0455 inline 001.png|RTENOTITLE]]
|=
+
| =
|number of components
+
| number of asphaltene monomers in phase ''L''<sub>1</sub>
 
|-
 
|-
|[[File:Vol1 page 0455 inline 001.png]]  
+
| [[File:Vol1 page 0455 inline 002.png|RTENOTITLE]]
|=  
+
| =
|number of asphaltene monomers in phase ''L''<sub>1</sub>  
+
| number of asphaltene monomers in phase ''L''<sub>2</sub>
 
|-
 
|-
|[[File:Vol1 page 0455 inline 002.png]]  
+
| [[File:Vol1 page 0455 inline 003.png|RTENOTITLE]]
|=  
+
| =
|number of asphaltene monomers in phase ''L''<sub>2</sub>  
+
| number of micelles in phase ''L''<sub>1</sub>
 
|-
 
|-
|[[File:Vol1 page 0455 inline 003.png]]  
+
| [[File:Vol1 page 0455 inline 004.png|RTENOTITLE]]
|=  
+
| =
|number of micelles in phase ''L''<sub>1</sub>  
+
| number of resin monomers in phase ''L''<sub>1</sub>
 
|-
 
|-
|[[File:Vol1 page 0455 inline 004.png]]  
+
| [[File:Vol1 page 0455 inline 005.png|RTENOTITLE]]
|=  
+
| =
|number of resin monomers in phase ''L''<sub>1</sub>  
+
| number of resin monomers in phase ''L''<sub>2</sub>
 
|-
 
|-
|[[File:Vol1 page 0455 inline 005.png]]
+
| ''p''
|=  
+
| =
|number of resin monomers in phase ''L''<sub>2</sub>  
+
| pressure, m/Lt<sup>2</sup>
 
|-
 
|-
|''p''  
+
| ''p''*
|=  
+
| =
|pressure, m/Lt<sup>2</sup>  
+
| reference pressure, m/Lt<sup>2</sup>
 
|-
 
|-
|''p''*
+
| ''p''<sub>''b''</sub>
|=  
+
| =
|reference pressure, m/Lt<sup>2</sup>  
+
| bubblepoint pressure, m/Lt<sup>2</sup>
 
|-
 
|-
|''p''<sub>''b''</sub>  
+
| ''p''<sub>''r''</sub>
|=  
+
| =
|bubblepoint pressure, m/Lt<sup>2</sup>  
+
| reservoir pressure, m/Lt<sup>2</sup>
 
|-
 
|-
|''p''<sub>''r''</sub>
+
| ''R''
|=  
+
| =
|reservoir pressure, m/Lt<sup>2</sup>
+
| gas constant
 
|-
 
|-
|''R''  
+
| ''T''
|=  
+
| =
|gas constant
+
| temperature, T
 
|-
 
|-
|''T''  
+
| ''T''*
|=  
+
| =
|temperature, T  
+
| reference temperature, T
 
|-
 
|-
|''T''*
+
| ''T''<sub>''c''</sub>
|=  
+
| =
|reference temperature, T
+
| temperature-dependent parameter
 
|-
 
|-
|''T''<sub>''c''</sub>  
+
| ''T''<sub>''f''</sub>
|=  
+
| =
|temperature-dependent parameter
+
| melting point temperature, T
 
|-
 
|-
|''T''<sub>''f''</sub>  
+
| ''T''<sub>''if''</sub>
|=  
+
| =
|melting point temperature, T  
+
| temperature of fusion (melting temperature) of component ''i'', T
 
|-
 
|-
|''T''<sub>''if''</sub>  
+
| ''u''<sub>''c''</sub>
|=  
+
| =
|temperature of fusion (melting temperature) of component ''i'', T
+
| critical speed required to mobilize surface deposit asphaltene, L/t
 
|-
 
|-
|''u''<sub>''c''</sub>  
+
| ''u''<sub>''o''</sub>
|=  
+
| =
|critical speed required to mobilize surface deposit asphaltene, L/t  
+
| oil velocity, L/t
 
|-
 
|-
|''u''<sub>''o''</sub>  
+
| ''v''<sub>''a''</sub>
|=  
+
| =
|oil velocity, L/t
+
| molar volume of pure asphaltene, L<sup>3</sup>/n
 
|-
 
|-
|''v''<sub>''a''</sub>  
+
| ''v''<sub>''ci''</sub>
|=  
+
| =
|molar volume of pure asphaltene, L<sup>3</sup>/n  
+
| critical volume of component ''i'', L<sup>3</sup>/n
 
|-
 
|-
|''v''<sub>''ci''</sub>  
+
| ''v''<sub>''cj''</sub>
|=  
+
| =
|critical volume of component ''i'', L<sup>3</sup>/n  
+
| critical volume of component ''j'', L<sup>3</sup>/n
 
|-
 
|-
|''v''<sub>''cj''</sub>  
+
| ''v''<sub>''ik''</sub>
|=  
+
| =
|critical volume of component ''j'', L<sup>3</sup>/n  
+
| partial molar volume of component ''i'' in phase ''k'' (''k'' = ''o'', ''s''), L<sup>3</sup>/n
 
|-
 
|-
|''v''<sub>''ik''</sub>  
+
| ''v''<sub>''io''</sub>
|=  
+
| =
|partial molar volume of component ''i'' in phase ''k'' (''k'' = ''o'', ''s''), L<sup>3</sup>/n  
+
| partial molar volume of component ''i'' in oil phase, L<sup>3</sup>/n
 
|-
 
|-
|''v''<sub>''io''</sub>  
+
| ''v''<sub>''is''</sub>
|=  
+
| =
|partial molar volume of component ''i'' in oil phase, L<sup>3</sup>/n  
+
| partial molar volume of component ''i'' in solid phase, L<sup>3</sup>/n
 
|-
 
|-
|''v''<sub>''is''</sub>  
+
| ''v''<sub>''jk''</sub>
|=  
+
| =
|partial molar volume of component ''i'' in solid phase, L<sup>3</sup>/n  
+
| partial molar volume of component ''j'' in phase ''k'' (''k'' = ''o'', ''s''), L<sup>3</sup>/n
 
|-
 
|-
|''v''<sub>''jk''</sub>  
+
| ''v''<sub>''jo''</sub>
|=  
+
| =
|partial molar volume of component ''j'' in phase ''k'' (''k'' = ''o'', ''s''), L<sup>3</sup>/n  
+
| partial molar volume of component ''j'' in oil phase, L<sup>3</sup>/n
 
|-
 
|-
|''v''<sub>''jo''</sub>  
+
| ''v''<sub>''js''</sub>
|=  
+
| =
|partial molar volume of component ''j'' in oil phase, L<sup>3</sup>/n  
+
| partial molar volume of component ''j'' in solid phase, L<sup>3</sup>/n
 
|-
 
|-
|''v''<sub>''js''</sub>  
+
| ''v''<sub>''''</sub>
|=  
+
| =
|partial molar volume of component ''j'' in solid phase, L<sup>3</sup>/n  
+
| molar volume of liquid, L<sup>3</sup>/n
 
|-
 
|-
|''v''<sub>''''</sub>  
+
| ''v''<sub>''m''</sub>
|=  
+
| =
|molar volume of liquid, L<sup>3</sup>/n  
+
| molar volume of mixture, L<sup>3</sup>/n
 
|-
 
|-
|''v''<sub>''m''</sub>  
+
| ''v''<sub>''r''</sub>
|=  
+
| =
|molar volume of mixture, L<sup>3</sup>/n  
+
| molar volume of resins, L<sup>3</sup>/n
 
|-
 
|-
|''v''<sub>''r''</sub>  
+
| ''v''<sub>''s''</sub>
|=  
+
| =
|molar volume of resins, L<sup>3</sup>/n  
+
| solid molar volume, L<sup>3</sup>/n
 
|-
 
|-
|''v''<sub>''s''</sub>  
+
| ''w''<sub>''i''</sub>
|=  
+
| =
|solid molar volume, L<sup>3</sup>/n
+
| weight fraction of component ''i'', m/m
 
|-
 
|-
|''w''<sub>''i''</sub>
+
| ''W''
|=  
+
| =
|weight fraction of component ''i'', m/m  
+
| weight percent of precipitated asphaltene, m/m
 
|-
 
|-
|''W''  
+
| ''y''<sub>''i''</sub>
|=  
+
| =
|weight percent of precipitated asphaltene, m/m
+
| mole fraction of component ''i'', n/n
 
|-
 
|-
|''y''<sub>''i''</sub>  
+
| ''y''<sub>''ik''</sub>
|=  
+
| =
|mole fraction of component ''i'', n/n  
+
| mole fraction of component ''i'' in phase ''k'' (''k'' = ''o'', ''s''), n/n
 
|-
 
|-
|''y''<sub>''ik''</sub>  
+
| ''y''<sub>''io''</sub>
|=  
+
| =
|mole fraction of component ''i'' in phase ''k'' (''k'' = ''o'', ''s''), n/n  
+
| mole fraction of component ''i'' in oil phase, n/n
 
|-
 
|-
|''y''<sub>''io''</sub>  
+
| ''y''<sub>''is''</sub>
|=  
+
| =
|mole fraction of component ''i'' in oil phase, n/n  
+
| mole fraction of component ''i'' in solid phase, n/n
 
|-
 
|-
|''y''<sub>''is''</sub>  
+
| ''y''<sub>''jo''</sub>
|=  
+
| =
|mole fraction of component ''i'' in solid phase, n/n  
+
| mole fraction of component ''j'' in oil phase, n/n
 
|-
 
|-
|''y''<sub>''jo''</sub>  
+
| ''y''<sub>''js''</sub>
|=  
+
| =
|mole fraction of component ''j'' in oil phase, n/n  
+
| mole fraction of component ''j'' in solid phase, n/n
 
|-
 
|-
|''y''<sub>''js''</sub>
+
| ''γ''
|=  
+
| =
|mole fraction of component ''j'' in solid phase, n/n
+
| shear rate, L/t
 
|-
 
|-
|''γ''
+
| ''γ''<sub>''i''</sub>
|=
+
| =
|shear rate, L/t
+
| activity coefficient of component ''i'' in a mixture
|-
 
|''γ''<sub>''i''</sub>  
 
|=  
 
|activity coefficient of component ''i'' in a mixture  
 
 
|-
 
|-
|''γ''<sub>''ik''</sub>  
+
| ''γ''<sub>''ik''</sub>
|=  
+
| =
|activity coefficient of component ''i'' in phase ''k'' (''k'' = ''o'', ''s'')  
+
| activity coefficient of component ''i'' in phase ''k'' (''k'' = ''o'', ''s'')
 
|-
 
|-
|''γ''<sub>''io''</sub>  
+
| ''γ''<sub>''io''</sub>
|=  
+
| =
|activity coefficient of component ''i'' in oil phase  
+
| activity coefficient of component ''i'' in oil phase
 
|-
 
|-
|[[File:Vol1 page 0456 inline 001.png]]  
+
| [[File:Vol1 page 0456 inline 001.png|RTENOTITLE]]
|=  
+
| =
|combinatorial free volume contribution  
+
| combinatorial free volume contribution
 
|-
 
|-
|[[File:Vol1 page 0456 inline 002.png]]  
+
| [[File:Vol1 page 0456 inline 002.png|RTENOTITLE]]
|=  
+
| =
|residual contribution  
+
| residual contribution
 
|-
 
|-
|''γ''<sub>''is''</sub>  
+
| ''γ''<sub>''is''</sub>
|=  
+
| =
|activity coefficient of component ''i'' in solid phase  
+
| activity coefficient of component ''i'' in solid phase
 
|-
 
|-
|''δ''<sub>''a''</sub>  
+
| ''δ''<sub>''a''</sub>
|=  
+
| =
|solubility parameter of asphaltene  
+
| solubility parameter of asphaltene
 
|-
 
|-
|''δ''<sub>''i''</sub>  
+
| ''δ''<sub>''i''</sub>
|=  
+
| =
|solubility parameter for component ''i''  
+
| solubility parameter for component ''i''
 
|-
 
|-
|''δ''<sub>''ik''</sub>  
+
| ''δ''<sub>''ik''</sub>
|=  
+
| =
|solubility parameter for pure component ''i'' in phase ''k'' (''k'' = ''o'', ''s'')  
+
| solubility parameter for pure component ''i'' in phase ''k'' (''k'' = ''o'', ''s'')
 
|-
 
|-
|[[File:Vol1 page 0456 inline 003.png]]  
+
| [[File:Vol1 page 0456 inline 003.png|RTENOTITLE]]
|=  
+
| =
|volume fraction average solubility parameter for phase ''k''  
+
| volume fraction average solubility parameter for phase ''k''
 
|-
 
|-
|''δ''<sub>''m''</sub>  
+
| ''δ''<sub>''m''</sub>
|=  
+
| =
|solubility parameter of mixture  
+
| solubility parameter of mixture
 
|-
 
|-
|''δ''<sub>''o''</sub>  
+
| ''δ''<sub>''o''</sub>
|=  
+
| =
|solubility parameter of oil phase  
+
| solubility parameter of oil phase
 
|-
 
|-
|[[File:Vol1 page 0456 inline 004.png]]  
+
| [[File:Vol1 page 0456 inline 004.png|RTENOTITLE]]
|=  
+
| =
|volume fraction average solubility parameter of oil phase  
+
| volume fraction average solubility parameter of oil phase
 
|-
 
|-
|''δ''<sub>''s''</sub>  
+
| ''δ''<sub>''s''</sub>
|=  
+
| =
|solubility parameter of solid phase  
+
| solubility parameter of solid phase
 
|-
 
|-
|[[File:Vol1 page 0456 inline 005.png]]  
+
| [[File:Vol1 page 0456 inline 005.png|RTENOTITLE]]
|=  
+
| =
|volume fraction average solubility parameter of solid phase  
+
| volume fraction average solubility parameter of solid phase
 
|-
 
|-
|Δ''C''<sub>''p''</sub>  
+
| Δ''C''<sub>''p''</sub>
|=  
+
| =
|heat capacity of fusion, mL<sup>2</sup>/nt<sup>2</sup>T  
+
| heat capacity of fusion, mL<sup>2</sup>/nt<sup>2</sup>T
 
|-
 
|-
|Δ''C''<sub>''pi''</sub>  
+
| Δ''C''<sub>''pi''</sub>
|=  
+
| =
|heat capacity of fusion of component ''i'', mL<sup>2</sup>/nt<sup>2</sup>T  
+
| heat capacity of fusion of component ''i'', mL<sup>2</sup>/nt<sup>2</sup>T
 
|-
 
|-
|Δ''C''<sub>''pij,tr''</sub>  
+
| Δ''C''<sub>''pij,tr''</sub>
|=  
+
| =
|heat capacity of ''j''th solid state transition of component ''i'', mL<sup>2</sup>/nt<sup>2</sup>T  
+
| heat capacity of ''j''th solid state transition of component ''i'', mL<sup>2</sup>/nt<sup>2</sup>T
 
|-
 
|-
|[[File:Vol1 page 0456 inline 006.png]]  
+
| [[File:Vol1 page 0456 inline 006.png|RTENOTITLE]]
|=  
+
| =
|expression for Gibbs free energy of formation of the micelle, mL<sup>2</sup>/nt<sup>2</sup>  
+
| expression for Gibbs free energy of formation of the micelle, mL<sup>2</sup>/nt<sup>2</sup>
 
|-
 
|-
|Δ''H''<sub>''f''</sub>  
+
| Δ''H''<sub>''f''</sub>
|=  
+
| =
|enthalpy of fusion, mL<sup>2</sup>/nt<sup>2</sup>  
+
| enthalpy of fusion, mL<sup>2</sup>/nt<sup>2</sup>
 
|-
 
|-
|Δ''H''<sub>''if''</sub>  
+
| Δ''H''<sub>''if''</sub>
|=  
+
| =
|enthalpy of fusion of component ''i'', mL<sup>2</sup>/nt<sup>2</sup>
+
| enthalpy of fusion of component ''i'', mL<sup>2</sup>/nt<sup>2</sup>
 
|-
 
|-
|Δ''U''<sub>''v''</sub>  
+
| Δ''U''<sub>''v''</sub>
|=  
+
| =
|molar internal energy of vaporization at the system temperature, mL<sup>2</sup>/nt<sup>2</sup>  
+
| molar internal energy of vaporization at the system temperature, mL<sup>2</sup>/nt<sup>2</sup>
 
|-
 
|-
|Δ''v''<sub>''i''</sub>  
+
| Δ''v''<sub>''i''</sub>
|=  
+
| =
|change of molar volume caused by fusion of component ''i'', L<sup>3</sup>  
+
| change of molar volume caused by fusion of component ''i'', L<sup>3</sup>
 
|-
 
|-
|[[File:Vol1 page 0457 inline 001.png]]  
+
| [[File:Vol1 page 0457 inline 001.png|RTENOTITLE]]
|=  
+
| =
|reference chemical potential of asphaltene component  
+
| reference chemical potential of asphaltene component
 
|-
 
|-
|''μ''<sub>''am''</sub>  
+
| ''μ''<sub>''am''</sub>
|=  
+
| =
|chemical potential of asphaltene in the mixture  
+
| chemical potential of asphaltene in the mixture
 
|-
 
|-
|''μ''<sub>''c''</sub>  
+
| ''μ''<sub>''c''</sub>
|=  
+
| =
|critical speed required to mobilize surface deposit asphaltene  
+
| critical speed required to mobilize surface deposit asphaltene
 
|-
 
|-
|''μ''<sub>''i''</sub>  
+
| ''μ''<sub>''i''</sub>
|=  
+
| =
|chemical potential of component ''i'', m/L<sup>2</sup>t<sup>2</sup>  
+
| chemical potential of component ''i'', m/L<sup>2</sup>t<sup>2</sup>
 
|-
 
|-
|''μ''<sub>''ik''</sub>  
+
| ''μ''<sub>''ik''</sub>
|=  
+
| =
|chemical potential of component ''i'' in phase ''k'', m/L<sup>2</sup>t<sup>2</sup>  
+
| chemical potential of component ''i'' in phase ''k'', m/L<sup>2</sup>t<sup>2</sup>
 
|-
 
|-
|''μ''<sub>''io''</sub>  
+
| ''μ''<sub>''io''</sub>
|=  
+
| =
|chemical potential of component ''i'' in the oil phase, m/L<sup>2</sup>t<sup>2</sup>  
+
| chemical potential of component ''i'' in the oil phase, m/L<sup>2</sup>t<sup>2</sup>
 
|-
 
|-
|''μ''<sub>''is''</sub>  
+
| ''μ''<sub>''is''</sub>
|=  
+
| =
|chemical potential of component ''i'' in the solid phase, m/L<sup>2</sup>t<sup>2</sup>  
+
| chemical potential of component ''i'' in the solid phase, m/L<sup>2</sup>t<sup>2</sup>
 
|-
 
|-
|''μ''<sub>''pi,k''</sub>  
+
| ''μ''<sub>''pi,k''</sub>
|=  
+
| =
|chemical potential of pure component ''i'' in phase ''k'' (''k'' = ''o'', ''s''), m/L<sup>2</sup>t<sup>2</sup>  
+
| chemical potential of pure component ''i'' in phase ''k'' (''k'' = ''o'', ''s''), m/L<sup>2</sup>t<sup>2</sup>
 
|-
 
|-
|''μ''<sub>''pi,o''</sub>  
+
| ''μ''<sub>''pi,o''</sub>
|=  
+
| =
|chemical potential of pure component ''i'' in oil phase, m/L<sup>2</sup>t<sup>2</sup>  
+
| chemical potential of pure component ''i'' in oil phase, m/L<sup>2</sup>t<sup>2</sup>
 
|-
 
|-
|''μ''<sub>''pi,s''</sub>  
+
| ''μ''<sub>''pi,s''</sub>
|=  
+
| =
|chemical potential of pure component ''i'' in solid phase, m/L<sup>2</sup>t<sup>2</sup>  
+
| chemical potential of pure component ''i'' in solid phase, m/L<sup>2</sup>t<sup>2</sup>
 
|-
 
|-
|''μ''<sub>''r,m''</sub>  
+
| ''μ''<sub>''r,m''</sub>
|=  
+
| =
|chemical potential of resins on the surface of the asphaltene micelle, m/L<sup>2</sup>t<sup>2</sup>  
+
| chemical potential of resins on the surface of the asphaltene micelle, m/L<sup>2</sup>t<sup>2</sup>
 
|-
 
|-
|''μ''<sub>''r,o''</sub>  
+
| ''μ''<sub>''r,o''</sub>
|=  
+
| =
|chemical potential of resins in the oil phase, m/L<sup>2</sup>t<sup>2</sup>  
+
| chemical potential of resins in the oil phase, m/L<sup>2</sup>t<sup>2</sup>
 
|-
 
|-
|''μ''<sub>''s''</sub>  
+
| ''μ''<sub>''s''</sub>
|=  
+
| =
|chemical potential of the solid, m/L<sup>2</sup>t<sup>2</sup>  
+
| chemical potential of the solid, m/L<sup>2</sup>t<sup>2</sup>
 
|-
 
|-
|''ρ''<sub>''o''</sub>  
+
| ''ρ''<sub>''o''</sub>
|=  
+
| =
|mass density of oil, m/L<sup>3</sup>  
+
| mass density of oil, m/L<sup>3</sup>
 
|-
 
|-
|''Φ''<sub>''ik''</sub>  
+
| ''Φ''<sub>''ik''</sub>
|=  
+
| =
|fugacity coefficient of component ''i'' in phase ''k''  
+
| fugacity coefficient of component ''i'' in phase ''k''
 
|-
 
|-
|[[File:Vol1 page 0457 inline 002.png]]  
+
| [[File:Vol1 page 0457 inline 002.png|RTENOTITLE]]
|=  
+
| =
|fugacity coefficient of pure component ''i'' in phase state ''k'' (''k'' = ''o'', ''s'')  
+
| fugacity coefficient of pure component ''i'' in phase state ''k'' (''k'' = ''o'', ''s'')
 
|-
 
|-
|''Φ''<sub>''io''</sub>  
+
| ''Φ''<sub>''io''</sub>
|=  
+
| =
|fugacity coefficient of component ''i'' in oil phase  
+
| fugacity coefficient of component ''i'' in oil phase
 
|-
 
|-
|''Φ''<sub>''a''</sub>  
+
| ''Φ''<sub>''a''</sub>
|=  
+
| =
|volume fraction of asphaltene in the mixture  
+
| volume fraction of asphaltene in the mixture
 
|-
 
|-
|''Φ''<sub>''cr''</sub>  
+
| ''Φ''<sub>''cr''</sub>
|=  
+
| =
|critical volume fraction of resins in the mixture  
+
| critical volume fraction of resins in the mixture
 
|-
 
|-
|''Φ''<sub>''ik''</sub>  
+
| ''Φ''<sub>''ik''</sub>
|=  
+
| =
|volume fraction of component ''i'' in phase state ''k'' (''k'' = ''o'', ''s'')  
+
| volume fraction of component ''i'' in phase state ''k'' (''k'' = ''o'', ''s'')
 
|-
 
|-
|''Φ''<sub>''r''</sub>  
+
| ''Φ''<sub>''r''</sub>
|=  
+
| =
|volume fraction of resins in the mixture  
+
| volume fraction of resins in the mixture
 
|}
 
|}
  
==References==
+
== References ==
<references>
+
 
<ref name="r1">Andersen, S.I. and Speight, J.G. 1999. Thermodynamic models for asphaltene solubility and precipitation. ''J. Pet. Sci. Eng''. '''22''' (1–3): 53-66. http://dx.doi.org/10.1016/S0920-4105(98)00057-6</ref>
+
<references />
<ref name="r2">Chung, T.-H. 1992. Thermodynamic Modeling for Organic Solid Precipitation. Presented at the SPE Annual Technical Conference and Exhibition, Washington, DC, 4-7 October. SPE-24851-MS. http://dx.doi.org/10.2118/24851-MS</ref>
+
 
<ref name="r3">Yarranton, H.W. and Masliyah, J.H. 1996. Molar mass distribution and solubility modeling of asphaltenes. ''AIChE J''. '''42''' (12): 3533-3543. http://dx.doi.org/10.1002/aic.690421222</ref>
+
== Noteworthy papers in OnePetro ==
<ref name="r4">Zhou, X., Thomas, F.B., and  Moore, R.G. 1996. Modelling of Solid Precipitation From Reservoir Fluid. ''J Can Pet Technol'' '''35''' (10). PETSOC-96-10-03. http://dx.doi.org/10.2118/96-10-03</ref>
 
<ref name="r5">Hirschberg, A., deJong, L.N.J., Schipper, B.A. et al. 1984. Influence of Temperature and Pressure on Asphaltene Flocculation. ''SPE J''. '''24''' (3): 283-293. SPE-11202-PA. http://dx.doi.org/10.2118/11202-PA</ref>
 
<ref name="r6">Soave, G. 1972. Equilibrium constants from a modified Redlich-Kwong equation of state. ''Chem. Eng. Sci''. '''27''' (6): 1197–1203. http://dx.doi.org/10.1016/0009-2509(72)80096-4</ref>
 
<ref name="r7">Buckley, J.S., Hirasaki, G.J., Liu, Y. et al. 1998. Asphaltene Precipitation and Solvent Properties of Crude Oils. ''Petroleum Science and Technology'' '''16''' (3-4): 251-285. http://dx.doi.org/10.1080/10916469808949783</ref>
 
<ref name="r8">Wang, J.X. and Buckley, J.S. 2001. An Experimental Approach to Prediction of Asphaltene Flocculation. Presented at the SPE International Symposium on Oilfield Chemistry, Houston, 13-16 February. SPE-64994-MS. http://dx.doi.org/10.2118/64994-MS</ref>
 
<ref name="r9">Burke, N.E., Hobbs, R.E., and  Kashou, S.F. 1990. Measurement and Modeling of Asphaltene Precipitation. ''J Pet Technol'' '''42''' (11): 1440-1446. SPE-18273-PA. http://dx.doi.org/10.2118/18273-PA</ref>
 
<ref name="r10">Kokal, S.L. and Sayegh, S.G. 1995. Asphaltenes: The Cholesterol of Petroleum. Presented at the Middle East Oil Show, Bahrain, 11-14 March. SPE-29787-MS. http://dx.doi.org/10.2118/29787-MS</ref>
 
<ref name="r11">Novosad, Z. and Costain, T.G. 1990. Experimental and Modeling Studies of Asphaltene Equilibria for a Reservoir Under CO2 Injection. Presented at the SPE Annual Technical Conference and Exhibition, New Orleans, 23-26 September. SPE-20530-MS. http://dx.doi.org/10.2118/20530-MS</ref>
 
<ref name="r12">Nor-Azlan, N. and Adewumi, M.A. 1993. Development of Asphaltene Phase Equilibria Predictive Model. Presented at the SPE Eastern Regional Meeting, Pittsburgh, Pennsylvania, USA, 2-4 November. SPE-26905-MS. http://dx.doi.org/10.2118/26905-MS</ref>
 
<ref name="r13">Rassamdana, H., Dabir, B., Nematy, M. et al. 1996. Asphalt flocculation and deposition: I. The onset of precipitation. ''AIChE J.'' '''42''' (1): 10-22. http://dx.doi.org/10.1002/aic.690420104</ref>
 
<ref name="r14">de Boer, R.B., Leeriooyer, K., Eigner, M.R.P. et al. 1995. Screening of Crude Oils for Asphalt Precipitation: Theory, Practice, and the Selection of Inhibitors. ''SPE'' ''Prod & Fac'' '''10''' (1): 55–61. SPE-24987-PA. http://dx.doi.org/10.2118/24987-PA</ref>
 
<ref name="r15">Hammami, A., Phelps, C.H., Monger-McClure, T. et al. 1999. Asphaltene Precipitation from Live Oils:  An Experimental Investigation of Onset Conditions and Reversibility. ''Energy Fuels'' '''14''' (1): 14-18. http://dx.doi.org/10.1021/ef990104z</ref>
 
<ref name="r16">Kawanaka, S., Park, S.J., and  Mansoori, G.A. 1991. Organic Deposition From Reservoir Fluids: A Thermodynamic Predictive Technique. ''SPE Res Eng'' '''6''' (2): 185-192. SPE-17376-PA. http://dx.doi.org/10.2118/17376-PA</ref>
 
<ref name="r17">Cimino, R., Correra, S., Sacomani, P.A. et al. 1995. Thermodynamic Modelling for Prediction of Asphaltene Deposition in Live Oils. Presented at the SPE International Symposium on Oilfield Chemistry, San Antonio, Texas, USA, 14-17 February. SPE-28993-MS. http://dx.doi.org/10.2118/28993-MS</ref>
 
<ref name="r18">Yang, Z., Ma, C.F., Lin, X.S. et al. 1999. Experimental and modeling studies on the asphaltene precipitation in degassed and gas-injected reservoir oils. ''Fluid Phase Equilib''. '''157''' (1): 143-158. http://dx.doi.org/10.1016/S0378-3812(99)00004-7</ref>
 
<ref name="r19">Cimimo, R., Correra, S., Del Bianco, A. et al. 1995. Solubility and phase behavior of asphaltenes in hydrocarbon media. In ''Asphaltenes: Fundamentals and Applications'', ed. E.Y. Sheu and O.C. Mullins, 97–130. New York: Plenum Press.</ref>
 
<ref name="r20">Won, K.W. 1986. Thermodynamics for solid solution-liquid-vapor equilibria: wax phase formation from heavy hydrocarbon mixtures. ''Fluid Phase Equilib''. '''30''' (0): 265-279. http://dx.doi.org/10.1016/0378-3812(86)80061-9</ref>
 
<ref name="r21">Schou Pedersen, K., Skovborg, P., and  Roenningsen, H.P. 1991. Wax precipitation from North Sea crude oils. 4. Thermodynamic modeling. ''Energy Fuels'' '''5''' (6): 924-932. http://dx.doi.org/10.1021/ef00030a022</ref>
 
<ref name="r22">Thomas, F.B., Bennion, D.B., Bennion, D.W. et al. 1992. Experimental And Theoretical Studies Of Solids Precipitation From Reservoir Fluid. ''J Can Pet Technol'' '''31''' (1): 22. PETSOC-92-01-02. http://dx.doi.org/10.2118/92-01-02</ref>
 
<ref name="r23">MacMillan, D.J., Tackett, J.E. Jr., Jessee, M.A. et al. 1995. A Unified Approach to Asphaltene Precipitation: Laboratory Measurement and Modeling. ''J Pet Technol'' '''47''' (9): 788-793. SPE-28990-PA. http://dx.doi.org/10.2118/28990-PA</ref>
 
<ref name="r24">Musser, B.J. and Kilpatrick, P.K. 1998. Molecular Characterization of Wax Isolated from a Variety of Crude Oils. ''Energy Fuels'' '''12''' (4): 715-725. http://dx.doi.org/10.1021/ef970206u</ref>
 
<ref name="r25">Fussell, L.T. 1979. A Technique for Calculating Multiphase Equilibria. ''Society of Petroleum Engineers Journal'' '''19''' (4): 203-210. SPE-6722-PA. http://dx.doi.org/10.2118/6722-PA</ref>
 
<ref name="r26">Nghiem, L.X. and Li, Y.-K. 1984. Computation of multiphase equilibrium phenomena with an equation of state. ''Fluid Phase Equilib''. '''17''' (1): 77-95. http://dx.doi.org/10.1016/0378-3812(84)80013-8</ref>
 
<ref name="r27">Godbole, S.P., Thele, K.J., and  Reinbold, E.W. 1995. EOS Modeling and Experimental Observations of Three-Hydrocarbon-Phase Equilibria. ''SPE Res Eng'' '''10''' (2): 101-108. SPE-24936-PA. http://dx.doi.org/10.2118/24936-PA</ref>
 
<ref name="r28">Peng, D.-Y. and Robinson, D.B. 1976. A New Two-Constant Equation of State. ''Industrial & Engineering Chemistry Fundamentals'' '''15''' (1): 59–64. http://dx.doi.org/10.1021/i160057a011</ref>
 
<ref name="r29">Nghiem, L.X., Hassam, M.S., Nutakki, R. et al. 1993. Efficient Modelling of Asphaltene Precipitation. Presented at the SPE Annual Technical Conference and Exhibition, Houston, Texas, 3-6 October. SPE-26642-MS. http://dx.doi.org/10.2118/26642-MS</ref>
 
<ref name="r30">86Nghiem, L.X. and Coombe, D.A. 1997. Modeling Asphaltene Precipitation During Primary Depletion. ''SPE J''. '''2''' (2): 170-176. SPE-36106-PA. http://dx.doi.org/10.2118/36106-PA</ref>
 
<ref name="r31">Nghiem, L.X., Coombe, D.A., and  Farouq Ali, S.M. 1998. Compositional Simulation of Asphaltene Deposition and Plugging. Presented at the SPE Annual Technical Conference and Exhibition, New Orleans, 27-30 September. SPE-48996-MS. http://dx.doi.org/10.2118/48996-MS</ref>
 
<ref name="r32">Nghiem, L.X., Kohse, B.F., Ali, S.M.F. et al. 2000. Asphaltene Precipitation: Phase Behaviour Modelling and Compositional Simulation. Presented at the SPE Asia Pacific Conference on Integrated Modelling for Asset Management, Yokohama, Japan, 25-26 April. SPE-59432-MS. http://dx.doi.org/10.2118/59432-MS</ref>
 
<ref name="r33">Nghiem, L.X., Sammon, P.H., and  Kohse, B.F. 2001. Modeling Asphaltene Precipitation and Dispersive Mixing in the Vapex Process. Presented at the SPE Reservoir Simulation Symposium, Houston, 11-14 February. SPE-66361-MS. http://dx.doi.org/10.2118/66361-MS</ref>
 
<ref name="r34">Nghiem, L.X., Kohse, B.F., and  Sammon, P.H. 2001. Compositional Simulation of VAPEX Process. ''J Can Pet Technol'' '''40''' (8): 54–61. JCPT Paper No. 01-08-05. http://dx.doi.org/10.2118/01-08-05</ref>
 
<ref name="r35">Li, Y.-K., Nghiem, L.X., and  Siu, A. 1985. Phase Behaviour Computations For Reservoir Fluids: Effect Of Pseudo-Components On Phase Diagrams And Simulation Results. ''J Can Pet Technol'' '''24''' (6): 29. PETSOC-85-06-02. http://dx.doi.org/10.2118/85-06-02</ref>
 
<ref name="r36">Kohse, B.F., Nghiem, L.X., Maeda, H. et al. 2000. Modelling Phase Behaviour Including the Effect of Pressure and Temperature on Asphaltene Precipitation. Presented at the SPE Asia Pacific Oil and Gas Conference and Exhibition, Brisbane, Australia, 16-18 October. SPE-64465-MS. http://dx.doi.org/10.2118/64465-MS</ref>
 
<ref name="r37">Leontaritis, K.J. and Mansoori, G.A. 1987. Asphaltene Flocculation During Oil Production and Processing: A Thermodynamic Colloidal Model. Presented at the SPE International Symposium on Oilfield Chemistry, San Antonio, Texas, USA, 4–6 February. SPE-16258-MS. http://dx.doi.org/10.2118/16258-MS</ref>
 
<ref name="r38">Pan, H. and Firoozabadi, A. 1997. Thermodynamic Micellization Model for Asphaltene Precipitation from Reservoir Crudes at High Pressures and Temperatures. Presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, USA, 5-8 October. SPE-38857-MS. http://dx.doi.org/10.2118/38857-MS</ref>
 
<ref name="r39">Pan, H. and Firoozabadi, A. 1998. A Thermodynamic Micellization Model for Asphaltene Precipitation: Part I: Micellar Size and Growth. ''SPE Prod & Oper'' '''13''' (2): 118-127. SPE-36741-PA. http://dx.doi.org/10.2118/36741-PA</ref>
 
<ref name="r40">Firoozabadi, A. 1999. ''Thermodynamics of Hydrocarbon Reservoirs''. New York: McGraw-Hill.</ref>
 
<ref name="r41">Srivastava, R.K., Huang, S.S., Dyer, S.B. et al. 1995. Quantification of Asphaltene Flocculation During Miscible CO2 Flooding In the Weyburn Reservoir. ''J Can Pet Technol'' '''34''' (8): 31. PETSOC-95-08-03. http://dx.doi.org/10.2118/95-08-03</ref>
 
</references>
 
  
==Noteworthy papers in OnePetro==
 
 
Use this section to list papers in OnePetro that a reader who wants to learn more should definitely read
 
Use this section to list papers in OnePetro that a reader who wants to learn more should definitely read
  
==External links==
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== External links ==
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Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro
 
Use this section to provide links to relevant material on websites other than PetroWiki and OnePetro
  
==See also==
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== See also ==
[[Thermodynamics and phase behavior]]
+
 
 +
[[Thermodynamics_and_phase_behavior|Thermodynamics and phase behavior]]
 +
 
 +
[[Phase_diagrams|Phase diagrams]]
 +
 
 +
[[Equations_of_state|Equations of state]]
  
[[Phase diagrams]]
+
[[Asphaltene_precipitation|Asphaltene precipitation]]
  
[[Equations of state]]
+
[[Remedial_treatment_for_asphaltene_precipitation|Remedial treatment for asphaltene precipitation]]
  
[[Asphaltene precipitation]]
+
[[Asphaltene_problems_in_production|Asphaltene problems in production]]
  
[[Remedial treatment for asphaltene precipitation]]
+
[[PEH:Asphaltenes_and_Waxes]]
  
[[Asphaltene problems in production]]
 
  
[[PEH:Asphaltenes and Waxes]]
+
[[Category:4.3.2 Precipitates]][[Category:5.2.2 Fluid modeling, equations of state]]

Latest revision as of 11:01, 2 June 2015

Thermodynamic models for predicting asphaltene precipitation behavior fall into two general categories: activity models and equation-of-state (EOS) models. This page provides the mathematics underlying the most commonly used models of each type.

Thermodynamic equilibrium

With the precipitated asphaltene treated as a single-component or multicomponent solid, the condition for thermodynamic equilibrium between the oil (liquid) and solid phase is the equality of component chemical potentials in the oil and solid phases. That is,

RTENOTITLE....................(1)

where μio and μ is are the chemical potential of component i in the oil and solid phases, respectively, and nc is the number of components. The application of activity coefficient models or EOS models gives different expressions for the chemical potential. In addition, not all components in the oil phase undergo precipitation; therefore, Eq. 1 applies only to those components that precipitate.

Activity models

Activity coefficients

Because asphaltenes are a solubility class that can be precipitated from petroleum by the addition of solvent, activity coefficient models have been applied to model the phase equilibrium phenomena. The introduction of activity coefficients in Eq. 1 yields

RTENOTITLE....................(2)

where:

  • RTENOTITLE = standard state fugacity of component i in phase k (k = o, s)
  • vik = partial molar volume of component i in phase k (k = o, s)
  • yik = mole fraction of component i in phase k (k = o, s)
  • γik = activity coefficient of component i in phase k (k = o, s)

Several approaches that use the activity coefficient model assume the oil and asphaltene as two pseudocomponents: one component representing the deasphalted oil and the other the asphaltenes. Andersen and Speight[1] provided a review of activity models in this category. Other approaches represent the precipitate as a multicomponent solid. Chung,[2] Yarranton and Masliyah,[3] and Zhou et al.[4] gave detailed descriptions of these models.

Flory-Huggins model

The solubility model used most in the literature is the Flory-Huggins solubility model introduced by Hirschberg et al.[5] Vapor/liquid equilibrium calculations with the Soave-Redlich-Kwong EOS[6] are performed to split the petroleum mixture into a liquid phase and a vapor phase. The liquid phase then is divided into two components: a component that corresponds to the asphaltene and a component that represents the remaining oil (deasphalted oil). When solvent is added into the oil, the second component represents the mixture of deasphalted oil and solvent. These two components are for modeling purposes and do not correspond to any EOS components used in the vapor/liquid calculations. It also is assumed that asphaltene precipitation does not affect vapor/liquid equilibrium.

Application of the Flory-Huggins solution theory gives the following expression for the chemical potential of the asphaltene component in the oil phase.

RTENOTITLE....................(3)

with RTENOTITLE....................(4)

where:

  • subscripts a, o, and m are used to denote the asphaltene component, the deasphalted oil, and the oil phase mixture, respectively
  • va = molar volume of pure asphaltene,
  • vm = molar volume of mixture,
  • δi = solubility parameter of component i,
  • δm = solubility parameter of mixture,
  • Φa = volume fraction of asphaltene in the mixture, μam = chemical potential of asphaltene in the mixture
  • RTENOTITLE = reference chemical potential of asphaltene component

Because the precipitated asphaltene is pure asphaltene, μs = RTENOTITLE. From the equality of chemical potential μam = μs, Eq. 3 gives

RTENOTITLE....................(5)

The molar volume, vm, of the oil mixture is calculated from the composition of the liquid phase obtained from vapor/liquid calculations that use the Soave-Redlich-Kwong EOS. The solubility parameter, δm, is calculated from

RTENOTITLE....................(6)

where ΔUv is the molar internal energy of vaporization at the system temperature, which also can be calculated from the Soave-Redlich-Kwong EOS. The remaining parameters are the molar volume of asphaltene, va, and the solubility parameter of asphaltene, δa, which are essential to the performance of this model. The molar volume of asphaltene can only be speculated on. Hirschberg et al.[5] used values of v a in the range of 1 to 4 m3/kmol. The solubility parameter of asphaltene can be estimated by measuring the solubility of asphaltene in different solvents of increasing solubility parameters. The asphaltene is assumed to have the solubility parameter of the best solvent. Hirschberg et al.[5] suggests that the solubility parameter of asphaltene is close to that of naphthalene. Eq. 5 gives the amount (volume fraction) of asphaltene soluble in the oil mixture. The amount of precipitation is determined by the difference between the total amount of asphaltenes present in the initial oil and the solubility of asphaltene under given conditions.

The solubility parameter can be correlated as a linear equation with respect to temperature as

RTENOTITLE....................(7)

where a and b are constants. parameter b is negative as the solubility parameter decreases with increasing temperature. Buckley et al.[7] and Wang and Buckley[8] showed that the measurements of the refractive index of crude oils can be used to determine the solubility parameters required for the Flory-Huggins model.

The Hirschberg et al.[5] approach also has been used with some degree of success by:

  • Burke et al.,[9]
  • Kokal and Sayegh,[10]
  • Novosad and Costain,[11]
  • Nor-Azian and Adewumi,[12]
  • Rassamdana et al.[13]

de Boer et al. used this model to screen crude oils for their tendency to precipitate asphaltene. They compared properties of some crudes in which asphaltene problems were encountered and properties of crudes that operated trouble free. They found that asphaltene problems were encountered with light crudes with high C1 to C3 contents, high bubblepoint pressures, large differences between reservoir pressure and bubblepoint pressure, and high compressibility. With an asphaltene molar volume of 1 m3/kmol, de Boer et al.[14] showed that the solubility of asphaltene in a light crude oil with Eq. 5 follows the curve shown in Fig. 1. Above the bubblepoint, the decrease in asphaltene solubility is caused by pressure effects. Below the bubblepoint, the increase in asphaltene solubility is caused by the variation in the oil composition. Clearly, a minimum asphaltene solubility occurs around the bubblepoint.

de Boer et al.[14] calculated the solubility of asphaltene with Eq. 5 for different values of in-situ crude oil densities and asphaltene-solubility parameters. They also introduced a maximum supersaturation at bubblepoint defined as

RTENOTITLE....................(8)

where pr and pb are, respectively, the reservoir pressure and the bubblepoint pressure at the reservoir temperature. Fig. 2 shows the maximum supersaturation at the bubblepoint as a function of the difference between reservoir and bubblepoint pressure, the in-situ oil density, and the asphaltene-solubility parameter. The influence of the asphaltene-solubility parameter is very small. Supersaturations are larger for lighter crudes. The boundary between problem and nonproblem areas lies at a maximum supersaturation of approximately 1. Although these results were derived with North Sea and Kuwait crudes, Hammami et al.[15] showed that they also are applicable to crudes from the Gulf of Mexico.

Extension of Flory-Huggins model

The Flory-Huggins model initially was developed for polymer solutions. The Hirschberg et al.[5] approach is based on the representation of asphaltene as a homogeneous polymer. Novosad and Constain[11] used an extension of the model that includes asphaltene polymerization and asphaltene-resin association in the solid phase. Kawanaka et al.[16] proposed an improvement whereby the precipitated asphaltene is treated as a heterogeneous polymer (i.e., a mixture of polymers of different molecular weights). The Scott-Magat theory was used to obtain a solubility model for a given molecular-weight distribution for asphaltene. Cimino et al.[17] also used the Flory-Huggins model with two components (solvent and asphaltene) but considered the solid phase to be a mixture of solvent and asphaltene instead of pure asphaltene as in Hirschberg et al. ‘s approach. Yang et al.[18] proposed a multicomponent Flory-Huggins model in which components are the same as the EOS components used in the oil/gas flash calculations.

Multicomponent activity coefficient models

These models are derived from methods for modeling wax precipitation.[19][20][21] Multicomponent solid/liquid K values are derived from Eq. 2 and then used with an EOS in a three-phase oil/gas/solid flash calculation. The solid/liquid K values are defined as

RTENOTITLE....................(9)

Eq. 2 gives:

RTENOTITLE....................(10)

with RTENOTITLE....................(11)

Eq. 11 is equivalent to[20][21]:

RTENOTITLE....................(12)

where:

  • Tif = fusion temperature of component i
  • Δ Cpi = Cpo,iCps,i, heat capacity change of fusion
  • ΔHif = heat of fusion of component i

ΔCpi is assumed to be independent of temperature in Eq. 12.

Starting with Eq. 12, methods were derived through the use of different models for activity coefficients. The earliest approach is from Won[20] in the modeling of wax precipitation. Won[20] suggested that the term involving ΔCpi and the integral involving Δvi are negligible and used regular solution theory to calculate the activity coefficients in Eq. 12 as follows.

RTENOTITLE....................(13)

RTENOTITLE....................(14)

RTENOTITLE....................(15)

where:

  • δik is the solubility parameter of component i in phase k (k = o, s)
  • vik is the molar volume of component i in phase k
  • Φik is the volume fraction of component i in phase k

Won gave solubility parameter values, δio and δis, for normal paraffins up to C40. Correlations also are provided to calculate ΔHif, vio, and vis. Although Won’s model was developed for wax precipitation, Thomas et al.[22] have applied it with some success in predicting asphaltene precipitation. However, they have developed their own correlations for solubility parameters. MacMillan et al.[23] also used Won’s model but kept all the terms in Eq. 12 instead of neglecting the terms involving ΔCpi and Δvi as Won did. They also included additional multiplication factors to the different terms in Eq. 12 to facilitate phase-behavior matching.

Hansen et al.[24] and Yarranton and Masliyah[3] used the Flory-Huggins model to calculate the activity coefficients in Eq. 12. Hansen et al.[24] applied their method to the modeling of wax precipitation, while Yarranton and Masliyah[3] modeled precipitation of Athabasca asphaltenes. Yarranton and Masliyah[3] proposed an approach for calculating the molar volumes and solubility parameters from experimental measurements of molar mass and density. Asphaltene density, molar volume, and solubility parameter are correlated with molar mass. Zhou et al.[4] used the Flory-Huggins polymer-solution theory with a modification to account for the colloidal suspension effect of asphaltenes and resins.

Equation of state models

These approaches model the oil, gas, and precipitate by an EOS, which is used to calculate the component fugacities in different phases. Cubic EOSs have been used to model petroleum reservoir fluids that exhibit vapor/liquid 1/liquid 2 behavior (see Fussell,[25] Nghiem and Li,[26] or Godbole et al.[27]). Godbole et al.[27] observed that the apparent second liquid phase could be approximated as a mixture of aggregated asphaltenes (solid phase) entrained in a portion of the other liquid phase in the modeling of mixtures of crude oil from the North Slope of Alaska and enriched gas. Under certain conditions, a phase-behavior program that includes a three-phase calculation with an EOS could be used to model some aspects of asphaltene precipitation; however, the prevailing approach consists of the use of a cubic EOS (e.g., Soave-Redlich-Kwong EOS[6] or Peng-Robinson EOS[28]) for the oil and gas phases and a solid model for the precipitate.

The simplest model for precipitated asphaltene is the single-component solid model. The precipitated asphaltene is represented as a pure solid, while the oil and gas phases are modeled with a cubic EOS. The fugacity of the pure solid is given by

RTENOTITLE....................(16)

where:

  • fs = solid fugacity
  • RTENOTITLE = reference solid fugacity,
  • p = pressure
  • p* = reference pressure
  • R = gas constant
  • vs = solid molar volume
  • T = temperature

The following fugacity equality equations are solved to obtain oil/gas/solid equilibrium.

RTENOTITLE....................(17a)

and RTENOTITLE....................(17b)

The oil and gas fugacities, fio and fig, for component i are calculated from an EOS. In Eq. 17b, subscript a denotes the asphaltene component in solution. Normally, this asphaltene component is the heaviest and last component of the oil (i.e., a = nc). The following simple stability test can be used to determine whether there is asphaltene precipitation: if faofs, asphaltene precipitation occurs, and if fao < fs, there is no precipitation.

Earlier applications of the single-component solid model for asphaltene precipitation were not successful.[22] Nghiem et al.[29] introduced a method for representing the asphaltene component in the oil that improves the capabilities of the single-component solid model to predict asphaltene precipitation. The method was subsequently refined by Nghiem et al.[30][31][32][33][34] The key to the approach is the split of the heaviest fraction of the oil into two pseudocomponents:

  • One that does not precipitate (nonprecipitating component)
  • One that can precipitate (precipitating component)

These two pseudocomponents have identical critical temperatures, critical pressures, acentric factors, and molecular weights. The differences are in the interaction coefficients. The interaction coefficients between the precipitating components and the light components are larger than those between the nonprecipitating component and the light components. The parameters of the model are the reference fugacity and the solid molar volume. The reference fugacity could be estimated from a data point on the asphatene precipitation envelope (APE), and a value for solid molar volume slightly larger than the EOS value for the pure component a is adequate.[29]

The following application of the model to a North Sea fluid from Nghiem et al.[31] illustrates the procedure. Table 1 shows the pseudocomponent representation of the reservoir fluid with the separator gas and separator oil compositions. The reservoir oil corresponds to a combination of 65.3 mol% separator oil and 34.7 mol% separator gas. The crucial step in the modeling of asphaltene is the split of the heaviest component in the oil (e.g., C32+) into:

  • A nonprecipitating component (C32A+)
  • A precipitating component (C32B+)

These two components have identical critical properties and acentric factors but different interaction coefficients with the light components. The precipitating component has larger interaction coefficients with the light components. With larger interaction coefficients, the precipitating component becomes more "incompatible" with the light components and tends to precipitate as the amount of light component in solution increases. Although C32B+ is called the precipitating component, the amount that precipitates is governed by Eq. 16. Normally, only a portion of the total amount of C32B+ will precipitate during a calculation. Hirschberg et al.[5] reports that the asphalt precipitate from a tank oil consists mainly (90%) of C30 to C60 compounds. For the purpose of modeling asphaltene precipitation, a heaviest component in the vicinity of C30+ is adequate. For this example, C32+ is used.

The Peng-Robinson EOS was used to model the oil and gas phases. The critical properties and acentric factors of the pseudocomponents in Table 1 are calculated as described in Li et al.[35] The interaction coefficients are calculated from

RTENOTITLE....................(18)

where:

  • dij = the interaction coefficient between component i and j
  • vci = the critical volume of component i,
  • e = an adjustable parameter

A value of e(C32A+) = 0.84 and a value of e(C32B+) = 1.57 were found to provide a good match of the saturation and onset pressure. The reference solid fugacity was obtained by calculating the fugacity of oil at one point on the APE (recombined oil with 69.9 mol% separator gas and 30 000 kPa) with the Peng-Robinson EOS and equating it to RTENOTITLE. The molar volume of the asphaltene precipitate was assumed equal to 0.9 m3/kmol.

Fig. 3 shows a good match of the experimental and calculated APE and saturation pressure curves at the reservoir temperature of 90°C. The model was able to predict precipitation conditions that are far from the reference conditions used to determine RTENOTITLE. Fig. 3 shows the amounts of precipitation calculated as constant weight percent of precipitate (similar to "quality lines" in oil/gas phase diagrams). As pressure decreases below the APE, the amount of precipitation increases and reaches a maximum at the saturation pressure. Below the saturation pressure, the amount of precipitation decreases with decreasing pressure. The results are consistent with the laboratory observations described here.

For nonisothermal conditions, Eq. 19 can be used to calculate the solid fugacity at (p, T) from the solid fugacity at a reference condition (p*, T*).[32][36]

RTENOTITLE....................(19)

where:

  • f = fugacity of the asphaltene component in the pure liquid state
  • Tf = melting point temperature
  • v = molar volume of liquid
  • ΔCp = heat capacity of fusion
  • ΔHf = enthalpy of fusion

Kohse et al.[36] used Eq. 19 to model the precipitation behavior of a crude oil with changes in pressure and temperature. Fig. 4 shows good agreements between the experimental and calculated APE and saturation-pressure curves. The measured data point of 1.6 wt% of precipitate also is close to the predictions.

The previous two examples illustrate the application of the single-component solid model to the modeling of precipitation behavior of crudes with changes in:

  • Pressure
  • Temperature
  • Composition

From a mechanistic point of view, the nonprecipitating component can be related to:

  • Resins
  • Asphaltene/resin micelles that do not dissociate
  • Heavy paraffins

The precipitating component corresponds to both the asphaltenes that dissociate and the asphaltene/resin micelles that precipitate unaltered. Because of identical critical properties and acentric factors, the nonprecipitating and precipitating components behave as a single component in solution. The larger interaction coefficients between the precipitating and the solvent components cause the precipitation of the former with the addition of solvent. The amount of precipitation depends on the solution of Eqs. 17a and 17b. Normally, only a portion of the precipitating component actually precipitates.

Solid precipitation with the previous model is reversible. Nghiem et al.[34] proposed an enhancement to the approach to obtain partial irreversibility. A second solid (Solid 2) is introduced that is obtained from the reversible solid (Solid 1) through a chemical reaction:

RTENOTITLE

If the forward reaction rate k12 is much larger than the backward reaction rate k21, Solid 2 behaves as a partially irreversible solid.

Thermodynamic models

Thermodynamic-colloidal model

Leontaritis and Mansoori[37] proposed a more mechanistic approach based on the assumption that asphaltenes exist in the oil as solid particles in colloidal suspension stabilized by resins adsorbed on their surface. This thermodynamic-colloidal model assumes thermodynamic equilibrium between the resins in the oil phase and the resins adsorbed on the surface of colloidal asphaltene (asphaltene micelle). The corresponding equilibrium equation is

RTENOTITLE....................(20)

Assuming that resins behave as monodisperse polymers and applying the Flory-Huggins polymer-solution theory gives the volume fraction of dissolved resins as

RTENOTITLE....................(21)

which is analogous to Eq. 5 for the asphaltene component in Hirschberg et al.‘s approach. In Hirschberg et al.‘s approach, the asphaltene component contains both resins and asphaltene, whereas Eq. 21 applies to the resins only. As in Hirschberg et al.‘s approach, EOS flash calculations with a multicomponent system are performed to obtain an oil/gas split and oil properties from which Φr is calculated. This value of Φr is compared with a critical resin concentration, Φcr, which is given as a function of pressure, temperature, molar volume, and solubility parameters.

Φcr is the key parameter of the model. If Φr > Φcr, the system is stable and no precipitation occurs. If ΦrΦcr, asphaltene precipitation occurs. The amount of precipitated asphaltene can be made a function of the asphaltene particle sizes.

Thermodynamic-micellization model

Pan and Firoozabadi[38][39] proposed the most mechanistic approach to model asphaltene precipitation by calculating the Gibbs free energy of formation of the asphaltene micelle and including it in the phase-equilibrium calculations. Details of the approach can be found in Firoozabadi.[40] Fig. 5 portrays schematically the system to be modeled. The species in the liquid phase (L1) are monomeric asphaltenes, monomeric resins, micelles, and asphalt-free oil species. The micelle consists of a core of n1 asphaltene molecules surrounded by a shell containing n2 resins molecules. The precipitate phase is considered as a liquid mixture (L2) of asphaltene and resin molecules. An expression for Gibbs free energy of formation of the micelle, RTENOTITLE, which includes n1, n2, and the shell thickness, D, was proposed. The Gibbs free energy of the liquid phase, L1, then is derived with:

  • An EOS for the asphalt-free oil species
  • Activity models for the monomeric asphaltenes and resins
  • Gibbs free energy of formation of the micelle, RTENOTITLE

Similarly, the Gibbs free energy of the precipitated phase, L2, which is a binary mixture of monomeric asphaltenes and resins, also is derived with the use of an EOS. The total Gibbs free energy of the system,

RTENOTITLE....................(22)

then is minimized with respect to:

  • n1 = number of asphaltene molecules in the micellar core
  • n2 = number of resin molecules in the micellar cell,
  • D = shell thickness of the micelle
  • RTENOTITLE = number of asphaltene monomers in liquid phase (L1)
  • RTENOTITLE = number of resin monomers in phase L1
  • RTENOTITLE = number of micelles in phase L1
  • RTENOTITLE = number of asphaltene monomers in precipitated phase L2
  • RTENOTITLE = number of resin monomers in phase L2

The minimization requires a robust numerical procedure.

The model was applied to predict precipitation from a tank oil with propane,[5] Weyburn oil with CO2,[41] and a North Sea oil with separator gas. Fig. 6 shows the predictions of Weyburn oil with CO2 obtained with the thermodynamic-micellization model. For comparison, the match obtained with the pure solid model[31] also is shown.

Nomenclature

a = constant
b = constant
dij = interaction coefficient between component i and j
D = shell thickness of the micelle, L
fao = fugacity of asphaltene component in oil phase, m/Lt2
fi = fugacity of component i, m/Lt2
RTENOTITLE = standard state fugacity of component i, m/Lt2
fig = fugacity of component i in the oil phase, m/Lt2
fik = fugacity of component i in phase k (k = o, s), m/Lt2
RTENOTITLE = fugacity of pure component i in phase state k (k = o, s), m/Lt2
RTENOTITLE = standard state fugacity of component i in phase k (k = o, s), m/Lt2
fio = fugacity of component i in the oil phase, m/Lt2
RTENOTITLE = fugacity of pure component i in oil phase, m/Lt2
RTENOTITLE = standard state fugacity of component i in oil phase, m/Lt2
fis = fugacity of component i in the solid phase, m/Lt2
RTENOTITLE = fugacity of pure component i in solid phase, m/Lt2
RTENOTITLE = standard state fugacity of component i in solid phase, m/Lt2
f = fugacity of the asphaltene component in the pure liquid state, m/Lt2
fp = porous medium particle transport efficiency factor
fs = solid fugacity, m/Lt2
RTENOTITLE = reference solid fugacity, m/Lt2
G = total Gibbs free energy of the system, m/L2t2
GE = total excess Gibbs energy for a phase, m/L2t2
Kis = solid/liquid K value for component i
nc = number of components
RTENOTITLE = number of asphaltene monomers in phase L1
RTENOTITLE = number of asphaltene monomers in phase L2
RTENOTITLE = number of micelles in phase L1
RTENOTITLE = number of resin monomers in phase L1
RTENOTITLE = number of resin monomers in phase L2
p = pressure, m/Lt2
p* = reference pressure, m/Lt2
pb = bubblepoint pressure, m/Lt2
pr = reservoir pressure, m/Lt2
R = gas constant
T = temperature, T
T* = reference temperature, T
Tc = temperature-dependent parameter
Tf = melting point temperature, T
Tif = temperature of fusion (melting temperature) of component i, T
uc = critical speed required to mobilize surface deposit asphaltene, L/t
uo = oil velocity, L/t
va = molar volume of pure asphaltene, L3/n
vci = critical volume of component i, L3/n
vcj = critical volume of component j, L3/n
vik = partial molar volume of component i in phase k (k = o, s), L3/n
vio = partial molar volume of component i in oil phase, L3/n
vis = partial molar volume of component i in solid phase, L3/n
vjk = partial molar volume of component j in phase k (k = o, s), L3/n
vjo = partial molar volume of component j in oil phase, L3/n
vjs = partial molar volume of component j in solid phase, L3/n
v = molar volume of liquid, L3/n
vm = molar volume of mixture, L3/n
vr = molar volume of resins, L3/n
vs = solid molar volume, L3/n
wi = weight fraction of component i, m/m
W = weight percent of precipitated asphaltene, m/m
yi = mole fraction of component i, n/n
yik = mole fraction of component i in phase k (k = o, s), n/n
yio = mole fraction of component i in oil phase, n/n
yis = mole fraction of component i in solid phase, n/n
yjo = mole fraction of component j in oil phase, n/n
yjs = mole fraction of component j in solid phase, n/n
γ = shear rate, L/t
γi = activity coefficient of component i in a mixture
γik = activity coefficient of component i in phase k (k = o, s)
γio = activity coefficient of component i in oil phase
RTENOTITLE = combinatorial free volume contribution
RTENOTITLE = residual contribution
γis = activity coefficient of component i in solid phase
δa = solubility parameter of asphaltene
δi = solubility parameter for component i
δik = solubility parameter for pure component i in phase k (k = o, s)
RTENOTITLE = volume fraction average solubility parameter for phase k
δm = solubility parameter of mixture
δo = solubility parameter of oil phase
RTENOTITLE = volume fraction average solubility parameter of oil phase
δs = solubility parameter of solid phase
RTENOTITLE = volume fraction average solubility parameter of solid phase
ΔCp = heat capacity of fusion, mL2/nt2T
ΔCpi = heat capacity of fusion of component i, mL2/nt2T
ΔCpij,tr = heat capacity of jth solid state transition of component i, mL2/nt2T
RTENOTITLE = expression for Gibbs free energy of formation of the micelle, mL2/nt2
ΔHf = enthalpy of fusion, mL2/nt2
ΔHif = enthalpy of fusion of component i, mL2/nt2
ΔUv = molar internal energy of vaporization at the system temperature, mL2/nt2
Δvi = change of molar volume caused by fusion of component i, L3
RTENOTITLE = reference chemical potential of asphaltene component
μam = chemical potential of asphaltene in the mixture
μc = critical speed required to mobilize surface deposit asphaltene
μi = chemical potential of component i, m/L2t2
μik = chemical potential of component i in phase k, m/L2t2
μio = chemical potential of component i in the oil phase, m/L2t2
μis = chemical potential of component i in the solid phase, m/L2t2
μpi,k = chemical potential of pure component i in phase k (k = o, s), m/L2t2
μpi,o = chemical potential of pure component i in oil phase, m/L2t2
μpi,s = chemical potential of pure component i in solid phase, m/L2t2
μr,m = chemical potential of resins on the surface of the asphaltene micelle, m/L2t2
μr,o = chemical potential of resins in the oil phase, m/L2t2
μs = chemical potential of the solid, m/L2t2
ρo = mass density of oil, m/L3
Φik = fugacity coefficient of component i in phase k
RTENOTITLE = fugacity coefficient of pure component i in phase state k (k = o, s)
Φio = fugacity coefficient of component i in oil phase
Φa = volume fraction of asphaltene in the mixture
Φcr = critical volume fraction of resins in the mixture
Φik = volume fraction of component i in phase state k (k = o, s)
Φr = volume fraction of resins in the mixture

References

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See also

Thermodynamics and phase behavior

Phase diagrams

Equations of state

Asphaltene precipitation

Remedial treatment for asphaltene precipitation

Asphaltene problems in production

PEH:Asphaltenes_and_Waxes