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Modeling of Asphaltene Onset Pressure from few Experimental Data: A Comparative Evaluation of the Hirschberg Method and the CPA Equation of State Fabio Pedro Nascimento, Marcos Miranda Souza, Gloria Meyberg Nunes Costa, and Sílvio Alexandre Beisl Vieira de Melo Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.8b03087 • Publication Date (Web): 27 Nov 2018 Downloaded from http://pubs.acs.org on November 27, 2018

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Modeling of Asphaltene Onset Pressure from few Experimental Data: A Comparative Evaluation of the Hirschberg Method and the CPA Equation of State Fábio P. Nascimento†, Marcos M. S. Souza†, Gloria M.N. Costa†, Silvio A.B. Vieira de Melo*,†,‡ †

Programa de Pós-Graduação em Engenharia Industrial, Escola Politécnica, Universidade

Federal da Bahia, Rua Aristides Novis, 2, Federação, Salvador, Bahia 40210-630, Brazil ‡

Centro Interdisciplinar em Energia e Ambiente, Campus Universitário da Federação/Ondina,

Universidade Federal da Bahia, Rua Barão de Jeremoabo, S/N, Ondina, Salvador, Bahia 40170115, Brazil KEYWORDS. Asphaltene; Precipitation; Onset Pressure; Equation of State; Hirschberg method; CPA

ABSTRACT. Asphaltene onset pressure (AOP) is a key parameter to determine flow assurance of live oils. In this study the capabilities of the Cubic-Plus-Association (CPA) equation of state (EoS) and the Hirschberg method of calculating the AOP of five oils are compared with a new approach using few experimental data. Two experimental data points of AOP and only one of bubble pressure (BP) are required for adequate parameterization of both models. The number of non-zero binary interaction parameters needed for the CPA EoS is reduced to only four binary pairs. In the Hirschberg method the BP calculations were performed with Soave-Redlich-Kwong EoS. For all oils, the CPA EoS provided lower deviations for both AOP and BP. To achieve a good correlation, the Hirschberg method requires the use of AOP experimental data to calculate the difference between the liquid phase solubility parameter and the asphaltene solubility parameter. Similarly,

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CPA EoS requires the calculation of the cross-associating energy parameter between asphaltene and resin.

1. Introduction Asphaltene is a not well-defined petroleum component. It is a polydisperse mixture whose composition may vary depending on the origin of the oil. Asphaltenes have a high molar mass and may contain heteroatoms, such as sulfur, oxygen and nitrogen. Therefore, they are commonly referred to as the most polar and heaviest part of petroleum.1,2,3 They are usually classified based on their solubility class and are insoluble in light n-alkanes (such as n-pentane or n-heptane) but soluble in aromatic solvents (such as toluene or benzene).1,3,4 During oil recovery changes in temperature, pressure and oil composition may induce the asphaltene precipitation. This can lead to deposition which, in turn, can clog reservoir wells and production lines, causing a loss of production and the need to take mitigating measures.1,2,5 Enhanced oil recovery by gas injection (such as natural gas, nitrogen or carbon dioxide) may cause asphaltene precipitation due to changes in oil composition. Hence, the accurate description of asphaltene precipitation onset is crucial to avoid deposition problems.3 Lack of knowledge about asphaltenes and how they interact with the other constituents of the oil have stimulated the development of different theories to explain their behavior in solution and the processes involved during the precipitation. A better understanding of the intermolecular forces between the asphaltene and the other constituents of the oil is fundamental as this directly affects the solubility of the asphaltene and its capacity to form aggregates.6 The two main approaches used to describe the phase behavior of asphaltenes are based on colloidal models and solubility models.6 In the colloidal model the asphaltenes are considered to be dispersed in the bulk oil in the form of aggregates stabilized by resin-type structures, which have a greater affinity with the oil. A reduction in the stabilizing capacity of the resin causes precipitation of asphaltene. In the solubility model the interactions between the asphaltene and the other constituents of the oil are thought to cause the asphaltene to be solubilized in the bulk oil. Equations of state and the Flory-Huggins7,8 theory are examples of solubility models.3,6 One of the first applications of the Flory-Huggins theory in the study of asphaltene precipitation was proposed by Hirschberg et al.9 The thermodynamic properties of the oil components are calculated considering asphaltene as a pure monodisperse polymer, according to the Flory-Huggins theory, combined with the Hildebrand and Scott10 regular solution theory. The Hirschberg method divides the liquid phase into two phases: the oil-rich phase and a second phase composed of pure


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asphaltene. The bubble pressure is calculated in the usual way using a cubic equation of state (EoS), such as Peng-Robinson (PR)11 or Soave-Redlich-Kwong (SRK).12 Novosad et al.13 used a modified approach of the Hirschberg method including asphaltene-resin association to study the asphaltene stability in the oil during depletion and due to CO2 injection. This approach was able to identify the onset of precipitation under several reservoir conditions. Later, Kawanaka and Mansoori14 used a statistical thermodynamic model proposed by Meyer15 to predict the onset of precipitation and the amount of asphaltene precipitated. In addition, asphaltene was considered as consisting of different components, similar to a polymer molecule. The calculations were performed considering a solid-liquid equilibrium. Rassamdana et al.16 and Yang et al.17 proposed a modification in the asphaltene solubility parameter calculation by defining this parameter as a function of specific gravity, molar mass and boiling point (unlike the Hirschberg method that depends only on temperature). Yang et al.17 also proposed a reformulation of the original equation considering that the oil phase contains dissolved gases and is as a multi-component mixture. They also introduced the binary interaction parameter ( lij ) in the Hirschberg equation, which improved the results obtained for the amount of precipitated asphaltene and the onset of precipitation. Rassamdana et al.16 applied the Hirschberg method (with SRK EoS) to predict the asphaltene precipitation onset due to the addition of an n-alkane in the reservoir. Several studies on asphaltene precipitation with different equations of state have been widely investigated and reported in the literature. The use of cubic or non-cubic equations of state to calculate the asphaltene precipitation has several applications because they can be used to calculate the complete asphaltene precipitation envelope and other derivative properties of interest, such as the isothermal compressibility and the bulk modulus.2,6 Among the non-cubic EoS, the two most commonly applied to predict asphaltene precipitation are the PC-SAFT (Perturbed Chain-Statistical Associating Fluid Theory)18 and the CPA (Cubic-Plus-Association)1 EoS. Arya et al.3 evaluated the performance of both CPA and PC-SAFT EoS in calculating the asphaltene precipitation envelope (APE) of seven fluids and concluded that the results provided by CPA EoS were more accurate than those obtained with the PC-SAFT EoS (with or without the association term). Asphaltene precipitation has been modeled with the CPA EoS in a series of studies.2,3,5,19,20,21,22 The CPA EoS is a thermodynamic model composed of a sum of two parts.5 The physical part, which describes short-range repulsions and dispersion attractions, is usually represented by a cubic EoS and makes use of three physical properties ( Pc , Tc , and  ) that can be treated as model parameters.3 The association part, which describes the site-site interactions due to hydrogenbonding, is based on Wertheim’s23,24,25,26 thermodynamic perturbation theory of first order and ACS Paragon Plus Environment

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makes use of two pure component parameters (  AA and  AA / k ). With the conventional mixing rules, CPA EoS is applicable to mixtures. CPA EoS performance is usually compared to that of equations of state such as Peng-Robinson and PC-SAFT. On the other hand, despite its simplicity, the Hirschberg method is commonly applied only to calculate the amount of asphaltene precipitated. Its capacity to correlate the asphaltene onset pressure (AOP) has not yet been fully stressed. Therefore, in this work the Hirschberg method, without any modification, and the CPA EoS are critically compared to correlate the AOP of five live oils from the literature. A new approach is proposed to significantly reduce both the number of binary interaction parameters of the CPA EoS and the number of experimental data required to adjust this EoS and calculate the AOP. The same number of experimental data points was used to estimate the parameters of Hirschberg method and CPA EoS, which are at least one bubble pressure and two AOP data points.

2. Thermodynamic modeling 2.1. The CPA equation of state All equilibrium calculations were performed following the procedures reported in previous works.5,27,28 The following assumptions were made for asphaltene precipitation using CPA EoS:3,20,28 (i) Asphaltenes are present in the oil as monomers with a single ring of polynuclear aromatics (ii) Asphaltene phase is modeled with CPA EoS as a liquid-dense phase. According to Shirani et al.5 precipitated asphaltenes look more like liquids than solids under reservoir conditions (iii) The liquid-dense phase is considered pure asphaltene (iv) Due to its low volatility, asphaltene only exists in the liquid and liquid-dense phases. Therefore, asphaltene cannot precipitate from the vapor phase (v) Only self-association between asphaltene molecules or cross-association between asphaltene and resin molecules is allowed (vi) The cross-association energy between asphaltene and resin (  A  / k ) is temperature dependent. In this work, the CPA EoS version from Li and Firoozabadi21 and Nasrabadi et al.22 was used. As already stated, the CPA EoS is defined as the sum of two contributions: (i) the physical part, which ACS Paragon Plus Environment

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describes the contribution of non-associating molecular interactions, such as repulsive and attractive short-range interactions and (ii) the association part, which describes the contribution of association effects, such as hydrogen-bonding. The CPA EoS is applicable to mixtures. Li and Firoozabadi21 version for this model makes use of one binary interaction parameter ( kij ) in the attractive shortrange term. The CPA EoS, in the form of the residual Helmholtz energy, is given in equation (1): A R  A ph  Aassoc

(1)

where A is the Helmholtz energy and superscripts R , ph and assoc stand for residual, physical contribution and association contribution, respectively. Li and Firoozabadi21 used the Peng-Robinson equation of state to represent the physical contribution of the CPA model. The detailed expression for A ph is presented in equations (2,3,4,5,6,7):

 1  (1  2 )bmix  amix A ph    ln(1  bmix )  ln  RT 2 2 RT  1  (1  2 )bmix  nc

(2)

nc

amix   xi x j ai a j (1  k ij )

(3)

i 1 j 1

nc

bmix   xi bi

(4)

i 1

 R 2Tc2i  T  1  ci 1   ai  0.45724   Pci  T c i   

2

 0.37464  1.54226i  0.26992i2 , i  0.5 ci   2 3 0.3796  1.485i  0.1644i  0.01667 i , i  0.5 bi  0.0778

RTci

(5)

(6)

(7)

Pci

where R is the universal gas constant, T is the temperature and  is the molar density of the mixture. xi is the mole fraction of component i and nc is the total number of components. Tci , Pci and i are the critical temperature, critical pressure and acentric factor of component i , respectively. kij is the binary interaction parameter.


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The association term contribution of the Helmholtz energy is obtained from the usual Wertheim’s23,24,25,26 first-order perturbation expression and is detailed in equation (8) proposed by Li and Firoozabadi.21 Each asphaltene molecule is assumed to have N A identical association sites and each resin molecule is assumed to have N  identical association sites. In this work we considered N A  N   4 .21

Aassoc 1  A  1       N A xA  ln  A    N x  ln    RT 2  2   

(8)

where xA and x are the mole fraction of asphaltene and resin, respectively, while A and  are, respectively, the mole fraction of asphaltene or resin molecule not bonded at one of the association sites. The association occurs between two sites. One site is necessarily on an asphaltene molecule and the other site could be either on an asphaltene or resin molecule. Therefore, as previously stated, self-association of resin molecules is not allowed. Due to these considerations, expressions for A and  are given in equations (9) and (10):21

A   

1 1  N A xA  A 

AA

 N  x   A 

1 1  N A xA  A A 

(9)

(10)

The association strength ( AA and A  ) is calculated from equations (11) and (12):

ij  g ij

g

bi  bj    ij   exp   1, i  A and j  A or  2   kT  

b  1  0.5 ,   mix 3 (1   ) 4

(11)

(12)

where  ij and  ij / k are the association volume and energy parameters, respectively. In this work we considered  AA   A   0.05 .3


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2.1.1. Oil characterization with CPA EoS Reservoir oils are a complex mixture of thousands of components, including not only well-defined components (N2, CO2, H2S, methane, ethane, propane, etc.) but also fractions of oil which are imprecisely defined. Therefore, a proper fluid characterization is needed to reduce the number of components required by equations of state.20,21 In this work a modification of the oil characterization proposed by Arya et al.3 and Li and Firoozabadi21 was applied. The live oil is divided into pure components [N2, CO2, H2S, methane (C1), ethane (C2), propane (C3), i-butane (iC4), n-butane (nC4), i-pentane (iC5), and n-pentane (nC5)], one pseudo-hydrocarbon component (C6), and the heptane plus fraction (C7+). The C6 pseudo-hydrocarbon component includes all components with normal boiling point between those of nC5 and nC6.21 The C7+ fraction is divided in two pseudo-components: (i) heavy component, and (ii) asphaltene. The heavy component contains the heavy alkanes, heavy aromatics, and all resins.21 The heavy component (  ) is allowed to cross-associate with asphaltene. The C7+ fraction splitting into heavy component and asphaltene is based on the molar mass ( MM ) of the C7+ fraction and on the asphaltene content in the dead oil (SARA29 analysis). The asphaltene MM is fixed at 750 g mol1 because asphaltene molecules are considered to be present as monomers in the crude oil.3 The heavy component molar mass is calculated so that the molar averaging of the heavy component and asphaltene molar masses, subject to the asphaltene content in the dead oil, equals the molar mass of the C7+ fraction. The CPA EoS physical properties ( Pc , Tc , and  ) for the well-defined components (N2, CO2, H2S, C1, C2, C3, iC4, nC4, iC5, and nC5) and for the pseudo-hydrocarbon (C6) component were taken from the literature.21 The physical properties of the asphaltene were defined following the procedure reported by Arya et al.,3 which is based on the assumption that the asphaltene solubility parameter varies from 60.08 to 72.73 bar0.5 at T  298 K and pressure ( P ) of 1.013 bar. The solubility parameter (  ) was calculated from equation (13)21 and initial estimates for Pc , Tc , and

 of the asphaltene were taken from Arya et al.3 [ Pc = 15.44 bar, Tc = 1040 K and  = 1.575] and from Li and Firoozabadi21 [ Pc = 6.34 bar, Tc = 1474 K and  = 2]. First, we chose to set the value of  as the mean of the initial estimates for this parameter. Then  was calculated at P = 1.013 bar and T = 298 K considering two different values for  AA / k (0 K and 3500 K). The values of Pc and Tc were adjusted so that the calculated  considering  AA / k = 0 K would approach 60.08 bar0.5 and  considering  AA / k = 3500 K would approach 72.73 bar0.5. This


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procedure was done based on the minimization of an objective function using a Nelder-Mead Simplex method,30 considering the initial estimates as boundaries. Although we have not ensured that the obtained final values for these parameters were at the global minimum of the objective function, the calculated values of  are satisfactory. The solubility parameter values calculated for asphaltene at T  298 K and P  1.013 bar are 60.69 bar0.5 (  AA / k  0 K) and 65.39 bar0.5 (  AA / k  3500 K) for Pc  9.25 bar, Tc  1390 .5 K and   1.79 . We considered a default value of 3500 K for  AA / k .3,31 Table 1 summarizes the CPA EoS physical parameters and MM values used in this work for the well-defined components, pseudo-hydrocarbon (C6) and asphaltene.

 AR    T   , x

   U R , U R  AR  T 

(13)

Table 1. CPA EoS physical parameters ( Pc , Tc ,  ) and molar mass ( MM ) for the well-defined components, pseudo-hydrocarbon (C6)21 and asphaltene Component

Pc /bar

Tc /K



MM /( g mol1 )

N2

33.90

126.21

0.039

28.0

CO2

73.75

304.14

0.239

44.0

H2S

89.40

373.20

0.081

34.1

C1

45.99

190.56

0.011

16.0

C2

48.72

305.32

0.099

30.1

C3

42.48

369.83

0.153

44.1

iC4

36.04

407.80

0.183

58.1

nC4

37.96

425.12

0.199

58.1

iC5

33.80

460.40

0.227

72.2

nC5

33.70

469.70

0.251

72.2

C6

30.12

507.40

0.296

86.2

Asphaltenea

9.25

1390.5

1.79

750

a

 AA  0.05 ;  AA / k  3500 K. Each oil has a specific value for the parameters Pc , Tc , and  for the heavy component because

the MM of this component is oil dependent. Initial values for these parameters were calculated using correlations from the literature. The critical pressure was calculated by equation (14):32


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log( Pc )  2.829  0.9412  103Tb  0.30475  105 Tb2  0.15141  108 Tb3  0.20876  104 Tb API

(14)

 0.11048 10 7 Tb2 API  0.1395 109 Tb2 API 2  0.4827 107 Tb API 2

where the critical pressure is in psi . The normal boiling point, Tb , in equation (14) is in °F. The API density is defined in equation (15):32

API 

141.5  131.5 SG

(15)

where SG is the dead oil (or C7+) specific gravity at atmospheric conditions. The normal boiling point is calculated from the Pedersen et al.32 correlation (equation 16):

Tb  97.58 MM 0.3323 m0.04609

(16)

where MM is in g mol1 , Tb is in K and m is the dead oil (or C7+) mass density in g cm3 in atmospheric conditions. The critical temperature was calculated from the Kesler and Lee33 correlation (equation 17):

Tc  341.7  811SG  (0.4244  0.1174 SG)Tb  (0.4669  3.2623SG)105 / Tb

(17)

where Tc and Tb are in °R. The acentric factor was calculated from equation (18):21



3  log( Pc / 1.01325 )    1 7 Tc / Tb  1 

(18)

where Pc is in bar and Tc and Tb are in K. Then the value of Pc for  was adjusted using the oil bubble pressure ( BP ) experimental data minimizing the objective function ( OF ) given in equation (19) through the Nelder-Mead Simplex method.30

1 OF  np

np

 BP i 1

calc i

 BPiexp

(19)


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where n p is the number of data points. Superscripts calc and exp stand for calculated and experimental, respectively. The kij values for the CPA model can be found in Table 2 and were taken from the literature.21 We decided to consider only the binary interaction parameters between  -N2,  -CO2,  -H2S and

 -C1. Other kij values were set to zero. In their original work, Li and Firoozabadi21 used kij values for all hydrocarbons present in the system (and not only for the  pairs) but in a preliminary evaluation it came to our attention that these four values coupled with the Pc estimation for the  are sufficient to provide a good correlation of the experimental bubble pressure data. Table 2. Binary interaction parameters ( kij ) for CPA EoS.21 Other kij values were set to zero Component

N2

CO2

H2S

C1



0.1

0.15

0.1

0.0289  1.633  10 4 MM

After the physical parameters of  are set, the cross-association energy parameter between asphaltene and  (  A  / k ) is calculated. Due to the low mole fraction of asphaltene in live oils and the fact that there is no vaporization of asphaltene, the value of  A  / k does not significantly affect the calculated bubble pressure. The  A  / k parameter is calculated based on the experimental AOP and is considered temperature dependent. For a given oil, a temperature is fixed and the value of

 A  / k is calculated using a quasi-Newton method34 so that the calculated AOP equals the experimental data within a tolerance ( 10 4 bar ). This procedure is performed for the lowest and highest temperatures evaluated for the oil and the calculated  A  / k values are interpolated as a function of temperature.34

2.1.2. Equilibrium calculations with CPA EoS The procedure to calculate the AOP was performed as follows: first, for a given oil overall composition and fixed temperature the bubble pressure is calculated using a vapor-liquid equilibrium algorithm27 based on the isofugacity criterion22 and assuming that only these two phases exist.28 Although the true bubble point involves a three-phase equilibrium35 the difference between bubble pressures calculated considering two-phase or three-phase21 equilibrium is negligible for the oils evaluated in this study. Then, for a pressure above the bubble pressure, a stability test is performed in the liquid phase using equation (20) to check if this phase is unstable and will form an additional asphaltene liquid-dense phase.28 Asphaltene precipitates if the fugacity ACS Paragon Plus Environment

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of asphaltene in the liquid phase ( fˆAl ) is greater than the fugacity of the asphaltene in pure state l

( f A ). The pressure is adjusted until the liquid phase is stable. Detailed expressions for the calculation of fugacity with CPA EoS are provided by Nasrabadi et al.22 l fˆAl  f A

(20)

2.2. The Hirschberg method The basic assumptions regarding the Hirschberg method are: (i) Asphaltenes are considered pseudo-pure components in the petroleum (ii) The precipitated asphaltene is assumed to be pure liquid and its chemical potential considered in the standard state (iii) The oil phase (solvent) is assumed to be liquid and the solubility parameter is calculated using a different approach from the original method36 (iv) In addition, there is a liquid-liquid equilibrium between the asphaltene and the solvent phase. Hirschberg et al.9 developed a model based on the solubility model using the Flory-Huggins approach. The chemical potential derived from the Flory-Huggins theory is given by equation (21):

i  ( i ) ref RT

 ln( Φi )  1 

Vi V 2  i  m   i  Vm RT

(21)

where Φ is the volume fraction, V is the molar volume,  is the solubility parameter,  is the chemical potential and the subscript i , m and ref refer to the component i , mixture and the reference state of pure liquid, respectively. The volume fraction is given by equation (22):

Φi 

xiVi Vm

(22)

The molar volume is obtained from equation (23):


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nc

Vm   xiVi

(23)

i 1

and the solubility parameter is given by equation (24):

nc

 m   Φi δi

(24)

i 1

where δi is the solubility parameter of component i and it can be calculated from equation (25):36



H  RT V

(25)

where H is the enthalpy of vaporization for a given temperature T and can be calculated according to the following steps: (i) Calculation of the normal boiling temperature from the critical temperature and acentric factor (  ) is given by:37 3    log( Pc )  1 7 1 

 

(26)

where   Tb / Tc is the ratio between the normal boiling temperature ( Tb ) in K and the critical temperature ( Tc ) in K. Pc is the critical pressure in atm.

(ii) Calculation of the enthalpy of vaporization of the component i at the normal boiling temperature (in BTU·lb-mole-1):37 (H i )Tbi  1.014[Tbi (8.75  4.571 log(Tbi ))]

(27)

(iii) Make a correction of the enthalpy of vaporization for a different temperature T from the normal boiling temperature, using Watson’s38 equation:

 Tc  T (H i )T  (H i )Tbi  i  Tc  Tb i  i

   

0.38

(28)


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Energy & Fuels

where Tbi is the normal boiling temperature and Tci is the critical temperature of component i .

2.2.1. Oil characterization with the Hirschberg method In the Hirschberg method a cubic EoS is first used to analyze the vapor-liquid behavior of the fluid. In this step asphaltene precipitation is ignored. Then, flocculation is taken into account. This method assumes that the liquid phase formed can split in the asphaltene phase and in the oil-rich phase. Moreover, an extension of the C7+ fraction (characterization) is necessary to proceed with the flash calculation. For the vapor-liquid calculation, good characterization is an important step and could directly influence the result. The characterization represents hydrocarbons as having seven or more carbon atoms in a reasonable number of pseudo-components. For the pseudo-components Pc , Tc , and  are not available. Thus, a characterization will be required to provide these parameters. Moreover, a lumping procedure must be performed to reduce the number of components in order to perform the flash calculation.37 Firstly, Pedersen et al.32 noted a pattern in the oil composition related to the number of carbons (equation 29): ln( xi )  A  B(CN) i

(29)

where A and B are constants adjusted from the molar fraction and the residue molar mass; CN is the carbon number. Similarly, C and D in equation (30) are estimated by density adjustment:

i  C  D ln[(CN) i ]

(30)

By using this extrapolation, the molar fraction and the molar mass may be assigned for each subfraction. The molar mass of each sub-fraction is calculated as follows (equation 31): MM i  14(CN) i  4

(31)

The next step is to calculate the critical properties and the acentric factor for each pseudocomponent using the correlations presented in equations (32)-(34):32


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Tci  d1  i  d 2 ln(MM i )  d 3 MM i 

 

ln Pci  d 5  d 6  i 

d4 MM i

d7 d8  MM i MM 12

Page 14 of 31

(32)

(33)

i  d9  d10 MM i  d11i  d12 MM i2

(34)

the twelve global constants ( d1 to d12 ) can be found in Pedersen et al.32 with the respective units. The last step is to group the pseudo-components. Pedersen et al.32 recommended a grouping based on the weight fraction where each grouped pseudo-component contains approximately the same weight. The Pc , Tc , and  values of the grouped pseudo-components are calculated as weight mean average values of Pc , Tc , and  of each pseudo-component. In this study a splitting/lumping procedure into 17 fractions was used. The characterization procedure is detailed in da Silva et al.37

2.2.2. Equilibrium calculations with the Hirschberg method The oil characterization and the bubble pressure calculations were performed in SPECS software39 using the SRK EoS considering the asphaltene content in the dead oil (SARA analysis). For a given oil, the temperature of the lowest BP is fixed and the value of the C7+ fraction density is calculated using SPECS software so that the calculated BP equals the experimental data within a tolerance ( 10 4 bar ). The kij values used with the SRK EoS were calculated by SPECS software using correlations from the literature.32 Then a stability test is performed in the liquid phase to check if there is precipitation of asphaltene. Taking into account that asphaltene is considered a pure component, the following condition should be satisfied at the AOP (equation 35):

A  ( A ) ref RT

0

(35)

The precipitation of asphaltene will begin from the point when the chemical potential of the pure asphaltene becomes smaller than the chemical potential of asphaltene in solution. From equation (21) it is possible to analyze the asphaltene stability by changing the pressure and calculating its inequality (equation 36):

F  ln( ΦA )  1 

VA VA  m   A 2  Vm RT ACS Paragon Plus Environment

(36)

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Energy & Fuels

where VA and Vm were calculated with SRK EoS in SPECS software. If F  0 , there is no precipitation. If F  0 , asphaltene is unstable and there is precipitation.17 As the pressure decreases at a constant temperature there will be an increase in the F value, which becomes positive after the AOP. The  m   A  value was calculated based on AOP experimental data following a procedure similar to that described for CPA EoS. The experimental AOP at the lowest temperature is used as input data and  m   A  is calculated so that the calculated AOP equals the experimental data within a tolerance ( 10 4 bar ). The same procedure is performed for the highest temperature and the calculated  m   A  values are interpolated as a function of temperature.

3. Results and discussion In this work the thermodynamic modeling of asphaltene precipitation with the Hirschberg method and CPA EoS was studied for five oils from the literature. Their compositions and other properties of interest ( MM , m , asphaltene content) are presented in Table 3. Although only five oils have been evaluated in this study, they have different characteristics that are commonly found in other oils, such as different asphaltene concentrations and different AOP behaviors (for some oils AOP only decreases with temperature while for others it can increase or decrease with temperature). Therefore, these five oils represent the most common behaviors found in the literature, and the results and conclusions can be extended to other oils with similar characteristics. As previously detailed in section 2, for CPA EoS all bubble pressure experimental data were used to fit the value of Pc of the heavy component (  ) of each oil. The physical parameters and MM of  , as well as the composition distribution of the C7+ fraction are presented in Table 4. For CPA EoS the available AOP data at the lowest and highest temperatures were used to calculate the crossassociation energy parameter between asphaltene and  (  A  / k ). The results are presented in Table 5.

As previously detailed, for the Hirschberg method the BP calculations were performed in SPECS software using the SRK EoS. One BP data was used to calculate the C7+ fraction density in SPECS software and the obtained values are presented in Table 6. The calculated values of  m   A  for the Hirschberg method are presented in Table 7 and were also obtained using the available AOP experimental data at the lowest and highest temperatures. It is worth mentioning that oil O5 has


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Page 16 of 31

only one AOP experimental data, therefore it is not possible to obtain a temperature dependency for

 A  / k or  m   A  . As a result, for this oil the value of these parameters were calculated using the only AOP data available and kept constant for all other temperatures. Also, for oil O4 the calculated values of  m   A  for the lowest and highest temperatures at which the AOP was measured were practically constant and therefore the value of 11.6470 has been kept for the other temperatures. For all temperatures  m   A  is first calculated using the T function given in Table 7. Secondly,  m is calculated using equations (25)-(28). Lastly, these two calculated values allow the determination of

 A that is used in the AOP calculation with the Hirschberg method. Table 3. Composition and properties of interest ( MM , m , asphaltene content) for the five oils evaluated in this study Component

Oil O121

O221

O321,40

O429

O529

mol % N2

0.09

0.09

0.47

0.48

0.80

CO2

1.02

1.02

1.59

0.92

0.05

H2S

0.05

0.05

1.44

-

-

C1

42.41

42.41

32.22

43.43

51.02

C2

10.78

10.78

12.42

11.02

8.09

C3

6.92

6.92

10.29

6.55

6.02

iC4

1.55

1.55

2.03

0.79

1.14

nC4

2.92

2.92

4.87

3.7

2.83

iC5

1.47

1.47

2.22

1.28

1.58

nC5

1.82

1.82

2.71

2.25

1.63

C6

2.86

2.86

4.12

2.7

2.67

C7+

28.11

28.11

25.62

26.88

24.17

C7+ MM /( g mol1 ) 209.50

209.50

284.36

228.07

368.87

m /( g cm3 )

0.852

0.852

0.8048

0.865

0.875

0.5

0.4

3.25

1.3

4.6

Asphaltenes (%, w/w)a a

SARA analysis.


16

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Energy & Fuels

Table 4. Composition distribution of the C7+ fraction, physical parameters for CPA EoS and MM of  for the five oils evaluated in this study Physical parameters and MM of 

C7+ fraction (mol %)

Oil

CPA EoS

MM



Asphaltene

Pc /bar

Tc /K



O1

28.07074

0.03926

15.72

748.47

0.693

208.74

O2

28.07859

0.03141

15.64

748.57

0.693

208.90

O3

25.30430

0.31570

13.83

771.71

0.935

278.55

O4

26.77374

0.10626

14.24

764.56

0.712

226.00

O5

23.62318

0.54682

9.68

838.08

0.968

360.05

( g mol1 )

Table 5. Cross-association energy parameter between asphaltene and  (  A  / k ) for CPA EoS of the five oils evaluated in this study. The cross-association volume parameter (  A  ) is set equal to 0.05 Oil

(  A  / k )/K

O1

 0.0125T  1406.15

O2

0.3556T  1320.34

O3

1.0076T  1151.50

O4

0.7714T  1288.10

O5

1986 .49

Table 6. Calculated values of the C7+ fraction density ( m ) in SPECS software for the Hirschberg method Oil

m /( g cm3 )

O1

0.831

O2

0.833

O3

0.908

O4

0.804

O5

0.800


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Page 18 of 31

Table 7. Calculated values of ( m   A ) for the five oils evaluated in this study with the Hirschberg method Oil

( m   A ) /bar0.5

O1

0.0226T  13.2640

O2

0.0052T  10.9889

O3

0.0116T  11.1610

O4

11.6470

O5

20.8740

It is important to remark that the differences between parameters describing the relationship between asphaltenes and resins change from fluid to fluid in both Tables 5 and 7. The main reason is the fact that not only the nature but also the amount and the characteristics of each fraction are unique for each oil. Therefore, it is not possible to obtain average parameters from a number of oils and extend their application to other oils even for a reduced number of properties. The average absolute deviations (AAD) between calculated and experimental results for all oils evaluated in this study were obtained from equation (37) and are presented in Table 8.

100 p Yi calc  Yi exp  AAD (%)   Y exp n p i 1 i n

(37)

where Y is the property of interest (AOP or BP).

Table 8. Calculated average absolute deviations (AAD) with Hirschberg method and CPA EoS for all the oils evaluated in this study

 AAD (%) , Hirschberg method

Oil

 AAD (%) , CPA EoS

AOP

BP

AOP

BP

O1

9.63

1.63

3.06

1.09

O2

12.83

0.83

8.07

1.05

O3

6.25

6.54

5.42

4.09

O4

0.34

0.86

0.29

0.74

O5

0.11a,b

0.73

0.01a,b

0.62

a

calc exp exp  PAOP ) / PAOP Absolute Deviation (AD) calculated as:  AD (%)  100 ( PAOP .

b

Not zero due to the truncation in the value of the calculated parameters. ACS Paragon Plus Environment

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Energy & Fuels

The results of AOP calculated by the Hirschberg method and the CPA EOS for oils O1 and O2 are presented in Figure 1. The experimental data were taken from graphs published by Li and Firoozabadi21 and Jamaluddin et al.29 using software 41 and are presented in Appendix-A. Both oils have almost the same composition differing only in the asphaltene content. For oil O1 the asphaltene content is 0.5 % (w/w) while oil O2 contains 0.4 % (w/w) asphaltene. The lower asphaltene content in oil O1 compared to oil O2 does not significantly affect the experimental BP. The deviations in the BP calculated with the CPA EoS and with the Hirschberg method, which uses the SRK EoS for the bubble pressure calculation, were very close despite the fact only one BP data was used in the parametrization of the Hirschberg method. The AOP values provided by CPA EoS indicated the lowest AAD values, which are different from those calculated with the Hirschberg method. At intermediate temperatures the calculated AOP with both models are predictive. CPA EoS shows better agreement with the experimental data than Hirschberg’s, particularly for oil O1. The main differences between the capabilities of both models were observed outside the experimental temperature range. As the temperature increases the CPA model predicts an AOP curve parallel to the BP curve21 while the Hirschberg method predicts a decrease in the system stabilization. According to previous works1,21 an increase in the temperature may strengthen or weaken the asphaltene precipitation. As the temperature rises, the solubility of the asphaltene in the bulk oil phase increases but the polar resin-asphaltene interactions decrease, which may induce asphaltene precipitation. At higher temperatures precipitated asphaltene is expected to melt and resolubilize in the oil producing a stable system.1

Figure 1. Asphaltene Onset Pressure (blue) and Bubble Pressure (red) for: (a) Oil O1 and (b) Oil O2.21 Symbols are experimental data for: □, AOP; ○, BP. Lines were calculated with: Hirschberg method;

,

, CPA EoS.

(a)

(b)


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Page 20 of 31

Figure 2 presents the results for oils O3 and O4. Again the lower deviations for the AOP and BP were obtained with the CPA EoS, however, the calculated results with the Hirschberg method were very close. For both models the predicted AOP at the intermediate temperatures were similar. However, at higher temperatures the CPA model predicts an asymptotic behavior of the AOP in relation to the BP curve while the Hirschberg method predicts a decrease in the system stabilization. It is worth mentioning that oils O1, O2 and O3 exhibit an experimental point of AOP that deviates from the trend of the other experimental points. According to Li and Firoozabadi21 an increase in the system temperature causes a decrease in the density, which consequently increases the entropy. These two effects compete among each other and, as a result, an increase in temperature may induce precipitation or resolubilization of asphaltene. This can explain the unusual behavior of the AOP at a specific temperature. In all the cases evaluated both models were not capable of predicting this singular behavior.

Figure 2. Asphaltene Onset Pressure (blue) and Bubble Pressure (red) for: (a) Oil O3 and (b) Oil O4.21,29,40 Symbols are experimental data for: □, AOP; ○, BP. Lines were calculated with: Hirschberg method;

,

, CPA EoS.

(a)

(b)

Figure 3 shows the results for oil O5. Like the other oils, the BP values calculated with the CPA EoS and with the Hirschberg method were very close. As there is only one experimental data of AOP the temperature dependency of  A  / k and (  m   A ) cannot be evaluated. Consequently, the deviations calculated with both models for this property are almost zero, due to the truncation in the value of the calculated parameters. Since there is no information about the temperature dependency of  A  / k , the CPA EoS predicts a slow decrease in the AOP as the temperature rises. As the ACS Paragon Plus Environment

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Energy & Fuels

temperature increases the CPA model predicts that the AOP tends to be parallel to the BP. The Hirschberg method predicts a quick decrease in the AOP for lower temperatures until the experimental temperature of AOP is reached. At this temperature, the curve changes behavior and a rapid reduction in system stabilization is predicted as the temperature increases further. As discussed in Arya et al.3 and Li and Firoozabadi21 two AOP data at different temperatures are needed so that CPA model can correctly reproduce the experimental behavior of the AOP. Our results are in agreement with their findings because, for oil O5 (in which  A  / k is T -independent) the calculated AOP slowly decreases as the temperature rises, while the expected behavior, based on experimental data of other oils, is a faster decrease of AOP. Figure 3. Calculated Asphaltene Onset Pressure (blue) and Bubble Pressure (red) for oil O5.29 Symbols are experimental data for: □, AOP; ○, BP. Lines were calculated with: method;

, Hirschberg

, CPA EoS.

The calculated  A  / k values for oils O1, O2, O3 and O4 for CPA EoS (Table 5) may increase or decrease depending on temperature, while the calculated values of ( m   A ) for the Hirschberg method (Table 7) are more dependent on temperature. In order to evaluate the influence of the number of experimental points used to calculate  A  / k and ( m   A ) , an additional study was performed with oils O1 to O4. In this case five of the experimental AOP of oils O1 and O2 were used in the coefficients regression of  A  / k and ( m   A ) as a function of temperature (the experimental point of AOP that deviates from the trend of the other experimental points was not considered in this regression). For oils O3 and O4 all of the experimental AOP were used in the coefficients regression. The correlations for  A  / k and ( m   A ) as a function of temperature are


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Page 22 of 31

presented in Table 9 while the calculated AAD with these new correlations are presented in Table 10. Table 9. Temperature dependency of ( m   A ) and  A  / k for oils O1 to O4 considering intermediate experimental AOP in the coefficients regression. Parameter Oil

(  A  / k )/K

( m   A ) /bar0.5

O1

 5.8512 ln(T )  1433.88

8.4286 ln(T )  28.160

O2

115.37 ln(T )  756.134

26.265 ln(T )  85.748

O3

306.31 ln(T )  290.597

19.028 ln(T )  57.093

O4

280.19 ln(T )  82.6232

21.778 ln(T )  64.539

Table 10. Calculated average absolute deviations (AAD) with Hirschberg method and CPA EoS for oils O1 to O4 considering intermediate experimental AOP in the coefficients regression

 AAD (%) , Hirschberg method

Oil

 AAD (%) , CPA EoS

AOP

BP

AOP

BP

O1

2.93

1.63

2.75

1.09

O2

5.51

0.83

6.62

1.05

O3

5.42

6.54

5.34

4.09

O4

0.29

0.86

0.25

0.74

Figure 4 shows the results of AOP and BP for oils O1 to O4 using the correlations given in Table 9. In all cases the calculated values show very good accuracy with the experimental AOP. The calculated AAD are presented in Table 10 and, as expected, are smaller than the results obtained in the first case evaluated when only two AOP experimental data points were used in the parameters calculation. In addition, a change in the calculated trend of AOP by the Hirschberg method is observed, which is very close to the results obtained with the CPA model. For lower temperatures, there is an increase in asphaltene stabilization as the temperature rises while at higher temperatures both models predict that the AOP curve tends to be parallel to the BP curve, i.e. the effect of temperature is more relevant for the asphaltene resolubilization outside the experimental temperature range. Therefore, more than two experimental AOP data should be used in the Hirschberg method so that the calculated results match the expected behavior of the AOP curve outside the experimental range of temperature. ACS Paragon Plus Environment

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Energy & Fuels

Figure 4. Asphaltene Onset Pressure (blue) and Bubble Pressure (red) for: (a) Oil O121, (b) Oil O221, (c) Oil O321,40 and (d) Oil O429. Symbols are experimental data for: □, AOP; ○, BP. Lines were calculated considering the coefficients given in Table 9 for:

, Hirschberg method;

, CPA EoS.

(a)

(b)

(c)

(d)

4. Conclusions The capability of the CPA EoS and the Hirschberg method to correlate the experimental AOP as well as the BP of five oils from the literature was critically evaluated. For CPA EoS only selfassociation between asphaltene molecules or cross-association between asphaltene and resin molecules was considered. The number of non-zero binary interaction parameters needed for the phase envelope calculation was reduced to only four binary pairs. At least two AOP and one BP


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Page 24 of 31

data are required for the correct parameterization of both models. For a fair comparison between the models, asphaltene was also considered to precipitate as a pure phase for CPA EoS. For the Hirschberg method, the difference between the liquid phase solubility parameter and the asphaltene solubility parameter ( m   A ) was calculated using two AOP points at different temperatures, providing a good representation the experimental data. Similarly, the crossassociating energy parameter between asphaltene and resin (  A  / k ) of the CPA EoS was also interpolated as a function of temperature. For all oils both models provided similar results for the AOP and BP but CPA EoS performed slightly better calculating lower values of AAD for both properties when only the AOP data at the highest and lowest temperatures are used in the parameterization of the models. The calculated values of BP with CPA EoS are very close to those obtained with Peng-Robinson EoS, which was used to describe the physical part of the CPA model, while the Hirchberg method uses the SRK EoS to calculate the BP. Furthermore, the consideration of asphaltene precipitating as a pure phase does not reduce the capability of CPA EoS to correlate the AOP. This observation is supported by the fact that the deviations obtained with this consideration are practically the same as those calculated when asphaltene is considered to precipitate with other components.21


24

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Energy & Fuels

APPENDIX-A Experimental data considered in this work for oils O1 to O5. All experimental data were taken from graphs published by Li and Firoozabadi21 and Jamaluddin et al.29 using software.41 Table A.1. Experimental data used for oil O1.21 AOP

BP

P /bar

T /K

P /bar

T /K

601.82

321.73

179.2

321.73

537.7

338.67

199.6

338.67

496.9

354.88

206.9

354.88

470.7

371.82

215.7

371.82

494

388.77

227.3

388.77

408

424.86

236.1

424.86

Table A.2. Experimental data used for oil O2.21 AOP

BP

P /bar

T /K

P /bar

T /K

441.97

322

177.5

322

471.35

338.6

197.9

338.6

388.32

355.15

205.7

355.15

371.7

372.06

214.6

372.06

358.9

388.97

224.8

388.97

302.7

425

235

425

Table A.3. Experimental data used for oil O3.21,40 AOP

BP

P /bar

T /K

P /bar

T /K

601.64

333.33

167.8

333.33

474.86

354.21

187.4

354.21

456.3

393.41

196.2

393.41

373.2

413.19

202.7

413.19


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BP

P /bar

T /K

P /bar

T /K

472.6

372.15

222.1

372.15

454.2

377.15

226.4

377.15

442.6

383.15

225.9

383.15

429.2

389.15

226.8

389.15


BP

P /bar

T /K

P /bar

T /K

365.4

361.15

293.8

355.15

293.7

361.15

ACKNOWLEDGEMENTS The authors acknowledge the support of ANP - Agência Nacional de Petróleo, Gás Natural e Biocombustíveis and Petrogal Brasil S.A., related to the grant from R&D investment rule.

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AUTHOR INFORMATION Corresponding author *S.A.B. Vieira de Melo (ORCID: 0000-0002-8617-3724). Tel: +55-71-32389802. Fax: +55-71-32839800. E-mail: [email protected]. Notes The authors declare no competing financial interest.


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Modeling of the Asphaltene Onset Pressure from ... - ACS Publications

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