Systematic Investigation of Asphaltene Deposition in the Wellbore and

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Systematic Investigation of Asphaltene Deposition in Wellbore and NearWellbore Region of a Deepwater Oil Reservoir Under Gas Injection. Part 1: Thermodynamic Modeling of the Phase Behavior of Polydisperse Asphaltenes Mohammed I. L. Abutaqiya, Caleb J. Sisco, Jianxin Wang, and Francisco M. Vargas Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.8b03234 • Publication Date (Web): 08 Jan 2019 Downloaded from http://pubs.acs.org on January 9, 2019

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Systematic Investigation of Asphaltene Deposition in Wellbore and NearWellbore Region of a Deepwater Oil Reservoir Under Gas Injection. Part I: Thermodynamic Modeling of the Phase Behavior of Polydisperse Asphaltenes Mohammed I. L. Abutaqiya1, Caleb J. Sisco1, Jianxin Wang2, Francisco M. Vargas1* 1

Department of Chemical and Biomolecular Engineering, Rice University, Houston, Texas77005, USA 2

Chevron Energy Technology Company, Houston, Texas, 77002, USA

Authors Email Address: [email protected], [email protected], [email protected], [email protected] *

Corresponding Author: email- [email protected], Phone- +1 (713) 348-2384 Abstract

Thermodynamic modeling is conducted for a high-asphaltene high-resins crude oil produced from a deepwater reservoir using the Perturbed-Chain Statistical Associating Fluid Theory (PCSAFT). The asphaltenes are characterized as a polydisperse fraction following a 3-parameter gamma distribution function with resins included as the lightest cut of the asphaltene distribution. Modeling results with 55 mol% injection indicate that the driving force of precipitation is sufficiently large that a significant amount of non-asphaltene components coprecipitate with asphaltenes. As pressure decreases from the upper asphaltene onset pressure (UAOP) to the bubble pressure (BP), the amount (by weight) of the asphaltene-rich phase surpasses the amount of asphaltene-lean phase. Interestingly as pressure decreases below BP, the asphaltene-lean phase dissolves into the asphaltene-rich phase until pressure reaches the lower onset pressure (LAOP) where the lean phase is completely dissolved. Due to the high driving force of precipitation, all asphaltenes precipitate out of solution before the bubble pressure and significant amounts of the other pseudo-fractions also co-precipitate. This causes the 1 ACS Paragon Plus Environment

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composition of asphaltenes in the asphaltene-rich phase to decrease as pressure decreases, and the asphaltene composition in the asphaltene-rich phase at the upper onset is 23.6 wt%, which is uncommonly low for a precipitating phase. Experimental images from high-pressure microscopy (HPM) show that the shape of the formed asphaltene-rich phase changes from rigid solid-like near the UAOP to soft liquid-like structure as pressure decreases. This indicates a decrease in asphaltene composition during depressurization, similar to the simulation results produced by PC-SAFT. A sensitivity analysis is performed to evaluate the assumption of polydisperse and monodisperse asphaltenes from a modeling perspective and the effect of the gamma distribution parameters. Keywords: PC-SAFT, Crude Oil Characterization, Deepwater Reservoir, Polydisperse Asphaltenes

Graphical Abstract

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1. Introduction Asphaltenes are a polydisperse mixture that constitutes the heaviest and most polarizable fraction of crude oil. They are defined as a solubility class that is insoluble in light paraffins (e.g. npentane or n-heptane) but soluble in aromatic solvents (e.g. toluene and benzene). Asphaltenes polydispersity can be characterized by separation into subcomponents using different light paraffins (e.g. C5-asphaltenes, C6-asphaltenes…etc). These subcomponents of the asphaltene fraction may exhibit markedly different phase and rheological behavior. It is well-known that asphaltenes can precipitate and deposit in production tubing under certain conditions of pressure, temperature, and composition1. Asphaltene precipitation is a necessary but not sufficient condition for deposition. Therefore, understanding the thermodynamic conditions at which asphaltenes precipitate is a key step in designing strategies for mitigating the deposition problem, and several thermodynamic variables are required as inputs to deposition models (e.g. asphaltene solubility, precipitation amounts, phase densities…etc). Various thermodynamic models have been deployed to quantify asphaltene phase behavior, which are broadly classified into two schools of thought: micellar and solubility theories. In the micellar theories, asphaltenes are assumed to be stabilized sterically by polar interactions with resins 2–8, while in the solubility theories, the phase behavior of asphaltenes is assumed to be dominated by dispersion forces. The solubility approach has now been supported by extensive experimental evidence9–17, thus refuting the micellar stabilization theory. Solubility models are mainly based on (1) regular solution theory of Scatchard and Hildebrand18,19 and Flory-Huggins20,21, and (2) equations of state (EOS). Regular solution models include Hirschberg et al.22, de Boer23, Flory-Huggins-Zuo24, and Yarranton and coworkers25–27.


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These models are limited to predicting only solubility limits and do not explicitly account for fluid compressibility. EOS models, on the other hand, can capture solubility limits and compressibility within the same thermodynamic framework, making them an attractive candidate for a comprehensive asphaltene deposition study. Alternatively, molecular simulation has been extensively implemented to understand and predict the phase behavior of asphaltenes and asphaltene-like molecules.11,28–30 Currently, the most popular EOS models for simulating asphaltene precipitation are the Statistical Associating Fluid Theory (SAFT)31,32 and the cubic plus association (CPA)33–35 equations of state. Of particular interest is the perturbed-chain variant of SAFT (PC-SAFT) developed by Gross and Sadowski36 which is readily available in commercial simulators. PCSAFT has been successfully applied for modeling asphaltene precipitation without including the association term37–49, while CPA requires use of the association term50–53. The capability of PCSAFT to capture asphaltene phase behavior without including association aligns with experimental evidence that asphaltene phase behavior is dominated by London dispersion forces, not by polar-polar interactions 9–17. The implementation of an equation of state for the modeling of PVT properties and asphaltene precipitation requires that the crude oil must be characterized into pseudo-components with representative simulation parameters. Whitson54 was the first to introduce the use of a gamma distribution to represent the molar composition of crude oils. Other crude oil characterizations followed, which can be considered as special cases of the gamma distribution: Pedersen method55 and Chi-Square method56. Additionally, methods based on the Saturates-AromaticsResins-Asphaltenes (SARA) analysis were developed57–59. Most of the crude oil modeling work done using PC-SAFT makes use of the SARA analysis to characterize the dead oil37,38,40–49,60. 4 ACS Paragon Plus Environment

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The asphaltenes pseudo-fraction is either treated as a monodisperse mixture38,40 or a polydisperse mixture48,49. For purposes of modeling the asphaltene onset pressure (AOP), treating asphaltenes as a monodisperse fraction is sufficient to capture the phase behavior. However, if the objective is to study the amounts of precipitated asphaltenes, it is necessary to characterize asphaltenes as a polydisperse mixture61,62. Yarranton and coworkers63–65 extended the use of the gamma-distribution to characterize the mass distribution of polydisperse asphaltenes and preformed the modeling using the Advanced Peng-Robinson66 EOS. Tavakkoli et al.49 extended the PC-SAFT characterization method of Punnapala and Vargas38 to account for asphaltene polydispersity through the implementation of a gamma distribution function. The authors applied the approach to a relatively light crude oil with resins content of 7 wt% and asphaltene content of 3 wt%. This paper is the 1st in a series of two publications for investigating the asphaltene deposition tendency of a high-asphaltene oil (Crude C2) under lean gas injection in a deep-water reservoir. The objective of this paper is to conduct thermodynamic modeling with PC-SAFT on Crude C2 using a polydisperse characterization procedure. A modified version of the polydisperse asphaltene characterization technique proposed by Tavakkoli et al.49 is implemented to account for the high amounts of resins and asphaltenes in Crude C2. The modeling results obtained in this work are used as an input to the deposition simulator (2nd paper of this series)67 to provide thermodynamic variables as a function of pressure, temperature, and composition along the production profile.


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2. Background on SARA-based Crude Oil Characterization The characterization procedure for polydisperse asphaltenes proposed by Tavakkoli et al.49 is an extension of the procedure proposed by Punnapala and Vargas38. In their procedure, Punnapala and Vargas implemented a monodisperse asphaltene characterization in which they require (1) the composition of flashed gas and stock tank oil (STO) of the original reservoir fluid, (2) the SARA analysis, (3) zero-flash GOR, (4) STO density and molecular weight, and (5) experimental high-pressure high-temperature (HPHT) bubble and asphaltene onset pressure measurements. The flashed gas analysis is used to define the mass composition of the light components, which are normally H2S, N2, CO2, C1, C2, C3, and a Heavy Gas pseudo-component (C4+). The SARA analysis defines the mass composition of the heavy components in the stock tank oil which is characterized as three pseudo-fractions: saturates, aromatics+Rresins (A+R), and asphaltene. The zero-flash GOR is used to mathematically recombine the flashed gas and STO compositions to form the live oil composition for the petroleum fluid. The STO density and molecular weight assist in converting the mass composition to moles. The PC-SAFT simulation parameters for pure components are taken from the work of Gross and Sadowski36. Parameters for the pseudo-components are calculated using correlations proposed by Punnapala and Vargas in terms of molecular weight and aromaticity (). The aromaticity is defined such that  = 0 for n-alkanes and  = 1 for polynuclear aromatics (PNA). These correlations are shown in Table 1. The aromaticity of the A+R fraction is tuned to match bubble pressure and density and the aromaticity and MW of asphaltenes are tuned to match experimental asphaltene onset pressure.


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Table 1. PC-SAFT parameters correlations for pseudo-components.38 Parameter

Pseudo-Components Correlations

𝑚

(1 − 𝛾𝑖 )(0.0257𝑀𝑖 + 0.8444 ) + 𝛾𝑖 (0.0101𝑀𝑖 + 1.7296)

𝜎(Å)

(1 − 𝛾𝑖 ) (4.047 −

𝜀⁄ (𝐾) 𝑘𝐵

(1 − 𝛾𝑖 ) exp (5.5769 −

4.8013 ln 𝑀𝑖 93.98 ) + 𝛾𝑖 (4.6169 − ) 𝑀𝑖 𝑀𝑖 9.523 234100 ) + 𝛾𝑖 (508 − ) 𝑀𝑖 𝑀𝑖 1.5

In the approach proposed by Tavakkoli et al.49, asphaltenes are treated as a polydisperse fraction that is assumed to follow a 3-parameter gamma distribution. The distribution parameters are tuned to asphaltene fractionation data obtained via the “Indirect Method”68, while the rest of the maltenes fraction is characterized as in the approach of Punnapala and Vargas. In the Indirect Method, asphaltene fractionation experiments using different n-alkane precipitants (i.e. nC5, nC6, nC7, and nC8) are performed to quantify the amount of asphaltenes precipitated as a function of precipitant volume. The asphaltene pseudo-components are then defined based on solubility cuts: C5-C6, C6-C7, C7-C8, and C8+ asphaltenes. For example, C5-C6 asphaltenes are those that are insoluble in nC5 but soluble in nC6. PC-SAFT parameters are tuned to precipitation amounts at 90 vol% of precipitant in order to remove kinetic effects on precipitation that are more likely to impact measurements performed at lower concentrations of the precipitant. 3. New Crude Oil Characterization Methodology The characterization methodology proposed in this work differs from the characterization methodology of Tavakkoli et al.49 in two key aspects: (1) the aromatics and resins are no longer combined in a single pseudo-component, and (2) resins constitute the lightest fraction of the polydisperse asphaltene distribution in the crude oil. This characterization procedure implies that resins, in addition to the four asphaltenes pseudo-components, are assumed to follow a gamma 7 ACS Paragon Plus Environment

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distribution. Figure 1 shows a schematic of the characterization methodology proposed in this work. To reduce the number of fitting parameters, the aromaticities (𝛾) of aromatics, resins, and asphaltenes pseudo-components are assumed to be equal. This value can be viewed as an average aromaticity for the non-paraffinic components of the crude oil. The molecular weight distribution of resins and asphaltenes is assumed to follow a gamma distribution described by Eq. (1)69–71:

𝑓(𝑟) =

𝛼 1 𝛼 𝛼(1 − 𝑟) [ ] (𝑟 − 1)𝛼−1 exp [ ] (𝑟̅ − 1) 𝑀𝑚 Γ(𝛼) (𝑟̅ − 1)

(1)

̅ /𝑀𝑚 , respectively, 𝑀 is the molecular weight of the where 𝑟 and 𝑟̅ are given by 𝑀/𝑀𝑚 and 𝑀 Asphaltene sub-fraction, 𝛼 is the shape parameters for the distribution, and 𝑀𝑚 is the molecular ̅ , along with weight of the pre-aggregated asphaltene monomer. The parameters  and 𝑀 aromaticity (𝛾) are simultaneously tuned to match STO density, bubble pressure, saturation density, and asphaltene precipitation data from Indirect Method measurements with nC5, nC6, nC7, and nC8 as precipitants. This global optimization is required since resins and asphaltenes constitute a large portion of the STO (~30 wt%) and variation in asphaltene parameters ( and ̅ ) can cause significant changes in oil density. 𝑀 The molecular weight of the asphaltene monomer (𝑀𝑚 ) is set to 400 g/mol, which is lower than the value of 1700 g/mol used by Tavakkoli et al.49 because resins are here included as the first cut of the distribution. Several other values of 𝑀𝑚 were tested and the value of 400 g/mol was found to yield the best representation of Indirect Method data. The resulting molecular weight of the resins fraction is 650 g/mol.


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Similar to the approach of Tavakkoli et al.49, the binary interaction parameters (𝑘𝑖𝑗 ) between the asphaltene fractions (including resins) and n-alkane precipitants (including saturates) is calculated using the correlation:

𝑘𝑖𝑗 = 𝐴 + 𝐵 ∗ 𝑀𝑛−𝑎𝑙𝑘𝑎𝑛𝑒 + 𝐶 ∗ 𝑀𝐴𝑠𝑝ℎ

(2)

With values of A, B, and C taken from the work of Tavakkoli et al.49: 8.27E-3, 2.05E-4, and 1.23E-5, respectively.

Figure 1. Schematic of the crude oil characterization procedure with polydisperse asphaltenes. 4. Characterization Results The compositional analysis, table of PVT properties, and asphaltene fractionation analysis for crude oil C2 is provided in Table 2, Table 3, and Table 4, respectively. The details of the experimental setup for measuring asphaltene precipitation amounts using the Indirect Method can be found elsewhere.68,72 The optimized aromaticity for aromatics-resins-asphaltenes is 0.56. ̅ are 1.014 for the resins+asphaltenes distribution is 989.3 g/mol, The optimized  and 𝑀 9 ACS Paragon Plus Environment

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̅ is a weight-average molecular weight. The respectively. Note that the optimized 𝑀 ̅ is 822 g/mol. The resulting set of PC-SAFT simulation corresponding mole-average 𝑀 parameters for crude oil C2 are shown in Table 5 with binary interaction parameters shown in Table 6. Simulation results for the Indirect Method with various n-alkane precipitants are shown in Figure 2. Note that only two asphaltene onset concentration (AOC) measurements were performed due to limited amount of crude oil sample. These measurements correspond to nC5 and nC7 precipitants. Comparison between experimental PVT data and modeling results from PC-SAFT is shown in Table 7. The modeling results shown in Table 7 indicate that the proposed methodology for the highasphaltene crude under investigation yield good match to experimental PVT data. Furthermore, the modeling results of precipitated amounts of asphaltenes at 90 vol% of different precipitants (Figure 2) are in good agreement with experimental data. PC-SAFT modeling results show a slight overprediction for the precipitated asphaltenes at 90 vol% of nC5 (the most volatile precipitant). It is possible that some vaporization of nC5 may have taken place during the experiment leading to more stable asphaltenes (lower precipitated amounts). Overall, the predictions are within a reasonable average absolute percent deviation of 1.7% from experimental data.


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Table 2. Compositional analysis for Crude C2. Component H2S N2 CO2 C1 C2 C3 i-C4 n-C4 i-C5 n-C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 C21 C22 C23 C24 C25 C26 C27 C28 C29 C30+ M (g/mol)

M (g/mol) 34.1 28.0 44.0 16.0 30.1 44.1 58.1 58.1 72.2 72.2 86.2 95.4 109.5 121.9 134.0 147.0 161.0 175.0 190.0 206.0 222.0 237.0 251.0 263.0 275.0 291.0 305.0 318.0 331.0 345.0 359.0 374.0 388.0 402.0 783.4

Gas (wt%) 0.520 1.195 27.706 15.748 19.877 3.689 11.427 4.279 5.452 4.986 2.883 1.416 0.562 0.192 0.043 0.005 0.000 0.000 0.020 30.8

Gas (mol%) 0.571 0.836 53.159 16.101 13.871 1.954 6.053 1.824 2.324 1.780 0.930 0.398 0.142 0.044 0.009 0.001 0.000 0.000 0.003 -

Liquid (wt%) 0.004 0.024 0.157 0.078 0.355 0.318 0.656 1.396 2.088 2.750 2.675 2.725 2.514 2.291 2.418 2.251 2.352 2.180 2.084 1.956 1.976 1.753 1.706 1.613 1.301 1.517 1.360 1.440 1.354 1.477 1.261 51.970 290.3

Liquid (mol%) 0.076 0.237 1.035 0.388 1.774 1.282 2.639 4.704 6.354 7.286 6.369 5.903 4.964 4.131 4.011 3.440 3.315 2.852 2.554 2.262 2.181 1.850 1.702 1.535 1.188 1.331 1.144 1.165 1.051 1.105 0.911 19.261

Overall (wt%) 0.049 0.113 2.620 1.508 2.019 0.419 1.401 0.692 1.108 1.735 2.163 2.622 2.476 2.485 2.280 2.075 2.190 2.039 2.132 1.975 1.888 1.771 1.789 1.587 1.545 1.460 1.178 1.374 1.231 1.305 1.226 1.337 1.142 47.065 161.6

Overall (mol%) 0.283 0.415 26.397 8.103 7.400 1.164 3.896 1.551 2.483 3.254 3.665 3.871 3.281 2.998 2.507 2.083 2.022 1.734 1.673 1.438 1.288 1.140 1.100 0.933 0.858 0.774 0.599 0.671 0.577 0.587 0.530 0.557 0.459 9.711


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Table 3. PVT properties for Crude C2. Property

Value

𝑃bub (psi) at 259 °F

1905

𝜌bub(g/cc) at 259 °F

0.761

Zero-flash GOR (scf/stb)

381

𝜌STO (g/cc) at 60 °F, 14.7 psi

0.9042

𝑀STO(g/mol)

290.3

Saturates (wt%)

49.9

Aromatics (wt%)

19.8

Resins (wt%)

19.2

C5 Asphaltenes (wt%)

11.1

Table 4. Asphaltene fractionation for Crude C2. Asphaltene Fraction

wt%

C5 Asph

11.1

C6 Asph

10.9

C7 Asph

10.1

C8 Asph

8.9

Table 5. PC-SAFT simulation parameters for Crude C2. Component 𝒛𝒊 (mol%) 𝑴(g/mol) H2S N2 CO2 C1 C2 C3 HG Sat Arom Resins Asph1 Asph2 Asph3 Asph4

0.000 0.280 0.410 26.079 7.899 6.805 7.584 36.424 9.069 4.371 0.030 0.115 0.163 0.771

34.08 28.01 44.01 16.04 30.07 44.10 69.54 202.62 322.93 649.61 998.98 1026.71 1085.72 1708

𝒎

𝝈(Å)

1.6517 1.2064 2.0729 1.0000 1.6069 2.0020 2.6315 6.0518 6.8182 12.3600 18.2869 18.7573 19.7583 30.3165

3.0737 3.3130 2.7852 3.7039 3.5206 3.6184 3.7541 3.9211 4.1654 4.2641 4.2989 4.3006 4.3041 4.3261

𝜺⁄ (𝑲) 𝒌𝑩 227.34 90.96 169.21 150.03 191.42 208.11 230.43 252.12 374.78 391.14 395.50 395.69 396.07 398.25


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Table 6. Non-zero binary interaction parameters for Crude C2. H2S

N2

CO2

C1

C2

C3

HG

Sat

H2S N2

0.09

CO2

0.0678

0

C1

0.062

0.03

0.05

C2

0.058

0.04

0.097

C3

0.05

0.06

0.10

0

0

HG

0.07

0.075

0.12

0.01

0.02

0.015

Sat

0.09

0.14

0.13

0.005

0.005

0.005

0

Arom

0.015

0.158

0.18

0.012

0.01

0.01

0.012

0.007

Resin

0.015

0.158

0.18

0.013

0.01

0.007

0.016

-0.025

Asph1

0.015

0.158

0.18

0.0173

0.0144

0.0115

0.0206

-0.021

Asph2

0.015

0.158

0.18

0.0176

0.0147

0.0119

0.0209

-0.021

Asph3

0.015

0.158

0.18

0.0183

0.0155

0.0126

0.0216

-0.020

Asph4

0.015

0.158

0.18

0.0260

0.0231

0.0202

0.0293

-0.012

0

Table 7. Comparison between experimental PVT data and PC-SAFT for Crude C2. APD* (%)

Property

Experimental PC-SAFT

𝐵𝑃 (psi) at 259 °F

1905

1908

0.1

𝜌bub(g/cc) at 259 °F

0.761

0.763

0.3

𝜌STO (g/cc) at 60 °F, 14.7 psi

0.9042

0.8996

0.5

*

APD: Absolute Percent Deviation


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100

Asphaltene Precipitated (wt/wt)%

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nC5 nC6

80

nC7 nC8

60 40 20 0 0

20

40 60 vol% n-alkane

80

100

Figure 2. Indirect Method measurements at 21 oC and 1 atm (markers) along with PC-SAFT modeling results (solid lines). Asphaltene onset concentrations (AOC) measured to be 31 vol% for nC5 and 34 vol% for nC7. Measurements are done with 1-day aging time. An important observation from Figure 2 is that the asphaltene onset concentrations (AOC) for nC5 and nC7 are underpredicted by PC-SAFT. This apparent discrepancy can be explained by considering the effect of aging time on Indirect Method measurements. As discussed by Tavakkoli et al.49, the AOC for a given precipitant is significantly affected by sample aging time. If the asphaltenes are aged longer, they become more unstable and require a lower amount of precipitant to drive asphaltenes out of solution (i.e. lower AOC), up until the point that equilibration occurs and precipitation is no longer a function of aging time. The authors also pointed out that aging time has no pronounced effect on the precipitated amounts at 90 vol% precipitant, suggesting that equilibration is nearly instantaneous at these conditions because of the high driving force. Thus, when driving force for precipitation is low, which is the case at the AOC, aging time has a significant effect on Indirect Method measurements, whereas the effect of aging time is negligible when driving force is high. Because PC-SAFT is a thermodynamic model, the simulation results are expected to be representative of only equilibrium conditions (i.e. infinite aging time). Because of the relatively short aging time for crude oil C2 (1 day) and 14 ACS Paragon Plus Environment

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differences in driving force across the range of precipitant concentrations, it is expected that measurements for 90 vol% precipitant represent equilibrium conditions while the AOC measurement is likely higher than it would be if asphaltenes were given a longer aging time. Thus, the PC-SAFT predictions showing good agreement at 90 vol% precipitant and underprediction of AOC is likely more representative of equilibrium behavior than the Indirect Method measurements performed with low driving force for precipitation. 5. Modeling of Asphaltene Phase Behavior and PVT properties at HPHT Reliable assessment of potential asphaltene deposition problems requires that the thermodynamic model can accurately predict the asphaltene phase behavior and PVT properties of crude oil at production conditions. Crude C2 is produced from a deepwater reservoir with injection of 55 mol% hydrocarbon gas. The composition of injection gas and production conditions for crude oil C2 are shown Table 8. Table 8. Production conditions for Crude C2. Preservoir (psi)

20000

Treservoir (°F)

259

Pwellhead (psi)

6000

Twellhead (°F)

140

Gas injection (mol%)

55

Gas composition H2S N2 CO2 C1 C2 C3 HG

(mol%) 0.00 0.23 0.26 84.59 6.64 5.03 3.25

Experimental HPHT UAOP measurements were performed by the service laboratory for the live oil at 120 °F and 259 °F and with 55 mol% gas injection at 165 °F and 259 °F. The live oil 15 ACS Paragon Plus Environment

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sample was found to be stable at the two studied temperatures as no asphaltene onsets were detected. For the case of 55 mol% injection, the service laboratory reported asphaltene onset pressures at 165 °F and 259 °F. PC-SAFT predictions for the HPHT phase envelope with 55 mol% injection is shown in Figure 3.a in a pressure-temperature (P-T) projection. The phase envelope at reservoir temperature (259 °F) is shown in Figure 3.b in a pressure-composition (Px) projection. The model shows good agreement to experimental AOP for the 55 mol% gas injection measurement performed at 259 °F. Additionally, the model predicts that asphaltenes are stable in the live oil sample, which is observed experimentally (the experimental data point at 0% injection is shown in Figure 3.b). However, the modeling results deviate significantly from the experimental AOP measured at 165 °F. The asphaltene phase envelope predicted by PC-SAFT shows a minimum upper critical solution temperature (MUCST) that is higher than 165 °F. Abutaqiya et al.73 reported that this behavior of the liquid-liquid phase boundary is observed for other oil samples and is a common feature of SAFT-type models. Regardless, the onset experiment performed at 165 °F was not conducted properly. This onset was obtained during sample cooling from reservoir temperature (259 °F) down to the original target temperature of 150 °F at fixed pressure (19,500 psi). During the cooling process, asphaltene particles were observed at 165 °F and this value was reported as the precipitation onset temperature at P = 19,500 psi. This experimental procedure is not the standard technique for measuring precipitation onset as thermal equilibration typically requires substantially more time than mechanical equilibration. For this reason, it is suspected that the reported onset is inaccurate and that the true asphaltene onset would be significantly higher than the reported value. It is worth mentioning, however, that it is possible to match both experimental data points using PC-SAFT if a temperature-dependent asphaltene molecular weight is used. This approach can be justified when


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considering that the molecular weight used in this work is that of an asphaltene aggregate, which can be significantly affected by temperature. Decreasing the molecular weight of asphaltenes has the consistent effect of shifting the asphaltene phase envelope in the bottom-left direction. Therefore, a lower asphaltene molecular weight can be used to capture the UAOP at 165 °F. However, given that the reported onset at 165 °F is unreliable, the modeling results at the lower temperature are not further improved. Also, note that if temperature-dependent asphaltene molecular weight is implemented, care must be taken to ensure that the model accuracy of Indirect Method results is not compromised. 40000

UAOP Exp UAOP PC-SAFT BP PC-SAFT P-T Trace

(a) + 55% injection

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Energy & Fuels

BP PC-SAFT

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Figure 3. Asphaltene phase envelope for crude oil C2 (a) P-T projection at 55 mol% gas injection, (b) P-x projection at reservoir temperature 259 °F. 17 ACS Paragon Plus Environment

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Although modeling asphaltene precipitation onsets is important for understanding the thermodynamic conditions at which asphaltenes are unstable, it is insufficient for the purpose of investigating asphaltene deposition. Asphaltene deposition is a complex multistep phenomenon that is related to various thermodynamic and rheological properties of the co-existing phases. The thermodynamic properties most relevant to deposition modeling include densities, compositions, phase amounts, and component partitioning between the precipitated and continuous phases as a function of the operating conditions. For the sake of demonstration, these properties are investigated as a function of pressure at fixed reservoir temperature. In this paper, the “bulk phase” is defined as that phase which is stable above the UAOP and the “onset phase” is that which forms at the UAOP. Figure 4 shows variations of phase fraction and phase density of the bulk and onset phases as a function of pressure for the case of C2 with 55% injection at reservoir temperature. The plots in Figure 4 simulate equilibrium fluid depressurization, moving from right-to-left, at fixed temperature, and the depressurization range is chosen such that the starting pressure is above the UAOP and the final pressure is below the LAOP. As shown in Figure 4.a, the fluid is a single liquid phase (L1) at high pressures. As pressure decreases, the first vertical dashed line is crossed, which corresponds to the UAOP. At the UAOP, the singlephase fluid splits into two liquid phases, a bulk phase (L1) that is lean in asphaltenes and a newly-formed liquid phase (L2) that is rich in asphaltenes. The asphaltene-rich onset phase is denser than the bulk phase as shown in Figure 4.b. As pressure decreases from the UAOP, more heavy fractions transfer from the bulk to the onset phase, causing the density of the bulk phase to decrease due to both volume expansion and composition changes. Interestingly, a significant amount of the asphaltene-rich phase is formed during depressurization to the point where it exceeds the amount of the bulk phase (below 9200 psi). This implies that not only do asphaltenes


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partition strongly to the onset phase, but also significant amounts of resins and aromatics coprecipitate. The unusually high precipitation amount predicted by PC-SAFT is caused by the high content of resins and asphaltenes present in the oil, the large driving force for precipitation induced by gas injection, and high operating pressure.

Phase Fraction (wt%)

100

L₁ L₂

(a)

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

0.8 0.7 0.6 0.5 0.4 0

5000


Figure 4. Variation of (a) phase fraction and (b) density for the bulk (L1) and onset (L2) phases during depressurization of C2 with 55 mol% gas injection at 250 °F. Dashed vertical lines denote saturation pressures (from right-to-left: UAOP, BP, LAOP).


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As the pressure is further decreased, the oil reaches the bubble point and a gas phase begins forming. Three phases exist between this pressure and the pressure corresponding to last vertical dashed line (LAOP). As gases liberate from L1, its density sharply increases as shown in Figure 4.b. At the LAOP, sufficient amounts of the light components have liberated from phase L1 that it becomes soluble in phase L2, causing the phases to combine and leaving one liquid and one gas phase below the LAOP. For most crude oils, the mass of the onset phase in the asphaltene instability region remains small relative to the bulk phase. In these cases, the bulk phase completely solubilizes the onset phase both above the UAOP and below the LAOP. However, for the 55% injection case shown in Figure 4, the bulk phase solubilizes the onset phase above the UAOP whereas the onset phase solubilizes the bulk phase below the LAOP. At all pressures below the LAOP, two phases exist: one liquid and one gas phase. As mentioned earlier, the large amount of precipitated asphaltene-rich phase implies that components other than asphaltenes are precipitating in significant proportions. Figure 5.a shows precipitation curves for each pseudo-component during the depressurization. The precipitation curves are expressed as the mass of each component in the newly-formed (onset) phase to the total mass of that component present in the existing liquid phases. For example, the precipitated Asphaltene fraction in Figure 5.a at T and P is calculated from:

Asph Precipitated (wt⁄wt)% = 100 ∗

wt. Asph in L2 wt. Asph in L1 + wt. Asph in L2

(3)


Page 21 of 39

Onset phase / Total liq (wt/wt)%

100

Asph (L₂) Resin (L₂) Arom (L₂) Sat (L₂)

(a)

80 60 40 20 0 0

100

Composition of Asph (wt%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

5000


15000

(b)

20000

Asph (L₁) Asph (L₂)

80 60 40 20 0 0

5000


15000

20000

Figure 5. (a) Ratio of the mass of each component in onset phase to its total mass in existing liquid phases and (b) composition of Asphaltenes in bulk and onset phases. Results are for depressurization of C2 at 250 °F and 55 mol% gas injection. Dashed vertical lines denote saturation pressures (from right-to-left: UAOP, BP, LAOP). At pressures above the UAOP, no onset phase exists and the precipitated fraction is zero for all components. As pressure decreases below the UAOP, asphaltenes partition most strongly into the onset phase, followed by resins, aromatics, and saturates. Below the UAOP but far from the bubble point, PC-SAFT predicts that nearly all asphaltenes precipitate out of the bulk phase. At the bubble point, almost 99 wt% of resins have also precipitated out, along with 90 wt% of 21 ACS Paragon Plus Environment

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aromatics and 47 wt% of saturates. These observations are indicative of the large driving force for precipitation in C2 induced by the high amount of gas injection. Below the bubble pressure, as described earlier, the bulk phase is solubilized by the onset phase. The precipitated fractions shown in Figure 5.a converge to 100 wt% at the LAOP because the onset phase is the only liquid phase that remains below the LAOP. It is worth mentioning that if the original approach of Tavakkoli et al.49 was used to characterize crude oil C2 (i.e. lumping aromatics and resins into a single fraction) it would not be possible to quantify the different partitioning of the aromatics and resins fractions. Because the driving force for precipitation is large and the heavy non-asphaltene fractions co-precipitate strongly with asphaltenes, more resolution is required to accurately capture component partitioning for the heavy fractions across the asphaltene instability region. Figure 5.b shows the variation of asphaltene composition in the bulk and onset phases as a function of pressure. The overall composition of asphaltenes in the system is 8.7 wt%, which is the value shown in phase L1 in Figure 5.b for pressures above the UAOP. At the UAOP, the newly-formed (onset) phase contains 23.6 wt% asphaltenes. Relative to the bulk phase, the onset phase is considered asphaltene-rich, although 23.6 wt% is a relatively low concentration compared to a typical asphaltene-rich phase50,74. As discussed before, nearly all asphaltenes partition out of the bulk phase far above the bubble pressure, followed by significant amounts of the other pseudo-components. Even though asphaltenes continue migrating to the onset phase below the UAOP, the asphaltene composition in the onset phase decreases with pressure because the other heavy components are also partitioning into the onset phase, which dilutes the asphaltene concentration.


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There is experimental evidence suggesting that the decreasing composition of asphaltenes in the onset phase predicted by PC-SAFT is observed in practice. The laboratory that performed experiments on C2 at 259 °F with 55% injection captured images of the oil sample during the depressurization experiment. Those images at pressures of 12868 psi and 9908 psi are shown in Figure 6 along with the PC-SAFT predictions of asphaltene weight composition in the onset phase.

100

Asph (L₁)

60 40

P 1 = 12868 psi

Asph (L₂)

80

20 0 3000

P2 = 9908 psi

PC-SAFT at 259 F

P 2 = 9908 psi

Composition of Asph (wt%)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

5000

7000 9000 11000 Pressure (psi)

13000

15000

P1 = 12868 psi

Figure 6. High pressure microscopy (HPM) images of crude oil C2 captured during depressurization experiments. The experiment was performed at 259 °F with 55 mol% gas injection. Corresponding modeling results using PC-SAFT are also shown.


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Predictions from PC-SAFT show that the asphaltene composition in the onset phase decreases during depressurization, going from 24 wt% at 12868 psi to 18 wt% at 9908 psi. In the HPM image at higher pressure, which is near the UAOP, the precipitated asphaltene particles look more rigid and solid-like, while at the lower pressure, the asphaltene particles are softer and more liquid-like, implying a decrease in asphaltene composition in the onset phase as predicted by PC-SAFT. Additionally, the brightness of the continuous phase shown in Figure 6 significantly increases at the lower pressure, indicating that a large fraction of the heavier components have precipitated out. PC-SAFT modeling results in Figure 5.a show that nearly 60 wt% resins and 90 wt% asphaltenes precipitate out of the bulk phase at P = 9908 psi. 6. Sensitivity analysis In this section the assumption of polydisperse asphaltenes is analyzed and compared to the monodisperse approach. Additionally, a sensitivity analysis is performed to understand the ̅ ) on the modeling results of Indirect influence of the gamma distribution parameters (𝛼 and 𝑀 Method. 6.1. Monodisperse vs polydisperse asphaltenes The assumption of a monodisperse or a polydisperse asphaltene fraction has varying implications on the simulation results and the appropriate choice depends largely on the objective of the modeling study. Figure 7.a shows a comparison of Indirect Method simulation results using the monodisperse (solid line) and polydisperse (dashed line) characterization approaches. Figure 7.b shows a comparison for modeling the HPHT asphaltene phase envelope and bubble curve. The monodisperse modeling results shown in Figure 7 are obtained by lumping the four asphaltene fractions generated as part of the newly proposed polydisperse asphaltene characterization 24 ACS Paragon Plus Environment

Page 25 of 39

procedure into a single fraction with the corresponding number-average molecular weight (1522 g/mol). resins are kept as a separate pseudo-fraction and no re-tuning of simulation parameters is performed.


100

(a)

80 60

40 nC5 nC6

20

nC7

nC8

0 0

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40 60 vol% n-alkane

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100

40000 UAOP Exp

(b)

UAOP PC-SAFT BP PC-SAFT


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Energy & Fuels

20000

10000

0 50

150

250 Temperature ( F)

350

450

Figure 7. Comparison between modeling results for Crude C2 using polydisperse (dashed line) and monodisperse (solid line) asphaltene characterization for (a) Indirect Method at 21 °C and (b) HPHT asphaltene phase envelop with 55 mol% injection. The BP curves in (b) for monodisperse and polydisperse approaches are overlapping.


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Page 26 of 39

From observation of Figure 7.a, PC-SAFT predicts that a higher concentration of precipitant is required to induce precipitation when asphaltenes are assumed monodisperse. This result is expected as the onset of precipitation is driven mainly by the heaviest asphaltene fraction. The heaviest fraction with the polydisperse approach (Asph4) has a molecular weight of 1708 g/mol whereas the heaviest fraction with the monodisperse approach has a molecular weight of 1522 g/mol. This implies that asphaltenes are more stable in solution when characterized as monodisperse, requiring a higher amount of precipitant to reach the onset of precipitation, which agrees with the modeling results. The amounts of precipitated asphaltenes as a function of precipitant volume are significantly overpredicted when a monodisperse approach is used. Because the monodisperse approach assumes that all asphaltenes partition equivalently, the precipitated amounts at 90 vol% precipitant shows a much wider range of values than what is experimentally observed (Figure 7.a). Even if the model parameters are re-tuned (𝛾 and 𝑀), the model is still not capable of reproducing the experimental data under the monodisperse asphaltene assumption. Incorporating asphaltene polydispersity is necessary to accurately predict precipitation amounts, which is important for the deposition study. The predictions of asphaltene onset pressure (AOP) for the live oil with 55% gas injection is shown in Figure 7.b using both the monodisperse and polydisperse approaches. As mentioned before, treating asphaltenes as a monodisperse fraction results in a more stable asphaltene as compared to a polydisperse fraction with the same average molecular weight. Therefore, the predicted asphaltene onset pressure at each temperature is consistently lower using a monodisperse approach, with a difference of approximately 1000 psi for AOP at reservoir temperature (259 °F). The predictions of bubble pressure, however, are very similar for the 26 ACS Paragon Plus Environment

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Energy & Fuels

monodisperse and polydisperse approaches because bubble point is mainly driven by the light fractions. Interestingly, the model still shows a minimum upper critical solution temperature (MUCST) that is higher than the temperature of the second UAOP. This further supports the statement that the MUCST behavior is a result of PC-SAFT and not the characterization procedure73. ̅ on Indirect Method modeling 6.2. Effect of 𝛼 and 𝑀 ̅ ) are studied for their effect on the modeling The parameters of the gamma distribution (𝛼 and 𝑀 results of the Indirect Method experiment. Figure 8 shows the effect of the shape parameter 𝛼 ̅ of the distribution. The while Figure 9 shows the effect of the average molecular weight 𝑀 dashed lines on the gamma distribution curves in Figure 8 and Figure 9 represent the molecular weight boundaries of the different cuts: Resins, Asph1, Asph2, Asph3, and Asph4. From observation of Figure 8, higher values of the shape parameter 𝛼 yields predictions of lower precipitated asphaltenes at 90 vol% for all precipitants. The effect is more pronounced with weaker precipitants (i.e. nC8) as compared to stronger precipitants (i.e. nC5). Additionally, the asphaltene onset concentration (AOC) consistently shifts to higher values for all precipitants as 𝛼 increases.


Energy & Fuels

100 Asphaltene Precipitated (wt/wt)%

Probability density function

𝛼 = 1.50 𝑀𝑚 = 400 𝑔/𝑚𝑜𝑙 ̅ = 989.3 𝑔/𝑚𝑜𝑙 𝑀

nC₅ nC₆

80

nC₇ nC₈

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𝛼 = 3.00 𝑀𝑚 = 400 𝑔/𝑚𝑜𝑙 ̅ = 989.3 𝑔/𝑚𝑜𝑙 𝑀

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𝛼 = 6.00 𝑀𝑚 = 400 𝑔/𝑚𝑜𝑙 ̅ = 989.3 𝑔/𝑚𝑜𝑙 𝑀

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Page 28 of 39

nC₅ nC₆

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Figure 8. Effect of 𝛼 on modeling results of Indirect Method. Left side shows the gamma distribution functions and the right side shows modeling results for Indirect Method corresponding to each distribution. Dashed lines on left side represent the boundaries of different ̅= cuts. Dashed lines in the right side correspond to optimized modeling results (𝛼 = 1.01 and 𝑀 989.3 𝑔/𝑚𝑜𝑙). The observed phenomena occur because an increasing 𝛼 yields a function shape that approaches ̅ . As 𝛼 increases for a fixed value of 𝑀 ̅ , the molecular a normal distribution centered around 𝑀 ̅ , yielding a lower weights of the lightest and heaviest fractions approach the value of 𝑀 molecular weight for the heaviest fraction than for lower values of 𝛼. Because the onset of precipitation is driven primarily by the heaviest fraction, more precipitant is required to induce


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Energy & Fuels

phase-splitting for higher values of 𝛼 and an increase in AOC is observed. The same mechanism can be used to explain the trends in precipitated amounts at 90 vol% precipitant because the heaviest asphaltene fraction constitutes the majority of asphaltenes in the distribution (~80 wt% excluding resins). The effect on precipitated amounts using the weaker asphaltene precipitants is more pronounced because these precipitants do not co-precipitate lighter asphaltene fractions as strongly. Because the molecular weight of the heaviest fraction is decreasing with increasing 𝛼, predictions for precipitated amounts using weaker precipitants are markedly affected. For the strongest precipitant (nC5), the precipitated amount is not significantly affected by changing 𝛼. This is seemingly because the average molecular weight of the distribution is fixed to a value large enough that nC5 can induce almost complete precipitation of the asphaltenes regardless of how the fractions are distributed. It is worth re-emphasizing that the AOC using nC5 is still affected by the shape parameter 𝛼 because AOC is driven primarily by the heaviest asphaltene fraction, not the average properties of the distribution. The sensitivities of the modeling results of the Indirect Method to changes in average molecular ̅ are shown in Figure 9. Increasing 𝑀 ̅ causes an increase in the predicted asphaltene weight 𝑀 precipitation amounts and a decrease in the predicted AOC for all precipitants, as higher molecular weights correspond to more unstable asphaltenes. The precipitation curves for ̅ because the asphaltene fractions are so different precipitants become similar for higher 𝑀 unstable that even a weak precipitant can force significant amounts out of solution. For lower molecular weight asphaltenes, the precipitation curves are more distinguishable and further apart. ̅ = 800 g/mol, no asphaltenes precipitate out of solution using nC8 as a precipitant. In fact, for 𝑀 The same observations also hold for AOC trends. It is worth mentioning that, regardless of the 𝛼 ̅ values used, the shape of the precipitation curves predicted by PC-SAFT are qualitatively and 𝑀 29 ACS Paragon Plus Environment

Energy & Fuels

similar: the precipitated amounts increase sharply after the onset then reach a limiting value or, in some cases, curve back down. This behavior has been experimentally reported by Tavakkoli et al.49 when asphaltenes are aged for a very long time. 100



𝛼 = 1.01 𝑀𝑚 = 400 𝑔/𝑚𝑜𝑙 ̅ = 800.0 𝑔/𝑚𝑜𝑙 𝑀

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𝛼 = 1.01 𝑀𝑚 = 400 𝑔/𝑚𝑜𝑙 ̅ = 900.0 𝑔/𝑚𝑜𝑙 𝑀

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𝛼 = 1.01 𝑀𝑚 = 400 𝑔/𝑚𝑜𝑙 ̅ = 1100.0 𝑔/𝑚𝑜𝑙 𝑀

0

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nC₅ nC₆

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nC₇ nC₈

60 40 20 0

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̅ on modeling results of Indirect Method. Left side shows the gamma Figure 9. Effect of 𝑀 distribution functions and the right side shows modeling results for Indirect Method corresponding to each distribution. Dashed lines on left side represent the boundaries of different ̅= cuts. Dashed lines in the right side correspond to optimized modeling results (𝛼 = 1.01 and 𝑀 989.3 𝑔/𝑚𝑜𝑙).


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Energy & Fuels

7. Conclusions and Future Work PC-SAFT modeling of a high-asphaltene high-resin content crude oil is performed using a polydisperse asphaltene characterization method. In this work, asphaltenes are characterized as four pseudo-fractions with weight composition obtained by n-alkane titration experiments. The molecular weights of the asphaltene pseudo-fractions are obtained using a 3-parameter gamma distribution function. In a previous work with polydisperse asphaltenes, resins were lumped with aromatics to form a single pseudo-fraction. In this work, resins are included in the asphaltene mass distribution and are assumed to be the lightest cut of the asphaltene fraction. From the mass distribution and measured resins weight content, the molecular weight of resins can be obtained, which allows for it to be treated as a separate pseudo-fraction within the modeling framework. In the methodology proposed in this work, three model parameters are required: aromaticity (𝛾), ̅ ), and distribution shape factor average molecular weight of the resins+asphaltene distribution (𝑀 (𝛼). These parameters are simultaneously optimized to match experimental bubble pressure of the live oil, density at saturation, stock tank oil density, and amounts of precipitated asphaltenes at 90 vol% of nC5, nC6, nC7, and nC8 precipitants obtained from the Indirect Method experiment. The optimized PC-SAFT parameters yielded simulation results that showed good agreement to PVT and asphaltene precipitation data. Using the optimized model parameters, several thermodynamic properties are investigated by simulated depressurization of crude oil C2 with 55 mol% injection at reservoir temperature. The model results indicate that a significant amount of asphaltene-rich phase precipitates within the asphaltene instability region. In fact, the precipitated phase exceeds the amount of the bulk phase above the bubble point and the system becomes dominated by the asphaltene-rich phase near the


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bubble point. At pressures below the bubble point – where light components are liberated to a gas phase – it is found that the original bulk phase dissolves into the asphaltene-rich phase. This interesting behavior is a result of the high driving force for precipitation induced by lean gas injection and the high content of resins and asphaltenes in the crude under investigation. Analysis of the percentage of precipitated fractions during depressurization indicate that all asphaltenes precipitate out well-before the bubble point, followed by resins, aromatics, and saturates. At the bubble point, almost 99 wt% of resins have precipitated out along with 90 wt% of aromatics and 47 wt% of saturates. Because all asphaltenes precipitate out before the bubble point, detailed analysis of the next pseudo-fraction lighter than asphaltenes (i.e. resins) is necessary in this case. This justifies de-lumping resins from aromatics and including them as part of the asphaltene distribution. PC-SAFT predicts that the asphaltene-rich phase formed at the upper onset pressure contains 23.6 wt% asphaltenes, as compared to 8.7 wt% of asphaltenes in the bulk phase. As pressure decreases, and due to the precipitation of significant amounts of other pseudo-fractions, the composition of asphaltenes in the asphaltene-rich phase decreases. High-pressure microscopy images show that the asphaltene-rich phase turns from rigid and solid-like near the onset to soft and liquid-like at lower pressure, which agrees with modeling results from PC-SAFT. 8. Acknowledgement The authors are thankful for the financial support provided by Chevron Energy Technology Company. The authors would also like to thank Mohammad Tavakkoli for fruitful discussions and valuable insight.


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9. Acronyms and Nomenclature

A+R

Aromatics + Resins

AOC

Asphaltene Onset Concentration

AOP

Asphaltene Onset Pressure

BP

Bubble Pressure

CPA

Cubic Plus Chain

EOS

Equation of State

GOR

Gas-to-Oil Ratio

HPHT

High-Pressure High-Temperature

m

number of segments per molecule

𝑀

Molecular Weight

𝑀𝑚

Monomer Molecular Weight

̅ 𝑀

Average Molecular Weight of the Distribution

PC-SAFT

Perturbed-Chain Statistical Associating Fluid Theory

PNA

Polynuclear Aromatics

PVT

Pressure Volume Temperature

SARA

Saturates Aromatics Resins Asphaltenes

STO

Stock Tank Oil

UAOP

Upper Asphaltenes Onset Pressure

zi

Mole fraction of component i

Greek Symbols 𝜀/𝑘𝐵

Segment-segment interaction energy

𝜌

Density

𝜎

Segment diameter

𝛾

Aromaticity factor


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Systematic Investigation of Asphaltene Deposition in the Wellbore and

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