Improvement of the Expanded Fluid Viscosity Model for Crude Oils

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Improvement of the Expanded Fluid Viscosity Model for Crude Oils: Effects of the Plus Fraction Characterization Methods and Density Victor Bouzas Regueira, Verônica Jesus Pereira, Gloria Meyberg Nunes Costa, and Sílvio Alexandre Beisl Vieira de Melo Energy Fuels, Just Accepted Manuscript • DOI: 10.1021/acs.energyfuels.7b03735 • Publication Date (Web): 22 Jan 2018 Downloaded from http://pubs.acs.org on January 22, 2018

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Improvement of the Expanded Fluid Viscosity Model for Crude Oils: Effects of the Plus Fraction Characterization Methods and Density Victor B. Regueira1, Verônica J. Pereira1, Gloria M.N. Costa1, Silvio A.B. Vieira de Melo1,2* 1

Programa de Engenharia Industrial, Escola Politécnica, Universidade Federal da Bahia, Rua Prof. Aristides Novis, 2, 6º. andar, Federação, 40210-630, Salvador, Bahia, Brasil. 2 Centro Interdisciplinar em Energia e Ambiente, Campus Universitário da Federação/Ondina, Universidade Federal da Bahia, 40170-115, Salvador, Bahia, Brasil.

ABSTRACT: The expanded fluid (EF) model is known for its capacity to calculate the viscosity of crude oil and its mixtures with solvents at high pressure and temperature, using a cubic equation of state (CEOS) with a minimum amount of experimental data. The main drawback of the EF model is usually the requirement of a proper plus-fraction characterization and accurate density input data. In this study, the sensitivity of the EF model related to the characterization methods and the density of the oils was evaluated against viscosity data of reservoir fluids. Oil viscosity was calculated above and below the saturation pressure in order to compare the EF model with other viscosity models available in the literature. The results confirmed that viscosity strongly depends on the quality of density predictions as well as on a good description of phase behavior below the saturation pressure. This demonstrates that a proper characterization is needed to calculate the oil viscosity accurately. Furthermore, a new tuning method with minimum experimental data improved the viscosity prediction as a function of pressure.

Keywords: density, viscosity model, cubic equation of state, characterization, crude oil

*

Corresponding author: [email protected] Tel. +55-71-32839802 Fax +55-71-32839800

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1. INTRODUCTION Proper estimates of the transport properties of reservoir fluids are the cornerstone of many projects in the oil and gas industry. For instance, the design and assessment of oil extraction, pipeline transportation or reservoir simulation requires a good knowledge of the viscosity of the oil in different conditions. Viscosity monitoring and control is crucial to achieve better reservoir profitability, as errors can lead to incorrect production rates1. Many methods can be used to measure the viscosity2, such as modern electromagnetic viscometers. However, laboratory measurements are expensive and tiresome, mainly when data is needed over a wide range of temperatures and pressures. In this case, models are used to overcome the experimental limitations. Models based only on theoretical approaches are most used for simple hydrocarbon gases, such as those from the Chapman-Enskog theory3. In addition, empirical correlations are commonly used to model the liquid viscosity due to their simplicity and relatively good accuracy within small temperature and pressure ranges4–6. The most known are based on Lohrenz-Bay-Clark (LBC)7, Little8, Andrade9, Walther6,10–12 or even the use of Artificial Intelligence techniques13,14. However, in most cases, theoretical models are better capable of predicting viscosity in a wider range of conditions than empirical correlations. In the half term, semi-empirical models adopt a theoretical framework and use experimental data in their development and further adjustment. Specific parameters are usually estimated for each component and the fluid parameters are calculated from mixing rules, used to determine the fluid viscosity15. One example is the friction theory (f-theory) model, which expresses the viscosity as the sum of a dilute gas viscosity and a residual friction term and uses at least one viscosity data to estimate one adjustable parameter in the equation16,17. Another semi-empirical


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method was developed by Pedersen18 and colleagues based on the principle of the corresponding states (CSP) and further improved by Lindeloff and colleagues19. In addition, an analogous correlation to a cubic equation of state was proposed by Guo and colleagues20 to calculate the viscosity and thermal conductivity of hydrocarbons and reservoir fluids. Very recently, Bonyadi and Rostami21 created a similar model based on SRK22 (Soave-Redlich-Kwong) EOS. However, applications of this last type of model21,23 are limited, they were developed for light reservoir fluids and their predictive accuracy with heavy oils has yet to be thoroughly tested24. The Expanded Fluid (EF) model developed by Yarranton and Satyro25,26 is based on the free volume theory, which states that as the fluid expands, its fluidity increases due to the greater distance between the molecules. The EF model was used to predict the viscosity of heavy oils and bitumen, which was a gap in the literature models. This model has been already tested for process simulation for many systems27. In the present study in particular, the EF model was tested, analyzed and extended for a wide range of lighter oils. It is important to highlight that some authors15,28 have already stated that the EF model exhibits a strong dependence on density predictions. As density experimental data is not always available, an equation of state (EOS) can be used to calculate the density input data required for the EF model. Cubic EOS is well known for its good capacity to describe phase equilibrium, but it fails to accurately calculate volumetric properties. Regarding the improvements to cubic EOS to calculate volumetric properties, Young and colleagues29 concluded that the best approach is the use of volume translation functions instead of more complex techniques such as modifying the co-volume functional form or complex mixing rules. Although a linear temperature-dependent volume correction was used by Motahhari and colleagues28 to improve the viscosity prediction using the EF model, the effect of density inputs were not deeply investigated. In the present


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work, other different volume-translation approaches are tested in order to further improve the viscosity predictions and to check if experimental density data is really required to obtain better results. Besides the density effect, viscosity prediction using the EF model is also affected by the plusfraction characterization method. Although the EF model has been already extended for characterized oils and EOS-predicted density28, the effect of the characterization methods on viscosity has not been evaluated in depth yet. Motahhari and colleagues28 also highlighted the high impact of oil characterization on viscosity predictions with the EF model. They used an exponential distribution with a single set of correlations. However, the choice of distribution to characterize the oil plus-fraction has a significant impact on the description of phase and volumetric behaviors30–32 and this issue should be well addressed. Despite this, we have observed in the literature that many studies neglect the effect of the characterization method on viscosity prediction without knowing that it really can lead to large errors20,33. The present study aims to better predict heavy and light oil viscosity through an improvement in the Expanded Fluid model. The effects of several volume-translation functions34,35 and characterization methods on viscosity prediction are compared in order to understand if density experimental data is really needed as an input and if a poor characterization can be offset by a further parameter estimation of the EF model. Finally, the whole viscosity curve for lighter oils, above and below the saturation pressure, is predicted by the EF model for the first time.

2. VISCOSITY MODEL This viscosity model investigated in this study is based on the approach developed by Motahhari and colleagues28, who extended the original EF model to characterized crude oils.


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These authors used an exponential distribution with a fixed number of pseudocomponents to characterize the oils, followed by Lee-Kesler36 and Kesler-Lee37 correlations to calculate the critical properties of pseudocomponents and the acentric factor. Equations 1 to 3 are used to calculate the viscosity of the fluid (μ), where μo is the dilute gas viscosity, ρ is the fluid density, ρs* is the density of the fluid in the compressed state, and c2 is a fitting parameter specific for each fluid. The dilute gas viscosity of components and pseudocomponents are calculated by empirical correlations3.

 -  o = 0.165(exp( c 2  ) - 1) 

(1)

1   *  0.65    exp  s   1  1    

 s* 

(2)

 so

(3)

exp(c3 P)

The EF model requires the fluid density (ρ) as a function of pressure and the dilute gas viscosity (μo), which are both temperature-dependent. For multicomponent fluids, prediction methods are used to calculate pure component specific properties (c2, c3 and ρso), which are temperatureindependent variables, along with the appropriate mixing rules. A simplified scheme to calculate the oil viscosity using improved EF model, including all experimental data needed, is presented in Figure 1. A more detailed explanation of how to apply the EF model to characterized oils can be found in Appendix I. Other conventional viscosity models such as LBC7, Little8, Lohrenz7, Pedersen CSP18, f-theory model30 were also tested in the present work and more details can be found elsewhere.


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Figure 1. Scheme for the calculation of oil viscosity proposed in the present study.

3. THERMODYNAMIC MODEL In order to perform the calculation of viscosity without density experimental data, it is necessary to use an EOS to predict it. An EOS is also used to calculate the liquid phase


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composition below the saturation pressure. In this work, the Soave-Redlich-Kwong (SRK)22 EOS is used (Equation 4). In this equation, P is the pressure, T is the temperature, R is the gas universal constant, v is the molar volume, a is the temperature-dependent attraction parameter and b is the covolume parameter. P

RT a(T )  v  b v(v  b)

(4)

Both parameters, a and b, are mixture specific and can be determined by the following Van der Waals mixing-rules: n

n

i

j

(5)

a   xi x j (ai a j ) 0.5 (1  k ij ) n

b   xi bi

(6)

i

where n is the number of components, xi and xj are the molar fraction of component i and j, kij is the binary interaction parameter between the components i and j, ai and bi are the pure component parameters obtained from the critical temperature (Tci), critical pressure (Pci) and acentric factor (ωi) of component i. kij between methane and the heaviest pseudocomponent is fitted to bubble pressure experimental data available. kij values for the SRK EOS were taken from literature38 for N2, CO2 and H2S, as shown in Table 1. Thus, only the kij between methane and the heaviest pseudocomponent was estimated to fit the model to the crude oil bubble pressure curve.


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Table 1. Initial non-zero binary interaction parameters N2

CO2

H2S

N2

0

0

0.12

CO2

0

0

0.135

C1

0.02

0.12

0.08

C2

0.06

0.15

0.07

C3

0.08

0.15

0.07

C4 and C5

0.08

0.15

0.06

C6

0.08

0.15

0.05

C7+

0.08

0.15

0.03

3.1. VOLUME-TRANSLATION METHODS For simplicity, efficiency and thermodynamic consistency29, we selected various approaches for comparison in order to improve density prediction, which strongly affects the viscosity calculated by the EF model15,28. These approaches are summarized in Table 2 and do not affect the description of gas-liquid phase equilibrium39.The corrected volume (VCOR) is obtained based on a volume translation in the calculated volume (VEOS) of the EOS, given by Equation 7, where the parameter c is calculated through a linear mixing rule, as expressed in Equation 8.

(7)

VCOR  VCEOS  c n

c   xi ci

(8)

i

As indicated in Table 1, the method M1 used the temperature independent volume translation proposed by Peneloux and requires the critical temperature (Tc), the critical pressure (Pc) and the


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acentric factor (ω) of each component, the molecular weight (MPSC), the density (ρPSC) and the molar volume of each specific pseudocomponent (VPSC), in the same conditions, using the SRK EOS. The method M2 is very similar to M1, but it also takes into account the thermal expansion effect. For this reason, it inserts a linear temperature dependent volume translation for pseudocomponents, as shown in Equation 9. The values of γ0 and γ1 for the defined components are calculated using SRK EOS, as described in the original method28, and the pseudocomponents are individually calculated for each oil using the liquid density at two temperatures (288.75 K and 423.15 K).

ci   0,i   1,i (T  288.75)

(9)

The method M3, which considers a high-temperature high-pressure volume-translation, can be applied only to defined components and the volume correction for the pseudocomponents is done as in method M2. Method M3 is based on a recently developed volume correction method with a strong thermodynamic consistency34 as it does not allow isotherm crossovers. The volume correction term uses the same inputs as methods M1 and M2 and is expressed as a linear function of the reduced temperature (Tr), given by Equation 10. The parameters A and B are given by a correlation of the molecular weight and the acentric factor of each component, as found in the original article34, and for mixtures they are given by a linear mixing rule.

ci  A  BTr

(10)


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In method M4, the volume translation of the defined components are determined as in method M1 but the pseudocomponents volume translation are adjusted with experimental data using a multiplier term (Mi), as expressed in Equation 11,where i represents the pseudocomponent, and ciN is the new volume correction term40. We selected the density experimental data of dead oils at either the atmospheric pressure or the saturation pressure to tune the parameter M.

ciN  ci  M i

(11)

Table 2. Volume-translation methods used in this work to improve density predictions Symbol Method description M1

Temperature independent volume translation proposed by Peneloux40,41

M2

Inclusion of a temperature dependent volume translation for pseudocomponents39

M3

High-temperature high-pressure volume-translation (HTHP VT-SRK)34,42

M4

Further adjustment of pseudocomponents volume translation with experimental data39

3.2. PLUS-FRACTION CHARACTERIZATION The plus-fraction characterization can have a large impact on the calculation of density and viscosity. To characterize the oils according to the original approach28 of the EF model, only Gamma38 and Soreide43 correlations are used to split the plus-fraction while Lee-Kesler36 and Kesler-Lee37 correlations are used to calculate the pseudocomponents critical temperature, critical pressure and acentric factor. However, Mottahari et al.28 did not study the effect of the number of pseudocomponents on the viscosity predictions. The present works seeks to investigate the effects of different characterization methods on the calculation of density and


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viscosity. The characterization schemes and their respective nomenclature are summarized in Table 3, and consist of the following steps: a) The first step is the splitting of the plus fraction, The Exponential40 and Gamma38 distributions are used with the plus fraction grouped as C7+ to make fair comparisons. The Exponential40 procedure used a logarithmic distribution for the SG distribution and the Soreide43 correlation was used always after the Gamma Distribution to estimate the Tb and SG distribution, similar to the original work. b) The lumping step is not critical for the present work because the calculation time is negligible. Thus, we grouped the plus fraction in a determined number of pseudocomponents based only on the equality of mass fractions and maintained the other components ungrouped. However, in some cases we changed the number of pseudocomponents to check its effect on the calculation of density and viscosity. c) The third step is the use of empirical correlations to calculate the pseudocomponents critical parameters and acentric factor. There are several correlations in the literature, but we selected only the most common and efficient ones, namely, i.e. Kesler-Lee37 and Cavett44 for the critical parameters, Lee-Kesler36 and Korsten45 for the acentric factor. The Exponential distribution was evaluated only with Pedersen’s correlation40 and its modifications for the heavier oils46. Other correlations were varied in combination with the Gamma distribution. Also, the Lee-Kesler correlation36 was used only when Tb/Tc > 0.8, while for lower values we used the Kesler-Lee correlation37 for the acentric factor.


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Table 3. Characterization schemes used in this work

Splitting scheme for MW and SG

Symbol C1 C2* C3 C4 C5

Correlations Critical parameters Acentric factor (Tc and Pc) (ω)

Exponential Pedersen40 Pedersen40 (Pedersen40) Gamma Kesler-Lee37 Lee-Kesler36 (Whitson38 and Soreide43) Gamma Kesler-Lee37 Korsten45 (Whitson38 and Soreide43) Gamma Cavett44 Korsten45 (Whitson38 and Soreide43) Exponential Pedersen modification Pedersen modification (Pedersen46) for heavy oils46 for heavy oils46 * Characterization used in the original approach by Mottahari et al.28

4. RESULTS 4.1. DENSITY PREDICTION In order to predict accurate values of liquid density, the volume translation methods given in section 3.1 were evaluated and compared. Note that M0 means no volume translation. Oils with different molecular weights were selected from the literature, as shown in Table 4, where Tmeas represents the temperature at which the data was measured and Pb is the oil saturation pressure at Tmeas. Oil density curves obtained from differential liberation tests were used to perform a preliminary study on the impact of the characterization and volume-translation methods. It is worth noting again that the kij between methane and pseudocomponents should be adjusted to fit SRK EOS to the bubble pressure experimental data.


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Table 4. Properties of the selected oils used to calculate the density. Components

O147

N2 CO2 H2 S C1 C2 C3 i-C4 n-C4 i-C5 n-C5 C6 C7+ MWC7+ (g/gmol) ρC7+ (g/cm3) Pb (atm) Tmeas (K)

0.25 3.60 2.32 47.64 6.50 4.50 1.17 2.87 1.40 1.90 2.74 25.12 212.7 0.830 269.1 394

O248 Composition (% mol) 0.16 0.91 0.00 36.47 9.67 6.95 1.44 3.93 1.44 1.41 4.33 33.29 218.0 0.850 179.3 378

O339 1.05 0.06 0.00 30.50 0.03 0.02 0.01 0.04 0.03 0.05 0.33 67.88 450.0 0.994 105.9 305

Figure 2 shows the effect of the number of pseudocomponents (PSC) that represent the oil plus fraction (C7+) on the density curve. Despite the good agreement between experimental and calculated density at this point, it is important to assure this agreement throughout the whole pressure range to allow accurate calculation of the viscosity even at pressures much lower than the bubble pressure. It is observed that the effect of the number of pseudocomponents (PSC) is minimal, causing deviations of less than 0.3%. It is also important to note that the density curves calculated with 5 PSC and 10 PSC are very similar and almost indistinguishable for those oils. As expected, a curve inflection occurs just at the bubble pressure. This results is coherent with those by Quinones-Cisneros and colleagues33 modeling. The most significant deviations are found closer to the atmospheric pressure, which is probably related to limitations of the EOS on


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describing the phase behavior, because a low number of pseudocomponents cannot properly represent changes in phase compositions.

(a)

(b)

(c)

Figure 2. Effect of the number of pseudocomponents (PSC) on the density curve for oil O1(a), O2(b) and O3(c).

Figure 3 illustrates the effect of volume-translation methods on the density curve for oils O1, O2 and O3. If no volume correction (M0) is adopted, it provides the largest average relative deviation (ARD). This confirms the low capacity of CEOS to model volumetric properties without any volume translation method. On the other hand, if volume correction is used, the best method for oil O1 (M1), O2 (M2) and O3 (M4) had a deviation of less than 0.8%. Overall, any


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of the volume-correction methods provided sufficiently accurate results for these three oils when compared to M0.

(a)

(b)

(c)

Figure 3. Effect of the volume-translation methods on the density curve for oils O1(a), O2(b) and O3(c).

The effect of the characterization methods on the density curve for oils O1, O2 and O3 is shown in Figure 4. In this case, fixed volume-translation methods were used: M2 for oil O1 and O2 and M1 for O3). The impact is not as high as for the volume-translation methods. For both oil O1 and O2, the characterization procedure C4 provided the lowest deviations, around 0.5 %, while for oil O3 it was the characterization procedure C1, but its modification for heavier oils did not improve the results. Nevertheless, the issue is to know if a small deviation in the predicted


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density can lead to big deviations in the predicted value of viscosity. This is discussed in the next sections.

(a)

(b)

(c)

Figure 4. Effect of the characterization methods on the density curve for oils O1(a), O2(b) and O3(c).

4.2. VISCOSITY PREDICTIONS As expected, the volume translation effect on the density prediction is higher than the effect of the characterization methods. Furthermore, a sensitivity analysis is required to analyze the magnitude of the density input and the characterization methods in the calculation of viscosity using the EF model, as shown in Figure 5 for oil O2. It is worth noting that the viscosity curve was evaluated below the bubble pressure, which is a new condition that has not been investigated using the EF model yet.


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(a)

(b)

Figure 5. Viscosity curves as a function of pressure using different density input (a) and characterization methods (b) for oil O2.

Figure 5a shows the strong effect of the volume-translation methods on the viscosity curves, as also seen in Figure 3. No volume-translation (M0) implies an inaccurate viscosity curve (ARD = 41.3 %), while any volume-translation methods (M2 and M4) provide good viscosity predictions (ARDC1-M2 = 9.5 % and ARDC1-M4 = 8.5 %), even without any further model tuning. Another interesting finding is that using the experimental density values (C1-ED) instead of the predicted density ones leads to slightly larger errors, such as the ARD for this case is 10.1%. This makes sense because the density experimental data also are subject to measurement errors. Furthermore, an inaccurate density prediction (C1-M0) can be improved with the EF model tuning (C1-M0-Tuned), which provided the lowest ARD (2.2 %). The tuning procedure is similar to that used in the Motahhari28 approach, where a multiplier (m) of the parameter c2 of the 5 pseudocomponents is adjusted to 1.42 to match the highest pressure oil viscosity. This preliminary result shows that one viscosity data point for the EF model tuning is more useful than a good fit of the whole experimental density curve. In Figure 5b, it is possible to separately observe the effect of changing the characterization methods on the viscosity curves for oil O2. In general, all the results have acceptable accuracy,


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probably due to efficient good volume translation. Furthermore, they show the slight effect of the correlation of the acentric factor (the difference between C2 and C3) and a more pronounced effect of the correlation of the critical parameters. However, it is important to point out methods C1 and C4, with 8.5 and 6.6 % of ARD. In addition, the effect of the number of pseudocomponents used to represent the plus-fraction on viscosity was also assessed. The results for 5 and 10 pseudocomponents were very similar (around 8% of ARD), but using only 1 pseudocomponent to represent the plus-fraction provided more than 18% of ARD. The performance of the EF model to calculate the viscosity was compared to other ones available in the literature, for 5 reservoir oils shown in Table 5. The same characterization methods and volume-translation techniques (C1-M1) were used and only data until C7+ was considered to characterize the oil. The CS and f-theory models were tested using the thermodynamic simulator SPECS (Technical University of Denmark). Table 5. Properties of the selected oils for EF model comparison with other viscosity models O433 Components N2 CO2 C1 C2 C3 i-C4 n-C4 i-C5 n-C5 C6 C7+ MWC7+ (g/gmol) ρC7+ (g/cm3) Pb (atm) Tmeas (K)

0.34 0.84 49.23 6.32 4.46 0.86 2.18 0.93 1.33 2.06 31.45 230.0 0.866 270.9 366

O533 O619 O719 Composition (mole %) 0.33 0.07 0.42 0.19 0.44 0.35 35.42 33.04 34.71 3.36 4.56 0.19 0.90 0.90 0.05 0.69 0.86 0.03 0.26 0.36 0.01 0.26 0.50 0.01 0.14 0.19 0.01 0.72 3.10 0.01 57.73 55.98 64.2 255 285.1 382.0 0.917 0.907 0.980 .0 156.8 146.0 127.6 344 345 314


O819 0.04 1.21 18.92 0.04 0.04 0.03 0.05 0.05 0.05 0.23 79.34 530.2 1.009 40.9 311

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Table 6 compares the EF-model and these models for oils O4 to O7. However, some of these models were not able to accurately calculate the viscosity for oil O8. The results show the best predictive performance of the EF-model over the other models, without any further adjustment and using just a simple characterization and volume correction technique. Although the deviations become larger as the oil gets heavier for all models, the magnitude of the deviation is lower for the EF model than the others. This demonstrates that the EF-model is more successful as a predictive tool than other viscosity models. Nevertheless, as is discussed in the next sections, this result can be improved with proper parameter estimation and characterization methods. The results of the EF-model to predict the viscosity curves were compared to those by Quinones-Cisneros and colleagues33 and Lindeloff and colleagues19. For oils O4 and O5, the results using the f-theory model and LBC model were obtained from Quinones-Cisneros and colleagues33 and the viscosity curve for oils O6 to O8 came from Lindeloff and colleagues19, using the improved CSP model. The predictive results for all oils under and above the bubble pressure are shown in Figure 6. The effect of the volume-translation and characterization methods of the EF model can be seen and this means that a small change in the density greatly impacts the viscosity prediction. Especially for lighter oils, the viscosity curve predicted with the EF-model is more accurate than that predicted with the LBC and f-theory models. On the other hand, larger deviations in viscosity were observed for all models and oils O6 to O8, but this changes when a proper tuning method is applied with minimum experimental data.


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Page 20 of 34

Table 6. Viscosity average relative deviation (%ARD) for oils O4 to O7, using C1-M1 technique.

Oils

EF model

f-theory

CS model

O4 O5 O6 O7

12.4 27.7 26.9 43.1

32.7 51.8 51.9 93.1

24.8 47.0 64.6 85.8

The viscosity curves obtained with the tuned models for oils O5, O6, O7 and O8 are shown in Figure 7. It was not necessary to tune the models to calculate the viscosity of oil O4 because the preliminary results were accurate enough (less than 2% ARD). The CS model was adjusted to fit the dead oil viscosity while the LBC and the f-theory models were adjusted using the viscosity data above the saturation pressure33. Both results were obtained from the original papers for a fair comparison. The EF-model was initially tuned by a fixed multiplier for c2, present in Equation 1 and labelled with the suffix Tuned. However, for some oils the viscosity was over predicted below the saturation pressure.

(a)

(b)


20

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Energy & Fuels

(c)

(d)

(e)

Figure 6. Viscosity curve predicted for oils O4 (a), O5 (b), O6 (c), O7 (d) and O8 (e) with different characterization and volume-translation methods. In order to achieve a more accurate viscosity curve, a new tuning procedure was performed, labelled with the suffix Tuned c2(P). This procedure was based on the previous tuning approach, but the multiplier m of the parameter c2 for the 5 pseudocomponents was considered as a function of pressure (P) and fitted to the viscosity experimental data of the oil using Equation 12. This procedure guarantees a well-adjusted viscosity curve, but instead of using one experimental data, at least three viscosity points are needed to estimate the parameters A, B and C. These parameters are constants and, when possible, should be estimated. Oils with similar characteristics need only a single equation. At least three viscosity experimental data points are required to calculate the constants A, B and C. In this case, a simple non-linear equation should be solved to find the constants. When more viscosity data is available, it is possible to perform parameter estimation of constants A, B and C. The results shown in Figure 7 indicate that in most cases 3 viscosity experimental data points are enough to achieve a good fit, but if more data points are used the accuracy increases. This approach is similar to that applied by Kumar et al.16 with the f-theory model, where a correction factor varying with pressure is used. This is


21

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Page 22 of 34

thoroughly justified by the high impact of heavier components on the oil viscosity below the saturation pressure. (12)

m  A P B + C

The results of EF-model for oil O5 are enough accurate by just tuning a fixed c2 parameter to match the oil viscosity at the saturation pressure. However, for oils O7 and O8, this procedure did not provide good results and 3 viscosity experimental data points were required to obtain the parameters of Equation 12 and accurately fit the curve, while for oil O6 it was necessary to use 5 viscosity data points to fit these same parameters.

(a)

(b)

(c) (d) Figure 7. Viscosity curve with tuned models for oils O5 (a), O6 (b), O7 (c) and O8 (d).


22

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Energy & Fuels

Furthermore, the EF-model was tested for a large oil databank collected from the literature, with more than 100 viscosity data points49. Due to the good results reached for the previous oils, the C4-M2 method was applied with the EF-model for this new test. ARD values are shown in Table 7 including the results of other models in the literature, given by Elsharkawy and colleagues49. Unlike the EF-model, the other methods have limited applications to viscous oils. Some are based on empirical correlations (Lohrenz7 and Little8) and other are semi-empirical models (PR49, MPR49 and CS Model). Only the predictive models cited by Elsharkawy and colleagues49 were considered for comparison in the present study. The following results show that for oils without experimental data, the EF model is a useful tool for viscosity prediction, if we consider that typical experimental accuracy of viscosity measurements is around 30 %28. This reinforces the improvement provided by an EOS capable of accurately predicting the density, if a proper characterization and volume correction is applied. The ARD increases for lower values of viscosity, but is approximately randomly distributed for the whole pressure range. This is probably due to experimental inaccuracy that occurs for low viscosity values, because the measurement of such small values leads to larger deviations than for high viscosity values. Another possible explanation is that the EF-model was not developed for lighters oils, with viscosity values lower than 1cP, causing more errors for these types of oils. However, the results have similar accuracy and are even better than those from the classical models found in the literature.


23

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Page 24 of 34

Table 7. Viscosity ARD (%) of EF model compared to those in the literature49. ARD%

Number of data points

µexp range (cP)

4949

0.25 - 3.52

73.4

35.8

1450

0.11 - 2.62

59.0

0.58 - 1.24

67.8

PR49 MPR49 MPR49 (Kij=0) (Kij=0) (Kij#0)

CS Model

Little

36.1

38.7

30.8

43.8

32.8

17.7

17.7

28.0

26.0

29.4

27.8

24.0

24.2

36.4

35.1

72.9

29.3

Lohrenz EF-Model

51

53

In this paper, the EF-model was tested and validated with a small number of oils. The effects of a variety of conditions were investigated and the results suggest the following guidelines for a good viscosity prediction with the EF-model: 

A higher number of pseudocomponents is not necessary, because it does not add significant improvement on the results. Five pseudocomponents for the higher fraction are enough to achieve a good accuracy;



When possible, select the best characterization method based on the oil properties or using the available equilibrium/density data, as it is important for viscosity prediction;



Using the experimental density data as an input to the EF-model does not increase the accuracy of the model, possibly due to the experimental uncertainty. Therefore, we recommend to use the simulated density data as an input to give more consistency on the thermodynamic/rheological models;



If satisfactory accuracy is not achieved for the viscosity prediction, adjust the parameter c2 based on the viscosity at the saturation pressure using a fixed multiplier m;


24

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Energy & Fuels



If a more accurate model is needed and there is available experimental data, use other viscosity data points (at least three) to estimate the parameters of Equation 12 and fit the viscosity curve.

It is important to note that these recommendations should be followed with care, because they are based on an evaluation using a small number of oils.

5. CONCLUSION The EF model (Expanded Fluid model) successfully predicts the viscosity of heavy oils. However, this model exhibits a strong dependence on the density predictions and a proper characterization is needed to obtain accurate results. In this work we studied the effects of the volume-translation method with the SRK EOS, in order to improve the density inputs to the EF model, as well as to identify the characterization effect on density predictions. The results show that the experimental density curve can be used to select the most adequate characterization and volume-translation method, which can change from oil to oil. Nonetheless, the density experimental data is not always necessary to model the viscosity because the results with the EOS-predicted density values are accurate for most cases. This work also expanded the applicability of the EF model for oils with different characteristics, obtaining predictive results of viscosity, below and above the saturation pressure, which are more accurate than those from most models in the literature. Moreover, our results showed that any characterization method or volumetric property can be used if a good parameter estimation of the EF model is done, mainly the multiplier of the c2 parameter for the pseudocomponents. If estimation of c2 parameter is not enough, a novel tuning approach can be successfully applied to fit the viscosity curve under and above the saturation pressure, adjusting the multiplier of c2 as a function of the pressure.


25

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Page 26 of 34

APPENDIX 1: EF MODEL The EF model was originally developed by Yarranton and Satyro25,26 and the approach used here is an extension of it, described in more detail in the original paper of the authors28. The first step to apply the EF model is to define fluid-specific parameters (µoi, c2i, c3i and ρsoi) for each pseudocomponent and defined component. For some of the defined components (N2, CO2, H2S), these fluid-specific parameters are presented in Table 852 and further calculated with Equation 1353, in the case of diluted gas viscosity (µoi).

 oi  p1  p 2  T  p 3  T 2  p 4  T 3

(13)

Table 8. Fluid-specific parameters for N2, CO2, H2S

Component N2

ρsoi

c2i

c3i (kPa-1)

p1

p2

p3

0.1147

0.1x10-6

11.8109

0.49838

-1.085 x10-4

4.4656

0.63814

-2.660 x10

-4

192.1463

-0.39855

9.65 x10-4

3

(kg/m ) 1012.39

CO2

1617.66

0.236

0.187x10

H2S

1194.59

0.188

0

-6

p4 0 5.4113 x10-8 -2.8172x10-7

For the pseudocomponents, Chung et al.3 approach is used to calculate the diluted gas-viscosity, based on Equations 14 to 17, with Hall-Yarborough Equation 18 for the critical volume (vc). For the defined hydrocarbons, Equation 14 is used with the parameters given in the literature53.

 o  4.0785  10 3

FC MW  T vc

2/3

(14)

 v


26

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Energy & Fuels

 v  (1.16145T * ) 0.14874  0.52487 exp(0.7732T * )  2.16175 exp(2.4378T * )

(15)

FC  1  0.2756

(16)

T* 

1.12593T Tc

(17)

vc  1.56MW 1.15 SG 0.7935

(18)

After this, the mixture dilute gas viscosity is calculated with Wilke’s method54, but the importance of this parameter can be negligible when compared to the value of the other parameters. The parameter c3 for the hydrocarbons is obtained from Equation 19. The parameters c2 and ρso are obtained from departure functions developed by Motahhari and colleagues28, using equations 20 to 22 and the parameters in Table 10, obtaining the reference value for each fluidspecific parameter for each pseudocomponent. c3 

2.8  10 7 1  3.23 exp(1.54  10 2 MW )

(19)

 a0,  so a o  a1,  o MW 2 ,  s s  MW 

(20)

 so ( ref )  

a4,  o  s  exp(a o MW )  3,  s  1  a5,  o exp(a6,  o MW )  s s

 a 2 ,c2  c2 ( ref )  (a0,c2  a1,c2 MW ) exp  a3,c2 MW   a 4,c2 ln( MW )  MW 

SG( ref )  a0,SG 

a1,SG MW

0.5



a2,SG MW



a3,SG MW

2



a4,SG MW

(21)

(22)

3


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Page 28 of 34

Table 9. Constants for fluid-specific reference functions. Property SG(ref)

a0

a1

a2

a3

a4

a5

a6

0.843593

0.1419

-16.6

-41.27

2535

-

-

-4775

3.984

0.4

-0.001298

938.3

0.08419

-0.00106

0.09353

0.000442

-333.4

-0.000166

0.0477

-

-

ρso(ref) c2(ref)

From the reference value, equations 23 and 24 are used with data from Table 11, to determine the fluid specific departure function for each component. b3  b1    X   b0  SG 2   b2  SG b4  MW  MW b4   

(23)

SG  SG  SG(ref )

(24)

Table 10. Constants for fluid-specific parameters departure functions Property 0

14640

739

0

0.67

0.4925

-191900

-0.371

83930

2.67

The mixture parameters c2, c3 and ρso are obtained from the appropriate mixing rules, given by Equations 25 to 27, the weight fraction of each component (xwi) and the component parameters (c2i, c3i and ρsoi). After this, EF model equations 1-3 can be applied to obtain the fluid viscosity.


28

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Energy & Fuels

 nc nc x wi x wj  1 1 o   s    o o 2   s ,i  s , j  i j

   

 nc nc xwi xwj  c2,i c2, j  c2      o o 2  i j s , i s, j  o s

x c3    wi i  c3,i nc

1

   

   

(25)

(26)

(27)

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.

ASSOCIATED CONTENT Supporting Information. Density and viscosity experimental data of oils O1 to O3 obtained in the literature are shown in Table S1; viscosity experimental data of oils O4 to O8 obtained in the literature are shown in Table S2 and S3; and estimated parameters of Equation 12 are shown in Table S4.

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Improvement of the Expanded Fluid Viscosity Model for Crude Oils

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