Models for Estimating the Viscosity of Paraffinic–Naphthenic Live

Article Cite This: Energy Fuels 2018, 32, 2622−2629

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Models for Estimating the Viscosity of Paraffinic−Naphthenic Live Crude Oils Luciana L. P. R. Andrade and Krishnaswamy Rajagopal* Universidade Federal do Rio de Janeiro, DEQ/Escola de Química/UFRJIlha do Fundão, CT, I-122, CEP 21949-900, Brazil ABSTRACT: Viscosity is an important property of live crude oil used in design and operation of production processes. Eleven widely used empirical correlations for estimating undersaturated oil viscosity were evaluated using measured undersaturated oil viscosity for live crude oils from primary separators of 10 Brazilian oil wells. The oil samples were characterized as paraffinic− naphthenic crudes. The empirical correlations were found to be inadequate to represent the measured data. The better literature models present average absolute percent relative errors of 1.18% and 1.40% but show wider scatter of data. For more accurate estimates of undersaturated oil viscosity of paraffinic−naphthenic live crudes oils, a new model based on Eyring theory is proposed, and this model correlates the experimental data with average absolute percent relative error of 0.97%.

1. INTRODUCTION

2. MODELS FOR ESTIMATING VISCOSITY OF CRUDE OILS AT DIFFERENT PRESSURES

Viscosity is an important physical property used in the design of transportation and storage systems as well as for planning the strategy of separation of live crude oil from reservoir fluids in production platforms. Along with density, the viscosity is an important parameter utilized for calculating the fluid flow through equipment and pipelines,1 selecting an optimum strategy of separation oil from reservoir fluids under pressure, transportation, and storage.2−4 The viscosity of crude oil is a function of temperature, pressure, and composition. The viscosity varies considerably with temperature. During production, the oil, gas, and water are separated from reservoir fluid at high pressure and at constant temperature in one or more stages in pressure vessels called separators. The crude oil with dissolved gas under pressure of the separator is commonly called live oil. When the pressure is reduced, the composition of oil does not change with decreasing pressure until bubble point pressure. The viscosity of residual live oil can vary with pressure differently for different classes of oils as the components are very different and the equilibrium composition will vary at operating pressure. It is necessary to estimate or measure experimentally the viscosity of live oils as a function of pressure. We have measured 51 viscosities of undersaturated oils from primary separators of 10 Brazilian oil wells. These oils were characterized as paraffinic−naphthenic crudes by studying the variation of kinematic viscosity with temperature using six representative samples. We measured the viscosities of live crudes above bubble point pressure. The measurements were realized at temperatures between 317 and 343 K, above wax appearance temperature (WAT) for oils with similar characteristics, to characterize these samples of crudes as Newtonian fluids. We have evaluated several frequently used literature correlations for estimating viscosity of live crude oils by comparing with our experimental data of undersaturated oil viscosity. In this work, Eyring theory is applied for modeling viscosity as a function of pressure. The model represents the experimental data satisfactorily. © 2018 American Chemical Society

2.1. Empirical Models. The viscosities of undersaturated oil should be measured preferably in the laboratory and correlated for accurate estimation at different pressures around the process or temperatures. While the viscosity of saturated oil at atmospheric pressure (dead oil) can be readily measured by several simple methods, the measurements of viscosity saturated oil and undersaturated oil can be expensive and time-consuming, especially at higher pressures. Several models are proposed in literature to estimate the viscosities of undersaturated oil. Models found in the literature to estimate oil viscosities were obtained empirically from oils with particular characteristics from different regions around the world. The dead oil viscosity models use temperature and oil API gravity as input parameters. Most of the undersaturated oil viscosity models use the pressure and bubble point pressure (differential pressure or ratio pressure) and the viscosity of saturated oil as input parameters. Some authors use also oil API gravity and/or dead oil viscosity as input parameters to estimate undersaturated oil viscosity. Most oil produced in off-shore fields of Brazil is paraffinic− naphthenic in nature, and there is a lack of experimental data in the literature to evaluate the published models for paraffinic− naphthenic crude oils. We obtain experimental data of viscosity of several live paraffinic−naphthenic crude oils from primary high-pressure separators in off-shore platforms and identify the models found in the literature best suited for representing the data obtained experimentally for undersaturated oil viscosity. We have evaluated the frequently used empirical models of Beal,5 Vazquez and Beggs,6 Khan et al.,7 Kartoatmodjo and Schmidt,8 Labedi,9 Petrosky and Farshad,10 Almehaideb,11 Elsharkawy and Alikhan,12 Elsharkawy and Gharbi,13 Hossain et al.,14 and Isehunwa et al.15 for undersaturated oil viscosity. Received: December 11, 2017 Revised: January 12, 2018 Published: January 27, 2018 2622

DOI: 10.1021/acs.energyfuels.7b03903 Energy Fuels 2018, 32, 2622−2629

Article

Energy & Fuels Table 1. Data Range and Reported Errors of Undersaturated Viscosity Models

a

correlation

crudes

P/kPa

μo / mPa s

ARE/%

AARE/%

Beal5 Vazquez and Beggs6 Khan et al.7 Kartoatmodjo and Schmidt8 Labedi9 Petrosky and Farshad10 Almehaideb11 Elsharkawy and Alikhan12 Elsharkawy and Gharbi13 Hossain et al.14 Isehunwa et al.15

US fields over the world Saudi Arabia South East Asia, North America Libya Gulf of Mexico UAE Middle East Kuwaiti fields over the world Niger Delta

10445−35542a 972−65603a 101.3−34577 170−41469 − 11031−70671 − 8873−68947 0−68258 2068−44126 2061−64858

0.2−315a 0.2−1.4a 0.13−71 0.168−517 − 0.224−4.090 − 0.2−5.7 − 3−517 0.08−4.3

2.7a −7.5a 0.094 −4.3a −3.1 −0.19 − −0.9 − − −

− − 1.915 6.9a − 2.6 2.885 4.9 − − 4.0

Reference 12

The Beal5 model was developed from crude oil data from California, Vazquez and Beggs6 and Hossain14 models were developed from the crude oil data from different regions of the world, the Khan et al.7 model was developed from the crude oil data from Saudi Arabian, the Kartoatmodjo and Schmidt8 model was developed from the crude oil data from South East Asia and North America using data bank, the Labedi9 model was developed from the crude oils data from Libya, the Petrosky and Farshad10 model was developed from the crude oil data from the Gulf of Mexico, the Almehaideb11 model was developed from the crude oil data from UAE, the Elsharkawy and Alikhan12 model was developed from the crude oil data from the Middle East, the Elsharkawy and Gharbi13 model was developed from the crude oil data from Kuwaiti, and the Isehunwa et al.15 model was developed from the light crude oil data from Niger Delta. Table 1 shows the source of crude oils, the pressure range, the viscosity range, and the error percentage for models evaluated. The empirical models found in literature were developed from different crude oils and using limited data. These models show large errors when applied for estimating viscosity of crude oil of different regions. These differences should be attributed to the origin of oil which determines the chemical base of oil (paraffinic, naphthenic, aromatic, or mixed) and not only to the data range of pressure, temperature, specific gravity, and relative amount of gas dissolving in oil. We evaluate literature correlations and observed that the models based on fundamental theories are the best models for estimating viscosities of oils at higher pressures. Models for estimating viscosities that were developed based on viscosity theories present good agreement with experimental values of viscosity of oils from different origins. The evaluated models have expressed undersaturated oil viscosity as a function of saturated oil viscosity and both bubble point pressure and oil pressure as input variables, and the Elsharkawy and Alikhan12 model and Elsharkawy and Gharbi13 model also have used calculated dead oil viscosity as input parameters. 2.2. Application of Eyring Theory Viscosity of Liquids. The viscosity of liquids can be estimated by Eyring theory16 according eq 1 ⎛ δ ⎞2 ⎛ Nh ⎞ μ = ⎜ ⎟ ⎜ ⎟e(−ΔG / RT ) ⎝a⎠ ⎝ V ⎠

distance between molecules of two adjacent layers, and a is the distance between two equilibrium positions of the molecule, considering that the molecules of the liquid at rest undergo continuous rearrangements, where at any moment a molecule overcomes this energy barrier by going to an adjoining position with new neighbors. This equation shows that viscosity depends on the size, density, and energy of activation of molecules of liquids. The viscosity of fluid in the reference state can be written as μ0 =

⎛ δ ⎞2 ⎛ Nh ⎞ (−ΔG0 / RT ) ⎜ ⎟ ⎜ ⎟e ⎝ a ⎠ ⎝ V0 ⎠

(2)

Combining eqs 1 and 2, we can obtain the following equation: V μ = 0 ·e(−ΔG −ΔG0 / RT ) μ0 V

(3)

From eq 3, we can represent the variation of viscosity with pressure replacing the ratio of molar volumes by the pressure using a thermodynamic relation which correlates volume of liquids to the pressure in an isothermal process, eq 4: 1 ⎛ ∂V ⎞ k T = − ·⎜ ⎟ V ⎝ ∂p ⎠ T

(4)

Considering isothermal compressibility, kT, nearly constant in the pressure range, the integration of eq 4 leads to ln

V0 = k T·(p − p0 ) V

(5)

Equation 5 can be rearranged to eq 6:

V0 = ek T(p − p0 ) V

(6)

Substitution of eq 6 into eq 3 gives eq 7, which correlates viscosity of liquids with temperature and pressure: μ = e(−ΔG −ΔG0 / RT )·ek T.(p − p0 ) μ0 (7) The references values, μ0 and p0, are replaced by the values of saturated oil viscosity, μob, and bubble point pressure, pb, respectively, and μ is replaced by values of undersaturated oil viscosity, μo. Equation 7 can be applied to the experimental undersaturated oil viscosities substituting c1 = (−ΔG − ΔG0)/ RT and c2 = kT as parameters for the paraffinic−naphthenic oils to be estimated from experimental data:

(1)

where N is Avogadro’s number, h is Planck’s constant, ΔG is the molar energy of activation for flow, V is molar volume, R is the gas constant, T is absolute temperature, δ is the shortest 2623

DOI: 10.1021/acs.energyfuels.7b03903 Energy Fuels 2018, 32, 2622−2629

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Energy & Fuels μo μob

Table 2. Experimental Values of Composition of Paraffinic− Naphthenic Oil

= ec1·ec2·(p − pb ) (8)

average mole fraction, mol %

The modification of the Eyring model results in the exponential variation of viscosity with pressure expressed by Bridgman.17 For Bridgman, the logarithm of the viscosity is proportional to pressure at higher pressures.

3. EXPERIMENTAL MEASUREMENTS 3.1. Experimental Method. The samples of live oil were taken from the primary separator of the oil wells. The temperature and pressure of separator were measured in the off-shore platforms when the live oil samples were taken. The separator samples are received in the laboratory in high-pressure sampling cylinders equipped with floating pistons. The viscosity of each live sample was measured at several pressures above bubble point and separator temperature with an oscillating piston viscometer (Cambridge Applied Systems, SPL 440), according to ASTM D7483-08.18 The temperature is controlled to a high precision by a circulating silicon oil bath within 0.2 K of the selected temperature using a Julabo F12 thermostatic bath and control system. The pressure meter is the Omega DP41-B ultrahigh precision input meter. The pressure measurement is accurate to 0.003 MPa or 0.005% full-scale, and the system can be readily controlled within 0.035 MPa of selected pressure. When the variance of measurements was small and constant, the values of the measured value and its standard deviation were recorded. The repeatability of the experiments was found in the range 0.1−1.0%, Rajagopal et al.19 3.2. Characterization of Crude Oil. Paraffinic−naphthenic oil samples were obtained from 10 different oil wells of the Santos Basin of Brazil. In order to obtain the composition of the live oil, the density of residual oil, molecular weight, and compositions of residual oil and liberated gas from the live oil were obtained. The composition of live oil is obtained by numerical recombination of the composition of oil with composition of gas using the density and molecular weight data of the oil. To obtain these properties and compositions, the sample received in high-pressure sampling cylinders is restored to original conditions and is collected from the sampling cylinder to the pycnometer. The gas dissolved in the oil is released from the pycnometer to the gasometer, and its composition is determinate. The compositions of residual oil and liberated gas are obtained separately at atmospheric pressure. The composition of the residual oil and liberated gas from the live oil were measured using chromatographs (Agilent Technologies, 7890A) following procedures ASTM D2887-0320 and ASTM D194503 (2010)21 at standard conditions. The density of residual oil was measured with densimeter (Anton Paar, DMA 4500 M) following procedure ASTM 5002-9922 at standard conditions, and the molecular weight of the oils was measured using depression of freezing point by cryoscopy (Gonotec Gmbh, Osmomat 010), following ASTM D2224-78 (1983).23 Table 2 shows representative compositions of the samples of paraffinic−naphthenic crude oils studied. The samples of crude oil were classified as paraffinic−naphthenic by the density−viscosity ratio proposed by Farah24 by means of a relationship (API/(A/B)) based on A and B parameters of the Walther-ASTM equation, which has been used for crude oil sample characterization by taking into account the temperature dependence of the crude oil kinematic viscosity. The values of parameters A and B were obtained graphically by plotting log10(log10(z)) vs log10(T), eq 9, where z was estimated using experimental kinematic viscosities as a function of temperature, eq 10

log10(log10(z)) = A − B log(T )

sample 1

sample 2

sample 3

CO2 H2S N2 CH4 C2H6 C3H8 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+

0.0147 0.0000 0.1816 4.8116 0.7524 0.7070 0.2768 0.5161 0.3498 0.4781 1.3414 2.9257 5.0351 5.1645 4.9100 4.3126 4.4236 4.5753 4.5469 4.5730 3.6357 3.2682 3.2131 2.8888 2.3558 2.0069 1.7959 1.5930 1.3178 1.1703 1.0665 0.9390 0.9179 0.7786 23.1562

0.0018 0.0000 0.2419 4.9133 0.7593 0.5776 0.1528 0.2274 0.0789 0.1032 0.7568 2.4181 5.3701 5.3733 5.2674 4.7658 4.8104 5.0524 5.0753 5.1085 4.1132 3.7507 3.6623 3.3190 2.6853 2.2954 2.1070 1.7354 1.4903 1.3326 1.2599 1.0073 0.9753 0.8155 18.3962

0.0376 0.0000 0.0396 6.1248 0.8998 0.7901 0.2835 0.5123 0.3396 0.4652 1.3193 2.8845 4.9607 5.0853 4.8348 4.2465 4.3558 4.5052 4.4772 4.5029 3.5800 3.2181 3.1638 2.8445 2.3197 1.9761 1.7684 1.5685 1.2976 1.1523 1.0502 0.9246 0.9039 0.7666 22.8012

e(13.0458−74.6851ν), G = e(37.4619−192.643ν), and H = e(80.4945−400.468ν). The parameters C, D, E, F, G, and H are equal to zero according to the following limits: C = 0 if 2 × 107 > ν > 2.0 mm2 s−1, D = 0 if 2 × 107 > ν >1.65 mm2 s−1, E = 0 if 2 × 107 > ν > 0.90 mm2 s−1, F = 0 and G = 0 if 2 × 107 > ν > 0.30 mm2 s−1, and H = 0 if 2 × 107 > ν > 0.24 mm2 s−1. The kinematic viscosities were measured for three different temperatures for crude oils samples, using capillary tube viscometer at different temperatures, following procedure ASTM D445-06.25 The values of relationship (API/(A/B)) calculated for our samples vary between 12 and 14 characterizing the oil as paraffinic−naphthenic, Table 3. Table 3 shows the classification of crude oils and their fractions based on the relationship of Walther-ASTM.24 The API of crude oil was calculated from measured liquid densities. The oil API gravity was calculated from standard density by eq 11:

⎛ 141.5 ⎞ API = ⎜ ⎟ − 131.5 ⎝ do ⎠

(9)

z = ν + 0.7 + C − D + E − F + G − H

component

(11)

where do is oil specific gravity (water = 1) at 288.71 K. The oils studied have API densities in the range 29−34 demonstrating that their wax appearance temperature (WAT), the lower limit for Newtonian flow, is around 291 K, according to Farah et al.26 It is expected therefore that the oils performed in a Newtonian manner at the temperatures used, 317−343 K.

(10)

2 −1

where ν is kinematic viscosity in mm s , T is temperature in K, and C, D, E, F, G, and H are the model fitted parameters: C = e(−1.14883−2.65868ν), D = e(−0.0038138−12.5645ν), E = e(5.46491−37.6289ν), F = 2624

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Table 4. Experimental Density, ρο, and Undersaturated Oil Viscosity, μo, as a Function of Pressure, p, for Undersaturated Oil Samples

Table 3. Classification of Crude Oils and Their Fractions Based on the Relationship API/(A/B), Farah24 API/(A/B)

type

14

aromatic−asphaltic aromatic−naphthenic aromatic−intermediate naphthenic paraffinic−naphthenic paraffinic

4. RESULTS AND DISCUSSION 4.1. Experimental Data. The viscosities of live oil samples were measured at several pressures above bubble point pressure and separator temperature, and 51 undersaturated oil viscosity experimental data values were obtained. The densities were also measured at the standard temperature of 288.71 K. The experimental density, ρο, as well as corresponding API value and measured undersaturated oil viscosity, μo, are shown in Table 4 as a function of pressure. 4.2. Evaluation of Literature Models. Eleven literature models for undersaturated oil viscosity were evaluated for their ability to represent our measured experimental values: Beal,5 Vazquez and Beggs,6 Khan et al.,7 Kartoatmodjo and Schmidt,8 Labedi,9 Petrosky and Farshad,10 Almehaideb,11 Elsharkawy and Alikhan,12 Elsharkawy and Gharbi,13 Hossain et al.,14 and Isehunwa et al.15 These models use differential pressure (p − pb) and saturated oil viscosity as input variables, while Vazquez and Beggs,6 Khan et al.,7 Labedi,9 and Almehaideb11 use pressure ratio (p/pb). The Elsharkawy and Alikhan12 and Elsharkawy and Gharbi13 models use dead oil viscosity like input parameters. These literature models are presented in Appendix I. We have used 51 undersaturated oil viscosity experimental data values to evaluate the correlation performances. For quantitative analyses, the statistical parameters used to compare the performance of the models are the average percent relative error, ARE, eq 12, average absolute percent relative error, AARE, eq 13, and the percent absolute standard deviation, SDA, eq 14. The lower value of ARE indicates a symmetrical distribution of experimental values around the correlation. ARE =

100 ND

ND

⎛ Xcalc − Xexp ⎞ ⎟⎟ Xexp ⎝ ⎠

∑ ⎜⎜ i=1

(12)

sample

T/K

ρο/kg m−3 (288.71 K)

API

p/kPa

μo /mPa s

1

326

879.70

2

326

878.88

3

326

881.68

4

317

861.97

5

343

882.24

6

320

885.14

7

321

859.19

8

319

873.04

9

320

879.66

10

318

861.03

30 30 30 30 30 30 30 30 30 30 30 30 29 29 29 29 29 29 33 33 33 33 33 29 29 29 29 29 29 29 29 29 34 34 34 34 34 31 31 31 31 30 30 30 30 30 33 33 33 33 33

13914 10487 7164 5654 4240 2979 14107 10397 7026 5599 4144 3496 6674 5654 4819 4123 3372 2785 5612 4454 3296 2785 2303 6895 5516 4137 2758 5662 4828 4205 3592 2907 10703 8000 6764 5288 4351 9549 8253 6219 5516 6067 5226 4289 3516 2841 9580 7413 6534 4903 3422

15.100 14.420 13.760 13.360 13.110 12.790 20.480 19.350 18.360 17.960 17.570 17.370 12.680 12.450 12.250 12.050 11.862 11.770 7.090 6.950 6.820 6.755 6.703 4.335 4.222 4.119 3.994 8.274 8.099 8.026 7.806 7.664 3.784 3.669 3.633 3.562 3.475 6.648 6.569 6.444 6.251 6.421 6.201 6.174 6.071 5.849 5.952 5.764 5.672 5.526 5.415

A lower value of AARE represents the better agreement between the estimated and experimental values. AARE =

100 ND

ND

⎛ Xcalc − Xexp ⎞ ⎟⎟ Xexp ⎝ ⎠

∑ ⎜⎜ i=1

(13)

where ND is the number of measurements, Xcalc is the correlation calculated value, and Xexp is the experimental value. Percent absolute standard deviation, SDA, is a measure of dispersion and is defined by eq 14. A smaller value of SDA indicates a smaller degree of dispersion and higher precision.

AAREmin, eq 16, are also calculated. The lower value of the AAREmax indicates higher accuracy.

2

SDA =

1 ∑ (AARE − AARE) ND − 1

⎛ Xcalc − Xexp AAREmax = max⎜⎜100 Xexp ⎝

(14)

Maximum average absolute percent relative error, AAREmax, eq 15, and minimum average absolute percent relative error, 2625

⎞ ⎟ ⎟ ⎠

(15)

DOI: 10.1021/acs.energyfuels.7b03903 Energy Fuels 2018, 32, 2622−2629

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Energy & Fuels Table 5. Statistical Accuracy of Undersaturated Oil Viscosity correlation

ARE/%

AARE/%

AAREmax/%

AAREmin/%

SDA/%

modified Eyring Beal5 Vazquez and Beggs6 Khan et al.7 Kartoatmodjo and Schmidt8 Labedi9 Petrosky and Farshad10 Almehaideb11 Elsharkawy and Alikhan12 Elsharkawy and Gharbi13 Hossain et al.14 Isehunwa et al.14

0.08 −0.98 2.24 −1.25 −0.61 4.09 −4.61 −56.39 −2.01 −8.36 2.15 −0.94

0.97 1.52 3.23 1.40 1.61 4.09 4.67 64.06 3.15 8.36 2.47 1.18

3.70 5.76 36.67 5.05 5.49 20.04 18.98 112.14 15.43 42.52 10.48 4.70

0.02 0.08 0.00 0.02 0.01 0.27 0.15 3.99 4.09 0.25 0.01 0.07

0.86 1.29 6.39 1.14 1.36 4.05 4.13 19.94 3.27 8.48 2.46 1.07

Figure 1. Cross plot for undersaturated oil viscosity.

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Energy & Fuels ⎛ Xcalc − Xexp AAREmin = min⎜⎜100 Xexp ⎝

⎞ ⎟ ⎟ ⎠

(16)

For the qualitative analyses, cross plots of estimated vs experimental undersaturated oil viscosity for each correlation are also presented. In the cross plot, the estimated values are plotted as a straight line with a slope of 45° indicating the accuracy of models. The value of AARE obtained is in the range from 1.18% for the Isehunwa et al.15 model to 8.36% for the Elsharkawy and Gharbi13 model and 64.06% for the Almehaideb11 model. Table 5 shows the statistical error analysis results of the undersaturated oil viscosity models. Figure 1 shows cross plots for undersaturated oil viscosity models. The undersaturated oil viscosity models gives low values of AARE indicating good accuracy, except for Almehaideb11 which presents 64.06%. However, the high values of AAREmax show wider scatter for these models. The Khan et al.7 and Isehunwa et al.15 models present the lowest values for AARE and AAREmax in comparison with other literature models, 1.18% and 4.70% for Isehunwa et al.15 and 1.40% and 5.05% for Khan et al.7 From Figure 1, it can be observed that the best estimations for the oils studied were obtained using Khan et al.7 and Isehunwa et al.,15 based on Eyring theory, besides the empirical correlations of Beal5 and Kartoatmodjo and Schmidt.8 The models proposed by Petrosky and Farshad,10 Elsharkawy and Alikhan,12 and Elsharkawy and Gharbi13 underestimate the viscosity values while the models proposed by Vazquez and Beggs,6 Labedi,9 and Hossain et al.14 overestimate the viscosity values. Almehaideb et al.11 showed high deviation from experimental values. Figure 2 shows that the undersaturated oil viscosity model proposed for paraffinic−naphthenic crudes have the smallest error, ARE %, and least scatter around the zero-error line. Khan et al.7 and Isehunwa et al.15 models show lower error and wider scatter around the zero-error line than do the others. The Isehunwa15 model was developed from light crude oils from Niger Delta that can show similarities with Brazilian oils. ́ The oils found in the Santos, Campos, and Espirito Santo basins, in the southeastern margin of Brazil, and from Angola to Cameroon in the West African margin are correlated with the same depositional environment.27 Other similarities can be found between oils from deepwater basins offshore “Golden Triangle” formed by Gulf of Mexico, West Africa (Angola e Nigeria), and South America (Campos Basin, Brazil).28 The models which show the larger errors are based on experimental data for Arabian crude oils or based on properties of crude oils from different regions of the world. 4.3. Application Eyring Theory. Fifty-one viscosity experimental data values of undersaturated oil viscosities were used to estimate the adjustable parameters of Eyring theory, eq 8. The values of saturated oil viscosity at bubble point pressure were extrapolated from experimental undersaturated oil viscosities, μo, for each sample of oil, observed in the Table 4. In eq 8, for viscosity μο to be μob at pressure pb, the constant c1 should be zero. The measurements of viscosity at the bubble point pressure have experimental errors much larger than measurements at higher pressures due to liberation of bubbles at the surfaces of the piston and cylinder. Forcing models to have exact value of viscosity at bubble point pressure increases the errors at higher pressures. We re-estimate the constant c1

Figure 2. Relative error for undersaturated oil viscosity models.

and its variation empirically by correlating viscosity measurements at higher pressures using Eyring theory. The bubble point pressures for the oils evaluated have been calculated by means of the Kartoatmodjo and Schmidt8 model which developed different models for calculation fluid properties by taking into account measured field surface data for all samples. The crude oils used in the present study are within the range of the Kartoatmodjo and Schmidt8 bubble point pressure correlation. The values of bubble point pressure and saturated oil viscosities are shown in Table 6. Table 6. Estimated Bubble Point Pressure, pb, Literature Correlation, and Saturated Oil Viscosities, μob

2627

sample

pb/kPa

μob / mPa s

1 2 3 4 5 6 7 8 9 10

1862 820 1856 1911 2060 2591 3229 4185 1806 2817

12.5897 16.5758 11.5211 6.6555 3.9416 7.6054 3.4506 6.1945 5.7454 5.3516 DOI: 10.1021/acs.energyfuels.7b03903 Energy Fuels 2018, 32, 2622−2629

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Energy & Fuels The parameters c1 and c2, eq 8, were estimated by means of minimization of the least-squares objective function, eq 17, using 51 experimental values of viscosities.

m = 2.6 × (p /6.894757)1.187 exp( −11.513 − 8.98 × 10−5(p /6.894757))

Khan et al.7

ND

F obj =

∑ ((Xexp − Xcalc)2 )

−5

μo = μob × e9.6 × 10

(17)

i=1

where i indicates an experimental point and ND indicates the number of experimental points. The estimated parameters are presented in eq 18: −3

Kartoatmodjo and Schimdt

·(p − pb )

1.8148 1.59 × ( −0.006517μob + 0.038μob )

(18)

When we apply the Eyring theory for undersaturated oil viscosity and estimate the parameters from the experimental data at higher pressures, we note that factor (−ΔG − ΔG0)/RT is different from zero and equal to 0.009 475 05. The proposed model is similar to Khan et al.7 and Isehunwa et al.15 models, all of which present better agreement with experimental data because of this correction. The Khan et al.7 and Isehunwa et al.15 models are also based on the Eyring theory and on Reynold’s suggestion that the viscosity of liquids is exponentially related to temperature. The present modification of Eyring theory shows the lowest error 0.97% in comparison with literature models, Table 5. To attribute lower weight to measured or estimated viscosity at bubble point, we have considered (ΔG − ΔG0) to be different from zero unlike in the correlations of Khan et al.7 and Isehunwa et al.15 based on Eyring theory. This increases the number of parameters to two. For comparing models with different number of parameters, we calculated Akaike’s information criterion, AIC, for our model and the models of Khan et al.7 and Isehunwa et al.15 Re-estimating the respective parameters using our experimental data, our model has a lowest value of AIC equal to −293 while the Khan et al.7 and Isehunwa et al.15 have value of AIC equal to −285.9. In the comparison between models, the best model will be the one with the lowest AIC value. Our model presents lower AARE and lower AAREmax.

Labedi

Petrosky and Farshad10 μo = μob + 1.3449 × 10−3((p − pb )/6.894757) × 10 A (I.6)

A = −1.0146 + 1.3322log(μob ) − 0.4876[log(μob )]2 − 1.15036[log(μob )]3

Almehaideb11 p μo = μob (0.134819 + 1.94345 × 10−4R S pb − 1.93106 × 10−9 × R S2)

Elsharkawy and Alikhan

(I.8)

Elsharkawy and Gharbi13 μo = μob + a((p − pb )/6.894757)

(I.9)

2 a = −5612 × 10−8 + 9481 × 10−8μod − 1459 × 10−8μod −3 + 81 × 10−8μod

Hossain et al.14 μo = μob + 0.004481((p − pb )/6.894757) × (0.555955μob1.068099 − 0.527737μob1.063547 )

Isehunwa et al.

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(I.10)

15 −4

Beal5

μo = μob × e1.02 × 10

× ((p − pb )/6.894757)

(I.11)

AUTHOR INFORMATION

Corresponding Author

μo = μob + 0.001((p − pb )/6.894757)

μo = μob (p /pb )m

(I.7)

12

1.19279 μo = μob + 10−2.0771((p − pb )/6.894757)(μod )

Literature Models for Undersaturated Oil Viscosity

Vazquez and Beggs

(I.5)

M = [10−2.488 + μod 0.9036 + (pb /6.894757)0.6151]/100.01976API

APPENDIX I

× (0.024μob1.6 + 0.038μob 0.56 )

(I.4)

9

⎡p ⎤ μo = μob + M ⎢ − 1⎥ ⎢⎣ pb ⎦⎥

5. CONCLUSIONS Several undersaturated oil viscosity models found in literature were evaluated by comparing the estimates with measured viscosities of live paraffinic−naphthenic crude oils. Only models based on Eyring theory, Khan et al. and Isehunwa et al., estimate the undersaturated oil viscosity of paraffinic− naphthenic crudes above bubble point pressure with higher accuracy. Isehunwa et al. shows better accuracy for paraffinic− naphthenic crudes. The proposed model based on Eyring theory can estimate undersaturated oil viscosity of paraffinic− naphthenic crudes above bubble point pressure with maximum error lower than 4%.

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(I.3)

8

μo = 1.00081μob + 0.001127((p − pb )/6.894757)

−5

μo = μob ·e9.47505·10 ·e1.50134·10

× ((p − pb )/6.894757)

*E-mail: [email protected]. Phone: +55 21 3938 7424. (I.1)

ORCID

Krishnaswamy Rajagopal: 0000-0002-0107-0954

6

Notes

The authors declare no competing financial interest.

(I.2) 2628

DOI: 10.1021/acs.energyfuels.7b03903 Energy Fuels 2018, 32, 2622−2629

Article

Energy & Fuels

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(15) Isehunwa, O. S.; Olamigoke, O.; Makinde, A. A. A Correlation to Predict the Viscosity of Light Crude Oils. Paper 105983; SPE 31st Annual International Technical Conference and Exhibition, Abuja, Nigeria, 2006. (16) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wiley & Sons, Inc: New York, 1960. (17) Bridgman, P. B. The viscosity of liquids under pressure. Proc. Natl. Acad. Sci. U. S. A. 1925, 11, 603−606. (18) ASTM D7483-08, Standard Test Method for Determination of Dynamic Viscosity and Derived Kinematic Viscosity of Liquids by Oscillating Piston Viscometer; ASTM International: West Conshohocken, PA, 2008, www.astm.org. (19) Rajagopal, K.; Andrade, L. L. P. R.; Paredes, M. L. L. Highpressure viscosity measurements for the binary system cyclohexane + n-hexadecane in the temperature range of (318.15 to 413.15) K. J. Chem. Eng. Data 2009, 54, 2967−2970. (20) ASTM D2887-03, Standard Test Method for Boiling Range Distribution of Petroleum Fractions by Gas Chromatography; ASTM International: West Conshohocken, PA, 2003, www.astm.org. (21) ASTM D1945-03 (2010), Standard Test Method for Analysis of Natural Gas by Gas Chromatography; ASTM International: West Conshohocken, PA, 2010, www.astm.org. (22) ASTM D5002-99, Standard Test Method for Density and Relative Density of Crude Oils by Digital Density Analyzer; ASTM International: West Conshohocken, PA, 1999, www.astm.org. (23) ASTM D-2224 (1983), Method of Test for Mean Molecular Weight of Mineral Insulating Oils by the Cryoscopic Method; ASTM International: West Conshohocken, PA, 1983, www.astm.org. (24) Farah, M. A. Caracterizaçaõ de frações de petróleo pela ́ viscosidade. Thesis; Escola de Quimica, Universidade Federal do Rio de Janeiro, Rio de Janeiro. 2006. (25) ASTM D445−06, Standard Test Method for Kinematic Viscosity of Transparent and Opaque liquids (and Calculation of Dynamic Viscosity; ASTM International: West Conshohocken, PA, 2006, www.astm.org. (26) Farah, M. A.; Oliveira, R. C.; Caldas, J. N.; Rajagopal, K. Viscosity of water-in-oil emulsions: Variation with temperature and water volume fraction. J. Pet. Sci. Eng. 2005, 48, 169−184. (27) Mello, M. R.; Koutsoukos, E. A. M.; Figueira, J. C. A. Brazilian and West African oils: generation, migration, accumulation and correlation. WPC-24119; 13th World Petroleum Congress; Buenos Aires, Argentina,1991. (28) Milani, E. J.; Brandão, J. A. S. L.; Zalán, P. V.; Gamboa, L. A. P. Petróleo na margem continental brasileira: geologia, exploraçaõ , resultados e perspectivas. Brazilian J. Geophys., 2000, 18 (3), 351−396. 10.1590/S0102-261X2000000300012.

ACKNOWLEDGMENTS The authors acknowledge the financial support of PETROBRAS/CENPES and ANP related to the grant from R&D investment rule and COPPETEC for the scholarship awarded to L.L.P.R.A. The authors thank Ian Hovell and Luis Augusto Medeiros Rutledge for help in the experiments and Rogério Fernandes de Lacerda for comments.

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NOMENCLATURE ρo = density of oil, kg m−3 μo = undersaturated oil viscosity, mPa s μob = saturated oil viscosity, mPa s μod = dead oil viscosity, mPa s API = oil API gravity, oAPI AARE = average absolute percent relative error do = oil specific gravity (water = 1) ND = number of data sets p = pressure, kPa pb = bubble point pressure, kPa SDA = percent absolute standard deviation T = temperature, K Xexp = experimental variable Xcalc = calculated variable

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DOI: 10.1021/acs.energyfuels.7b03903 Energy Fuels 2018, 32, 2622−2629