Extension of the Expanded Fluid Viscosity Model to Characterized Oils

Article pubs.acs.org/EF

Extension of the Expanded Fluid Viscosity Model to Characterized Oils H. Motahhari,† M. A. Satyro,‡ S. D. Taylor,§ and H. W. Yarranton*,† †

Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta T2N 1N4, Canada Virtual Materials Group, Calgary, Alberta T3G 2A7, Canada § DBR Schlumberger, Edmonton, Alberta T6N 1M9, Canada ‡

S Supporting Information *

ABSTRACT: The Expanded Fluid (EF) viscosity model for Newtonian fluids is extended to crude oils characterized as mixtures of defined components and pseudo-components. The EF models take the fluid density, dilute gas viscosity, pressure, and fluid composition as inputs and requires three fluid-specific parameters, c2, c3, and ρos , for the fluid or its components. Generally, experimental viscosity data are required to determine these values for each component. In this study, an internally consistent estimation method was developed to predict the fluid-specific parameters of the model for hydrocarbons when no experimental viscosity data are available. The method uses n-paraffins as the reference system and correlates the fluid-specific parameters for hydrocarbons as departures from the reference system. The method was evaluated against viscosity data of over 250 pure hydrocarbon compounds and petroleum distillation cuts. The model predictions were within the same order of magnitude of the measurements, with an overall average absolute relative deviation of 31%. The method was then used to calculate the correlation parameters for the pseudo-components of nine dead and live oils characterized on the basis of their gas chromatography (GC) assays. The viscosities of the crude oils were predicted within a factor of 3 of the measured values using the measured density of the oils as the input. The applicability of the EF model was also demonstrated using the densities determined with the Peng− Robinson equation of state. A simple method was proposed to tune the model to available viscosity data using a single multiplier to the c2 parameter (and also to c3 and ρos if necessary) of the pseudo-components. Single-parameter tuning of the model improved the viscosity prediction for the characterized oils to within 30% of the measured values.

1. INTRODUCTION Reliable liquid viscosity estimates for crude oils are required over a wide range of temperatures, pressures, and compositions for reservoir and process engineering calculations, including flow in porous media, pressure drops in pipelines and heat exchangers, and determination of heat- and mass-transfer coefficients. The viscosity is usually calculated simultaneously with phase equilibrium using viscosity models embedded in reservoir and process simulators. A suitable viscosity model for this purpose must (1) trace continuously the full range of single-phase properties in the gas, liquid, critical, and supercritical regions, (2) be fast, (3) predict both purecomponent and mixture viscosities, (4) be compatible with the fluid characterization used for the phase behavior model. It is common practice to characterize a crude oil as a mixture of some defined components and pseudo-components.1 The defined components are components identifiable with analytical assays and typically include light hydrocarbons up to normal hexane and certain non-hydrocarbons, such as carbon dioxide and hydrogen sulfide. Components heavier than hexane, usually termed the C7+ fractions, are represented by a limited number of pseudo-components based on assays, such as true boiling point (TBP) distillation or gas chromatography.1 Each pseudocomponent represents the mass fraction and average properties of either a boiling point or a molecular weight range within the overall distribution. Physical properties, such as molecular weight (MW), specific gravity (SG) and normal boiling point (NBP), are measured or defined for each fraction, and the © 2013 American Chemical Society

critical properties and acentric factor are then estimated using well-known correlations. Although several viscosity models are available in the literature, as reviewed by Poling et al.2 and Monnery et al.,3 few models are suitable to model the viscosity of oils characterized using pseudo-components. Suitable models include corresponding state methods, friction theory, and empirical correlations. Corresponding state models relate the reduced viscosity of the fluid to the reduced viscosity of a reference fluid at the same reduced temperature and density coordinates.4 The non-correspondence of most fluids to the reference fluid is corrected using shape factors, which are generally density- and temperature-dependent. Variations of Ely and Hanley’s method5 are often used in property calculators6 and process simulators7 to model the viscosity of the light fraction of the crude oils. Baltatu and co-workers8,9 showed that this model can be used for heavier hydrocarbons, such as crude oil distillation cuts, using aromaticity corrections or mass shape factors. Data for physical boiling point cuts of several crude oils were used to formulate the corrections. Special Issue: 13th International Conference on Petroleum Phase Behavior and Fouling Received: September 25, 2012 Revised: January 10, 2013 Published: January 10, 2013 1881

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Pedersen and co-workers10,11 developed a corresponding state viscosity model, in which the reduced pressure is one of the coordinates instead of the reduced density. This model uses methane as the reference fluid and was developed for characterized reservoir fluids. The model also provides reliable viscosity predictions for pure hydrocarbon compounds. The model is tuned with an effective molecular weight of the mixtures and the rotational coupling coefficient, a parameter replacing the shape factors. Empirical correlations for these parameters were developed from a database of experimental viscosity data of the pure components, characterized reservoir fluids, and petroleum distillation cuts. The model is implemented in most commercial reservoir simulators12,13 and provides reliable viscosity predictions for most conventional reservoir fluids.14 However, it can be challenging to apply this model to heavy petroleum fluids15 because these fluids often correspond to methane at temperatures below its freezing point. The “friction theory” (F theory) viscosity model16 has also been successfully applied to characterized crude oils. This model relates the viscosity of the dense fluid to the friction between fluid layers. The friction force is related to the contributions of the repulsive and attractive internal pressures of the fluid. The cubic Peng−Robinson (PR) equation of state (EoS) and Soave−Redlich−Kwong (SRK) EoS are used to calculate the van der Waals pressure terms.16 A version of the model was developed for hydrocarbons with only one adjustable parameter17 for each fluid, using a characteristic critical viscosity. This version of the model was extended to crude oils characterized into pseudo-components.18 The critical viscosity values of the pseudo-components of the oil were related to their molecular weight and critical pressure and temperature through one common proportionality factor. The parameter must be determined by tuning the model to crude oil viscosity data. The model also uses one other tuning parameter19 for the characterized oils with molecular weights higher than 200 g/mol. This new parameter is used to correct the estimation of the repulsive and attractive pressure terms calculated with the EoS. One- or two-parameter tuned models successfully fitted experimental data above the saturation pressure and provided predictions at lower pressures to within experimental uncertainties.18−20 Zuo et al.21 introduced temperature dependence into these two parameters to more accurately model the viscosity of the characterized oils at different temperatures. Empirical viscosity correlations are usually specific to the gas or liquid phase22,23 and have been used to estimate kinematic viscosity of distillation cuts of crude oils, characterized by their average normal boiling point and specific gravity. These correlations are widely used in the process simulators to predict viscosity of petroleum fractions and heavy hydrocarbons.7,24 One of the earliest gas−liquid consistent correlations is a modification of the empirical residual method,25 known as the Lohrenz−Bray−Clark (LBC) correlation.26 This correlation relates the dense fluid viscosity to the reduced density by a fourth-degree polynomial. Although the correlation is simple and fast and is implemented in most reservoir simulators,12,13 its application to heavy hydrocarbon fluids can be unreliable.14 It is common to tune the correlation to the available experimental data by adjusting the critical volumes of the C7+ fractions. The expanded fluid (EF) model27,28 is a recently developed empirical correlation designed for modeling the viscosity of

hydrocarbons. This model satisfies all of the criteria previously set for a viscosity model suitable for simulation purposes. Each fluid is characterized for the model with three fluid-specific parameters and uses the density of the fluid as the input. The EF model was successfully applied to determine the viscosity of the pure hydrocarbons, including n-alkanes, branched alkanes, aromatics, and cyclics, as well as the common non-hydrocarbons in the natural gas processing industry over a wide range of pressures and temperatures in gas and liquid phases.27−29 The model also predicted the viscosity of over 100 mixtures of hydrocarbon and non-hydrocarbon components, with overall average deviations less than 8%, in both gas and liquid phases.30 The EF model has also been used to determine the viscosity of heavy oils/bitumen and their mixtures with solvents.27,28,31,32 However, the model is not predictive for hydrocarbon fluids and requires experimental viscosity data to determine the fluidspecific parameters. Its application to crude oils characterized using pseudo-components has not yet been investigated. The objective of this study is to extend the EF viscosity model to crude oils characterized using pseudo-components. First, a simple and internally consistent estimation method is developed to predict the fluid-specific parameters of the model for hydrocarbons when no experimental viscosity data are available. The model parameters are correlated as departures from a n-paraffin reference system. The validity of the viscosity predictions from this method are tested against viscosity data of pure heavy hydrocarbons and distillation cuts of the crude oils. Second, the parameter correlations are used to calculate the parameters of the EF model for the pseudo-components of the characterized crude oils. The viscosities of the crude oils are predicted using the measured densities as an input. A simple approach is proposed to tune the predictions to measured viscosity data. Finally, the application of the EF viscosity model to characterized crude oils using densities estimated by the PR EoS is demonstrated.

2. EF VISCOSITY MODEL The EF viscosity model is based on the empirical observation that, as the fluid expands, its viscosity decreases. The reduction of the viscosity is related to the increased fluidity of the EF because of the greater distance between the molecules.33 When the model is based on measured densities,27 the viscosity of the fluid (μ) in mPa s is given by

μ − μo = 0.165(exp(c 2β) − 1) β=

(1)

1 ⎧ exp⎨ ⎩

0.65 ρs*

() ρ

⎫ − 1⎬ − 1 ⎭

(2)

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 compressed state of the fluid is, by definition, a hypothetical state of the fluid, beyond which the fluid cannot be compressed without incurring a solid−liquid phase transition. It is assumed that the viscosity of the fluid approaches infinity as the fluid density approaches the compressed state density.27 The compressed state density is a function of pressure as follows:

ρs* =

ρso exp(−c3P)

(3)

is the compressed state density in vacuum, c3 is a fitting where constant in kPa−1, and P is the pressure in kPa. For many pure hydrocarbons, the parameters c2 and c3 were related27 to the viscosity of the fluid at 25 °C and molecular weight, respectively. However, in general, the model has three temperatureρos

1882

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independent parameters for each fluid: c2, c3, and ρos . The inputs to the model are the measured fluid density, pressure, and dilute gas viscosity. Note that the effect of the temperature on viscosity is introduced through the density and dilute gas viscosity, both of which are temperature-dependent. The model is only valid for Newtonian fluids. 2.1. Dilute Gas Viscosity. The dilute gas viscosity of the fluid is the viscosity of the fluid in the zero-density limit. Yaws34 used the following empirical correlation to express the dilute gas viscosity for several hydrocarbons and non-hydrocarbons:

μo = A + BT + CT 2 + DT 3

μo,mix =

i

xiμo, i ∑j xjφij

(13)

where φij is given by φij =

0.25 2 [1 + (μo, i /μo, j )0.5 (MWj/MW) ] i 0.5 [8(1 + MW/MW)] i j

(14)

and xi and μo,i are the mole fraction and dilute gas viscosity of the component “i” of the mixture, respectively. 2.3. Application to Oils Characterized Using Pseudocomponents. Figure 1 shows the proposed algorithm for the

(4)

where A, B, C, and D are fluid-specific fitting parameters. The values of these parameters are obtained from Yaws’ handbooks34,35 for several compounds to calculate the dilute gas viscosity. Dilute gas viscosity for compounds not listed in Yaws’ handbooks is calculated using the method by Chung et al.,36 a modification of the Chapman−Enskog theory, given by

μo = 4.0785 × 10−3

∑

Fc MWT vc 2/3Ω υ

(5)

where μo is in mPa s, MW is the molecular weight in g/mol, T is the temperature in K, and vc is the critical volume in cm3/mol. Ωυ is the viscosity collision integral given by

Ω υ = 1.16145(T *)−0.14874 + 0.52487 exp(− 0.7732T *) + 2.16178 exp( −2.43787T *)

(6)

where T* is a dimensionless temperature defined using the critical temperature (Tc) as follows: T * = 1.2593

T Tc

(7) Figure 1. Flow diagram of the algorithm for application of the EF viscosity model to characterized oils.

Fc in eq 5 is an empirical correction factor defined as follows:

Fc = 1 − 0.2756ω + 0.05903μr 4 + κ

(8) application of the EF model to characterized crude oils. First, the crude oil is represented as the mixture of defined components and pseudocomponents. The numerical values of the fluid-specific parameters of the EF model are known for the defined components. Estimation methods are required to relate numerical values of the fluid-specific parameters of the EF model to the basic physical properties of the pseudo-components, such as molecular weight, normal boiling point, and specific gravity. Second, the fluid-specific parameters for the oil are calculated with the mixing rules, and the oil viscosity is then predicted at any given pressure and temperature. The input oil density can be the measured value or the value estimated from a density model, such as an EoS. Finally, the model can be tuned against the experimental viscosity data if available. Single common multipliers (αX) are applied to the estimated fluid-specific parameters of the pseudo-components as follows:

where ω is the acentric factor, μr is the dimensionless dipole moment, and κ is the special correction for a highly polar and associating compound, such as alcohols and acids. For mostly nonpolar hydrocarbons, eq 8 reduces to Fc = 1 − 0.2756ω

(9)

2.2. Mixing Rules. The EF model treats a mixture as a single fluid. The fluid-specific parameters of the mixture are calculated with the following mixing rules:30 o ρs,mix

c 2,mix o ρs,mix

c3,mix

⎞−1 ⎛ nc nc ww ⎛ ⎞ 1⎟ i j 1 ⎟ ⎜ ⎜ = ⎜∑ ∑ ⎜ o + ρo ⎟(1 − βij)⎟ s, j ⎠ ⎠ ⎝ i = 1 j = 1 2 ⎝ ρs, i nc

=

nc

∑∑ i=1 j=1

⎛ ww c 2, j ⎞ i j c 2, i ⎟(1 − β ) ⎜ o + ij ⎜ 2 ⎝ ρs, i ρs,oj ⎟⎠

⎛ nc w ⎞−1 = ⎜⎜∑ i ⎟⎟ ⎝ i = 1 c3, i ⎠

(10)

Xi = αX Xi(est)

(11)

(15)

where X is any of the viscosity characterization parameters c2, c3 or ρos . Because the mixing rules of the EF model are linear when βij = 0, the common multiplier also applies directly to the calculated fluid-specific parameters of the crude oil. The method used to characterize crude oils is presented in section 3, and the models used to predict density are provided in section 4. The main focus of this study is on the estimation of model parameters for pseudo-components, which is presented in section 5. Section 6 illustrates the application of the model to a number of examples.

(12)

where nc is the number of components in the mixture, wi is the mass fraction of the component “i” in the mixture, and βij is a binary interaction parameter with a default value of zero. The binary interaction parameter can be used to tune the model when experimental viscosity data for the mixture are available. It was shown30 that, in almost all cases, predictions with βij set to zero are within the experimental uncertainties. Therefore, βij was set to zero in this study. The mixture dilute gas viscosity is calculated using Wilke’s method37 as follows:

3. OIL CHARACTERIZATION Experimental viscosity and density data of nine crude oils were used in this study to evaluate the extension of the EF model to characterized oils. Table 1 gives a summary of the MW and SG 1883

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Table 1. Summary of the Crude Oils and Conditions Evaluated in This Study oil description WC-B-B2 WC-B-HO1 WC-HO5 EU1 ME3a ME1 ME2 EU2

AS1

dead dead dead dead live live live live dead live lived lived dead live

mass fraction of C30+

MWa (g/mol)

SG

temperature (K)

pressure (kPa)

viscosity (mPa s)

0.7 0.68 0.67 0.46 0.45 0.33 0.13 0.66 0.59 0.58 0.58 0.58 0.47 0.39

520 424 484 327 238 186 73 373 362 302 303 279 271 186

1.018 0.997 1.005 0.936 0.900b 0.831c 0.667c 0.921c 0.903 0.886b 0.886b 0.879b 0.875 0.827b

293−448 298−448 298−473 298−318 325 334−448 386 331−423 313−423 313 313 313−423 354 354

100−10000 100−10000 100−8270 100 9900−34480 4200−13900 27600−37900 500−10000 1600−2100 4100−15100 3100−35100 7600−20100 100 6900−55200

13−160000 8.5−7700 7.9−36000 83−285 28−46 0.6−3.3 0.19−0.22 8.9−184 7−346 132−166 137−214 5.9−166 2.9 1.5−2.6

Estimated after GC assay extrapolation, except for WC-B-B2, which was measured with vapor pressure osmometry in toluene at 50 °C. bLive oil SG calculated with eqs 22 and 23 with Cf determined from dead oil SG and the assumption of no gas separation. cLive oil SG calculated with eqs 22 and 23 assuming Cf = 0.29 and no gas separation. dLive crude oils enriched with light n-alkanes. a

at standard conditions of the dead crude oils as well as the pressure and temperature ranges at which data are available. Note that the SG of the dead oil (or corresponding live oil) is required to apply the model. In this study, the SGs of dead oils were used to characterize the pseudo-components of the dead oils and the corresponding live oils, except for crudes ME3a, ME1, and ME2, where the SG of the dead oil was not available. For these crudes, the live oil SG was calculated from the GC assay, as reported in Table 1. Viscosity and density for bitumen WC-B-B2 and WC-B-HO1 were measured at the University of Calgary using a capillary viscometer with an inline Anton-Paar density meter. Data for WC-HO5 were provided by the DBR Technology Centre and were collected using a similar methodology. Data for all other oils were obtained from commercial PVT studies provided by a sponsor and are from different geographical areas. The GC assays for the crude oils are given in the Supporting Information. Gas chromatography (GC) assays are the basis for the oil characterizations prepared in this study. The GC assay provides a quantitative relative amount of different compounds in the crude, commonly in terms of the mass fraction and MW of the standard carbon number (SCN) fractions. On the basis of the GC assay technique and the style of the report, SCN data are reported up to carbon number n − 1 and the heavier hydrocarbons of the oil are usually lumped and reported as n plus fraction; for example, C7+, C11+, and C30+. The oils in this study were characterized up to the SCN fraction of C29 by GC, and the residue is reported as C30+. The C30+ fraction comprised 13−70% of the oils in this data set (Table 1). 3.1. GC Assay Extrapolation. Given that a significant amount of the crude is lumped into the plus fraction, the GC assay must be extrapolated to estimate the heavier carbon number (CN) fractions and completely characterize the oil. For this purpose, molar distribution of the carbon number fractions in C30+ is assumed to follow an exponential distribution. The exponential distribution is a special form of the general threeparameter gamma distribution with shape factor α = 11 given by f (MW) =

where MW is the molecular weight, MWC30+ is the molecular weight of the C30+ fraction, and MWmin is the minimum molecular weight found in the C30+ fraction. The mole fraction (xi) of carbon number fraction “i”, which includes the compounds with molecular weights between MWbi−1 and MWbi, is then given by xi = xC30+[f0 (MWbi) − f0 (MWbi − 1)]

(17)

and f 0(MWbi) is defined as follows: ⎛ MW − MW ⎞ bi min ⎟ f0 (MWbi) = −exp⎜⎜ − ⎟ MW − MW ⎝ C30 + min ⎠

(18)

The mole fraction of the C30+ fraction is generally unknown and is estimated as follows: xC30+ =

wC30+ MWoil MWC30+

(19)

where wC30+ is the mass fraction of the C30+ fraction and MWoil is the average molecular weight of the oil. The average molecular weight of the corresponding fraction is given by MWi =

f1 (MWbi) − f1 (MWbi − 1) f0 (MWbi) − f0 (MWbi − 1)

(20)

where f1(MWbi) is defined as follows: f1 (MWbi) = (MWbi + MWC30+ − MWmin)f0 (MWbi)

(21)

The GC assay was extrapolated by defining 63 carbon number fractions with molecular weights evenly distributed between the molecular weight of the standard C29 fraction and the maximum molecular weight found in the oil. The value of the maximum molecular weight was set to 4000 g/mol. The value for MWmin is set as equal to the upper bound molecular weight of the last reported SCN fraction, i.e., C29. Thereafter, the value of MWC30+ is adjusted to match the calculated average molecular weight of the oil from the reported measured MW of the oil. If the MW of oil is not measured, the MWC30+ is adjusted to have a smooth

⎛ MW − MW ⎞ 1 min ⎟ exp⎜⎜ − MWC30+ − MWmin ⎝ MWC30+ − MWmin ⎟⎠ (16) 1884

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densities from the PR EoS, instead of measured densities. Details of the density models are provided below. 4.1. Modified Rackett Correlation. Spencer and Danner’s39 well-known modification to the Rackett correlation40 was used to predict the density of the petroleum distillation cuts. The modified correlation is given by

transition of the molar distribution of the SCN fractions from C29 to higher SCNs. The specific gravities of the carbon number fractions were determined from the following correlation:38 SGi = 0.2855 + Cf (MWi − 66)0.13

(22)

Cf is a tuning parameter and determined by matching the calculated specific gravity of the oil to the measured value. The specific gravity of the crude is calculated as follows: −1 ⎛ wk ⎞ ⎟⎟ SGoil = ⎜⎜∑ ⎝ k SGk ⎠

vs =

(25)

where vs is the molar volume of the saturated liquid at temperature T, Tc and Pc are critical constants of the fluid, R is the universal gas constant, and Z RA is the Rackett compressibility. The numerical values of the Rackett compressibility have been regressed against saturated liquid molar volume data and are tabulated for pure components.2 There are also correlations relating this parameter to other properties of the fluids, such as the acentric factor.41 However, in this study, the Rackett parameter is used as a tuning parameter to match the reported specific gravity of the petroleum distillation cuts. Note that the correlation is only valid for saturated liquids. Saturated liquid densities are sufficient to test the performance of the EF correlations for petroleum cuts, which are generally studied in the liquid state at atmospheric pressure, where compression corrections to the saturated liquid volume are very small. For fluids subjected to very high pressures, a correction to Rackett-estimated densities may be applied using the Tait equation. 4.2. PR EoS. The PR EoS42,43 is a cubic EoS and is widely used in process and reservoir simulators for the phase behavior modeling of the characterized oils, along with cubic SRK EoS.44 However, both lack the capability to accurately predict the liquid molar volume. This deficiency is usually corrected using volume translation.45 The detailed formulation of the PR EoS is given in Appendix 1: PR EoS. Volume translation45 is introduced to improve the molar volume (i.e., density) predictions of the PR EoS as follows: v = vEoS − c (26)

(23)

where k in this summation includes all of the components of the oil; that is, the C1−C29 fractions from the GC assay, the carbon number fractions from the extrapolation of the GC assay, and the defined non-hydrocarbon components, such as carbon dioxide, hydrogen sulfide, and nitrogen. Note that eq 22 was originally proposed for the carbon number fractions heavier than C6. Therefore, the recommended values of SG by the American Petroleum Institute (API)23 are used for the standard carbon number fractions from C1 to C6 (the values from C1 to C4 are hypothetical). 3.2. Pseudo-component Definition. Once the complete description of the oil was constructed as the molar distribution of the carbon number fractions and their specific gravities, the oil was divided into a number of defined components and pseudo-components. Defined components include defined non-hydrocarbons and standard carbon number fractions up to C6, which are modeled as the corresponding normal paraffin. Each pseudo-component represents a specific molecular weight range above C7. A total of 7−13 pseudo-components were defined for each crude oil depending upon the maximum molecular weight. A set of consecutive carbon number fractions was lumped into each pseudo-component. The average properties for the pseudo-components were calculated as follows: −1 ⎛ wz ⎞ ⎜ ⎟ θj̅ = ⎜∑ ⎟ ⎝ z θz ⎠

2/7 RTc Z RA[1 + (1 − T / Tc) ] Pc

where v is the corrected molar volume, vEoS is the EoScalculated molar volume, and c is the fluid-specific volume shift. To improve the calculated molar volumes over a broad range of temperatures, a linear temperature-dependent volume shift is introduced as follows:

(24)

where θ is the molecular weight or specific gravity and z are the carbon number fractions lumped into the pseudo-component “j”.

c = γ0 + γ1(T − 288.75)

(27)

where T is the temperature in K, γ0 is the fixed volume shift, and γ1 is the temperature dependency term. The values of γ0 and γ1 must be determined for each component by matching the predicted liquid molar volumes by the EoS to the actual measured molar volumes of the saturated liquids at two given temperatures. Numerical values of γ0 and γ1 for the defined components of the characterized oils are given in Table 2. These values were determined by matching the predicted molar volumes of these compounds by EoS to the actual molar volumes of the saturated liquids35 at two temperatures, 288.75 K (60 °F) and the reduced temperature (Tr) of 0.85. For the components with 0.85Tr below 288.75 K, the saturated liquid molar volume at the normal boiling point was used, instead of at 288.75 K. The values of γ0 and γ1 for the pseudo-components of each characterized oils must be individually determined. The coefficients for each pseudo-component were adjusted to fit

4. DENSITY MODELING The development and testing of the estimation methods for the fluid-specific parameters of the EF model were based on measured density data. However, density predictions were required to use the EF model to predict the viscosity of the petroleum fluids lacking density data. Any density model can be used to provide the input densities for the EF viscosity model; however, for general application, the density model must apply to liquid, gas, and fluid phases. Also, because the viscosity model is sensitive to density, particularly near the compressed state density conditions, the density model must provide sufficiently accurate estimations of the density of the fluids, especially for saturated liquids. The modified Rackett correlation39 was used to estimate the density of petroleum distillation cuts. In addition, to demonstrate the application of the viscosity model in simulation, the model was tested with 1885

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X = X(ref) + ΔX

Table 2. Constants of Temperature-Dependent Volume Translation, eq 27, for Defined Components of the Characterized Oils component

γ0 (cm3/mol)

γ1 (cm3 mol−1 K−1)

carbon dioxide methane ethane propane n-butane n-pentane n-hexane

0.0929 2.3608 −1.8087 −2.6827 −4.2303 −3.3612 −0.3511

0.0228 0.0364 0.0252 0.0343 0.0380 0.0384 0.0339

where X is c2 or The reference n-paraffin has the same molecular weight as the hydrocarbon compound or pseudocomponent of interest. The departure from the paraffinic values because of the aromaticity of the hydrocarbon is formulated by a quadratic polynomial approximation given by ΔX = AX ΔSG2 + BX ΔSG

1

ΔSG = SG − SG(ref)

c3 =

where ρT1 and ρT0 are liquid densities in temperatures T1 and T0 (288.75 and 423.15 K, respectively) and A is given by A=

613.9723 ρT 2

2.8 × 10−7 1 + 3.23 exp( −1.54 × 10−2MW)

(33)

The above correlation approximates the correlation by Yarranton and Satyro27 and converges to a fixed value of 2.8 × 10−7 kPa−1 for higher molecular weight hydrocarbons. The reference system and the departure functions for c2 and ρos are discussed individually below. 5.1. n-Paraffin Reference System. Experimental viscosity and density data for the family of n-paraffins from methane (nC1) to n-tetratetracontane (n-C44) were compiled from the National Institute of Standards and Technology (NIST) database52 and API Project 42.53 The data for the n-paraffins up to n-C16 include measurements at higher pressures, whereas the data for heavier n-paraffins are at atmospheric conditions. The dilute gas viscosity of n-paraffins was calculated using eq 4 with parameters obtained from Yaws’ handbook.34 Following the regression approach described in detail by Satyro and Yarranton,28 the parameters of the EF viscosity model for these components were calculated from the experimental data. Then, the parameters were correlated as functions of the molecular weight of n-paraffin components. Also, specific gravities of reference n-paraffins were correlated to the molecular weight. 5.1.1. SG. The SG values of the reference n-paraffin components from C1 to C20 were obtained from the API handbook.23 SG values for heavier n-paraffins were estimated on the basis of the experimental density data of these components. The density of water at 15.6 °C is 999.022 kg/ m3 23 and was used to convert density to SG at standard conditions of 15.6 °C and 1 atm. Note that the n-paraffins from methane to n-butane (n-C4) are in the gaseous state at standard conditions; hence, their SG were based on the standard liquid densities recommended by the API.23

(29)

0

(32)

In this formulation, the molecular weight captures the effect of the size of the compound, while the specific gravity adds the contribution of the chemical family type of the hydrocarbon. Note that the development of a comprehensive estimation method for parameter c3 of the EF model is not feasible at this time because of limited availability of high-pressure viscosity and density data of the heavy hydrocarbons. Therefore, the following adaptation of the original development by Yarranton and Satyro27 is proposed to use for the purpose of this study:

(28)

0

(31)

where AX and BX are fitting parameters and ΔSG is given by

the EoS-estimated liquid densities to independently determined values at two temperatures, 288.75 and 423.15 K. Following the approach by Pedersen et al.,46 the liquid density at 288.75 K was set to the defined specific gravity of the pseudocomponent, while the density at 423.15 K was calculated using the ASTM 1250-80 correlation for the thermal expansion of the stable oils as follows: ρT = ρT exp[−A(T1 − T0)(1 + 0.8A(T1 − T0))]

(30)

ρos .

5. ESTIMATION METHODS FOR EF MODEL PARAMETERS Estimation methods are required to relate the fluid-specific parameters of the EF model to the physical properties of the pseudo-components. The obvious candidates to represent the pseudo-components of the characterized oils are petroleum distillation cuts. However, experimental viscosity data for the cuts are scarce in the literature, and the reported data do not include corresponding density measurements. Instead, pure heavy hydrocarbon compounds are selected as the model components for the pseudo-components because of their availability in the literature. The physical properties commonly used to develop the estimation methods are the normal boiling point and specific gravity,47 which roughly characterize the molecular energy and size, respectively. The normal boiling point is not readily available for the heavy hydrocarbons, but the molecular weight is. Therefore, the molecular weight and specific gravity were selected as the correlating properties.48 Note that the molecular weight and normal boiling point of the petroleum fluids are generally correlated through some well-known methods.38,49−51 The estimation methods of the parameters of the EF model were formulated as departure functions from a reference system. The family of n-paraffins was chosen as the reference system. The form of the departures is given by

Table 3. Constants of the Parameter Correlations (eqs 34, 36, and 37) for the Reference n-Paraffin System from C1 to C44 and beyond property

a0

a1

a2

a3

a4

a5

a6

SG ρos (kg/m3) c2

0.843593 −4775 9.353 × 10−2

0.1419 3.984 4.420 × 10−4

−16.60 0.4000 −333.4

−41.27 −1.298 × 10−3 −1.660 × 10−4

2535 938.3 4.770 × 10−2

8.419 × 10−2

−1.060 × 10−3

1886

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Figure 2. (a) Correlated and measured specific gravity of n-paraffins from C1 to C44 and (b) relative deviation of correlated SGs [(SGcorr − SGexp)/ SGexp] versus molecular weight from C1 to C44.

Figure 3. (a) ρos of n-paraffins determined from fitting EF model to viscosity data (symbols) and correlated ρos (lines); and (b) observation of the ratio of ρos to the density of n-paraffins at 15.6 °C approaches to a plateau.

The specific gravities of the n-paraffins were correlated as a function of the MW as follows: SG(ref) = a0SG +

a1SG MW

0.5

+

where K is a constant. The compressed state density of the reference n-paraffin family was then correlated to the molecular weight as follows:

a a a 2SG + 3SG2 + 4SG3 MW MW MW

ρso

(34)

(ref)

where a0−a4 are constants of the correlation, which were determined by regression and are given in Table 3. The value of a0, the limiting specific gravity for n-paraffin with infinite molecular weight, was set to 0.843 593, as previously used by Twu49 in the development of the correlation for the specific gravity of n-paraffins as a function of the normal boiling point. The fitted correlation is shown in Figure 2. The average absolute relative deviation (AARD) and maximum absolute relative deviation (MARD) of the correlated values from the experimental values are 0.5 and 2.5%, respectively. 5.1.2. Compressed State Density. The compressed state density values of n-paraffins up to n-C44 (618 g/mol) were obtained from experimental data (Figure 3a). To extrapolate to higher molecular weights, it was noted that the ratio of the compressed state density to the density of the n-paraffins at 15.6 °C reaches an asymptote at molecular weights higher than ∼300 g/mol (Figure 3b). Hence, the assumption was made to extrapolate the compressed state density of the n-paraffins to higher molecular weights subject to the following constraint: limMW →∞

ρso

(ref)

ρ(ref) at 15.6 ° C

⎛ a 0ρ o ⎞ o = ⎜ s + a1ρso MW a2ρs ⎟exp(a3ρso MW) ⎝ MW ⎠ a4ρso + 1 + a5ρso exp(a6ρso MW)

(36)

where a0−a6 are the constants of the correlation and are listed in Table 3. Figure 3a compares the correlated compressed state density values against the values obtained from fitting the EF model to the measured data. The regressed numerical value of constant K in eq 35 was determined to be 1.12, Figure 3b. 5.1.3. Parameter c2. The values of the parameter c2 of nparaffins up to n-C44 were obtained from experimental data, Figure 4; but, no experimental data are available to extrapolate c2 to molecular weights higher than 618 g/mol. Instead, the atmospheric viscosity data of the WC-B-B2 bitumen were used to guide the extrapolation of c2. The detailed characterization of the bitumen is discussed later. Briefly, the extrapolated c2 correlation for MW higher than 618 g/mol, coupled with the departure function (described later), was set to predict the correct viscosity of the characterized WC-B-B2 bitumen. The final correlation of parameter c2 to molecular weight for the reference n-paraffin family is given by

=K (35) 1887

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Information. The EF model was fitted to the data of the heavy hydrocarbons to determine the numerical values of the compressed state density and parameter c2 for each compound. Then, the departures of c2 and ρos for each compound from the corresponding reference n-paraffin were calculated using the reference equations (eqs 36 and 37, respectively). Note that the calculated departures of c 2 were independent of the extrapolation of c2 for the reference n-paraffin system because the molecular weights of the pure heavy hydrocarbons in the data set are all less than 618 g/mol. The departures of ρos (Δρos ) increased almost monotonically with increasing ΔSG (Figure 5a). However, the calculated departures of c2 (Δc2) were significantly scattered (Figure 5b). The departures of c2 were found to depend upon not only specific gravity but also the structure of the molecules, that is, the number and types of branches, the relative positioning of the fused rings, as well as the relative positioning of the branches and non-fused rings on the main chain of the molecule. It is not practical to construct a correlation including structural parameters for pseudo-components that represents a mixture of unknown structures. Therefore, the departure functions were fitted to data for 24 single- and fused-ring naphthenic and aromatic hydrocarbons (the development group). The departures of c2 and ρos for the development group are highlighted as the open circles on panels a and b of Figure 5. The remaining 148 heavy hydrocarbons (test group) were used to evaluate the departure functions and were not used in their development. Note that adding more components to the development group does not improve the data fitting because the correlations do not account for structural effects. Equation 31 was expanded as follows to correlate the departure functions:

Figure 4. c2 parameter of n-paraffins determined from fitting the EF model to viscosity data (symbols) and the c2 parameter correlation (line).

⎛ a 2c ⎞ c 2(ref) = (a0c2 + a1c2 MW)exp⎜ 2 + a3c2 MW⎟ + a4c2 ⎝ MW ⎠ ln(MW)

(37)

where a0 to a4 are the constants of the correlation and are listed in Table 3. Figure 4 shows the extrapolation of c2 to molecular weights above 618 g/mol. Note that the molecular weight of the heaviest pseudo-component of the characterized WC-B-B2 bitumen is 2791 g/mol. Although eq 37 smoothly extrapolates to molecular weights as high as 4000 g/mol, its validity for pseudo-components with MW beyond 2791 g/mol was not studied. The application of eq 37 to higher molecular weights is unnecessary as long as the high-molecular-weight SCN fractions are lumped into a pseudo-component with an average molecular weight lower than 2791 g/mol. As a test, the correlations for parameters c2 and ρos were used in the EF model to predict the viscosity of the n-paraffins from n-C1 to n-C44. The AARD and MARD were 7.4 and 65%, respectively. Note that the correlations will not be used to predict n-paraffin properties, only in the prediction of pseudocomponent properties. 5.2. Departure Functions for Heavy Hydrocarbons. Experimental viscosity and density data for over 170 diverse hydrocarbons, including branched paraffins, aromatics, and naphthenes, were compiled from the NIST database,52 API Project 42,53 and the Thermodynamic Research Center (TRC) handbook.54 Details of their chemical families are provided later in section 6.2, and the compounds are listed in the Supporting

⎛ ⎛ b3 ⎞ b1 ⎞ ⎟ΔSG2 + ⎜b2 + ⎟ΔSG ΔX = ⎜b0 + b ⎝ ⎝ MW 4 ⎠ MW b4 ⎠

(38)

where ΔX is the desired departure (Δρos or Δc2) and b0−b4 are the fitting parameters. The values of fitting parameters determined by simultaneous regression for both parameters and their numerical values are given in Table 4. The objective of the least-squares regression was to minimize the deviation between the predicted viscosity (using parameters calculated from reference and departure functions) and the measured viscosity for the hydrocarbons in the development group. The AARD and MARD of the viscosity predictions for the development group were 24 and 180%, respectively. The

Figure 5. Correlated and obtained from data departures of the parameters of the EF model for pure heavy hydrocarbons for (a) compressed state density and (b) parameter c2. 1888

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viscosities for the limited higher pressure data (up to 500 MPa) available for seven of these components52,53 were within the same overall AARD and MARD. Figure 6a shows that, for a fixed molecular weight, the viscosity model correctly predicts the viscosity of the reference n-paraffin as well as the increasing trend of the viscosity versus specific gravity at the temperatures of 310.9 and 372.0 K. Note that the correlation does not predict the abrupt increase of the viscosity for the three components (9-n-octyl-perhydronaphthacene, 2-n-octyl-perhydrotriphenylene, and 2-n-octyl-perhydrochrysene) with the approximately similar SG of ∼0.943. Although these three components are isomers with the same SG, their viscosities are significantly different because of the relative positioning of the fused saturated rings in their structures, and these structural effects are not taken into account. Similar trends are observed versus MW at fixed SG (Figure 6b). These predictions are satisfactory for our purpose, which is not to obtain exact predictions for pure hydrocarbons but reasonable approximations for ill-defined pseudo-components. 6.2. Petroleum Distillation Cuts. The validity of the proposed correlations for the parameters of the EF model to the pseudo-components of the crude oils was assessed against the viscosity data of petroleum fractions. A total of over 500 data points were compiled from the literature55−58 for the viscosity of 117 petroleum cuts at different temperatures (Table 6). Most of the reported data are kinematic viscosities versus temperature at atmospheric pressure for the petroleum cuts, which are characterized by their average boiling point and specific gravity. The boiling point and specific gravity of these cuts vary from 327 to 762 K and from 0.660 to 1.112, respectively. To characterize the cuts for the EF model, the molecular weight of the cuts were calculated using Twu’s correlation49 as a function of the boiling point and specific gravity. The required densities and dilute gas viscosities of the cuts versus temperature were estimated by the modified Rackett correlation39 (eq 25) and method by Chung et al.36 (eqs 5−9) using critical properties calculated by Twu’s method,49 respectively. The modified Rackett correlation was tuned for each cut to match the reported specific gravity. The viscosity model parameters were then calculated from the reference and departure functions. The MARD and AARD of the predicted viscosities are provided in Table 6. The overall AARD is 27%, and the largest MARD was 100%. Note that the largest underpredictions were within 1 order of magnitude below the measured values and were observed for cracked materials. These results are not surprising considering the completely predictive approach of the model based on pure hydrocarbon

Table 4. Constants of the Departure Functions for the Parameters of the EF Model, eq 38 property

b0

b1

b2

b3

b4

Δρos (kg/m3) Δc2

0 0.4925

14640 −191900

739 −0.371

0 83930

0.67 2.67

main source of error is the neglected structural dependence of the c2 parameter. The black solid symbols in panels a and b of Figure 5 represent the departure values calculated from eq 38 for all 172 hydrocarbons in the test and development groups. Note that the ρos departures have little dependence upon the molecular weight, while the departures for c2 depend somewhat upon the molecular weight at low molecular weights but not at high molecular weights. The molecular weight dependence appears as a departure from the main trend of correlation points in panels a and b of Figure 5. The results for the test group are discussed later.

6. APPLICATION AND TESTING OF THE MODEL The model was applied to a number of examples following the procedure shown in Figure 1. The fluid was first divided into pure components and pseudo-components as appropriate for the fluid description. Except for the pure hydrocarbon test described below, pure hydrocarbons were modeled using previously determined EF model parameters.27 For pseudocomponents, the EF model ρso and c2 parameters were calculated from the departure functions (eq 38, with constants in Table 4, coupled with eqs 36 and 37). Their c3 parameters were calculated from eq 33. At this point, the model parameters have been defined for all components. Then, the EF model parameters for the whole fluid are calculated from the mixing rules (eqs 10−12). The dilute gas viscosity was calculated from eqs 4, 5, 13, and 14 using critical properties and acentric factor values calculated from the Lee−Kesler50,51 and Hall− Yarbrough59 correlations. Finally, the fluid viscosity was calculated from eqs 1−3. Example calculations are given in Appendix 2: Example Calculations. 6.1. Pure Hydrocarbons. As a test of the departure functions, they were used to predict the viscosity of the pure heavy hydrocarbons in the test group. Table 5 reports the AARD and MARD of the predictions for each family of compounds. The predictions are evaluated against experimental atmospheric data at temperature ranges of 253−383 K for alkylcyclopentanes, alkylcyclohexanes, and alkylbenzenes and 273−373 K for the rest. The predicted viscosities are within 1 order of magnitude of the measured values, with overall AARD and MARD of 31 and 334%, respectively. The predicted

Table 5. Summary of Errors of the Viscosity Predictions for Pure Heavy Hydrocarbons

a

hydrocarbon family

number of compounds

MW range (g/mol)

SG range

MARDa (%)

AARDb (%)

branched paraffins non-fused aromatics fused aromatics non-fused naphthenics fused naphthenics alkylcyclopentanes alkylcyclohexanes alkylbenzenes

16 12 20 16 36 16 16 16

198−451 168−351 184−427 138−393 194−433 84−294 98−308 92−302

0.767−0.817 0.860−1.103 0.890−1.104 0.826−0.944 0.863−1.031 0.754−0.827 0.775−0.832 0.858−0.874

84 334 100 137 99 75 32 134

17 59 29 26 35 13 6.8 49

MARD = maximum of |(μcorr(i) − μexp(i))|/μexp(i) × 100, for i = 1−N. bAARD = (100/N)∑i N= 1|(μcorr(i) − μexp(i))|/μexp(i). 1889

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Figure 6. Increasing trend of viscosity with (a) SG for heavy hydrocarbons of MW ∼ 360 g/mol and (b) MW for heavy hydrocarbons of SG ∼ 0.935. Data are from the API Project 42.53

Table 6. Summary of Errors of the Viscosity Predictions for Petroleum Distillation Cuts petroleum fraction description d

Arabian light crude cuts Arabian berri crude cutse Arabian medium crude cutse Arabian heavy crude cutse Oklahoma crude cutse Boscan crude cutse California crude cutse Pennsylvania crude cutse Wyoming crude cutse Minas crude cutse Iranian export crude cutse stabilized Arabian crude cutse midway special crude cutse Safania crude cutse light valley crude cutse waxy crude cutse midcontinent distillatesf smackover distillatesf distillates from residue of cracking midcontinent gas oilf Pennsylvania distillatesf Pennsylvania lube oilf cracked residuum cutsf cuts from pressure distillateg,h cuts from cycle stockg,h gas oils and kerosenesg,h virgin midcontinent naphtha cutsg,h miscellaneous cutsg,h

ABPa (K)

SG range

viscosityb (mPa s)

MWc (g/mol)

MARD (%)

AARD (%)

429−560 422−672 422−672 422−672 411−511 455−563 411−461 411−511 411−511 356−583 363−496 391−469 373−518 417−474 433−526 398−490 452−729 505−676 508−691 518−705 685−744 678−762 327−631 366−603 502−580 405−439 534−656

0.740−0.867 0.755−0.888 0.762−0.900 0.755−0.902 0.758−0.828 0.814−0.888 0.782−0.818 0.746−0.797 0.764−0.822 0.697−0.829 0.719−0.801 0.732−0.789 0.750−0.870 0.746−0.785 0.791−0.870 0.762−0.825 0.792−0.905 0.860−0.928 0.925−1.059 0.809−0.866 0.875−0.887 0.998−1.112 0.660−1.013 0.709−0.919 0.816−0.851 0.756−0.781 0.852−0.907

0.3−3.8 0.4−14 0.4−15 0.4−17 0.3−4 0.7−4 0.3−0.9 0.3−1.5 0.3−1.6 0.3−1.1 0.3−1.2 0.4−0.9 0.4−1.9 0.5−1.0 0.6−2.1 0.4−1.2 0.4−102 0.7−39 1.8−329 0.7−32 4.9−156 205−1920 0.2−102 0.3−8.2 0.6−4.7 0.5−0.7 1.1−21

129−212 126−311 125−305 126−304 119−176 139−201 117−142 120−181 118−176 93−237 95−170 110−152 98−171 123−156 129−178 111−161 140−379 165−296 153−256 184−368 335−417 229−332 78−331 96−272 187−266 118−141 193−337

53 31 29 29 40 20 24 25 26 27 39 38 20 26 42 36 70 56 91 47 79 100 86 47 37 41 40

27 18 17 16 23 11 16 13 17 23 28 29 15 16 22 33 25 22 39 18 56 95 31 32 30 36 29

a

ABP = average boiling point. bCalculated from reported kinematic viscosity. cCalculated by Twu’s method.49 dData are from ref 56. eData are from ref 55. fData are from ref 58. gData are from ref 57. hFrom midcontinent gas oil.

data and the uncertainties associated with the estimated density and molecular weight. Note that the model performed similarly for the different cuts of every studied crude oil regardless of the geographical origin of the crude (panels a and b of Figure 7). The model tended to overpredict the viscosity of the low boiling point cuts and underpredict the viscosity of the high boiling point cuts (Figure 8). The distillation cuts are complex mixtures of the hydrocarbons with the same vapor pressure but are polydisperse in molecular weight. The number average molecular weight or the molecular weight from property correlations may not be the most suitable value for the viscosity model parameter correlations. An effective molecular weight (for example, a mass average molecular weight) may be required. Alternatively, use of the normal boiling point and

specific gravity as the characterizing properties for the estimation methods of the parameters of the EF model may result in better predictions for the petroleum cuts. However, this option was not studied further because of limited reliable density and viscosity data for the petroleum cuts and normal boiling points for the pure heavy hydrocarbons. 6.3. Oils Characterized Using Pseudo-components. A total of nine different crude oils were included in this study to evaluate the application of the viscosity model with the proposed parameter correlations to characterized oils (Table 1). Among the fluids are three live (with solution gas) oils without data for their corresponding dead (gas free) oils. The remaining six crude oils are dead oils, three of which have data for the corresponding live oil. The WC-B-B2 bitumen was 1890

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Figure 7. Experimental and predicted viscosity of the consecutive boiling point cuts of (a) Arabian heavy oil and (b) Boscan oil from Venezuela. Data are from Beg et al.55

Table 7. GC Assay Data of WC-B-B2, the Only Oil in the Development Group

Figure 8. Dispersion plot of the predicted viscosity for petroleum cuts versus measured values.

selected as the development oil to guide the extrapolation of the c2 correlation for the reference n-paraffin system to molecular weights higher than 618 g/mol. The remaining oils were used as the test group to evaluate the modeling methodology. 6.3.1. WC-B-B2 Development Oil. The WC-B-B2 oil is a west Canadian bitumen with a specific gravity of 1.018 and viscosity of 89 000 mPa s at atmospheric pressure and 20 °C. The viscosity and density of this bitumen were measured at temperatures and pressures up to 175 °C and 10 MPa. Details on the experimental method and data are provided elsewhere.32 The GC assay of the bitumen (Table 7) is an average assay based on multiple measurements on several batches of the crude oil by different laboratories. The bitumen was characterized on the basis of the extrapolated GC assay as described previously. The bitumen was represented by 13 pseudo-components (Table 8). Note that the characterized bitumen does not include any defined components. The EF model parameters were then estimated using the procedure outlined in Figure 1. Using the measured density data as input, the viscosity of the bitumen was calculated at temperatures up to 175 °C and atmospheric pressure (Figure 9a) and higher pressures (Figure 9b). Recall that the extrapolation of the c2 values for the reference n-paraffin system to MW higher than 618 g/mol was constrained to fit the atmospheric data points of this bitumen. The MARD and AARD of the fit are 11 and 8.1%, respectively. The viscosity predictions at higher pressure conditions are within the same

component

MW (g/mol)

mass fraction

CO2 H2S N2 methane ethane propane isobutane n-butane isopentane n-pentane 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+

44.01 34.08 28.01 16.04 30.07 44.10 58.12 58.12 72.15 72.15 86 100 114 121 134 147 161 175 190 206 222 237 251 263 275 291 305 318 331 345 359 374 388 402 909 (adjusted)

0 0 0 0 0 0 0 0 9.20 2.91 9.76 7.07 6.21 1.41 3.70 6.27 8.91 1.24 1.48 1.71 1.70 1.87 1.89 1.86 1.87 1.96 1.81 1.66 1.48 1.55 1.48 1.43 1.40 1.44 7.00

× × × × × × × × × × × × × × × × × × × × × × × × × × ×

10−6 10−6 10−5 10−4 10−4 10−3 10−3 10−3 10−3 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−2 10−1

range of error with MARD and AARD of 19 and 8.9%, respectively. 6.3.2. Test Group Predictions. The five dead and eight live crude oils in this group were characterized by extrapolation of their GC assays. The specific gravities of the carbon number fractions of the dead and live oils were determined using eq 22, 1891

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Table 8. Properties of the Pseudo-components of WC-B-B2 pseudo-component

mass fraction

MW (g/mol)

SG

c2

ρos (kg/m3)

1 2 3 4 5 6 7 8 9 10 11 12 13

0.102 0.094 0.076 0.075 0.093 0.088 0.115 0.119 0.066 0.049 0.036 0.039 0.048

189 276 343 424 551 697 875 1124 1385 1603 1822 2100 2791

0.885 0.928 0.951 0.974 1.002 1.026 1.051 1.078 1.101 1.117 1.132 1.148 1.183

0.248 0.290 0.323 0.361 0.418 0.479 0.549 0.638 0.722 0.786 0.845 0.912 1.050

978.1 1004.0 1020.3 1036.9 1058.1 1077.6 1096.9 1118.3 1136.2 1148.8 1159.9 1172.3 1198.4

c3 (kPa−1) 2.38 2.68 2.76 2.79 2.80 2.80 2.80 2.80 2.80 2.80 2.80 2.80 2.80

× × × × × × × × × × × × ×

10−7 10−7 10−7 10−7 10−7 10−7 10−7 10−7 10−7 10−7 10−7 10−7 10−7

Figure 9. Measured and calculated viscosity of the characterized WC-B-B2 at (a) atmospheric pressure and (b) higher pressures.

and the calculated SG (eq 23) was matched to the reported SG of the corresponding dead oil by adjustment of parameter Cf. For the live oils with no dead crude oil data (crude oils ME3a, ME2, and ME1), the specific gravities were determined using the default value of 0.29 for the parameter Cf in eq 22. The viscosities of the crude oils were predicted using the measured density as the input at different pressure and temperature conditions. Table 9 reports the MARD and

AARD of the viscosity predictions for all of the oils of Table 1. Note that only a single data point was available for the AS1 dead oil; hence, no AARD was calculated. The predicted viscosities are well within a factor of 3 of the measured values (Figure 10). These predictive results based on the characterization of the oil by the GC assay are remarkable considering that no tuning of data was performed. We stress that the model is only valid for Newtonian fluids. Heavy oils and bitumens at lower temperatures (typically below room temperature) or crude oils below their cloud point are non-Newtonian, and the viscosity must be modeled differently. The EF model can still be used to calculate the viscosity of the continuous phase in these situations. 6.3.3. Model Tuning. The model can be tuned to fit experimental data by adjusting the parameters ρos , c2, and c3 for the pseudo-components. Up to three multipliers can be used (one for each parameter); however, a single multiplier applied to parameter c2 was sufficient to fit the data for the test group oils with MARD no greater than 45% and with AARD less than 25% (Table 9). Recall that the departures of c2 for pure heavy hydrocarbons were correlated to the specific gravity and molecular weight with less certainty than the other parameters, and therefore, the c2 parameter is the most suitable candidate for adjustment when matching experimental values. The applied multiplier (αc2) to the c2 values of the pseudocomponents ranged from 0.836 to 1.192. Note that, when dead oil viscosities were available, the model was tuned only to the dead oil data, typically at a single pressure and temperature. The live oil viscosities predicted after dead oil tuning were within 30% of the measurements. The effect of an additional

Table 9. Summary of the Viscosity Prediction Errors for the Test Group of Characterized Oils Using the Measured Density as Inputa non-tuned model AARD (%)

αc2

MARD (%)

AARD (%)

dead

18

11

1.011

32

8

dead dead live live live live dead live live live dead live

73 68 48 42 11 28 70 71 74 72 80 107

55 61 47 34 10 14 57 71 73 61

1.113 1.163

45 10 17 7 2 32 14 19 28 22 0 17

20 7 12 2 1 11 5 16 23 18

oil description WC-BHO1 WC-HO5 EU1 ME3a ME1 ME2 EU2

AS1

a

tuned model

MARD (%)

1.206 1.098 1.019 1.192

0.836 81

9

The model was tuned with a single multiplier to fit the dead oil data. 1892

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Figure 10. (a) Dispersion plot of predicted viscosity values of the oils in the development and test groups versus measured values. The dashed lines indicate a factor of 3 deviations from the measurements. (b) Relative deviations of the predicted viscosity values versus measured values.

Figure 11. Improvement of the predictions of the non-tuned model (solid line) by one-parameter tuning (dotted line) and two-parameter tuning (dashed line). Comparison to experimental data of WC-B-HO1 at (a) atmospheric pressure, (b) high-pressure conditions, and (c) relative deviation versus temperature.

tuning. Application of the multipliers as 1.005 and 1.076 to c2 and ρos reduced the AARD and MARD to 6 and 11%, respectively. Of course, this option is only available if there are sufficient viscosity data to justify using two adjustable parameters. 6.3.3.2. Effect of GC Assay Characterization. The performance of the model for heavy oil WC-HO5 is less satisfactory than for WC-B-HO1. The predictions are considerably less than the measurements with AARD and MARD of 55 and 73%, respectively. The GC assay of this crude oil was extrapolated by adjusting the average molecular weight of C30+ to 809 g/mol to obtain a smooth molar distribution of molecular weights. The average molecular weight of the heavy oil from this extrapolation was 483 g/mol, considerably lower than the measured MW of crude oil as 570 g/mol using a freezing point depression method. Therefore, the heavy oil was recharacterized as follows to obtain an average molecular weight of 556

tuning parameter, the GC extrapolation method, and a conventional oil tuning example are discussed below. 6.3.3.1. Two-Parameter versus One-Parameter Tuning. As an example, consider the viscosity of WC-B-HO1 (panels a and b of Figure 11), which was predicted with AARD and MARD of 11 and 18%, respectively. The model predicted the viscosity of the heavy oil at elevated pressures with the same accuracy of the atmospheric conditions. Hence, the model can be tuned for this oil by the applying multiplier of αc2 = 1.011 to the c2 values of the pseudo-components (panels a and b of Figure 11). The objective function was to reduce the absolute deviation of the modeled viscosity at atmospheric pressure. Although the tuned model predicted the viscosity with lower AARD of 8%, the MARD increased to 32% because of higher deviations of the tuned model at 25 °C. A better approach is to fine-tune the model by applying multipliers to parameters ρos and c2 of the pseudo-components simultaneously, that is, two-parameter 1893

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Figure 12. Effect of the average molecular weight of the characterized oil on viscosity predictions. Comparison to experimental data of WC-HO5 at (a) atmospheric pressure, (b) high-pressure conditions at 323 K, and (c) relative deviation versus temperature.

Figure 13. Improvement of the viscosity prediction with one-parameter tuning to dead EU1 oil data and two-parameter tuning to dead and live EU1 oil data. Comparison to experimental data of (a) dead oil at atmospheric pressure, (b) live oil at 325 K, and (c) relative deviation versus temperature.

parameters. Then, the viscosity of the crude was predicted versus temperature and pressure (panels a and b of Figure 12). The predictions based on the second characterization are considerably improved with AARD and MARD of 15 and 55%, respectively.

g/mol: (1) the mass fraction of the C30+ residue was assumed to be 10% higher than the reported value in the GC assay (see the Supporting Information for “generated” GC assay), and (2) the average molecular weight of C30+ was set equal to that of the WC-B-B2 heavy oil (908 g/mol). The pseudo-components of the crude oil were redefined, including their EF model 1894

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Table 10. Summary of the Errors of the Density Predictions with PR EoS and Viscosity Predictions with the EF Model Using EoS-Estimated Density as Inputa viscosity prediction density prediction oil description WC-B-B2 WC-B-HO1 WC-HO5 EU1 ME3a ME1 ME2 EU2

AS1 a

dead dead dead dead live live live live dead live live live dead live

non-tuned model

tuned model

MARD (%)

AARD (%)

MARD (%)

AARD (%)

αρos

0.11 0.18 0.29 0.19 0.19 1.40 7.88 2.16 0.95 0.81 1.17 1.01 0.50 0.69

0.35 0.39 1.24 0.23 0.29 1.86 9.37 2.35 1.41 0.83 1.84 1.61

53 38 79 71 55 52 39 73 82 81 83 88 58 75

16 20 61 65 52 44 34 63 68 80 82 76 58 57

0.999 1.002 0.997 0.989

1.051 1.086 1.045

0.971 1.118 0.996 0.998

1.104 1.098 1.194 1.262

1.049

1.114

0.89

αc2

αc3

MARD (%)

AARD (%)

0.17 0.29 0.33

27 13 7.0 4.0 2.7 7.9 3.6 5.6 12 21 33 49 13 4.6

13 6.0 3.3 2.6 1.4 2.4 2.0 2.5 5.2 17 21 29

̀

2.8

The viscosity model was tuned to the dead oil data.

Figure 14. Less satisfactory density prediction by PR EoS for (a) live oil ME2 because of the lack of dead oil SG data and (b) dead WC-B-B2 bitumen at lower temperature and higher pressure conditions.

The sensitivity of the viscosity predictions to the characterization of the oil is unavoidable because the characterization affects the molecular weight of pseudo-components, which, in turn, determines the model parameters. The molecular weight of the plus fraction is one of the least reliable property values and is commonly used as one of the adjustable parameters in regression of the phase behavior of the reservoir fluids.60 Hence, it is not surprising that there are significant deviations in predicted viscosities that rely indirectly on C30+ molecular weights. One option for tuning is to adjust the C30+ molecular weight as above. However, the single-parameter model tuning also compensates for the effects of molecular weight errors. For instance, the viscosity of WC-HO5 with the original characterization was tuned to the atmospheric pressure data by applying a single multiplier of 1.113 to the c2 parameter (Figure 12a). The tuned model is almost as accurate as the model with the modified characterization at all pressures, with AARD and MARD of 20 and 45%, respectively. In general, singleparameter tuning is recommended because it is straightforward and easily made part of a consistent characterization methodology. 6.3.3.3. Conventional Oil Viscosity Modeling Example. Thus far, all of the examples presented in detail have been heavy oils. The final example is the EU1 conventional oil. The model underpredicted the viscosity of both dead and live EU1

crude oil (panels a and b of Figure 13). The AARD and MARD are 61 and 68% for the dead oil and 47 and 48% for the live oil, respectively. The calculated viscosity values for the dead oil were improved by applying the single multiplier of 1.163 to the c2 values of the pseudo-components (Figure 13a). Using the same value of the multiplier improved the predictions for the live oil considerably (Figure 13b). The AARD and MARD for the tuned model are 7 and 10% for the dead oil and 12 and 17% for the live oil, respectively. The model slightly overpredicts the live oil viscosity, possibly because of discrepancies in the GC assay data of the dead and live oils, especially for the light ends. Note that the single-parameter tuning is already within the accuracy of most viscosity measurements and two-parameter tuning based on three dead oil data points is difficult to justify. However, two-parameter tuning of the model to collective viscosity data of the dead and live oils improved the model fit, reducing the AARD and MARD to 4 and 5% for the dead oil and to 3 and 5% for the live oil, respectively (panels a and b of Figure 13). In general, single-parameter tuning is sufficient to fit the viscosity data to within typical experimental accuracy (±30%), but two-parameter fine-tuning is advantageous when sufficient viscosity data are available. 6.3.3. EoS Applications. The EF model with the parameter estimation methods can also provide viscosity predictions for the characterized oils using the densities estimated with an EoS. 1895

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7. CONCLUSION The EF viscosity model was extended to characterized crude oils. Correlations based on molecular weight and specific gravity were developed to calculate the fluid-specific parameters of the pseudo-components. The model parameters for the crude oil were calculated from the pseudo-component parameters using mass-based mixing rules. The AARD and MARD of the viscosity predictions for over 150 pure hydrocarbons were 31 and 334%, respectively. The calculated viscosities of over 100 petroleum distillation cuts were within 1 order of magnitude of the measurements, with AARD and MARD of 27 and 100%, respectively. The proposed method was then tested on nine crude oils characterized on the basis of GC C30+ assays. Both measured and EoS-estimated densities of the oils were used as the inputs to the EF model. The non-tuned predictions of the model with measured densities were within a factor of 3 of the measured values, with AARD and MARD of 48 and 107%, respectively. The non-tuned predictions with the EoS densities were within a factor of 5 of the measurements. A simple procedure was developed to tune the model to available viscosity data by applying up to three (one for each parameter) single multipliers to the fluid-specific parameters of the pseudo-components. The following are the recommendations for the tuning of the model for the characterized oils: (1) Parameter c2 is the first candidate for adjustment to match the experimental data because of higher uncertainties in its correlation. In general, tuning only the c2 parameter was sufficient to predict viscosities (using measured densities as an input) to within 30%. (2) Tuning the c2 parameter can compensate for the effect of inaccurate oil characterizations, for example, inaccurate molecular weights of the crude oils. (3) Fine tuning is possible when sufficient data are available to justify two-parameter tuning of the model, that is, the simultaneous application of single multipliers to parameters c2 and ρos of the pseudo-components. (4) Two-parameter tuning is most likely required when EoS-estimated densities are used as the input to the EF model. Adjustment of ρos compensates for the effect of the less accurate input densities on the predicted viscosity. (5) The adjustment of the parameter c3 is essential if the EoS-estimated densities at higher pressures affect the viscosity predictions. This situation is unavoidable when cubic EoS is used for extra heavy hydrocarbons at lower temperatures. The EF viscosity model with the parameter correlations is a simple yet powerful tool for viscosity predictions for hydrocarbons and characterized oils with no experimental viscosity data. When data are available, the model parameters can be tuned by applying up to three single multipliers. The main shortcoming of the EF model is the need for accurate densities as input. Therefore, if the EF model is to be used with a cubic EoS, temperature-dependent volume translation must be applied to the EoS.

This application is demonstrated below using the PR EoS, as implemented in the VMGSim process simulator. Note that temperature-dependent volume translations were implemented externally to the PR EoS in VMGSim to improve the density predictions, as described previously. The required parameters for the EoS modeling of the characterized oils are the critical pressure, critical temperature, and acentric factor of the defined components and pseudocomponents. The numerical values for the defined components were taken from the pure component database of VMGSim,61 which is partially based on the data from Yaws’ handbook35 and the NIST database. 62 The Lee−Kesler50,51 and Hall− Yarbrough59 correlations were used to estimate the critical properties and acentric factor of the pseudo-components. The inputs to these correlations are the normal boiling point and specific gravity. The normal boiling points for the pseudocomponents were calculated using the correlation by Soreide,38 with SG and MW as the inputs. The densities predicted with the EoS model are in good agreement with the measured values, with AARD and MARD of 1.3 and 9.4%, respectively (Table 10). Note that the volume translation of the EoS model was not tuned to the measured density of the oils, except to the input specific gravity. Higher deviations occurred for the characterized live oils lacking the dead oil SG data, including oils ME3a, ME1, and ME2, as shown in Figure 14a. The specific gravities of the pseudocomponents of these oils were calculated using eq 22, with the Cf parameter set equal to the default value of 0.29. The less satisfactory density predictions likely reflect the inaccuracy of using the default Cf value. Note that the calculated compressibility of the oils at lower temperatures is slightly lower than the actual value because of the inherent limitation of cubic EoS. Therefore, the highpressure densities are consistently lower than the measurements. The high-pressure deviations were most significant for the dead extra heavy oils and bitumen. As shown in Figure 14b, the density of the WC-B-B2 heavy oil at lower temperatures was underpredicted because the oil compressibility was underpredicted. The AARD and MARD of the viscosity predictions using the EoS densities as an input are 55 and 88%, respectively (Table 10). The predictions are less satisfactory in comparison to the predictions made using the measured density as the input (Table 9) because of the errors in the predicted density. The adjustment of ρos in addition to the c2 adjustment then becomes necessary mainly to compensate for the effect of the less accurate input density. Note that the tuning of the EF model for the extra heavy oils and bitumen required the application of a single multiplier to the parameter c3 for the pseudocomponents (Table 10). The adjustment of c3 compensates for the poor compressibility predictions from the EoS model. Note that parameter c2 for WC-B-B2 was not adjusted because it was already used to guide the extrapolation of the reference system correlation. Also note that the tuning for all oils was performed on the basis of only the dead oil data and the viscosity of the corresponding live oils was predicted. The only exception is for crude oil AS1, whereas the model was tuned to the live oil data because of limited data for the dead oil. The AARD and MARD of the model after tuning are 8.4 and 49%, respectively. Multi-parameter tuning of the model to the dead and live oil viscosity data simultaneously is also an option, as discussed previously for the conventional oil example.

■

APPENDIX 1: PR EOS The PR EoS is cubic EoS and defined as follows:42 P=

RT a − 2 v−b v + 2bv − b2

(A1)

where b is the co-volume given by 1896

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0.0777969RTc Pc

(a) Sample calculation of the fluid-specific parameters for pseudo-component 1 with MW = 189 g/mol and SG = 0.885: MW = 189 g/mol → SG(ref) = 0.7653, c2(ref) = 0.2794, and ρos (ref) = 883.42 kg/m3. SG = 0.885 → ΔSG = 0.1197 → Δc2 = −0.0313 and Δρos = 94.72 kg/m3. Then, c2 = 0.2481, ρos = 978.14 kg/m3, and c3 = 2.38 × 10−7 kPa−1. (b) Sample calculation of dilute gas viscosity of pseudocomponent 1 (MW = 189 g/mol and SG = 0.885) at T = 293 K: MW = 189 g/mol and SG = 0.885 → vc = 713.8 cm3/mol and NBP = 949.7 K from the Hall−Yarbrough59 and Soreide38 correlations, respectively. NBP = 949.7 K and SG = 0.885 → Tc = 725.5 K and ω = 0.558 from the Lee−Kesler50,51 correlations. Then, μo = 0.0045 mPa s from eqs 5−9. (c) Calculation of fluid-specific parameters for WC-B-B2: ρos , c 2 , and c 3 of the pseudo-components determined as demonstrated in “a” and reported in Table 8. ρos mix = 1074.7 kg/m3, c2,mix = 0.5149, and c3,mix = 2.73 × 10−7 kPa−1 from eqs 10, 11, and 12, respectively, with βij = 0. (d) Calculation of the viscosity at P = 5 MPa and T = 293 K: dilute gas viscosities of the pseudo-components determined at 293 K as demonstrated in “b”. Then, μo,mix = 0.0028 mPa s from eqs 13 and 14. ρ*s = 1076.2 kg/m3 from eq 3. ρ = 1017 kg/m3 → β = 26.1975 from eq 2 → μ = 119 000 mPa s. Experimental viscosity32 of WC-B-B2 is 120 000 mPa s.

(A2)

and a is the attractive term given by a = acα(T )

ac =

(A3)

0.457235R2Tc 2 Pc

(A4)

where α(T) is an empirical dimensionless scaling function of the temperature. This empirical function is correlated to the acentric factor for nonpolar or slightly pure components, such as hydrocarbons, as follows: α = [1 + fω (1 −

Tr )]2

(A5)

where Tr is the reduced temperature and fω is given by42,43 for ω < 0.5: fω = 0.37464 + 1.54226ω − 0.26992ω 2

(A6)

for ω ≥ 0.5: fω = 0.3796 + 1.4850ω − 0.1644ω 2 + 0.01666ω3

■

(A7)

The parameters a and b are calculated for the mixtures using the following mixing rules: a= b=

(A8)

List of pure hydrocarbon compounds used in this study and GC assay data of the crude oils used in this study. This material is available free of charge via the Internet at http://pubs.acs.org.

(A9)

Corresponding Author

∑ ∑ (1 − kij) aiaj xixj i

j

∑ xibi i

ASSOCIATED CONTENT

S Supporting Information *

■

AUTHOR INFORMATION

*Telephone: (403) 220-6529. Fax: (403) 282-3945. E-mail: [email protected].

where xi is the mole fraction of component i and kij is the interaction parameter between components i and j and determined on the basis of experimental vapor−liquid equilibrium data.

Notes

The authors declare no competing financial interest.

■

■

ACKNOWLEDGMENTS The authors are grateful for financial support from the sponsors of the Natural Sciences and Engineering Research Council of Canada (NSERC) Industrial Research Chair in Heavy Oil Properties and Processing, including NSERC, Schlumberger, Shell Canada Energy, Ltd., and Petrobras. We thank Schlumberger for sharing the WC-HO5 data set. We also thank the Virtual Materials Group for providing VMGSim process simulation software.

APPENDIX 2: EXAMPLE CALCULATIONS (A) Estimate viscosity of n-heptylcyclohexane at P = 20 MPa and T = 293 K. Data: MW = 182.35 g/mol and SG = 0.8148 from the API handbook23 and measured63 density = 822.8 kg/ m3. (a) Calculation of the fluid-specific parameters of the EF correlation for the reference n-paraffin with MW = 182.35: SG(ref) = 0.7622, c2(ref) = 0.2755, and ρos (ref) = 881.98 kg/m3 from eqs 34, 36, and 37, respectively. (b) Calculation of the departures: ΔSG = 0.0526 from eq 32. Then, Δc2 = −0.0146 and Δρos = 40.11 kg/m3 from eq 38, with constants from Table 4. (c) Calculation of the fluid-specific parameters for nheptylcyclohexane: c2 = 0.2609 and ρos = 922.09 kg/m3 from eq 30. c3 = 2.34 × 10−7 kPa−1 from eq 33. (d) Calculation of the viscosity at P = 20 MPa and T = 293 K: μo = 4.58 × 10−3 mPa s from eq 4, with constants from Yaws’ handbook.34 ρ*s = 926.42 kg/m3 from eq 3. ρ = 822.8 kg/ m3 → β = 11.9831 from eq 2 → μ = 3.60 mPa s. Experimental viscosity63 of n-heptylcyclohexane is 3.53 mPa s. (B) Estimate viscosity of WC-B-B2 at P = 5 MPa and T = 293 K. Data: the crude is characterized as 13 pseudocomponents (Table 8) based on GC assay data. Measured32 density = 1017 kg/m3.

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