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Journal of Petroleum Science and Engineering

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Effective viscosity prediction of crude oil-water mixtures with high waterfraction

Jiangbo Wena,b, Jinjun Zhanga,⁎, Min Weia

a National Engineering Laboratory for Pipeline Safety/MOE Key Laboratory of Petroleum Engineering/Beijing Key Laboratory of Urban Oil & GasDistribution Technology, China University of Petroleum, Beijing 102249, Chinab College of Petroleum Engineering, Liaoning Shihua University, Fushun 113001, China

A R T I C L E I N F O

Keywords:Crude oil-water mixtureEmulsionHigh water fractionEffective viscosity modelFrictional loss

A B S T R A C T

In the petroleum industry, a flowing crude oil-water system with high water fraction usually does not exist as astable and homogeneous emulsion, but in the form of a mixture of oil and free water with some water beingemulsified into the oil and thus, hydrodynamically, acting as part of the oil. Therefore, the existing viscositymodels developed for stable and homogeneous emulsions cannot accurately predict viscosity of this kind ofliquid-liquid mixtures. In the present work, the effective viscosity of crude oil-water mixtures with high waterfraction was measured by using a stirring method. It was found that the effective viscosity of the mixturesdecreased with increasing shear rate, water fraction and temperature. Based on the Taylor viscosity model andcombined with our previous study on the emulsification behaviors of crude oil-water systems, a viscosity modelwas developed for crude oil-water mixtures with high water fraction, which is characterized in covering theeffects of shear rate, emulsified water fraction and crude oil compositions. Validation was done by using 40 datapoints which were obtained from mixtures of two crude oils and not used for model development, and theresults showed that the proposed model gave accurate predictions of the mixture viscosity, with an averagerelative deviation of 7.4%. Approach is proposed to application of this model in the prediction of the frictionalloss of oil and water two phase flow in pipes.

1. Introduction

The crude oil-water two-phase or crude oil-gas-water multiphaseflow is quite common in the petroleum industry. Due to the naturalsurfactants such as resins, asphaltenes and wax particles containing inthe crude oil, crude oil and water emulsify easily in the process ofproduction. The emulsification of crude oil and water has significantinfluences on multiphase flow (Grassi et al., 2008; Nädler and Mewes,1997; Wang et al., 2011; Yusuf et al., 2012). Actually, a flowing crudeoil-water system usually does not exist as a stable and homogeneousemulsion, but in the form of an unstable oil-water mixture composed ofW/O emulsion and free water, significantly differing from a stableemulsion in dispersity and static stability.

A large number of viscosity models for stable emulsions have beendeveloped in the previous studies. In 1906, Einstein (1906) derived anequation for the viscosity of dilute dispersed systems.

η ϕ= (1 + 2.5 )r d (1)

where ηr is the relative viscosity, defined as the ratio between emulsionviscosity (ηe) and continuous phase viscosity (ηc), i.e., η η η= /r e c; ϕd is the

volume fraction of the dispersed phase.Later researchers (Schramm, 1992) added more terms into the

Einstein's equation to extend its applicability to emulsions, with thegeneral form as follows.

η k ϕ k ϕ k ϕ= 1 + + + + ...r 1 d 2 d2

3 d3

(2)

where k1, k2, k3 are parameters associated with particular emulsionsystems.

Brinkman (1952) proposed a model for the emulsion with sphericaldroplets.

η ϕ= (1 − )r d−2.5 (3)

Richardson (1933) firstly developed an exponential relationshipbetween the relative viscosity and the volume fraction of the dispersedphase.

η e= kϕr d (4)

where k is an undetermined parameter, if ϕd < 0.74, k=7, if ϕd > 0.74,k=8. The Richardson equation has been found suitable for manyemulsion systems. And it has been modified by many researchers to

http://dx.doi.org/10.1016/j.petrol.2016.09.052Received 20 April 2016; Received in revised form 25 September 2016; Accepted 28 September 2016

⁎ Corresponding author.E-mail address: zhangjj@cup.edu.cn (J. Zhang).

Journal of Petroleum Science and Engineering 147 (2016) 760–770

0920-4105/ © 2016 Elsevier B.V. All rights reserved.Available online 30 September 2016

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adapt more emulsion systems (Broughton and Squires, 1938; Marsden,1977; Mooney, 1951).

The above-mentioned viscosity models correlate the relative visc-osity with the volume fraction of the dispersed phase only, with theeffect of other factors implied in the undetermined parameters. WhileTaylor (1932) assumed that the dispersed phase was extremely smallspherical droplets, and derived an equation for dilute emulsions whichcontains the dispersed phase viscosity ηd

⎛⎝⎜

⎞⎠⎟η

η ηη η

ϕ= 1 ++ 2.5

+rc d

c dd

(5)

Based on the Taylor model and the continuum theory, Phan-Thienand Pham (1997) developed an equation for emulsion viscosity.

⎡⎣⎢

⎤⎦⎥η

η KK

ϕ2 + 52 + 5

= (1 − )r2/5 r

3/5

d−1

(6)

where K is the ratio of dispersed phase viscosity and the of continuousphase viscosity, i.e., K η η= /d c.

After Phan-Thien and Pham, Pal (2000) proposed an improvedmodel by considering the effect of surfactants on the viscosity.

⎡⎣⎢

⎤⎦⎥η

η KK

K ϕ2 + 52 + 5

= 1 −r−2/5 r

−3/5

s d (7)

where Ks is a coefficient associated with surfactants, which needs to bedetermined by fitting experimental data.

Emulsions often exhibit non-Newtonian flow behaviors, such as theshear-rate-dependence of viscosity. Therefore, some works had beendone to cover this non-Newtonian property.

Based on the Brinkman's equation (Brinkman, 1952), Pal andRhodes (1989) developed a model for predicting the shear-rate-dependence of emulsion viscosity.

η K K γ ϕ= [1 − ( ) ]r 0 f d−2.5 (8)

where K0, K γ( )f are defined as the non-Newtonian coefficients; K0represents the influence of hydration, which associates with theemulsion;K γ( )f represents the influence of droplets coalescence, whichassociates with shear rate.

Based on the Richardson's equation (Richardson, 1933),Rønningsen (1995) proposed an empirical equation which correlatesthe emulsion viscosity with shear rate and temperature.

η k k T k ϕ k Tϕln = + + +r 1 2 3 d 4 d (9)

where T is temperature; k1, k2, k3, k4 are the parameters associatedwith shear rate, which can use the following empirical values:k1=0.0412, k2=−0.002605, k3=0.03841, k4=0.0002497.

After Pal and Rhodes (1989), Dou and Gong (2006) developed anequation for water/heavy crude oil (W/O) emulsion, which includes theeffect of the water fraction on the non-Newtonian coefficient.

η K γ K ϕ ϕ= [1 − ( ) ( ) ]r f f d d−2.5 (10)

K ϕ( ) =

η γ ϕϕ

η γ ϕϕ

f d

1 − ( , )

1 − ( , )

r−0.4

d

d

r−0.4

max

max (11)

where K ϕ( )f d is the coefficient of water fraction, representing the effectof water fraction on the non-Newtonian rheological behavior; ϕmax ismaximum water fraction of the emulsion.

It has been known that the droplet size and size distribution of thedispersed phase has significant effect on emulsion viscosity. Wang(2009) incorporated the effect of droplet size into the Dou's equationand derived the following viscosity model.

⎛⎝⎜

⎞⎠⎟η

η KK

K ϕ K N ϕ+ 2.5

1 + 2.5= [1 − ( )⋅ ( )⋅ ]r

2/5 r3/5

f f Re,p d−1

(12)

where K N( )f Re,p is the Reynolds number factor of the droplets asso-ciated with shear rate and droplet size.

By taking into the consideration of the effects of interfacialrheology, Pal (2011) developed a new model for the viscosity ofconcentrated emulsions.

⎡⎣⎢

⎤⎦⎥η

η ε εε ε

f ϕ2 + 5 /(1 − )2 + 5 /(1 − )

= ( )rr m m

m m

3/2

d(13)

f ϕ ϕ( ) = (1 − )−5/2 (14)

where εm is the interfacial mobility parameter.Based on the factors of shear rate, oil concentration, emulsifier

concentration and temperature affecting viscosity, Azodi and Nazar(2013) developed a rheological equation for predicting the viscosity ofoil in water emulsion.

⎛⎝⎜

⎞⎠⎟η k γ C ϕ C

TC ω= exp + +n

1 1 d2

3(15)

where k1, n, C1, C2 and C3 are the constants and depend on theemulsion system; ω is the emulsifier concentration, wt%.

Strictly speaking, the classic viscosity definition for homogeneousfluids cannot be applied to unstable and inhomogeneous oil-watermixtures. That is to say, existing viscosity models developed for stableemulsions are no longer applicable for this mixture. Moreover,according to their principle of measurement of the traditional visc-ometers, such as coaxial cylinder viscometer and cone-plate visc-ometer, are not suitable for measuring “viscosity” of this inhomoge-neous and unstable mixture.

At the earliest stage, the effective viscosity of the oil-water mixturewas estimated by using the weighed mean method to meet the need ofthe industry (Govier and Aziz, 1972):

η η ϕ η ϕ= (1 − ) +m o w w w (16)

where ηmis the viscosity of an oil-water mixture; ηo is the oil viscosity;ηwand ϕw are the viscosity and the volume fraction of water, respec-tively. Obviously, this simple weighted rule does not give accurateprediction of the mixture viscosity.

It is common practice in the industry to obtain the effectiveviscosity of an oil-water mixture on the basis of pressure drop vs. flowrate relation and flow loop data. Pan (1996) studied the stratified flowand dispersed flow in a flow loop, and developed a correlation forcalculating the viscosity of oil-water mixture by considering theinfluence of flow-induced mixing degree on viscosity.

η c ϕ η ϕ η c η ϕ= (1 − )[ + ] + (1 − )m m o o w w m c d−2.5 (17)

where cmis the mixing coefficient representing the mixing degree of theoil-water two phase flow; ϕo and ϕw are the volume fraction of oil phaseand water phase, respectively. However, Pan's model covers neither theeffect of the emulsified water in the oil phase, nor the effect of oilcompositions; it is hard to be applicable to broad crude oil-watersystems.

An oil-water system may become homogeneous if sufficient stirringis applied to it. Effective viscosity of a stirred fluid system may bedetermined according to the shaft torque and stirring speed. In thepresent study, the effective viscosities of crude oil-water mixtures ofeight crude oils with high water fractions were measured by using thisstirring method, and the effects of shear rate, water fraction andtemperature on the effective viscosity of mixtures were studied.Moreover, a model for predicting the effective viscosity of the crudeoil-water mixtures with high water fraction was developed andvalidated by using experimental data of mixtures prepared with othertwo crude oils. Finally, approach is proposed to application of thismodel in the prediction of the frictional loss of oil and water two phaseflow in pipes.

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2. Experimental section

2.1. Materials

Ten crude oils produced in different oilfields in China were used inexperiments, which were named as oils #1–#10. The physical proper-ties of these oils are listed in Table 1, and the measurements are carriedout on stabilized stock tank oils. Besides the crude oil compositions, thebrine salinity also affects the emulsification tendency. However, thiswork focuses on the effect of oil compositions on the viscosity of crudeoil-water mixture, so the water used in experiments was ultrapurewater. The experimental data of oils #1–#8 are used to develop theviscosity model, and the experimental data of oils #9 and #10 are usedto validate the model.

The pour point of crude oil was tested according to ASTM D5853-

11. The wax appearance temperature (WAT) and the concentration ofthe precipitated wax of crude oil were determined according to ASTMD4419-90. The viscosity of crude oil was determined according toASTM D445-09. The contents of asphaltenes, resins, and wax of crudeoil were determined according to ASTM D4124-09. The total acidnumber of crude oil was determined according to ASTM D664-11a. Thecontent of the mechanical impurities of crude oil was determinedaccording to ASTM D473-07. The carbon number distributions ofcrude oil were determined using the gas chromatograph Agilent 6890according to SY/T 5779-2008, and the average carbon number wascalculated by using the data of carbon number distributions and themass fraction of alkanes, which can be learned in the literature (Yi andZhang, 2011) for details.

Take oils #9 and #10 as examples. The viscosity and the concen-tration of precipitated wax vs. temperature curves are shown in Figs. 1

Table 1Physical properties of oils.

oils #1 #2 #3 #4 #5 #6 #7 #8 #9 #10

Density at 20 °C (kg/m3) 879.5 863.8 839.4 872.0 864.9 856.6 835.9 818.7 864.3 868.9Pour point (°C) −7 36 −1 −9 1 29 18 17 11 13Wax appearance temperature (°C) 20 45 20 34 23 45 33 22 33 25Viscosity at 40°C (mPa·s) 71.25 24.97 3.40 11.91 17.26 16.30 7.65 3.06 17.00 28.72Asphaltenes (wt%) 3.72 2.92 0.81 3.96 2.18 1.10 6.79 5.11 2.38 3.10Resins (wt%) 7.98 7.94 0.69 1.61 5.75 8.04 1.46 0.11 9.45 10.20Wax (wt%) 3.50 22.01 6.52 3.78 7.53 12.60 10.63 9.39 14.13 14.29Total acid number (mg KOH/g) 3.32 0.24 0.27 0.15 0.38 0.07 0.64 0.17 0.30 0.26Mechanical impurities (wt%) 0.25 0.24 0.30 0.41 0.34 0.05 0.19 0.03 0.28 0.23The average carbon number of crude oil 18.8 20.3 17.8 16.8 18.2 19.0 18.3 17.9 18.5 17.6

Fig. 1. Viscosity vs. temperature of oils #9 and #10.

Fig. 2. Concentration of precipitated wax vs. temperature of oils #9 and #10.

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762

and 2, respectively...

2.2. Viscosity measurement apparatus for crude oil-water mixtures

A stirring apparatus is used to measure the effective viscosity ofcrude oil-water mixtures. As shown in Fig. 3, the experimentalapparatus is mainly composed of three parts: (1) A flat impeller,75 mm in height and 75 mm in diameter, and with six diversion holeson it; A cylindrical vessel, 120 mm in diameter and 180 mm in height,and connected with a circulating water bath for temperature control. Atotal of 1 L of oil-water mixture was used once in the experiment. (2) Acirculating water bath (HAAKE model AC200), with the effectivetemperature ranging from 5 to 90 °C, and temperature control preci-sion of 0.1 °C. (3) A stirring power system composed of a stirrer, atachometer, and a torque meter (IKA), with the effective stirring speedranging from 10 to 1600 rpm, and the accuracy of the torquemeasurement of 0.01 N cm..

2.3. Stirring method for determining the viscosity of crude oil-watermixture

The crude oil-water mixture with high water fraction contains bothemulsified water, as a W/O emulsion, and free water. Because of theinhomogeneity of the system and the tendency of phase separationunder static conditions, the traditional viscometers, such as coaxialcylinder viscometer and cone-plate viscometer, cannot used to measurethe effective viscosity of the mixture. However, the mixture can keephomogeneous under the condition of sufficient stirring; and the shafttorque of the stirrer should be related to the effective viscosity of themixture. Based on this consideration, Yu et al. (2013) developed amethod for determining the effective viscosity of unstable oil-watermixtures. The basic principle and main procedures are as follows:

In the flow field of a stirred vessel, the relationship between fluidviscosity η and shaft torqueM of the stirrer at a certain stirring speed is

Fig. 3. Schematic of the viscosity measurement apparatus for crude oil-water mixture.

Table 2Correlations between viscosity and torque.

Stirring speed (r/min) Correlations

200 η M= 414.62 3.0346

250 η M= 154.75 3.4332

300 η M= 43.37 3.4181

350 η M= 17.51 3.3831

400 η M= 7.32 3.7238

Note: the unit of η is mPa s.; the unit of M is N·cm.

Fig. 4. Effective viscosity of crude oil-water mixture vs. shear rate.

J. Wen et al. Journal of Petroleum Science and Engineering 147 (2016) 760–770

763

as follows:

η aM= b (18)

where a and b are fitting parameters.Eighteen Newtonian fluids with the viscosities ranging from 5 to

1024 mPa·s were used to obtain the shaft torques at specific speeds.Then, the values of a and b were determined by fitting the torque vs.viscosity data at each stirring speed, and correlations shown in Table 2were thus obtained.

The effective viscosity of crude oil-water mixtures with high waterfraction is usually shear-dependent. In order to characterize this shear-dependence, the shear rate in the stirred vessel at a specific speed hasto be determined. It should be noted that the fluid at different positionsof the vessel experiences different shear rates. For example, the shearrate near the impeller must be higher than that away from the impeller.Taking advantage of the relationship between the energy dissipationrate and the shear rate, Zhang et al. (2003) presented a method forestimating the mean shear rate in a stirred vessel, as shown in Eq. (19).

γ πMN ηV = 2 /( ) (19)

where M is the shaft torque of the stirrer, N m; N is the stirring speed,r/s; η is fluid viscosity, Pa s; V is fluid volume, m3.

For a particular stirring system (stirred vessel, impeller, andvolume of fluid), the correlation relating the mean shear rate to thestirring speed and viscosity were determined by using fluids withknown viscosities and measuring the shaft torques at various stirringspeeds. For the stirring system used in this study, the correlation thusobtained for the mean shear rate is as follows.

γ η N = 0.0775 −0.3618 1.4616 (20)

Based on the above principle, following procedures are carried out.Firstly, the oil-water mixture is stirred at a chosen speed, and the shaft

torques is recorded; then a correlation corresponding to that stirringspeed listed in Table 2 is used to obtain the viscosity; and further, Eq.(20) is used to determine the corresponding mean shear rate.Consequently, one set of effective viscosity vs mean shear rate data isdetermined. By duplicating the above procedures at different stirringspeeds, effective viscosities at different mean shear rates are thusobtained.

3. Experimental results

3.1. Effect of shear rate on the effective viscosity of crude oil-watermixtures

The measured effective viscosity of crude oil-water mixtures atvarious shear rates is shown in Fig. 4..

It can be seen that the effective viscosity decreases with the increaseof shear rate, i.e. taking on shear thinning behavior. Moreover, theshear thinning behavior weakens with the increase of the water fraction(see oils #1 and #2), and the increase of temperature (see oils #3 and#4).

When a crude oil-water mixture is stirred at low shear rate, theemulsification degree of oil and water is low, and the oil phase is notentirely dispersed. Therefore, the oil phase has greater influence on theeffective viscosity of the mixture than the water phase does, leading to ahigher viscosity. With the increase of shear rate, the emulsificationdegree becomes higher and the oil phase is dispersed to a greaterextent, forming a dispersion system with the oil droplets containingemulsified water are disperse phase, while the free water is continuousphase. Therefore, the free water has more effect on the effectiveviscosity than the oil, leading to a lower viscosity.

With the increase of water fraction, the effect of oil viscosity on theeffective viscosity of mixture becomes weaker, while the effect of water

Fig. 5. Effective viscosity of crude oil-water mixture vs. water fraction (300r/min).

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764

on the effective viscosity increases. Therefore, although the crude oil-water mixture still shows shear thinning behavior under a higher waterfraction, this shear thinning behavior is weaker than that at lower waterfraction.

Similarly, with the increase of temperature, the effect of oil viscosityon the effective viscosity weakens due to the decreasing oil viscosity.Therefore, the shear thinning behavior of a crude oil-water mixture athigher temperature is weaker than that at lower temperature.

3.2. Effect of water fraction on the effective viscosity of crude oil-water mixtures

The measured effective viscosity of crude oil-water mixtures atvarious water fractions is shown in Fig. 5..

As expected, the effective viscosity decreases with the increase ofwater fraction. Moreover, the trend of the viscosity varying with waterfraction becomes gentle with the increase of temperature.

3.3. Effect of temperature on the effective viscosity of crude oil-watermixtures

The measured effective viscosity of crude oil-water mixtures atvarious temperatures is shown in Fig. 6..

It can be seen that the effective viscosity decreases with the increaseof temperature. Moreover, the trend of the viscosity variation withtemperature becomes gentle with the increase of water fraction.

4. Viscosity model of crude oil-water mixtures with highwater fraction

4.1. Model development

A flowing crude oil-water system may exist in different patternsdependent on the fraction and properties of the oil and water phase,and particularly on the shear intensity. There have two extreme mixingpatterns for a high water fraction system, i.e., the complete stratifica-tion of oil and water and the complete dispersion of oil and water (O/W), as shown in Fig. 7. The other mixing patterns must be fallen

Fig. 6. Effective viscosity of crude oil-water mixture vs. temperature (300 r/min).

Fig. 7. Two extreme mixing patterns.

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between these two extreme patterns..Let ηSF denotes the effective viscosity of the complete stratification

of flow, while ηDF represents the effective viscosity of the completedispersion flow. When a crude oil-water mixture flows in any patternbetween the two extreme cases, assume that the effective viscosity ofthe oil-water mixture can be expressed as the weighted average of theviscosity of the two extreme patterns.

η η c η c= (1 − ) +m SF m DF m (21)

where cm is defined as the mixing coefficient with the value between 0and 1, with cm=0 representing the complete stratified flow, and cm=1representing complete dispersion flow.

For the complete stratified flow, assume that the effective viscosity

of crude oil-water mixture (ηSF) could be represented by the weightedaverage rule.

η η ϕ η ϕ= (1 − ) +SF o w w w (22)

where ηo and ηw are the oil and water viscosity in mPa·s, respectively; ϕwis the volume fraction of water.

For the complete dispersion flow, the oil and water are mixeduniformly, and the mixture can be regarded as a homogeneousemulsion. Therefore, in this case, the effective viscosity of the crudeoil-water mixture (ηDF) can be calculated by using viscosity modeldeveloped for stable emulsions. We’ll use the Taylor model for furtherderivation:

Fig. 8. Mixing coefficient cm vs. entropy production rate.

Fig. 9. Relative deviation of calculated value of a using Eq. (34). Fig. 10. Relative deviation of calculated value of b using Eq. (35).

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766

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥η η

η ηη η

ϕ= 1 ++ 2.5

+DF cc d

c dd

(23)

where ηc and ηd are the viscosity of the continuous phase and dispersedphase respectively, mPa s; ϕd is the volume fraction of the dispersedphase.

As stated above, most of the emulsion viscosity models weredeveloped for stable emulsions, and the crude oil-water mixture withhigh water fraction is an unstable and inhomogeneous system, contain-ing both emulsified water and free water. In this system, the emulsifiedwater acts as part of the oil phase hydrodynamically. Therefore, theeffective volume fraction of the oil phase should be actually greaterthan the real volume fraction of the pure oil. The volume fraction of the“effective oil phase” is actually the sum of the volume fraction of pureoil and the volume fraction of the emulsified water.

Given the above, for the crude oil-water mixture with high waterfraction, water is the continuous phase and the “effective oil phase” isthe dispersed phase. Based on this fact, we modify Eq. (23) by using thevolume fraction of the “effective oil phase”, and obtain the followingequation:

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥η η

η ηη η

ϕ ϕ= 1 ++ 2.5

+( + )DF w

w o

w oo E

(24)

where ϕo is the volume fraction of crude oil; ϕE is the emulsified waterfraction.

Furthermore, we substitute Eqs. (22) and (24) into Eq. (21), andobtain a new model for calculating the effective viscosity of crude oil-water mixture with high water fraction:

⎡⎣⎢⎢

⎛⎝⎜

⎞⎠⎟

⎤⎦⎥⎥η η ϕ η ϕ c η

η ηη η

ϕ ϕ c= [ (1 − ) + ](1 − ) + 1 ++ 2.5

+( + )m o w w w m w

w o

w oo E m

(25)

In application of Eq. (25), it is necessary to determine theemulsified water fraction (ϕE) and the mixing coefficient (cm).

4.2. Calculation of emulsified water fraction (ϕE)

In our previous work (Wen et al., 2014), it was found that theemulsified water fraction under flowing conditions can be calculated bythe following correlation.

ϕ k S= kE 1 g

2 (26)

where Sg is the entropy production rate of viscous flow, W/(kg K); k1, k2are the parameters related to the composition of crude oil.

According to thermodynamics, the entropy production rate is ameasure of the irreversibility of a thermodynamic process, and so theentropy production rate of viscous flow describes the irreversibility ofthe viscous flow (Jou et al., 2010). The entropy production rate ofviscous flow can be calculated as follows.

S φρT

=gK (27)

where φ is the energy dissipation rate per unit volume of fluid (White,2011), W/m3; ρ is the density of fluid, kg/m3; TK is the absolutetemperature of the fluid, K.

The mean energy dissipation rate (φ ) in a stirred vessel can becalculated according to the shaft power of stirring23:

φ πNMV

= 2(28)

Fig. 11. Comparison between measured data and calculated value by the proposed model (oil #9).

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767

The mean energy dissipation rate of pipe flow can be calculated asfollows (Zhang and Yan, 2002):

φ λρvd

=2

3

(29)

where λ is the hydraulic friction factor (Darcy friction factor); v issectional average velocity of the fluid, m/s; d is the pipe diameter, m.

According to our previous work (Wen et al., 2016), k1 and k2 arerelated to crude oil compositions and can be calculated by using thefollowing empirical correlations:

k c c TAN CN= 0.0311 a+r0.125

w+m0.017 0.030

oil0.246 (30)

k c c TAN CN= 0.9982 a+r0.036

w+m0.024 −0.027

oil−0.351 (31)

where ca+r is the content of asphaltenes and resins, wt%; cw+m is thesum of the concentration of precipitated wax and the content ofmechanical impurities, wt%; TAN is the total acid number of the crudeoil, mg KOH/g; CNoil is the average carbon number of the crude oil.

4.3. Calculation of mixing coefficient (cm)

By rearranging Eq. (21), the mixing coefficient cm can be obtainedas follows:

cη ηη η

=−−m

m SF

DF SF (32)

By using effective viscosity data ηm of the mixtures of the eight crudeoils, the values of cm can be obtained by Eq. (32), where ηSF and ηDF arecalculated by using Eq. (22) and Eq. (24), respectively.

As mentioned above, the mixing degree of the oil and water, i.e. themixing coefficient cm, depends on flow, too. Therefore, it is a reasonableassumption that cm is a function of the entropy production rate ofviscous flow, Sg. Fig. 8 shows how cm correlates with the Sg..

According to the tendency of the curves in Fig. 8, we use a linearequation to correlate cm and Sg:

c aS b= +m g (33)

where a and b may be determined by fitting experimental data.As to the determination of a and b, we assume that they are related

to water fraction, temperature and crude oil composition according tothe analysis to the factors influencing the mixing degree of oil andwater. By fitting 160 sets of experimental data from the mixtures ofeight crude oils (oil #1 to oil # 8), correlations were obtained betweena, b and water fraction, temperature and crude oil compositions.

Fig. 12. Comparison between measured data and calculated value by the proposed model (oil #10).

Fig. 13. Calculated viscosities compared with measured viscosities.

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768

a T ϕ c c TAN CN= 0.637 0.292w0.665

a+r−0.041

w+m−0.006 −0.073

oil−0.207

(34)

b T ϕ c c TAN CN= 0.041 0.072w0.344

a+r−0.417

w+m−0.235 0.229

oil0.545

(35)

The average relative deviations of Eq. (34) and Eq. (35) arerespectively 7.0% and 15.2%, as shown in Fig. 9 and Fig. 10...

According to the conditions under which this model was developed,we recommend that it is better to use the model for: (1) light tomedium crude oil; (2) water fraction ranging from 0.70 to 0.90; (3)temperature above the pour point of crude oil.

5. Model validation

The measured effective viscosities of crude oil-water mixtures ofoils #9 and #10 were used to validate the proposed viscosity model.Fig. 11 and Fig. 12 show the comparison between the experimentalresults and the predicted values.

For the forty sets of experimental data used for model validation,the proposed model gave prediction of an average relative deviation of7.4%, as shown in Fig. 13, indicating that the proposed viscosity modelaccurately predicts the effective viscosity of crude oil-water mixtures..

In addition, the model prediction was compared with the resultsfrom other published emulsion viscosity model, as shown in Fig. 14.For the 40 sets of experimental data, the average relative deviations ofthese model predictions are shown in Table 3.

It can be seen that the proposed viscosity model gives much moreaccurate results compared to other emulsion viscosity models. In theEinstein model, Brinkman model and Richardson model, only thevolume fraction of the dispersed phase was taken into account. In theTaylor model, the volume fraction of the dispersed phase and theviscosities of dispersed phase and continuous phase were taken intoaccount. In the Rønningsen model, the volume fraction of the dispersedphase and temperature were taken into account. All these five emulsionviscosity models do not take into account of the effect of emulsifiedwater fraction under flowing conditions (ϕE) on the viscosity ofmixture, and this may result in their worse performance comparedwith the proposed model.

6. Approach to model application

Because the viscosity model developed in this study contains the

entropy production rate of viscous flow, which is a function of theenergy dissipation rate, and the latter in turn must be calculated fromthe frictional loss of flow, the application of the model in the predictionof the frictional loss of multiphase flow in pipes are based on a trial-anderror procedure as follows:

(1) Assume a pressure drop of the flow, denoted as ΔPe.(2) Calculate the hydraulic friction factor, denoted as λe, by using a

multiphase flow calculation equation such as the Beggs-Brill'sequation and based on the assumed pressure drop.

(3) Calculate the mean energy dissipation rate φ , and the entropyproduction rate of viscous flow Sg according to the hydraulicfriction factor obtained in step 2 by using Eq. (29) and Eq. (27).

(4) Calculate the emulsified water fractionϕEby using Eq. (26) andEqs. (30)–(31), the mixing coefficient cmby using Eqs. (33)–(35),and finally the effective viscosity of crude oil-water mixture μm.

(5) Calculate a new pressure drop by using the multiphase flowcalculation method according to the effective viscosity of crudeoil-water mixture obtained in step 4, denoted as ΔPc.

(6) Evaluate if ΔP i ΔP i ε( ) − ( ) <c e (convergence threshold value), thenend the calculation procedure, and thus the pressure drop is takenas ΔPc; if ΔP i ΔP i ε( ) − ( ) >c e , return to step 2 and repeat the abovecalculation procedure starting from ΔPc. Here i denotes the ithiteration.

7. Conclusion

The effective viscosity of crude oil-water mixtures with high waterfraction was measured by using the stirring method, and the effect ofshear rate, water fraction and temperature on the effective viscosity ofthe mixtures was studied. Combined with our previous study on theemulsification behaviors of crude oil-water systems with high waterfraction, an effective viscosity model was developed. The followingconclusions could be drawn from the present study:

(1) The effective viscosity of crude oil-water mixtures with high waterfraction decreases with the increase of shear rate, water fraction,and temperature.

(2) A model was developed for predicting the effective viscosity ofcrude oil-water mixtures with high water fraction. This model ischaracterized in three aspects: First, it may describe the shear-

Fig. 14. Comparison between the proposed model and other emulsion viscosity models.

Table 3Average relative deviation of viscosity models.

Model The proposed model Einstein Taylor Brinkman Richardson RønningsenAverage relative deviation (%) 7.4 59.6 59.8 54.2 41.3 72.6

J. Wen et al. Journal of Petroleum Science and Engineering 147 (2016) 760–770

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rate-dependence of the effective viscosity of the crude oil-watermixtures with high water fraction. Second, it takes the effect ofemulsified water into account. Third, it associates with thecomposition of crude oil.

(3) Forty sets of the effective viscosity data from mixtures of two crudeoils, which were not used in development of the model, were usedto validate the model. The results show that the proposed modelgives accurate prediction, with an average relative deviation of7.4%.

Conflict of imterest

The authors declare no competing financial interest.

Acknowledgments

Support from the National Natural Science Foundation of China(Grant 51134006), the National Natural Science Foundation of China(Grant 51534007) and the Science Foundation of China University ofPetroleum Beijing (Grant LLYJ-2011-55) is gratefully acknowledged.

Appendix A. Supporting information

Supplementary data associated with this article can be found in theonline version at doi:10.1016/j.petrol.2016.09.052.

References

Azodi, M., Nazar, A.R.S., 2013. An experimental study on factors affecting the heavycrude oil in water emulsions viscosity. J. Petrol. Sci. Eng. 106, 1–8.

Brinkman, H.C., 1952. The viscosity of concentrated suspensions and solutions. J. Chem.Phys. 20, 571.

Broughton, G., Squires, L., 1938. The viscosity of oil-water emulsions. J. Phys. Chem. 42,253–263.

Dou, D., Gong, J., 2006. Apparent viscosity prediction of non-Newtonian water-in-crudeoil emulsions. J. Petrol. Sci. Eng. 53, 113–122.

Einstein, A., 1906. Eine neue bestimmung der moleküldimensionen. Ann. der Phys. 324,289–306.

Govier, G.W., Aziz, K., 1972. The Flow of Complex Mixtures in Pipes. Van NostrandReinhold Company, New York.

Grassi, B., Strazza, D., Poesio, P., 2008. Experimental validation of theoretical models intwo-phase high-viscosity ratio liquid–liquid flows in horizontal and slightly inclined

pipes. Int. J. Multiph. Flow. 34, 950–965.Jou, D., Casas-Vázquez, J., Lebon, G., 2010. Extended irreversible thermodynamics.

Springer, London.Marsden, S., 1977. Stability of concentrated crude oil-in-water emulsions as a function of

shear rate, temperature and oil concentration. J. Can. Petrol. Technol. 16, 54–59.Mooney, M., 1951. The viscosity of a concentrated suspension of spherical particles. J.

Colloid Sci. 6, 162–170.Nädler, M., Mewes, D., 1997. Flow induced emulsification in the flow of two immiscible

liquids in horizontal pipes. Int. J. Multiph. Flow. 23, 55–68.Pal, R., Rhodes, E., 1989. Viscosity/concentration relationships for emulsions. J. Reheol.

33, 1021–1045.Pal, R., 2000. Viscosity-concentration equation for emulsions of nearly spherical

droplets. J. Colloid Interface Sci. 231, 168–175.Pal, R., 2011. Influence of interfacial rheology on the viscosity of concentrated emulsions.

J. Colloid Interface Sci. 356, 118–122.Pan, L., 1996. High pressure three phase (gas/liquid/liquid) Flow (Ph.D. thesis).

Imperial College.Phan-Thien, N., Pham, D.C., 1997. Differential multiphase models for poly dispersed

suspensions and particulate solids. J. Non-Newton. Fluid 72, 305–318.Richardson, E.G., 1933. Über die Viskosität von Emulsionen. Kolloid-Z. 65, 32–37.Rønningsen, H.P., 1995. Correlations for predicting viscosity of W/O emulsions based on

North Sea crude oils. In: SPE International Symposium on Oilfield Chemistry, SanAntonio.

Schramm, L.L., 1992. Emulsions: Fundamentals and Applications in the PetroleumIndustry. American Chemical Society, Washington D.C..

G.I., Taylor, 1932. The viscosity of a fluid containing small drops of another fluid. Proc.R. Soc. Lond. Ser. A 138, 41–48.

Wang, W., 2009. Investigation of Oil and Water Two Phase Flow and Properties ofWater-in Crude Oil Emulsion (Ph.D. thesis). ,China University of Petroleum, Beijing.

Wang, W., Gong, J., Angeli, P., 2011. Investigation on heavy crude-water two phase flowand related flow characteristics. Int. J. Multiph. Flow. 37, 1156–1164.

Wen, J., Zhang, J., Wang, Z., Zhang, Z., Zheng, F., Zhu, Y., Han, S., 2014. Full and partialemulsification of crude oil-water systems as a function of shear intensity, waterfraction, and temperature. Ind. Eng. Chem. Res. 53, 9513–9520.

Wen, J., Zhang, J., Wang, Z., Zhang, Y., 2016. Correlations between emulsificationbehaviors of crude oil-water systems and crude oil compositions. J. Petrol. Sci. Eng.146, 1–9.

White, F.M., 2011. Fluid Mechanics. McGraw-Hill, Rhode Island.Yi, S., Zhang, J., 2011. Relationship between waxy crude oil composition and change in

the morphology and structure of wax crystals induced by pour-point-depressantbeneficiation. Energy Fuels 25, 1686–1696.

Yu, C., Zhang, J., Ding, Z., Huang, Q., 2013. Measuring the flow behavior of oil-watermixtures with the stirring method. Shiyou Xuebao 34, 574–579.

Yusuf, N., Al-Wahaibi, Y., Al-Wahaibi, T., Al-Ajmi, A., Olawale, A.S., Mohammed, I.A.,2012. Effect of oil viscosity on the flow structure and pressure gradient in horizontaloil–water flow. Chem. Eng. Res. Des. 90, 1019–1030.

Zhang, J., Yan, D., 2002. Calculation of average shear rate in pipe flow based on energydissipation rate. Shiyou Xuebao 23, 88–90.

Zhang, J., Huang, Q., Yan, D., 2003. Estimation of average shear rate in stirred vesselsfor pipelining shear simulation. Shiyou Xuebao 24, 94–96.

J. Wen et al. Journal of Petroleum Science and Engineering 147 (2016) 760–770

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