Determination of Water-in-Oil Emulsion Viscosity in Porous Media

Mohamed Arhuoma, Mingzhe Dong, Daoyong Yang,*, and Raphael Idem

Faculty of Engineering, UniVersity of Regina, Regina, SK, Canada S4S 0A2, and Department of Chemical andPetroleum Engineering, UniVersity of Calgary, Calgary, AB, Canada T2N 1N4

Experiments have been conducted to determine the viscosities of water-in-oil (W/O) emulsions in porousmedia. W/O emulsions were first prepared for different volume fractions of the dispersed phase and thencharacterized for their properties and rheological parameters including flow index and consistency constant.All prepared W/O emulsions with volume fractions between 6.78% and 33.48% were found to behave asnon-Newtonian shear-thinning fluids at fairly high viscosities. The viscosities of the emulsions were measuredduring emulsion flow in three types of sandpacks. A correlation of the viscosities of the W/O emulsions inporous media was developed by performing a regression on the experimentally measured data. The newlydeveloped correlation was validated, and a sensitivity analysis was performed to examine the effects of tortuosityand emulsion quality. The emulsion quality has a dominant effect on the viscosity of the W/O emulsions andhas been included in the correlation for the first time to achieve accurate predictions of the viscosities ofW/O emulsions in porous media. The existing correlations for oil-in-water (O/W) emulsions provideunderestimated predictions for the viscosities of W/O emulsions, whereas the droplet size distribution doesnot have a significant impact on the viscosity of the W/O emulsions tested in this study.

1. Introduction

Because heavy oils are often produced from subsurfacereservoirs with water in the form of water-in-oil (W/O)emulsions, it is of great interest for petroleum engineers tounderstand how these mixtures behave during flow withinporous media. Extensive efforts have been made to determineand quantify the viscosity of oil-in-water (O/W) emulsions inporous media both theoretically and experimentally. Fewattempts, however, have been made to determine the viscosityof W/O emulsions, even though almost two-thirds of crude oilworldwide is mainly produced in the form of W/O emulsions.1Therefore, it is of fundamental and practical importance toaccurately determine and evaluate the viscosity of W/O emul-sions in porous media.

As an enhanced oil recovery (EOR) technique, alkalineflooding has been extensively studied for conventional oils,including numerous laboratory experiments and some field tests.Laboratory experiments showed that caustic flooding couldsignificantly improve oil recovery of waterflood at a concentra-tion of 0.1% NaOH.2 It is well-accepted that in situ O/Wemulsions tend to plug growing water fingers and channels and,thus, divert flow to improve sweep efficiency. Recent researchshowed that waterflood recovery of Western Canadian heavyoils with viscosities from 1000 to over 10000 mPa s could beimproved considerably by alkaline flooding.3-7 This is ascribedto the fact that alkaline solutions can penetrate into heavy oilin porous media by forming W/O emulsions in situ. Becauseof the high viscosity of W/O emulsions, the resistance to waterflow in the high water saturation zone can be increasedsignificantly to improve sweep efficiency and thus oil recovery.

In the literature, there exist two major groups of studies fordetermining emulsion viscosities. One determines the viscosityof an emulsion as a function of emulsion quality and theviscosities of the internal and external phases without consider-ing porous media.1,8,9 The other considers the porous media

and emulsion properties, most of which are related to O/Wemulsions rather than W/O emulsions.1,10,11 The power-lawmodel is the simplest representation of the viscosity of non-Newtonian fluids. The O/W emulsion flow in glass beadpacksof several different meshes has been studied.10 A viscosity modelhas been developed to simulate the viscosity of an O/Wemulsion, assuming that the emulsion is a single-phase andhomogeneous fluid,12,13 and a model has been formulated todetermine the effective viscosity of non-Newtonian fluids.1 Also,experiments have been conducted to study the flow mechanismof emulsions in porous media and to investigate the emulsionrheology and the blocking or capture effect of emulsions duringdisplacement process.11 So far, no efforts have been madeavailable to thoroughly study the viscosity of W/O emulsionsin porous media. In addition, because experimentally determin-ing the viscosities of emulsions is time-consuming, accurate andwell-constructed correlations need to be developed for charac-terizing W/O emulsion flow in porous media.

In this study, experiments were performed to determine theviscosities of W/O emulsions. The emulsions were first preparedby the so-called agent-in-water technique.14-16 Then, emulsionflow tests in sandpacks were performed at different flow rates,and the corresponding differential pressures were recorded whenthe flow reached the steady state so that the emulsion viscositieswere determined accordingly. Subsequently, the experimentalemulsion viscosity together with the other dependent parameters,namely, porosity, permeability, flow index, consistency constant,tortuosity, and flow rate, were used to develop a correlation.

2. Experimental Section

2.1. Materials. An oil sample from an Alberta heavy oilreservoir was used for the experiments; its properties are listedin Table 1. Oil viscosity was measured using a viscometer (DV-II+Pro, Brookfield, Middleboro, MA) with a heating/coolingwater bath. The viscometer was equipped with two spindles(CPE-42 and CPE-52) for low- and high-viscosity measure-ments, respectively. The crude oil used in this study wasconsidered as a Newtonian fluid because the viscosity of thecrude oil remains almost constant while being measured with a

* To whom correspondence should be addressed. Tel.: 1-306-337-2660. Fax: 1-306-585-4855. E-mail: tony.yang@uregina.ca.

University of Regina. University of Calgary.

Ind. Eng. Chem. Res. 2009, 48, 709271027092

10.1021/ie801818n CCC: $40.75 2009 American Chemical SocietyPublished on Web 07/02/2009

viscometer at different rotation speeds. Brine with 1.0 wt %NaCl was used as the water phase for all experiments. Thealkaline solution used in this study was prepared by mixingdistilled water with 1.0 wt % NaCl and 0.2 wt % NaOH. Sands(U.S. Silica Company, Berkeley Springs, WV) of 40-60,60-100, and 120-170 meshes were used to make threedifferent sandpacks. After numerous trials of emulsion prepara-tion with a wide range of concentrations of NaCl and NaOH inthis study, the above-mentioned concentration combination wasfound to lead to consistent and stable emulsions. The emulsionequality (i.e., water content in volume percentage) was measuredusing a Dean-Stark distiller (Style A, Kimax) (Kimble Glass,Vineland, NJ).

2.2. Experimental Setup. Figure 1 illustrates a block diagramof the emulsion flow test. The experimental setup consisted of asyringe pump, three cylinders, a sandpack, a sample cylinder, anda pressure transducer that was connected to a desktop computerfor continuous recording of the pressure drop. The syringe pump(500D, Teledyne ISCO, Lincoln, NE) was used to inject fluids ata desired rate. All emulsion flow tests in this study were conductedat the ambient temperature of 22 C.

Figure 2 shows the sandpack holder used for all emulsionflow tests. The holder was composed of a stainless-steel pipe,two caps, two distributors, and two rubber O-rings. The internalsmooth surface of the holder was roughened by gluing a layerof sand to it to avoid the bypassing of fluid during flow tests.The distributors were equipped with a very fine screen to preventsand production, and the O-rings offered a tight seal on bothends. The sandpacks were 60 mm in length and 43 mm indiameter. A vibration unit was used in preparing sandpacks toensure consistent and well-packed porous media.

A mixer (Arrow-850, Arrow Engineering, Hillside, NJ) wasused to agitate the oil and water to make stable emulsions. Amicroscope (ME600, Nikon, Tokyo, Japan) equipped with adigital camera and software was used to determine the dropletsize distributions of the emulsions.

2.3. Experimental Procedures.2.3.1. Sandpack Preparation and Property Determina-

tion. Sandpacks were prepared at the ambient temperature of22 C, and fresh sand was used for each test to ensure similarwettability. Three major steps involved in preparing each porousmedium included seizing the sand, packing the core holder, anddetermining properties of the porous medium.

First, the silica sand was classified into three categories,namely, A, B, and C, each having different meshes: 40-60,60-100, and 120-170, respectively. The sieving process wasundertaken using a sieve shaker (Rotap CW, Martin Engineer-ing, Neponset, IL) with different sieves varying from 40 to 220meshes. Second, the core holder was placed in the vibrationunit in the vertical position and filled with 1.0 wt % NaCl brine,and then the desired sand was gradually added into the coreholder. A perfect packing was needed for the preparation ofthe porous media. Finally, the porosity was measured by twodifferent methods, namely, the weight and volumetric methods,and the permeability measurement was conducted using Darcys

law. Table 2 summarizes the properties of the three types ofporous media used for the emulsion flow tests.

2.3.2. Emulsion Preparation. Each emulsion was preparedusing the so-called agent-in-water technique; that is, the agentwas first mixed thoroughly with water and then agitated withcrude oil using a mixer.14-16 To start, 1.0 wt % NaCl brineand 0.2 wt % NaOH were well mixed and ready for use in theexperiments. During the experiments, crude oil and causticsolution were mixed with known volume proportions (6.78%,12.61%, 16.52%, 24.36%, and 33.48% of water) and thenagitated at 200 rpm for 60 min, which were found to be theoptimum conditions for making emulsions from the causticsolution and the given crude oil. The prepared emulsions werefound to be of the W/O type with a minimum stability time of3 days. Thus, such prepared emulsions could be used in theemulsion flow tests because of their fair stability. All emulsionswere prepared at the ambient temperature of 22 C.

2.3.3. Determination of Emulsion Viscosity. The detailedprocedure for determining the viscosity of the emulsions isdescribed as follows: An emulsion flow test was initiated afterthe properties of the porous medium and the emulsion werecharacterized. Once the sandpack was already fully saturatedwith brine, it was placed horizontally. Then, the heavy oil wasinjected into the sandpack at a flow rate of 15 cm3/h (frontalvelocity ) 0.80 m/day) for a total of 1.0 pore volume (PV) atwhich water production was negligible and thus the irreduciblewater saturation was reached. At this point, the pressure dropwas usually very stable. Subsequently, the injection flow rateof crude oil was increased to 20 cm3/h (frontal velocity ) 1.06m/day) and continued at this rate until the pressure drop becamestable. Accordingly, the same procedure was repeated for othertwo flow rates, 25 cm3/h (frontal velocity ) 1.33 m/day) and30 cm3/h (frontal velocity ) 1.60 m/day).

At the end of each oil injection, the prepared emulsion wasinjected at the same flow rate and continued until the followingtwo conditions were met: a stable pressure drop and equalityof the inlet and effluent emulsion qualities. During the experi-ments, the pressure drop stability was checked, and then asample of the effluent was taken and the quality test wasconducted. As soon as the emulsion quality of the effluent wasfound to be equal to that of the inlet, the injection was switchedto a higher flow rate. It should be noted that the injected andproduced emulsion qualities had to be equal before a higherflow rate was selected. Also, the pressure drop at this point wasused to calculate the viscosity of the emulsion.

3. Results and Discussion

3.1. Droplet Size Distribution. It is accepted that thedroplet size distribution is one of the most importantcharacteristics of O/W emulsions.14 A Nikon microscope andimaging software were used to determine the droplet sizedistribution for all emulsions used in the experiments. Thedroplet size distribution for the 6.78%, 12.61%, 16.52%,24.36%, and 33.48% emulsions were measured and graphi-cally illustrated. Emulsions with lower water contents, suchas the 6.78% emulsion, have smaller drops and fewer largedrops (see Figure 3a) compared to other emulsions withhigher quality, as shown in Figure 3b. This means that thepercentage of the large drops in an emulsion increases withincreasing water content and vice versa for the small drops.The droplet size distribution for the 6.78% emulsion had adroplet size ranging from 0 to 5.35 m, as illustrated in Figure4a, whereas Figure 4b shows the droplet size distributionfor the 33.48% emulsion, in which the droplet size ranges

Table 1. Physical Properties of the Pelican Oil Sampleproperty value

density at 15 C (kg/m3) 970.9density at 25 C (kg/m3) 964.2viscosity at 15 C (mPa s) 2440viscosity at 22 C (mPa s) 1360viscosity at 25 C (mPa s) 1020water content (%)

from 0 to 15.50 m. The average droplet sizes for the 6.78%,12.61%, 16.52%, 24.36%, and 34.48% emulsions were foundto be 0.961, 1.265, 1.513, 1.618, and 2.931 m, respectively.Based on the laboratory experiments, it was found that thedroplet size distribution does not have a significant impacton the viscosity of W/O emulsions. This is contrary to theprevious findings for O/W emulsions. This difference mightbe due to the fact that all of the emulsions used in this studywith different droplet size distributions showed no trappingof the droplets in the sandpacks because the viscosity of crude

oil is much higher than that of water in W/O emulsions. Thisis experimentally indicated by the facts that the emulsion qualitiesat the inlet and outlet reached equality and the pressure drop becamestable and remained almost constant after emulsion flow in the

Figure 1. Block diagram used to conduct experiments for emulsion flow in sandpacks.

Figure 2. Photograph of the stainless-steel sandpack holder.

Table 2. Physical Properties of Sandpackssandpack

property A B C

k (darcy) 12.0 6.0 2.9 (%) 31.6 31.0 30.6length (mm) 60 60 60diameter (mm) 43 43 43area (mm2) 1451.5 1451.5 1451.5

Figure 3. Photograph of droplet distribution under the microscope: (a) 6.78%and (b) 33.48% emulsions.

7094 Ind. Eng. Chem. Res., Vol. 48, No. 15, 2009

sandpack for a certain time. Therefore, the droplet size has less ofan effect on the flow of W/O emulsions than on that of O/Wemulsions in porous media and is excluded from the correspondingcorrelations described herein.

3.2. Tortuosity. Because fluid flows through porous mediawithin a network simple channel structure, the parametertortuosity (R) is used to characterize the fluid flow in the porousmedia, which depends on the path structure of the porousmedium.13 In this study, a conventional procedure developedfor determining tortuosity was used to characterize the porousmedia used in the experiments.13 This was done by bringingthe flow rheograms of the capillary and porous media intosuperposition for determining the tortuosity.

Figure 5 shows the superposition curves used to determine thetortuosities of the three different sandpacks, A-C. The tortuositywas found to be 1.820, 1.678, and 1.635 for sandpacks A, B, andC, respectively. The tortuosities of the sandpacks are plotted againstthe sandpack grain size in Figure 6. It can be seen from this figurethat tortuosity is weakly dependent on the grain size.

3.3. Viscosities of W/O Emulsions. Figure 7 shows thepressure drop profiles of the flow tests for the oil and the16.52%, 24.36%, and 33.48% emulsions through sandpack A.At a given flow rate, the corresponding pressure drop increasesas the emulsion quality increases. This is due to the differencesin viscosity between the injected emulsions.17 Figures 8 and 9show the pressure drop profiles of flow tests for the oil and the6.78%, 12.61%, 16.52%, and 24.36% emulsions flowing throughsandpacks B and C, respectively.

In principle, Darcys law can be generalized for bothNewtonian and non-Newtonian fluids.9,18 In this study, theeffective viscosity of an emulsion through a porous mediumwas determined using Darcys law for both emulsion flowand oil flow in the same sandpack and comparing themeasured pressure drop of the emulsion flow with that ofthe oil flow at the same flow rate. Then, eq 1 can be easily

Figure 4. Droplet size distribution: (a) 6.78% and (b) 33.48% emulsions.

Figure 5. Superposition of the sandpacks and capillary rheograms.

Figure 6. Tortuosity vs grain size.

Figure 7. Pressure drop profiles for oil and 16.52%, 24.36%, and 33.48%emulsions flowing through sandpack A.

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obtained to calculate the viscosity of an emulsion (e) as afunction of the oil viscosity (o) and the pressure drops ofemulsion flow (Pe) and oil flow (Po) at the same flowrate through the same sandpack

The calculated viscosities of the emulsions flowing throughsandpacks A-C are listed in Tables 3-5, respectively. As foreach sandpack, the emulsion viscosity at a given emulsion

quality decreased as the flow rate increased. This is an indicationof shear-thinning non-Newtonian behavior.19,20

4. Correlation of Emulsion Viscosity

4.1. Mathematical Model. The introduction of the pro-posed model for calculating the viscosity of W/O emulsionswas initiated from the Ostwald-de Waele model or thepower-law model.9 The power-law model is used to explainthe relationship between the shear rate and shear stress withthe viscosity. The general form of the power-law model fornon-Newtonian fluids is9

where K is the consistency constant (Pa sn), n is the flowindex, is the shear rate (s-1), and is the shear stress (Pa).

The parameters n and K represent the degree of non-Newtonian behavior. The fluid is considered to be a non-Newtonian fluid if n * 1. In addition, the degree ofnon-Newtonian behavior increases as the flow index, n,deviates from unity. Then, the power law has the same formas that for the Newtonian fluid (n ) 1 and K ) ).9 Becausethe Pelican crude oil was found to be a Newtonian fluid, n) 1 and K ) . Traditionally, correlations for determiningthe viscosity of emulsions in porous media are based on thefollowing equation (refer to the Appendix for a detailedmathematical derivation)

where k is the permeability of the porous medium (m2), Vpis the average velocity (m/s), eff is the effective emulsionviscosity (mPa s), R is the tortuosity, and is the porosity(fraction).

The detailed procedure for developing the correlation isprovided in the Appendix. It was found from the preliminaryresults in this study that the existing correlations generated fromeq 3 failed to predict the viscosity of W/O emulsions, and thus,modifications had to be made. Tables 3-5 demonstrate thatthe viscosity of an emulsion is strongly proportional to itsquality. This trend is also reported in the literature for O/Wemulsions.17,20,21 Therefore, eq 3 was modified to thefollowing form to take emulsion quality into account

The constants C, a, and b in eq 4 are to be determined througha regression procedure.

Figure 8. Pressure drop profiles for oil and 6.78%, 12.61%, 16.52%, and24.36% emulsions flowing through sandpack B.

Figure 9. Pressure drop profiles for oil and 6.78%, 12.61%, 16.52%, and24.36% emulsions flowing through sandpack C.

eo

)PePo

e )oPePo

(1)

Table 3. Emulsion Viscosity for the W/O Emulsions in Sandpack Aflow rate, Q oil pressure drop, Po emulsion pressure drop, Pe

emulsion quality, (%) cm3/h 10-7 m3/s mmH2O Pa mmH2O Pa emulsion viscosity, e (mPa s)16.52 15 2.50 1876 18398 3191 31294 2313.3

20 3.33 2528 24792 4174 40934 2245.525 4.17 3080 30206 5013 49162 2213.530 5.00 3725 36531 5926 58116 2163.6

24.36 15 2.50 1876 18398 3924 38483 2844.720 3.33 2528 24792 5118 50192 2753.425 4.17 3080 30206 6112 59940 2698.830 5.00 3725 36531 7248 71081 2646.2

33.48 15 2.50 1876 18398 4903 48084 3554.920 3.33 2528 24792 6533 64069 3514.525 4.17 3080 30206 7882 77299 3480.730 5.00 3725 36531 9497 93137 3467.2

) Kn (2)

eff ) K( 4VpRk)n-1 (3)

eff ) CaKb( 4VpRk)n-1 (4)

7096 Ind. Eng. Chem. Res., Vol. 48, No. 15, 2009

4.2. Emulsion Rheological Behavior. The rheological prop-erties (n and K) of emulsions with qualities of 6.78%, 12.61%,16.52%, 24.36%, and 33.48% were examined and are plottedin Figure 10. The parameters n and K are the flow index andthe consistency constant, respectively. It should be noted thatthe values of flow index for the emulsions are less than unity.This means that all of the emulsions behaved as non-Newtonianfluids.1,13 Also, the flow index decreased with increasingemulsion quality. As this parameter increases toward unity, thefluid moves closer to the Newtonian region, and it becomes a

Newtonian fluid when n equals unity. The consistency constantincreased with increasing emulsion quality. This implies thatthe viscosity increases as K increases. Thus, the fluid with ahigher K value has a higher viscosity.10 The relationship betweenthe rheological behavior parameters, n and K, and the emulsionquality is illustrated in Figure 11. The flow index decreases asthe emulsion quality is increased, whereas the consistencyconstant increases as the emulsion quality is increased. This isbecause the emulsion behaves as a non-Newtonian fluid as itsquality increases.

Table 4. Emulsion Viscosity for the W/O Emulsions in Sandpack Bflow rate, Q oil pressure drop, Po emulsion pressure drop, Pe

emulsion quality, (%) cm3/h 10-7 m3/s mmH2O Pa mmH2O Pa emulsion viscosity, e (mPa s)6.78 15 2.50 4185 41042 5030 49329 1634.6

20 3.33 5592 54841 6709 65795 1630.825 4.17 6850 67178 8190 80319 1626.130 5.00 8397 82349 9997 98041 1619.1

12.61 15 2.50 4185 41042 5935 58205 1928.720 3.33 5592 54841 7864 77122 1912.525 4.17 6850 67178 9571 93863 1900.130 5.00 8397 82349 11642 114173 1884.9

16.52 15 2.50 4185 41042 6970 68355 2265.120 3.33 5592 54841 8992 88185 2185.725 4.17 6850 67178 10905 106945 2164.130 5.00 8397 82349 12975 127246 2100.5

24.36 15 2.50 4185 41042 8472 83085 2753.620 3.33 5592 54841 11187 109711 2720.125 4.17 6850 67178 13394 131355 2659.130 5.00 8397 82349 16054 157442 2604.5

Table 5. Emulsion Viscosity for the W/O Emulsions in Sandpack Cflow rate, Q oil pressure drop, Po emulsion pressure drop, Pe

emulsion quality, (%) cm3/h 10-7 m3/s mmH2O Pa mmH2O Pa emulsion viscosity, e (mPa s)6.78 15 2.50 8485 83212 9361 91803 1504.9

20 3.33 11390 111702 12520 122784 1495.125 4.17 14226 139514 15570 152695 1488.730 5.00 16987 166592 18483 181263 1478.2

12.61 15 2.50 8485 83212 11067 108534 1774.120 3.33 11390 111702 14758 144732 1762.325 4.17 14226 139514 18329 179753 1752.530 5.00 16987 166592 21404 209909 1711.5

16.52 15 2.50 8485 83212 12720 124745 2040.320 3.33 11390 111702 16914 165876 2019.225 4.17 14226 139514 20980 205751 2006.330 5.00 16987 166592 24920 244390 1992.9

24.36 15 2.50 8485 83212 15275 149802 2448.620 3.33 11390 111702 20106 197180 2405.325 4.17 14226 139514 24904 244234 2381.130 5.00 16987 166592 29364 287973 2349.8

Figure 10. Rheological behavior of the emulsions.Figure 11. Flow index (n) and consistency constant (K) vs emulsionquality.

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4.3. Sensitivity Analysis. A sensitivity analysis was per-formed to evaluate the newly developed eq 4 for predicting theviscosity of W/O emulsions with seven parameters, namely, theexperimentally determined flow index (n), the consistencyconstant (K), the emulsion quality (), the average velocity (Vp),the tortuosity (R), the permeability (k), and the porosity ().The emulsion viscosity correlation was generated using thefollowing procedure:

(1) Experiments were conducted to determine the above-mentioned seven parameters, which are included in eq 4.

(2) The nonlinear regression technique was used to generatethe regression for the experimental data. Equation 5 is theregressed correlation

(3) The regression was repeated excluding the emulsionquality, giving eq 6 as the regressed correlation. A differentform of the correlation exists, with a different absolute averagerelative error and R2 value

(4) The regression was repeated for all other parametersexcluding the tortuosity, and eq 7 was obtained as the correlation

(5) The three correlations were compared in terms of theabsolute average relative errors, R2 values, and parity charts foreach correlation.

The aforementioned three correlations were used to calculatethe viscosities of emulsions. Correspondingly, a parity chart wasgenerated for each correlation with the absolute average relativeerror. The absolute average relative error is defined as the sumof the relative difference between the experimental and calcu-lated values of the emulsion viscosity divided by the number

of measurements. In this study, the absolute average relativeerror can be written as22,23

where n is the number of data points, cal is the calculatedemulsion viscosity (mPa s), exp is the experimental emulsionviscosity (mPa s), and a is the absolute average relativeerror (%).

Figure 12 illustrates the viscosities of emulsions obtainedusing eq 5, which represents the viscosity of an emulsion whenall of the interdependent parameters are included. The abso-lute average relative error and R2 value were found to be 3.11%and 0.9729, respectively, with an overestimated level of 3.30%and an underestimated level of 3.00%. As a result, the effectiveemulsion viscosity (eff) was ultimately found to be a function

Figure 12. Parity chart for the model including emulsion quality (),consistency constant (K), flow index (n), average flow velocity (Vp),permeability (k), porosity (), and tortuosity (R).

eff ) 3.211 1030.366K0.44( 4VpRk)n-1 (5)

eff ) 0.76 103K1.31( 4VpRk)n-1 (6)

eff ) 3.64 1030.41K0.43(4Vpk)n-1 (7)

Figure 13. Parity chart for the model excluding the emulsion quality.

Figure 14. Parity chart for the model excluding tortuosity.

Table 6. Validation of the Newly Developed Correlation for anEmulsion of 33.48% in Sandpack A

flow rateemulsion viscosity

(mPa s)cm3/h 10-7 m3/s measured calculated

relative error[(exp - cal)/exp] 100%

15 2.50 3554.9 3610.4 -1.5620 3.33 3514.5 3496.8 0.5025 4.17 3480.7 3411.2 1.9930 5.00 3467.2 3342.8 3.58

a )1n | i)1n exp,i - cal,iexp,i | 100% (8)

7098 Ind. Eng. Chem. Res., Vol. 48, No. 15, 2009

of emulsion quality (), consistency constant (K), flow index(n), average flow velocity (Vp), permeability (k), porosity (),and tortuosity (R).

Figure 13 is the parity chart for the experimental andcorrelated emulsion viscosities generated using eq 6 with allof the parameters except the emulsion quality. In this case,the absolute average relative error and R2 value were foundto be 5.30% and 0.9121, respectively, with an overestimatedlevel of 7.10% and an underestimated level of 4.00%. Thisis mainly due to the contribution of emulsion quality and itssignificant impact on the viscosity of emulsions. Emulsionviscosity was found to rapidly increase with increasingemulsion quality. This explains the deformation in the overall

trend when emulsion quality is excluded from the emulsionviscosity correlation.

Figure 14 is the parity chart for the experimental andcorrelated emulsion viscosities based on eq 7, which excludestortuosity. The absolute average relative error and R2 value weredetermined to be 3.27% and 0.9675, respectively, with anoverestimated average error of 3.70% and an underestimatedaverage error of 3.00%. The tortuosity has a minor effect onthe viscosity of emulsions, as demonstrated in the experiments.In practice, it is difficult and time-consuming to experimentallydetermine the tortuosity. The tortuosity can be excluded whendetermining the viscosity of an emulsion because the correlationis still sufficiently accurate for determining the viscosity of W/O

Table 7. Comparison between the Newly Developed Correlation and the Existing Models for Determining Emulsion Viscosity in Sandpack Aemulsion viscosity, mPa s

flow rate this studyemulsion quality (%) cm3/h 10-7 m3/s measured calculated Uzoigwe and Marsden10 Christopher and Middkeman24 Gregory and Grisky25

16.52 15 2.50 2313.3 2235.6 609.3 752.1 736.820 3.33 2245.5 2186.7 595.9 735.5 720.525 4.17 2213.5 2149.5 586.8 724.3 709.630 5.00 2163.6 2119.5 578.7 714.3 699.7

24.36 15 2.50 2844.7 2785.6 563.0 698.6 681.720 3.33 2753.4 2714.4 548.4 680.5 664.125 4.17 2698.8 2660.5 538.9 668.6 652.530 5.00 2646.2 2617.2 529.9 657.5 641.6

Table 8. Comparison between the Newly Developed Correlation and the Existing Models for Determining Emulsion Viscosity in Sandpack Bemulsion viscosity, mPa s

flow rate this studyemulsion quality (%) cm3/h 10-7 m3/s measured calculated Uzoigwe and Marsden10 Christopher and Middkeman24 Gregory and Grisky25

6.78 15 2.50 1634.6 1562.9 791.6 954.3 951.120 3.33 1630.8 1557.1 788.7 950.6 947.525 4.17 1626.1 1552.5 786.6 948.1 945.030 5.00 1619.1 1548.9 784.5 945.7 942.6

12.61 15 2.50 1928.7 1892.9 666.6 815.0 803.920 3.33 1912.5 1864.8 656.4 802.5 791.625 4.17 1900.1 1843.2 649.3 793.9 783.130 5.00 1884.9 1825.8 642.4 785.4 774.7

16.52 15 2.50 2265.1 2160.2 585.9 723.3 708.520 3.33 2185.7 2112.9 573.6 708.1 693.625 4.17 2164.1 2076.9 564.5 696.8 682.530 5.00 2100.5 2048.0 556.3 686.7 672.7

24.36 15 2.50 2753.6 2676.6 538.4 668.0 651.920 3.33 2720.1 2608.2 523.8 649.9 634.225 4.17 2659.1 2556.3 514.5 638.4 623.030 5.00 2604.5 2514.7 505.4 627.1 612.0

Table 9. Comparison between the Newly Developed Correlation and the Existing Models for Determining Emulsion Viscosity in Sandpack Cemulsion viscosity, mPa s

flow rate this studyemulsion quality (%) cm3/h 10-7 m3/s measured calculated Uzoigwe and Marsden10 Christopher and Middkeman24 Gregory and Grisky25

6.78 15 2.50 1504.9 1554.3 789.3 951.4 948.320 3.33 1495.1 1548.5 786.3 947.8 944.725 4.17 1488.7 1544.0 784.0 945.1 942.030 5.00 1478.2 1540.4 782.3 942.9 939.8

12.61 15 2.50 1774.1 1851.7 658.3 804.8 793.920 3.33 1762.3 1824.1 648.0 792.2 781.525 4.17 1752.5 1803.0 640.3 782.8 772.230 5.00 1711.5 1786.0 634.9 776.2 765.7

16.52 15 2.50 2040.3 2091.0 575.9 710.9 696.420 3.33 2019.2 2045.2 562.4 694.2 680.025 4.17 2006.3 2010.4 552.4 681.8 667.930 5.00 1992.9 1982.4 544.5 672.1 658.4

24.36 15 2.50 2448.6 2576.6 528.1 655.3 639.520 3.33 2405.3 2510.8 514.0 637.7 622.325 4.17 2381.1 2460.9 503.2 624.4 609.330 5.00 2349.8 2420.8 495.1 614.3 599.5

Ind. Eng. Chem. Res., Vol. 48, No. 15, 2009 7099

emulsions. The final form of the developed correlation can beexpressed as in eq 5.

4.4. Correlation Validation. After the correlation wasconstructed, we conducted a validation test to ensure itsapplicability for out-of-range predictions. The emulsion with33.48% quality, which was not included in the data forgenerating the correlation, was used to validate the correla-tion. Equation 5, the newly developed correlation, was usedto predict the viscosity for the 33.48% emulsion in sandpackA. At four different flow rates, namely, 15, 20, 25, and 30cm3/h, the emulsion viscosities were calculated correspond-ingly. Table 6 lists the experimental emulsion viscosities andthe data calculated using eq 5 for the 33.48% emulsion. Themaximum relative error was found to be 3.58%. This indicatesthat the newly developed correlation can predict the viscosityof W/O emulsions flowing through porous media with a smallrelative error. It was found that the emulsion viscositydecreases as the flow rate increases, as indicated in Table 6.Thus, the W/O emulsions tested in this study behave as non-Newtonian fluids.

4.5. Correlation Comparison. The newly developed cor-relation in this study was compared with some avail-able models.10,24,25 Tables 7-9 include comparisons of theexperimentally measured W/O emulsion viscosities and thecalculated data from the aforementioned models for all threesandpacks, A-C. It can be seen that the emulsion viscositiesare underestimated by all three existing models. This ismainly due to the fact that all of the existing models weregenerated for O/W emulsions rather than W/O emulsions.The first model was developed for O/W emulsions with verylow viscosities.10 The second model was developed usingpacked tubes and not a true porous medium,24 and the thirdmodel was developed for molten polymers rather than foremulsions.25 In particular, the fluids used in all of theexperiments were tap water and Soltrol with a viscosity of1.3 mPa s, and glass beads of five different ranges of meshsize rather than actual sands were used.

5. Conclusions

Experiments have been conducted in this study to determinethe viscosities of W/O emulsions using three types of sand asporous media at different flow rates. The major conclusions thatcan be drawn from this study are as follows:

(1) The droplet size distribution did not have a significantimpact on the viscosity of the W/O emulsions prepared in thisstudy. This is contrary to the previous findings for O/Wemulsions. This difference can be ascribed to the fact that allemulsions used in this study with different droplet size distribu-tions showed no trapping of the droplets in the sandpacksbecause the viscosity of crude oil is much higher than that ofwater in W/O emulsions. This is experimentally indicated bythe emulsion qualities at the inlet and outlet reaching equalityand by the pressure drop becoming stable and remaining almostconstant.

(2) The measured emulsion viscosity decreased as the flowrate increased. The degree of reduction was higher for emulsionsof higher quality compared with those of lower quality. Thereduction is mainly due to the fact that all of the emulsionsshowed shear-thinning behavior. The reduction was stronger inlow-permeability porous media because the shear rate is higherin lower-permeability porous media at the same flow rate.

(3) A correlation was developed for determining the viscosityof W/O emulsions in porous media. The emulsion quality hasa dominant effect on the viscosity of W/O emulsions and was

therefore included in the correlation for the first time to achieveaccurate predictions of the viscosities of W/O emulsions inporous media.

(4) Emulsion quality was found to be a significant parameteraffecting the overall accuracy of the correlation. The absoluteaverage relative error and R2 value were found to be 3.11%and 0.9725, respectively, with overestimated and underestimatedlevels of 3.30% and 3.00%, respectively. A sensitivity analysisshowed that the tortuosity has a minor effect on the viscosityof emulsions for the homogeneous porous media used in thisstudy and can be excluded from the correlation.

(5) The new correlation was validated using experimentallymeasured emulsion viscosities for a 33.48% emulsion. Themaximum relative error was found to be 3.58%. This meansthat the correlation can be used to accurately predict theviscosities of W/O emulsions.

(6) Compared with the newly developed model, the existingmodels for O/W emulsions usually provide underestimatedpredictions for the viscosities of W/O emulsions.

Acknowledgment

We thank the Petroleum Technology Research Centre (PTRC),the Natural Sciences and Engineering Research Council (NSERC)of Canada, and the Canada Foundation for Innovation (CFI)for financial support of this work.

NomenclatureVariablesd ) capillary diameter, mk ) permeability of porous medium, m2K ) consistency constant, Pa snn ) flow indexQ ) flow rate, m3/src ) capillary tube radius, mVc ) flow velocity for capillary tube, m/sVp ) average velocity in porous media defined in eq 3, m/sVjc ) average velocity defined in eq (A-5), m/sVjp ) average velocity defined in eq (A-7), m/sGreek SymbolsR ) tortuosity ) shear rate, s-1c ) shear rate for capillary tube, s-1p ) shear rate for porous medium, s-1Lc ) capillary tube length, mLp ) length of porous medium, mP ) pressure drop, PaPc ) pressure drop across capillary tube, PaPe ) pressure drop of emulsion flow, PaPo ) pressure drop of oil flow, PaPp ) pressure drop across porous medium, Pa ) relative average error, %a ) absolute average relative error, % ) emulsion quality, %cal ) calculated emulsion viscosity, mPa se ) viscosity of external phase or emulsion, mPa seff ) effective emulsion viscosity, mPa sexp ) experimental emulsion viscosity, mPa so ) viscosity of suspending medium or oil phase, mPa s ) shear stress, Pac ) shear stress for capillary tube, Pap ) shear stress for porous medium, Pa ) porosity, fraction

7100 Ind. Eng. Chem. Res., Vol. 48, No. 15, 2009

Subscriptsa ) absolutec ) capillary tubecal ) calculatede ) emulsioneff ) effectiveexp ) experimentalo ) oilp ) porous medium

AppendixThe general form of the power-law model for a non-

Newtonian fluid is9

The first step for developing the proposed model was todetermine the shear rate and shear stress formulas and theirrelationships with viscosity. Equations A-2 and A-3 can be usedto calculate the shear rate and shear stress, respectively, for acapillary tube13

According to Hagen-Poiseulle theory, the volumetric flow ratefor laminar flow in a capillary tube, Q, can be expressed as26

Dividing both sides by the cross-sectional area gives

Furthermore, Darcys law can be arranged in the following form

Applying the Dupuit-Forchheimer correlation, which states thatthe average pore velocity equals the average velocity dividedby the porosity, transforms eq A-6 into13

Combining eqs A-5 and A-7 yields

Equation A-8 implies that a parameter can be used to super-impose the rheological properties of the capillary and sandpack,which is termed the tortuosity (R).13 Consequently, eq A-8 canbe rearranged as

Substituting rc in eqs A-2 and A-3 by its equivalent from eqA-9 results in eqs A-10 and A-11, respectively

Because the viscosity is equal to the shear stress divided by theshear rate, it can be written in the form

This form of the viscosity is known as the effective viscosity,which is mainly a function of the shear rate in the case of non-Newtonian fluids. In other words, the effective viscosity is theviscosity of a fluid at a specific shear rate.18

Finally, substituting eq A-10 into eq A-12 yields theexpression

In addition, as discussed previously, the emulsion quality ()significantly affects the emulsion viscosity and is added intoeq 3 to yield the final form

The constants C, a, and b in eq 4 are to be determined throughthe regression procedure as described previously.

Literature Cited(1) Abou-Kassem, J. H.; Farouq Ali, S. M. Modelling of Emulsions Flow

in Porous Media. J. Can. Pet. Technol. 1995, 34, 30.(2) Jennings, H. Y., Jr.; Johnson, C. E., Jr.; McAuliffe, C. D. A

Caustic Waterflooding Process for Heavy Oils. J. Pet. Technol 1974,26, 1344.

(3) Ma, S. Enhanced Oil Recovery by Dilute Alkaline Flooding. M.Sc.Thesis, University of Regina, Regina, SK, Canada, 2005.

(4) Liu, Q. Interfacial Phenomena in Enhanced Heavy Oil Recovery byAlkaline Flood. Ph.D. Dissertation, University of Regina, Regina, SK,Canada, 2006.

(5) Liu, Q.; Dong, M.; Yue, X.; Hou, J. Synergy of Alkali andSurfactant in Emulsification of Heavy Oil in Brine. Colloids Surf. A2006, 273, 219.

(6) Ma, S.; Dong, M.; Li, Z.; Shirif, E. Evaluation of the Effectivenessof Chemical Flooding Using Heterogeneous Sandpack Flood Test. J. Pet.Sci. Eng. 2007, 55, 294.

(7) Liu, Q.; Dong, M.; Ma, S.; Tu, Y. Surfactant Enhanced AlkalineFlooding for Western Canadian Heavy Oil Recovery. Colloids Surf. A 2007,293, 63.

(8) Mooney, M. The Viscosity of a Concentrated Suspension of SphericalParticles. J. Colloid Interface Sci. 1951, 6, 162.

(9) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena;Wiley: New York, 2002.

(10) Uzoigwe, A. C.; Marsden, S. S., Jr. Emulsion Rheology and FlowThrough Unconsolidated Synthetic Porous Media. Presented at the SPE-AIME 45th Annual Fall Meeting, Houston, TX, Oct 4-7, 1970; Paper SPE3004.

(11) Khambharatana, F. Flow of Emulsions in Porous Media. Ph.D.Dissertation, University of Alberta, Edmonton, AB, Canada, 1993.

(12) Alvarado, D. A. The Flow of Macroemulsions Through PorousMedia. Ph.D. Dissertation, Department of Petroleum Engineering, StanfordUniversity, Stanford, CA, 1975.

(13) Alvarado, D. A.; Marsden, S. S., Jr. Flow of Oil-in-Water EmulsionsThrough Tubes and Porous Media. SPE J. 1979, 19, 369.

(14) Bennett, H. Practical Emulsions; Chemical Publishing Company:Brooklyn, NY, 1967.

(15) Becher, P. Emulsions: Theory and Practice; American ChemicalSociety: Washington, DC, 2001.

) Kn (A-1)

c )8Vcd (A-2)

c )rcPc2Lc

(A-3)

Q ) rc4

8 (PcLc) (A-4)Vjc )

12(rc2 )2( PcLc) (A-5)

Q ) kA (PL ) (A-6)

Vjp )k( PpLp) (A-7)

k( PpLp) ) 12(rc2 )2( PcLc) k ) 12(rc2 )2 (A-8)

rc ) Rk/ (A-9)

p )4VpRk (A-10)

p )Rk/Pp

2Lp(A-11)

eff )) K

n

) Kn-1 (A-12)

eff ) K( 4VpRk)n-1 (3)

eff ) CaKb( 4VpRk)n-1 (4)

Ind. Eng. Chem. Res., Vol. 48, No. 15, 2009 7101

(16) Rastogi, M. C. Surface and Interfacial Science: Applications toEngineering and Technology; Alpha Science: Oxford, UK, 2003.

(17) Sjoblom, J. Encyclopedic Handbook of Emulsion Technology;Marcel Dekker: New York, 2001.

(18) Sadowski, T. J.; Bird, R. B. Non-Newtonian Flow through PorousMedia. I. Theoretical. J. Rheol. 1965, 9, 243.

(19) Darley, H. C. H.; Gary, G. Composition and Properties ofDrilling and Completion Fluids; Gulf Professional Publishing: Houston,TX, 1988.

(20) Schramm, L. L. Emulsions: Fundamentals and Applications inthe Petroleum Industry; American Chemical Society: Washington, DC,1992.

(21) Sherman, P. Emulsion Science; Academic Press: New York, 1968.(22) Behkish, A. Hydrodynamic and Mass Transfer Parameters in Large-

Scale Slurry Bubble Column Reactors. Ph.D. Dissertation, University ofPittsburgh, Pittsburgh, PA, 2004.

(23) Sahinoglu, M. Trustworthy Computing: Analytical and QuantitatiVeEngineering EValuation; John Wiley & Sons: New York, 2007.

(24) Christopher, R. H.; Middleman, S. Power-Law Flow through aPacked Tube. Ind. Eng. Chem. Fundam. 1965, 4, 422.

(25) Gregory, D. R.; Grisky, R. G. Flow of Molten Polymers ThroughPorous Media. AIChE J. 1967, 13, 122.

(26) Papanastasiou, T.; Georgiou, G.; Alexandrou, A. Viscous FluidFlow; CRC Press: Boca Raton, FL, 2000.

ReceiVed for reView November 26, 2008ReVised manuscript receiVed June 9, 2009

Accepted June 10, 2009

IE801818N

7102 Ind. Eng. Chem. Res., Vol. 48, No. 15, 2009