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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:
[email protected].
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
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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 (%)
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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
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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.
Ind. Eng. Chem. Res., Vol. 48, No. 15, 2009 7095
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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
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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.
Ind. Eng. Chem. Res., Vol. 48, No. 15, 2009 7097
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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
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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.
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) 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)
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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