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Petroleum Science and Technology, 29:649–663, 2011
Copyright © Taylor & Francis Group, LLC
ISSN: 1091-6466 print/1532-2459 online
DOI: 10.1080/10916460903451977
The Role of Throat Orientation on Dispersion of
Solvent in Crude Oil-Saturated Porous Media
A. R. REZAEIPOUR,1 R. KHARRAT,1
M. H. GHAZANFARI,1;2 AND E. YASARI1;2
1Petroleum Research Center, Petroleum University of Technology, Tehran, Iran2Chemical and Petroleum Engineering Department, Sharif University of
Technology, Tehran, Iran
Abstract In this work a series of hydrocarbon solvent injection experiments was
performed on glass micromodels with different throat orientations that were initiallysaturated with crude oil at several fixed flow rate conditions. The solvent concentration
as a function of location and time was measured using image analysis of colorintensity of continuously provided pictures during the injection process. The provided
concentration calibration curve of solvent in crude oil was used for back-calculatingthe solvent concentration along the dispersion zone. The longitude and transverse
dispersion coefficients were determined by fitting the results of the mathematical modelto the experimental data. It was found that the longitude dispersion decreased when the
throat orientation angle increased. In contrast, the transverse dispersion increased.In addition, two trends were observed in variation of longitudinal and transversal
dispersion versus Pe. For Pe < 50, the longitude and transverse dispersivities wereequal to 2.94e-7 and 1.03e-6 cm2/sec, respectively, and for 50 < Pe < 400, they were
equal to 1.34e-6 and 3.02e-7 cm2/sec. For the range of experiments performed here,it was found that the longitudinal-transversal dispersion ratio varied from 3.5 to 1.5
when the throat orientation angle changed from 0 to 60ı. The results of this workwill be helpful in designing successful miscible enhanced oil recovery processes.
Keywords crude oil, dispersion, experimental, hydrocarbon solvent, micromodel,numerical, porous medium, throat orientation
1. Introduction
Miscible displacement in fluid flowing through a porous medium plays an important role,
which is relevant to many environmental and industrial applications; for instance, miscible
displacements in enhanced secondary oil recovery operations, pollutant spreading in
groundwater, clean-up of contaminated sites, filtration, chromatography, and fluid–solid
catalytic and noncatalytic reactions. The dispersion coefficients, which are key parameters
for simulation of miscible flooding processes, are affected by many factors such as
flow field and pore structure characteristics (Sahimi, 1995). However, the role of throat
orientation on dispersion of a substance like hydrocarbon solvent in crude oil–saturated
porous medium remains a topic of debate in the literature. In addition, there is a lack of
sufficient data for longitudinal and transversal dispersion of solvent in crude oils.
Address correspondence to A. R. Rezaeipour, Tehran Petroleum Research Center, ShahidGhasemzadian St., South Khosrowst., Sattarkhan Ave., P.O. Box 1453953111, Tehran, Iran. E-mail:[email protected]
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If two miscible fluids are in contact, with an initially sharp interface, they will slowly
diffuse into one another. As time passes, the sharp interface between the two fluids will
change to a diffused mixed zone grading from one pure fluid to the other. This diffusion
arises because of random motion of the molecules. If fluids are flowing through the
porous medium, dispersion may be greater than that due to diffusion alone (Gamanis
et al., 2005).
Dispersion in a porous medium occurs as a consequence of two different processes:
(1) molecular diffusion, which originates from the random molecular motion of solute
molecules; and (2) mechanical dispersion, which is caused by nonuniform velocities and
flow path distribution. Molecular diffusion and mechanical dispersion cannot be separated
in a flow regime (Bear et al., 1993).
The Peclet number expresses roughly the ratio of the diffusion time (required for
solvent molecules to move from one streamline to another one) to the convection time
(required for molecules to move along streamlines at the mean fluid velocity). In general,
the dispersion coefficients have been found to be nonlinear functions of Peclet number,
and five dispersion regimes have roughly been distinguished depending on the prevailing
transport mechanism (Sahimi, 1995): (1) the diffusion regime P e < 0:3; (2) the transition
regime 0:3 < P e < 5; (3) the power law regime 5 < P e < 300; (4) the pure convection
or mechanical dispersion regime 300 < P e < 105; and (5) the turbulent dispersion
regime P e > 105.
A great deal of experimental work has been focused on the determination of the hy-
drodynamic dispersion coefficients as function of the Peclet number by using a variety of
techniques such as acoustic, nuclear magnetic resonance (NMR), computed tomography
(CT) scan, radioactive tracers, etc. (Bacri et al., 1987; Sahimi, 1995; Ding and Candela,
1996; Peters et al., 1996; Drazer et al., 1999; Manz et al., 1999) for the measurement of the
transient changes of solute concentration. However, the unanswered question remaining
is what the variation of dispersion coefficients with the throat orientation would be.
Micromodels are transparent artificial models of porous media that can be used to
simulate transport processes at the pore scale (Wilson, 1994) and therefore are very
attractive for these types of studies. The types of micromodel studies reported in the
literature (Buckley, 1991) vary widely in their methods and applications. Mattax and Kyte
(1961) made the first etched glass network, but this approach was significantly improved
by application of photo etching techniques (Davis and Jones, 1968). Micromodels are
mostly fabricated by etching the desired pore network pattern on two plates of mirror
glass, which are then fused together. Using this method, highly intricate and detailed
patterns can be etched with the dimensions of pores and throats as low as a few
micrometers. The models are provided with an inlet and an outlet at the two ends.
Details of the model production procedure are given elsewhere (McKellar and Wardlaw,
1982). Most researchers have used the micromodel as a qualitative tools, and a few have
conducted quantitative studies, especially when the problem is dispersion of solvent in
crude oil.
In this work a series of miscible hydrocarbon solvent injections was conducted on
micromodels that were initially saturated with the crude oil at several fixed flow rate
conditions. The porous models were designed such that the angles of throat orientation
and mean flow direction of 0, 30, 45, and 60ı were provided. The solvent concentra-
tion contours as a function of time and location were measured, and the longitudinal
and transversal dispersion coefficients were determined by fitting the predicted solvent
concentration profile from numerical solution of the convection–diffusion equation to the
experimental data. The role of throat orientation on dispersion was investigated.
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Throat Orientation and Dispersion of Solvent 651
2. Mathematical Modeling
2.1. Governing Equations
Transport of a nonreacting fluid in saturated porous media is generally modeled using
the convection–diffusion equation, mass conservation law (Bacri et al., 1987; Buckley,
1991) as follows:
@
@x
�
DL
@C
@x
�
C@
@y
�
DT
@C
@y
�
�@.uxC/
@x�
@.uyC/
@yD
@C
@t(1)
where C is solute concentration, usually defined as mass of solute per unit volume
of solution; t is time; ux and uy are the components of flow velocity; and DL and
DT are hydrodynamic longitude and transverse dispersion coefficients, respectively. The
governing equations of incompressible flow in an isentropic and homogeneous porous
media are described by the continuity and momentum conservation equations as fol-
lows:
@ux
@xC
@uy
@yD 0 (2)
ux Dk
'�
�
�@p
@x
�
(3)
uy Dk
'�
�
�@p
@y
�
(4)
where k and ' are the permeability and porosity of the medium, and � is the viscosity
of fluid. The effects of molecular diffusion and mechanical dispersion are usually added
together in a single diffusive flux term where the value of the hydrodynamic disper-
sion coefficient equals the sum of the molecular diffusion and mechanical dispersion
coefficients. So the hydrodynamic dispersion coefficients in Eq. (1) could be defined
as:
DL D De C ˛LP e (5)
DT D De C ˛T P e (6)
where De , u, ˛L, and ˛T are the effective molecular diffusion, average flow velocity,
and longitude and transverse dispersivity, respectively. The longitude and transverse
dispersivity were determined by fitting the predicted solute concentration profile from
numerical solution of the convection–diffusion equation to experimental data.
2.2. Numerical Solution
There are different methods to numerically solve the set of Eqs. (1) to (6) (De Arcangelis,
1986; Sahimi, 1995; Lowe and Renkel, 1996; Souto and Moyne, 1997; Whitaker, 1999;
Oppenheimer, 2000; Ahmadi et al., 2001; Bruderer and Bernabe, 2001; Drazer and
Koplik, 2001; Huseby et al., 2001). But the solution is restricted by many factors such
as required memory volume and computer CPU performance. Some refinement in size
of grids in both domains of time and space is required to achieve more accurate results
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652 A. R. Rezaeipour et al.
Figure 1. Concentration profiles at y D 0 for the case of DL D 5e �5 and DT D 5e �6 cm2/sec.
Flow rate is equal to 2e-4 mL/min.
because of sharp changes in flow field velocity and concentrations. Any incompatible
changes in step sizes may lead to divergence or instability in solutions.
Here, a computer code was developed to simultaneously solve Eqs. (1) to (6). Initial
and boundary conditions are C.0; x; y/ D 0, C.t > 0; 0; 0/ D 1, .@C=@y/.t; x; 0/ D 0,
.@C=@y/.t; x; W / D 0, and .@C=@x/.t; 1; y/ D 0. These special boundary conditions
cause a stiff problem and a proper approach to the solution should be chosen. The sample
results of the mathematical model for the case of flow rate equal to 2e-4 mL/min and
longitudinal and transversal dispersion coefficients of 5e-4 and 5e-6 cm2 /sec are shown
in Figure 1.
3. Experimental Descriptions
3.1. Micromodel
The glass micromodel is fabricated by etching the desired pattern on a plate of mirror
glass and then fusing it to another one. Using this method, detailed patterns can be etched
which the dimensions of pores and throats as low as a few micrometers.
3.2. Setup
The micromodel setup is composed of a micromodel holder, which is placed on a platform
and includes a camera equipped with a video recording system, a precise pressure
transmitter, and a precise low rate pump, which is used to control the flow rate of
fluids through micromodel. The pump is used to inject working fluid depending on the
request from a low of 1.5e-5 mL/min to a maximum of 10 mL/min in the range. The
schematic of experimental setup is shown in Figure 2.
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Throat Orientation and Dispersion of Solvent 653
Figure 2. Schematic of micromodel setup.
3.3. Fluids
Fluids used in the experiments are n-heptane as hydrocarbon solvent and a sample crude
oil from Iranian Sarvak reservoir with 19.8 ıAPI. In order to elevate the visibility, the
color of hydrocarbon solvent was changed to red using Sudan red. Colored hydrocarbon
solvent was filtered using fine filter paper to remove any dissolved dye particles.
3.4. Image Analysis
Continuous imaging from the injection process was performed by a digital camera
connected to a computer. An image analysis code was developed to process the images
and to determine the concentration contour of solvent during the experiments. Colors in
the images are a combination of red, green, and blue character with a certain intensity
from 0 to 255. The red characteristic of the images was chosen to represents the
concentration of solvent in the micromodel, because it is the characteristic most sensitive
to the solvent concentration in a wide range.
3.5. Test Procedure
Before beginning the experiments, the micromodels were cleaned using toluene and
propanol and then saturated with the crude oil. Hydrocarbon solvent was injected into
the micromodel through the inlet port at a preselected flow rate using a high accuracy
flow-controlled injection pump. The outlet was at atmospheric pressure.
3.5.1. Injection Rate Effects. In a porous rock, fluid must move on the average at
about 45ı to the net direction of flow (Davis and Jones, 1968); therefore, this angle was
selected for investigating the effect of injection rate on dispersion coefficients. Injection
was performed at six different flow rates of 2e-4, 4e-4, 8e-4, 16e-4, 32e-4, and 64e-4
mL/min. Figures 3a–3d typically shows the successive images of solvent flooding into
the crude oil saturated micromodel at a flow rate of 1e-4 mL/min.
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Figure 3. Visualization of dispersion of solvent in crude oil at different times: (a) 1 min, (b) 20
min, (c) 40 min, and (d) 60 min. (This figure can be viewed in color online)
3.5.2. Throat Orientation Effects. Some homogeneous porous media with different
angles of 30, 45, 60, 90ı between throat orientation and mean flow direction were
generated to investigate the throat orientation effect on dispersion of hydrocarbon solvent
in crude oil. Generated throat orientations are shown in Figure 4.
Figures 5a–5d show images of flooding experiments at 0.015 PV solvent injection
into the micromodels with different throat orientation angles at fixed flow rate of 0.0004
mL/min.
3.6. Calibration Curve
To quantify the concentration of solvent in micromodel using an image analysis technique,
a series of calibration tests by known volume hydrocarbon solvent and crude oil mixtures
was performed to correlate the R� to volumetric concentration of hydrocarbon solvent. To
avoid the effect of any probable changes in light conditions, dimensionless red intensity,
R�, is defined for each image as below:
R DR � Ro
Rs � Ro
(7)
Figure 4. Schematic of micromodel patterns used in the experiments. The throat orientation angles
are (a) 0ı, (b) 30ı, (c) 45ı, and (d) 60ı.
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Throat Orientation and Dispersion of Solvent 655
Figure 5. Solvent flooding in crude oil–saturated micromodel with different throat orientations at
flow rate of 4e-4 mL/min and 0.15 PV injection: (a) 60ı, (b) 45ı, (c) 30ı, and (d) 0ı. (This figure
can be viewed in color online)
where R is the red intensity. Indices s and o refer to pure solvent and oil, respectively.
The results are shown in Figure 6. Correlation that relates R� to the concentration is
C �D �2:167R�3
C 3:4696R�2D 0:3091R�
D 3 � 10�5 (8)
where C �D VS =V , VS , and V are the solvent and total mixture volume, respectively.
3.7. Micromodel Properties
Absolute permeability of the micromodel was determined using Darcy’s law by measuring
the pressure drop/flow rate response. Distilled water at room temperature was injected
into the micromodel at desired injection flow rate, and the pressure drop between the
injection port and the exit was measured by a pressure transmitter. Four different pressure
drop and flow rate data sets were obtained and the best-fitting straight lines passed the
origin. The porosity of micromodel was measured using an image analysis technique. The
Figure 6. Calibration curve of dimensionless red color intensity to solvent concentration.
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656 A. R. Rezaeipour et al.
Table 1
Physical and hydraulic properties of micromodel
Length, cm 10 Width, cm 6
Absolute permeability, D 1.15 Average areal porosity 0.485
Average depth, �m 46 Pore volume, cm3 0.134
Table 2
Peclet numbers in accordance with experiment conditions
Flow rate, mL/min 2e-4 4e-4 0.8 1.6 3.2 6.4
Pe 12 23 46 92 185 370
etched depths of the pores in the micromodel are relatively uniform, so the ratio of the
area occupied by colored fluid to the total area of the micromodel is equal to porosity.
During the injection of colored water into the micromodel with known flow rate, for
a desired incremental time, the incremental area occupied by the colored water was
measured using an image analysis technique. Incremental time multiplied by injection
flow rate divided by incremental area is equal to average etched height of that region. The
pore volume is equal to areal porosity multiplied by average etched height. The physical
and hydraulic properties of the micromodel are shown in Table 1.
4. Results and Discussion
4.1. Peclet Number
The ratio between the time needed for fluid to traverse a characteristic length l by
advection, tadv , and the time needed for particles to travel the same length by diffusion,
tdif , is defined as Peclet number, P e D tadv=tdif D .ut= l/=.Dmt= l2/ D ul=Dm, where
l is grain diameter or pore length, and Dm is the molecular diffusion. For the mixture
of hydrocarbon solvent and crude oil, Dm was calculated by using the correlation (Perry
et al., 1999) and is equal to 5e-6 cm2/sec. At relatively low P e values, the diffusion
time is comparable to the convection time. At increasing values of P e, the diffusion time
exceeds the convection time, so that the forward convective transport of solute is favored
against its diffusion and lateral transport. Table 2 shows the calculated Peclet number for
each flow rate.
4.2. Effect of Flow Rate
To quantify longitudinal dispersion, concentration profiles at y D 0 (central axis of the
pattern) were fitted to the results of mathematical model using the least squares method.
An objective function is defined as the sum of square of difference between experimental
and corresponding numerical values as:
I D
nX
kD1
.C �
k;exp � C �
k;num/2 (9)
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Throat Orientation and Dispersion of Solvent 657
Figure 7. Typical location of C�
k;expand C�
k;numat experiments and numerical solution. (This
figure can be viewed in color online)
where C �
k;exp and C �
k;num are experimental and numerically predicted concentrations,
respectively. In the parameter estimation process, one parameter was changed at a time,
and the lowest value of the objective function was determined for that parameter at the
fixed value of the other. This provided insight on how a change in the parameters affected
the fit to the experimental data (Bard, 1974). Figure 7 shows the typical procedure for
determining the C �
k;exp and C �
k;num, and Figure 8 indicates the fitting procedure to find the
longitudinal dispersion coefficient. Figure 9 also shows the concentration profile from
experimental data and concentration profile by numerical solution by some different
dispersion coefficients.
To find the transverse dispersion coefficient, images of 0.15 and 0.25 pore volumes
of solvent injections were selected for curve fitting. Concentration profiles obtained by
numerical calculations were fitted to experimental data at the same values of injected pore
volumes. To reduce the effect of local error, three different cross sections were selected
to minimize the objective value I . In this case, the objective value I was defined as
the summation of three sums of squares for difference between numerical results and
experimental data. Figure 10 schematically shows the fitting procedure, and Figure 11
shows experimental data and predicted numerical results for two different values of
transverse dispersion coefficient.
Figures 12 and 13 show the estimated longitudinal and transverse dispersion coeffi-
cients versus Peclet number. In Figure 12 it is obvious that the longitudinal dispersion
coefficient is increasing with Peclet number. Two trends can be seen in this figure, the
first at P e < 50 and the second at 50 < P e < 400, in which the slope of the second
trend is greater than the first one. When P e < 50 the longitudinal dispersivity, ˛T , was
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Figure 8. Typical fitting procedure to find longitudinal dispersion coefficient. (This figure can be
viewed in color online)
Figure 9. Experimental concentration profile in longitudinal direction and numerical results for
two different longitudinal dispersion coefficients.
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Throat Orientation and Dispersion of Solvent 659
Figure 10. Typical fitting procedure to find transverse dispersion coefficient. (This figure can be
viewed in color online)
Figure 11. Sample experimental and numerical concentration profile in the transverse direction.
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Figure 12. Longitudinal dispersion coefficients versus Peclet number.
found 2.94e-7 cm2/sec and for 50 < P e < 400 it was equal to 1.34e-6 cm2 /sec. Figure 13
shows that the transverse dispersion coefficient increased with the Peclet number. Also,
two trends can be seen for P e < 50 and 50 < P e < 400, in which at the first region
the slope was greater than the second region. For P e < 50, transversal dispersivity, ˛T ,
was equal to 1.03e-6 cm2/sec and for 50 < P e < 400 it was equal to 3.02e-7 cm2 /sec.
Figure 13. Transverse dispersion coefficient versus Peclet number.
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Throat Orientation and Dispersion of Solvent 661
Figure 14. Longitudinal dispersion coefficient versus throat orientation angle.
Generally, different behaviors can be seen in two regions. At P e < 50, the transverse
dispersion coefficient grows faster than the longitude dispersion coefficient with Peclet
number and ˛L=˛T < 1. Conversely, at 50 < P e < 400, the longitude dispersion
coefficient increase with is more rapid than the transverse dispersion coefficient and
˛L=˛T > 1.
4.3. Effects of Throat Orientation
In this part, the effect of angle between throat orientation and mean flow direction in
the micromodel is investigated. The method of analysis and calculation of dispersion
coefficients is as stated in section 4.2. All conditions in the experiments were the same
and a flow rate of 4e-4 mL/min was selected to comprising dispersion in porous model
with different throat orientations.
Figure 14 and 15 show the results of longitudinal and transversal dispersion coeffi-
cients versus throat orientation angles. Figure 14 shows that the longitudinal dispersion
coefficient decreased when the angle between throat orientation and direction of flow
changed from 0 to 60ı. The trend of variations for transverse dispersion coefficient is
shown in Figure 15. When the angle between throats changes from 0 to 60ı the transverse
dispersion coefficient increases. It is obvious from Figures 14 and 15 that the longitudinal-
transversal dispersion ratio decreases when the throat orientation angle increases. For the
range of experiments performed here, it was found that this ratio varied from 3.5 to 1.5
when the throat orientation angle changed from 0 to 60ı.
5. Conclusion
It was demonstrated that the dispersion of hydrocarbon solvent in crude oil can be
visually observed and quantified from micromodel experiments. Using inverse modeling,
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662 A. R. Rezaeipour et al.
Figure 15. Transverse dispersion coefficient versus throat orientation angle.
the longitudinal and transversal dispersion coefficients of hydrocarbon solvent in crude
oil were determined as a function of Peclet number. The role of throat orientation angle
as well as flow field on dispersion of hydrocarbon solvent in crude oil was investigated.
The main outlines of this work are as follows:
� Longitudinal and transversal dispersivity of hydrocarbon solvent in crude oil were
successfully measured for different throat orientation angles.
� At low values of Peclet number the order of magnitude of longitudinal and transversal
dispersion coefficients was the same, whereas at high values they were not.
� Two trends were observed for the variation of longitudinal dispersion versus P e.
When P e < 50 the longitudinal dispersivity was found to be 2.94e-7 cm2/sec and for
50 < P e < 400 it was equal to 1.34e-6 cm2/sec.
� Two trends were observed for the variation of transversal dispersion versus P e. For
P e < 50, transverse dispersivity was equal to 1.03e-6 cm2/sec and for 50 < P e < 400
it was equal to 3.02e-7 cm2/sec.
� The longitudal dispersion decreased when the throat orientation angle increased from
0 to 60ı. In contrast, the transverse dispersion increased when the throat orientation
angle increased.
� For the range of experiments performed here, it was found that the longitudinal–
transversal dispersion ratio varied from 3.5 to 1.5 when the throat orientation angle
changed from 0 to 60ı.
� The results of this work can be helpful for designing a successful miscible enhanced
oil recovery processes.
Acknowledgment
The authors are grateful for the technical support of the Tehran Petroleum Research
Center.
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Throat Orientation and Dispersion of Solvent 663
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