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:rahim.rezaeipour@gmail.com

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650 A. R. Rezaeipour et al.

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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658 A. R. Rezaeipour et al.

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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