Numerical Approach to Mobilization of an Oil Slug in a Capillary Tube Subjected to Various External Excitations
C a: Capillary Number;
Re : Reynolds Number;
A : Amplitude of the External Excitation, mm;
f : Frequency o f External Excitation, Hz
The topic of enhanced oil recovery (EOR) by external vibratory excitation is of a great interest and considerable research has been done on the subject in the past years.Various researchers have completed experimental, numerical, and theoretical work on an oil drop ganglion inside capillary tubes. Vigil  showed in his research by using numerical simulations that vibrations can help entrapped oil ganglions move through a constriction. With the help of experiment, Li  showed a relationshipbetween mobilization of oil and the amplitude and the frequency of the waves. Beresnev  has completed theoretical analysis and shown that oil ganglions stuck in the pore throat and requiring a small push, maybe mobilized by an external vibration. Graham [4,5] has shown that non sinusoidal waves can achieve greater ganglion mobilization in narrow constricted tubes. Iassonov  found from the theoretical model that vibrations can significantly decrease the value of the minimum pressure gradient required to mobilize an entrapped oil ganglion.
Averbakh  studied the motion of a liquid droplet inside a capillary tube under the influence of a static force and an acoustic field. The numerical results state that vibrations attenuate the influence of the capillary force and the droplet motion in a capillary of constant cross-section starts at a lower value of the static force. Pride  have done numerical simulations using the lattice-boltzmann method in two dimensions and found that seismic stimulation will mobilize the trapped oil, when two important dimensionless criteria are met.
Oil recovery studies via vibrations on core samples have also been investigated by various authors. Kouznetsov  found that in the presence of vibro-seismic sources, the rate of oil displaced by water is increased, and the amount of oil remaining decreases. Nikolaevskii  has attributed the positive effect of vibration upon the restoration of permeability for the dispersed oil phase and shown the limitations of surface-based vibrators employed. Through experiments, Naderi  has shown that ultrasound radiation leads to higher oil recovery.
The use of seismic vibration to recover oil has been proposed as a low-cost method, with reports of several successful field results . Through Field experiments, Nikolaevskiy  has found that vibrations have a positive impact upon oil recovery and oil production is increased near the residual oil saturation. Westermark  has found that downhole stimulation is limited in its effectiveness in oil recovery and surface vibro-seismic stimulation have the most success. An excellent literature review regarding the latest research in the field of oil recovery by seismic vibrations is given by Huh .
Although much has been done in the field of EOR by vibratory stimulation, the research in this field remains ongoing. It has been shown by members of our research group through indirect experiments  and through simulations  that formation of water film decreases pressure drop needed to mobilize the oil slug. Past member of our research group has shown that vibratory stimulations have a positive effect on oil slug mobilization . It was also shown that the pressure drop needed to mobilize the oil slug with vibratory stimulation is between pressure drop needed to move the oil slug after it has been in a tube for 24 hours and pressure drop needed to move the oil slug as soon as it is placed in the tube. From this it was hypothesized that the vibrations help form a thin water film around the tube. A visual proof of this hypothesis was not presented due to the lack of high quality expensive visualization equipment.
In this study, the interest is in understanding how external excitations aid in the formation of the water film and how that in turn, leads to the mobilization of the oil slug at a pore scale. Simulations are carried out to mobilize trapped oil slugs in a capillary tube filled with water. To ensure stability of the simulations, Capillary number is fixed at 0.00192 and Reynolds number is fixed at 25.72 for all the simulations. Excitations in form of different types of waves such as square wave, sine wave, and saw tooth wave are to be applied to the tube to see which one has the most significant effect on the mobilization. The amplitudes and the frequency of the waves are varied to study their impacts. Effects of external excitation are measured andevaluated in the following manner: External excitation is applied to the tube containing an oil slug with no water film and without any external water injection. Following the vibration, water injection is then applied and pressure drop vs. time is monitored to see the variation in the pressure drop due to external excitations.
Geometry and Boundary conditions
The modeling and simulations are conducted using a software called Ansys Fluent. Fluent employs a VOF method for the tracking of the fluid-fluid interface. The experimental work being done by our research group members utilizes a straight capillary tube and for that reason, a straight capillary tube is also used in the simulations of the present research. The geometry used in all the simulations is an inner 5mm long tube with an inner radius of 0.25 mm, as shown in Figure 1. The reason for choosing a small length and diameter is because a typical microflow problem is studied in the tubes with a diameter in the micrometer range . Another reason is that a short tube with small diameter requires less meshing and thus lesscomputational power is required. The cylindrical capillary tube is modeled by a two-dimensional axisymmetric option in Fluent. Use of the axisymmetric option enables the employment and meshing of only half of the actual flow domain. This results in a mesh with fewer elements and saves computation time. The tube is assumed to be a rigid body and no deformation is considered in the simulations.
Figure 2. Fine mesh near the wall to capture the thin films accurately.
The advantage of this type of meshing is that the thin films can be captured accurately without increasing the meshingelements in the rest of the domain.
Figure 3. Boundary conditions employed in the simulations.
The boundary conditions are velocity inlet for the inlet, pressure outlet for the outlet, and a no slip boundary condition at the wall as shown in Figure 3. The contact angle for all the simulations is set at 180o. An oil slug is placed inside the capillary tube without a surrounding water film. External excitation such as a sine wave, a square wave, and a saw tooth wave are applied as the moving wall boundary condition by using a user defined function (UDF) in Fluent. No water is injected during the period in which external vibrations are turned on. The vibrations are stopped after the oil drop is vibrated for about three periods and then the water is injected through the tube inlet. It is found that the duration of vibration has no impact on the water film formation and pressure drop profile. Hence, it is safe to stop the vibrations after about three periods or earlier. The oil slug should be vibrated for at least two periods so to make sure proper water film formation. It is not possible to give the real time in seconds for the completion of three periods due to the computational speed being dependent on the machine beingemployed. The pressure difference between the inlet and outlet of the tube, as a function of time, are recorded and compared to the simulations in which no external vibrations are applied. The amplitude and frequency of the external vibrations are varied by changing the actual values in the UDF code.
The meshing used in the simulations is shown in Figure 2. The mesh is very fine near the tube wall and coarser away from the tube wall as for capturing the thin water film near the wall. Two different meshes have been tried and most accurate of the two has been used in the simulations. The verification of the meshing and time step for thin film multiphase flow is done using core-annular flow. Simulations at two different capillary numbers are run with Mesh 1 and Mesh 2 and the results are compared with experimental results of Table III in  as shown in Table 1. See the author’s thesis  for detailed description of the meshing used and the verification of the meshing.
Table 1. Grid test for two different meshes to determine the most accurate mesh, where S is simulation.
The properties of the liquids employed in the simulations are given in Table 2.
Table 2. Properties of the two immiscible liquids used in the simulations.
Establishing upper and lower bound pressure drop limits
It is expected that vibrations will help form a thin water film around the oil slug which will help lower the pressure drop needed to push the oil slug out of the tube . In order to quantify the effect of vibrations, a pressure drop profile obtained from a simulation with external excitation needs to be compared with the pressure drop profile obtained from a simulation without external excitations. All the parameters between the two simulations should be same except for the presence and absence of the external excitations. All simulations with external excitations in this paper are conducted with Ca = 0.00192 and Re = 25.72
to ensure a multiphase flow that is dominated by surface tension effects rather than viscous or inertial effects. Figure 4 shows the pressure profile for Ca = 0.00192 and Re = 25.72 in the absence of external excitation. The maximum pressure drop is 538 Pa (A) and the steady state pressure drop is 110 Pa (B). The upper bound for all simulations with vibrations should be smaller or equal to the upper bound (A) in Figure 4 due to the presence of the water film. The steady state pressure drop (B) for all vibration simulations will be same as the lower bound in Figure 4 because the steady state pressure drop is unaffected by vibrations. The pressure drop profile obtained from all simulations in the absence of external excitations show a similar trend as seen in Figure 5. The figure shows pressure drop vs. time for Ca = 0.00192, Re = 6 .67. Comparing Figure 4 and Figure 5, it can be seen that the trend is identical. Figure 6 – Figure 10 show snapshots of the oilslug for Ca = 0.00192, Re = 69.67 to see how the change in oil slug shape affects the pressure drop in Figure 5. The oil slug is initially placed in the capillary tube without any
water film between the tube wall and the oil slug. Water injection is initiated to mobilize the oil slug. As seen in Figure 5 and Figure 6 – Figure 10, as the water is continuously injected, the pressure drop reaches a maximum value just before the water film starts forming. The pressure drop decreases linearly during the water film formation and the rear meniscus of the oil slug deforms. Then a big drop in the pressure drop is observed as the water film formation process is completed and the rear meniscus of the oil slug gradually returns back to its original shape.
Figure 4. Presure Drop Vs. Time graph for C a = 0.00192 and Re = 25.72 with no inital water film. (A) represents themaximum pressure drop and (B) represents the steady state pressure drop.
Once the entire oil slug is covered by the water film, the pressure drop becomes a constant. The enveloping of the entire oil slug by the water film and constant pressure dropis defined as the completion of the mobilization of the oil slug.
Effect of external excitations on water film development
In this section, the qualitative effect of external excitation on an oil slug will be shown via pictures obtained from postprocessing of the simulation data. Figure 11 – Figure 15 show an oil slug under the effect of square wave externalexcitation, at different times. It can be observed that the vibrations cause the oil to move back and forth.
Figure 6. Shape of the oil slug at t = 0.00067184s (original in color).
Figure 7. Shape of the oil slug at t = 0.0106018s (original in color).
Figure 8. Shape of the oil slug at t = 0.0110018s (original in color).
Figure 9. Shape of the oil slug at t = 0.0140018s (original in color).
Figure 10. Shape of the oil slug at t = 0.0150018s (original in color).
Figure 11. Oil slug with external square wave excitation, ampitude = 0.04026 mm, frequency = 20 Hz, t =0.109908s(original in color).
in a simple harmonic motion in the absence of any water injection. The back and forth motion of the oil slug leads to the constant formation and drainage of the water film on the opposite end of the oil slug. The water film initially takes form at one end of the oil slug, reaches its maximum length, and then begins to drain from that end and starts forming on the opposite end. Formation and drainage of water film on both sides of the oil slug is a completion of one cycle of vibration. All vibrations of different types have the same type of effect on the oil slug and the only difference would be in the length of the water film established as will be shown later in Table 3.
The three different types of vibrations employed in this section are: the square wave, the sine wave, and the saw tooth wave. It is important to study which wave, all things being equal, has the highest impact in terms of water film development. The effect can be qualitatively studied by looking at the figures of water film development around the oil slug. Figure 16 – Figure 18 show water film development around the oil slug for the square wave, sine wave, and saw tooth wave at an amplitude of 0.04026 mm and a frequency of 20 Hz at the end of a one and half cycle of vibration. From the visual inspection of the pictures and from Table 3, it can be noted that at any given frequency, the water film around the oil slug vibrated with the square wave being the greatest, followed by the sine wave and finally the sawtooth wave.
The rear meniscus of the oil slug for the square wave experiences the maximum deformation followed by the sine wave and then the sawtooth wave. It can also be observed that as frequency is increased, the water film surrounding the oil slug decreases for all types of waves. As frequency increases, the deformation of the rear meniscus of the oil slug also decreases.
Table 3. Approximate maximum water film length surrounding the oil slug for different types of waves at a fixed amplitude and varying frequencies.
Figure 12. Oil slug with external square wave excitation, ampitude = 0.04026 mm, frequency = 20 Hz, t =0.115733s (original in color).
Figure 13. Oil slug with external square wave excitation, ampitude = 0.04026 mm, frequency = 20 Hz, t = 0.125378s(original in color).
Figure 14. Oil slug with external square wave excitation, ampitude = 0.04026 mm, frequency = 20 Hz, t =0.136077s (original in color).
Figure 16. Oil slug with external sawtooth wave excitation, ampitude = 0.04026 mm, frequency = 20 Hz (original in color).
Effect of external excitations on a pressure drop
In the previous section, it was noted that applying external excitation to an oil slug, with no initial water film, helps form a water film around the oil slug. It was also observed that different amplitudes and frequencies lead to different amounts of water film surrounding the oil slug. In this section, the effect of vibration upon the pressure drop due to water film formation is analyzed. Specifically, the impact of amplitude at constant frequency and the impact of frequency at constant amplitude for a square wave will be analyzed. Table 3 and Table 4 depict the numerical results.
Figure 17. Oil slug with external sine wave excitation, ampitude = 0.04026 mm, frequency = 20 Hz (original in color).
Table 4. Approximate maximum water film len gth surrounding the oil slug for square wave at a varying amplitude and fixedfrequency.
Figure 18. Oil slug with external square wave excitation, ampitude = 0.04026 mm, frequency = 20 Hz (original in color).
Figure 19 shows Pressure drop vs. Amplitude for fixed frequency of a square wave based on Table 4. The graph depicts both maximum pressure drop and steady state pressure drop obtained from the simulations. It can be observed that as the amplitude is increased, the maximum pressure drop needed to mobilize the oil slug decreases. The steady state pressure drop stays constant and its numerical value is equal to 110 Pa, as obtained in Figure 4, without vibration. Figure 20 shows Pressure drop vs. Frequency for the fixed amplitude of a square wave based on Table 3. As noted, as frequency is increased, the maximum pressure drop increases while the steady state pressure drop remains constant for all frequencies. According to the graph, the maximum pressure drop at a frequency of 10 Hz, is almost equal to the steady state pressure drop at a frequency of 10 Hz. The reason for this is that by applying a frequency of 10 Hz leads to the entire oil slug being covered by the water film and this would lead to the same pressure drop value as the pressure drop at steady state. From the Table 3, it can be observed that any type of vibration at any frequency and amplitude leads to a lower maximum pressure drop than the case without any external excitation.
Effect of different types of waves on pressure drop
The previous section analyzed the effect of different amplitudes and frequencies of a square wave upon a maximum pressurse drop. In this section, different types of waves are compared with square waves at the same frequency and amplitude to see how the maximum pressure drop changes with different types of vibrations. The three waves used in this section are the square wave, sawtooth wave, and sine wave. Figure 21, shows the maximum pressure drop and steady state pressure drop for different vibrations at a fixed amplitude and two different frequencies based on Table 3. As noted for all vibrations, the maximum pressure increases with an increase in frequency as viewed for the square waves in section 5. Also, the square wave leads to the highest reduction in maximum pressure drop, followed by the sine wave and the sawtooth wave, respectively. This
is due to the fact that square waves form the longest water film around an oil slug, followed by sine waves and sawtooth waves, respectively, as observed in section 4. These results can also be observed in Table 3, which also shows that the higher the water film length, the lower the maximum pressure drop and the smaller the water film length, the higher the maximum pressure drop required to mobilize the oil slug.
It is demonstrated in the numerical simulation results that under the influence of external vibratory stimulation, a thin water film is formed around the oil slug. It is this thin water film that reduces the maximum pressure drop needed to move the oil slug in the subsequent water injection step.
Figure 19. Pressure Drop Vs. Square Wave Amplitude, at a frequency of 20 Hz (original in color).
Figure 20. Pressure Drop Vs. Square Wave Frequency, at an amplitude of 0.04026 mm (original in color).
Square Wave, A = 0.04026 mm Sawtooth wave, A = 0.04026 mm Sine Wave, A = 0.04026 mm
Different types of waves, along with frequencies and amplitudes are found to have a definitive affect upon the formation of a water film, as well as the maximum pressure drop required to move the oil slug.
Figure 21. Pressure Drop Vs. Wave Frequency, for Square wave, Sine wave, and Sawtooth wave. The amplitude for all the waves is fixed at 0.04026 mm (original in color).
It is known that in case a highly viscous liquid surrounded by a less viscous liquid, the pressure drop and thus the energy of pumping the heavy viscosity liquid can be reduced . This same phenomenon brought upon by vibrations is thought to help an entrapped oil slug in a capillary tube. Due to the lack of observational equipment, this has not been directly observed in the laboratory and very limited simulations have been done due to computationally expensive nature of multiphase flow simulations. Three different types of waves: square wave, sine wave, and sawtooth wave were utilized in this research as external excitations. It is found that any type of external excitation helps form a thin water film around the oil slug. At a fixed amplitude and frequency, the length of this water film is greatest for square wave, followed by sine wave and sawtooth wave respectively. For all three different waves, it is noticed that increase in amplitude at a fixed frequency will result in longer water film and decrease in amplitude will result in shorter water film. A similar statement can be made for frequency at a fixed amplitude in that higher frequency will reduce the length of the water film and lower frequency will increase the length of the water film. The effect of the increase/decreases in water film length has a direct correlation with the maximum pressure dropas observed from the Pressure drop Vs. Time graph. It is found that increasing the amplitude will lower the maximum pressure drop and increasing the frequency increase the maximum pressure drop. At fixed amplitude and frequency, square wave excitation will lead to lowest maximum pressure drop value followed by sine wave and sawtooth wave respectively. As can be seen that higher amplitude and lower frequency are beneficial in generating a longer water film and reducing the maximum pressure needed to mobilize the oil slug. It is also found that square wave outperforms sine wave and sawtooth wave in terms on water
film development which in turn leads to lower maximum pressure drop.
The main message of this study is that vibrations have a positive impact on the oil slug mobilization. Usually, authors of different studies conclude that vibrations are capable of acting as the extra external force needed to push the oil slug out of the constriction of the tube (throat). The results presented here have shown that in the absence of any throat, vibrations still have a positive effect on the oil slug mobilization by a formation of thin water film.
The authors would like to thank with great appreciation the National Sciences and Engineering Research Council of Canada(NSERC) for its support to this research. The authors are also grateful to International Performance Assessment Centre for Geologic Storage of Carbon Dioxide (IPAC–CO2) for allowingus to use that high performance cluster for the computations. We would also like to thank Patrick Mann of IPAC–CO2 for his help with the setup of Fluent on the cluster and for his continual help with the remote operation of Fluent.
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