Effects of transient injection-suction on interfacial dynamics in immiscible Newtonian fluid-fluid radial displacement in Hele-Shaw cell

doi:10.21203/rs.3.rs-1730992/v1

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Effects of transient injection-suction on interfacial dynamics in immiscible Newtonian fluid-fluid radial displacement in Hele-Shaw cell

https://doi.org/10.21203/rs.3.rs-1730992/v1

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In this experimental study, the instabilities created at the contact surface of two Newtonian immiscible fluids i.e., water and base oil during transient radial injection-suction inside Hele-Shaw cell were investigated. Base oil with viscosities of 150, 920 and 1506 cSt was used for injection and suction purposes. The maximum injection and suction fluid flow rates that were pseudo-sinusoidal and percussive time-dependent were considered to be 10000, 20000 and 30000 µl/min. The amount of fluid injected or sucked in all three methods was considered equal. The results were compared with the results of constant velocity injection. The effects of injection and suction frequency on contact surface instability were also studied. Injection and suction were performed for at least six complete cycles. One of the most important objectives of this study was to investigate the ability to eliminate instabilities and return the contact surface of two fluids to the initial circular state at the end of each cycle. It was observed in the methods of percussive and constant-velocity that it is not possible to return to the initial circle after the first step. It was also observed that the growth rate of instabilities in different cycles of the pseudo-sinusoidal method is slower than the percussive and constant-velocity methods. The lower the viscosity and flow rate of the injected or suction fluid, the less instability occurs. The results showed that with increasing the flow rate, the number of oil packets in the injection-suction of constant-velocity method has increased, but in the reverse percussive method, this is the case. The results showed that in the injection-suction by pseudo-sinusoidal method, which do not cause instabilities such as water droplets, fingers and oil packets, but in two methods of injection-suction at a constant-velocity and percussive method, such instabilities are observed. It can be concluded that considering the conditions of the experiments performed, the pseudo-sinusoidal method is the most suitable method with the ability to return to the initial steady state among the three methods used. Of course, it should be noted that the greater the number of repetitions of cycles, the more instabilities created and the more difficult it is to return to the initial state.

Fluid-Fluid immiscible displacement

Hele-Shaw cell

Unstable radial injection

Pseudo-sinusoidal and percussive injection and suction

Instability of surfaces between two fluids

When one fluid moves another fluid, the displacement can be stable or unstable. If the displacement is unstable, fingers of the displaced fluid can penetrate the displaced fluid. For example in horizontal flow, a finger occurs when a fluid with a lower viscosity moves a fluid with a higher viscosity. One of the most common devices for evaluating fluid displacement in a homogeneous porous medium is the Hell-Shaw cell. In this context, different types of Hele-Shaw cells are used for many experiments. The reason for using this device instead of the actual porous media is that the flow characteristics in both systems are very similar, while the use of a homogeneous Hell-Shaw cell facilitates experiments as well as modeling. Figures (1) shows a three-dimensional schematic of the Hele-Shaw cell where r_b shows the base radius or minimum radius as well as r the maximum radius of the fingers.

The first experimental study to understand the instability at the contact surface of two fluids was conducted by Hill [1], and Slobod and Caudlel [2] in 1952. But the first analysis of the motion physics of the contact surface of two fluids was presented by Saffman and Taylor [3] in 1958. For fluid-fluid displacement within the Hele-Shaw cell, characteristics such as surface stress behavior [4, 5], non-Newtonian fluid morphology [6, 7], surface roughness [4], media anisotropy [8, 9], ratio Viscosity [10], spacing between plates [11], distribution of densities and viscosities [12] and gravity [13] have been investigated. Extensive studies on fingers instability have been conducted by Homsy [14], Yortsos [15], and McCloud & Maher [16]. Bataille [17] in 1968 and later Paterson [18] in 1981 examined the controlling factors associated with increased instability in the early stages of non-miscible flow displacement. The effects of surface stress were investigated by Cardoso and Woods [19] in 1995 and it was shown that the injection radius directly affects the stability of the contact surface of two fluids. Hornof and Baig [20] studied the effect of oil viscosity in 1995 when water was injected from the corner of the Hale-Shaw cell and showed that the oil recovery efficiency increases when the mobility ratio decreases. They also found that adding reactants to oil and water increased oil recovery. Although most practical processes are time dependent, most research has been done on fluid-fluid displacement at constant injection rates. The effects of oscillation on injected air pressure at different wavelengths were tested by Pisarenko and Gland [21] in 2003. They used air to move the oil in the Hell-Shaw cell and concluded that pressure fluctuations reduced finger development. In another study, Brailovsky [22], by numerically simulating radial displacement in the Hele-Shaw cell, showed that a long pause in the injection process causes stability and contraction of the finger structures. Dias, Miranda and Pariso [23] used linear stability analysis in 2010 to test unstable fluid displacements in the Hele-Shaw cell, and proposed the patch injection method as a way to reduce instability. Dias, Miranda and Pariso [24] showed in 2010 that if the injection rate changed by ${t}^{-\frac{1}{3}}$, it could significantly reduce finger bifurcation during nonlinear injection. Subsequent studies by Reis and Miranda [25] in 2011 used linear stability analysis for time-dependent injection to control the unstable finger phenomenon during radial injection into the Hele-Shaw cell. Later, Dias [26] in 2012 showed that in non-miscible radial displacement with constant injection rates method, there is the ability to reduce more instability than other injection methods. The linear injection presented by Dias [27] in 2013 confirmed that constant injection rates reduces instability in non-miscible displacements, but the same method of injection in miscible displacements leads to increased instability. In the model proposed by Dias and et al., It is possible to reach the desired morphology in the radial injection method in the Hele-Shaw cell by experimental control methods and by selecting a specific fixed injection rate. Studies on miscible fluid-fluid displacement during fluid injection as a nonlinear function of time were conducted by Yuan and Azaiez [28, 29] in 2014 and 2015. Also, a study on the displacement of the reaction flow in the periodic injection by Yuan and Azaiez [28] was conducted in 2014 and showed that stable constant injection can be made unstable by converting it to time-dependent injection. Finally, unstable flow complicates the fingers and increases the reactivity due to mixing of the two solutions. The effects of inertia on time-dependent radial injection were investigated by Yuan and Azaiez [29] in 2015, and it was found that the degree of instability may vary according to the period and range of the injection cycle. Chen [30] in 2015 proposed a method to increase the mixing rate of radial injection between two miscible fluids with different viscosities. In 2015, Ahmadlouydarab and et al., [31] studied on immiscible radial displacement in the presence of phase change and concluded that in small values of the capillary number, the behavior of condensed and uncondensed fluid is similar in the formation of instabilities, but in higher values of capillary number are different. Research on time-dependent injections for immiscible fluids by Tiago, Lins and Azaiez [32] was conducted in 2016. The effect of various parameters such as phase change, amplitude, period as well as the role of mobility and surface tension ratio were discussed and a set of injection methods was observed that reduced the instabilities. They proposed a new method called controlled injection. In this method, the flow is constantly adjusted to the growth rate of the instabilities, which reduces the growth of the fingers. In 2017, Azaiez and Lins [33] studied the pseudo-resonance motions of the radial injection flow of immiscible fluids in a homogeneous porous media and obtained the critical value of the fluid injection period. Also, the dynamic control simulations of the states of pockets and droplets in immiscible displacements in porous media were performed by Lins and Azaiez [34] in 2018, and the results of this study showed that time-dependent cyclic plans can be used to control the formation of droplets / pockets in The flow, and in particular the delay of their appearance, through the proper combination of amplitude and period of displacement plan. In one study by Surguchev [35], the effect of water flow cycle injection on the simulation of a heterogeneous reservoir was investigated and it was observed that the efficiency in the constant injection process was improved. The development of finger instabilities at the contact surfaces between water and oil was also studied by Brailovsky [22] using the Buckley-Leverett model for radial displacement. Numerical simulations of radial flow in Hele-Shaw with various ways showed that a long pause in the injection process causes the stability and contraction of finger structures that occur due to surface tension. Li and et al., [36] studied finger control by adjusting the injection rate and concluded that in this case, it is very difficult to predict and control finger development patterns in situations where the system is non-local and non-linear. The results showed that by controlling the fluid injection rate, growing instabilities can be reduced. Huang and Chen [37] performed experiments on immiscible, slightly miscible, and miscible radial currents in the Hale-Shaw cell in which the smallest instability between contact surfaces was described linearly based on time-dependent injection and for non-miscible and slightly miscible currents showed that the linear injection method can significantly reduce finger instability and the length of contact surfaces between two fluids compared to the fixed injection method. The most common method of quantitative stability assessment is to follow the length of the contact surfaces. One way to show more instability is to have a longer contact length (finger-shaped environment). According to Huang and Chen opinions [37], this parameter is useful for situations where the level of contact between fluids, for example where there is a reaction. They studied the intermittent injection method and observed that different layers of fluid formed and the contact surfaces between the two fluids increased, and for situations where instability was controlled, the length of the fingers or the position of the fingertips and fingers base could be studied. A study by Leshchiner and et al., on the growth process of air bubbles inside a layer of oil in a Hele-Shaw cell resulted in an equation for harmonic moments. Due to the large number of data involved in this phenomenon, so numerical simulation is very difficult and in most cases can not even describe the structure of the fingers well [38]. This research shows how time-dependent injection can affect the formation and growth of instabilities. Although the effect of injection rate has been reported in many studies, limited studies have been performed to quantitatively describe the fingers. Generally, in the study of sources, it can be concluded that the contact surface behavior of two fluids, only injection or only unstable suction is investigated and the effect of injection with suction and its effect on the possibility of reversibility of two fluids contact surface stability both theoretically and the experimental form has not been studied. For this purpose, and in order to increase the existing knowledge in this field, in the current experimental study we will pursue the following main goals during injection and suction at constant velocity, pseudo-sinusoidal and percussive at different flow rates and viscosities:

Investigation of the dynamic behavior of two immiscible fluids contact surfaces in time-dependent radial injection and suction.

Investigation of the effect of injection and suction cycle repetition on the morphology and stability of the contact surface of two fluids.

Investigating the possibility of reversibility of the contact surface of two fluids from unstable conditions to the initial stable conditions during injection and suction.

2.1. Experimental Setup

In this research, an automatic syringe pump (dispenser), Hele-Shaw cell, video camera, three types of oil, water and ink have been used. The operation of the equipment is shown schematically in Figure (2). The setup used in this experimental study can also be seen in Figure (3). The Hele-Shaw cell consists of two perfectly transparent Plexiglas plates spaced 0.56 cm apart. A hole is made in the center of the top plate to insert the tip of the syringe. Before starting the injection, the two plates are filled with one of the oils introduced in Table (1). Injectable fluid is water to which a droplet of ink (Artline) has been added. Experiments have shown that due to the very small amount, the ink has no effect on the viscosity, density and surface tension of water. Table (2) shows the surface tension information obtained for different fluids.

Table 1

Physical characteristics of oils used in research.
Properties Number of oil	density (kg/m³)	ASTM Viscosity (25 ° C) (cSt)	ASTM Viscosity (40 ° C) (cSt)	ASTM Viscosity (100 ° C) (cSt)
1	894.5	150.776	70.52	10
2	869.2	920.322	421.038	50
3	866.2	1506.27	720.168	77

Table 2

Surface tension obtained between fluids used in this research.
Fluids	Surface tension σ(N/m)
Injection fluid and air	0.0665
Oil No. 1 with air	0.03089
Oil No. 2 with air	0.03328
Oil No. 3 with air	0.0347
Injection fluid and oil No. 1	0.03561
Injection fluid and oil No. 2	0.03322
Injection fluid and oil No. 3	0.0318

2.2. Injection methods

For the experiments, injection and suction methods were considered in three forms: pseudo-sinusoidal, percussive and constant velocity. Table (3) shows the viscosity characteristics and the amount of flow rates selected for each injection method. More complete explanations of the methods will be provided in the results section. It should be noted that all experiments were performed in vitro conditions at a temperature of 25 ° C and a pressure of 1 atmosphere. To measure the area and circumference of the fingers created in the Hele-Shaw cell, first the images of the desired moments were recorded from the recorded videos of each test and then the area and circumference of the fingers were obtained using Digimizer software.

Table 3

Injection and suction methods with different flow rates and viscosities
Injection and suction method	Viscosity (cSt)	Flow Rate (µl/min)
Pseudo-Sinusoidal	10	20000
	50	10000,20000,30000
	77	20000
percussive	10	20000
	50	10000,20000,30000
	77	20000
Constant Velocity	50	2666
	50	5333
	50	8000

2.3. Mathematical Equations

Assuming that both displacement and displaced fluids are incompressible, the dimensionless equations of motion and continuity can be written in the form of Equations (1) and (2):

$$Re\left[\frac{\partial {u}^{*}}{\partial {t}^{*}}+{\rho }{u}^{*}.\nabla {u}^{*}\right]=\frac{Ca}{{\alpha }^{2}}(-\nabla {p}^{*}+{\mu }^{*}{\nabla }^{2}{u}^{*})$$

$$\nabla {u}^{*}=0$$

The dimensionless injection rate equation can also be written as follows:

$${\text{Q}}^{\text{*}}\left(\text{t}\right)=D\text{sin}\left(\frac{2{\pi }{\text{t}}^{\text{*}}}{{\text{T}}^{\text{*}}}\right)$$

The dimensionless Reynolds, Capillary, and Womersley numbers are also defined by the following equations:

$$Re= \frac{VH}{v}=\frac{\rho Q\left(t\right) H}{\mu A\left(t\right)}$$

$$Ca=\frac{{R}^{2}Q\left(t\right)}{\sigma \beta A\left(t\right)}$$

$${\alpha }^{2}=\frac{{2\pi H}^{2}{\rho }_{2}\sigma Q\left(t\right)}{T{{\mu }_{2}}^{2}{u}_{ave}A\left(t\right)}$$

In the above equations H, D, A(t), Q(t), µ, ρ, R, σ and T, respectively, are the distance between the two Hele-Shaw plates, the instantaneous injection range, the instantaneous surface of the injected fluid, the instantaneous flow rate, viscosity, fluid density, surface tension and periodicity of each injection cycle. Also mentioned in the equations:${Q}^{*}=Q\sigma /\left({\mu }_{2}H{\pi }{R}_{0}^{2}\right),$ ${{ \sigma }^{*}=\beta }_{2}{\sigma }^{2}/\left({\mu }_{2}H{R}_{0}^{3}\right),$ ${ p}^{*}=p{{\beta }}_{2}\sigma /\left({\mu }_{2}H{R}_{0}^{2}\right),$ ${ t}^{*}=t\frac{{\mu }_{2}H}{\sigma }$, ${ U}^{*}=\frac{U\sigma }{{\mu }_{2}H{R}_{0}},$ r^*=r/R₀ and R^*=R/R₀ .

Subtitle (1) refers to injectable fluid and subtitle (2) refers to displaced fluid. R₀ is the initial radius of the injected fluid. On the other hand, in the Hele-Shaw cell, mobility is defined as ${\beta }=\frac{{H}^{2}}{12{\mu }}$. In addition, the radial position of the contact surfaces of the two fluids is defined by the function ℛ (θ, t), according to the angle θ and time t, as a result of which the number of fingers is calculated accordingly.

3.1. Pseudo-sinusoidal method

A complete injection and suction cycle with a maximum injection flow rate of 20000 µl/min as shown in Figure (4). The total duration of a complete cycle with the mentioned flow rate is 180 seconds and is divided into 16 stages (sections) of injection and suction. Each cycle has 8 injection stages and 8 suction stages. Each injection and suction step has an injection or suction volume of 1000 µl. A cycle begins with step 1 injection. Step 1 involves injecting a fluid with a flow rate of 2500 µl/min for 24 seconds (total volume 1000 µl). Step 2 with injection rate of 5000 µl/min and injection time of 12 seconds. Step 3 is 10000 µl/min with an injection time of 6 seconds. Finally, in stage 4, the fluid is injected with a flow rate of 20000 µl/min for 3 seconds. From stage 5 to stage 8, the flow rates from 20000 µl/min to 2500 µl/min continue in descending order according to the process shown in Figure (4) of the injection diagram. At the end of stage 8, the cycle entered the suction phase and the eight stages of suction were performed. Suction steps with negative flow rates are shown in Figure (4).

For injection and suction with flow rates of 30000 µl/min and 10000 µl/min, the number of injection and suction steps is the same as flow rate of 20000 µl/min in 16 stages. Also, the injection volume of each step is 1000 µl. The difference between different flow rates is in the duration of the stages and the cycle period, which depends on the flow rate. The general trend in all flow rates of the Pseudo-sinusoidal method is as shown in Figure (4). The periodicity of a complete injection and suction cycle for the maximum different flow rates is shown in Table (4). In each experiment, 6 complete 16-step cycles are performed and the results of different cycles are compared.

Table 4

Periodicity of a complete cycle for maximum different flow rates in Pseudo-sinusoidal injection-suction method
Maximum flow rates (µl/min)	10000	20000	30000
The periodicity or time of a complete cycle (s)	360	180	120

In order to investigate and analyze the morphological changes of the two fluid contact surfaces in the Pseudo-sinusoidal injection-suction method, as a basic model, the results related to the maximum injection flow rate of 20000 µl/min are examined. Before starting the injection of the first stage of the first cycle, a circle with a radius of 1 cm was formed. It should be noted that the shape of the initial circle affects how the fingers are formed in the later stages of the injection and suction operations, so the initial circle must be created with great precision.

In stage 1, by starting the injection within 24 seconds, the length of the largest finger increases from 0.000 to 0.523 cm. The growth rate of finger length in this stage is less than the next stages because the injection speed is slower. As a result, the capillary number is low.

In stage 2, as the flow rate increases, the instability of the contact surfaces between the two fluids increases and the growth of the fingers increases so that the number of fingers has increased from 1 to 2. At this stage, for 12 seconds, the length of the largest finger starts at 0.523 cm and continues to 1.24 cm. The changes in finger length at this stage are greater than at stage 1 because as the flow rate increases, the growth rate of the fingers increases. The morphology of the two-phase contact surface at the moment t = 36s which is the end of stage 2 and the beginning of stage 3 is shown in Figure (5).

In step 3, in 6 seconds the largest finger increases from 1.24 cm to 1.884 cm and the number of fingers increases from 2 to 4. The base radius changes less than the maximum finger radius, so this indicates a greater tendency to increase finger length. And the capillary number is higher than the previous stage due to the increased injection flow.

In stage 4, the injection is done with the maximum flow rate and in 3 seconds, the length of the largest finger starts from 1.884 cm and reaches about 2.529 cm. The fingers grow faster than the rest of the steps, so the number of fingers increases from 4 to 7. Due to the fact that at this stage the amounts of flow rates are maximum and the effective radius is moderate, the amount of capillary number is more than the previous stages.

Step 5 is a repetition of step 4. The injection time of both stages is equal to the difference that the number of fingers reaches from 7 to 8 and the increase rate of the length of the largest finger is less than the previous stage because most fingers become wider in this stage and because the effective radius increases compared to the previous stage, the number of capillaries decreases.

In stage 6, the fingers expand as the flow rate decreases. In 6 seconds, the length of the largest finger increases from 2.815 cm to 3.554 cm. The number of fingers has increased from 8 to 10 and the fingers are longer than in the 5th stage. Due to the flow rate has been decreased and the effective radius has been increased compared to the previous stage, although this causes to reduce the capillary number, but the capillary number is still high and causes the number of fingers to increase.

In step 7, the length of the largest finger increases from 3.554 cm to 4.069 cm in 12 seconds. Compared to the previous step, the changes in finger length are less and the fingers become wider, but the number of fingers is fixed and equal to 10. With decreasing flow rate and increasing radius, the capillary number decreases compared to the previous stage.

At the end of the 8th stage, the injection phase of the first cycle ends. The duration of the injection in the eighth stage is 24 seconds, which is the longest finger with a length of 4.454 cm. The minimum flow rate and the maximum effective radius value occur at this stage. This stage is the stage that number of fingers remains constant and is contained the minimum value of the capillary number for the first cycle.

The difference between the suction phase and the injection phase is in the order of fluid displacement. In the injection phase, the water moves the oil outwards of Hele-Shaw, but in the suction phase, the water pushes the oil inwards of Hele-Shaw. The suction phase begins in stage 9 with the lowest flow rate and the length of the largest finger decreases but the number of fingers remains constant. All parameters of step 9 are the same as step 8. In the ninth step, it is noteworthy that the base radius is increased but the maximum finger radius is reduced. Within 24 seconds, the length of the largest finger decreases from 4.515 cm to 4.179 cm.

In step 10, the finger lengths are reduced again. In 12 seconds, the length of the largest finger decreases from 4.179 cm to 3.896 cm. The total volume of the injected fluid and the flow rate in step 10 are equal to step 7. As the length of the fingers decreases, the base radius continues to increase, and the capillary number is higher than the previous step.

In step 11, unlike the previous two steps, the base radius decreases with the maximum finger radius. Within 6 seconds, the length of the largest finger has decreased from 3.896 cm to 3.482 cm.

In step 12, the maximum suction rate is established and within 3 seconds the length of the largest finger decreases from 3.482 cm to 2.949 cm. The changes in the value of the base radius are so small that they can be assumed to be constant.

In step 13, step 12 is repeated and the base radius is almost constant as in the previous step. Within 3 seconds, the length of the largest finger decreases from 2.949 cm to 2.226 cm. Also, the base radius and the maximum finger radius are reduced.

In stage 14, the injection flow rate decreases and in 6 seconds the length of the largest finger reduces with a sharp droplet from 2.226 to 1.317 cm. At this stage the fingers became very small and their number is reduced from 6 to 2.

In stage 15, the intensity of the decrease in finger length increases due to the sharp decrease in the amount of water inside the cell, and the length of the largest finger decreases from 1.317 to 0.322 cm. Also at this stage, the changes in the base radius are much greater and the number of fingers decreases from 2 to 0.

By entering stage 16, the size of the base radius reaches the radius of the initial circle at the beginning of stage 1, with the difference that the instabilities created in stages 1 to 16 of the first injection and suction cycle cause irregularities in the periphery of the circle.

Figure (5) shows snapshots of how the two-fluid contact surface develops in the 16 steps mentioned in the first cycle. In table (5) the start and end time of each step, number of fingers, minimum and maximum base radius size in each step, maximum instantaneous radius size, number of injected fluid droplets, and number of oil bags created in each step of the first cycle, has been shown.

Table 5

Results from the steps of the first cycle for some important quantities
step	Start and end time of steps	Fingers		Number of droplets	Number of pockets	Minimum and maximum largest finger	minimum and maximum r (cm)	minimum and maximum r_b (cm)
step	Start and end time of steps	Number	Branched	Number of droplets	Number of pockets	Minimum and maximum largest finger	minimum and maximum r (cm)	minimum and maximum r_b (cm)
1	0–24	0–1	0	0	0	0-0.523	1.078–2.999	1.078–2.476
2	24–36	1–2	0	0	0	0.523–1.240	2.999–4.13	2.476–2.89
3	36–42	2–4	0	0	0	1.240–1.884	4.13–4.851	2.89–2.967
4	42–45	4–7	0	0	0	1.884–2.529	4.851–5.621	2.967–3.092
5	45–48	7–8	0	0	0	2.529–2.815	5.621–5.956	3.092–3.141
6	48–54	8–10	0	0	0	2.815–3.554	5.956–6.783	3.141–3.229
7	54–66	10–10	0	0	0	3.554–4.069	6.783–7.343	3.229–3.274
8	66–90	10–10	0	0	0	4.069–4.515	7.343–7.882	3.274–3.367
9	90–114	10–10	0	0	0	4.515–4.179	7.882–7.591	3.367–3.412
10	114–126	10–10	0	0	0	4.179–3.896	7.591–7.332	3.412–3.478
11	126–132	10 − 9	0	0	0	3.896–3.482	7.332–6.920	3.478–3.438
12	132–135	9 − 8	0	0	0	3.482–2.949	6.920–6.385	3.438–3.436
13	135–138	8 − 6	0	0	0	2.949–2.226	6.385-5.600	3.436–3.334
14	138–144	6 − 2	0	0	0	2.226–1.317	5.600-4.599	3.334–3.282
15	144–156	2 − 0	0	0	0	1.317–0.322	4.599–3.182	3.282–2.860
16	156–180	0–0	0	0	0	0.322-0	3.182–1.187	2.860–1.187

With the beginning of the second cycle and the repetition of the 16 steps mentioned, from step 1 to 16, the length of the fingers increases compared to the first cycle. It can be said that in the injection and suction process of this cycle, more instabilities are observed in all stages than in the previous cycle. Among these instabilities is the creation of more branches in the fingers, which starts from stage 6 and is observed up to stage 13. Figures (6) shows a snapshot of how the contact surface of the two fluids developed during the second cycle using the pseudo-sinusoidal method. By comparing the results in Figure (6) with the results presented in Figure (5), we can see an increase in the amount of instabilities.

In the fifth cycle, more instabilities are expected than in the previous cycles. From the third stage in the fifth cycle, the number of fingers reaches the maximum possible number during the cycle. The length of the largest finger has reached 5.709 cm in step 8, indicating that the length of the fingers has also become longer than in previous cycles. In steps 9, 10 and 14, the base radius increases during suction. It should be noted that the maximum number of branches in this cycle has reached 4. Figures (7) shows a snapshot of how the contact surface of two fluids develops during the fifth cycle using the pseudo-sinusoidal method. In the fifth cycle, the fingers became wider and more symmetrical. It is also noteworthy that the finger base radius increases in steps 9 to 11 and 13 despite the suction.

To compare the results of the tricycles, Table (6) compares the number of fingers, the number of branched fingers, and the length of the largest finger in the first, second, and fifth cycles. As can be seen, with increasing number of cycles, the length of the largest fingers has increased in all stages. In stage 6 of the first cycle, in stage 4 of the second cycle and in stage 2 of the fifth cycle, the number of fingers reaches a maximum. It can be concluded that as the number of cycles increases, the number of fingers reaches their maximum and minimum number in the cycle faster. In the first cycle, none of the fingers will branch. In the second cycle, the number of branched fingers reaches a maximum of 3, and in cycle 5, this number reaches 4. As a result, as the number of cycles increases, both the number of branches in the fingers increases and at the same time the formation and disappearance of branches becomes faster.

Table 6

Number of fingers, number of branched fingers, and length of the largest finger in the first, second, and fifth cycles.
step	Start and end time of steps	Number of fingers			Number of branched fingers			The length of the largest finger
step	Start and end time of steps	first cycle	second cycle	fifth cycle	first cycle	second cycle	fifth cycle	first cycle	second cycle	fifth cycle
1	0–24	0–1	0–1	0–5	0	0–0	0–0	0-0.523	0-1.102	0-1.763
2	24–36	1–2	1–4	5–10	0	0–0	0–0	0.523–1.240	1.102–2.034	1.763–2.836
3	36–42	2–4	4–6	10–10	0	0–0	0–0	1.240–1.884	2.034–2.942	2.836–3.520
4	42–45	4–7	6–9	10–10	0	0–0	0–1	1.884–2.529	2.942–3.459	3.520–3.998
5	45–48	7–8	9–9	10–10	0	0–0	1–4	2.529–2.815	3.459–4.069	3.998–4.601
6	48–54	8–10	9–9	10–10	0	0–2	4–4	2.815–3.554	4.069–4.491	4.601–5.036
7	54–66	10–10	9–9	10–10	0	2–2	4–4	3.554–4.069	4.491–4.801	5.036–5.519
8	66–90	10–10	9–9	10–10	0	2–3	4–4	4.069–4.515	4.801–5.087	5.519–5.709
9	90–114	10–10	9–9	10–10	0	3–3	4–4	4.515–4.179	5.087–5.062	5.709–5.404
10	114–126	10–10	9–9	10–10	0	3–3	4 − 3	4.179–3.896	5.062–4.604	5.404–5.204
11	126–132	10 − 9	9–9	10–10	0	3–3	3 − 0	3.896–3.482	4.604–4.353	5.204–4.827
12	132–135	9 − 8	9–9	10–10	0	3 − 2	0–0	3.482–2.949	4.353–3.779	4.827–4.470
13	135–138	8 − 6	9 − 8	10–10	0	2 − 0	0–0	2.949–2.226	3.779–3.241	4.470–3.838
14	138–144	6 − 2	8 − 4	10–10	0	0–0	0–0	2.226–1.317	3.241–3.399	3.838–2.921
15	144–156	2 − 0	4 − 1	10 − 5	0	0–0	0–0	1.317–0.322	3.399–1.179	2.921–1.85
16	156–180	0–0	1 − 0	5 − 0	0	0–0	0–0	0.322-0	1.179-0	1.85-0

3.2. percussive method

In this method, each cycle starts with the injection phase with the maximum flow rate, then decreases and then in the following increases, and with the start of the suction phase, the flow rate decreases and increases again. The volume of fluid injected or sucked in each step is 1000 µl. In step 1, fluid injection is started with a flow rate of 20000 µl/min for 3 seconds. The second stage of injection is performed with a flow rate of 10000 µl/min in 6 seconds, the third stage of injection with a flow rate of 5000 µl/min in 12 seconds and the fourth stage, injection with a flow rate of 2500 µl/min in 24 seconds. Then the fifth to eighth steps in an increasing trend of injection flow rate again reaches 20000 µl/min. Then, in the ninth step, it suddenly enters the suction phase and the injected fluid with a flow rate of 20000 µl/min is sucked from inside the cell. Figures (8) shows the 16 steps of injection and suction by percussive method.

Table (7) shows the start and end time of each step, the number of fingers, the minimum and maximum size of the base radius in each step, the size of the maximum instantaneous radius, the number of droplets and the size of some important quantities in each step of the first cycle. As the results show, the length of the largest finger in the first cycle of the percussive method has reached 5.362 cm and the length of each finger is longer than the first cycle of the pseudo-sinusoidal method. The maximum number of fingers has reached 12 and the maximum number of fingers with branched has reached 2. No pockets or droplets are observed in this cycle.

Figure (9) shows snapshots of how the two fluid contact surfaces develop in the 16 steps mentioned in the first cycle. Due to the start of the injection phase with the maximum flow rate, a significant number of fingers (6) is observed during 9 seconds. Following the injection phase until the end of step 8, the fingers become larger and wider. Starting the suction phase with the maximum flow rate in the ninth step, the lower part of the fingers are pulled inwards. This is due to the increase in capillary forces applied to the lower part of the fingers as a result of the increased suction speed towards the injection center. In the 135 and 171 second images in Figure (9) is visible that, the fingers are stretched inward (toward the injection center).

Table 7

Number of fingers, minimum and maximum size of base radius, size of maximum instantaneous radius, number of droplets in different stages of the first cycle of percussive method.
step	Start and end time of steps	Fingers		Number of droplets	Number of pockets	Minimum and maximum largest finger	minimum and maximum r (cm)	minimum and maximum r_b (cm)
step	Start and end time of steps	Number	Branched	Number of droplets	Number of pockets	Minimum and maximum largest finger	minimum and maximum r (cm)	minimum and maximum r_b (cm)
1	0–3	0–6	0–0	0	0	0-1.693	0.927–3.317	0.927–1.624
2	3–9	6–11	0–2	0	0	1.693–2.698	3.317–4.353	1.624–1.655
3	9–21	11–11	2–2	0	0	2.698-3.400	4.353–5.195	1.655–1.795
4	21–45	11–12	2–2	0	0	3.400-3.923	5.195–5.870	1.795–1.947
5	45–69	12–12	2 − 1	0	0	3.923–4.383	5.870–6.372	1.947–1.988
6	69–81	12–12	1–1	0	0	4.383–4.682	6.372–6.695	1.988–2.013
7	81–87	12–12	1–1	0	0	4.682–5.059	6.695–7.124	2.013–2.064
8	87–90	12–12	1–1	0	0	5.059–5.362	7.124–7.446	2.064–2.083
9	90–93	12–12	1–1	0	0	5.362–5.264	7.446–7.338	2.083–2.074
10	93–99	12–12	1–1	0	0	5.264–4.842	7.338–6.969	2.074–2.126
11	99–111	12–12	1–1	0	0	4.842–4.553	6.969–6.671	2.126–2.117
12	111–135	12 − 11	1–1	0	0	4.553–4.133	6.671–6.319	2.117–2.186
13	135–159	11–11	1 − 0	0	0	4.133–3.669	6.319-5.800	2.186–2.130
14	159–171	11 − 9	0–0	0	0	3.669–2.962	5.800-5.081	2.130–2.119
15	171–177	9 − 3	0–0	0	0	2.962–2.025	5.081–4.103	2.119–2.078
16	177–180	3 − 0	0–0	0	0	2.025-0	4.103–1.113	2.078–1.113

In the second cycle, the number of fingers, the droplets formed from the injected fluid, the oil pockets, the branching in the fingers and their length increase compared to the first cycle. The number of fingers and the number of branched fingers reach a maximum of 21 and 3, respectively, and the number of oil pockets created and the droplets reach 1 and 10, respectively. In stages 11 to 15, the base radius increases despite the presence of fluid suction in the center of the cell. This is due to the effect of reducing the length of the fingers and the movement of fluid inside them towards the center of the cell. Table (8) shows the start and end time of each step, the number of fingers, the minimum and maximum size of the base radius in each step, the size of the maximum instantaneous radius, the number of droplets of injected fluid, and the number of oil pockets created in each step of the second cycle.

Table 8

Results of the second cycle’s steps for important quantities.
step	Start and end time of steps	Fingers		Number of droplets	Number of pockets	Minimum and maximum largest finger	minimum and maximum r (cm)	minimum and maximum r_b (cm)
step	Start and end time of steps	Number	Branched	Number of droplets	Number of pockets	Minimum and maximum largest finger	minimum and maximum r (cm)	minimum and maximum r_b (cm)
1	0–3	0–21	0–0	0–4	0–1	0-2.294	1.113–3.811	1.113–1.517
2	3–9	21–21	0–0	4–8	1–1	2.294–3.240	3.811–4.938	1.517–1.698
3	9–21	21–21	0–1	8–9	1–1	3.240–3.833	4.938–5.614	1.698–1.780
4	21–45	21 − 18	1–2	9–10	1–1	3.833–4.358	5.614–6.314	1.780–1.955
5	45–69	18 − 17	2–3	10–10	1–1	4.358–4.803	6.314–6.804	1.955–2.001
6	69–81	17–17	3–3	10–10	1–1	4.803–5.177	6.804–7.264	2.001–2.086
7	81–87	17–17	3–3	10 − 8	1–1	5.177–5.512	7.264–7.643	2.086–2.130
8	87–90	17–17	3–3	8–8	1–1	5.512–5.817	7.643–7.967	2.130–2.150
9	90–93	17–17	3–3	8–8	1–1	5.817–5.679	7.967–7.795	2.150–2.115
10	93–99	17–17	3–3	8–9	1–1	5.679–5.389	7.795–7.431	2.115–2.041
11	99–111	17–17	3–3	9–9	1–1	5.389–4.994	7.431–7.113	2.041–2.118
12	111–135	17 − 16	3–3	9–10	1–1	4.994–4.612	7.113–6.743	2.118–2.130
13	135–159	16–16	3 − 2	10 − 8	1–1	4.612–4.177	6.743–6.318	2.130–2.141
14	159–171	16–16	2–2	8–8	1–1	4.177–3.498	6.318–5.585	2.141–2.087
15	171–177	16–16	2 − 1	8–8	1–1	3.498–2.446	5.585–4.614	2.087–2.167
16	177–180	16 − 0	1 − 0	8 − 2	1 − 0	2.446–0.473	4.614–1.480	2.167–1.006

Figure (10) shows a number of morphologies related to the 16 stages in the second cycle. As in the first cycle, the injection process begins with the maximum flow rate. The fingers in this cycle are narrower and longer than the first cycle and from the very beginning, oil pockets are seen inside the injection fluid, which appear at the end of the sixteenth stage of the first cycle and the beginning of the first stage of the second cycle and remain in the injection fluid until the end of the cycle. According to Table 9 in the fifth cycle, the number of fingers and the number of droplets have reached a maximum of 17 and 5, respectively, both of which are less than in the second cycle, but the number of pockets has increased significantly from 1 to 38. Also, the number of branched fingers has reached 5. Similar to the second cycle, in most suction stages, the size of the base radius increases.

Table 9

Results for some quantities in the 16 stages of the fifth cycle.
step	Start and end time of steps	Fingers		Number of droplets	Number of pockets	Minimum and maximum largest finger	minimum and maximum r (cm)	minimum and maximum rb (cm)
step	Start and end time of steps	Number	Branched	Number of droplets	Number of pockets	Minimum and maximum largest finger	minimum and maximum r (cm)	minimum and maximum rb (cm)
1	0–3	0–17	0–5	4–5	6–21	1.253–2.761	1.973–4.563	0.720–1.801
2	3–9	17–17	5–5	5 − 4	21–34	2.761–3.676	4.563–5.648	1.801–1.972
3	9–21	17 − 16	5–5	4 − 3	34–38	3.676–4.295	5.648–6.377	1.972–2.081
4	21–45	16 − 14	5 − 4	3–3	38–38	4.295–4.748	6.377–6.974	2.081–2.226
5	45–69	14–14	4 − 3	3–3	38–38	4.748–5.255	6.974–7.585	2.226–2.330
6	69–81	14–14	3–3	3–3	38–38	5.255–5.586	7.585–7.931	2.330–2.345
7	81–87	14–14	3–3	3–3	38–38	5.586–5.951	7.931–8.310	2.345–2.358
8	87–90	14–14	3–4	3–3	38–38	5.951–6.182	8.310–8.583	2.358–2.401
9	90–93	14–14	4–4	3–3	38–38	6.182–6.091	8.583–8.474	2.401–2.383
10	93–99	14–14	4 − 3	3–3	38–38	6.091–5.836	8.474–8.244	2.383–2.408
11	99–111	14–14	3–3	3–3	38–38	5.836–5.582	8.244–7.944	2.408–2.362
12	111–135	14–14	3–3	3–3	38–38	5.582–5.031	7.944–7.433	2.362–2.401
13	135–159	14–14	3–3	3–3	38–38	5.031–4.246	7.433–6.832	2.401–2.485
14	159–171	14 − 13	3–3	3–3	38–38	4.246–3.832	6.832–6.268	2.485–2.435
15	171–177	13 − 8	3 − 1	3–4	38–38	3.832–3.097	6.268–5.484	2.435–2.387
16	177–180	8 − 0	1 − 0	4–6	38 − 5	3.097–1.397	5.484–2.154	2.387–0.757

In figure (11), that is snapshots of how the contact surface of two fluids is created during the 16 stages of injection and suction in the fifth cycle, dislocation is seen in the fingers along with a large number of pockets of oil and droplets of displacement fluid.

The data in Table (10), which presents the results of the first, second, and fifth cycles for some quantities, show that the number of fingers does not follow a specific pattern, but by comparing Figs. 9 to 12 with each other, it can be seen that the number of fingers is directly related to the narrowness or width of the fingers, so in the second cycle, due to the narrowness of the fingers, the number of fingers is more than in the first and fifth cycles. But in general, the length of the fingers increases with increasing number of cycles, so that at the end of stage 8 of cycle 1 (end of injection phase) the length of the largest finger from 5.36, at the end of this stage in the second cycle to 5.81 and for the fifth cycle reaches 6.18 cm. By increasing the number of cycles, the number of branched fingers will increase. Also, the time of branching of the fingers in the injection phase is faster than the time of disappearance of the finger branching in the suction phase of the cycles. Generally, in the percussive injection-suction method, the results show that the number of oil pockets is strongly dependent on the number of injection cycles, so that reaches from zero pockets in the first cycle to 38 pockets in the fifth cycle. In addition, in this injection-suction method, droplets are formed from the second cycle. The reason for this could be a sudden decrease in the injection rate from its maximum to its minimum value during a cycle, which does not give enough time for a slow change in the morphology of the fingers so that during the rapid process of finger contraction (moving to the center cell), a part of the fingers is cut off and makes a drop.

Table 10

Number of fingers, number of branched fingers, and length of the largest finger in the first, second, and fifth cycles.
step	Number of fingers			Number of branched fingers			The length of the largest finger			Number of droplets			Number of pockets
step	first cycle	second cycle	fifth cycle	first cycle	second cycle	fifth cycle	first cycle	second cycle	fifth cycle	first cycle	second cycle	fifth cycle	first cycle	second cycle	fifth cycle
1	0–6	0–21	0–17	0–0	0–0	0–5	0-1.69	0-2.29	1.25–2.76	0	0–4	4–5	0	0–1	6–21
2	6–11	21–21	17–17	0–2	0–0	5–5	1.69–2.69	2.29–3.24	2.76–3.67	0	4–8	5 − 4	0	1–1	21–34
3	11–11	21–21	17 − 16	2–2	0–1	5–5	2.69–3.40	3.24–3.83	3.67–4.29	0	8–9	4 − 3	0	1–1	34–38
4	11–12	21 − 18	16 − 14	2–2	1–2	5 − 4	3.40–3.92	3.83–4.35	4.29–4.74	0	9–10	3–3	0	1–1	38–38
5	12–12	18 − 17	14–14	2 − 1	2–3	4 − 3	3.92–4.38	4.35–4.80	4.74–5.25	0	10–10	3–3	0	1–1	38–38
6	12–12	17–17	14–14	1–1	3–3	3–3	4.38–4.68	4.80–5.17	5.25–5.58	0	10–10	3–3	0	1–1	38–38
7	12–12	17–17	14–14	1–1	3–3	3–3	4.68–5.05	5.17–5.51	5.58–5.95	0	10 − 8	3–3	0	1–1	38–38
8	12–12	17–17	14–14	1–1	3–3	3–4	5.05–5.36	5.51–5.81	5.95–6.18	0	8–8	3–3	0	1–1	38–38
9	12–12	17–17	14–14	1–1	3–3	4–4	5.36–5.26	5.81–5.67	6.18–6.09	0	8–8	3–3	0	1–1	38–38
10	12–12	17–17	14–14	1–1	3–3	4 − 3	5.26–4.84	5.67–5.38	6.09–5.83	0	8–9	3–3	0	1–1	38–38
11	12–12	17–17	14–14	1–1	3–3	3–3	4.84–4.55	5.38–4.99	5.83–5.58	0	9–9	3–3	0	1–1	38–38
12	12 − 11	17 − 16	14–14	1–1	3–3	3–3	4.55–4.13	4.99–4.61	5.58–5.03	0	9–10	3–3	0	1–1	38–38
13	11–11	16–16	14–14	1 − 0	3 − 2	3–3	4.13–3.66	4.61–4.17	5.03–4.24	0	10 − 8	3–3	0	1–1	38–38
14	11 − 9	16–16	14 − 13	0–0	2–2	3–3	3.66–2.96	4.17–3.49	4.24–3.83	0	8–8	3–3	0	1–1	38–38
15	9 − 3	16–16	13 − 8	0–0	2 − 1	3 − 1	2.96–2.02	3.49–2.44	3.83–3.09	0	8–8	3–4	0	1–1	38–38
16	3 − 0	16 − 0	8 − 0	0–0	1 − 0	1 − 0	2.02-0	2.44–0.47	3.09–1.39	0	8 − 2	4–6	0	1 − 0	38 − 5

3.3. Constant-velocity injection-suction method

In the constant-velocity injection-suction method, there are no changes in the injection-suction fluid flow rate over time. The only common point of this method with the two previously mentioned methods is the volume of fluid injected-suction and the period of rotation is equal to the previous methods. The average injection-suction fluid flow rate in the two methods with a maximum flow rate of 20000 µl/min with a 180-second period for the constant-velocity injection-suction method is 5333 µl/min. The injection and suction phases each take a volume of 8000 µl in 90 seconds at a flow rate of 5333 µl/min. For convenience as well as analysis and comparison of the results of this method with previous methods, injection and suction phases were divided into equal time intervals of 15 seconds so that each cycle was divided into 12 stages, including 6 stages of injection and 6 stages of suction.

Table (11) shows the results of the first injection-suction cycle at a constant-velocity. As the results show, the length of the largest finger increases to 7.377 cm. No droplets, pockets or branches are observed in this cycle and the number of fingers reaches a maximum of 9. The length of the base radius as well as the length of the largest finger increases in step 7 despite being in the suction phase. The reason for this is the inertia of the injected fluid in the 6-stages of injection phase, which still tends to move outward radially. Figures (13) shows snapshots of how the two fluid contact surfaces develop during the 12 stages of injection and suction in the first cycle using the constant-velocity method with injection flow rate of 5333 µl/min.

Table 11

Results for some quantities in the 12 stages of the first injection-suction cycle at constant- velocity method.
step	Start and end time of steps	Fingers		Number of droplets	Number of pockets	Minimum and maximum largest finger	minimum and maximum r (cm)	minimum and maximum r_b (cm)
step	Start and end time of steps	Number	Branched	Number of droplets	Number of pockets	Minimum and maximum largest finger	minimum and maximum r (cm)	minimum and maximum r_b (cm)
1	0–15	0–1	0	0	0	0-0.667	1.015–3.205	1.015–2.538
2	15–30	1–3	0	0	0	0.667–1.248	3.205–4.622	2.538–3.373
3	30–45	3–5	0	0	0	1.248–2.031	4.622-5.600	3.373–3.569
4	45–60	5–6	0	0	0	2.031–2.608	5.600–6.270	3.569–3.661
5	60–75	6–7	0	0	0	2.608–3.058	6.270–6.891	3.661–3.833
6	75–90	7–9	0	0	0	3.058–3.462	6.891–7.377	3.833–3.914
7	90–105	9 − 7	0	0	0	3.462–3.178	7.377–7.130	3.914–3.951
8	105–120	7–7	0	0	0	3.178–2.827	7.130–6.773	3.951–3.945
9	120–135	7 − 6	0	0	0	2.827–2.459	6.773–6.260	3.945–3.801
10	135–150	6 − 4	0	0	0	2.459–1.753	6.260–5.292	3.801–3.538
11	150–165	4 − 1	0	0	0	1.753 − 0.714	5.292–3.680	3.538–2.966
12	165–180	1 − 0	0	0	0	0.714 − 0.289	3.680–1.303	2.966–1.014

Similar results were obtained in the second cycle. No oil pockets or drops were observed. But in the fifth cycle of this method, different types of phenomena were observed. As shown in Table (12), the length of the largest finger at the end of the sixth stage is 6.487 cm and the number of droplets, pockets and branches in the fingers has reached a maximum of five, eight and two, respectively. Also, the number of fingers reaches a maximum of thirteen, which have a fixed number from the end of the first stage to the beginning of the last stage. In the initial stage of the suction phase (seventh stage), due to the fingers inward effect, the length of the base radius increases. Figures (14) shows snapshots of how the contact surface of two fluids develops during the 12 stages of injection and suction in the fifth cycle while using the constant-velocity method with injection flow rate of 5333 µl/min. The presence of jackets and created fluid droplets is obvious.

Table 12

Results for some quantities in the 12 stages of the fifth injection-suction cycle at constant-velocity method.
step	Start and end time of steps	Fingers		Number of droplets	Number of pockets	Minimum and maximum largest finger	minimum and maximum r (cm)	minimum and maximum r_b (cm)
step	Start and end time of steps	Number	Branched	Number of droplets	Number of pockets	Minimum and maximum largest finger	minimum and maximum r (cm)	minimum and maximum r_b (cm)
1	0–15	0–13	0–1	4–5	0–8	0.529–3.059	1.760–4.870	1.231–1.811
2	15–30	13–13	1–1	5 − 4	8–8	3.059–4.462	4.870–6.323	1.811–1.860
3	30–45	13–13	1–1	4 − 3	8–8	4.462–5.137	6.323–7.211	1.860–2.073
4	45–60	13–13	1–2	3–3	8–8	5.137–5.760	7.211–7.853	2.073–2.093
5	60–75	13–13	2–2	3–3	8–8	5.760–6.295	7.853–8.382	2.093–2.087
6	75–90	13–13	2–2	3–3	8–8	6.295–6.635	8.382–8.799	2.087–2.164
7	90–105	13–13	2–2	3–3	8–8	6.635–6.487	8.799–8.668	2.164–2.180
8	105–120	13–13	2–2	3–3	8–8	6.487–6.287	8.668–8.426	2.180–2.139
9	120–135	13–13	2–2	3–3	8–8	6.287–5.751	8.426–7.903	2.139–2.152
10	135–150	13–13	2 − 1	3–3	8–8	5.751–4.971	7.903–7.108	2.152–2.136
11	150–165	13–13	1–1	3–3	8–8	4.971–3.705	7.108–5.903	2.136–2.197
12	165–180	13 − 0	1 − 0	3–3	8 − 0	3.705 − 0.527	5.903–1.782	2.197–1.254

During injection or suction of the fluid radially inside the Hell-Shaw cell, the formation of instabilities such as the formation of air packets, droplets or fingers are the factors that suffer problems with returning the contact surface of the two fluids to their initial state before the injection-suction process. In this experimental study, the instabilities created at the contact surface of two Newtonian immiscible water and base oil during unstable injection and suction in Hele-Shaw cell were investigated radially. Base oil with viscosities of 150, 920 and 1506 cSt was used for injection and suction. The maximum injection and suction fluid flow rates that were pseudo-sinusoidal and percussive time-dependent were considered to be 10000, 20000 and 30000 µl/min. The results were compared with the results of constant-velocity injection. The effects of injection and suction frequency on contact surface instability were also studied. Injection and suction were performed for at least six complete cycles. One of the most important objectives of this study was to investigate the ability to eliminate instabilities and return the contact surface of two fluids to the initial circular state at the end of each cycle. It was observed in the methods of percussive and constant-velocity that it is not possible to return to the initial circle after the first step. It was also observed that the growth rate of instabilities in different cycles of the pseudo-sinusoidal method is slower than the percussive and constant-velocity methods. The lower the viscosity and flow rate of the injected or suction fluid, the less instability occurs. The results showed that with increasing the flow rate, the number of oil packets in the injection-suction of constant-velocity method has increased, but in the reverse percussive method, this is the case. The results showed that in the injection-suction by pseudo-sinusoidal method, which do not cause instabilities such as water droplets, fingers and oil packets, but in two methods of injection-suction at a constant-velocity and percussive method, such instabilities are observed. It can be concluded that considering the conditions of the experiments performed, the pseudo-sinusoidal method is the most suitable method with the ability to return to the initial steady state among the three methods used. Of course, it should be noted that the greater the number of repetitions of cycles, the more instabilities created and the more difficult it is to return to the initial state.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
All authors have participated in conception, analysis and interpretation of the data, drafting the article and revising it, and approval of the final version.
This manuscript has not been submitted to, nor is under review at, another journal or other publishing venue.
Results of the study have been presented clearly, honestly, and without fabrication and data manipulation.
Proper acknowledgements to other works have been given.

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No competing interests reported.

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Effects of transient injection-suction on interfacial dynamics in immiscible Newtonian fluid-fluid radial displacement in Hele-Shaw cell

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Abstract

Figures

1. Introduction

2. Experimental Procedure

2.1. Experimental Setup

2.2. Injection methods

2.3. Mathematical Equations

3. Results And Discussion

3.1. Pseudo-sinusoidal method

3.2. percussive method

3.3. Constant-velocity injection-suction method

4. Conclusion

Declarations

References

Additional Declarations

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