Introduction

In a partially penetrated oil/gas well, the oil/water contact locally rises toward the perforated interval by a process known as water coning. This phenomenon happens when the vertical component of the viscous force is greater than the net gravitational force (Abass and Bass 1988). Since 1935, a number of scholarly studies have addressed the subject of water cones and provided analytical mathematical models to pinpoint the critical flow that prevents the formation of the water cone (Høyland et al. 1989; Farmen et al. 1999; Safari et al. 2018; Guo et al. 1992; Tabatabaei et al. 2012). Numerous creative solutions have been developed to deal with the problem of water cones, such as downhole water sink (DWS) technology, which relies on producing water separately from oil to prevent water from flooding the oil area (Wojtanowicz et al. 1991; Arslan et al. 2003, 2004; Ould-amer et al. 2004; Arslan 2005; Johns et al. 2005; Zeidani et al. 2008; Utama 2008). Moreover, downhole water loop (DWL) technology relies on re-injecting the water produced from the drainage region to prevent low pressure in the reservoir (Wojtanowicz and Xu 1992; Jin and Wojtanowicz 2010b, 2011; Jin et al. 2010). Despite the DWL technique's positive outcomes, the issue of oil droplets escaping with water can damage the injection region (Jin and Wojtanowicz 2013, 2014; Bedrikovetsky et al. 2005). This problem, which is brought on by the reverse cone phenomenon, was addressed by Jin et al. (2010), who provided an analytical approach to identify the critical flow values that prevent the formation of a reverse cone. However, because of the varied geological context and the inaccuracy of the used data, analytical solutions might not be accurate. To separate the oil droplets that might emerge with the water circulation, Jin and Wojtanowicz (2013) suggested employing a well separator, which depends on the density difference to separate the oil from the water. No numerical analysis of this type of separator was available, and in particular, the impact of mixture inlet velocity and flow rates on separation efficiency was not examined, despite the vast number of publications that examined separators in the oil industry, such as Ghaffarkhah et al. (2019); Al-Kayiem et al. (2019); Refsnes et al. (2019). In this work, the influence of the inlet velocities and flow rates of well separators utilized with DWL is studied numerically, and a design to lessen the impact of the inlet velocity of the mixture on separator performance is suggested.

Mechanisms of DWL and DOWS

To prevent cone growth, Widmyer developed the downhole water sink (DWS) technique in 1955 (Widmyer 1958). In this technique, oil and water are produced separately through two production tubes. According to his suggestion, the top (the oil zone) and bottom (the water zone) perforations must be separated (Swisher and Wojtanowicz 1996; Bowlin et al. 1997; Shirman 1998). For lowering the amount of water produced in oil wells as a result of applying the DWS technique, Wojtanowicz and Xu (1992) suggested the downhole water loop (DWL) technology. In this technology, in addition to the traditional completion in the oil zone, the water loop equipment is added in the water zone below the oil–water contact (OWC) as shown in Fig. 1C. This technology had the advantage of becoming a solution to the environmental compliance problem associated with the disposal of produced water (Jin and Wojtanowicz 2010a; Jin 2009). In other words, formation water might be kept away from oil-producing perforations, allowing for increased oil recovery per well with reservoir pressure maintenance.

Fig. 1
figure 1

A Oil saturation profile of the reverse cone (Inikori and Wojtanowicz 2001); B Oil saturation profile of the cone; C Schematic of the DWL compilation (Jin 2009)

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The main problem is the occurrence of the reverse cone phenomenon (Fig. 1A), which occurs in the opposite direction of the water cone (Fig. 1B). This phenomenon occurs when the viscous force for the drainage completion (water) exceeds the viscous force for the top (oil) completion.

To ascertain the critical flow values at which no cone and no reverse cone formation occur, analytical models were proposed by Jin et al. (2010); in other words, only oil is produced by the higher perforation and only water is produced by the lower perforation, as follows:

$${Q}_{\mathrm{opc}}=\frac{{Q}_{\mathrm{wd}}{B}_{\mathrm{w}}{h}_{\mathrm{o}}}{M {B}_{\mathrm{o}}{h}_{\mathrm{w}}}\left(1-{D}_{\mathrm{di}}\right)+\frac{0.003066\;{K}_{\mathrm{o}}{h}_{\mathrm{o}}\left({\gamma }_{\mathrm{w}}-{\gamma }_{\mathrm{o}}\right)}{{\mu }_{\mathrm{o}}{B}_{\mathrm{o}}\mathrm{ln}({r}_{\mathrm{e}}/{r}_{\mathrm{w}})}\left[{Z}_{\mathrm{op}}-{Z}_{\mathrm{wd}}+{Z}_{\mathrm{wd}}{D}_{\mathrm{di}}\right]$$
(1)
$${Q}_{\mathrm{wdc}}=\frac{{Q}_{\mathrm{op}}M{B}_{\mathrm{o}}{h}_{\mathrm{w}}}{{B}_{\mathrm{w}}{h}_{\mathrm{o}}} \frac{1}{\left(1-{D}_{\mathrm{di}}\right)}+\frac{0.003066\;{K}_{\mathrm{w}}{h}_{\mathrm{w}}\left({\gamma }_{\mathrm{w}}-{\gamma }_{\mathrm{o}}\right)}{{\mu }_{\mathrm{w}}{B}_{\mathrm{w}}\mathrm{ln}({r}_{\mathrm{e}}/{r}_{\mathrm{w}})} \frac{\left[{Z}_{\mathrm{wd}}-{Z}_{\mathrm{op}}+{Z}_{\mathrm{wd}}{D}_{\mathrm{di}}\right]}{\left(1-{D}_{\mathrm{di}}\right)}$$
(2)

where \({Q}_{\mathrm{opc}}\) is the critical oil production rate (Stb/d), \({Q}_{\mathrm{wd}}\) is the water drainage/injection rate (Stb/d), \({Q}_{\mathrm{wdc}}\) is the critical water drainage/injection rate (Stb/d), and \({Q}_{\mathrm{op}}\) is the oil production rate (Stb/d).

The analytical models, however, become unreliable under conditions of heterogeneous geological environments or the unreliability of the geological data used in the studies, making it possible for oil droplets to escape with the water produced by the bottom perforation due to the reverse cone phenomenon. To solve this problem, Jin and Wojtanowicz (2013) suggested applying the DOWS approach, which uses the density difference concept to separate oil droplets from a water stream. A simplified schematic of the separator concept used with the DWL technique can be seen in Fig. 2B, where water and oil droplets enter from the inlet and separate inside the separator before the oil comes out from the top and the water from the bottom and are injected into the injection area using a pump (P).

Fig. 2
figure 2

A Forces acting on an oil droplet moving through a water stream (Jin and Wojtanowicz 2013); B DOWS schematic

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The oil droplet is susceptible to three forces: gravity (\({F}_{\mathrm{G}}\)), buoyancy (\({F}_{\mathrm{B}}\)) and drag (\({F}_{\mathrm{D}}\)), as shown in Fig. 2A. Analytical solutions for the critical water flow values that enable the oil droplet to rise and exit toward the surface can be determined by evaluating these forces. Mathematical solutions, including the Drift-Flux model in Eqs. (3–7), were presented by Jin and Wojtanowicz (2013), where he conducted a laboratory experiment on a separator with a diameter of 57 mm and a distance between the oil and water inlets of 229 mm to determine the values of some constants such as \({\mathrm{c}}_{\mathrm{o}}\).

$${V}_{\mathrm{d}}=1.53\frac{ {\left(1-{f}_{\mathrm{o}}\right)}^{n}}{{{\upmu }_{\mathrm{D}}}^{0.977}}{\left[\frac{{\upsigma }_{\mathrm{ow}} g \left({\rho }_{\mathrm{w}}-{\rho }_{\mathrm{o}}\right)}{{{\rho }_{\mathrm{w}}}^{2} }\right]}^{0.25}$$
(3)
$${f}_{\mathrm{o}}= \frac{{q}_{\mathrm{o}}}{A\left[{1.2 v}_{\mathrm{m}}+1.53{\left[\frac{{\sigma }_{\mathrm{ow}} g \left({\rho }_{\mathrm{w}}-{\rho }_{\mathrm{o}}\right)}{{{\rho }_{\mathrm{w}}}^{2} }\right]}^{0.25}\right]}$$
(4)
$${V}_{\mathrm{o}}=1.53\frac{ {\left(1-{f}_{\mathrm{o}}\right)}^{n}}{{{\upmu }_{\mathrm{D}}}^{0.977}}{\left[\frac{{\sigma }_{\mathrm{ow}} g \left({\rho }_{\mathrm{w}}-{\rho }_{\mathrm{o}}\right)}{{{\rho }_{\mathrm{w}}}^{2} }\right]}^{0.25}-{c}_{\mathrm{o}} {v}_{\mathrm{w}}$$
(5)
$${V}_{\mathrm{wc}}=1.53\frac{ {\left(1-{f}_{\mathrm{o}}\right)}^{n}}{{c}_{\mathrm{o}} {{\upmu }_{\mathrm{D}}}^{0.977}}{\left[\frac{{\sigma }_{\mathrm{ow}} g \left({\rho }_{\mathrm{w}}-{\rho }_{\mathrm{o}}\right)}{{{\rho }_{\mathrm{w}}}^{2} }\right]}^{0.25}$$
(6)
$${\upmu }_{\mathrm{D}}=\frac{2\left(2{\upmu }_{\mathrm{w}}+3{\upmu }_{\mathrm{o}}\right)}{{\upmu }_{\mathrm{w}}+{\upmu }_{\mathrm{o}}}$$
(7)

where \({V}_{\mathrm{d}}\) is the drift velocity (when, \({v}_{\mathrm{w}}\)= 0, m/s), \({\mathrm{f}}_{\mathrm{o}}\) is the oil volume fraction,\({V}_{\mathrm{o}}\) is the fraction in situ oil droplet velocity (m/s),\({V}_{\mathrm{wc}}\) is the critical in situ water velocity (m/s),\({\upmu }_{\mathrm{D}}\) is the dimensionless viscosity group, dimensionless, n is the drift-flux exponent, dimensionless (n = 1), \({\mathrm{c}}_{\mathrm{o}}\) is the drift-flux profile parameter, dimensionless (\({c}_{\mathrm{o}}\)=1.235).

Although the movement of the oil droplets within the water stream inside the separator is accurately described by these equations, they do not account for the inlet velocity, where the inlet velocity into the separator differs in value and direction from the velocity of the mixture inside the separator. As a result, we will use numerical solutions in this work to illustrate the impact of the inlet velocity into the separator and the flow rates.

Numerical study

Numerical simulations were performed using commercial CFD software, which provides the widest range of turbulence and physical models to accurately simulate. This work was done in two stages, the first of which was to demonstrate the commercial simulator's suitability for simulating this kind of separator. In the second stage, the impact of mixture inlet velocities and flow rates on separation efficiency will be discussed, and a design that addresses this issue will be presented. It is known that the flow pattern (laminar and turbulent) is determined by the Reynolds number \({\mathrm{Re}}_{\mathrm{D}}\) (Jin and Wojtanowicz 2013):

$${\mathrm{Re}}_{\mathrm{D}}=\frac{{\rho }_{\mathrm{w}} \left|{V}_{\mathrm{w}r}\right|{\mathrm{ d}}_{\mathrm{o}}}{{\mu }_{w}}$$
(8)
$$\left|{V}_{\mathrm{wr}}\right|=\left|\left|{V}_{\mathrm{o}}\right|-\left|{V}_{\mathrm{w}}\right|\right|$$
(9)

where \({\rho }_{\mathrm{w}}\) is the water density (kg/m3), \({\mathrm{V}}_{\mathrm{o}}\) is the in situ oil droplet velocity (m/s), \({V}_{\mathrm{w}}\) is the in situ water velocity (m/s), \({\mu }_{w}\) is the water viscosity (pa·s),\({\mathrm{d}}_{\mathrm{o}}\) is the oil droplet diameter (m).

The flow is laminar when the Reynolds numbers are less than 2000 and turbulent when they are greater than 4000 (Nayyar 2000). In this paper, turbulence was modeled using the Reynolds-averaged Navier-Stokes K-epsilon model, which has been applied to various investigations of gravity separators. The results it provided were quite close to actual field observations (Ghaffarkhah et al. 2018; Refsnes et al. 2019). Two extra conservation equations are included in K-epsilon model, one for the turbulent kinetic energy \(\mathrm{k}\) and the other for the turbulent dissipation rate \(\upvarepsilon\):

$$\frac{\partial ({\rho }_{\mathrm{m}}.k)}{\partial t}+\frac{\partial }{\partial {x}_{\mathrm{i}}}\left({U}_{\mathrm{i}}.{\rho }_{\mathrm{m}}.k\right)=\frac{\partial }{\partial {x}_{\mathrm{i}}}\left[\left(\frac{{\mu }_{\mathrm{i}}}{{\sigma }_{\mathrm{k}}}+{\mu }_{\mathrm{m}}\right)\frac{\partial \mathrm{k}}{\partial {x}_{\mathrm{j}}}\right]-{\rho }_{\mathrm{m}}.\varepsilon +{P}_{\mathrm{k}}$$
(10)
$$\frac{\partial ({\rho }_{\mathrm{m}}.\varepsilon )}{\partial t}+\frac{\partial }{\partial {x}_{\mathrm{i}}}\left({U}_{\mathrm{i}}.{\rho }_{\mathrm{m}}.\varepsilon \right)=\frac{\partial }{\partial {x}_{\mathrm{j}}}\left[\left(\frac{{\mu }_{\mathrm{t}}}{{\sigma }_{\varepsilon }}+{\mu }_{\mathrm{m}}\right)\frac{\partial \varepsilon }{\partial {x}_{j}}\right]+\frac{\varepsilon }{k}.({C}_{\varepsilon 1}.{P}_{\mathrm{k}}-{C}_{\varepsilon 2}.{P}_{\mathrm{k}} .{\rho }_{\mathrm{m}} .\varepsilon )$$
(11)

In these equations are, \({C}_{\varepsilon 1}\)=1.44, \({C}_{\varepsilon 2}\)=1.92, \({\sigma }_{\varepsilon }\)=1.3, and \({\sigma }_{k}\)=1.

$${\mu }_{\mathrm{i}}={C}_{\mu } . {\rho }_{\mathrm{m}} .\frac{{k}^{2}}{\varepsilon }$$
(12)

where \({C}_{\mu }\)=0.09. \({P}_{\mathrm{k}}\) is the turbulence production due to viscous forces.

$${P}_{\mathrm{k}}={\mu }_{\mathrm{i}} . \left(\frac{\partial {U}_{\mathrm{i}}}{\partial {x}_{\mathrm{j}}}+\frac{\partial {U}_{\mathrm{j}}}{\partial {x}_{\mathrm{i}}}\right).\frac{\partial {U}_{\mathrm{j}}}{\partial {x}_{\mathrm{j}}}$$
(13)

Stage 1: numerical model validation

The capability of the commercial CFD simulator utilized in this work will be confirmed by following a procedure similar to the laboratory experiment that was used to develop the analytical model Eqs. (3–7) and then contrasting the numerical findings with the analytical. To simulate the behavior of oil droplets in a water stream inside the separator, a geometric model for the studied separator, as shown in Fig. 3A, was built wherein the bottom inlet is used for oil and the upper inlet is used for water Fig. 3C. The oil inlet area is 3.14 \({\mathrm{mm}}^{2}\), the water inlet area is 1256 \({\mathrm{mm}}^{2}\), the bottom outlet area around the oil inlet is 1252.86 \({\mathrm{mm}}^{2}\), the separator’s diameter is 57 mm, and the distance between the two inlets (oil and water) is 229 mm. Tetrahedron-type meshes, with a total of 1,061,733 cells, are widely dispersed in the oil inlet, as depicted in Fig. 3A, B. We used a variety of water flow values in the separator, as given in Table 1, for three different types of oil, whose attributes are shown in Table 2 (Jin and Wojtanowicz 2013). Then, we calculated the oil velocity values (\({\mathrm{V}}_{\mathrm{o}}\)) for each water flow value by measuring the time at which the oil crossed between two fixed sites (the first is located at an altitude of 20 mm, and the second is at an altitude of 30 mm).

Fig. 3
figure 3

A 3D shape of DOWS meshes; B Bottom view of meshes; and C A simplified diagram of the procedure applied in stage 1 (model validation)

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Table 1 The water velocity values and oil flow rates applied in stage 1 (model validation)
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Table 2 Specifications of primary (water) and secondary (oils) phases
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The outcomes are displayed in Fig. 4, where we observe the convergence with the analytical model's outcomes (Eq. 5). Table 3 displays the boundary condition that was employed. Because the VOF model is appropriate for immiscible liquids like oil and water, it was utilized during this work to represent the oil–water multiphase flow in the separator. The primary (continuous) phase was water, while the secondary (dispersed) phase was oil. The viscosity model was also laminar at this stage. The coupled algorithm was used to achieve pressure–velocity coupling between continuity and momentum. Based on the least squared cell-based approach, gradients were discretized. PRESTO scheme was selected. QUICK was used to discretize momentum.

Fig. 4
figure 4

Comparison of droplet velocities between the numerical and drift-flux models: A light mineral oil; B heavy mineral oil; C #11 oil

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Table 3 Boundary conditions used in stage 1 (model validation)
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As we previously stated, the analytical model does not account for mixture inlet velocities, which, in our opinion, significantly affect the separator's performance. In order to understand how the mixture inlet velocities affected the process, we looked at stage 2.

Stage 2: effect of mixture inlet velocities and flow rates

At this stage, we will demonstrate the impact of the separator's inlet velocities and flow rates by running a simulation that is more similar to what occurs in the reverse cone where water and oil enter from the side inlet, causing separation to take place inside the separator, resulting in oil coming out of the upper outlet and water coming out of the lower outlet. We utilized the same software settings used in the first stage in all the cases investigated at this stage, with some alterations, whereas, in the first stage, the laminar model produced results close to the analytical calculations due to the low Reynolds number value, by substituting in Eq. 8, where the diameter of the oil droplet does not exceed 10 mm and the largest relative velocity does not exceed 0.15 m/s, i.e., the Reynolds number is roughly 1500. At this stage, the oil droplets are significantly larger, as we will see, and the mixture is moving at a high velocity of up to 0.4 m/s at the entrance area. Therefore, we used the turbulent model (K-epsilon model) at this stage. A contact angle (θ = 170 degree) is taken to account for the wetting behavior of the wall with the fluids. The time step size used is 0.004 s. We conducted all of our investigations at a processing time of 8 s because the time needed for the mixture to completely traverse the separator in all cases is between 2.5 and 5 s (by dividing the separator's length by mix velocity). We think that this period can provide a clear understanding of the separation capacity. All the cases studied in this stage will be done on light mineral oil, and the velocity inside the separator will be less than the critical velocity specified in the first stage (\({{\mathrm{V}}_{\mathrm{mix}}<\mathrm{V}}_{\mathrm{wc}}\)).

Case 1 The mixture enters from the inlet, as shown in Fig. 5A, B, according to the inlet velocity of 0.4 m/s (flow rate of 246.3 \({\mathrm{mm}}^{3}/\mathrm{s}\)), the oil volume fraction of 0.2, the outlet velocity from the top of 0.04 m/s (flow rate of 50.24 \({\mathrm{mm}}^{3}/\mathrm{s}\)), and the outlet from the bottom under a pressure of 2501.55 pa. Meaning that the percentage of exit from the top is 20.39% and the percentage of exit from the bottom (injection) is 79.6%.

Fig. 5
figure 5

DOWS geometry of case 1: A Dimensions and boundary conditions of DOWS geometry; B 3D geometry of DOWS in the simulator

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The mesh validation check was performed at four values of the number of elements, as shown in Fig. 6; it can be seen that the changes are simple. So, at this stage, we used mesh number 795861 to analyze all cases.

Fig. 6
figure 6

Mesh independency for case 1

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Here the separation efficiency for case 1 is 29.05% (the proportion of oil exiting the top outlet to oil entering the separator). Appendix 1 (A–I) shows the water saturation profile for a plane located in the center of the separator during the separation process, and Fig. 7 depicts the water saturation profile to indicate the impacts of inlet velocity during separation. In Fig. 7 (number 1), we can note that the main reason for the decrease in the separation efficiency is the inability of the oil droplets to reach the upper outlet, where the oil droplets collide with the mixture stream, which returns them downward. And it can be observed that the upper outlet is impacted by the inlet velocity Fig. 7(number 2), which scatters the oil droplets and prevents them from exiting, as becomes clearer in the velocity profile in Appendix 1 (J).

Fig. 7
figure 7

Contour of the water phase during the separation process; number 1 denotes the region where the stream of the mixture and oil droplets contact; number 2 denotes the impact region of the input mixture on the top outlet area

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Case 2 In this case, we doubled the inlet section area and cut the velocity in half (i.e., the same flow rate); the separation efficiency was 20.16%. Appendix 2 (A–F) illustrates the stages of separation in this case. It is clear that Case 1 had better results than this case. This can be explained by the fact that the mixture's collision with the separator's walls due to its high velocity encourages the separation of oil droplets from the water stream. This behavior is similar to the centrifugation principle, which is the basis for many separators, where high velocity facilitates efficient separation of the oil particles from the water stream.

Case 3 As we said, the main problem with the conventional design in case 1 is that the oil droplets could not get out of the upper outlet. To solve this problem and improve the results of the separator, we proposed a new design, which is a simple modification of the design of Case 1, shown in Fig. 8A–D. The design concept is based on enlarging the size of the mixture inlet by 15 mm, as shown in Fig. 8B, in order to enable the oil droplets to reach the outlet. In addition, to make the upper outlet area quieter, we lowered the height of the mixture inlet so that the distance between the inlet and the upper outlet is 36 mm and moved the upper outlet as far as possible from the place where the mixture hits the separator walls, as shown in Fig. 8C. Under the same boundary conditions as in Case 1, simulations for Case 3 revealed a separation efficiency of 40.28%, i.e., 38.65% more than for Case 1.

Fig. 8
figure 8

The proposed design: A 3D geometry; B geometry dimensions; C upper view; D side view

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Cases 4, 5, 6, and 7 To understand the effect of the inlet flow rates and the upper outlet flow rates on the separation efficiency, the flow values on the proposed design in Case 3 are studied as follows: Case 4 where the inlet flow is 184.72 \({\mathrm{mm}}^{3}/\mathrm{s}\), i.e., the reduction rate is 25%; Case 5 where the inlet flow is 123.15 \({\mathrm{mm}}^{3}/\mathrm{s}\), i.e., the reduction rate is 50%; Case 6 where the upper outlet flow rate is 62.8 \({\mathrm{mm}}^{3}/\mathrm{s}\), i.e., the increase rate is 25%; and Case 7 where the outlet flow rate is 75.36 \({\mathrm{mm}}^{3}/\mathrm{s}\), i.e., the increase rate is 50%. The results of all cases are shown in Table 4.

Table 4 The separation efficiency results of all cases
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According to the results of cases 4, 5, 6, and 7, regulating the separator's inlet flow rate improves separator performance more than regulating the upper outlet flow.

Conclusions and future work

This work gives a numerical assessment of the separators used with the DWL approach in order to better understand how well they can separate oil from water and prevent re-injection of oil, which damages the injection region where the effect of flow rates and mixture inlet velocities on separation efficiency was studied, and a novel design that successfully reduces the effect of the mixture inlet velocities was provided. The results of this research can be summarized as follows:

  1. 1.

    At the model validation stage, the outcomes of the commercial numerical simulator used in this work were contrasted with those of the analytical models. The findings demonstrated that when designing two-phase separators, it is viable to rely on commercial simulation to create numerical solutions that are close to the outcomes of the actual separators.

  2. 2.

    In the study of high inlet velocity, the separation efficiency was unsatisfactory, which can be explained by two points. The first is the inability of the oil droplets to exit through the upper outlet, as the oil droplets collide with the mixture stream and are forced to return to the bottom. The second is manifested by the currents generated by the high velocity near the upper outlet, which also prevents oil droplets from leaving.

  3. 3.

    The separation efficiency of low inlet velocity was very poor where the outcomes demonstrate that decreasing the velocity by expanding the inlet section area decreases the separation efficiency. This can be explained by the fact that high-velocity collisions between the mixture and the separator walls aid in the separation of the mixture's oil particles, whereas low-velocity collisions are weak and cause the droplet particles to remain suspended (scattered) in the water stream, making it challenging to form large droplets that can ascend to the upper outlet.

  4. 4.

    The problem of oil droplets not reaching the upper outlet was effectively reduced by the innovative new design, which included an expansion of the mixture entry area and changes to the inlet and outlet positions.

  5. 5.

    According to the findings of the flow rates studied, regulating the separator's inlet flow rate has a more significant impact on separator performance than does regulating the upper outlet flow.

  6. 6.

    In all cases studied in this work, the velocity inside the separator is less than the critical velocity (\({\mathrm{V}}_{\mathrm{wc}}\)) calculated analytically, but the complete separation of the oil droplets did not take place due to the influence of the inlet velocity, which is different in value and direction from the velocity within the separator. So, the analytical models are insufficient to construct this kind of separator because the inlet velocity was not taken into account.

  7. 7.

    The outcomes in all cases are poor. One could argue that gravity separation is ineffective when used alone; thus, in our upcoming work, we'll propose coupling a hydro-cyclone separator to a gravity separator (separation in two stages) to improve separation efficiency.