Introduction

Horizontal separators are essential in the oil and gas industries, as they play a critical role in separating the mixture of oil, gas, and water that flows out of production wells into distinct phases. The effectiveness of the separator used and the extent of phase separation achieved during the oil production process directly impact the final quantity of oil that can be produced in a stock tank. By properly maintaining the separator level, efficient separation of oil, gas, and water can be ensured, resulting in the production of high-quality oil.

A separator is a piece of equipment that is used to separate the oil, water, and gas components produced from an oil well. The degree of phase separation refers to how effectively these components have been separated from each other. The efficiency of the separator and the extent of phase separation determine the amount of oil that can be stored in the stock tank.

As oil is produced, the pressure in the reservoir decreases, causing changes in the hydrocarbon components. To ensure better separation of the produced phases, it is crucial to adjust the dimensions of the separator appropriately. However, the dimensions of the separator cannot be modified during production. Therefore, to enhance phase separation, the level of the separator needs to be adjusted. This adjustment also helps prevent issues like carry-over, foaming, and emulsion formation, which can disrupt the production process and pose safety hazards. Thus, it is essential to maintain the separator level at the desired level consistently.

The pipelines experience various flow regimes, including slug flow. Slug flow causes pressure and flow rate fluctuations, which can impact the performance of the separator and its associated equipment (Luo et al. 2014). Slug flow is a complex phenomenon that can be further classified into different subtypes based on characteristics such as slug length, frequency, and velocity of the gas and liquid phases (Luo et al. 2014).

The efficiency of phase separation relies on the liquid level inside the separator. To guarantee optimal phase separation when reservoir pressure fluctuates, it is essential to have an intelligent system that can regulate and maintain the liquid level at the desired point.

This study proposes a new correlation method to effectively measure the desired liquid level of the separator using novel separator pressures. The results have shown that the new correlation technique successfully achieved accurate measurements of the desired liquid level.

To test this method, we built and tested an intelligent system under various operational conditions in the laboratory. The laboratory study conducted on phase separation revealed that the required level of liquid for the separation process varies depending on the pressure and temperature of the separator.

The intelligent system effectively maintained the desired liquid level of the separator by utilizing a new correlation technique. This was accomplished by acquiring new separator pressure readings through installed sensors, which helped mitigate the negative effects of the slug flow regime and minimized issues such as foam formation and excessive flushing of the separator.

The system was tested in a lab under various operating conditions before being implemented in an integrated production model with an actual oil field in southern Iran. The model was subsequently simulated for a period of 840 days. During this time, the intelligent system successfully increased oil production in the storage tank by 13% and prevented the separator from over-flushing during the formation of slug flow.

After conducting laboratory tests on the new correlation method and separator level controller, the model was then integrated into a comprehensive production model that included a real oil reservoir model.

This study aims to create and evaluate a new correlation that can measure the desired level for the separator across different operational scenarios. Additionally, we plan to design and test a smart control system that can address this issue across various operating conditions.

The newly developed smart control system is an innovative solution designed to regulate the liquid level of the separator. It is capable of accurately measuring the liquid level by monitoring the pressure and temperature within the separator. Once measured, the system can then adjust the liquid level to the desired value.

This innovative system simultaneously regulates the liquid level of the separator and eliminates slug flow effects on the separator performance in real-time. Some of the novelties of this study are as follows:

  • A new correlation has been developed for measuring the desired liquid level in the separator under different operational conditions.

  • Designing and building a smart control system coupled with a gas–liquid flow loop to test the novel correlation with the horizontal gas–liquid separator and its PID level controller.

  • Investigating the effect of operational pressure and temperature of the separator on flow regimes before the separator

  • Determining the effectiveness of separator liquid height on phase separation efficiency and its performance

  • Utilizing a smart control system with novel correlation to improve oil production in a real integrated production model.

This section aims to clarify the differences between this study and previous studies.

Yadigaroglu et al. (2018) has explained the different regimes of multiphase flows, along with their effect on the pressure drop of multiphase flow. When the velocities of the liquid and gas are very low, the liquid phase remains continuous while the gas phase is dispersed in the form of bubbles. This type of flow is referred to as bubble flow. As the flow rate of the liquid increases, the bubbles have the potential to grow larger through coalescence (Yadigaroglu et al. 2018). In pipes, there is a multiphase flow regime characterized by the presence of large gas bubbles dispersed in a continuous liquid. These bubbles can occupy a significant portion of the pipe. Furthermore, there are smaller bubbles dispersed within the liquid, although many of them have merged to form the larger bubbles, also known as plugs. The flow regime known as multiphase fluid flow is characterized by the presence of liquid plugs, also known as slugs, separated by large gas pockets. In vertical flow, these gas pockets take the shape of symmetrical bullets that occupy nearly the entire cross-sectional area of the tubing (Yadigaroglu et al. 2018). Figure 1 illustrates different flow regimes observed in horizontal pipes (Perez 2007).

Fig. 1
figure 1

Different flow regimes observed in horizontal pipes (Perez 2007)

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Wang et al. (2002) proposed a new optimization technique for allocating production and lift-gas rates to wells subject to multiple constraints. Their approach handles flow interactions and applies to various complexities. However, their optimization technique was simple and had few constraints, such as flow regimes and changes in separator pressure and temperature. They did not consider the negative impacts of slug flow on production or investigate the effects of separator liquid level and operational pressure. The previous study also failed to consider the integrated production model. In contrast, this study optimizes production in the integrated production model by using a real oil field reservoir in southern Iran. It regulates the liquid level of the separator and eliminates the effects of slug flow on the separator simultaneously.

Nnabuife et al. (2022) and Luo et al. (2014) investigated slug flow control in offshore production facilities. Nnabuife et al. eliminated slug flow at the separator inlet by using a choke valve opening as a control input, while Luo et al. used separator liquid control to prevent overflow when slug flow forms before the separator. However, both studies by Luo et al. (2014) and Nnabuife et al. (2022) have several shortcomings. These include not investigating horizontal separators in the lab, not examining various operational conditions in the lab, not providing a smart control system, not integrating the model into the production model with a real reservoir, and not using key separator parameters such as pressure and temperature to regulate the liquid level. In contrast, this study has improved the production process in the integrated production model by utilizing a real oil field reservoir in southern Iran. To achieve this, the study simultaneously regulated the liquid level of the separator and eliminated the effects of slug flow on the separator.

Song et al. (2023) conducted a study on controlling the liquid level and pressure of an offshore separator using a PI controller. However, their study had several shortcomings. Firstly, they did not develop a smart control system in the lab. Secondly, their model only included a pipeline and a separator, without considering the effects of the oil reservoir, vertical wells, and surface chokes and manifolds. Additionally, they did not take into account changes in the separator pressure and temperature, as well as the flow regimes prior to the separator. This paper addresses all of the deficiencies identified in the analysis conducted by Song et al. (2023).

Krishnamoorthy et al. (2019) employed a feedback control system to optimize oil and gas production by regulating GOR and gas-lift rates. However, their study (Krishnamoorthy et al. 2019) suffered from certain limitations. These included the absence of a smart control system in the laboratory and the failure to incorporate an integrated model. Their model solely consisted of a pipeline and a separator, disregarding the impact of the oil reservoir, vertical wells, and surface chokes and manifolds. Furthermore, the study overlooked changes in separator pressure and temperature, as well as flow regimes preceding the separator.

Backi and Skogestad (2017) conducted a study on liquid and pressure control for a laboratory-scale horizontal separator. However, they did not take into account the flow regimes or the impact of operational pressure on the separation process. The study by Backi and Skogestad (2017) had a few limitations. These included the absence of a smart control system and the lack of consideration for an integrated model. Furthermore, the researchers did not consider the variations in separator pressure and temperature, as well as the flow regimes preceding the separator.

Paula et al. (2015) designed a sliding control for the liquid level at a subsea separator. However, they did not investigate the control system in the lab pilot or the flow regimes before the separator. The study conducted by Paula et al. (2015) had several shortcomings. They did not develop a smart control system or consider an integrated model. Additionally, they did not account for changes in separator pressure, temperature, or flow regimes before the separator.

Elhagar et al. (2019) lacked an integrated model and flow regime analysis prior to the separator. Additionally, they did not investigate the control system in the laboratory pilot. The study conducted by Elhagar et al. (2019) had several shortcomings. They did not design a smart control system in the lab, nor did they consider an integrated production model. Furthermore, they did not take into account the impact of changes in separator pressure, temperature, or flow regimes on the separator performance prior to the separator.

Kharoua et al. (2013) proposed a simulation model for three-phase gravity separators using CFD. However, their study had some limitations. Specifically, they did not include a smart control system and did not investigate the impact of different flow regimes on separator performance. In contrast, this study addresses these shortcomings by incorporating a smart control system and conducting laboratory experiments.

Sausen et al. (2012) conducted a study to examine the impact of hydrodynamic slug flow on separator performance and to develop a real-time smart control system. The researchers focused on severe slugging, disregarded hydrodynamic slugging, and made certain assumptions. In addition to investigating the separation efficiency through experiments, this study designed an automatic smart control system to enhance separation efficiency. Unlike Sausen et al. (2012), the laboratory did not examine the separation efficiency of the separator.

Aimacaña-Cueva et al. (2022) conducted a study to test two control strategies on a three-phase horizontal separator with a physical control device. However, they did not consider the impact of flow regime or separator pressure on the preferred liquid level in the separation process. The main focus of this study is to analyze how slug flow in the hydrodynamics affects the efficiency of the separator. Additionally, a smart control system was developed to manage operational parameters in real time, and the experimental results were analyzed. Although Aimacaña-Cueva et al. (2022) did not investigate the separation efficiency of the separator in the laboratory, this study aimed to design an automatic smart control system to enhance and control the operation of the device during experiments.

Medeiros et al. (2021) proposed an adaptive control strategy for an oil and gas separator using a gain scheduling technique. However, they did not investigate the effect of separator pressure on the liquid level. This study aims to investigate how hydrodynamically slug flow affects the efficiency of the separator. Additionally, a real-time intelligent control system was designed to manage operational parameters, and the experimental results were analyzed.

Wang et al. (2016) developed a mathematical model to simulate severe slugging characteristics. However, they did not take into account slug flow in horizontal pipelines or the removal of the hydrodynamic slug flow regime. This research aims to explore the impact of hydrodynamic slug flow on the efficiency of the separator. In this study, we designed a real-time smart control system capable of managing operational parameters and analyzing experimental results. While Wang et al. (2016) did not investigate the separation efficiency of the separator in the laboratory, our research developed an automatic smart control system to enhance and manage its operation during experiments.

Li et al. (2023) conducted a simulation of a control system for a three-phase oil separator. Their objective was to develop a safe and efficient alternative to traditional controls. However, they did not examine the impact of operational pressure or liquid level on the separation process. In a similar vein, Wu et al. (2022) developed a computer program to forecast fluid behavior and assess separator performance using different operating parameters. Yet, they too failed to investigate the influence of liquid level on the separation process.

Wang et al. (2021) proposed a model for a control and data acquisition system that aims to enhance the production and operation of an oilfield in China. The authors refined the thermal and hydraulic models using collected data and optimized the key parameters related to thermal characteristics and critical temperature gradient. The model was then employed to forecast the temperature and pressure drop of multiphase flow throughout the pipeline. To integrate the pipeline, surface choke, production well, and oil reservoir into a cohesive system, we also introduced a smart control system that effectively mitigates the adverse effects of slug flow on separator performance.

Hong et al. (2023) conducted a study with the goal of advancing the petroleum industry by examining the impact of interface structure and behavior on fluid flow and phase interactions. They also analyzed the use and regulation of interfaces in the industry, investigated methods for characterizing interface properties, explored factors affecting interface formation and stability through phase interactions, and summarized approaches for inhibiting and destroying interfaces. In addition, we studied the significance of flow regimes and integrated the pipeline, surface choke, production well, and oil reservoir into a unified system. This research specifically focuses on investigating the effect of slug flow on the efficiency of separators.

This study has developed an automated intelligent control system that utilizes a unique correlation method to improve the separation efficiency between different phases. This is accomplished by measuring the desired liquid level of the separator under various operational conditions. The next section will describe the innovative correlation method, the two-phase flow loop, the liquid level controller of the separator, the software discretization, and the coding for system simulation.

Methods

This section provides a step-by-step explanation of the research method, equipment, and materials required. The beta parameter represents the liquid level divided by the separator height. In “Calculation of the beta ratios” section, we present a new correlation for measuring the desired liquid level (beta ratio). In “Two-phase flow loop” section, we describe the two-phase flow loop and its equipment. In “Designing the liquid level controller of the separator” section, we explain the PID control equation used in designing the liquid level controller of the separator. “Software discretization” and “Coding for system simulation” sections showcase the equations of the controller types, software discretization, and coding for system simulation, along with the flowchart of the controller system.

Calculation of the beta ratios

Equations 1 and 2 are used to design horizontal gas–liquid separators for field applications (Fadaei et al. 2021; Stewart and Arnold 2011). In oil production, the pressure in the reservoir decreases, causing a change in the composition of the produced hydrocarbons.

When there are changes in the reservoir pressure and composition of the produced hydrocarbons, it is necessary to adjust the design of the separator. This entails making appropriate changes to its length and diameter. However, in practical terms, it is not always possible to alter the dimensions of the separator. Consequently, the most viable solution is to modify the liquid level of the separator. To determine the new liquid level (beta ratio) of the separator for new separator pressures during production, Eq. (6) can be employed.

$$d \times L_{eff} = k \times \left( {\frac{{TZQ_{ g} }}{P}} \right)\left[ {\left( {\frac{{\rho_{g} }}{{\rho_{l} - \rho_{g} }}} \right)\frac{{C_{D} }}{{d_{m} }}} \right]^{0.5}$$
(1)

The liquid capacity constraint equation for designing horizontal separators with a half liquid level is shown in Eq. (2) (Fadaei et al. 2021; Stewart and Arnold 2011).

$$d^{2} \times L_{eff} = 42,441 \cdot t_{r} \cdot Q_{L}$$
(2)

The beta coefficient is the ratio of separator liquid level to separator height (Eq. 3) (Fadaei et al. 2021; Stewart and Arnold 2011).

$$\upbeta = \frac{h}{d}$$
(3)

The alpha coefficient is defined by Eq. (4) as the ratio of the liquid cross-sectional area to the total cross-sectional area of the horizontal separator.

$$\upalpha = \frac{{A_{Liquid} }}{{A_{total} }}$$
(4)

If the liquid level is the only variable that changes in a horizontal separator, it also affects the separation percentage between the liquid and gas phases, leading to suboptimal performance. Additionally, when the separator pressure changes while all other operational parameters remain constant, the β ratio needs to be adjusted in order to enhance the separation percentage between the liquid and gas phases. The derivation process of Eq. 6 is fully explained in Appendix 1. It is important for the liquid droplet’s transit time to be equal to the settlement time.

$$t_{g} = t_{b}$$
(5)

Equation 6 describes the relationship between alpha, beta, and the θ angle as shown in Fig. 2.

$$\alpha = \frac{{\left( {\uptheta - sin\uptheta } \right)}}{2\pi }\& \beta = \frac{{1 - cos\frac{\uptheta }{2}}}{2}$$
(6)
Fig. 2
figure 2

Cross-sectional area of the separator

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Indices 1 displays the initial separator state, whereas Indices 2 displays the updated separator state. The desired liquid level (β2) is obtained by calculating β2 using Eq. (7) for new separator pressures. all the parameters of the Eq. (7) are described in the Nomenclatures table.

$$\left( {1 + cos\frac{{\uptheta _{2} }}{2}} \right)(2\pi -\uptheta _{2} + sin\uptheta _{2} )^{ - 1} = A$$
(7)

where parameter A is equal to \(\frac{{\left( {1 + cos\frac{{\uptheta _{1} }}{2}} \right)}}{{\left( {2\pi -\uptheta _{1} + sin\uptheta _{1} } \right))}} \times \frac{{p_{2} }}{{p_{1} }} \times \left[ {\frac{{\rho_{g1} }}{{\rho_{g2} }} \times \frac{{\rho_{l} - \rho_{g2} }}{{\rho_{l} - \rho_{g1} }}} \right]^{0.5}\) where all the parameters are described in the Nomenclatures table. The derivation process of Eq. 7 is fully explained in Appendix 1.

Two-phase flow loop

To evaluate the performance of the gas–liquid separator with liquid level control, we designed a two-phase flow loop. We mixed a single-phase water flow with the gas flow using a static mixer, resulting in a two-phase flow. This two-phase flow then moves to form a developed flow. Meanwhile, we measured the air flow rate using an air Rota meter and the pressure using a pressure gauge.

The gas–liquid separator underwent testing with different liquid levels and water/gas flow rates. To capture liquid droplets as small as 20 microns, a filter was placed at the gas outlet. Liquid hold-up was measured using a weighting method, and the separation percentage was determined by calculating the weight of water droplets trapped in the filter. Figure 3 illustrates the two-phase flow loop at the laboratory pilot.

Fig. 3
figure 3

Gas–liquid flow loop

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Designing the liquid level controller of the separator

It is necessary to establish a relationship between the variable h and the changes in liquid volume in the separator. In order to calculate the liquid volume accurately, the cross-sectional area of the liquid needs to be determined.

$$\frac{dh}{{dt}} = - 0.0003526 \times h\left( t \right) - 2.7 \times x\left( t \right)$$
(8)

After using the Laplace transform, the transform function of the system is determined as follows.

$$T\left( s \right) = { }\frac{H\left( s \right)}{{X\left( s \right)}} = \frac{ - 2.7}{{s + 0.0003526{ }}}$$
(9)

If the evaluation of the system using the state space is considered, its state space equations will be as follows.

$$\left\{ {\begin{array}{*{20}l} {\frac{dl}{{dt}} = - 0.0003526 l\left( t \right) - 2.70l x\left( t \right)} \hfill \\ {y = x\left( t \right)} \hfill \\ \end{array} } \right.{ }$$
(10)

Software discretization

In discrete-time control algorithms, the z-transform is used, while in continuous-time, the Laplace transform and s-domain are used. The z-transform plays a role similar to that of the Laplace transform in continuous-time systems. Sampling time is a crucial concept in discrete time control. Inappropriate sampling time can destabilize a digital controller system, even if the analog controller system is stable. The transformation function of a PID controller is as follows (Yang and Zhou 2022; Stanislawski et al. 2022).

$$G\left( s \right) = { }\frac{U\left( s \right)}{{E\left( s \right)}} = { }K_{p} + { }\frac{{K_{i} }}{s} + { }K_{d} s$$
(11)

The above equation is expressed in the simplified z space as follows.

$$G\left( z \right) = \frac{U\left( z \right)}{{E\left( z \right)}} = { }K_{p} + { }\frac{{K_{i} }}{{1 - z^{ - 1} }} + { }K_{d} \left( {1 - z^{ - 1} } \right)$$
(12)

By arranging and doing the addition, the fraction will be as follows.

$$U\left( z \right) = { }\left[ {\frac{{({ }K_{p} + K_{i} { } + K_{d} ) + \left( { - { }K_{p} - 2K_{d} } \right){ }z^{ - 1} { } + { }K_{d} z^{ - 2} }}{{\left( {1 - z^{ - 1} } \right)}}} \right]{ }E\left( z \right)$$
(13)

The above relationship can be written as follows.

$$U\left( z \right) - z^{ - 1} U\left( z \right) = \left[ {(K_{p} + K_{i} + K_{d} ) + \left( { - K_{p} - 2K_{d} } \right){ }z^{ - 1} { } + { }K_{d} z^{ - 2} } \right]{ }E\left( z \right)$$
(14)

By converting to the discrete time, a simplified relationship for the PID controller will be obtained, which can be used in coding programs (Yang and Zhou 2022; Stanislawski et al. 2022).

$$u\left[ k \right] = u\left[ {k - 1} \right] + (K_{p} + K_{i} + K_{d} )e\left[ k \right] - \left( {K_{p} + 2K_{d} } \right)e\left[ {k - 1} \right] + K_{d} e\left[ {k - 2} \right]$$
(15)

As can be seen, the value of the control variable at each moment (u[k]) is equal to its value at the last moment (u[k-1]), the error value at the exact moment (e[k]), the last moment (e[ k-1]) and two moments before (e[k-2]) and depends on the controller coefficients\({(K}_{p}, {K}_{i }, {K}_{d}\)). It should be noted that the effects of sampling time (\({T}_{s}\)) have been neglected for the sake of simplicity.

The sampling time was 1 s. The above calculates a simple PID method in coding. Three common methods to discretize continuous time are forward Euler, backward Euler, and trapezoidal method. The Laplace domain transform function is mentioned to discretize the PID controller.

$$G\left( s \right) = { }\frac{U\left( s \right)}{{E\left( s \right)}} = { }K_{p} + { }\frac{{K_{i} }}{s} + { }K_{d} s$$
(16)

One of the points that can cause the instability of the above controller is the sensitivity of the process and the control system to the derivative term (\({K}_{d}s\)). To reduce this issue, the derivative expression is often written as a low-pass filter according to the following equation so that the system gets less noise.

$$G\left( s \right) = { }\frac{U\left( s \right)}{{E\left( s \right)}} = { }K_{p} + { }\frac{{K_{i} }}{s} + { }\frac{{N{ }K_{d} }}{{1 + { }\frac{N}{s}}}$$
(17)

Considering the three usual methods of discretization above, the equivalent of the integral term \(\frac{{K}_{i}}{s}\) in each method will equal the following equation.

$$\frac{{K_{i} { }T_{s} }}{z - 1}\quad {\text{Forward}}\;{\text{Euler}}\;{\text{method}}$$
(18)
$$\frac{{K_{i} { }T_{s} { }z{ }}}{z - 1}\quad {\text{Backward}}\;{\text{Euler}}\;{\text{method}}$$
(19)
$$\frac{{K_{i} { }T_{s} }}{2}\;\frac{z + 1}{{z - 1}}\quad {\text{Trapezoidal}}\;{\text{method}}$$
(20)

Similarly, the equivalent of the derivative expression \(\frac{{N K}_{d}}{1+ \frac{N}{s}}\) for each discretization, method can be written as follows.

$$\frac{N}{{1 + N{ }T_{s} /\left( {z - 1} \right)}}\quad {\text{or}}\quad \frac{{N\left( {z - 1} \right)}}{{z - 1 + N{ }T_{s} }}\quad {\text{Forward}}\;{\text{Euler}}\;{\text{method}}$$
(21)
$$\frac{N}{{1 + N{ }T_{s} z/\left( {z - 1} \right)}}\quad {\text{or}}\quad \frac{{N\left( {z - 1} \right)}}{{\left( {1 + N{ }T_{s} } \right)z - 1}}\quad {\text{Backward}}\;{\text{Euler}}\;{\text{method}}$$
(22)
$$\frac{N}{{1 + N{ }T_{s} \left( {z + 1} \right)/2\left( {z - 1} \right)}}\quad {\text{or}}\quad \frac{{N\left( {z - 1} \right)}}{{\left( {1 + N{ }T_{s} /2} \right)z + N{ }T_{s} /2 - 1}}\quad {\text{Trapezoidal}}\;{\text{method}}$$
(23)

Then, according to the general relationship of the PID controller (Eq. (16)) for discretization using the above methods, it can be written as follows.

$$G\left( z \right) = \frac{U\left( z \right)}{{E\left( z \right)}} = K_{p} + { }\frac{{K_{i} { }T_{s} }}{z - 1} + { }\frac{{N\left( {z - 1} \right)}}{{z - 1 + N{ }T_{s} }}\quad {\text{Forward}}\;{\text{Euler}}\;{\text{method}}$$
(24)
$$G\left( z \right) = \frac{U\left( z \right)}{{E\left( z \right)}} = K_{p} + { }\frac{{K_{i} { }T_{s} { }z{ }}}{z - 1} + { }\frac{{N\left( {z - 1} \right)}}{{\left( {1 + N{ }T_{s} } \right)z - 1}}\quad {\text{Backward}}\;{\text{Euler}}\;{\text{method}}$$
(25)
$$G\left( z \right) = \frac{U\left( z \right)}{{E\left( z \right)}} = K_{p} + { }\frac{{K_{i} { }T_{s} }}{2}{ }\frac{z + 1}{{z - 1}} + { }\frac{{N\left( {z - 1} \right)}}{{\left( {1 + N{ }T_{s} /2} \right)z + N{ }T_{s} /2 - 1}}\quad {\text{Trapezoidal}}\;{\text{method}}$$
(26)

On the other hand, according to the relationship obtained in Eq. (13), each of the relationships 24–26 is considered as follows (Yang and Zhou 2022; Stanislawski et al. 2022).

$$G\left( z \right) = \frac{U\left( z \right)}{{E\left( z \right)}} = \frac{{b_{0} + b_{1} z^{ - 1} + b_{2} z^{ - 2} }}{{a_{0} + a_{1} z^{ - 1} + a_{2} z^{ - 2} }}$$
(27)

By equating Eq. (27) with Eqs. (24) to (26), we can calculate the coefficients of Eq. (27) for each method. After solving the problem, we obtain the following results for each method. The coefficients resulting from discretization using the forward method are shown below.

$$b0{ } = { }Kp{ } + { }Kd{ * }N$$
(28)
$$b1 = { }Kp{ * }\left( {{ }N{ * }Ts{ } - { }2{ }} \right) + { }Ki{ * }Ts{ } - { }2{ * }Kd{ * }N$$
(29)
$$b2{ } = { }\left( {{ } - Kp{ } + { }\left( {Ki{ * }Ts{ }} \right)} \right){* }\left( {\left( {{ }N{ * }Ts{ }} \right) - { }1{ }} \right) + { }Kd{ * }N$$
(30)
$$a0{ } = { }1$$
(31)
$$a1{ } = { }N{ * }Ts{ } - 2$$
(32)
$$a2{ } = { }1 - { }\left( {{ }N{ * }Ts{ }} \right)$$
(33)

Coefficients resulting from discretization by backward method:

$$b0{ } = { }\left( {{ }Kp{ } + { }\left( {Ki{ * }Ts{ }} \right)} \right){* }\left( {\left( {{ }N{ * }Ts{ }} \right) + { }1{ }} \right) + { }Kd{ * }N$$
(34)
$$b1{ } = { } - Kp{ * }\left( {{ }N{ * }Ts{ } + { }2{ }} \right) - { }Ki{ * }Ts{ } - { }2{ * }Kd{ * }N$$
(35)
$$b2{ } = { }Kp{ } + { }Kd{ * }N$$
(36)
$$a0{ } = { }1{ } + { }N{ * }Ts$$
(37)
$$a1{ } = { } - 2{ } - { }N{ * }Ts$$
(38)
$$a2{ } = { }1$$
(39)

The coefficients resulting from the trapezoidal discretization method are shown as follows.

$$b0{ } = { }Kp{ * }\left( {{ }1{ } + { }\left( {{ }N{ * }\frac{Ts}{2}{ }} \right)} \right) + { }\left( {\left( {{ }Ki{ * }\frac{Ts}{2}{ }} \right){* }\left( {{ }1{ } + { }\left( {{ }N{ * }\frac{Ts}{2}{ }} \right)} \right)} \right) + { }Kd{ * }N$$
(40)
$$b1{ } = { }2{* }\left( {{ } - Kp{ } + { }\left( {{ }Ki{ * }N{ * }Ts{ * }\frac{Ts}{4}} \right) - { }Kd{ * }N{ }} \right)$$
(41)
$$b2 = { }\left( {{ } - Kp{ } + { }\left( {Ki{ * }\frac{Ts}{2}{ }} \right)} \right){* }\left( {\left( {{ }N{ *}\frac{{{ }Ts}}{2}{ }} \right) - { }1{ }} \right) + { }Kd{ * }N{ }$$
(42)
$$a0{ } = 1{ } + { }\left( {{ }N{ * }\frac{Ts}{2}{ }} \right)$$
(43)
$$a1{ } = { } - 2$$
(44)
$$a2{ } = { }1{ } - { }\left( {{ }N{ * }\frac{Ts}{2}{ }} \right)$$
(45)

The control variable is calculated using the discretization methods and conversion to discrete-time domain equations.

$$u\left[ k \right] = - \frac{{a_{1} }}{{a_{0} }}u\left[ {k - 1} \right] - \frac{{a_{2} }}{{a_{0} }}u\left[ {k - 2} \right] + \frac{{b_{0} }}{{a_{0} }}e\left[ k \right] + \frac{{b_{1} }}{{a_{0} }}e\left[ {k - 1} \right] + \frac{{b_{2} }}{{a_{0} }}e\left[ {k - 2} \right]$$
(46)

This equation is used to calculate the control variable for backward, forward, and trapezoidal PID methods. It includes present and past error values as well as past control variable values that affect u[k].

Coding for system simulation

The algorithm for simulating the control system is shown in Fig. 4.

Fig. 4
figure 4

Algorithm of the simulation of the control system

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The algorithm selects fixed and variable values, obtains the liquid height for the new pressure using equations and relationships, and applies it as input to the PID controller for feedback. This section involves opening the control valve to achieve the desired situation. By applying the calculated value to the valve, we can measure the changes in liquid flow rate, volume inside the separator, and water height. These values are then used to repeat the calculations. The simulation is repeated until the end, parameters are saved, and graphs are drawn. Accurate control parameter selection leads to desired results.

Results and discussions

Interpretation of laboratory results, Interpretation of simulation results, Simulating the experimental model using different methods, Simulation results with different inlet liquid flow rates, and Simulation of an integrated realistic production model.

To summarize, in “Interpretation of laboratory results” section emphasizes the importance of designing a control system for the separator level to maintain the desired liquid level, especially when there are fluctuations in pressure. It is also crucial to determine the optimal separator pressure for maximizing oil production in the stock tank, assuming the inlet flow regime is not a slug.

The performance of the liquid level controller was first evaluated on the pilot gas–liquid separator in the laboratory. Afterwards, the proposed integrated model, which incorporates liquid level control, will be utilized in a real reservoir simulation process. The objective is to effectively eliminate slug flow prior to the separator and increase the total oil barrels in the stock tank.

In “Interpretation of simulation results” section focuses on different forms of the system controller, namely (P), (I), (D), (PI), (PD), and (PID), which were utilized in simulations to analyze the error curve of the system. Out of these controllers, the PD and PID controllers demonstrated superior performance, unlike the P controller, which displayed steady-state error in the system response. In order to determine the optimal controller, a simulation was conducted using various input flow values for the separator, resulting in the following obtained results. The PID controller was chosen for controller design due to its enhanced response. The control parameters obtained at this stage are as follows.

In “Simulating the experimental model using different methods” section, it is explained that the trapezoidal method is faster in the initial stage of control and demonstrates less error and overshoot when reaching the desired value, as shown in Fig. 14. In summary, it has been observed that the trapezoidal method is superior to other models for level controller design in both laboratory and field applications.

In “Simulation results with different inlet liquid flow rates” section, it was demonstrated that the trapezoidal PID controller, as shown in Table 6, has a minor average error over time and effectively maintains the liquid level at the desired value for different input flow rates and the input slug flow regime. Additionally, the trapezoidal controller has a separation percentage of 99.1%, which is considerably higher than other controller types, as stated in “Two-phase flow loop” section. Based on these results, it can be concluded that the trapezoidal controller successfully controls and maintains the liquid level of the separator at the desired levels during slug flow regime, as well as other flow regimes like bubble, smooth, wavy stratified, and plug. After successful validation of the intelligent model in the laboratory, an integrated model, as depicted in Fig. 20, was implemented. This integrated model utilizes both the level controller model and the separator model.

In “Simulation of an integrated realistic production model” section, we demonstrated that implementing PID trapezoidal level control, coupled with the integrated production model in a hardware-in-the-loop environment, resulted in a total oil production of 1428 barrels. In comparison, production without level control yielded 1263.3 barrels. It is worth noting that there was no slug flow observed before the separator. By utilizing PID trapezoidal level control, in conjunction with the integrated production model in a hardware-in-the-loop environment, we can potentially increase the stock tank’s total oil production by approximately 13 percent over a span of 840 days.

In “Simulation results with different inlet liquid flow rates” section, the simulation results show that the control system can operate effectively under different inlet liquid flow rates and flow regimes. It is clear from this section that the control system can successfully minimize fluctuations in the liquid level when the slug flow regime is present. This ensures optimal phase separation by kee** the liquid level at its desired value.

Interpretation of laboratory results

Fadaei et al. (2021) proposed a new relationship for designing multiphase separators. They found different coefficient values for varying liquid levels (β ratios) in the separator. Table 1 shows the correlation between the β ratio and gas constraint coefficient (k) in Eq. (1). It demonstrates that the β ratio and gas constraint coefficient have a varied relationship (Fadaei et al. 2021). The maximum coefficient is 65.5 at a β ratio of 0.05, while the minimum is 6.9 at a β ratio of 0.9. The values in Table 1 are used in both relationships 1 and 2. These two relationships are then utilized to determine the optimal level of the separator under new temperature and pressure conditions, as described in Eq. 6. It is important to note that the results presented in Table 1 have not been reported by any previous researcher. The findings in Table 1 can be applied to the design of multiphase separators and the modification of pressure and surface control equations used in industrial applications (Fadaei et al. 2021).

Table 1 The gas constraint coefficient of the semi-empirical correlation for various liquid levels of the separator
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Table 2 displays the laboratory investigation results, including the parameters for Eqs. 1 to 6, which are measured under different operating conditions of the laboratory separator. Furthermore, the table illustrates the necessary change in the liquid level in order to improve the separation efficiency of the separator.

Table 2 Results of the Eqs. (1) and (6) for various separator pressures
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Table 2 demonstrates the results of the experiments carried out in this study. For a separator pressure of 101.22 kPa, with a gas flow rate of 8 m3/h, a water flow rate of 1 m3/h, and a β ratio of 0.1, the separation percentage for the liquid and gas phases is 87.3%, with a bubble flow regime before the separator, as depicted in Fig. 5a.

Fig. 5
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Flow regimes before the separator in the laboratory pilot

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However, as the separator pressure increases to approximately 151.8 kPa, the β ratio should also increase to around 0.14 to maintain the maximum separation percentage. The increased liquid level inside the separator traps more liquid drops, increasing the separation percentage, while the flow regime before the separator remains bubble flow.

For a separator pressure of 202.45 kPa, with a gas flow rate of 23.6 m3/h, a water flow rate of 1 m3/h, and a β ratio of 0.1, the separation percentage for the liquid and gas phases is 88.1%. As the separator pressure increases to around 405 kPa, the β ratio should also increase to around 0.17 to maintain the maximum separation percentage and the flow regime before the separator changes from stratified wavy (Fig. 5c) to stratified smooth (Fig. 5d).

Furthermore, for a separator pressure of 101.22 kPa, with a gas flow rate of 18.9 m3/h, a water flow rate of 1 m3/h, and a β ratio of 0.5, the separation percentage for the liquid and gas phases is 99.6%, and the flow regime before the separator is Stratified wavy (Fig. 5e). However, as the separator pressure increases to around 151.8 kPa, the β ratio should increase to around 0.56 to maintain the maximum separation percentage and the flow regime before the separator changes to Stratified smooth (Fig. 5f). Conversely, as the separator pressure decreases to around 90 kPa, the β ratio should decrease to around 0.48 to maintain the maximum separation percentage and the flow regime before the separator changes to Stratified wavy (Fig. 5g).

At a gas flow rate of 17 m3/h and separator pressure of 220 kPa, with a water flow rate of 1 m3/h and a β ratio of 0.54, the liquid and gas phases exhibit a separation percentage of 99.96, and the flow regime before the separator is slug (Fig. 5j). To maintain this maximum separation percentage with an increase in separator pressure to approximately 300 kPa, the β ratio must be increased to 0.58, and the flow regime before the separator changes to stratified wavy. It was found that increasing the separator pressure can eliminate the slug flow regime at the separator inlet.

As indicated by Table 2, it can be inferred that while kee** all operational parameters of the horizontal separator constant except the separator pressure, the suitable liquid level for achieving maximum separation percentage can be influenced. Specifically, if the separator pressure increases, the suitable liquid level must be raised to ensure maximum separation percentage. Conversely, if the separator pressure decreases, the liquid level should be lowered to preserve separator performance and achieve maximum separation percentage. Notably, the liquid constraint specified in Eq. (2) is satisfied in all cases. None of the researchers have yet mentioned the correlation between the liquid level of the separator and the pressure and temperature of the separator with the percentage of phase separation, as indicated in the literature review.

According to the expressed experimental results, the integrated modeling of the separator with before facilities like pipeline, choke, and down-hole equipment is necessary because of changes in the reservoir pressure. Another novelty in this paper is investigating the slug flow formation and eliminating it by changing the separator pressure. Changing the separator pressure can eliminate or intensify the slug flow at the separator inlet and increase or decrease the suitable liquid level for maximum separation. Determining the best separator pressure and liquid level at each time step is necessary to maximize the total oil production in the stock tank.

The results from the experiments show that as the gas flow rate increases, there is a transition in the flow regime from stratified-smooth to stratified-wavy, as depicted in Fig. 5c. On the other hand, when the separator pressure increases, the flow regime transitions from stratified-wavy (Fig. 5c) to stratified-smooth. Additionally, the flow regime changes from stratified-wavy to slug flow as the gas flow rate increases, as seen in Fig. 5i.

On the other hand, if the separator pressure is decreased while kee** the water flow rate and gas flow rate constant, the flow regime transitions from a slug flow to a more severe slug flow, as shown in Fig. 5j. Based on the experimental results, when the slug flow is dominant at the separator inlet, increasing the separator pressure eliminates it, while decreasing the separator pressure intensifies it.

Designing a control system for the separator level is crucial for maintaining the desired liquid level, especially when the pressure in the separator fluctuates. Additionally, it is important to identify the optimal separator pressure for maximizing oil production in the stock tank, provided that the inlet flow regime is not a slug.

The performance of the designed liquid level controller was initially investigated on the pilot gas–liquid separator at the laboratory scale. Subsequently, the proposed novel integrated model, which includes liquid level control, will be applied in a real reservoir simulation process. The aim is to effectively eliminate slug flow before the separator and enhance the total oil barrels in the stock tank.

Interpretation of simulation results

This section provides a thorough explanation of all simulation results. In order to better understand the impact of the controller on a system or process, the behavior of the system in open-loop mode is first examined. Then, the effects of a preliminary closed-loop controller are explored, followed by the effects of the optimal and appropriate controller. The open-loop model of the obtained relationship was simulated and shown in Fig. 6.

Fig. 6
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Open loop model of the control system

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The equation of the Fig. 5 is dh/dt\(=-0.0003526 h\left(t\right)-2.7 x\left(t\right)+0.0795 {Q}_{in}\left(t\right)+1.474\). The separator system only works properly for specific initial values. It has a low output error when the control valve opening ratio is 0.55, but for other values, there is little to no control. The closed-loop model of the obtained relationship was simulated and shown in Fig. 7.

Fig. 7
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Closed loop model of the control system

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The system controller was simulated using various forms such as (P), (I), (D), (PI), (PD), and (PID), and the error curve of the system was determined. The PD and PID controllers demonstrated superior performance compared to the other controllers, exhibiting steady state error in the system response with the P controller. In order to identify the optimal controller, a simulation was conducted with different input flow values for the separator, yielding the obtained results. The PID controller was selected for designing the controller due to its improved response. The control parameters obtained at this stage are as follows.

$$\begin{aligned} K_{p} & = { } - 0.00233 \\ K_{i} & = { } - 2.04{\text{ exp}}\left( { - 6} \right) \\ K_{d} & = { } - 0.032 \\ \end{aligned}$$

Simulating the experimental model using different methods

The PID controller system consists of the computational system, actuators, and automatic valves installed on the horizontal separator. Next, the horizontal separator was placed in the gas–liquid flow loop and analyzed. The system was then simulated, allowing for the testing of various control parameters for different states, in order to determine the optimal controller design through a trial and error approach. The selected control values will be applied during practical tests as the values and parameters of the control system, enabling the measurement of the simulation’s validity and results in real-world scenarios. In the system simulation, four different models of PID controllers (simple PID, forward PID, trapezoidal PID, and backward PID) were used to select control parameters for each model, which were then applied in actual tests. The control parameters obtained for each model are described in Table 3.

Table 3 Parameters obtained from simulation for different control models
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As a fundamental topic in control theory, the optimal control system for a plant is the inverse of the system’s transfer function. This implies that if the system has a substantial integral component, the controller will have a small integral component, and the same applies to other components. The outcomes of each type of controller are illustrated below.

Simple PID controller

According to Fig. 8, the simple PID controller could control the liquid level of the separator and change the valve opening as soon as possible, which has a slight delay time. The value of the control variable at each moment (u[k]) is obtained using Eq. (15) (Fig. 9).

Fig. 8
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The results of the test with a simple PID controller

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Fig. 9
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The results of the test with the forward PID controller

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Forward PID controller

The coefficients for the transfer function shown in Eq. (27) are obtained from Eqs. (28) to (33) for the forward PID controller.

Backward PID controller

The coefficients of the transfer function shown in Eq. (27) are obtained from Eqs. (34) to (39) for the backward PID controller.

Trapezoidal PID controller

As shown in Fig. 11, the height of the pilot gas–liquid separator is 25 cm and it is initially empty. The desired liquid level, or set point, was determined using Eq. 6. The coefficients of the transfer function, as presented in Eq. (27), were obtained from Eqs. (40) to (45) for the trapezoidal PID controller. After inputting these values into the simulation codes, Fig. 10 compares the results of all four models used in the test. From this comparison, the following conclusions can be drawn (Fig. 11).

Fig. 10
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The results of the test with the backward PID controller

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Fig. 11
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The results of the test with trapezoidal PID controller

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In the simple method, the system’s average error over time is significant, and it reaches the desired value faster than other methods. However, due to its significant overshoot, the desired response cannot be evaluated in this application (Fig. 12).

Fig. 12
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Comparison of test results with different controller models

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As shown in Fig. 13, the forward method demonstrates acceptable response speed and behavior when reaching the desired value. However, the system’s behavior is even better when using the trapezoidal method to reach the desired value.

Fig. 13
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System behavior at the beginning of the system control process

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In the system control process, initially, the trapezoidal response closely resembles the forward response. However, over time, the trapezoidal response gradually becomes more similar to the backward response. When comparing various methods, it is generally observed that the trapezoidal method exhibits the speed of the forward response at the start of control, along with the low error and minimal overshoot of the backward response when reaching the desired value, as depicted in Fig. 14.

Fig. 14
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The average error rate of the system

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The trapezoidal method is faster in the initial stage of control and exhibits less error and overshoot when reaching the desired value, as depicted in Fig. 14. In conclusion, it has been observed that the trapezoidal method outperforms other models for level controller design in both laboratory and field applications.

Simulation results with different inlet liquid flow rates

To analyze the capability of the liquid level controller in the hardware-in-the-loop simulation for the gas–liquid flow loop in the laboratory pilot, we changed the flow rate of the inlet water flow temporarily. Table 4 displays the flow rates observed during the 6000-s test. If the liquid level controller of the separator demonstrates excellent performance in controlling the level of different inlet liquid flow rates, the separator will not experience over-flushing during slug flow. Instead, its level will remain consistent and near the optimum determined level.

Table 4 Water flow rates during the simulation
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The graphs obtained for each test method are based on the simulation. All other specifications and parameters remain the same. The values of the control parameters depend on the physics and nature of the system being designed. To obtain these values, the following points should be considered.

Comparison of methods in simulation with different inlet flow rates

To test the effectiveness of the smart control system in minimizing the negative impacts of slug flow entering the separator, the inlet flow rate was deliberately adjusted, both increased and decreased. This was done to measure the system’s ability and efficiency in maintaining and controlling the liquid level at a desired value. The results obtained for each type of controller are shown below.

According to Fig. 15, the system is stable, and there are multiple changes associated with the level control. The position of the control valve corresponds to abrupt changes and fluctuations. The liquid level is significantly affected by a dramatic change in the liquid inlet flow rate. According to Fig. 16, the system is stable, and level control is related to changes. The position of the control valve is also related to changes and fluctuations. The liquid level is affected by alterations in the flow rate of liquid input.

Fig. 15
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Test result of 6000 s with variable flow rates for simple PID controller

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Fig. 16
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Test result of 6000 s with variable flow rates for forward PID controller

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According to Fig. 17, the system is stable, and there is a correlation between level control and changes. The position of the control valve is also linked to changes and fluctuations. The liquid level is impacted by variations in the flow rate of the liquid input. According to Fig. 18, the system is stable, and level control is linked to quantitative changes. The position of the control valve is not related to sudden fluctuations. Making slight changes in the liquid inlet flow rate has a slight impact on the liquid level.

Fig. 17
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Test result of 6000 s with variable flow rates for backward PID controller

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Fig. 18
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Test result of 6000 s with variable flow rates for trapezoidal PID controller

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The comparison of the system’s response to the simulation by applying flow rate changes is as follows.

It can be seen from Fig. 19 that the average error over time of the simple method is higher at lower flow rates. In Table 5, the summary of test results is shown.

Fig. 19
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The response of different models to changes in the inlet flow to the separator

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Table 5 Summary of test results
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As shown in Table 6, the trapezoidal PID controller demonstrates a minor average error over time and effectively maintains the liquid level at the desired value for various input flow rates and the input slug flow regime. The separation percentage, as indicated in “Two-phase flow loop” section, for the trapezoidal controller is 99.1%, considerably higher than other controller types. Based on the results, the trapezoidal controller successfully controls and maintains the liquid level of the separator at the desired levels during slug flow regime, as well as other flow regimes such as bubble, smooth, wavy stratified, and plug. Following the successful validation of the intelligent model in the laboratory, an integrated model, as shown in Fig. 20, was implemented. The level controller model, along with the separator model, was employed in this integrated model.

Table 6 Summary of the results of the tests
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Fig. 20
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The integrated production model

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Simulation of an integrated realistic production model

In this section, an integrated model of the oil reservoir, well, surface choke, pipeline, and separator was developed. The goal was to assess the potential increase in oil production in the storage tank with the implementation of the proposed control system. Subsequently, an integrated model encompassing all the essential components, including the oil reservoir, production wells, surface chokes, production manifold, and horizontal separator, was created. Additionally, an accurate surface control model of the separator and stock tank was included. It is worth noting that, due to the availability of comprehensive information and a realistic model, uncertainty was not taken into account during the modeling process.

The actual onshore oil field in Iran contains a 45 ft thick layer of sand, which is covered by a layer of shale. Initially, the reservoir was completely filled with gas and a thin layer of oil. Through geophysical analysis, a gas–water contact was determined. The field consists of three appraisal wells and two production wells. Originally, the gas-oil contact was discovered at a depth of 8344.3 ft and the water–oil contact was found at a depth of 9213 ft TVD.

The field was simulated using a black-oil model. The oil gravity, oil formation volume factor, oil viscosity, and undersaturated compressibility was 35° API, 1.4 RB/STB, 0.29 cp and 1.34×\({10}^{-5}\) \({psia}^{-1}\), respectively. Also, initial bubble point: 3885 psi, initial reservoir pressure: 4650 psi, GOR: 1 Mscf/STB in oil rim. The Vogel correlation was used as the reservoir sub-model, and its output feeds the well sub-model. The Duns-Ross correlation was used as a well sub-model. Its output feeds the choke sub-model. The output of the choke sub-model feeds the pipeline sub-model. The Beggs–Brill correlation was used as a pipeline sub-model, and its output feeds the separator with the level controller sub-model. The surface pipelines were 4 inches in diameter and 1335 m long in total. Figure 20 shows the integrated production model.

The model was run for 20,000 h to determine the amount of oil that was produced and stored in the tank. It is worth mentioning that the trapezoidal model, which performed better than other control models in laboratory mode, also showed superior performance in the modeling mode. Table 6 displays variations in valve position, steady-state error, and overshoot values for different types of controllers.

Table 6 shows that the trapezoidal PID control type yields the best results. The PID gains for the trapezoidal model were − 15, − 0.2 and − 7 for \({k}_{p}, {k}_{i} and {k}_{d}\), respectively. Using the trapezoidal discretization method is recommended for the horizontal separator level controller as it provides the best response with minimum activity from the control valve, resulting in a longer lifetime for the control valve and other controlling devices. During 2×\({10}^{4}\) hours of production, Fig. 21 illustrates the changes in liquid level and valve opening percentage.

Fig. 21
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The test result with the trapezoidal PID method

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The system is stable, with relatively small changes in the liquid level. The position of the control valve experiences minimal changes. The results of all the aforementioned methods are summarized in Table 6. Table 7 displays the total amount of oil produced and stored in the stock tank.

Table 7 Total produced oil in the stock tank
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Based on Table 7, the stock tank’s total oil production with PID trapezoidal level control coupled to the integrated production model in the hardware-in-the-loop environment is 1428 barrels for simulation with level control and 1263.3 barrels for production without level control. It is important to note that there was no slug flow before the separator. By using PID trapezoidal level control coupled with an integrated production model in the hardware-in-the-loop environment, it is possible to increase the stock tank’s total oil production by approximately 13 percent.

Conclusions

This study presents a novel approach to optimize oil production from oil reservoirs by introducing a new correlation for real-time adjustment of the liquid level in the separator. This approach offers significant economic benefits as it eliminates the need to modify the separator size during oil production.

By combining the proposed correlation with a PID separator controller, slug flow can be eliminated before the separator, while simultaneously maximizing oil production in the stock tank.

Other notable achievements of this study include:

  • Elimination of the slug flow regime before the separator through an increase in separator pressure.

  • Development of a smart control system capable of effectively operating under varying inlet liquid flow rates and flow regimes in both operational and laboratory conditions.

  • Demonstration of the control system’s ability to greatly reduce liquid level fluctuations during slug flow.

  • Evidence of the smart control system’s effectiveness in increasing oil production in real oil fields. For instance, by simulating the integrated production model coupled with the smart control system for approximately 840 days, a 13% increase in oil production was observed.