A novel analytical technique for determining inflow control devices flow area in CO2-EOR and CCUS projects

CO2-EOR is one of the principal techniques for enhanced oil recovery (EOR). The CO2 injection not only promotes oil recovery but also leads to greenhouse gas discharge reduction. Nonetheless, a key challenge in the CO2 flooding process is a premature CO2 breakthrough from highly permeable zones. In recent years, Inflow Control Devices, ICDs, have been used as a potential solution to mitigate an early gas breakthrough. The key and important parameter in ICDs installation is obtaining its opening flow area. The common ways to obtain the ICD flow area such as utilizing optimization algorithms are very complicated and time-consuming, and further these methods are not analytical. The aim of this work is to solve the mentioned challenges—postpone the breakthrough time in gas injection and present an easy, fast, and analytical technique for obtaining ICDs flow area. This paper presents a new analytical method for obtaining inflow control devices flow area for injection wells in an oil reservoir under CO2-EOR in order to balance the injected CO2 front movement in all layers. Then, in order to compare the advantages and disadvantages of the presented technique with other methods such as optimization algorithms, a case study has been done on a real reservoir model under CO2 injection. Later, the results of studied scenarios in the case studied are given and compared. The results show that by utilizing the proposed method recovery factor is raised by improving sweep efficiency, and the breakthrough time is more postponed compared to the other methods about 400 days. Further, the ICD flow area calculation takes 2 min by presented analytical techniques, but the optimization algorithm takes 4040 min to run the simulation model to find the ICD flow area. In the end, the findings of the presented analytical formula can help to set the ICD flow area very fast without the simulation and help researchers for a better quantitative understanding of parameters affecting the ICD flow area by the given formula such as reservoir permeability.


CO 2 -EOR mechanism
The CO 2 injection into oil reservoirs is a commonly used approach for carbon capture, utilization, and storage (CCUS) projects in order to reduce greenhouse gases and enhanced oil recovery. CO 2 injection efficiency is reliant on CO 2 miscibility in oil (Zhang et al. 2018). In an oil reservoir containing a significant amount of light hydrocarbons during the injection of CO 2 , the oil light hydrocarbons dissolve within the CO 2 , and CO 2 dissolves in the oil. Therefore, the oil viscosity reduces significantly (Zhang et al. 2015).
Oil viscosity reduction causes an improvement in oil mobility, which decrease the residual oil saturation in the reservoir and enhanced oil recovery (Li et al. 2013). The CO 2 dissolution in oil at specific reservoir conditions such as oil compositions, temperature, and pressure, provokes the oil to swell, which plays an essential role in attaining better oil recovery. Swelled oil droplets force oils-initially unable to produce-to get out of the pores and swipe toward the production well. Therefore, the residual oil saturation decreases (Perera et al. 2016).

CO 2 -EOR challenges
Although the injection of CO 2 reduces greenhouse gases and increases oil recovery, it is typically utilized in carbonate reservoirs, which usually have low permeability. However, the reservoirs usually include zones with high permeability (Siqueira et al. 2017) and fractures in reservoir layers (Dejam andHassanzadeh 2018b). Therefore, utilizing CO 2 injection in EOR projects have significant problems and challenges with CO 2 short-circuiting between injection and production wells and the early breakthrough of CO 2 in high permeable, thief zones, and aquifer (Dejam and Hassanzadeh 2018a). Hence, injected CO 2 production prevents oil production from the remaining layers, and significant volumes of oil will remain in the reservoir (Yang et al. 2019). Thus, having a well-balanced injected CO 2 influx is required to maximize oil production.
Controlling CO 2 injection in the high permeable and thief zones, breakthrough layers, where the CO 2 early production happens, yields a better CO 2 distribution in the reservoir, which improves the oil recovery (Yu et al. 2014). To manage the breakthrough time in reservoirs is required to control each layer individually in the well; therefore, advanced completion with inflow control devices can be employed (Mohammadpourmarzbali et al. 2019).

Inflow control device
Inflow Control Devices are a conventional type of advanced completions that present passive inflow control. ICDs are broadly utilized and can be a perfected well completion technology (Ugwu and Moldestad 2018). Inflow Control Devices have been utilized to balance the injected fluid influx by making extra backpressure in the layers produce excess fluid at tremendous rates (Ratterman et al. 2005). Employing ICDs can delay CO 2 breakthroughs and maintain a balanced flow. The ICDs have to be designed based on the reservoir properties (Rahimbakhsh and Rafiei 2018) to manage CO 2 flow and make a better CO 2 distribution in the reservoirs, providing high-quality CO 2 storage in the reservoir during the oil production time. Brouwer et al. (2001) managed the injected fluid in a high-degree heterogeneous field with a horizontal injection well by using control valves to prevent injection fluid breakthrough and maximize recovery by a simple algorithm in two scenarios, constant flow scenario and constant pressure scenario (Brouwer et al. 2001). Brouwer et al. (2004) developed a closed-loop method to optimize the flooding process by maximizing the net present value (NPV) (Brouwer et al. 2004). Naus et al. (2004) formed an operational approach for commingled production with infinitely changeable ICV using sequential linear programming for short-term production optimization (Naus et al. 2004). Alhuthali et al. (2010) proposed a rate control technique for optimizing water flood in an intelligent well containing ICVs (Alhuthali et al. 2010). Essen et al. (2010) propose a workflow based on a gradient-based optimization technique in order to predict the production and injection rate of inflow control valves in horizontal wells (Van Essen et al. 2010). Hassanabadi et al. (2012) adjusted the ICD flow area by using particle swarm optimization and the neural smart system to maximize the cumulative oil production and minimize cumulative water production. In this study, the algorithm has been implemented separately for all valves (Hassanabadi et al. 2012). Fonseca et al. (2015) studied ensemble multi-objective production optimization of on-off inflow control devices on a real-field case by a modified net present value (Fonseca et al. 2015). Chen and Reynolds (2017) optimize ICV settings and well controls concurrently to maximize NPV. In this study, the NPV achieved by this method was compared with two other scenarios NPVs, NPV achieved by only well control optimization, and NPV achieved by only ICV settings optimization (Chen and Reynolds 2017). In 2018, Aakre investigated the performance of the autonomous inflow control valves in injecting CO 2 into the reservoir. This is the first time in the world that autonomous inflow control valves are used in CO 2 -EOR operations (Aakre et al. 2018). Cao et al. (2019) presented a novel well fluid modeling for heterogeneous reservoirs in order to accurate the simulation result (Cao et al. 2019). Salvesen et al. (2020) simulated CO2-EOR utilizing the OLGA in combination with ROCX by employing autonomous inflow control valves in wells (Salvesen Holte et al. 2020). Safaei presented a new method in order to accurate CO2 and brine interfacial tension modeling (Safaei-Farouji et al. 2022b) and investigated the CO2 trapping via machine learning (Safaei-Farouji et al. 2022a) to improve and enhance CO2 storage efficiency in underground reservoirs.
Literature review on the CO 2 -EOR and use of ICDs in the CO 2 injection process reveals that the CO 2 short cycle and premature breakthrough time is an essential problem that can be mitigated by inflow control devices. Furthermore, an important question is what will be the required flow area of the ICDs in order to control fluid injection or production in different layers in which most of the techniques are based on optimization algorithms. To do so, a reliable system model is required which is not available most of the time. In addition, solving optimization algorithms for reservoirs with many wells and complex system models is very time-consuming. Likewise, there is not much effort to develop analytical techniques to calculate injected fluid rate and flow area at the same time in order to obtain ICDs flow area only by using properties of the system independent of the system's complex structure. In other words, previously performed studies were not based on an analytical formula; hence, the investigation and effect of parameters such as reservoir parameters were difficult. As a result, the current research represents a novel analytical technique to calculate injected CO2 rate and flow area in order to maximize breakthrough time and improve oil production.
This research presents a new analytical formula for obtaining the ICDs flow area. First, the design and simulation section is given in which the new formula is derived. Then, scenarios are defined in order to compare the advantages and disadvantages of the proposed methods with other methods. Finally, the results section is presented.

Design and simulation
This study initially aims to develop a new analytical technique to determine the ICDs flow area in order to balance the gasfront movement in reservoir layers with different permeability to maximize breakthrough time. In this method, at first in the part 1, by utilizing a fluid flow equation in porous media, an equation is developed in which the gasfront velocity in a high permeable layer reduces to the gas front velocity of the injected gas in a low permeable layer. Second in the part 2 and part 3, this equation is combined by 2 different equations which are calculated pressure drop due to the ICD. Finally, by considering these 3 equations, the analytical technique is developed for calculating the ICD flow area in which gasfront moves by the same velocity in low and high permeable layers.

Part 1
Assuming a reservoir with 2 layers each having (Fig. 1) different permeability and thickness, the ICD will set up at a high permeable layer (layer 1 in this example). The injection inflow equation (for CO 2 gas injection) from the well to the reservoir for each layer is as follows; assumed that injection is a piston-like gas flooding, constant reservoir properties during the injection periods, the injection well is a vertical well, and steady-state flow (Fetkovich 1975): where q i is the injection rate in layer i; h i and K i are the thickness and permeability of layer i respectively;s , i is the sum of total Darcy skin and non-Darcy flow skin of layer i; µ is CO 2 viscosity; P e is reservoir pressure; and P wi is CO 2 injection pressure at sand face for layer i.
Front fluid flow velocity for each layer can be obtained dividing the flow rate (Eq. 1) by the cross-section area for each layer: where V i is the front velocity in layer i, and r i is the injected CO 2 front radius from the injection well in layer i (Fig. 2).
In order to delay the CO 2 breakthrough time in a layer with higher permeability, the CO 2 front velocity in the high permeable layer should reduce to the CO 2 front velocity in the low permeable layer in order to make the CO 2 front in each layer reach the production well at the same time. Hence, the front velocity in each layer should be equal (Eq. 4), this means that the front radius at each time in both layers is the same (Eq. 5).
By substituting Eq. (2) in Eq. (4): where P w2 is sand face pressure for layer 2 which is equal to P bh2 (as there is no ICD in layer 2) , and P w1 is ICD outlet pressure, or sand face pressure, in layer 1.
Since it is important to know P w1 , by solving Eq. (6) for P w1 : To obtain the cross-section area in Eq. (7) which is unknown, Eqs. (3) and (5) is combined to obtain the cross-section:

Fig. 2
Injected gas front radius in a two-layer reservoir Now, Eq. (7) can be re-write in order to eliminate the cross-section:

Part 2
As shown in Fig. 3, the pressure loss across the ICD is as follows.
where P bh1 is bottom hole injected pressure in layer 1, and P w1 is ICD outlet pressure in layer 1 calculated from Eq. (9), so:

Part 3
Also, the pressure loss equation for ICD is as follows (Geo-Quest 2014): where C ′ u is unit constant, C v is ICD constant, A c is the ICD cross-section, A p is the well cross-section, D is well diameter, f is the Fanning friction factor, and ρ is the fluid density.
By the combination of Eqs. (11) and (12): By solving Eq. (13) for A c : For simplifying the above equation: where: where B can represent sandface pressure, D can represent acceleration term, and C can represent friction term.
To maximize the CO 2 breakthrough time, the ICDs flow area can be adjusted according to Eq. (15) in which the CO 2 front moves in the high permeable layer at the same velocity as in the low permeable layer.

Case study
In order to compare the advantages and disadvantages of the proposed method to other methods, four scenarios have been studied in this research. At first, a base scenario is defined. Then two scenarios are defined with ICD installation in a high permeable layer-in these two scenarios, the ICD flow area is calculated with the optimization algorithm and the proposed method, respectively. Finally, the sensitivity analysis is done on the ICD flow area. And, the results of all scenarios are compared in the result section. It should be noted that in all scenarios initial condition, reservoir property, injection rate, and production rate are the same-only the ICD flow area is different.
The reservoir model under this study for all scenarios consists of 1 producer and 1 injector. It has 2 appropriate reservoir layers for CO 2 injection (layers 7 and 8).
The location of the wells has been shown in Fig. 4, and the properties of the reservoir are given in Table 1. Reservoir fluid is heavy oil (Table 2) and is suitable for CO2 flooding. Furthermore, For the simulation of the CO 2 injection, the compositional simulator is utilized.

Base scenario
In this scenario, the production well was producing with constant wellhead pressure, and constant CO 2 injected rate at the injection well as shown in Table 3.

Optimization algorithm
Numerous algorithms which are classified into three classes, including approximate, exact, or heuristic/metaheuristic, are utilized to determine the optimal solutions. During the past decade, optimization problems have been resolved by metaheuristic algorithms.
GWO is a recently advised swarm-based metaheuristic. It was developed and suggested by Mirjalili (Mirjalili et al. Fig. 4 Reservoir model in this study (colours representing oil saturation)   Well head pressure = 250 psia CO 2 injection rate = 6000 Mscf/day 2014). It is stimulated by the hunting behavior and leadership of grey wolves in the environment. The population is classified into four types, including alpha (α), beta (β), delta (δ), and omega (ω) in this algorithm. The three most appropriate wolves are recognized as alpha, beta, and delta leading other wolves, or omega to the suitable search space areas. The wolves update their locations around alpha, beta, or delta during optimization (Amar et al. 2018). Also, the GWO algorithm's general steps are shown in Fig. 5: The objective function was maximizing the NPV (which can be expressed by Eq. 19) by changing the ICD flow area. In this scenario, the injection and production conditions were the same as the base scenario (Table 3), and GOW is used to find the optimum ICD flow area. The determined values of the ICDs flow area can be seen in Table 4. where Q o, Q w, and Q g are the oil, water, and gas production rate respectively; Q i denotes the gas injection rate; r o is the oil price; r w and r g are the cost due to the water and gas handling respectively; r i is the injected gas cost; b is the annual discount rate, and C o is constant cost.

Proposed method
In this scenario injection and production scenario was the same as the base scenario (Table 3), but ICD has been used in the high permeable layer (layer 8). Then, the developed analytical technique is employed to obtain the ICDs flow area.
The CO 2 injection rate in Table 3 and the Parameters in Table 5 have been used to calculate the ICDs flow area using Eq. (15). The determined values of the ICDs flow area can be seen in Table 6.

Further investigation
For further and better investigation and analysis, a variety of different ICD flow areas, including 0.00001, 0.0001, 0.0002, 0.0003, 0.0004, 0.0007, 0.001, and 0.005 ft 2 have been used in high permeable layer (layer 8). In these scenarios, injection and production conditions were the same as in the base scenario (Table 3). Figure 6 compares the CO 2 injection profile between both layers-high and low permeable layers-in all scenarios at the same time interval (after 9 years of injection). The   proposed analytical method has been able to balance the velocity of the fluid front in each of the two layers, and CO 2 breakthrough time in layers 8 and 7 happens at the same time. In fact, the proposed method was able to balance the injection fluid distribution in both layers by reducing the injection flow in the high permeability layer and increasing the injection flow in the low permeability layer. It means that more oil was displaced toward the production well in layer 7 and left less oil behind in layer 8. So, sweep efficiency was improved, and more oil was produced. Meanwhile, in other Fig. 6 Comparison of CO 2 front radius for all scenarios after 9 years of injection in layers 7 and 8 (colours representing CO 2 Fluid-in-place) scenarios, the short-circuiting of CO 2 between injection and production wells happens. In other words, CO 2 cannot be stored in the reservoir for the CCUS and EOR applications. For instance, to clarify and illuminate the CO 2 breakthrough, consider the distribution of injected CO 2 in a scenario with Ac = 0.0001 ft 2 . The orange line, as shown in the top-right subplot, represents the injected CO 2 front, where the CO 2 fluid-in-place value suddenly drops to zero. As can be seen, injected CO 2 front reached the production well and moved much faster in layer 7 compared to layer 8, and CO 2 breakthrough happened.

Result and discussion
As can be seen in Fig. 7, before breakthrough time, cumulative oil production at the base scenario and other scenarios are a bit higher compared to the proposed method scenario. On the other hand, after the CO 2 breakthrough time, it can be seen that more cumulative oil is produced in the proposed analytical method. In other words, although at the beginning other scenarios apparently worked better, the proposed method scenario produces more oil at the end and indicated that the proposed method can be applied for better management and improvement, which is a signature of improvement in the CO 2 injection efficiency (Fig. 8).
Furthermore, gas production ( Fig. 9) suddenly increases in all scenarios (this is the time when the CO 2 breakthrough has started). However, the breakthrough time in the proposed method has been delayed compared to the other scenarios. It means that the cross-section area obtained from the analytical method has helped the ICD Fig. 7 Comparison of Cumulative oil production for all scenarios perform much better than other methods in controlling breakthrough time and controlling gas production. Also, as shown in Fig. 10, the plateau period for the proposed method is longer than other scenarios due to the efficient CO 2 allocation in both layers and postponing CO 2 breakthrough time.
Finally, economic analysis performs to see how net present value can be improved by the proposed method compared to the other scenarios. The result shows that the proposed method has appropriate NPV between other scenarios -only GWO algorithm very slightly has a better NPV compared to the proposed method about 0.05 present (Fig. 11).
To sum up the results, the results of the simulation are shown great improvement in CO 2 flood performance after employing the proposed method and GWO scenario compare to the other scenarios. A better daily oil production profile was achieved and more cumulative oil was produced and gas production decreased. Also, there is a significant

Advantages and disadvantages
Although both the optimization method (GWO) and the proposed method have been able to control the fluid front movement in the two layers very well, but the advantage of the proposed method over the optimization is that, first the simulation model is not required for the proposed method and only reservoir layer properties and total injection rate is needed. Second, the desired valve opening can be obtained very fast with the given equation, while for optimization, complete information and all the details and complexity of the reservoir for simulation are required, and the calculation time is very long.

Summary and conclusions
• The utilization of the proposed method for setting inflow control devices flow area in the reservoir containing thief and high permeable zones assists in improving CO 2 flood performance not only by reducing gas production, but also by improving oil recovery and extending the production plateau period. • The new analytical technique can effectively set an ICD flow area in keeping with its dynamic and petrophysical properties in order to delay and maximize CO 2 breakthrough time. • The proposed method only requires reservoir layer properties and total rate of injection data, while all the details and complexity of the reservoir for simulation are needed for other methods, which are based on reservoir simulation. • The ICD flow area can be obtained very fast with the given formula, while other methods, especially optimization algorithm is time-consuming. • Utilizing the proposed formula for setting ICDs flow area for a cost-effective and efficient CO 2 injection in underground reservoirs is an application of the developed methodology for CCUS projects. • The findings of the presented analytical formula can help researchers for a better understanding of parameters affecting ICD flow area such as reservoir permeability.
Funding The authors declare that they have no relevant financial or non-financial interests to disclose.
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