Introduction

Energy demand is increasing in the world, and fossil fuel is still one of the main energy resources. As a vast majority of oil fields are in the second half of their life, enhancing oil recovery in such fields is of great importance, which can be carried out using leading-edge technologies in this regard. Water injection is one of the most popular and common options for improving oil recovery (Kalam et al. 2022) because of its availability, relative ease of injection, good dispersity in oil formations, and productivity associated with oil displacement (Asadollahi 2012). This, in turn, contributes to reservoir pressure maintenance together with oil displacement toward the producing interval. Having said that, water injection may cause early water breakthrough in high-permeability layers as well (Rafiei 2014). Discrepancy in layer properties of a multi-layer reservoir may impede water flooding performance. Speedy movement of injected water through the layer with higher permeability causes premature water breakthrough and excessive water production. Thus, after some years, production will no longer be economically viable, and producing well will be shut down (Mohammadpourmarzbali et al. 2019). In this work, a mathematical procedure is developed to design oil wells in an optimum way with the end goal of increasing oil production and postponing water breakthrough.

Smart wells are unconventional types of wells with downhole devices including sensors and specific valves controlling the flow rate known as control valves. These wells also allow real-time monitoring of fluid flow rate, pressure, and adjustment of downhole valves. Smart well technology provides high flexibility in the operation of multi-layer reservoirs so that each layer can be controlled individually. This technology also integrates regional production management by virtue of control valves and proper flow monitoring equipment to improve well and oilfield performance. Regional production management is capable of increasing oil production and/or reducing unnecessary fluid production. Intelligent well control instruments are divided into two categories—passive inflow control devices (ICDs) and active interval control valves (ICVs) (Ebadi and Davies 2006; Gao et al. 2007). ICDs are generally preferred to ICVs owing to their lower expense, lower complexity, and long-term durability (Rahimbakhsh and Rafiei 2018). However, in general, ICV's applicability has been investigated to be successful in real-case scenarios (Ahn et al. 2023). Moreover, autonomous flow control devices (AFCDs) are introduced as a means of controlling unwanted phases’ production (gas/water) (Eltaher et al. 2019). An autonomous inflow control device (AICD) takes advantage of both active and passive control, providing a pressure drop dependent on flowing fluid properties and rate. Also, the autonomous inflow control valve (AICV) poses an increased pressure drop (due to flow into the by-pass module), controlling the unwanted fluid flow rate. According to experience, smart valves' ability to increase production depends on factors such as porosity and permeability distribution in the reservoir layer; hence, heterogeneous reservoirs, which are broadly observed with the early breakthrough of the injection fluid, are the best candidates for intelligent well completions (Ebadi et al. 2005). In case of permeability differences between different layers, fluid flow will be partitioned across them and lower permeability region will be bypassed, which is a concern in various areas of engineering involving stratified porous media such as tracer dispersion. For example, Dejam and Hassanzadeh derived a reduced-order model for tracer dispersion in stratified porous media by generalizing the Taylor dispersion theory and stokes flow in porous media (Dejam and Hassanzadeh 2022). Shale reservoirs are great examples of heterogeneous structures (Li et al. 2017). By their local climate, deposition, and diagenesis variety, shale reservoirs have inhomogeneous characteristics in the case of internal properties and spatial distribution (Xiaorong et al. 2016). Also shale layers due to their dense texture and low permeability can introduce severe heterogeneities when they are in the vicinity of a more permeable layer (Dan et al. 2015).

As mentioned earlier, one of the severe challenges in water injection scenarios is early water breakthrough in highly conductive layers. To tackle this issue, ICDs have been employed as a potentially viable solution to curb early water breakthroughs from so-called “guilty” intervals. Improper design of the ICD not only fails to control the fluid flow rate but also increases operating costs. For this purpose, controlling and determining the flow cross-sectional area is one of the most important factors in ICDs’ design.

Regarding implementing intelligent well completions in petro physically heterogeneous reservoirs, several research studies have been done which are reviewed as follows. Brouwer et al. (2001) managed water injection into a field with a high degree of heterogeneity using a horizontal injection well by employing a genetic algorithm (GA) method. Pressure and rate data for each well segment were taken as input data for the GA working based on optimizing the productivity index for each well section. Downhole control valves were also used during water injection operations for the sake of preventing early water breakthrough and reaching optimal production possibilities. The only function of the downhole control valve used was to open and close logical operators. Meum et al. (2008) investigated the ICV adjustment by using the single shooting multi-step quasi-Newton (SSMQN) method in which the objective function was oil recovery. Additionally, Al-Ghareeb, (2009) applied GA to optimize ICVs’ settings in a naturally fractured reservoir. Objective functions included minimizing water cut, extending production plateau, and maximizing net present value (NPV). Wang et al. (2016) developed a coupled model of reservoir-ICD-wellbore to take account of bottom water in heterogeneous reservoirs. They found water coning to be delayed using ICD. Zhang et al. (2019), also, proposed a new AICD model accounting for bottom water as well as solid particles in loose sands using computational fluid dynamics (CFD). In our study, the heterogeneity considered is the layered nature of the reservoir in which reservoir layers can be of their petrophysical properties.

As for quantifying the smart wells’ profitability, several ICD/ICV size determination methods have been introduced and proved to be helpful in practice. Cullick and Sukkestad )2010) investigated the purpose served by smart wells in a multi-branch horizontal production well. They made use of Perkins )1993) model (sub-critical pressure drop across ICV) to find the size of the ICV and predicted pressure drop across the valve. Cullick and Sukkestad assumed that the inflow control valve acts as a choke, and set the goal of maximizing oil production. Their methodology was based on simulation analysis and automated procedures to optimize the objective function. Meshioye et al. )2010) found the optimal injection rate to each layer using the set-point optimization method combined with numerical reservoir simulation. The optimization process was conducted in three sample cases of a three-layer reservoir with different characteristics. ICV valves were used to delay water breakthrough time, and the objective function was to reach the maximum oil recovery or the highest NPV. Li et al. (2011) generated a flow profile of a horizontal well by using an inflow-outflow method through which additional frictional pressure drops created by ICDs were also considered. They used a general, empirical equation for the ICD pressure drop, and skin factor was included in the ICD pressure drop as well. Hassanabadi et al. (2012) optimized ICDs’ port sizes with a particle swarm optimization algorithm via neural network modeling to maximize oil production. Behrouz et al. (2016) proposed a workflow to determine the cross-sectional area of the ICV based on its effect on predefined target performance. Their workflow starts with a predefined ICV area at a fully open position. Based on the trial-and-error method, the workflow selects the size of the cross-sectional area of the valve. The target considered in the study by Behrouz et al. was an increment in total oil output and diminishing water production. Chen and Reynolds (2017) ran the EnOpt algorithm (an ensemble-based optimization technique) to simultaneously optimize well control (rate/pressure) and ICV setting in a case of water alternating gas (WAG) injection with NPV as the objective function. Lee et al. (2017) studied the effect of ICD regulation on flux equalization and well productivity reduction. They used various cases of well productivity index and permeability changes enabling the operator to calculate the required ICD strength or number of ICD joints along the well to maximize the recovery factor. Being a time-consuming simulation process, only the fast close wellbore simulation was used to design ICD completions. Ugwu and Moldestad (2018) numerically simulated a case study with reservoir conditions alike the Troll offshore Norway using the ICD via Eclipse reservoir simulator. The early breakthrough time for water and gas in their study was delayed and gas production was reduced by 51% with ICD completion. Prakasa et al. (2019) developed a fully analytical workflow (assisted by type curve matching) for ICD completions of both homogeneous and heterogeneous reservoirs producing via horizontal wells. Although, in terms of computational efficiency, such a fully analytical method proves beneficial, its practicality for real-world scenarios, which are broadly in need of numerical reservoir simulation, may be questionable. Also, in some cases of their study, their proposed model had no improvement in the oil production rate compared to empirical models. Moradi and Moldestad (2020) simulated near-wellbore oil production of a horizontal well by applying ICD and AICD completions using OLGA and ROCX packages. An empirical equation was utilized to calculate the ICD pressure drop dependent on the flow rate, valve geometry, and fluid density. By using these downhole control devices, water breakthrough time was delayed, accumulated water production was reduced, and cost-effective oil production was achieved. As for handling massive computations relevant to multilateral well completion fluid dynamics, Aljubran and Horne (2020) showed how a machine learning approach (coupled with reservoir simulation) can optimally design ICV settings over time to maximize well production for both homogenous and heterogeneous reservoirs. Zhang et al. (2021) employed a multi-objective optimization algorithm to obtain optimal ICD configuration for a horizontal well. Determining the ICD location and nozzles’ number and size yielded production rate and water breakthrough time to occur optimally. Sabet et al. (2022), took advantage of both CFD and empirical correlations to describe ICD characteristics. Using correlations based on CFD results, their work can provide operators with easy-to-use formulae flexible about well/reservoir conditions. Recently, Rezvani and Rafiei (2023) presented an analytical approach for ICD flow area using the velocity equalization concept for injection host layers for CO2-EOR applications. They showed their model is capable of postponing injected CO2 breakthrough time. However, their study was not accompanied by layer-based gas breakthrough insights, and it was shown that total gas breakthrough was delayed by their proposed method. We, also, published premier findings of our study on ICD cross-sectional area quantification using an analytical formulation in conference proceedings (Mostakhdeminhosseini et al. 2020). In that study, the obtained analytical formulation ran for a field case and was compared to a no-control scenario. The results showed its suitability for improving oil recovery as well as delaying early well water production.

Having discussed the relevant literature to our study, distinctions of this study in comparison with the literature should be outlined. Firstly, no research to date has considered the concept of equalizing velocity between oil-producing layers in a water flooding scenario. This leads to the delaying of water breakthrough which will be discussed later in detail. Previous studies for ICD/ICV size determination either use empirical equations (Li et al. 2011; Moradi and Moldestad 2020) or iterative numerical simulations (Behrouz et al. 2016; Chen and Reynolds 2017; Lee et al. 2017; Meshioye et al. 2010; Moradi and Moldestad 2020; Sabet et al. 2022; Ugwu and Moldestad 2018; Zhang et al. 2021) to optimize oil production or NPV. Iterative numerical simulations require a trial and error technique for selecting a valve size through an optimization task which is indeed time-consuming, especially for real-world field scenarios. This determines how an easy-to-use analytical tool for downhole control devices’ size determination can be useful for operators and engineers. Such an analytical tool is to be time-saving for further reservoir simulation computations. Secondly, most of the reviewed studies lack observations on all 3 factors of oil production, water breakthrough time, and NPV. Cullick and Sukkestad (2010), Hassanabadi et al. (2012), and Prakasa et al. (2019) had no study on water breakthrough time, (Aljubran and Horne 2020; Cullick and Sukkestad 2010; Lee et al. 2017; Zhang et al. 2021) did not consider NPV in their optimizations. In this study, we present our findings not only on oil production optimization but also on water breakthrough time and NPV maximization.

In this study, an analytical method is proposed for computing proper ICDs’ flow cross-sectional area. This exact solution, needing only basic reservoir properties of each layer together with well operational parameters (more importantly surface injection flow rate), is able to determine the ICD flow area in a fast way. On the other hand, there are other approaches available such as artificial-intelligence (AI) -based and sensitivity-analysis-based methods which are relatively computationally demanding. Both of these approaches are compared with our analytical model in this study. The workflow suggested in this study starts with reservoir simulation of the concerned injection-production scenario coupled with either our proposed analytical model or AI-based (GA here) methods (see Fig. 1) for ICD cross-sectional area determination. Having been determined, a comparison can be made by the user to select the most efficient method in terms of simulating production and recovery factor increase, water breakthrough delay, and water cut reduction based on reservoir simulation results. Recall that neither GA nor sensitivity analysis cases were covered in our study presented before (Mostakhdeminhosseini et al. 2020). The core of the analytical formulation of this study originates from equalizing fluid velocity in high- and low-permeability layers of a two-layer production interval, and deriving the ICD cross-sectional area based on that. By reducing the fluid velocity in the highly permeable layer, the water breakthrough time in this layer is reduced to the water breakthrough time of the low-permeability layer. This way, water breakthrough times in these two layers are synchronized, which is advantageous for oil production and the NPV of the field. The novelty of our work lies in the analytical method that is based on the idea of eliminating the early water breakthrough issue by adjusting the water injection rate into different layers with different petrophysical properties which was never done to the extent of our knowledge. This analytical solution is advantageous in the case of computational time which is much faster than the other methods alongside its ability to rectify early water production which strongly determines the success of the water injection technique. This advantageous analytical solution is applicable in oil reservoirs and in the case of other types of reservoirs needs some adjustments which can be named as its limitation which also can be a potential extension to this work that can broaden its applicability.

Fig. 1
figure 1

Workflow proposed for determining ICDs cross-sectional area in this study

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Methodology

Analytical methods to determine the cross-sectional area of ICDs as well as a GA approach are introduced in this section. Analytical methods are widely accepted owing to their computational efficiency and flexibility compared to numerical counterparts (Chu et al. 2023; Dutt and Mandal 2012; Wang et al. 2021). A schematic workflow of the proposed methodology for determining ICDs’ flow area is as Fig. 1. According to this flowchart, numerical reservoir simulation is set to run for any consecutive injection-production scenario while coupled with either analytical method or GA to determine ICDs’ flow area through which the simulation results can be compared together. Entries of GA from reservoir simulation are oil and water production rate plus water injection rate, whereas the latter works iteratively to reach the optimum ICD size and requires multiple simulation runs, the former is quite straightforward and runs only one time considering the ICD size from analytical formulation. This way, one can investigate to what extent each of these methods is successful for ICD flow area determination in terms of increasing hydrocarbon production and tackling water breakthrough and early production issues.

Analytical technique

To begin with, the physical concept behind the formulae of the analytical method derivation should be clarified. To delay water breakthrough time—an important monitoring parameter of water flooding (Huang et al. 2019)—, the fluid front movement must be balanced in layers with different permeability values. Fluid velocity in the high-permeability layer should be reduced to that of the low-permeable layer. Accordingly, a piston-like movement is achieved and the injection fluid front in both layers reaches the production well at the same time leading to delaying the water breakthrough.

According to Eq. (1), layer velocity is equal to the flow rate divided by the cross-sectional area of the corresponding layer.

$$V_{1} = V_{2} \Rightarrow \frac{{q_{1} }}{{A_{1} }} = \frac{{q_{2} }}{{A_{2} }}$$
(1)

where V1, V2, q1, q2, A1, and A2 are the velocity, flow rate, and cross-sectional area of the fluid flow in layer 1 and layer 2, respectively.

The general form of the injection inflow equation from well to reservoir is as Eq. (2) (Ahmed 2018):

$$q_{i} = \frac{{k_{i} h_{i} }}{{141.2 \times \mu B \times \left( {\ln \left( {\frac{{r_{e} }}{{r_{{\text{w}}} }}} \right) + S_{i} } \right)}}\left( {\Delta P_{i} + \Delta P_{{{\text{VALVE}}}} } \right)$$
(2)

where qi is the injection flow rate in layer i, hi is the thickness of layer i, ki permeability of layer i, Si is the skin factor of the wellbore in contact with layer i, and ∆Pi and ∆Pvalve denote pressure difference in layer i and pressure-drop across the ICD, re and rw are reservoir and wellbore radius, µ is fluid viscosity and B is formation volume factor, respectively. As for conciseness, units of all computational parameters are summarized in the nomenclature section.

The term ∆Pi is calculated as the difference between reservoir external pressure (Pe) and wellbore pressure (Pwi) for layer i. The pressure-drop term across the ICD is as Eq. (3):

$$\Delta P_{{{\text{VALVE}}}} = P_{{w_{i} }} - P_{{{\text{eq}}_{i} }}$$
(3)

where Pwi is the bottom-hole injection pressure for layer i and \(P_{{{\text{eq}}_{i} }}\) is wellbore pressure after ICD insertion.

In a hypothetical two-layer reservoir with different thicknesses and permeabilities (see Fig. 2), ICD is placed in the layer with higher permeability (layer 1). The injection inflow equation from well to reservoir is written down in Eq. (4) for layer 1 and in Eq. (5) for layer 2.

$$q_{1} = \frac{{k_{1} h_{1} }}{{141.2 \times \mu B \times \left( {\ln \left( {\frac{{r_{e} }}{{r_{{\text{w}}} }}} \right) + S_{1} } \right)}}\left( {P_{{{\text{eq}}_{1} }} - P_{e}^{{}} } \right)$$
(4)
$$q_{2} = \frac{{k_{2} h_{2} }}{{141.2 \times \mu B \times \left( {\ln \left( {\frac{{r_{e} }}{{r_{{\text{w}}} }}} \right) + S_{2} } \right)}}\left( {P_{{{\text{w}}_{2} }}^{{}} - P_{e}^{{}} } \right)$$
(5)
Fig. 2
figure 2

Schematic profile view of a two-layer reservoir with ICD installed in the high-permeability layer

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By substituting Eqs. (4) and (5) into Eq. (1) one can have Eq. (6) as:

$$V_{1} = V_{2} \Rightarrow \frac{{q_{1} }}{{A_{1} }} = \frac{{q_{2} }}{{A_{2} }} \Rightarrow \frac{{k_{1} h_{1} }}{{A_{1} }}\left( {P_{{{\text{eq}}_{1} }} - P_{e} } \right)^{{}} = \frac{{k_{2} h_{2} }}{{A_{2} }}\left( {P_{{{\text{w}}_{2} }}^{{}} - P_{e}^{{}} } \right)^{{}}$$
(6)

where A for both layers 1 and 2 can be quantified as shown in Eq. (7).

$$A_{1} = 2\pi r_{1} h_{1} \;{\text{and}}\;A_{2} = 2\pi r_{2} h_{2}$$
(7)

A1 and A2 are the cross-sectional areas of two layers in which injected waterfront moves, and r1 and r2 are the front radius of injected water in two layers.

These two front radii of injected water in two layers (r1 and r2) must be equal since the fluid velocity has to be identical in these layers. As such, Eq. (8) derives how the flow rate in these two layers can be related:

$$V_{1} = V_{2} \Rightarrow r_{1} = r_{2} \Rightarrow q_{1} = \frac{{h_{1} }}{{h{}_{2}}}q_{2}$$
(8)

Considering this relationship between the two layers’ flow rate, the total flow rate is determined as follows in Eq. (9).

$$q_{{{\text{total}}}} = q_{1} + q_{2} = \left( {\frac{{h_{1} + h_{2} }}{{h_{2} }}} \right)q_{2}$$
(9)

According to Eq. (9) and having the surface injection rate (qtotal) known, one can determine the downhole injection rate for each layer (q1 and q2).

Determining \(P_{{{\text{eq}}_{i} }}\) from Eq. (6) combined with Eq. (3), the ICD pressure difference can be derived as Eq. (10) below:

$$\Delta P_{{{\text{VALVE}}}} = P_{{w_{1} }} - \left( {\frac{{k_{2} }}{{k_{1} }}(P_{{w_{2} }}^{{}} - P_{e}^{{}} ) + P_{e}^{{}} } \right)$$
(10)

The ICD pressure-drop equation is also calculated from the following equation (Eq. 11) (Schlumberger 2010):

$$\Delta P_{{{\text{VALVE}}}} = C_{u} \frac{{\rho \left( {\frac{{q_{i}^{{}} }}{{A_{c}^{{}} }}} \right)^{2} }}{{2C_{V}^{2} }} + 2C_{u} f\frac{L}{D}\rho \left( {\frac{{q_{{{\text{total}}}} }}{{A_{P}^{{}} }}} \right)^{2}$$
(11)

where Cu is a constant coefficient, CV is ICD constant, Ac is ICD cross-sectional area, AP is well cross-sectional area, D is well diameter, f is fanning friction factor, and ρ is fluid density.

Therefore, Eq. (12) in the following is obtained to calculate the cross-sectional area of ​​the ICD:

$$A_{c} = \sqrt {\frac{{\frac{{C_{u} \rho q_{i}^{2} }}{{2C_{V}^{2} }}}}{{P_{{w_{1} }} - \left( {\frac{{k_{2} }}{{k_{1} }}(P_{{w_{2} }}^{{}} - P_{e}^{{}} )^{{}} + P_{e}^{{}} } \right) - 2C_{u} \frac{{fL\rho q^{2} }}{{DA_{p}^{2} }}}}}$$
(12)

Genetic algorithm approach

Genetic algorithm (GA) is accounted as an evolutionary AI-based algorithm in terms of evaluating complex engineering problems. This algorithm initiates by producing an initial random solution (chromosomes) and reproducing it iteratively to converge to the optimum solution of the problem (Mohammadi Behboud et al. 2023). Reproduction is implemented through crossover and mutation operations. The crossover factor (CF) is defined as the probability of offspring production by chromosomes’ pairs, and the mutation factor (MF) is the probability of binary chromosome pairs if they change from 0 to 1. Optimization through GA can be achieved following the below procedure (Alam et al. 2015; Karkevandi-Talkhooncheh et al. 2017):

  1. 1.

    Setting initial chromosomes, CF and MF.

  2. 2.

    Quantifying objective function (fitness of chromosomes), and finding the best chromosomes.

  3. 3.

    Selecting chromosomes, parents’ crossover and offspring mutation to have new chromosomes.

  4. 4.

    Evaluating objective function whether the propriety criteria are fulfilled.

  5. 5.

    If the solution is not proper, the iteration process reverts to step 3; if the solution is proper (or the maximum iteration number is reached), the optimum solution is obtained.

Here, GA works in parallel with the reservoir model running in the commercial reservoir simulator to optimize NPV—the difference between present cash inflow/outflow (Taghavinejad et al. 2022). Equation (13) serves as the objective function of the optimization problem using GA, and the cross-sectional area of the valve is the main variable in this problem.

$${\text{NPV}} = \mathop \sum \limits_{K = 1}^{M} \frac{{Q_{o} r_{o} - Q_{w} r_{w} - Q_{g} r_{g} - Q_{i} r_{i} }}{{(1 + b)^{{t_{k} }} }} - C_{0}$$
(13)

where Qo, Qw, Qg, and Qi refer to oil production rate, water production rate, gas production rate, and water injection rate, respectively; ro, rw, rg, and ri denote oil price, cost of water handling, cost of gas handling, and cost of injected water, respectively; b is the annual discount rate, C0 is constant cost, and tk is time in a year.

Considering the workflow shown in Fig. 1, reservoir simulation computations are coupled with the GA to select the best ICD flow area among a range for that. In this respect, the injection-production scenario (compared with other scenarios discussed later in this study) is run, and the optimal amount of cross-sectional area of the valve is calculated via GA. For this goal and according to Eq. (13) as the objective function of GA optimization, the oil price is taken as 50 $/STB, the annual discount rate is taken as 0.1, and the overall water handling/injection price is taken as 5 $/STB. It is noteworthy that there is no gas in the production system, the constant cost is zero to make comparisons easier, and production rates and time are taken from the results of reservoir simulation.

GA is implemented considering 50 iterations with CF = 0.8, MF = 0.05, and a population size of 4. Using a commercial genetic algorithm toolbox and a text editor for coupling that with reservoir simulator input (for manipulation of the ICD flow area), reservoir simulation is run and production/injection rates as well as run time are obtained for NPV calculation (Eq. 13). Until reaching a maximum for the NPV, GA progresses to change the ICD flow area in the reservoir simulation input. The GA optimization result for the field case described later is the ICD flow area equal to 0.00034 ft2.

Field application

The field case oil reservoir of this study is in southwestern Iran, and its properties are shown in Table 1. This reservoir is formed from 13 layers, and production occurs from layers 8 and 9. Permeability of layers 8 and 9 are 400 and 800 mD, respectively, as shown in Table 1 and illustrated in Fig. 3. The valve is located in the layer with higher intrinsic permeability (layer 9). The flow rate of the water injected in layer 9 has been controlled by installing ICD to delay the water breakthrough time. The cross-sectional area of the fluid flow has been calculated by the presented analytical equation (Eq. 12) which is equal to 0.000238ft2. Also, following the GA approach, the optimum cross-sectional area is determined as 0.00034 ft2. Table 1 encompasses the basic reservoir properties of this case study. Also, Fig. 3 illustrates a schema of the 3D reservoir model for simulation tasks associated with an injection-production scenario via one injection well and one production well located in the field. The accompanying graphical legend of the reservoir model shows the oil saturation value illustrated by colors (from blue to red) throughout the reservoir grid blocks. In the following, base case scenarios together with other scenarios for ICD flow area determination are introduced.

Table 1 Properties of the studied reservoir model
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Fig. 3
figure 3

The reservoir model in this study

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Base case scenario

A base case scenario is defined based on the maximum performance of the production well and maintaining field pressure. However, in this case, no management of layer-scale water injection is applied, and injection takes place only by its surface control moderation. Production and injection scenarios of this base case are introduced as follows.

Production scenario

As for production from this reservoir, and according to the vertical flow performance (VFP) table of the production well, the minimum allowable wellhead pressure is 150 psi. As a result, the maximum production of the well occurs when the wellhead pressure is 150 psi. A sensitivity analysis of the production rate was performed on the reservoir simulation task to determine the maximum production potential rate. Sensitivity analysis was performed on constrained flow rates of 2000, 2700, and 6000 STB/day. As shown in Fig. 4, the maximum capacity of the well for production proves to be 2700 STB/day whether the production constraint is 2700 STB/day or more.

Fig. 4
figure 4

Sensitivity analysis on field oil production ratio

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Injection scenario

The injection flow rate is controlled by voidage replacement meaning that the injection and production volumes would remain identical. Sensitivity analysis was performed on 1500, 2500, 2700, and 3500 STB/day flow rates and 2700 STB/day was selected as a suitable injection rate for the injection well. A comparison of field pressure diagrams plotted in Fig. 5 shows that reservoir pressure falls for cases with injection flow rates of 1500 and 2500 STB/day. Reservoir pressure reduction for the case with 1500 STB/day injection flow rate is remarkably steeper than that of the other one.

Fig. 5
figure 5

Comparison of field pressure at different injection rates

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Flow-control scenarios

Table 2 presents the field parameters of our analytical equation for moving the injection waterfront at the same velocity as different layers based on the cross-sectional area of the ICDs (0.000238 ft2 value as mentioned earlier).

Table 2 Parameters for valve control using analytical technique
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Further discussion on the work will be based on the results of the reservoir simulation using various scenarios of layer-scale injection control. The first scenario would be the base case in which no layer-scale control is considered for water injection. Comparison of all other scenarios with the base case would be meaningful in terms of indicating to what extent they are fruitful for the production system. Analytical technique and GA scenarios are also crucial parts of the workflow suggested in Fig. 1 thereby one can determine which of these computational methods works better for a specific case. In addition to the three mentioned scenarios, three other scenarios are also considered with fixed values for the ICDs flow area. As a matter of fact, these three cases (cases 4–5 according to Table 3), are sub-cases of a sensitivity analysis by which another meaningful comparison can be made between their results and those of other scenarios. Making use of sensitivity analysis is quite common in engineering studies of such a case, and benchmarking them along with analytical technique and GA can provide a deep understating of this field study. All considered scenarios whose results are fully discussed in the following section are listed in Table 3.

Table 3 Injection-production scenarios considered
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Results and discussion

As regards evaluating the efficiency of the analytically derived equation for ICDs’ flow area determination (Eq. 12), and for assessing improvement of the water injection process amid the oil production period, results of the proposed analytical method should be compared with results of other scenarios (listed in Table 3). Based on Eq. (12), for calculating the cross-sectional area of the valve, as it was mentioned earlier, the optimum opening of the valve to control the injection flow rate is determined to be 0.000238 square feet for this field case. Figure 6 shows that oil recovery obtained using the analytical model of our study is virtually matched by the GA method, and it is higher than those of other scenarios. It can be implied that by installing a flow control valve with the flow cross-sectional area optimally designed obtained from the analytical equation, the oil recovery factor can significantly increase over 20 years rather than those of base case and sensitivity analysis scenarios.

Fig. 6
figure 6

Comparison of oil recovery in all scenarios

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Figure 7 depicts that using the analytical method for designing the cross-sectional area of the valve delays the water breakthrough time much more than other scenarios. Cumulative water production using this analytical technique, at the end of the simulation period, was reduced by 90.7% compared to the worst scenario (the base case), and 56.5% in comparison with the GA’s scenario.

Fig. 7
figure 7

Comparison of water breakthrough time in two scenarios

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As implied from Fig. 8, the injection flow rate in the highly permeable layer is well controlled by installing ICD with the cross-sectional area of ​​the fluid flow obtained from the analytical technique. The injection rate to two layers (layers 8 and 9 of the field) was balanced by the proposed method which is quite comparable with the results obtained from GA, particularly in the second half of the reservoir’s production life.

Fig. 8
figure 8

Comparison of water injection rate in different layers in all scenarios

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Based on the managerial approach of equalizing the velocity of the fluid in layers 8 and 9 of the field, breakthrough times for each layer in all 6 scenarios are plotted in Fig. 9. The time the water cut becomes greater than zero is the breakthrough time. As shown in Fig. 9, the breakthrough for the high-permeability layer (layer 9) occurs sooner for all scenarios, except in scenario 4 which is due to the extreme restriction of valve flow area (0.0001 ft2). For the high-permeability layer, breakthrough time increases from the base case scenario to scenarios 5 and 6, and then GA and analytical technique scenarios have higher amounts of water breakthrough time, respectively. For the low-permeability layer (layer 8), scenario 4 has the sooner water breakthrough time, and then the analytical technique and GA scenarios have higher breakthrough time values, respectively. After these three scenarios, there are scenarios 6, 5, and base case with higher breakthrough time values, respectively. It can be inferred that the three latter scenarios as well as scenario 4 are undesirable scenarios in which there is a considerable gap between the water breakthrough time of layers 8 and 9. On the other hand, in GA and analytical techniques, water breakthrough time values of layers 8 and 9 become closer together, which is advantageous for production and water handling.

Fig. 9
figure 9

Comparison of water breakthrough time of each layer in all scenarios

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As illustrated by black lines (solid and dotted-solid), the breakthrough time of the highly permeable layer is better delayed by using the analytical method in comparison with all other scenarios (except that of scenario 4 which is due to its extremely small valve size). Since the surface injection flow rate is constant, the injection rate in the low-permeability layer increases compared with the highly permeable layer (for equalizing velocities in two layers). As a result, the breakthrough time of the two layers becomes closer to each other (roughly synchronized), which is evident in the black curves of Fig. 9. Admittedly, the proposed method can effectively adjust the valve cross-sectional area so that water breakthrough time occurs simultaneously in different layers with different petrophysical characteristics. For cases other than analytical technique, breakthrough times for two layers are distinctly apart from each other, leading to early water production of the whole production system. In other words, although one layer contributes to water breakthrough later than the other one, the analytical technique proves to be a more promising scenario for water production control (as confirmed in Fig. 7) owing to synchronizing the water breakthrough time of the high- and low-permeability layers. Early breakthrough of water even from one of the layers can put a stop to oil production from the other layer due to the emersion of a direct and conductive path for water from the injection well to the production one which mitigates the ability of the water to sweep oil from the less conductive layer. By delaying and synchronizing the breakthrough time of water of both layers, oil production would be higher, less water would be produced and the cost of water handling and separation would be reduced.

Figure 10 illustrates the NPV bar chart for all scenarios. To calculate NPV values, the oil price is considered as $50 per barrel, injected water cost together with water handling expense are $5 per barrel of oil produced, and the discount rate is 0.1. As demonstrated, GA shows the maximum NPV obtained for this study working based on its objective function which is NPV maximization. As a matter of fact, although the proposed analytical method, by delaying the water breakthrough time, is of a significantly great amount of NPV (above $274.8 million), that of the GA is marginally higher (less than $276.6 million). The difference between the analytical and GA, NPVs seemingly is because the objective function of the latter is to optimize the NPV and that of the former is to delay water breakthrough time.

Fig. 10
figure 10

NPV calculated for all the scenarios

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At last, all scenarios’ results discussed above are collected in Table 4. In this table, oil recovery, cumulative water production, water breakthrough time, and NPV are shown for all scenarios of this study. All results were previously shown and discussed graphically, and this tabular form provides better insights for comparing each scenario’s results. As shown, the most desirable value of each parameter is italicized. In the water breakthrough time column, results are classified for both the low-permeability layer (#8) and high-permeability layer (#9) together with the ratio of results for layer 8 to layer 9 (Table 5).

Table 4 Results for oil recovery, cumulative water production, water breakthrough time, and NPV of all scenarios
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Table 5 General comparison of different cases results
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It is outlined that the genetic algorithm approach provides the most optimum results for the NPV and oil recovery, and the most desirable result for controlling water production and breakthrough time is for the analytical technique. GA and analytical method result in approximately 1.5% higher oil recovery compared with the worst-case scenario (#4). It is also evident that the analytical technique has been successful in synchronizing water breakthrough times in layers 8 and 9 with a near unity ratio of their water breakthrough time values (1.06). For the worst scenario (base case), the ratio of water breakthrough times in layers 8 and 9 is 1.76, which is not desirable for water production handling. As the analytical formulation of this study is not an optimization tool in nature and is a mathematical tool for fast ICD flow area determination, it can be inferred that this analytical technique outweighs all other scenarios. The reason is that it provides desirable results for water production control compared to all other scenarios as well as time-efficient, favorable results for oil recovery and NPV comparable with those of the GA scenario.

Conclusions

A workflow was suggested based on reservoir simulation calculations which is coupled with a computational method of ICDs cross-sectional area quantification. An analytical formulation was presented in this study for ICD cross-sectional area determination. Based on this formulation, fluid movement in layers of a multi-layer reservoir with different properties was set to be balanced; accordingly, oil recovery increased, water breakthrough time was delayed, and cumulative water production was outstandingly reduced. Additionally, a genetic algorithm (GA) was utilized as an alternative approach, and a base case scenario (with no layer-scale inflow management of the valve) was employed alongside a sensitivity analysis regarding ICD’s flow area for the sake of the analytical method and GA efficiency assessment in terms of oil production, water production, water injection rate, well water cut, and net present value (NPV). Accordingly, the most important conclusions drawn are as:

  • Analytical technique determines ICD flow area with no reservoir simulation run and merely a closed-form formulation while other methods such as GA optimization and sensitivity analysis require iterative reservoir simulation runs.

  • Analytical technique scenario provides desirable results with only one reservoir simulation run for obtaining field production and recovery results. In contrast, GA and sensitivity analysis scenarios obtain such results after multiple reservoir simulation runs for ICD flow area determination.

  • The analytical method proposed outperforms the GA as it is less computationally demanding and provides better results for delaying water breakthrough time and lowering water production for the field data.

  • The NPV calculated using the analytical technique and GA scenarios is greater than that of the base case scenario, which reveals the economic benefit of the analytical technique and GA methods to determine the cross-sectional area of the ICDs.