Introduction

The global surge in oil demand has created a pressing need to optimize the production of existing oil fields, given the decreasing number of new discoveries. Due to the maturity of most of these large fields, it is essential that reservoir management and development strategies are implemented with care to maximize recovery factor (RF) and economic return, and minimize the negative impact on the environment. By employing innovative technologies and implementing robust reservoir management practices, the industry can unlock untapped potential and extend the lifespan of existing oil fields. Such strategies not only contribute to meet the growing energy demands but also promote sustainable resource utilization and minimize the need for exploration in environmentally sensitive areas.

Mega oilfield development projects must consider a variety of objectives given the interdependence of various factors and the need to strike a balance between financial, environmental, and sustainability concerns. By considering multiple objectives, comprehensive decision-making can be achieved, thus minimizing potential trade-offs and maximizing overall project outcomes. Integrating diverse objectives enhances the overall viability of the project, satisfies the interests of stakeholders, and contributes to long-term success. By carefully balancing financial gains with environmental responsibility, oilfield development can be carried out in a sustainable and responsible manner. This approach ensures that the economic benefits derived from the project are not achieved at the expense of environmental degradation. Through careful consideration of multiple objectives, the oil and gas industry can substitute development practices that harmonize economic gain, environmental protection, and long-term sustainability.

This study aims to investigate how prioritizing the optimization of EMV as the primary objective function impacts other objective function s in the oilfield development and management plan, with a specific focus on evaluating the behavior of other objective functions across various stages of the development process. Furthermore, an analysis of the NPV response for each RM is undertaken during the optimization process, considering economic uncertainties. This approach contributes significantly to enhancing the understanding of the importance of integrating diverse objective functions into the entirety of the field development optimization plan.

Background

The optimization of oil and gas field development involves challenges posed by subsurface uncertainty, integrating multiple objectives, and balancing financial gains with environmental responsibility. Since the late 1970s, scholars have been constantly pursuing approaches to effectively meet the needs of various oilfield development projects. Over the course of extensive research on production optimization, numerous approaches have been proposed to assess the performance of different development programs. For instance, Rosenwald and Green (1974) aimed to minimize the discrepancy between the production-demand curve and the actual flow curve. Babayev (1975) focused on achieving the minimum total cost per unit of output. Similarly, Lasdon et al. (1986) sought to maximize the deliverability of a gas reservoir within a specified timeframe, minimize the shortfall in total gas withdrawal between the demand schedule and the actual deliverable gas amount on a monthly basis, and explore the optimization of weighted combinations of these objectives. These early studies were the first to demonstrate the wide range of objectives considered in the pursuit of effective oilfield development optimization strategies.

In recent years, the field of production optimization has witnessed a surge in research interest, with particular emphasis on optimizing the NPV using various optimization approaches (Brouwer and Jansen 2004; Sarma et al. 2005; Bangerth et al. 2006; Kraaijevanger et al. 2007; Zandvliet et al. 2008; Suwartadi et al. 2009; Onwunalu and Durlofsky 2010; Afshari et al. 2011; Capolei et al. 2012; Forouzanfar and Reynolds 2013; Oliveira and Reynolds 2014; Bukshtynov et al. 2015; Jesmani et al. 2016; Naderi and Khamehchi 2017; Karkevandi-Talkhooncheh et al. 2018; Bertini Junior et al. 2019; Santos(D.R.) et al., 2020; Ng et al. 2021; Bertini et al. 2022; Santos(D.R.) et al., 2023. However, it is essential to acknowledge the inherent uncertainty associated with characterizing reservoirs, leading to the adoption of multiple geological realizations to account for geological uncertainties within the reservoir model. This approach is usually referred to as robust optimization (Mirzaei-Paiaman et al. 2021). Research endeavors aim to develop optimization strategies that consider uncertainties and variations in reservoir characteristics, operational parameters, and economic factors to ensure more reliable and resilient decision-making processes in the context of oilfield development and management (Jansen et al. 2009; Wang (H.) et al., 2010; Ogunyomi et al. 2011; Dilib and Jackson 2012; Wang (H.), 2013; Valestrand et al. 2014; Shirangi and Durlofsky 2015; Morosov and Schiozer 2016; Gupta and Grossmann 2017; Jahandideh and Jafarpour 2018; Annan Boah et al. 2019; Loomba et al. 2021; Mirzaei-Paiaman et al. 2022, 2023). In the context of robust production optimization, the primary objective is to maximize the expected NPV (i.e., EMV) by considering a set of reservoir realizations, with the expected NPV estimated through the averaging of NPV values.

Various objective functions have been explored in research studies pertaining to the optimization of oil production, apart from commonly used measures like NPV and EMV. Some alternative objectives include maximizing COP, RF, profit, and others (Ariadji et al. 2014; Rahim and Li 2015; Zhang et al. 2016; Borzouie and Borzouie 2016; Ranjith et al. 2017; Ogbeiwi et al. 2018; Simonov et al. 2019; Jia et al., 2020; Silva & Guedes Soares, 2021; Loomba et al. 2022a; Koray et al. 2023). Each method of optimization has its advantages and disadvantages. NPV-based optimization is widely used in the industry as it provides a measure of the project’s economic viability. However, it is not directly proportionate to elements such as RF, COP, or environmental impacts, which are crucial for a sustainable project. Maximizing COP as an objective function can be advantageous as it directly targets the extraction of the maximum amount of oil from the reservoir. However, it may neglect economic considerations and fail to account for project profitability. Optimizing the RF focuses on enhancing the efficiency of hydrocarbon recovery. Nevertheless, it might not capture other important factors like economic risks or uncertainties associated with NPV.

More recently, researchers have focused on considering multiple objective functions simultaneously instead of relying on a single objective function. However, the predominant objective function considered in such multi-objective optimization studies has been NPV, along with the reduction of risk and uncertainty associated with NPV (Pinto et al. 2019; Alpak et al. 2022). In addition to NPV, some researchers have considered other objectives in conjunction with NPV, such as RF, COP, and water production (Rostamian 2017; Rostamian et al. 2019a a, b; You et al. 2020; Wang (L.) et al., 2021; de Moraes and Coelho 2022; de Moraes et al. 2023). Considering multiple objective functions simultaneously offers a more comprehensive approach to optimization. It allows decision-makers to balance economic, technical, and environmental aspects. Incorporating NPV, RF, COP, and other relevant factors is an approach that provides a more holistic view of the project. However, it can be more challenging to implement due to the complexity of simultaneously optimizing multiple objectives and managing trade-offs between conflicting goals. In this paper, the relationship between different objective functions in a case where the primary objective function in the robust optimization is EMV, is investigated.

Recent advancements in reservoir management optimization have seen a shift from traditional objective functions, with researchers exploring streamline-based approaches for enhanced field development strategies. Streamlines, depicting the convective flow trajectory between injection and production points, offer comprehensive insights into the time-of-flight, phase distribution, and static characteristics of traversed cells. Taware et al. (2012) and Roshandel and Siavashi (2023) utilized streamlines and total time of flight to identify regions of un-swept and undrained oil. Safarzadeh et al. (2015) introduced a novel objective function, Well Assessment based on Time-of-Flight (WATOF), leveraging multi-objective genetic algorithms and streamline simulations to optimize water injection rates. Siavashi et al. (2016) emphasized the efficiency of WATOF, contrasting it with traditional COP calculations. Naderi et al. (2021) proposed the proportionally distributed streamlines (PDSLs) objective function as an alternative to COP or NPV functions for waterflooding projects. Chen et al. (2020) tested PDSLs in a real field at Mangala Field, India.

Moreover, in recent years, there has been a notable advancement in field development and well placement techniques through the application of artificial intelligence and machine learning. Mousavi et al. (2024) systematically evaluated various advanced computational techniques, including extreme gradient boosting (XGBoost), light gradient boosting machine (LightGBM), gradient boosting with categorical features support (CatBoost), support vector regression (SVR), and multilayer perceptron (MLP), for estimating NPV in reservoir engineering. Similarly, Esfandi et al. (2024) explored the use of boosting algorithms for precise well placement in Carbon Dioxide-Enhanced Oil Recovery (CO2-EOR) in light oil carbonate reservoirs.

This study addresses the shortcomings of prior research on petroleum asset management by considering a broader range of objectives, including economic and environmental factors. A systematic approach to optimizing field development decisions that considers the interplay of these objectives is proposed. The results demonstrate that a multidimensional approach can lead to more sustainable and profitable field development strategies.

Methodology

In this study, the optimization results from Mirzaei-Paiaman et al. (2023) are used, where the goal was to obtain an optimal field development plan in a water-alternating-CO2 (CO2-WAG) operation. In their work, an integrated optimization system was utilized, comprising the iterative discrete Latin hypercube (IDLHC) algorithm, a reservoir simulator (IMEX by CMG), and in-house software to calculate the economic indicators. The primary objective function that was maximized in their robust optimization workflow was EMV, where a set of nine RMs and three economic scenarios were considered. Their optimization process was structured into four sequential stages.

  1. 1.

    In the initial stage, the focus was on optimizing the quantity, position, and type of wells.

  2. 2.

    The second stage involved fine-tuning the well locations to further enhance the optimization process.

  3. 3.

    The third stage of optimization was dedicated to finding the optimal platform capacity.

  4. 4.

    The fourth stage dealt with optimizing the placement of internal control valves (ICV).

Knowing that the primary objective function in the robust optimization performed byMirzaei-Paiaman et al., (2023) was EMV, the behavior and changes of the other aforementioned objective functions, namely COP, RF, CWP, CGP, and NPV of each RM, are investigated under different economic scenarios. For a more comprehensive and detailed understanding of this workflow, it is highly recommend referring to the research conducted by Mirzaei-Paiaman et al. (2023).

Application

Reservoir Model description

The optimization study of Mirzaei-Paiaman et al. (2023) employed the open-access UNISIM-II-D benchmark model. In this benchmark, uncertainties are based on the preliminary stage of field development, which uses different data sources (well logs, well testing, seismic, etc.) derived from a reference or ground truth model referred to as the UNISIM-II-R model. The UNISIM-II-R model, constructed based on real data from Brazilian pre-salt reservoirs (Correia et al. 2015), addresses the complexities posed by light oil fractured carbonates containing substantial amounts of CO2-rich associated gas, thereby presenting various gas treatment and recovery challenges. Table 1 presents an overview of the characteristics of a UNISIM-II-R reservoir. These characteristics include factors such as reservoir depth, initial pressure, temperature, oil viscosity, and associated gas CO2 content. Analyzing and understanding these attributes is crucial for obtaining the best recovery mechanism.

Table 1 UNISIM-II-R Reservoir characteristics
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The UNISIM-II-D model is based on a corner-point grid consisting of 46 × 69 × 30 cells, with average dimensions of 100 × 100 × 8 m and comprising a total of 65,000 active cells. There are two UNISIM-II-D versions, namely UNISIM-II-D-BO (representing black-oil simulation) and UNISIM-II-D-CO (representing compositional simulation), which utilize distinct fluid models. UNISIM-II-D-BO employs a PVT table, whereas UNISIM-II-D-CO employs a Peng-Robinson Equation of State (EOS), incorporating seven pseudo-components.

Considering the objective of field development optimization, UNISIM-II-D-BO was used. The selection of UNISIM-II-D-BO as the low-fidelity model was based on its favorable simulation runtime. Employing the DLHG (discretized Latin hypercube combined with geostatistical realizations) sampling technique, an ensemble of models was generated to account for geological and reservoir engineering uncertainties (Schiozer et al., 2017). A comprehensive range of uncertainties were taken into consideration, including matrix porosity, matrix permeability, fracture porosity, fracture permeability, fracture spacing, net-to-gross ratio, rock type, reservoir top and bottom, relative permeability, and rock compressibility. By assimilating 1.5 years of production data from a wildcat well, an initial ensemble of 500 scenarios was reduced to 199 models. Incorporating operational uncertainties through DLHC led to an additional set of 199 simulation models, considering factors such as group, platform, producer, injector availability, and well index multiplier (Santos (S.) et al., 2020).

For efficient production optimization in a CO2-WAG project, a method introduced by Meira et al. (2020) was employed by Santos (S.) et al., (2020), which further reduced the ensemble to nine RMs. The authors employed a systematic approach to select and rank RMs for ensemble-based production optimization in carbonate reservoirs subject to WAG injection. Nine RMs were selected based on criteria focused on maximizing the well economic indicator for producers and minimizing it for injectors. The selection process involved evaluating various system outputs, including field indicators and well indicators, using a probabilistic optimization method executed multiple times to ensure robustness. A higher weight was given to parameters related to risk curve representation, and the representativeness of RMs was assessed by comparing errors between RM risk curves and ensemble risk curves. The authors considered different approaches for RM selection, analyzing errors, and conducting qualitative analyses of risk curves to assess uncertainty representation. Furthermore, they evaluated the impact of using a high number of wells and system outputs on RM selection and production optimization. Figure 1 displays the porosity maps corresponding to the 15th layer of the RMs. It serves as a sample to illustrate the distinction among these nine RMs. Table 2 provides the mean and standard deviation of the nine selected RMs. The arrangement of entries in this table progresses from the most optimistic to the most pessimistic case.

Table 2 Mean and Standard deviation of 9 selected RMs
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In this study, all wells except a Wildcat well are intelligent (i.e., equipped with ICV). Producers and injectors are equipped with 3 and 2 ICVs, respectively. When the ratio of gas to oil exceeds 2,000 m3/m3, an open valve is switched to a shut-off setting. Producers operate under a maximum production rate of 3,000 m3/day of liquid and a minimum bottomhole pressure of 275 Kgf/cm2. A water-cut above 0.95 is used as a closing constraint. Maximum injection rates of 5,000 m3/day water and 2 million m3/day gas, and a maximum bottomhole pressure of 480 Kgf/cm2 are used. Six months was used as WAG half-cycle duration. In this benchmark, the maximum size of a platform has processing capacities of 28,618, 28,618, and 19,079 m3/day for the produced oil, liquid, and water, respectively. The maximum allowed processing capacities for water and gas injection are 38,157.0 and 8 million m3/day, respectively.

Optimization algorithm

The optimization algorithm employed was IDLHC. This is an optimization method used to efficiently explore and search a high-dimensional parameter space to find the optimal or near-optimal solutions for a given problem. This methodology adequately treats posterior frequency distributions of discrete random variables and maximizes non-necessarily monotonic objective functions within discontinuous search spaces and many local optimums. It is particularly useful when evaluating the objective function is computationally expensive or time-consuming (Sheikholeslami and Razavi 2017).

The method combines the principles of discrete Latin hypercube sampling (DLHS) and iterative optimization techniques. IDLHC has been successfully applied in the oil industry, where production strategy optimization problems are characterized by many discrete random variables in discontinuous search spaces with non-necessarily monotonic objective functions, usually NPV or RF, with many local maximums within a maximization problem (von Filho et al. 2016). Population-based optimization using DLHS best suits this methodology, with consistent convergence to the global optimum, few objective function evaluations, and simultaneous multiple numeric reservoir simulation runs. This easy-to-use, reliable methodology with low computational time and costs is an interesting option for optimization methods in problems of production strategy design related to the oil industry (von Filho et al. 2016).

At the beginning of a given optimization stage, the optimization attributes were defined for the decision variable, such as well location, and corresponding optimization levels were established with prior user-defined probabilities. To initiate the optimization algorithm on the desired problem, a Latin hypercube sampling (LHS) should be performed on the discrete parameter space. LHS ensures that the parameter space is evenly and randomly sampled, preventing any clustering or bias in the initial sample points. Each point in the LHS represents a set of parameter values, forming a population of potential solutions. The objective function, in this study, EMV, is then evaluated for each of these points to determine their fitness. 100 samples in each iteration of the algorithm are evaluated. In the iterative phase, IDLHS refines the initial LHS sample by selecting the top 20% of best-performing solutions and creating a new population based on them. This process continues for a specified number of iterations or until a convergence criterion is met. The refinement of the sample is done using various optimization algorithms like genetic algorithms, simulated annealing, or particle swarm optimization, which guide the search towards better solutions in the parameter space. The combination of LHS and iterative optimization ensures a more efficient exploration of the high-dimensional space, leading to the identification of better solutions while reducing the computational cost compared to traditional exhaustive search methods (Loomba et al. 2022b). Once an optimization stage is completed (i.e., an optimal value of a decision variable is found), the next stage starts, which addresses the optimization of the next decision variable. This sequential optimization framework continues until all decision variables are optimized. As mentioned earlier, this study deals with four optimization stages, where in each stage one, well quantity, location, and type are optimized. In stage two, further tuning of well locations is made. In stage three, the capacity of the platform is optimized, and finally, in stage four, the placement of ICVs is optimized.

In stage one, well type is implicitly optimized as the number and location of wells are optimized, meaning that it does not need to be regarded as an individual optimization variable. For optimizing the well quantity, the vector of optimization attributes has 55 elements, which is equal to the number of candidate wells. Each attribute is of the binary categorical type, with levels of 0 or 1 (0: do not drill, 1: drill). The search space for optimizing the well quantity is 255 in size. After finding the optimal well quantity, the well location is optimized. A relatively large individual region around each well is considered as the search space. Aiming at reducing computations, for wells with large search spaces, only a fraction of the grids is explored. Then, if necessary, unexplored grids are examined in a second stage. In well location optimization, the vector of decision variables contains 2\(\:{\text{N}}_{\text{w}}\) integer variables, where \(\:{\text{N}}_{\text{w}}\) is the number of wells undergoing location optimization in a given stage. In stage three, the decision vector to be optimized contains four optimization attributes, each of which is represented by a set of discretized real values. Platform capacity was optimized for the produced water and liquid, as well as the injected water and gas. In stage four, each well is given a set of feasible ICV configurations. Each configuration represents a default valve location arrangement. 10 and 6 configurations are considered for each producer and injector, respectively. As a result, the length of the decision vector is equal to the number of intelligent wells, and each optimization attribute associated with a producer has ten discrete categorical levels. Similarly, each optimization attribute associated with an injector has six discrete categorical levels. The search space is 10Np × 6Ni, where Np and Ni are the number of intelligent producers and injectors, respectively. The number of sample evaluations in optimization of well quantity, well location, well location fine-tuning, platform capacity, and internal control valve placement were 1314, 1396, 884, 389, and 1014, respectively. Since each sample was tested on nine reservoir models, the corresponding number of simulations is 11,826, 12,564, 7956, 3501, and 9126, respectively. The simulation time for each RM ranged from 124 s to 158 s. In this study, we use the UNISIM group’s cluster. The UNISIM cluster system boasts an approximate combined capacity, including 9 TFlops of processing power, 30 TB of storage, and 3 TB of memory. Additionally, the system automatically allocates CPU and memory resources for simulations based on various factors, such as the level of congestion within the cluster due to other users’ activities. For a more comprehensive and detailed understanding of this optimization algorithm and sequential procedure, it is highly recommend referring to Mirzaei-Paiaman et al. (2023).

Objective functions

NPV and EMV

In this research, the primary objective is to maximize the EMV as the key performance indicator. The EMV represents the weighted average of potential NPVs associated with the project. NPV, in turn, signifies the NPV of future cash flows. This study extensively examines the influence of economic and geological uncertainties in a comprehensive manner. Specifically, three economic scenarios categorized as most-likely, pessimistic, and optimistic, alongside the integration of nine diverse geological reservoir models based on UNISIM-II-D is investigated. Each economic scenario represents a unique set of economic conditions that may influence the project’s outcomes. The optimistic economic scenario embodies a favorable projection, indicating an environment with potential for high economic growth and positive market conditions. The likelihood of this scenario materializing is captured by its assigned probability. Conversely, the pessimistic economic scenario portrays a more adverse outlook, characterized by economic challenges and potential downturns. Its occurrence most-likely quantifies the likelihood of facing such unfavorable conditions. Additionally, the most-likely economic scenario reflects a balanced and moderate outlook, acknowledging the inherent uncertainty in economic conditions. Its probability represents the likelihood of experiencing this intermediate scenario. Table 3 provides the data related to each economic scenario.

Fig. 1
figure 1

The porosity maps corresponding to the 15th layer of the RMs (The black cells represent the faults in these models)

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Mathematically, the EMV is formulated as the summation of NPVi multiplied by the probability of occurrence (pi) for each RMi (Eq. (1)(Mirzaei-Paiaman et al. 2023).

$$\:\text{E}\text{M}\text{V}\:=\:\sum\:_{\text{i}\:=\:1}^{\text{m}}\sum\:_{\text{j}=1}^{\text{k}}{{\text{p}}_{\text{i}}\times\:\text{p}}_{\text{k}}\times\:{\text{N}\text{P}\text{V}}_{\text{i},\text{k}}$$
(1)

In Eq. (1), m represents the number of RMs, pi denotes the likelihood of each RM’s occurrence. Additionally, pk refers to the probabilities of optimistic, pessimistic, and the most-likely economic scenarios. Finally, NPVi, k represents the NPV associated with a specific RM(i) for each economic scenario (k). In this study an objective function known as the NPV considering economic uncertainty (NPVeco), which is associated with each RM is investigated. Equation 1 can be reformulated to incorporate NPVeco as follows:

$$\:\text{N}\text{P}\text{V}\text{e}\text{c}\text{o}\:=\:\sum\:_{\text{k}\:=\:1}^{\text{k}}{\text{p}}_{\text{j}}{\text{N}\text{P}\text{V}}_{\text{i},\text{k}}$$
(2)

The NPV is a financial metric used to evaluate the profitability of an investment or project. It represents the sum of the present values of all expected net cash flows (NCF) associated with the investment over a specific time period. The NPV calculation considers the time value of money, which accounts for the fact that money received in the future is worth less than the same amount received today due to inflation and opportunity costs.

$$\:\text{N}\text{P}\text{V}\:=\:\sum\:_{\text{s}=1}^{\text{T}\text{s}}\frac{{\text{N}\text{C}\text{F}}_{\text{s}}}{{(1+\text{b})}^{\text{t}\text{s}}}$$
(3)

NPV is calculated using net cash flow (NCFs), discount rate (b), and time period (ts). NCFs is the cash flow difference during each period, covering revenues, expenses, etc. Discounting NCFs with a rate (b) adjusts for the time value of money. (b) represents the required return or cost of capital. (ts) is the duration from investment to a point in time. NPV sums discounted cash flows over the total time of the project, indicating the investment profitability. Positive NPV means favorable returns, while negative NPV suggests potential loss. NPV aids investment decisions and accounts for the time value of money.

The NCF is given by:

$$\:\text{N}\text{C}\text{F}=\left(\left(\text{R}-\text{R}\text{o}\text{y}-\text{S}\text{T}-\text{O}\text{C}\right)\times\:\left(1-\text{T}\right)\right)-\text{I}\text{n}\text{v}-\text{A}\text{C}$$
(4)

where R is gross revenue, Roy is royalties, ST is social taxes, OC is operational costs, T is the corporate tax rate, Inv is the platform and wells investment, and AC is the abandonment cost. The platform investment (IP) is calculated as:

$$\begin{aligned}\:&\text{I}\text{P}=417+12.2\times\:{\text{C}}_{\text{p}\text{o}}+3.15\\&\times\:\left({\text{C}}_{\text{p}\text{l}}+{\text{C}}_{\text{p}\text{w}}+{\text{C}}_{\text{i}\text{w}}\right)+\\&9.61\times\:{\text{C}}_{\text{p}\text{g}}+0.1\times\:{\text{n}}_{\text{w}}\end{aligned}$$
(5)

Here, Cpo, Cpl, Cpw, and Ciw are processing capacities, and Cpg is the gas processing capacity, all in specific units per day. nw denotes the number of well slots (Hayashi 2006). Table 3 presents the economic parameters associated with each economic uncertainty.

Production based objective function

Production-based objective functions in reservoir engineering play a crucial role in optimizing oil and gas field development. These functions include the maximization of oil RF, CGP, and COP and the minimization of CWP. Each objective serves as a valuable indicator of reservoir performance and environmental impact. The oil RF is a fundamental metric that assesses how efficiently the reservoir can produce oil over time. It represents the percentage of original oil in place that can be recovered. A higher RF indicates a more productive and sustainable reservoir. COP is a key factor directly related to the recovery factor. It indicates the total volume of oil extracted from the reservoir over time. High COP signifies efficient reservoir management and successful extraction techniques, contributing to the overall recovery factor.

CGP measures the total amount of gas produced from the reservoir over its lifetime. While gas production can be economically beneficial, it may also pose environmental challenges. If not utilized or re-injected efficiently, excess gas may need to be flared, leading to harmful emissions and environmental consequences. Managing gas production to minimize flaring and optimize utilization is vital for sustainable operations.

Table 3 Data for three economic strategies
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CWP represents the total volume of water produced alongside oil and gas. Water production is often an unavoidable byproduct of hydrocarbon extraction. If not managed properly, it can lead to environmental contamination due to the potential toxic material and chemical content. Effective water management practices, such as water re-injection into wells or water treatment, are essential to mitigate environmental impacts and maintain sustainable operations. It is necessary to emphasize that these production-based objective functions are critical in reservoir engineering as they provide valuable insights into reservoir performance, recovery efficiency, and environmental impact. By addressing the economic and operational limitations associated with gas and water production, reservoir engineers can optimize production strategies while ensuring environmental responsibility and sustainable resource management.

In this study, the objective functions were investigated, with a primary focus on the EMV as the main objective function. The changes and behaviors of each objective were studied during different stages of field development optimization. Throughout the research, the variations and interactions of each objective function, including the oil RF, CGP, COP, and CWP, were examined.

Results and discussion

Stage A: well quantity, type, and placement optimization

In this section, we investigate the impact of different objective functions along with the EMV within the context of optimizing the number and location of wells. In this case, where the number and location of wells are considered as decision variables, the solution space for the EMV becomes substantially larger compared to situations where the number of wells is fixed. The optimization procedure is detailed in the work of Mirzaei-Paiaman et al. (2023). The resulting optimal well arrangement consists of 11 production wells and 10 injection wells.

It is essential to highlight that the EMV experienced significant growth during the optimization process, starting from an average value of 0.01 billion dollars in the preliminary iterations and reaching the optimal value of 1.17 billion dollars. This increase in the EMV demonstrates the effectiveness of the proposed methodology.

Figure 2A and B present the RF and COP trends throughout the 13 iterations, respectively, concerning the EMV optimization. Notably, Fig. 3 demonstrates a strong consistency between EMV, RF, and COP, indicating that improvements in EMV result in simultaneous enhancements in RF and COP. In other words, optimizing the EMV in this specific case leads to concurrent optimizations of the RF and COP.

Fig. 2
figure 2

(A) Oil RF versus EMV (B) COP versus EMV (Stage A)

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This observed correlation among EMV, RF, and COP carries significant scientific implications. By evaluating and optimizing the EMV, improvements in both the COP and the RF are inherently driven. The achieved best COP value of 104.8 million m3 and the corresponding 42.2% RF highlight the effectiveness of the optimization process in maximizing oil production while enhancing the overall recovery efficiency. Such consistent and interrelated behavior among the key metrics reinforces the validity of this approach and underscores the significance of EMV as a robust objective function in decision-making for well placement and production strategies.

Figure 3(A) illustrates the relationship between EMV and CGP over 13 optimization iterations. The figure clearly demonstrates the substantial variation in the solution space during the optimization process. CGP ranges from a minimum of 3.22 × 1010 m3 to a maximum of 6.24 × 1010 m3. Several crucial factors must be considered by decision-makers when evaluating the feasibility of gas production in the studied field. These factors include available surface and subsurface equipment, global gas prices, the need for gas production, and compliance with environmental regulations. It is essential to note that the CGP corresponding to the highest EMV obtained in the last iteration was 5.77 × 1010 m3. In cases where there is no facility to re-inject gas into the reservoir or when the gas price is not substantial enough for export, the goal should be to minimize gas production. Point A is the minimum achievable CGP at the highest EMV. This point in the 9th iteration reveals the lowest CGP value of 4.12 × 1010 m3, coupled with a reasonably high EMV of 1.1 billion dollars. Decision makers may choose to sacrifice 6.2% of the EMV value (1.1 billion dollars) to reduce the total gas production (4.12 × 1010 m3) by 29%, which holds significant importance, particularly due to its potential impact on vulnerable environments like offshore fields or those near preservation areas.

Figure 3(B) presents the CWP alongside the evolution of the EMV throughout the optimization process. The generation of excess water during petroleum production has consistently posed considerable challenges. These challenges encompass various issues, including infrastructure deterioration both on the surface and subsurface, elevated levels of toxicity that require costly specialized treatment, and the need for robust pumps to reinject the water into the reservoir with substantial energy consumption. Analyzing Fig. 3 (B) reveals that the peak EMV corresponds to a CWP of 5.67 × 107 m3. Point B (the minimum achievable CWP at the highest EMV) in Fig. 3 (B) is characterized by an EMV of 1.104 billion dollars and a CWP of 2.95 × 107 m3. This configuration achieves a 47.9% reduction in water production at the cost of a 6.2% reduction in EMV. Crucial determinants in selecting between these alternatives include company policies and adherence to environmental regulations. These points merely represent a sampling of the multitude of solutions attainable through the optimization process. Consequently, informed decision-making hinges on careful consideration of these facets to strike a balance between economic gains and environmental responsibilities.

Fig. 3
figure 3

(A) CGP versus EMV, (B) CWP versus EMV (Stage A)

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In this study, as mentioned earlier, robust optimization is conducted by employing nine distinct reservoir models that represent the total sub-models of the UNISIM-II-D reservoir. These models capture the intricate geological complexities inherent in reservoirs and, as a result, yield differing NPVs. It is worth noting that each of these reservoir models is subjected to three distinct economic scenarios. Figure 4 graphically presents the NPV values under economic uncertainty, termed NPVeco. As illustrated in the figure, the behavior of NPVeco for each reservoir model in response to EMV showcases intriguing divergence. This divergence is attributable to the unique geological attributes of each reservoir model. In panels Fig. 4-RM1, RM2, RM5, RM6, RM7, RM8, and RM9 the reservoir model consistently aligns with the EMV trend. In other words, an increase in EMV corresponds to an elevation in NPVeco. Conversely, in panels Fig. 4-RM3 and RM4 (the two most pessimistic scenarios for UNISIM-II-D), the relationship between NPVeco and EMV is conflicting. In these cases, an upswing in EMV does not necessarily entail a parallel increase in NPVeco.

Given the perpetual uncertainty surrounding the true reservoir model and its associated properties, it becomes crucial to incorporate the pessimistic reservoir model scenario in the context of NPVeco. Embracing this worst-case scenario as a RM within this optimization process is imperative. Such an approach bolsters one’s confidence in the potential outcomes of the optimization endeavors. Consequently, the strategy of maximizing outcomes under the worst-case scenario, which is inherently pessimistic, ensures a comprehensive exploration of potential outcomes. This includes ensuring that the NPVeco of the worst-case reservoir models is also maximized. Consequently, one of the primary focal points for future studies is the optimization of NPVeco for this worst-case reservoir model scenario, concurrently considering the EMV. Illustrated in Fig. 4, the worst-case reservoir models (RM3 and RM4) present the highest NPVeco, reaching 0.826 and 0.508 billion dollars, respectively. Meanwhile, the EMV corresponding to this optimal value is 1.027 billion dollars for RM3 and 1.037 billion dollars for RM4. It is noteworthy that when exclusively focusing on EMV, the NPVeco value corresponding to the worst-case scenarios diminishes by 25% for RM4 and 15% for RM3.

The decision-making process becomes notably intricate in this scenario. Decision-makers are confronted with the challenge of choosing between agreeing upon the final EMV value and embracing the highest NPVeco value derived from the worst-case scenario. Each value holds its own distinct significance within the context of substantial projects, and both offer unique insights. In essence, this situation poses a profound scientific and strategic challenge. The balance between the quantitative metrics of EMV and NPVeco requires careful consideration and deliberation.

Fig. 4
figure 4

NPV of each RM considering economic uncertainty (Stage A)

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Stage B: well placement optimization (fine-tuning)

In the preceding stage, the optimal number of wells required to maximize the EMV was determined. The position of each well was established using the Mirzaei-Paiaman et al. (2023) methodology. The optimization process in this stage focused on refining the position of 21 production and injection wells. The outcomes of this optimization revealed a notable 16.4% increase in the final EMV, which consequently surged to $1.370 billion. This stage involves a comprehensive analysis of the behavior exhibited by each individual objective in relation to the EMV. This exploration aims to determine whether the pursuit of maximizing the EMV leads to favorable outcomes for other objectives. By delving into these interactions, the broader implications and potential benefits of maximizing the EMV can be better assessed.

Figure 5 illustrates the relationship between EMV and both RF and COP. Notably, both objective functions exhibit consistent behaviors with respect to EMV variations. It is important to note that, due to the fixed number of wells, the range of EMV observed in this figure is not as expansive as in the previous step. This reduction in range consequently leads to a more constrained solution space for both objective functions.

Analyzing Fig. 5(A) and 5(B), the optimal values for RF and COP, specifically 41.67% and 9.47 × 107 m3, respectively, are identified.

Fig. 5
figure 5

(A) Oil RF versus EMV; (B) COP versus EMV (Stage B)

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The CGP is illustrated in Fig. 6(A). As discussed earlier, in accordance with the production company’s policy, one potential strategy involves minimizing gas production while achieving the highest EMV. In Fig. 6(A), Point A (the minimum achievable CGP at the highest EMV) highlights the reasonable minimum of CGP at 5.58 × 1010, yielding an EMV of 1.34 billion dollars. The corresponding CGP value for the EMV-maximizing point is 5.69 × 1010 m3. Opting for the minimum gas production strategy results in a 2.1% EMV reduction, translating to a 2.2% decrease in gas production. The decision-making process should be meticulous, particularly when this strategy is chosen.

CWP, illustrated in Fig. 6(B), holds intrinsic importance in oil and gas projects due to its environmental and operational implications. While focusing on optimizing EMV, minimizing CWP emerges as another important objective, considering its far-reaching effects on surface facilities, environmental stability, and project sustainability. Figure 6(B) demonstrates that, at the maximum EMV strategy, CWP stands at 3.62 × 107 m3. However, an alternative route considering Point B, the minimum achievable CWP at the highest EMV (3.29 × 107 m3), triggers an 8.9% reduction in CWP and lowers EMV by 1.6%, transitioning from 1.37 billion dollars to 1.35 billion dollars.

This intricate interplay between CWP and EMV requires meticulous evaluation to discern the superior strategy for overall project advancement. Crucial variables involve the ecological ramifications of water extraction, the economic load of water treatment and disposal, the toll on surface infrastructure, and the financial benefits resulting from higher EMV. Additional points of consideration involve potential environmental degradation from water contamination and habitat disturbance, the considerable financial burden associated with water treatment and disposal, and the potential corrosive and scaling impact on surface facilities due to increased water production. The strategic choice between minimizing CWP or maximizing EMV constitutes a multifaceted decision, contingent on specific project circumstances. By undertaking a comprehensive assessment of all pertinent factors, decision-makers can aptly navigate this choice, steering the project toward optimal outcomes for stakeholders and the environment.

As discussed in the previous section, the NPV of each RM was examined relative to the EMV, considering economic uncertainty. This is shown in Fig. 7 for stage B. It is evident that only the two most pessimistic RMs (RM3 and RM4) exhibit conflicting behavior, while the other RMs (RM1, RM2, RM5, RM6, RM7, RM8, and RM9) exhibit consistent behavior towards EMV. As shown in Fig. 7, if EMV is maximized, the value of RM3 and RM4 does not necessarily maximize, as is the case for the other RMs.

More importantly, given the unknown and complex behavior of the true reservoir, full confidence can be placed in the accurate representation of the reservoir by all the other RMs. Therefore, it is crucial to obtain the highest value for these two pessimistic RMs while also achieving the maximum EMV. This decision may improve the reliability of the selected strategy. In other words, it can be certain that this project will be profitable if a maximum value for the pessimistic case is available.

Fig. 6
figure 6

(A) CGP versus EMV, (B) CWP versus EMV (Stage B)

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Fig. 7
figure 7

NPV of each RM considering economic uncertainty (Stage B)

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Stage C, platform liquid capacity

In this stage, four platform capacity parameters were taken into consideration in relation to produced and injected fluids, encompassing liquids and gases, for the purpose of optimization. To perform the optimization, the liquid production capacity was systematically divided into a range of 21,118 to 28,618 cubic meters per day. In a similar manner, water production capacity was discretized between 10,079 and 19,079 cubic meters per day. For water injection capacity, the discretization spanned 20,157 to 38,157 cubic meters per day, while gas injection capacity was discretized from 6 × 106 m3/day to 8 × 106 m3/day. The comprehensive optimization procedure is outlined in the work by Mirzaei-Paiaman et al. (2023). Compared to the earlier location optimization stage, the EMV increased by 8.01% to $1.48 billion dollars in this phase.

Since the type, number, and location of the wells remain constant, the fluctuations in EMV do not appear to be of significant magnitude. Examining Fig. 8(A) and 8(B), it is evident that the upward trajectory of the RF and COP surpasses that of EMV variations. This trend is similarly reflected in Fig. 8(C) and 8(D), where instances of peak gas and water production coincide with high points in the EMV.

When considering a strategy to decrease gas production, Fig. 8(C) offers valuable insights. To achieve a reduction of more than 13.6%, reducing it from 5.64 × 1010 m3 to a minimum of 4.87 × 1010 m3, a concession of just over 5% of the EMV is required, as illustrated by Point A (the minimum achievable CGP at the highest EMV) in Fig. 8(C). Notably, opting for point B (the minimum achievable CWP at the highest EMV) in Fig. 8(D) results in a substantial 37% reduction in CWP, from 3.78 × 107 m3 to 2.38 × 107 m3. This strategic adjustment only requires a marginal EMV concession, approximately 4.7%, which means a decrease from 1.48 billion to 1.41 billion dollars.

The significance of adopting a gradual decision-making approach is underscored by the impact observed. A modest reduction of 5% in the peak EMV results in substantial cascading effects. Specifically, a reduction of over 37% in CWP and an approximate 13.6% decrease in CGP are brought about. This highlights the effectiveness of a deliberate and selective strategy, wherein minor adjustments to the EMV lead to considerable gains in resource conservation. This outcome reflects a management approach characterized by astuteness and informed decision-making.

Fig. 8
figure 8

(A) Oil Recovery Percentage versus EMV (Stage C), (B) COP versus EMV (Stage C), (C) CGP versus EMV (Stage C), (D) CWP versus EMV (Stage C)

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In contrast to the initial two stages, where a relatively high degree of variation in the EMV and NPV of each RM was observed, a dissimilar trend in the NPV of each RM is now being noted during this stage (Fig. 9). It is worth noting that, as EMV exhibits decreased variation in this stage, a corresponding reduction in NPV variation is seen in each RM. This trend is illustrated in Fig. 10, except for RM3 and RM1. Out of the nine RMs analyzed, seven display a consistent pattern of behavior aligning with EMV optimization. However, RM3 and RM1 stand apart in this context. Their NPV values do not undergo as significant alterations as the other RMs. This suggests that adjustments in the liquid production capacity of the platform have a relatively minor impact on NPV within these two reservoir sub-models. This phenomenon can be attributed to the unique characteristics and properties of these reservoirs.

Fig. 9
figure 9

NPV of each RM considering economic uncertainty (Stage C)

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Stage D, ICV location optimization

For optimization of ICV location, each well is given a set of feasible ICV configurations, each of which represents a predetermined ICV location arrangement. More intricate details regarding the optimization procedure can be found in the work of Mirzaei-Paiaman et al. (2023). Notably, even less variation in EMV is observed in this stage compared to the prior three stages, as depicted in Fig. 10. The EMV range spans from a minimum of 1.21 to a maximum of 1.51 billion dollars. Like stage C, the RF and COP exhibit consistent behaviors with EMV during optimization (Fig. 10(A) and (B).

Once again, for CGP, by accepting a reduction in EMV to 1.50 (a 0.6% reduction from 1.51 billion dollars), a 4% decrease in gas production, equating to 5.51 × 1010 m3 from 5.69 × 1010 m3 (Fig. 10(C)-Point A (minimum achievable CWP at the highest EMV) can be achieved. Similarly, in this stage, notable reductions in water production by making a modest 3.3% sacrifice in EMV, transitioning from 1.51 to 1.46 billion dollars can be achieved. This results in a substantial 13.8% reduction in water production, decreasing it from 3.96 × 107 m3 to 3.42 × 107 m3 (Fig. 10(D)-Point B (minimum achievable CWP at the highest EMV)). This reduction in both water and CGP holds significant implications as it mitigates environmental impact, reduces maintenance costs, and curtails water treatment expenses.

As expected, the variation of NPV in each RM is insignificant compared to the previous three stages, given the small variation in EMV. In Fig. 11, it becomes evident that all NPV solutions are clustered closely around a single solution across all RMs. Therefore, no interpretation of the variation of NPV can be established in this case.

Fig. 10
figure 10

(A) Oil Recovery Percentage versus EMV (Stage D), (B) COP versus EMV (Stage D), (C) CGP versus EMV (Stage D), (D) CWP versus EMV (Stage D)

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Based on the results, investigations indicate that there is a trade-off between EMV and CGP. In the initial stage (Stage A), a judicious reduction of 6.2% in the maximum achievable EMV can result in a substantial reduction of 29% in CGP. In the subsequent stage (Stage B), a more modest reduction of 2.1% in EMV yields a 2.2% reduction in CGP. Moving forward to Stage C, a 5% reduction in EMV is associated with a noteworthy 13.7% decrease in CGP. Finally, in Stage D, a nuanced approach to decision-making allows for a minimal 0.6% reduction in EMV to be coupled with a consequential 4% reduction in CGP. Table 4 provides a summary of the reductions in EMV and CGP at each stage.

This study underscores the significant findings across the four stages, highlighting their importance in the context of CWP. In Stage A, analysis reveals that by implementing a strategic reduction of merely 6.2% in EMV, it can be realized a remarkable 47.9% reduction in CWP. Transitioning to Stage B, a more modest 1.6% reduction in EMV is associated with an appreciable 8.9% reduction in CWP. It is shown that an alternative strategy in Stage C results in a sacrifice of only 4.7% in EMV, yielding a substantial 37% reduction in CWP. Finally, in Stage D, a nuanced strategy that entails a minor 3.3% reduction in EMV leads to a substantial 13.8% reduction in CWP (Table 5).

Fig. 11
figure 11

NPV of each RM considering economic uncertainty (Stage D)

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Table 4 Reduction of EMV and CGP in each stage based on alternative strategy
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Table 5 Reduction of EMV and CWP in each stage based on alternative strategy
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Conclusions

This study underscores the importance of incorporating multiple objective functions alongside EMV to provide a comprehensive framework for guiding decision-making processes in oilfield development. By examining various objective functions in conjunction with EMV as the primary criterion, we gain valuable insights into the complexities of well placement and production strategies. We underscore the importance of balancing economic gains with environmental responsibilities to ensure sustainable practices. Furthermore, findings emphasize the potential benefits of minimizing CGP and CWP to mitigate environmental impacts and optimize resource utilization. The results can be summarized as follows:

  • This study examines the sensitivity of objective functions to optimization variables across different stages of the optimization process.

  • A strong correlation is identified between EMV and COP, emphasizing EMV’s role in driving improvements in COP and RF.

  • The study advocates for a balanced approach that considers both economic benefits and environmental responsibilities, proposing modified strategies to reduce CGP and CWP. The study demonstrates that strategically prioritizing CGP and CWP reduction can be achieved with minimal sacrifice to EMV. Significant reductions in CGP (ranging from 4 to 29%) were achieved across all stages with minimal EMV reduction (between 0.6% and 6.2%). Similar to CGP, substantial reductions in CWP (ranging from 8.9 to 47.9%) were obtained with minor EMV sacrifices (between 1.6% and 6.2%).

  • Insight is provided into the potential environmental benefits of minimizing gas and water production, including reductions in greenhouse gas emissions and the preservation of groundwater resources.

For future studies, one can consider multi-objective optimization, which is a more robust and sustainable approach to oilfield development than solely relying on EMV. Multi-objective optimization allows decision-makers to consider multiple objectives simultaneously, such as maximizing EMV, maximizing the NPV of the worst-case reservoir model, and minimizing water and gas production.