Reduced-Order Modelling for Production Optimisation

Summary The determination of optimal well settings is very demanding computationally because the simulation model must be run many times during the course of the optimization. For this reason, reduced-order modeling procedures, which are a family of techniques that enable highly efficient simulations, may be very useful for optimization problems. In this paper, we describe a recently developed reduced-order modeling (ROM) technique that has been used in other application areas, the trajectory piecewise linearization (TPWL) procedure, and incorporate it in production-optimization computations. The TPWL methodology represents solutions encountered during the optimization runs in terms of Taylor-series expansions around previously simulated states. This requires a small number of preprocessing (training) simulations using the full (high-fidelity) model, during which pressure and saturation states and Jacobian matrices are saved. These states and matrices are then projected into a low-dimensional space using proper orthogonal decomposition (POD). Simulations in this reduced space can be performed very efficiently; in this work, we observe runtime speedups of a factor of 450. Overall speedups are, however, less because of the preprocessing overhead. We assess the TPWL representation for simulations of waterflood in a heterogeneous 3D model containing more than 20,000 gridblocks and six wells. The high degree of accuracy of the TPWL model is first demonstrated for several testing simulations in which producer- and injector-well settings differ from those used in the training runs. The TPWL representations are then used in optimizations involving the determination of optimal bottomhole pressures (BHPs) for a reservoir model with four production wells and two injection wells. A gradient-based algorithm is applied for the optimizations. In the first case, the BHPs of the producers and injectors are optimized at six different times (36 control variables) and in the second case at 15 different times (90 control variables). Results for optimized net present value (NPV) using TPWL are shown to be in consistently close agreement with those computed using high-fidelity simulations. Most significantly, when the optimal well settings obtained using the TPWL procedure are applied in high-fidelity models, the resulting NPVs are within approximately 0.5% of the values determined using the high-fidelity simulations. Our overall conclusion is that the TPWL representation may be quite useful in production-optimization problems.

SummaryIn this work, we investigate the efficient estimation of the optimal design variables that maximize net present value (NPV) for the life-cycle production optimization during a single-well carbon dioxide (CO2) huff-n-puff (HnP) process in unconventional oil reservoirs. A synthetic unconventional reservoir model based on Bakken Formation oil composition is used. The model accounts for the natural fracture and geomechanical effects. Both the deterministic (based on a single reservoir model) and robust (based on an ensemble of reservoir models) production optimization strategies are considered. The injection rate of CO2, the production bottomhole pressure (BHP), the duration of injection and the production periods in each cycle of the HnP process, and the cycle lengths for a predetermined life-cycle time can be included in the set of optimum design (or well control) variables. During optimization, the NPV is calculated by a machine learning (ML) proxy model trained to accurately approximate the NPV that would be calculated from a reservoir simulator run. Similar to the ML algorithms, we use both least-squares (LS) support vector regression (SVR) and Gaussian process regression (GPR). Given a set of forward simulation runs with a commercial compositional simulator that simulates the miscible CO2 HnP process, a proxy is built based on the ML method chosen. Having the proxy model, we use it in an iterative-sampling-refinement optimization algorithm directly to optimize the design variables. As an optimization tool, the sequential quadratic programming (SQP) method is used inside this iterative-sampling-refinement optimization algorithm. Computational efficiencies of the ML proxy-based optimization methods are compared with those of the conventional stochastic simplex approximate gradient (StoSAG)-based methods. Our results show that the LS-SVR- and GPR-based proxy models are accurate and useful in approximating NPV in the optimization of the CO2 HnP process. The results also indicate that both the GPR and LS-SVR methods exhibit very similar convergence rates, but GPR requires 10 times more computational time than LS-SVR. However, GPR provides flexibility over LS-SVR to access uncertainty in our NPV predictions because it considers the covariance information of the GPR model. Both ML-based methods prove to be quite efficient in production optimization, saving significant computational times (at least 4 times more efficient) over a stochastic gradient computed from a high-fidelity compositional simulator directly in a gradient ascent algorithm. To our knowledge, this is the first study presenting a comprehensive review and comparison of two different ML-proxy-based optimization methods with traditional StoSAG-based optimization methods for the production optimization problem of a miscible CO2 HnP.

Naturally fractured reservoirs present large development risks due to the uncertainty nature of complex subsurface fracture distributions and heterogeneous petrophysical properties. therefore, it is important to consider geological uncertainties when performing field development optimization for fractured reservoirs. The objective of this study is to propose a robust optimization method capable of considering multiple fractured model realizations while performing bi-objective optimizations focused on maximizing field Net Present Value (NPV) and minimizing specific risk metrics. In this study, the embedded discrete fracture model (EDFM) was used to perform forward simulations for fracture reservoirs, and the stochastic simplex approximate gradient (StoSAG) algorithm was adopted as the optimizer. With respect to the optimization target, we considered both the mean NPV of multiple realizations and a specified risk metric of NPV from the worst-case scenario, the standard deviation, or conditional value at risk (CvaR). The Lexicographic method was employed to handle the bi-objective optimization. This method was applied to the EGG base model, which is commonly used in the field of reservoir development optimization. Fractures with different distribution patterns were embedded in the model to represent fractured reservoirs, and 10 permeability fields were used to consider permeability distribution uncertainties. The optimization results for the three selected risk metrics showed that the range of decreasing in the mean NPV was from 0.04% to 0.31 %, with the worst scenario final NPV increasing by $1.26×106, the standard deviation of the set decreasing by $2.84×106, and the CVaR at 20% increasing by $3.33×105. Although the range of change in each metric for this case was not significant compared to their original magnitudes, the changes in each optimization curve showed a positive improvement trend. This study introduced a bi-objective robust production optimization algorithm, based on the StoSAG and Lexicographic methods, designed to address production optimization of complex fractured reservoirs considering geological uncertainties. Three different risk metrics were investigated to minimize development risk while optimizing the expected field NPV, which can be critical to minimize field development risks when performing production optimization tasks.

Reduced-order flow modeling and geological parameterization for ensemble-based data assimilation

Comparison of SQP and AL algorithms for deterministic constrained production optimization of hydrocarbon reservoirs

Summary This paper presents a realistic deepwater application of a deep learning (DL)-based reservoir surrogate model in forecasting reservoir performance (via the prediction of state variables and well outputs) and nonlinearly constrained production optimization under geological uncertainty by utilizing an efficient gradient-based method. The reservoir surrogate model is based on the multimodel embed-to-control observe (E2CO) architecture, consisting of four blocks of neural networks (NNs): encoder, transition, transition output, and decoder. In this work, the surrogate model is coupled with the powerful stochastic-gradient-based line-search sequential quadratic programming (LS-SQP) workflow to handle robust production optimization in the presence of nonlinear state constraints. This is the first time the E2CO-based production optimization method driven by a novel nonlinear optimization is applied to a reservoir model with very realistic characteristics of a deepwater reservoir (not a toy reservoir model) in terms of structural architecture and flow property distribution in an efficient data-driven manner using the multimodel E2CO DL surrogate as detailed by Nguyen and Onur (2025). Five realizations of the synthetic realistic Reservoir X model with different channelized heterogeneous permeability distributions under waterflooding are used for showcasing the prediction and optimization performances of the DL-based framework. The results from this framework are directly and quantitatively compared with those simulated using a commercial high-fidelity reservoir simulator (HFS).

Summary In the process of waterflood development in oil and gas reservoirs, the injection/production ratio is one of the key indicators affecting both the development effectiveness and economic benefits. A reasonable injection/production ratio directly impacts the development parameters, such as reservoir pressure maintenance levels and water-cut rise rates, and it helps maintain high production capacity in the oil field, ultimately achieving a higher recovery factor. However, existing constrained-waterflood reservoir production optimization methods often face challenges such as time-consuming calculations and the risk of getting trapped in local optima when dealing with such nonlinear constraint problems, making it difficult to effectively achieve a balance between constraints and economic benefits. To address this issue, we propose an injection/production ratio–constrained production optimization method based on the worst-case soft actor-critic (WCSAC) algorithm. This method effectively handles the constraint issue by introducing a safety critic, ensuring the injection/production ratio constraint is met, and maximizing economic benefits throughout the entire life cycle. Specifically, this method models the injection/production ratio–constrained production optimization problem as a constrained Markov decision process (CMDP). On the one hand, it uses conditional value at risk (CVaR) as a safety metric to assess the safety risks of the policy and achieves a balance between reward and safety by adjusting adaptive safety weights. On the other hand, it uses the maximum entropy mechanism to enhance policy exploration and improve the efficiency of global optimization. This results in the training of a stable and reliable safe reinforcement learning (RL) agent to achieve injection/production ratio–constrained production optimization. The agent can adjust and determine the production optimization plan that satisfies the injection/production ratio constraint in real time based on the current reservoir development status, without the need for retraining, thus reducing computational time. To validate the performance of the proposed method, tests were conducted in two reservoir models. The analysis results indicate that, compared with other optimization methods, the proposed method can achieve a higher net present value (NPV) while satisfying the injection/production ratio constraints, demonstrating good model applicability.

Linearized reduced-order models for subsurface flow simulation

It is challenging to make optimal field development and reservoir management decisions with diminishing resources and low-emission requirements. For an optimal exploitation of the reservoir fluids, it is necessary to introduce advanced digital tools and account for geological uncertainty when making decisions. This paper discusses an ensemble-based probabilistic decision-making workflow for closed-loop reservoir management. The reservoir model is a synthetic but realistic model with oil, gas, and water. We show how to use ensemble methods in a workflow for integrated uncertainty prediction, history matching, and robust optimization. The workflow uses advanced ensemble-based history-matching techniques to update a reservoir model in the annual maturation process. After the history-matching update, an optimization process decides the best wells to drill next to gain the highest net present value. An additional robust decision step evaluates the wells over the ensemble of reservoir models to ensure they lead to a positive ensemble-averaged net present value. The approach allows for up-to-date predictions, including uncertainty estimates, leading to improved decision support for field development, well-planning, production strategies, and reservoir management.

This paper presents a physics-based data-driven method to learn predictive reduced-order models (ROMs) from high-fidelity simulations and illustrates it in the challenging context of a single-injector combustion process. The method combines the perspectives of model reduction and machine learning. Model reduction brings in the physics of the problem, constraining the ROM predictions to lie on a subspace defined by the governing equations. This is achieved by defining the ROM in proper orthogonal decomposition (POD) coordinates, which embed the rich physics information contained in solution snapshots of a high-fidelity computational fluid dynamics model. The machine learning perspective brings the flexibility to use transformed physical variables to define the POD basis. This is in contrast to traditional model reduction approaches that are constrained to use the physical variables of the high-fidelity code. Combining the two perspectives, the approach identifies a set of transformed physical variables that expose quadratic structure in the combustion governing equations and learns a quadratic ROM from transformed snapshot data. This learning does not require access to the high-fidelity model implementation. Numerical experiments show that the ROM accurately predicts temperature, pressure, velocity, species concentrations, and the limit-cycle amplitude, with speedups of more than five orders of magnitude over high-fidelity models. Our ROM simulation is shown to be predictive 200% past the training interval. ROM-predicted pressure traces accurately match the phase of the pressure signal and yield good approximations of the limit-cycle amplitude.

Successfully implementing polymer flooding and maximizing benefits requires selecting best options of variables such as polymer concentration and slug, and number and location of new wells. Option-decisions combined generate thousands of scenarios. Therefore, even using smart algorithm optimizers to efficiently find the maximum of a business objective mathematical function can be a very time-consuming process. The objective of this article is to demonstrate a methodology to improve the chances of finding the maximum net present value (NPV) solution for planning an offshore polymer-flooding; this includes finding, for example, the minimum injection volume to more easily offshore operations. The reservoir and economy models included here were automatically coupled with software that encloses a smart numerical algorithm for searching complex maximum/minimum functions. The options of the previously mentioned decision variables were selected to maximize the NPV of the inverted five-spot polymer-flooding project under constrained rig availability. The process was conducted in three stages: Stage 1: potential value estimation Stage 2: narrowing options through deterministic numerical simulation Stage 3: A) numerical optimization of all decision variable options except the drilling sequence; B) numerical optimization of the drilling sequence A total of 379 scenarios were numerically forecasted in just a few months. The best scenario showed three times the NPV of the nonflooding case. Compared to the reference water-flooding scenario (i.e., all the same options but with the fluid injected), the NPV was 1.3 times greater, the water-oil ratio (WOR) was 0.45 times lower, and Np was 1.25 times greater. Unobvious scenarios, such as reducing the yearly drilling rig availability but extending drilling by four years, were revealed. A comparison of the working time for Stage 2 with Stages 3A and 3B showed that the numerical optimization is six times faster per scenario generated. This study demonstrates that the use of numerical algorithms of polymer flooding yields a significant incremental value over traditional deterministic simulations in a much shorter time frame and with fewer costs compared to previous steps related to building a reservoir model. It is expected to be applicable to all types of enhanced oil recovery (EOR) processes.

Summary In the process of oil and gas reservoir development, the natural heterogeneity of the reservoir makes the development performance extremely sensitive to the matching of the injection-production well control scheme. Reasonable production optimization is the key to achieving balanced displacement and efficient development. Due to the complexity of oil and gas reservoirs and the large number of variables, traditional production optimization methods often face problems, such as complex and time-consuming calculations, hindering their ability to obtain global optimal solutions. Reinforcement learning (RL), with its intrinsic strengths in sequential decision-making and high-dimensional problem solving, offers a promising alternative for production optimization. Nevertheless, existing RL approaches exhibit limited exploration capabilities in complex reservoir environments, often becoming trapped in local optima. Additionally, they typically fail to incorporate well pattern information effectively, overlooking the location information and the connectivity between injection and production wells. To address these challenges, we propose a dual-reward coupled RL production optimization algorithm based on a global perspective. Specifically, the production optimization problem will be modeled as a Markov decision process. Through the intrinsic reward mechanism, the agent is encouraged to conduct deeper exploration to overcome the optimization limitations in complex environments, and the well network production information is integrated into the model in combination with the graph neural network (GNN) to enhance the modeling ability of the dynamic characteristics of the reservoir. In this way, the agent can effectively avoid relying only on local objectives for optimization and then achieve a more balanced injection and production strategy. The proposed method adaptively learns by engaging in intrinsic-extrinsic directional interactions with the uncertain reservoir environment, effectively leveraging accumulated well control experience in a manner that is similar to real-world field operating mode. Simulation results derived from two reservoir models demonstrate that, in comparison with other optimization methods, the proposed approach achieves higher net present value (NPV) and exhibits outstanding performance in oil displacement.

Field S has undergone through few stages of static modelling in order to improve the quality of the reservoir model as more data became available to the team. The most recent static model was deemed necessary as to add value of 1. New seismic acquisition of 3D 4C OBC 2. New Environment of Depositional (EOD) concepts 3. A higher resolution approach to well log correlation 4. Updated inputs including petrophysics, FMI and well tests This paper aims to demonstrate the improvement in reservoir modelling through the recent acquisition of a 4-Component 3-Dimension Ocean Bottom Cable (4C 3D OBC) seismic survey, new sedimentological study, a higher resolution approach to well log correlation and updated petrophysical inputs including FMI and well test data. In essence all available subsurface data should be integrated to produce the most optimum reservoir model to ensure the best technical result possible. This effort requires collaborative and proactive team work that has proven to be successful in Filed S where the Delta Group reservoir has now been managed to have a reliable history match compared to previous reservoir models

Aquaculture in Floating Net Cage (FNC) is one of the reliable aquaculture technologies that is used to optimize the utilization of waters on rivers, lakes and reservoirs. The types of fish that is usually preserved in Floating Net Cage are Gold Fish ( Cyprinus carpio ) and Nile Tilapia ( Oreochromis niloticus ). Survey method is applied in this research and data sample is drawn from the farmer population of floating net cages in Kecamatan Pinoh Utara. The objective of this study is to determine the financial feasibility of aquaculture with Floating Net Cage (FNC) system. This achieved by considering Net Present Value (NPV), IRR (Internal Rate of Return), Net Benefit-Cost Ratio (B/C Ratio), Analysis of PP (Payback Periods) and Sensitivity Analysis. The result indicates that aquaculture of Nile Tilapia and Gold-Fish with Floating Net Cage (FNC) system was financially significant shown by Net Presents Value (NPV) 227.246.769,84 rupiahs, Internal Rate of Return (IRR) 190,97%, Net Benefit Cost ratio (B/C Ratio) 4,34 and Payback Periods (PP) 0,53 years (6 months 10 days). The sensitivity analysis indicates that if there was an increase in operational costs, the amount should not be more than 304,19% and if there was a decrease in profits, the amount should not be more than 44,11%. Keywords: Financial Analysis, Aquaculture, Floating Net Cage, Net Present Value (NPV), Internal Rate of Return (IRR), Net Benefit-Cost Ratio (B/C Ratio).

In reservoir management, many decisions are made considering model-based production forecasts and optimization processes. These approaches can generate biases and the actual production and economic return may be overestimated. One of the reasons for these biases is the optimization process itself (procedure bias). Thus, the objective of this work is to investigate biases caused by model-based optimization processes using synthetic benchmark cases, analyzing the magnitude and the impact on future decisions. We use synthetic benchmarks composed of: (1) an ensemble of data-assimilated simulation models; (2) a subset of this ensemble, named Representative Models (RMs); (3) a reference case, used as the real response of the reservoir (ground truth). Two case studies are analyzed: one focused on design variables (development phase), and the other on control variables (management phase). We demonstrate how specialized and robust strategies (resulting from nominal and robust optimizations, respectively) behave in relation to the ensemble of models and in relation to the reference case, using Net Present Value (NPV) and Expected Monetary Value (EMV) as objective functions. The results confirm the presence of bias and overestimated forecasts caused by optimization processes. In Case Study 1 (development phase), the robust strategy showed an expected return improvement of 45% due to optimization, while the actual gain was only 6%. Specialized strategies presented differences between expected and actual economic gains ranging from 38% to 179% (with an average of 79%). In Case Study 2 (management phase), the robust strategy yielded a 4.1% expected increase in economic return compared to a 2.5% actual gain, with specialized strategies showing an average overestimation of 38% for the specialized strategies. The bias was stronger in Case Study 1 due to the greater impact of development variables on reservoir performance. Risk curve and boxplot analyses showed that strategies tend to become overly specialized to the model in which they were optimized, may leading to suboptimal decisions when applied to the real field. By employing synthetic benchmarks with known reference cases, this work quantifies the overestimation introduced by optimization processes, providing valuable insights to help practitioners recognize and account for procedure bias, reducing the risk of overconfident model-based decisions in real-field applications.