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The main objective of this work is to investigate efficient estimation of the optimal design variables that maximize net present value (NPV) for the life-cycle production optimization during a single-well CO2 huff-n-puff (HnP) process in unconventional oil reservoirs. This work extends our previous work where we considered only well control variables such as injection rate and production BHP, and duration of injection and production periods as the optimal design variables using a single, simple unconventional reservoir model ignoring the effects of double permeability and geomechanical effects in life-cycle production optimization. In this work, we also add length of each cycle as a design variable into set of our design variables. A more realistic unconventional reservoir model is considered, where Bakken oil composition is used as reservoir fluid, and natural fractures and geomechanical effects are considered. In addition, applications of robust life-cycle optimization treating uncertainty in reservoir model by a set (ensemble) of reservoir models and maximizing NPV over a suite of reservoir models are given. 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. As ML algorithms we used both least-squares support vector regression (LS-SVR) and Gaussian process regression (GPR). Given forward simulation results with a commercial compositional simulator that simulates miscible CO2 HnP process a proxy is built based on the ML method chosen. Having the proxy model, we use it in the iterative training-optimization algorithm directly to optimize the design variables. As an optimization tool the sequential quadratic programming (SQP) method is used inside this iterative training-optimization algorithm. Computational efficiencies of the ML proxy-based optimization methods are compared with that of the conventional stochastic simplex approximate gradient (StoSAG) method and/or simplex gradient method. Our results show that the LS-SVR and GPR based proxy models prove to be accurate and useful in approximating NPV in optimization of the CO2 HnP process. The results also indicate that both the GPR and LS-SVR methods exhibit very similar convergence rates and require similar computational time for optimization. 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 the best of our knowledge, this work is the first presenting a detailed investigation of LS-SVR and GPR applications in comparison with StoSAG and simplex to the optimal well-control problem for a complex miscible CO2 HnP process in unconventional oil reservoirs. We provide insight and information on proper training of the SVR and GPR proxies for this type life-cycle production optimization problem.
The main objective of this work is to investigate efficient estimation of the optimal design variables that maximize net present value (NPV) for the life-cycle production optimization during a single-well CO2 huff-n-puff (HnP) process in unconventional oil reservoirs. 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. The ML proxy model can be obtained with either least-squares support vector regression (LS-SVR) or Gaussian process regression (GPR). Given forward simulation results with a commercial compositional simulator that simulates miscible CO2 HnP process in a simple hydraulically fractured unconventional reservoir model with a set of design variables, a proxy is built based on the ML method chosen. Then, the optimal design variables are found by maximizing the NPV based on using the proxy as a forward model to calculate NPV in an iterative optimization and training process. The sequential quadratic programming (SQP) method is used to optimize design variables. Design variables considered in this process are CO2 injection rate, production BHP, duration of injection time period, and duration of production time period for each cycle. We apply proxy-based optimization methods to and compare their performance on several synthetic single-well hydraulically fractured horizontal well models based on Bakken oil-shale fluid composition. Our results show that the LS-SVR and GPR based proxy models prove to be accurate and useful in approximating NPV in optimization of the CO2 HnP process. The results also indicate that both the GPR and LS-SVR methods exhibit very similar convergence rates and require similar computational time for optimization. Both ML based methods prove to be quite efficient in production optimization, saving significant computational times (at least 5 times more efficient) than using a stochastic gradient computed from a high fidelity compositional simulator directly in a gradient ascent algorithm. The novelty in this work is the use of optimization techniques to find optimum design variables, and to apply optimization process fast and efficient for the complex CO2 HnP EOR process which requires compositional flow simulation in hydraulically fractured unconventional oil reservoirs.
Summary We design a new and general work flow for efficient estimation of the optimal well controls for the robust production-optimization problem using support-vector regression (SVR), where the cost function is the net present value (NPV). Given a set of simulation results, an SVR model is built as a proxy to approximate a reservoir-simulation model, and then the estimated optimal controls are found by maximizing NPV using the SVR proxy as the forward model. The gradient of the SVR model can be computed analytically so the steepest-ascent algorithm can easily and efficiently be applied to maximize NPV. Then, the well-control optimization is performed using an SVR model as the forward model with a steepest-ascent algorithm. To the best of our knowledge, this is the first SVR application to the optimal well-control problem. We provide insight and information on proper training of the SVR proxy for life-cycle production optimization. In particular, we develop and implement a new iterative-sampling-refinement algorithm that is designed specifically to promote the accuracy of the SVR model for robust production optimization. One key observation that is important for reservoir optimization is that SVR produces a high-fidelity model near an optimal point, but at points far away, we only need SVR to produce reasonable approximations of the predicting output from the reservoir-simulation model. Because running an SVR model is computationally more efficient than running a full-scale reservoir-simulation model, the large computational cost spent on multiple forward-reservoir-simulation runs for robust optimization is significantly reduced by applying the proposed method. We compare the performance of the proposed method using the SVR runs with the popular stochastic simplex approximate gradient (StoSAG) and reservoir-simulations runs for three synthetic examples, including one field-scale example. We also compare the optimization performance of our proposed method with that obtained from a linear-response-surface model and multiple SVR proxies that are built for each of the geological models.
Summary In the context of production optimization, we consider the general problem of finding the well controls that maximize the net present value (NPV) of life-cycle production, where the well controls are either the bottomhole pressure (BHP) or a rate (oil, gas, water, or total liquid) at each well on a set of specified control steps (time intervals), with the limitations on surface facility considered as nonlinear-state constraints [e.g., field-liquid-production rates (FLRs), field-water-production rates (FWRs), and/or field-gas-production rates]. If the reservoir simulation used for reservoir management has sufficient adjoint capability to compute gradients of the objective function and all state constraints, we show that one can develop a significantly more computationally efficient procedure by replacing the adjoint-enhanced reservoir simulator by a proxy model and optimizing the proxy. Our methodology achieves computational efficiency by generating a set of output values of the cost and constraint functions and their associated derivative values by running the reservoir simulator for a broad set of input design variables (well controls) and then using the set of input/output data to train a proxy model to replace the reservoir simulator when computing values of cost and constraint functions and their derivatives during iterations of sequential quadratic programming (SQP). The derivation of the equations for computing the proxy-based model that uses both function and gradient information is similar to that of least-squares support vector regression (LS-SVR). However, this method is referred to as gradient-enhanced support vector regression (GE-SVR) because, unlike LS-SVR, the method uses derivative information, not just function values, to train the proxy. Similar to LS-SVR, improved (higher) estimated optimal NPV values can be obtained by using iterative resampling (IR). With IR, after each proxy-based optimization, one evaluates the cost and constraint functions and their derivatives at the estimated optimal controls using reservoir-simulator output, and then adds this new input/output information to the training set to update the proxy models for predicting NPV and constraints. Using the updated proxies, one applies SQP optimization again. IR continues until the simulator and proxy evaluated at the latest estimate of the optimal well controls give the same value of NPV within a specified percentage tolerance and the constraints evaluated by reservoir simulator at the latest optimal well controls are such that the constraints are satisfied within some small specified tolerance. Our results indicate that proxy-based optimization with iterative resampling might require up to an order of magnitude less computational time than pure reservoir-simulator-based optimization. By comparing the results generated with an LS-SVR proxy with the GE-SVR results, we find that GE-SVR is roughly an order of magnitude more computationally efficient than LS-SVR but also provides a better approximation of a complex cost-function surface so that it is possible to locate multiple optima in cases where LS-SVR fails to identify the multiple optima.
Guo, Zhenyu (University of Tulsa) | Chen, Chaohui (Shell Exploration and Production Company Incorporated) | Gao, Guohua (Shell Global Solutions US Incorporated) | Cao, Richard (Shell Exploration and Production Company Incorporated) | Li, Ruijian (Shell Exploration and Production Company Incorporated) | Liu, Chunlei (Shell Exploration and Production Company Incorporated)
Summary Reservoir model parameters generally have very large uncertainty ranges, and need to be calibrated by history matching (HM) available production data. Properly assessing the uncertainty of production forecasts (e.g., with an ensemble of calibrated models that are conditioned to production data) has a direct impact on business decision making. It requires performing numerous reservoir simulations on a distributed computing environment. Because of the current low-oil-price environment, it is demanding to reduce the computational cost of generating multiple realizations of history-matched models without compromising forecasting quality. To solve this challenge, a novel and more efficient optimization method (referred to as SVR-DGN) is proposed in this paper, by replacing the less accurate linear proxy of the distributed Gauss-Newton (DGN) optimization method (referred to as L-DGN) with a more accurate response-surface model of support vector regression (SVR). Resembling L-DGN, the proposed SVR-DGN optimization method can be applied to find multiple local minima of the objective function in parallel. In each iteration, SVR-DGN proposes an ensemble of search points or reservoir-simulation models, and the flow responses of these reservoir models are simulated on high-performance-computing (HPC) clusters concurrently. All successfully simulated cases are recorded in a training data set. Then, an SVR proxy is constructed for each simulated response using all training data points available in the training data set. Finally, the sensitivity matrix at any point can be calculated analytically by differentiating the SVR models. SVR-DGN computes more-accurate sensitivity matrices, proposes better search points, and converges faster than L-DGN. The quality of the SVR proxy is validated with a toy problem. The proposed method is applied to a real field HM example of a Permian liquid-rich shale reservoir. The uncertain parameters include reservoir static properties, hydraulic-fracture properties, and parameters defining relative permeability curves. The performance of the proposed SVR-DGN optimization method is compared with the L-DGN optimizer and the hybrid Gauss-Newton with a direct-pattern-search (GN-DPS) optimizer, using the same real field example. Our numerical tests indicate that the SVR-DGN optimizer can find better solutions with smaller values of the objective function and with a less computational cost (approximately one-third of L-DGN and 1/30 of GN-DPS). Finally, the proposed method is applied to generate multiple conditional realizations for the uncertainty quantification of production forecasts.