Comparative study of kernel- and deep learning-based proxy models for nonlinearly constrained life-cycle production optimization

Abstract

This study presents a comparative study for the application of deep learning–based and kernel-based proxy models in hydrocarbon production optimization with nonlinear constraints, comparing their performance against high-fidelity simulators (HFS) in terms of computational efficiency and the quality of the results achieved. One of the proxy models employed is referred to as embed to control and observe (E2CO), a deep learning–based model. The other model is a kernel-based proxy model known as least-squares support-vector regression (LS-SVR). Both proxy models can predict production/injection outputs for each well. We use the sequential quadratic programming (SQP) method for nonlinearly constrained production optimization. Our optimization objective revolves around the net present value (NPV), while the nonlinear state constraints involve field liquid production rate (FLPR) and field water production rate (FWPR). NPV, FLPR, and FWPR are estimated using two different types of proxy models. The gradient of the objective function and the Jacobian matrix of constraints are computed analytically for LS-SVR. In contrast, the method of stochastic simplex approximated gradient (StoSAG) is employed for optimization with E2CO and HFS. The reservoir model considered in this study is a two-phase, three-dimensional reservoir characterized by channelized and heterogeneous permeability, sourced from the SPE10 benchmark case. We optimize well controls to maximize NPV in an oil-water waterflooding scenario. All proxy models can achieve optimal NPV results like those obtained through HFS but with significantly lower computational resources. Both proxy models are found to be approximately 15-fold more efficient than using HFS when the number of simulation runs is compared. Forward prediction results show that both proxy models outperform numerical simulator, and the E2CO can predict up to 2400 times faster than HFS whereas speedups gained over HFS go up to 5000-fold with LS-SVR. The training time required for E2CO is at least an order of magnitude longer than that for LS-SVR. The main contributions of the paper are (i) development of a novel methodology that uses an E2CO proxy model within a gradient-based iterative retraining proxy approach for life-cycle production optimization with nonlinear state constraints, (ii) comparison of this methodology in terms of computational cost and accuracy with an LS-SVR proxy model, and (iii) new insights into the accuracy and prediction performances of these machine learning-based proxy models for 3D oil-water systems as well as their efficiency in nonlinearly constrained production optimization for waterflooding applications.