Life-Cycle Production Optimization with Nonlinear Constraints Using a Least-Squares Support-Vector Regression Proxy

Abstract

In this work, we develop computationally efficient methods for deterministic production optimization under nonlinear constraints using a kernel-based machine learning method where the cost function is the net present value (NPV). We use the least-squares support-vector regression (LSSVR) to maximize the NPV function. To achieve computational efficiency, we generate a set of output values of the NPV and nonlinear constraint functions, which are field liquid production rate (FLPR) and water production rate (FWPR) in this study, by running the high-fidelity simulator for a broad set of input design variables (well controls) and then using the collection of input/output data to train LS-SVR proxy models to replace the high-fidelity simulator to compute NPV and nonlinear constraint functions during iterations of sequential quadratic programming (SQP). To obtain improved (higher) estimated optimal NPV values, we use the existing so-called iterative sampling refinement (ISR) method to update the LSSVR proxy so that the updated proxy remains predictive toward promising regions of search space during the optimization. Direct and indirect ways of constructing LSSVR-based NPVs as well as different combinations of input data, including nonlinear state constraints and/or the bottomhole pressures (BHPs) and water injection rates, are tested as feature space. The results obtained from our proposed LS-SVR-based optimization methods are compared with those obtained from our in-house StoSAG-based line-search SQP programming (LS-SQP-StoSAG) algorithm using directly a high-fidelity simulator to compute the gradients with StoSAG for the Brugge reservoir model. The results show that nonlinear constrained optimization with the LSSVR ISR with SQP is computationally an order of magnitude more efficient than LS-SQP-StoSAG. In addition, the results show that constructing NPV indirectly using the field liquid and water rates for a waterflooding problem where inputs come from LSSVR proxies of the nonlinear state constraints requires significantly fewer training samples than the method constructing NPV directly from the NPVs computed from a high-fidelity simulator. To the best of our knowledge, this is the first study that shows the means of efficient use of a kernel-based machine learning method based on the predictor information alone to perform efficiently life-cycle production optimization with nonlinear state constraints.