Learning reservoir flow dynamics is of primary importance in creating robust predictive models for reservoir management including hydraulic fracturing processes. Physics-based models are to a certain extent exact, but they entail heavy computational infrastructure for simulating a wide variety of parameters and production scenarios. Reduced-order models offer computational advantages without compromising solution accuracy, especially if they can assimilate large volumes of production data without having to reconstruct the original model (data-driven models). Dynamic mode decomposition (DMD) entails the extraction of relevant spatial structure (modes) based on data (snapshots) that can be used to predict the behavior of reservoir fluid flow in porous media. In this paper, we will further enhance the application of the DMD, by introducing sparse DMD and local DMD. The former is particularly useful when there is a limited number of sparse measurements as in the case of reservoir simulation, and the latter can improve the accuracy of developed DMD models when the process dynamics show a moving boundary behavior like hydraulic fracturing. For demonstration purposes, we first show the methodology applied to (flow only) single- and two-phase reservoir models using the SPE10 benchmark. Both online and offline processes will be used for evaluation. We observe that we only require a few DMD modes, which are determined by the sparse DMD structure, to capture the behavior of the reservoir models. Then, we applied the local DMDc for creating a proxy for application in a hydraulic fracturing process. We also assessed the trade-offs between problem size and computational time for each reservoir model. The novelty of our method is the application of sparse DMD and local DMDc, which is a data-driven technique for fast and accurate simulations.

Koopman operator theory enables a global linear representation of a given nonlinear dynamical system by transforming the nonlinear dynamics into a higher dimensional observable function space where the evolution of observable functions is governed by an infinite-dimensional linear operator. For practical application of Koopman operator theory, various data-driven methods have been developed to derive lifted state-space models via approximation to the Koopman operator. Based on approximate models, several Koopman-based model predictive control (KMPC) schemes have been proposed. However, since a finite-dimensional approximation to the infinite-dimensional Koopman operator cannot fully represent a nonlinear dynamical system, plant-model mismatch inherently exists in these KMPC schemes and negatively influences the performance of control systems. In this work, we present offset-free Koopman Lyapunov-based model predictive control (KLMPC) framework that addresses the inherent plant-model mismatch in KMPC schemes using an offset-free control framework based on a disturbance estimator approach and ensures feasibility and stability of the control system by applying Lyapunov constraints to the optimal control problem. The zero steady-state offset condition of the developed framework is mathematically examined. The effectiveness of the developed framework is also demonstrated by comparing the closed-loop results of the proposed offset-free KLMPC and the nominal KLMPC.