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Integrated reservoir studies for performance prediction and decision-making processes are computationally expensive. In this paper, we develop a novel linearization approach to reduce the computational burden of intensive reservoir simulation execution. We achieve this by introducing two novel components: (1) augment the state-space to yield a bi-linear system, and (2) an autoencoder based on a deep neural network to linearize physics reservoir equations in a reduced manifold employing a Koopman operator. Recognizing that reservoir simulators execute expensive Newton-Raphson iterations after each timestep to solve the nonlinearities of the physical model, we propose "lifting" the physics to a more amenable manifold where the model behaves close to a linear system, similar to the Koopman theory, thus avoiding the iteration step. We use autoencoder deep neural networks with specific loss functions and structure to transform the nonlinear equation and frame it as a bilinear system with constant matrices over time. In such a way, it forces the states (pressures and saturations) to evolve in time by simple matrix multiplications in the lifted manifold. We also adopt a "guided" training approach: our training process is performed in three steps: we initially train the autoencoder, then we use a "conventional" MOR (Dynamic Mode Decomposition) as an initializer for the final full training when we use reservoir knowledge to improve and to lead the results to physically meaningful output. Many simulation studies exhibit extremely nonlinear and multi-scale behavior, which can be difficult to model and control. Koopman operators can be shown to represent any dynamical system through linear dynamics. We applied this new framework to a two-dimensional two-phase (oil and water) reservoir subject to a waterflooding plan with three wells (one injector and two producers) with speed ups around 100 times faster and accuracy in the order of 1-3 percent on the pressure and saturations predictions. It is worthwhile noting that this method is a non-intrusive data-driven method since it does not need access to the reservoir simulation internal structure; thus, it is easily applied to commercial reservoir simulators and is also extendable to other studies. In addition, an extra benefit of this framework is to enable the plethora of well-developed tools for MOR of linear systems. This is the first work that utilizes the Koopman operator for linearizing the system with controls to the author's knowledge. As with any ROM method, this can be directly applied to a well-control optimization problem and well-placement studies with low computational cost in the prediction step and good accuracy.

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Economic pressure to improve production efficiency in unconventional reservoirs has met a stiff challenge to scale up traditional reservoir modeling methods to the entire field for quantifying well performance. The main reasons are lack of availability of key reservoir and well parameters and difficulty to setup and maintain models because of the large well count and rapid pace of operations. As a result, decline curve analysis is still the prevailing method for large scale evaluations, which does not consider routine pressure variations and operational constraints. Analytical rate transient (RTA) models warrant identification of flow regimes and geometrical assumptions (well and fractures) to apply discrete analytical models for various flow segments. This inherent limitation of RTA makes it interpretive and not conducive to fieldscale application, besides often lacking necessary inputs for all wells. It is desirable to have better understanding through a robust and consistent well performance analysis method at field scale to unlock significant production optimization opportunities with existing field infrastructure and investment. We have applied a reduced physics formulation based on Dynamic Drainage Volume (DDV) using commonly measured data for most wells (namely, flowback data, daily production rates, and wellhead pressure) to calculate continuous pressure depletion, transient productivity index (PI) and inflow performance relationship (IPR). This transient well performance (TWP) method eliminates the surface and wellbore operational impacts to extract the true reservoir signal that can be used for robust well performance analysis and forecasting. We applied the TWP method in multiple basins with large well counts (more than 1000 wells) producing under a variety of methods. In this paper, we present several case studies illustrating various production optimization opportunities, focusing on naturally flowing and gas-lifted wells. The fluid properties and bottomhole pressure estimated using data-driven methods for all wells provided excellent match with blind data (PVT lab reports and downhole gauge data). The TWP method normalizes reservoir and completion quality to extract valuable insights on effectiveness of well and completions design in the presence of varying geological and fluid properties. The transient PI and dynamic IPR results provided valuable insights on how and when to select various artificial lift systems. During gas lift, we identified several wells that were over-injecting gas volumes at higher compressor discharge head, with line of sight to significant operational cost savings and reduced energy consumption. The proposed methodology combines pragmatic use of physics and data-driven methods to solve a critical need for analyzing unconventional reservoirs. Field application of the novel DDV method on large well population has been quite successful in identifying various optimization opportunities that would not have been possible, timely, or repeatable with other traditional methods.

This work aims to obtain reduced-order models for fluid flows in porous media that can be used for optimal well-control design and are they are equipped with input-output tracking capabilities. Meeting the net-zero emission paradigm will require a realignment of hydrocarbon production strategies with other forms of energy production, such as hydrogen and geothermal. Profiting from all these energy sources is only possible if accurate and timely predictions of the injection-production behavior of fluids, including geomechanics issues in the subsurface, can be attained. High-fidelity reservoir simulation provides accurate characterizations of complex flow dynamics in the subsurface. Still, it is unsuitable for production or uncertainty quantification due to its prohibitive computational complexity. Balanced truncation (BT) is a well-known model reduction technique for linear systems. It is input-output invariant and does not require a training phase once the system can be written in a linear state-space form, unlike other methods (Proper Orthogonal Decomposition - POD, Deep learning, among others). However, reduced-order models are unsuitable for long-term simulations as these simulations exhibit highly nonlinear behavior. This paper builds upon the bilinear formulation of dynamical systems to construct a suitable reduced-order model. A combination of data-driven model reduction strategies and machine learning (deep-neural networks ANN) will be used to simultaneously predict state and the best correlated input-output matching. We remove the states that are hard to control and observe in the bilinear space by introducing a loss function to the Artificial Neural Network (ANN) training process based on the variational interpretation of the controllability and observability gramians. Both these matrices are related respectively to the energy demanded to control a state (i.e., how hard is it to change a gridblock pressure by controlling the injector wellťs bottom-hole pressure) and to the energy produced by a state (i.e., if we can infer the pressure in a gridblock by measuring the rate of a producing well). We applied this new framework to a two-dimensional two-phase (oil and water) reservoir under waterflooding with three wells (one injector and two producers). The proposed method is a non-intrusive data-driven method as it does not need access to the reservoir simulation's internal structure; thus, it can be easily applied with any commercial reservoir simulator and is extensible to other studies. Although state information is well preserved during truncation, the output, e.g., cumulative production, presents a slightly worse response than simply applying POD. This is because we also identify the output matrices (C and D) and enforce the orthogonalization of the projection matrices through a loss function and not by construction. As far as we know, it is the first attempt to apply balanced truncation of bilinear reservoir models to solve well-control problems. It has the potential to lead the trend of generating robust reduced-order (proxy) models. In this paper, we propose a novel data-driven framework to construct the proxy model while using as much physical information as possible to guide the neural network to best correlates the input well-control and output well-response, making it ideal for well-control optimization.