Input-Output Invariant Fast Proxy Models for Production Optimization
Authors: Dall'Aqua Marcelo J. , Coutinho Emilio J. R. , Gildin Eduardo , Guo Zhenyu , Zalavadia Hardik , Sankaran Sathish 
Published: June 2023

Abstract
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.