Guided Deep Learning Manifold Linearization of Porous Media Flow Equations

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.