Machine Learning for Scientific Computing: Applications to Reservoir Simulation

Abstract

For decades, reservoir management has witnessed a growing dependence on reservoir simulation due to its remarkable flexibility and modularity to empower reservoir engineers. The ability to test numerous ideas and scenarios, assimilate data, quantify uncertainty, and make informed decisions has positioned simulators as invaluable tools for mitigating risk and avoiding costly endeavors. However, despite the capabilities of state-of-the-art simulators, a critical challenge persists— their effective utilization demands a cadre of experienced engineers. Full-field reservoir simulation tasks are known to be time-consuming, tedious, and expensive. Addressing these challenges is paramount to unlocking the full potential of reservoir simulation in optimizing decision-making processes and resource utilization. Motivated by the recent achievements in machine learning, particularly in domains such as speech recognition, image generation, and language models, our aim here is to leverage this success and harness the momentum within the machine learning community to build better solution methodologies for scientific problems. By building upon the foundation laid by recent breakthroughs in machine learning we address some of the challenges in scientific computing. This dissertation revolves around the burgeoning field of scientific machine learning (SciML). Our primary objective is to develop and employ machine learning algorithms tailored for scientific applications, especially in the field of reservoir simulation. These algorithms are either informed of physics or inspired by it and they can be applied to a wide spectrum of challenges associated with stochasticity, data assimilation, uncertainty quantification, and parameter optimization within the context of reservoir simulation. Our contributions incorporate two main ideas; Physics-Informed Neural Network (PINN) for deterministic PDEs, and neural operator learning for parametric PDEs. Specifically, we make the following contributions: 1. We apply Physics-Informed Neural Network (PINN) to the classical Buckley-Leverett problem for multiphase transport in porous media and highlight the challenges that PINN face in solving hyperbolic conservation laws with shocks. We show that these types of problems can be solved accurately using PINN by embedding the PDE residual with the Oleinik entropy condition. We then refine the PINN approach by encoding the initial and boundary conditions in a hard manner in the network architecture instead of soft enforcement through the loss function. We show that this modification leads to better and faster convergence. We further develop this algorithm by utilizing an attention inspired neural network architecture to improve the robustness and accuracy of our approach. 2. We develop Physics-Informed Deep Operator Network (PI-DeepONet) to learn the mapping between the space of flux functions of the Buckley-Leverett PDE and the space of solutions. We use Physics-Informed PI-DeepONets to achieve this mapping without any paired input-output observations, except for a set of given initial or boundary conditions. As such, we obtain a parametric solution of the Buckley-Leverett PDE which was not previously possible using traditional methods. The trained PI-DeepONet model can predict the solution accurately given any type of f lux function (concave, convex, or non-convex) while achieving up to four orders of magnitude speedup over traditional numerical solvers. Moreover, the trained PI-DeepONet model demonstrates excellent generalization qualities that render it a promising tool for accelerating the solution of transport problems in porous media. 3. We introduce U-Net enhanced DeepONet (U-DeepONet) for learning the solution operator of highly complex CO2-water two-phase flow in heterogeneous porous media. The U-DeepONet is a novel neural operator learning algorithm that can generalize accurately to a wide range of parameters for a given PDE. We show that the U-DeepONet has state-of-the-art performance when benchmarked against other operator learning algorithms such as the U-Net Enhanced Fourier Neural Operator (U-FNO) and the Fourier-Multiple Input Neural Network (Fourier-MIONet). 4. We introduce the U-U-Net enhanced Deep Operator Network (UU-DeepONet), a novel operator learning framework specifically designed for temporal extrapolation and real-time inference. UU-DeepONet represents a significant advancement in machine learning for scientific computing and to the best of our knowledge, it is the first operator learning architecture that is capable of both generalization to new PDE instances and extrapolation to unseen time steps.

Date of Award	20 Jul 2024
Original language	American English
Supervisor	Mohammed Al Kobaisi (Supervisor)