Dissertação de Mestrado - Átila Luna Ambrósio da Silva

Physics Informed Neural Network Applied to Fractional Flow Equations

Physics Informed Machine Learning is the strategy of developing a neural network with physical constraints, commonly expressed in partial differential equations (PDEs) and their initial and boundary conditions. In this approach, the main idea is to incorporate underlying physical laws expressed in these PDEs as prior information for the neural network. In this work, we investigate the applicability of this technique to the direct problem of two-phase fluid transport in porous media, particularly in the context of gas injection in an oil reservoir, whose physical constraints are described using nonlinear first-order hyperbolic PDEs, subject to specific initial and boundary conditions. Initially, we develop the equations governing the problem without considering the fluid volume change factor to study the convergence of the solutions to these PDEs. Based on the obtained results, we introduce the volume change equations to capture the gas phase's behavior better. The fractional flux functions used in our examples were chosen as non-convex to include shock and refraction phenomena in the solutions. We also incorporate a diffusive factor, transforming the hyperbolic PDEs into parabolic ones. Through this approach, the neural network could learn consistent approximate solutions. Consequently, this effect smoothens the solution curves at the points of shock.