Physics Based Deep Learning for Nonlinear Two-Phase Flow in Porous Media

Abstract

Summary There is growing interest in employing Machine Learning (ML) strategies to solve forward and inverse computational physics problems. The physics-informed machine learning (PIML) frameworks developed by Raissi et al.[ 1 ] and Zhu et al.[ 2 ] are prominent examples. The basic idea is to encode the partial differential equations (PDE) that govern the flow physics into the neural network. This encoding is achieved by enriching the loss function with the governing conservation equation. Using the initial and boundary conditions, the network is then able to learn the solution of the forward problem without any labeled data. The scarcity of site-specific “labeled” data presents serious challenges to modeling of Enhanced Oil Recovery (EOR) processes. Thus, if PIML approaches can be used to model the nonlinear flow and transport that govern EOR processes, then they could change the practice of reservoir simulation. In this work, we explore the application of a particular PIML approach to solve the nonlinear hyperbolic equation that describes nonlinear immiscible two-phase flow in porous media. Specifically, we are concerned with the forward solution of a Riemann problem - a nonlinear conservation law together with piecewise constant data having a single discontinuity. It is well known that it is hard to solve this nonlinear transport problem, especially with a non-convex flux function, due to emergence of saturation shocks in the domain. The focus is on the pure forward problem, i.e., the absence of previously simulated (so-called labeled) data in the interior of the domain. The PIML framework breaks down for this nonlinear hyperbolic problem with non-convex flux function. We have found that it is essential to add a diffusion term to the underlying nonlinear PDE. That is, we used the parabolic form of the equation with a finite Peclet number. When the loss function includes a finite amount of diffusion, the neural network can actually produce reasonable approximations of the forward solution when shocks and mixed waves (shocks and rarefactions) are present. For the obtained neural networks we also analyze the training process and provide 2-D visualizations of the loss landscape, then we discuss possible reasons for the observed behavior.