Physics Informed Machine Learning for the Buckley-Leverett Problem. Application to Flow and Transport in Porous Media.

A new field in deep learning has recently emerged: the so-called "physics informed deep learning"¹ is a framework for solving general nonlinear partial differential equations (PDEs) using data, a promising alternative to traditional numerical methods for solving a PDE such as finite difference and finite volume methods. In this physics-informed approach, developed neural networks encode underlying physical laws as prior information and then leverage that information during training.

As machine learning is becoming a widely accepted tool in the geosciences and specifically in the study of subsurface transport, it is critical to understand its limitations when applied to subsurface problems. In this work we investigate the application of a new hybrid machine learning/physics-based approach to the reservoir modeling and two-phase transport problems that are known to be challenging to solve with standard numerical methods.

The methodology relies on a deep neural network architecture that matches any available experimental data with physics-based regularization. The network is used as a discrete representation of the physical quantities (i.e. pressure, saturation, composition) subject to a set of governing laws (e.g. mass conservation, Darcy's law). We apply this approach to a two-phase immiscible transport problem (Buckley-Leverett). From a limited dataset, the model learns the parameters of the governing equation (the fractional flow curve) and is able to provide an accurate physical solution, both in terms of shock and rarefaction. The use of these models for the inverse problem (history matching) is also presented, where the model simultaneously learns the physical laws and determines the key uncertainty subsurface parameters.

The proposed technique is a simple and elegant way to instill physical knowledge into machine learning algorithms. This alleviates the two most significant shortcomings of machine learning algorithms: the requirement for large datasets and the reliability of extrapolation. The principles presented in this work can be generalized in innumerable ways in the future and should lead to a new class of algorithms to solve both forward and inverse physical problems.

[1] Raissi, Perdikaris, and Karniadakis. Physics informed deep learning.