Learning Generic Solutions for Multiphase Transport in Porous Media via the Flux Functions Operator

TL;DR

This paper introduces a physics-informed DeepONet model that learns to predict solutions of multiphase transport PDEs in porous media, significantly speeding up simulations without requiring extensive data.

Contribution

The work develops a novel PI-DeepONet architecture that learns the operator mapping flux functions to solutions, eliminating the need for data generation and enabling rapid, accurate predictions.

Findings

Achieves up to four orders of magnitude speedup over traditional solvers.

Successfully generalizes across different flux function types.

Accurately predicts solutions without paired input-output data.

Abstract

Traditional numerical schemes for simulating fluid flow and transport in porous media can be computationally expensive. Advances in machine learning for scientific computing have the potential to help speed up the simulation time in many scientific and engineering fields. DeepONet has recently emerged as a powerful tool for accelerating the solution of partial differential equations (PDEs) by learning operators (mapping between function spaces) of PDEs. In this work, we learn the mapping between the space of flux functions of the Buckley-Leverett PDE and the space of solutions (saturations). We use Physics-Informed DeepONets (PI-DeepONets) to achieve this mapping without any paired input-output observations, except for a set of given initial or boundary conditions; ergo, eliminating the expensive data generation process. By leveraging the underlying physical laws via soft penalty constraints during model training, in a manner similar to Physics-Informed Neural Networks (PINNs), and a unique deep neural network architecture, the proposed PI-DeepONet model can predict the solution accurately given any type of flux function (concave, convex, or non-convex) while achieving up to four orders of magnitude improvements in speed over traditional numerical solvers. Moreover, the trained PI-DeepONet model demonstrates excellent generalization qualities, rendering it a promising tool for accelerating the solution of transport problems in porous media.