Online learning to accelerate nonlinear PDE solvers: applied to multiphase porous media flow

TL;DR

This work develops an online/adaptive learning framework to accelerate nonlinear PDE solvers by dynamically tuning a relaxation parameter in Picard iterations for multiphase flow in porous media. An offline phase trains a machine learning model on a simple 2D reservoir using dimensionless inputs, then online updates adapt the model during realistic 2D/3D simulations, integrating it directly into the open-source IC-FERST solver. The method achieves substantial performance gains, reducing walltime by up to 85% in offline analyses and around 37% on average in coupled tests, with additional improvements expected from online updates; it generalizes beyond multiphase flow to other solvers employing relaxation-based convergence control.

Abstract

We propose a novel type of nonlinear solver acceleration for systems of nonlinear partial differential equations (PDEs) that is based on online/adaptive learning. It is applied in the context of multiphase flow in porous media. The proposed method rely on four pillars: (i) dimensionless numbers as input parameters for the machine learning model, (ii) simplified numerical model (two-dimensional) for the offline training, (iii) dynamic control of a nonlinear solver tuning parameter (numerical relaxation), (iv) and online learning for real-time improvement of the machine learning model. This strategy decreases the number of nonlinear iterations by dynamically modifying a single global parameter, the relaxation factor, and by adaptively learning the attributes of each numerical model on-the-run. Furthermore, this work performs a sensitivity study in the dimensionless parameters (machine learning features), assess the efficacy of various machine learning models, demonstrate a decrease in nonlinear iterations using our method in more intricate, realistic three-dimensional models, and fully couple a machine learning model into an open-source multiphase flow simulator achieving up to 85\% reduction in computational time.

Figures (16)

• Figure 1: Sequential nonlinear solver loops. Reprinted from salinas:17.
• Figure 2: Adaptive learning acceleration for nonlinear PDE solvers.
• Figure 3: Two-dimensional homogeneous layered reservoir used for offline training. It shows the saturation of the displaced phase in one point in time during the numerical simulation. Blue represents the injected phase and red the displaced phase.
• Figure 4: Saturation of the displaced phase in one point in time during the numerical simulation. Blue represents the injected phase and red the displaced phase. In all cases, we injected one phase on the left and produce both phases on the right.
• Figure 5: Test case 1 (2D) permeability field in m$^2$.