Enhancing Extrapolation of Buckley–Leverett Solutions with Physics-Informed and Transfer-Learned Fourier Neural Operators

Accurate modeling of multiphase flow in porous media remains challenging due to the nonlinear transport and sharp displacement fronts described by the Buckley–Leverett (B-L) equation. Although Fourier Neural Operators (FNOs) have recently emerged as powerful surrogates for parametric partial differential equations, they exhibit limited robustness when extrapolating beyond the training regime, particularly for shock-dominated fractional flows. This study aims to enhance the extrapolative performance of FNOs for one-dimensional B-L displacement. Analytical solutions were generated using Welge’s graphical method, and datasets were constructed across a range of mobility ratios. A baseline FNO was trained to predict water saturation profiles and evaluated under both interpolation and extrapolation conditions. While the standard FNO accurately reconstructs saturation profiles within the training window, it misestimates shock positions and saturation jumps when extended to longer times or higher mobility ratios. To address these limitations, we develop Physics-Informed FNOs (PI-FNOs), which embed PDE residuals and boundary constraints, and Transfer-Learned FNOs (TL-FNOs), which adapt pretrained operators to new regimes using limited data. Comparative analyses show that both approaches markedly improve extrapolation accuracy, with PI-FNOs achieving the most consistent and physically reliable performance. These findings demonstrate the potential of combining physics constraints and knowledge transfer for robust operator learning in multiphase flow systems.