A Systematic Analysis of Physics-Informed Neural Networks for Two-Phase Flow with Capillarity: The Muskat–Leverett Problem

This work develops and systematically evaluates a physics-informed neural network (PINN) solver for the fully coupled, time-dependent Muskat–Leverett system with capillarity modeled in the pressure equation. A single shallow–wide multilayer perceptron jointly predicts wetting pressure and water saturation; physical capillary pressure regularizes the saturation front, while a small numerical diffusion term in the saturation residual acts as a training stabilizer rather than a shock-capturing device. To guarantee admissible states in stiff regimes, we introduce a saturation soft-clamping head enforcing 0<Sw<1 and activate it selectively for stiff mobility ratios. Using IMPES solutions as reference, we perform a sensitivity study over network depth and width, interior collocation and boundary data density, mobility ratio, and injection pressure. Shallow-wide networks (10 layers × 50 neurons) consistently outperform deeper architectures, and increasing interior collocation points from 5000 to 50,000 reduces mean saturation error by about half, whereas additional boundary data have a much weaker effect. Accuracy is highest at an intermediate mobility ratio and improves monotonically with higher injection pressure, which sharpens yet better conditions the front. Across all regimes, pressure trains easily while saturation determines model selection, and the PINN serves as a physics-consistent surrogate for what-if studies in two-phase porous-media flow.