Flow physics-informed learning of relative permeability in porous media

Predicting relative permeability (kr) curves directly from three-dimensional (3D) digital rock images remains a significant challenge in digital rock physics (DRP). Existing methods either rely on computationally expensive pore network modeling (PNM) or direct numerical simulation (DNS) during inference, or use purely data-driven networks that lack physical consistency. This study introduces a permeability-field physics-informed neural network (PF-PINN), a coupled deep learning surrogate that maps a 3D binary rock image, through its local thickness transform, to drainage kr curves for both wetting and non-wetting phases. This approach embeds two-phase Darcy flow physics into the training loss, ensuring physically admissible predictions without requiring a simulation during inference. The model architecture integrates a 3D U-Net encoder to extract a spatially varying permeability field and a global morphological feature vector, a pressure multilayer perceptron (MLP) that satisfies Darcy boundary conditions, and a relative permeability MLP conditioned on pore geometry. Ground truth data were generated using a PNM workflow applied to 350 sub-volumes from seven rock types, with voxel resolutions ranging from 4.94 to 17.17 μm/voxel. Two evaluation protocols were employed, namely a stratified random split with a 20% holdout per rock and 70 test samples, and a rock-type holdout in which Bentheimer and Monte Gambier were fully withheld, resulting in 100 test samples. Under the stratified protocol, the PF-PINN achieved Rmean2 = 0.952 (Rw2  = 0.963, Rnw2 = 0.941). Under the rock-type holdout, the model achieved Rmean2 = 0.927 on unseen geological classes, demonstrating physics-informed cross-lithology generalization. The wetting-phase R2 difference between the two protocols was only 0.013, suggesting that embedding Darcy mass conservation enables an effective transfer of flow behavior across the geological families.