Data-driven discovery of interpretable water retention models for deformable porous media

The water retention behavior—a critical factor of unsaturated flow in porous media—can be strongly affected by deformation in the solid matrix. However, it remains challenging to model the water retention behavior with explicit consideration of its dependence on deformation. Here, we propose a data-driven approach that can automatically discover an interpretable model describing the water retention behavior of a deformable porous material, which can be as accurate as non-interpretable models obtained by other data-driven approaches. Specifically, we present a divide-and-conquer approach for discovering a mathematical expression that best fits a neural network trained with the data collected from a series of image-based drainage simulations at the pore-scale. We validate the predictive capability of the symbolically regressed counterpart of the trained neural network against unseen pore-scale simulations. Further, through incorporating the discovered symbolic function into a continuum-scale simulation, we showcase the inherent portability of the proposed approach: The discovered water retention model can provide results comparable to those from a hierarchical multi-scale model, while bypassing the need for sub-scale simulations at individual material points.