Structure-Preserving Physics-Informed Neural Networks for Anisotropic Porous Media with Pressure Dependent Viscosity

Abstract

Modeling flow through porous media with realistic physical constraints remains a longstanding challenge in subsurface engineering. Anisotropy in permeability, pressure-dependent viscosity, and non-negativity requirements on pressure fields introduce mathematical complexity and numerical instability, especially in mesh-free learning frameworks. This thesis presents a structure-preserving Physics-Informed Neural Network (PINN) formulation for simulating nonlinear Darcy flow in anisotropic porous domains governed by Barus-type viscosity laws.

To enforce discrete maximum principles (DMP) and ensure physically admissible pressure fields, two constraint strategies are developed. A hard enforcement mechanism is implemented via output transformations that restrict predictions to within prescribed bounds. In parallel, a soft enforcement strategy augments the loss function with penalization terms that discourage DMP violations. These approaches are systematically evaluated within both strong-form PINNs and variational PINNs, the latter based on Galerkin and Variational Multiscale (VMS) formulations.

A series of numerical studies demonstrates the performance of the proposed methods across several settings. A one-dimensional benchmark using manufactured solutions validates convergence. In a square reservoir with a central borehole, the effect of permeability anisotropy is analyzed by sweeping the directional contrast ratio. It is observed that hard constraints are essential to maintain DMP adherence under strong anisotropy. In a separate case involving localized central forcing, the impact of nonlinear viscosity is assessed by varying the Barus coefficient. Increasing nonlinearity results in larger DMP violations unless physically motivated constraints are imposed. Sensitivity studies also reveal the influence of boundary condition density, penalty weights, and network depth on stability and accuracy.

The results indicate that while both soft and hard constraints improve physical fidelity, hard enforcement consistently outperforms in preserving maximum principles. Among all tested configurations, the VMS PINN with hard constraints yields the most robust performance, maintaining zero violations across anisotropy sweeps and producing stable velocity and pressure fields.

All models are implemented using the DeepXDE library and use physically meaningful parameter ranges relevant to oil recovery and geologic carbon storage. This work demonstrates that constraint-aware PINNs can serve as scalable, reliable solvers for complex porous media problems without sacrificing physical realism.