Data-Driven Derivation of Governing Equations for Fluid Flow in Porous Media

Accurate estimation of fluid flow in porous media has significant impact on a wide range of applications, including water management, the oil & gas industry, CO₂ sequestration, and environmental cleanup technology. For subsurface flow modeling, current approaches rely heavily on the Darcy equation. However, it is believed that the simplified nature of the Darcy equation, in conjunction with the need for accurate geological representation, may be a potential source of discrepancies between numerical and experimental results. This highlights the need for a refined fluid flow model in porous media. The present study reexamines fluid flow in heterogeneous porous media based on data-driven modeling. Starting with the canonical problem of flow over a cylinder in a channel, we gradually increased the complexity of the problem by adding more cylinders in varying configurations, approximating heterogeneous porous media. Numerical simulations, based on the finite volume method to solve the continuity and Navier-Stokes equations, were used to collect data. Subsequently, the governing equations were inferred from the spatiotemporal snapshots of the vorticity field using a sparse regression algorithm.   The results indicate that the convective terms of the vorticity transport equation vanish while quadratic terms emerge. Moreover, as the flow configuration approaches a more heterogeneous medium representation, the coefficients of the discovered PDEs become time-dependent, exhibiting increasing regularity and periodic dependency. Therefore,  the study paves the way for improvements to the Darcy equation as the asymptotic limit for fluid flow in complex porous media, and questions its ability to accurately capture the underlying physics responsible for the growth of small perturbation within the context of hydrodynamic instabilities.