A unified macroscopic equation for creeping, inertial, transitional, and turbulent fluid flows through porous media

Abstract

Over the course of several decades, numerous model equations of the macroscopic fluid flow through porous media have been proposed. The application of such equations is, however, often complicated due to the requirement of variant specifications of parameters and empirical factors for different flow regimes. It is, therefore, necessary and desirable to have a unified fundamental equation that is capable of predicting porous media flows for the entire spectrum of flow regimes that are practically encountered. This work aims to fulfill that requirement. With the aid of a hypothesis-based analysis, finite-element simulations, and published experimental data, a new macroscopic transport equation has been proposed to predict statistically stationary single-phase incompressible flows through a non-deformable stationary porous medium. The new model may be written as a drag law associated with a dimensionless resistance parameter that is a function of the porous medium geometry and the flow forces. Though complex, this resistance parameter may be modeled as a power function in terms of three predictable parameters. Overall, the proposed transport equation has been found to be a more extensive form of other key models in existence. Using approximately 6000 analytical, numerical, and experimental data points, the equation has been validated as an excellent model for creeping, inertial, transitional, and turbulent porous media flows. The results show that the proposed equation is applicable to simple and complex porous media of 30%–90% porosity. Moreover, a dimensionless group in terms of the equation's resistance parameter has been established as useful for scaling.