Macroscopic Orientation of Inertial Flows in Porous Media

Macroscopic models of inertial flows in porous media have many practical applications where direct numerical simulations are not feasible. The Forchheimer equation describes macroscopic momentum transport accounting for inertial effects at the pore scale through a nonlinear correction tensor \({\textbf{F}}_\beta\) to the permeability. The goal of this work is to study the effects of inertial flow orientation on the Forchheimer correction. Using up-scaling approaches such as the volume averaging method, \({\textbf{F}}_\beta\) can be determined. However, the procedure requires to deal with a nonlinear problem for the deviations of the local velocity field. This is commonly tackled by assuming that the inertial convective velocity is decoupled from the velocity deviations. Here, we propose an alternative approach based on regular perturbation expansion leading to a series of linear closure problems. The values of \({\textbf{F}}_\beta\) predicted by both approaches are compared for various values of the Reynolds number and flow orientation. Compared to the local inertial–convection approach, the proposed linearized closure problem has the advantage of being self-consistent, independent of the pore Reynolds number and of flow orientation. It is, however, limited in validity by Reynolds number below one and requires the solution of closure problems of higher dimensions. Then, macroscopic simulations are performed to evaluate the importance of varying pressure gradient orientation on the macroscopic inertial flow. Numerical results of the general macroscopic model obtained by the volume averaging method highlight the necessity to account for extra-diagonal terms as well as macroscopic gradient orientation in the determination of the Forchheimer tensor.