Origin of the inertial deviation from Darcy's law: An investigation from a microscopic flow analysis on two-dimensional model structures

Abstract

Inertial flow in porous media occurs in many situations of practical relevance among which one can cite flows in column reactors, in filters, in aquifers, or near wells for hydrocarbon recovery. It is characterized by a deviation from Darcy's law that leads to a nonlinear relationship between the pressure drop and the filtration velocity. In this work, this deviation, also known as the nonlinear, inertial, correction to Darcy's law, which is subject to controversy upon its origin and dependence on the filtration velocity, is studied through numerical simulations. First, the microscopic flow problem was solved computationally for a wide range of Reynolds numbers up to the limit of steady flow within ordered and disordered porous structures. In a second step, the macroscopic characteristics of the porous medium and flow (permeability and inertial correction tensors) that appear in the macroscale model were computed. From these results, different flow regimes were identified: (1) the weak inertia regime where the inertial correction has a cubic dependence on the filtration velocity and (2) the strong inertia (Forchheimer) regime where the inertial correction depends on the square of the filtration velocity. However, the existence and origin of those regimes, which depend also on the microstructure and flow orientation, are still not well understood in terms of their physical interpretations, as many causes have been conjectured in the literature. In the present study, we provide an in-depth analysis of the flow structure to identify the origin of the deviation from Darcy's law. For accuracy and clarity purposes, this is carried out on two-dimensional structures. Unlike the previous studies reported in the literature, where the origin of inertial effects is often identified on a heuristic basis, a theoretical justification is presented in this work. Indeed, a decomposition of the convective inertial term into two components is carried out formally allowing the identification of a correlation between the flow structure and the different inertial regimes. These components correspond to the curvature of the flow streamlines weighted by the local fluid kinetic energy on the one hand and the distribution of the kinetic energy along these lines on the other hand. In addition, the role of the recirculation zones in the occurrence and in the form of the deviation from Darcy's law was thoroughly analyzed. For the porous structures under consideration, it is shown that (1) the kinetic energy lost in the vortices is insignificant even at high filtration velocities and (2) the shape of the flow streamlines induced by the recirculation zones plays an important role in the variation of the flow structure, which is correlated itself to the different flow regimes.

On the Inertial Single Phase Flow in 2D Model Porous Media: Role of Microscopic Structural Disorder

“…Based on the above considerations, the present study is a continuation of previous investigations (Abbasian Arani 2006;Lasseux et al 2011;Agnaou 2015;Agnaou et al 2017) taking into account six different types of 2D globally disordered (GD) structures with solid inclusions of circular or square cross sections. The focus is laid upon 2D configurations, keeping in mind that 3D cases are also of major interest, in particular with the objective of comparing the two situations and analyzing whether the fact that the inertial correction departs from the quadratic (Forchheimer) form can be reasonably attributed to 3D effects as suggested in Fourar et al (2004).…”

“…The incipient departure from Darcy's law due to the contribution of inertia has been observed by many researchers (Edwards et al 1990;Koch and Ladd 1997;Andrade et al 1999;Lasseux et al 2011;Agnaou et al 2017) in the laminar flow regime prior to the appearance of turbulence. The flow becomes unsteady for a Reynolds number larger than a critical value in the laminar regime.…”

“…In this way, they studied the effects of locally disordered (LD) structures on the inertial correction to Darcy's law. Similar structures were adopted by Agnaou (2015), Agnaou et al (2016Agnaou et al ( , 2017 to study the first Hopf bifurcation for one-phase flow or to investigate the origin of inertial effects through a detailed investigation of the flow structure. Although many studies have been dedicated to the analysis of flow in different types of pore structures, a more systematic study on the effects of the microstructure on the onset and on the role of inertia, as well as on the occurrence of unsteadiness, is still lacking.…”

“…The component of the average velocity orthogonal to the mean flow was monitored to capture flow unsteadiness. Then, steady numerical simulations were carried out to identify the different inertia regimes corresponding to weak, strong and beyond strong inertia (Agnaou et al 2017) as well as to further assess the effects of pore structure disorder on the inertial contributions to the deviations from Darcy's law.…”

Modeling transport of charged species in pore networks: Solution of the Nernst–Planck equations coupled with fluid flow and charge conservation equations

“…This work considers single-phase, isothermal, incompressible flow of a dilute electrolytic solution, treated as a Newtonian fluid, in a non-deformable porous medium. Assuming flow in the viscousdominated regime [1,2], the movement of the electrolytic solution can be described using the following steady-state momentum and mass conservation (Stokes) equations…”