Investigation of Post-Darcy Flow in Thin Porous Media

Abstract

We present numerical simulations of post-Darcy flow in thin porous medium: one consisting of staggered arrangements of circular cylinders and one random distribution of cylinders bounded between walls. The simulations span a range of Reynolds numbers, 40 to 4000, where the pressure drop varies nonlinearly with the average velocity, covering nonlinear laminar flow to the fully turbulent regime. The results are compared to those obtained by replacing the bounding walls with symmetric boundaries with the aim to reveal the effect of bounding walls on microscopic characteristics and macroscopic measures, i.e., pressure drop, hydrodynamic dispersion and Reynolds stresses. We use large eddy simulation to directly calculate the Reynolds stresses and turbulent intensity. The simulations show that vortical structures emerge at the boundary between the cylinders and the bounding walls causing a difference between the microscopic flow in the confined and non-confined porous media. This affects the averaged values of pressure drop, the hydrodynamic dispersion and the Reynolds stresses. Finally, the distance between the bounding walls is altered with the particle Reynolds number kept constant. It is observed that the difference between results calculated in confined and non-confined cases increases when the bounding walls are narrower.

Carreau law for non-newtonian fluid flow through a thin porous media

“…Passing to the limit by (29), we deduce the first equality in (45). The second equality is proved analogously.…”

“…Also, the Bingham plastic behavior in the thin porous media Ω ε has been studied in [8,9]. For other studies concerning thin porous media, we refer to Anguiano [3,4,5,6,7], Anguiano and Suárez-Grau [11,12,14], Jouybari and T. S. Lundström [29], Prat and Agaësse [33], Suárez-Grau [35], Yeghiazarian et al [36] and Zhengan and Hongxing [37]. However, as far as we know, in the previous literature there is no study for the homogenization of three-dimensional incompressible stationary Stokes system with a non-linear viscosity following the Carreau law in a thin porous media, as we consider in this article.…”

Sharp Pressure Estimates for the Navier–Stokes System in Thin Porous Media

“…As the previous case, we have to consider an exponent r ≥ 2. Due to the L 2 -estimates for velocity, given in (25), the value of r must be as close as possible to 2. Its value can be deduced after estimating the intertial term.…”

“…The proof follows the lines of Step 2 by taking into account estimates of the velocity (25), estimates of the restricted operator (57) and just changing the value of δ by 1, so we omit it. As a result, we can deduce that for r satisfying (55) imposing δ = 1, it holds…”

“…This issue is widely studied in porous media, where critical Reynolds numbers are obtained above which Darcy's laws are no longer valid, because the inertial term of the Navier-Stokes system must be taken into account, see for instance Dybbs and Edwards [23], Bourgeat and Mikelić [20], Boettcher et al [18] or Zeng and Grigg [35]. In this sense, the only study related to this question in thin porous media is recently obtained by Jouybari and Lundström [25], where they present direct numerical simulations of post-Darcy flow in thin porous media. However, as far as the authors are aware, there are no theoretical studies where a critical Reynolds number is obtained below which Darcy's law is valid for each type of thin porous media described above, which is the goal and main novelty of this paper.…”