Derivation of a quasi-stationary coupled Darcy-Reynolds equation for incompressible viscous fluid flow through a thin porous medium with a fissure

Abstract

We consider a non‐stationary Stokes system in a thin porous medium of thickness ε that is perforated by periodically distributed solid cylinders of size ε, and containing a fissure of width ηε. Passing to the limit when ε goes to zero, we find a critical size ηε≈ε23 in which the flow is described by a 2D quasi‐stationary Darcy law coupled with a 1D quasi‐stationary Reynolds problem. Copyright © 2017 John Wiley & Sons, Ltd.

Theoretical derivation of Darcy's law for fluid flow in thin porous media

“…We denote 𝑌 = (−1∕2, 1∕2) 3 the unitary cube in ℝ 3 as the reference cell and 𝑇 an open connected subset of 𝑌 with a smooth boundary 𝜕𝑇 such that 𝑇 ⊂ 𝑌. We denote 𝑌 𝑓 = 𝑌 ⧵ 𝑇.…”

“…We denote by 𝜏 ( 𝑇 𝑘,𝑎 𝜀 ) the set of all translated images of 𝑇 𝑘,𝑎 𝜀 . The set 𝜏 ( 𝑇 𝑘,𝑎 𝜀 ) represents the obstacles in ℝ 3 .…”

“…and calcule the right hand side of (5.7) by using (4.2), which gives with 𝐹 𝜀 = ∇𝑃 𝜀 . Next, we get for every φ ∈ 𝐻 1 0 (Ω) 3 where φ( 𝑥 ′ , 𝑧 3 ) = 𝜑 ( 𝑥 ′ , 𝜀𝑧 3 ) , using the change of variables (2.3), that…”

“…where R𝜀 φ = 𝑅 𝜀 𝜑 for any φ ∈ 𝐻 1 0 (Ω) 3 where φ( 𝑥 ′ , 𝑧 3 ) = 𝜑 ( 𝑥 ′ , 𝜀𝑧 3 ) . Finally, we estimate the right-hand side of (5.9) and give the estimate of the extended pressure P𝜀 .…”

“…This result is proved in Fabricius et al [19] by using the multiscale expansion method, which is a formal but powerful tool to analyze homogenization problems, and later rigorously developed in Anguiano and Suárez-Grau [10] by using an adaptation of the periodic unfolding method, see Arbogast et al [13] and Cioranescu et al [15][16][17], which is introduced in Anguiano and Suárez-Grau [8]. For other related studies concerning Newtonian fluids in thin porous media such as the derivation of coupled Darcy-Reynolds for fluid flows in thin porous media including a fissure and the modelling of fluid flows in thin porous media with non-homogeneous slip boundary conditions on the cylinders, we refer to Anguiano [2,3] and Anguiano and Suárez-Grau [9,12]. In addition, for the case of non-Newtonian power-law fluid flows in thin porous media, we refer to Anguiano [4,5] and Anguiano and Suárez-Grau [11] and for the case of non-Newtonian Bingham fluid flows in thin porous media, we refer to Anguiano and Bunoiu [6,7].…”