Derivation of a coupled Darcy–Reynolds equation for a fluid flow in a thin porous medium including a fissure

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

“…where w ξ denotes the unique solution of the local Stokes problem given by (11), see [20,Theorem 2]. Thus, (û, π) takes the form…”

“…where the permeability function U and the corresponding cell problem are respectively given by ( 10) and (11).…”

“…Such assumption is realistic in applications like enhanced oil recovery, where the flow, frequently described by the Carreau law in the engineering literature, is mainly driven by the pressure difference between the injection point and the well [18,Chapter 4]. Since f is constant, one gets that P ≡ 0 and V is also constant, given by V = Y f w f (y) dy where w f is the solution to system (11) with ξ = f . In order to compute the solution to system (11), we rely on a mixed formulation of the problem and solve it by a finite element method, using FreeFem++ software [28].…”

“…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.…”

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

“…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].…”