Stokes–Brinkman–Darcy Solutions of Bimodal Porous Flow Across Periodic Array of Permeable Cylindrical Inclusions: Cell Model, Lubrication Theory and LBM/FEM Numerical Simulations

Abstract

An analytical study is devised for the problem of bimodal porous flow across a periodic array of permeable cylindrical inclusions. Such a configuration is particularly relevant for porous media systems of dual granulometry, an idealization often taken, e.g. in the modelling of membranes and fibrous applications. The double-porosity system is governed by the Stokes-Brinkman-Darcy equations, the most general description in this class of flow problems characterized by the permeabilities of the surrounding matrix and inclusions, their porosities and the relative volume fraction. We solve this problem with the Kuwabara cell model and lubrication approach, providing analytical solutions for the system effective permeability in closed analytical form. The ensemble of results demonstrates the self-consistency of the bimodal solutions in eight possible limit configurations and supports the validity of the Beavers-Joseph interface stress jump condition for transmission from the open Stokes flow to low-permeable Darcy region. At the same time, these solutions bring further insight on the relative significance of the governing parameters on the effective permeability, with a focus on the role of the effective viscosity (porosity) distribution. Furthermore, although the cell model is restricted to relatively small volume fractions in open flow, its validity extends in lesspermeable background flow inside Brinkman/Brinkman description. In turn, the lubrication approximation remains more adequate in the opposite limit of the dense impermeable inclusions. These conclusions are drawn from comparisons with the numerical solutions obtained with the developed lattice Boltzmann model and the standard finite element method. The two methods principally differ in the treatment of the interface conditions: implicit and explicit, respectively. The purpose of this task is therefore twofold. While the numerical schemes help quantifying the validity limits of the theoretical approach, the analytical solutions offer a non-trivial benchmark for numerical schemes in highly heterogeneous soil.

The permeability and quality of velocity field in a square array of solid and permeable cylindrical obstacles with the TRT–LBM and FEM Brinkman schemes

“…For the purpose of the present study, the permeability range of inclusion and matrix regions are assumed to be k 1 > 0 and k 2 > 0, respectively. Their zero limits, towards a very impermeable Darcy regime, have been studied in [21], with focus on the link between the interface condition (Eq. (3)) and the discontinuous Beavers-Joseph interface condition [1,4].…”

“…In fact, the first problem of the Stokes flow around solid cylinders has been tackled in this way in a number of works, e.g., [24][25][26][27]. The Brinkman solution was recently extended in [21], employing two complementary theoretical approaches: the cell model [25,26], operated in the low volume fraction of inclusions, and the lubrication theory [27,28], applied in the opposite limit of high volume fraction. Hereafter, the accuracy of numerical schemes is evaluated on the basis of the permeability error:…”

“…The reference theoretical value k th is found in [21,24], and k num is computed from < u >, which is the numerical solution for the mean velocity in a periodic cell driven by the mean pressure gradient < ∇ p >.…”

“…A suitable candidate for this purpose is the study of a periodic array of permeable cylinders, so far restricted to specific configurations, e.g., [6,16]. Recently, this problem has been analytically approached for the most general configuration [21], bimodal porous system, at both low and high concentration of obstacles. The work [21] provides reference solutions to this problem class, applicable over an extended range of bimodal permeabilities and volume fractions.…”

“…Recently, this problem has been analytically approached for the most general configuration [21], bimodal porous system, at both low and high concentration of obstacles. The work [21] provides reference solutions to this problem class, applicable over an extended range of bimodal permeabilities and volume fractions.…”