Momentum transfer at the boundary between a porous medium and a homogeneous fluid-I. Theoretical development

In this paper we illustrate how the method of volume averaging can be used to derive Darcy's law with the Forchheimer correction for homogeneous porous media. Beginning with the Navier-Stokes equations, we find the volume averaged momentum equation to be given by 
$$\langle v_\beta \rangle = - \frac{K}{{\mu _\beta }} \cdot (\nabla \langle p_\beta \rangle ^\beta - \rho _\beta g) - F\cdot \langle v_\beta \rangle .$$

 The Darcy's law permeability tensor, K, and the Forchheimer correction tensor, F, are determined by closure problems that must be solved using a spatially periodic model of a porous medium. When the Reynolds number is small compared to one, the closure problem can be used to prove that F is a linear function of the velocity, and order of magnitude analysis suggests that this linear dependence may persist for a wide range of Reynolds numbers.

The Forchheimer equation : A theoretical development

Analysis of fluid flow and heat transfer interfacial conditions between a porous medium and a fluid layer

Dissolution of a porous medium creates, under certain conditions, some highly conductive channels called wormholes. The mechanism of propagation is an unstable phenomenon depending on the microscopic properties at the pore scale and is controlled by the injection rate. The aim of this work is to test the ability of a Darcy-scale model to describe the different dissolution regimes and to characterize the influence of the flow parameters on the wormhole development. The numerical approach is validated by model experiments reflecting dissolution processes occurring during acid injection in limestone. Flow and transport macroscopic equations are written under the assumption of local mass non-equilibrium. The coupled system of equations is solved numerically in two dimensions using a finite volume method. Results are discussed in terms of wormhole propagation rate and pore volume injected.

On the ability of a Darcy-scale model to capture wormhole formation during the dissolution of a porous medium

Direct numerical simulations (DNS) have been performed of turbulent flow in a plane channel with a solid top wall and a permeable bottom wall The permeable wall is a packed bed, which is characterized by the mean particle diameter and the porosity The main objective is to study the influence of wall permeability on the structure and dynamics of turbulence The flow inside the permeable wall is described by means of volume-averaged Navier–Stokes equations Results from four simulations are shown, for which only the wall porosity (, the wall is classified as a highly permeable wall near which viscous effects are of minor importance It is observed that streaks and the associated quasi-streamwise vortices are absent near a highly permeable wall This is attributed to turbulent transport across the wall interface and the reduction in mean shear due to a weakening of, respectively, the wall-blocking and the wall-induced viscous effect The absence of streaks is consistent with a decrease in the peak value of the streamwise root mean square (rms) velocity normalized by the friction velocity at the permeable wall Despite the increase in the peak values of the spanwise and wall-normal rms velocities, the peak value of the turbulent kinetic energy is therefore smaller Turbulence near a highly permeable wall is dominated by relatively large vortical structures, which originate from a Kelvin–Helmholtz type of instability These structures are responsible for an exchange of momentum between the channel and the permeable wall This process contributes strongly to the Reynolds-shear stress and thus to a large increase in the skin friction

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J. Alberto Ochoa-Tapia

Papers

Momentum transfer at the boundary between a porous medium and a homogeneous fluid-I. Theoretical development

Momentum transfer at the boundary between a porous medium and a homogeneous fluid—II. Comparison with experiment

Diffusive transport in two-phase media: spatially periodic models and maxwell's theory for isotropic and anisotropic systems

Bulk and surface diffusion in porous media: An application of the surface-averaging theorem

Facilitated transport in porous media