Generalized Darcy’s Law in Filtration Theory

Yuri P Rybakov,Natalya V Semenova,Gh Adam,J Buša,D Podgainy,M Hnatič

doi:10.1051/epjconf/201817302017

Yuri P Rybakov, Natalya V Semenova + Show 4 more

Open Access

https://doi.org/10.1051/epjconf/201817302017

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Abstract

We study the hydrodynamics of flow in a porous medium modeling the grain filling in filters. Using the lattice approximation, we derive the structure of the current in porous media and obtain the transverse diffusion coefficient D which proves to be proportional to the diameter d of the grains as constituents of the medium. We consider the axially-symmetric stationary flow in a cylindrical filter and show that the vertical velocity takes its maximal value at the wall, this effect being known as the “near-wall” one. We analyze the solution to the Euler equation with the modified Darcy force, which depends not only on the velocity but also on the gradient of the pressure included in the Darcy coefficient. Finally, within the scope of the perturbation method, we derive the main filtration equation and discuss the influence of modifying the Darcy’s law on the efficiency of the filtration process.

Highlights

Lattice approximation for the flow in porous mediaThe main difficulty in the filtration process seems to be the so-called “near-wall” effect, that is an anomalously large value of the flow velocity near the wall due to larger values of the gaps between the wall and the grains, the effectiveness of the filtration being decreasing [1]
In order to take this effect into account, let us first consider the discrete variant of the mass conservation equation and number the lattice vertexes by the indexes i, j and k, the corresponding Cartesian coordinates being x, y and z, respectively
One can rewrite the equation (3) in the form of the stationary conservation law: div j = 0, where the components of the current j in cylindrical coordinates ρ, z read: jρ = uρ − D ∂ρuz, jz = ruz, and the transverse diffusion coefficient is introduced: D = p d. It is worth-while to stress that the effect of the transverse diffusion in porous media is widely discussed in literature [2, 3]

Summary

Introduction

Lattice approximation for the flow in porous mediaThe main difficulty in the filtration process seems to be the so-called “near-wall” effect, that is an anomalously large value of the flow velocity near the wall due to larger values of the gaps between the wall and the grains, the effectiveness of the filtration being decreasing [1]. In order to take this effect into account, let us first consider the discrete variant of the mass conservation equation and number the lattice vertexes by the indexes i, j (transverse to the flow ) and k (along the flow), the corresponding Cartesian coordinates being x, y and z, respectively. One can rewrite the equation (3) in the form of the stationary conservation law: div j = 0, where the components of the current j in cylindrical coordinates ρ, z read: jρ = uρ − D ∂ρuz, jz = ruz, and the transverse diffusion coefficient is introduced: D = p d.

Results

Conclusion