A New Momentum Equation for Gas Flow in Porous Media: The Klinkenberg Effect Seen Through the Kinetic Theory
Résumé
In this paper, an original set of transport equations for gas in porous media is developed. As far as low pressure gases or very fine grained porous media are concerned, molecular effects are likely to promote a dependency of the permeability on the pressure. These phenomena are usually modelled using the Klinkenberg correction to the Darcy's law. The retained methodology brings a new interpretation of this particular problem. The approach is based on a volume averaging scale-change methodology applied to the Boltzman equation taking into account the presence of walls. It leads to a homogenized kinetic equation describing the problem at the macroscale. A proper closure is then applied following the strategy proposed by Levermore to obtain a hydrodynamic description. The hydrodynamic force applied by the porous structure on the gas exhibits a strong non-linearity with the gas velocity. However, a linearization is proposed, recovering formally the classical Darcy's law. The validity of the resulting permeability tensor is finally discussed. As its dependency with pressure is concerned, it opens an original interpretation of the nature of the Klinkenberg effect.