A Two-Parametric Model for Gas Flow in Low-Permeable Porous Materials

Abstract

Capillary flow at low pressure gradients is determined by the pore structure of the matrix and the viscosity of the fluid. Traditionally, the conductivity of porous materials is characterized by a single parameter—the coefficient of permeability—which includes all geometric and structural resistances of the capillary structure. The fluid is described by its dynamic viscosity, which is independent of pressure in liquids. Gases, on the other hand, are highly compressible which means that their coefficient of permeability is becoming pressure dependent. The Klinkenberg effect takes account of this behaviour by introducing the notion of intrinsic permeability; nevertheless, it remains applicable only to a limited pressure range. At low- and high-pressure gradients, the Klinkenberg linearization deviates considerably from the real flow behaviour. This paper demonstrates that microporous capillary resistance to gas flow cannot be adequately described by a single material parameter. On the basis of ideal gas law conditions, the mass flow is derived by a two-parametric power law. Pore pressure decline is characterized by a permeability exponent n and a reference velocity v₁, replacing both dynamic viscosity of the fluid and permeability. The actual pressure distribution over the flow path is expressed by means of an equation of motion which leads to a logarithmic linearization of the material parameters. Velocity and acceleration vectors are derived and discussed in relation to the actual pressure distribution. The two parameters n and v₁ remain constant throughout and independent of pore pressure and are thus used to describe the flow rates over the entire pressure range. This model is then validated by means of flow analyses based on data given in the literature and compared to the classical theory of permeability.

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