Different models can define the capacity of the porous medium to transmit flow. Some use the permeability, κ (SI unit: m2), and the viscosity of the fluid, μ (SI unit: Pa·s); others use the hydraulic conductivity, K (SI unit: m/s). Several relationships have been established to describe Darcian and non-Darcian flow through different porous materials.

The capacity of the porous medium to transmit flow cannot be described by porosity alone, as the shape and orientation of the pores is also an important factor. This is described by the permeability κ. For an isotropic medium κ is a scalar quantity. Often porous materials are anisotropic and κ then is a tensor.

Equation 2-1 changes to

The Kozeny–Carman equation describes the flow through granular soils and packed beds by estimating the permeability of the porous medium from the porosity εp and average particle diameter dp (see, for example, Ref. 5).

The Ergun equation is an extension of Kozeny–Carman equation to higher Reynolds numbers (Re > 10), where Darcy’s linear relation between pressure drop and velocity is no longer valid. In Ergun equation the linear relation between pressure drop and velocity is augmented by a quadratic term. The pressure drop (Ref. 6) in Ergun’s equation is given by:

Here, dp is the effective (average) particle diameter in the porous medium and εp is the porosity. With definitions of permeability κ and parameter β (SI unit 1/m) as

we get Equation 2-2 and Ergun’s velocity can then be calculated as

Using the Reynolds number Re and the friction factor fp for packed beds

From this expression, we observe that for low Reynolds numbers Re < 1 (or equivalently, high friction factors) the flow can be described by the linear Darcy’s Law (Equation 2-1).

For increasing filter velocities (Re > 10), inertial effects and turbulent friction forces become significant. In the Forchheimer drag this is considered by adding an additional term to Darcy’s law:

Here, the Forchheimer parameter cF is dimensionless. The permeability κ is specified by the user and is not computed from porosity nor particle diameter as for Ergun’s equation. If the parameter β is defined as

Equation 2-6 becomes equivalent to Equation 2-1.

The Burke-Plummer equation is applicable in flow regimes with Reynolds number Re > 1000. Here, the pressure drop is proportional to the square of the velocity:

here, L is the length of the packed bed, Δp is the pressure drop dp is the effective (average) particle diameter in the porous medium, εp is the porosity, and v is the velocity magnitude.

Using the same definition for the parameter β (SI unit 1/m) as in Ergun’s equation (Equation 2-5) the pressure drop is related to the velocity as

As flow travels from areas of higher pressure to areas of lower pressure, Equation 2-2 can be more generally written as

Note that the viscosity is not used to define the velocity as in Darcy’s law, but only the fluid density, the porosity and particle diameter. Note also that the analytical friction factor for Burke-Plummer equation is fp  = 1.75. A small parameter is added inside the square-root to avoid division by zero when the pressure gradient approaches zero.

The Klinkenberg effect becomes important for gas flows in porous media where the mean free path of the gas molecules is comparable to the pore dimensions. In this case, molecular collisions with the pore walls occur more often than collisions between molecules. To take this effect into account, Klinkenberg (Ref. 7) derived the following expression for the permeability:

here, κ∞ is the permeability of the gas at high pressure and density (Knudsen number 0.001 < Kn < 0.1), bK is the Klinkenberg parameter (also called Klinkenberg slip factor, or gas slippage factor), whose default value is set to 1 kPa according to Ref. 8, and pA is the absolute pressure as defined in the Darcy’s Law interface (pA = p + pref).