Different models can define the capacity of the porous medium to transmit flow. Some use the permeability, κ, and the viscosity of the fluid, μ, and others use the hydraulic conductivity, K (SI unit: m/s).

Equation 8-33 changes to

The hydraulic potential in the equation comes from the pore pressure, p, and gravity, ρ gD. COMSOL Multiphysics solves for the pressure, p. By default, g is the predefined acceleration of gravity (a physical constant), and D is the vertical coordinate. The choice of D has a significant impact on results and the physics involved. For example, if D is the vertical coordinate z and if the flow is entirely horizontal within the xy-plane, then the gradient in D vanishes and the driving force is caused by pressure gradients alone.

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)

If the model is defined using the Kozeny-Carman equation, the expression for Darcy’s velocity in Equation 8-33 changes to

In case of relatively fast flow (Re > 10) or relatively high Knudsen numbers (Kn > 0.1) Darcy’s linear relation between velocity and pressure drop is no longer valid. Therefore, different permeability models have been introduced to capture these effects.

The flow regime through a packed bed may be identified from the Reynolds number of the bed. According to textbooks, for Reynolds numbers Re < 10 the flow can be described by Kozeny-Carman equation (Darcian flow). For 10 < Re < 1000, which is sometimes called the transitional regime, the flow is better described by the non-Darcian Ergun equation, and for Re > 1000 Ergun’s equation can be approximated by the Burke-Plummer equation for turbulent flows.

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 10), where Darcy’s linear relation between pressure drop and velocity is no longer valid.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

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.

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 8-26 becomes equivalent to Ergun’s equation (Equation 8-22)

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

As flow travels from areas of higher pressure to areas of lower pressure, Equation 8-22 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).