Effective Permeability Equations

The effective permeability through a material will vary directionally, particularly if the material has a layered structure. If we consider a simple layered material as illustrated below, we can use 2 equations to define the effective permeability horizontally and vertically through the material.

Equation for Horizontal Permeability:

Equation for Vertical Permeability:

Before describing what each equations means, it is important to understand the parameters or symbols they use.

k = permeability (the ability of a material to transmit fluid)

• In these equations we are solving for keff, which is the effective permeability. The term effective denotes that we want the permeability for the entire material, not just one or a few of the layers within it.

• When ki is used in the equations, we are referring to the permeability of a specific layer, i. We are using "i" as a placeholder for the actual position of the layer, where "i" can be any number. For example, if we are referring to the 2nd layer in a material, "i" would equal 2 and the permeability would be expressed as k2. We will use the subscript "i" here, because we will need to consider individual permeability values for each layer in order to get the total keff.

h = layer height

• These equations use h for the height of the layers in the material. As with the permeability, the subscript "i" will also be used with h as hi to denote the height of the individual layers, as the layers will typically vary in height.

• The h with the overlying bar is used to represent the total height of the material or the sum of h1 through the h of the last layer of the material.

In the above diagrams, you will also see that an l term is included which represents the length of each layer. This term is not included in the permeability equations, and is negligible if all the layers are the same length.