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Permeability of a medium
Storyboard 
When working with larger scale media rather than small samples, it is useful to work with quantities similar to densities rather than magnitudes associated with limited volumes. Therefore, the focus is not on flow through a limited volume, but rather on flow densities. Similarly, hydraulic resistances of a finite volume are not considered; instead, permeability is used, which serves a similar role to a density of hydraulic resistance.
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Flow density between columns
Concept
As the volume flow ($J_V$), involving the tube radius ($R$), the viscosity ($\eta$), the pressure difference ($\Delta p$), and the tube length ($\Delta L$), is modeled using the Hagen-Poiseuille equation:
| $ J_V =-\displaystyle\frac{ \pi R ^4}{8 \eta }\displaystyle\frac{ \Delta p }{ \Delta L }$ |
It is possible to calculate using the section Flow ($S$) and the tube radius ($R$) with the following equation:
| $ S = \pi r ^2$ |
Additionally, the flux density ($j_s$) which is defined by
| $ j_s = \displaystyle\frac{ J_V }{ S }$ |
and the definition of the hydrodynamic permeability ($k$) is
| $ k = \displaystyle\frac{ R ^2}{8}$ |
from which it follows:
| $ j_s = -\displaystyle\frac{ k }{ \eta }\displaystyle\frac{ \Delta p }{ \Delta L }$ |
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Volume Flow and its Speed
Equation
A flux density ($j_s$) can be expressed in terms of the volume flow ($J_V$) using the section or Area ($S$) through the following formula:
 $ j_s = \displaystyle\frac{ J_V }{ S }$ |
Flow is defined as the volume the volume element ($\Delta V$) divided by time the time elapsed ($\Delta t$), which is expressed in the following equation:
| $ J_V =\displaystyle\frac{ \Delta V }{ \Delta t }$ |
and the volume equals the cross-sectional area the section Tube ($S$) multiplied by the distance traveled the tube element ($\Delta s$):
| $ \Delta V = S \Delta s $ |
Since the distance traveled the tube element ($\Delta s$) per unit time the time elapsed ($\Delta t$) corresponds to the velocity, it is represented by:
| $ j_s =\displaystyle\frac{ \Delta s }{ \Delta t }$ |
Thus, the flow is a flux density ($j_s$), which is calculated using:
| $ j_s = \displaystyle\frac{ J_V }{ S }$ |
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Surface of a disk
Equation
The surface of a disk ($S$) of ($$) is calculated as follows:
| $ S = \pi r ^2$ |
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Hydraulic permeability
Equation
The remaining factor is called the hydrodynamic permeability ($k$) and can be calculated using the tube radius ($R$) with the following formula:
| $ k = \displaystyle\frac{ R ^2}{8}$ |
If we examine the hydraulic conductance ($G_h$), we can notice that the numerator contains the cross-sectional area of the tube, represented as $\pi R^2$. Here, the tube radius ($R$) corresponds to a property of the liquid, the viscosity ($\eta$) is related to the viscosity of the fluid, and the tube length ($\Delta L$) refers to the generated pressure gradient.
| $ G_h =\displaystyle\frac{ \pi R ^4}{8 \eta | \Delta L | }$ |
Thus, the factor specific to the geometry of the pores can be defined as the hydrodynamic permeability ($k$) using the following formula:
| $ k = \displaystyle\frac{ R ^2}{8}$ |
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Flow under hydrostatic pressure
Equation
The Hagen Poiseuille equation can be rewritten as a function of the flux density ($j_s$) in terms of the hydrodynamic permeability ($k$), the viscosity ($\eta$) and the gradient of the pressure difference ($\Delta p$) > and the tube length ($\Delta L$):
| $ j_s = -\displaystyle\frac{ k }{ \eta }\displaystyle\frac{ \Delta p }{ \Delta L }$ |
As the volume flow ($J_V$), involving the tube radius ($R$), the viscosity ($\eta$), the pressure difference ($\Delta p$), and the tube length ($\Delta L$), is modeled using the Hagen-Poiseuille equation:
| $ J_V =-\displaystyle\frac{ \pi R ^4}{8 \eta }\displaystyle\frac{ \Delta p }{ \Delta L }$ |
It is possible to calculate using the section Flow ($S$) and the tube radius ($R$) with the following equation:
| $ S = \pi r ^2$ |
Additionally, the flux density ($j_s$) which is defined by
| $ j_s = \displaystyle\frac{ J_V }{ S }$ |
and the definition of the hydrodynamic permeability ($k$) is
| $ k = \displaystyle\frac{ R ^2}{8}$ |
from which it follows:
| $ j_s = -\displaystyle\frac{ k }{ \eta }\displaystyle\frac{ \Delta p }{ \Delta L }$ |
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Pressure difference between columns
Equation
The height difference, denoted by the height difference ($\Delta h$), implies that the pressure in both columns is distinct. In particular, the pressure difference ($\Delta p$) is a function of the liquid density ($\rho_w$), the gravitational Acceleration ($g$), and the height difference ($\Delta h$), as follows:
| $ \Delta p = \rho_w g \Delta h $ |
If there is the pressure difference ($\Delta p$) between two points, as determined by the equation:
| $ \Delta p = p_2 - p_1 $ |
we can utilize the water column pressure ($p$), which is defined as:
| $ p_t = p_0 + \rho_w g h $ |
This results in:
$\Delta p=p_2-p_1=p_0+\rho_wh_2g-p_0-\rho_wh_1g=\rho_w(h_2-h_1)g$
As the height difference ($\Delta h$) is:
| $ \Delta h = h_2 - h_1 $ |
the pressure difference ($\Delta p$) can be expressed as:
| $ \Delta p = \rho_w g \Delta h $ |
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