If we consider the area in the section that does not contain pores, subtracting the pore section ($S_p$) from the section Flow ($S$) and dividing it by the area of a generic grain with a radius of ERROR:10129,0, we obtain the number of visible grains in the cross-section:

$\displaystyle\frac{S-S_p}{\pi r_0^2}=\displaystyle\frac{(1-f)S}{\pi r_0^2}$

where we use the relation for the porosity ($f$):

If the number of grains is equal to ERROR:6039,0 with the expression:

where the radius is the tube radius ($R$). With this, we obtain the relation:

$\displaystyle\frac{(1-f)S}{\pi r_0^2}=\displaystyle\frac{fS}{\pi R^2}$

resulting in:

(ID 109)

With the height the infinitesimal distance ($ds$), the volume of the section Flow ($S$) is

$S ds$

and the volume of the pores with the pore section ($S_p$) is

$S_p ds$

Therefore, the porosity ($f$) is calculated as

$f = \displaystyle\frac{S_p ds}{S ds} = \displaystyle\frac{S_p}{S}$

resulting in the following equation:

(ID 938)

(ID 977)

The volume of the capillary can be calculated from the tube radius ($R$) and the capillary length ($l$), which is equal to the volume of a chain of grains of the radius of a generic grain ($r_0$) and the tube length ($\Delta L$) multiplied by the generic own porosity ($q_0$):

$\pi R^2 l = q_0 \pi r_0^2 \Delta L$

This, in conjunction with the porosity ($f$) in the relationship

results in the following relationship:

(ID 2215)

(ID 4362)

As the porosity ($f$), calculated with the pore section ($S_p$) and the section Flow ($S$) using

together with the equation for the calculation of the pore section ($S_p$) based on the number of capillaries ($N_p$) and the tube radius ($R$) through

results in

$f = \displaystyle\frac{N_p\pi R^2}{S}$

can be solved for the number of capillaries ($N_p$), yielding

(ID 4363)

(ID 4364)

To calculate the total flow ($J_{Vt}$) using the number of capillaries ($N_p$) and the volume flow ($J_V$) for each capillary through

we obtain the number of capillaries ($N_p$) with the porosity ($f$), the section Flow ($S$), and the tube radius ($R$) through

and the Hagen-Poiseuille law using the viscosity ($\eta$), the pressure difference ($\Delta p$), and the capillary length ($l$) is calculated with

$J_v = - \displaystyle\frac{\pi R^4}{8\eta}\displaystyle\frac{\Delta p}{l}$

Using the relationship for the tube radius ($R$) in terms of the radius of a generic grain ($r_0$)

and for the capillary length ($l$), the generic own porosity ($q_0$), and the sample length ($\Delta L$)

we obtain

(ID 4365)

In the case of capillaries through which a liquid with the liquid density ($\rho_w$) flows due to the pressure difference ($\Delta p$) generated by ERROR:5446.1 under the influence of gravity represented by the gravitational Acceleration ($g$) and calculated with the equation:

this can be applied in the Hagen-Poiseuille equation, in terms of the total flow ($J_{Vt}$), which in turn depends on the radius of a generic grain ($r_0$), the generic own porosity ($q_0$), the porosity ($f$), the viscosity ($\eta$), the section or Area ($S$), and the sample length ($\Delta L$) as described in the equation:

Together with the definition of the flux density ($j_s$):

$j_s = \displaystyle\frac{J_{Vt}}{S}$

We have:

$j_s=\displaystyle\frac{J_{Vt}}{S}=\displaystyle\frac{ r_0 ^2 }{8 q_g }\displaystyle\frac{ f ^3 }{(1- f )^2}\displaystyle\frac{ \rho_w g }{ \eta }\displaystyle\frac{ \Delta h }{ L }$

resulting in:

(ID 4366)

Since the flux density ($j_s$) is related to the radius of a generic grain ($r_0$), the porosity ($f$), the liquid density ($\rho_w$), the gravitational Acceleration ($g$), the viscosity ($\eta$), the generic own porosity ($q_0$), the height difference ($\Delta h$), and the sample length ($\Delta L$) through the equation:

We can define a factor that we'll call the hydraulic conductivity ($K_s$) as follows:

This factor encompasses all the elements related to the properties of both the soil and the liquid that flows through it.

(ID 4739)

With Darcy's law, where the pressure difference ($\Delta p$) equals the hydraulic resistance ($R_h$) and the total flow ($J_{Vt}$):

Thus, with the equation for the soil with the section Flow ($S$), the radius of a generic grain ($r_0$), the viscosity ($\eta$), the generic own porosity ($q_0$), the porosity ($f$), the pressure difference ($\Delta p$), and the sample length ($\Delta L$):

Therefore, the hydraulic resistance ($R_h$) is:

(ID 10594)

(ID 15100)

(ID 15101)