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Porous medium hydraulic resistance
Storyboard 
Darcy's Law considers hydraulic resistance, which in its basic form corresponds to that of a tube with a given length and radius. However, in many situations, the fluid flows through a medium containing pores rather than a single cavity. These pores act as capillaries, and their hydraulic resistance can be modeled as that of small tubes. The sum of these multiple hydraulic resistances in parallel forms the total hydraulic resistance of a porous material.
ID:(2071, 0)
Porous medium hydraulic resistance
Storyboard
Darcy's Law considers hydraulic resistance, which in its basic form corresponds to that of a tube with a given length and radius. However, in many situations, the fluid flows through a medium containing pores rather than a single cavity. These pores act as capillaries, and their hydraulic resistance can be modeled as that of small tubes. The sum of these multiple hydraulic resistances in parallel forms the total hydraulic resistance of a porous material.
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Equations
The volume flow ($J_V$) can be calculated from the hydraulic conductance ($G_h$) and the pressure difference ($\Delta p$) using the following equation:
Furthermore, using the relationship for the hydraulic resistance ($R_h$):
results in:
Since the hydraulic resistance ($R_h$) is equal to the hydraulic conductance ($G_h$) as per the following equation:
and since the hydraulic conductance ($G_h$) is expressed in terms of the viscosity ($\eta$), the tube radius ($R$), and the tube length ($\Delta L$) as follows:
we can conclude that:
If there is the pressure difference ($\Delta p$) between two points, as determined by the equation:
we can utilize the water column pressure ($p$), which is defined as:
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:
the pressure difference ($\Delta p$) can be expressed as:
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:
and the volume equals the cross-sectional area the section Tube ($S$) multiplied by the distance traveled the tube element ($\Delta s$):
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:
Thus, the flow is a flux density ($j_s$), which is calculated using:
Examples
If the porous medium is modeled as a network of the hydraulic resistance ($R_h$) identical elements connected in parallel in groups of the number of equal hydraulic resistances in parallel ($N_p$), which are then summed in series as the number of identical hydraulic resistors in series ($N_s$):
In this way, the general parallel sum, which results in the total hydraulic resistance in parallel ($R_{pt}$) according to
is transformed into
Similarly, the general series sum, which yields the total hydraulic resistance in series ($R_{st}$) according to
is transformed into
By using the definition of the hydraulic resistance ($R_h$) with the values the viscosity ($\eta$), the tube length ($\Delta L$), and the tube radius ($R$) according to
and calculating the number of equal hydraulic resistances in parallel ($N_p$) from the porosity ($f$) and the section Tube ($S$) using
as well as the number of identical hydraulic resistors in series ($N_s$) with the tube length ($\Delta L$) and the capillary length ($l$) through
the final result is obtained as
The number of identical hydraulic resistors in series ($N_s$) can be obtained by dividing the tube length ($\Delta L$) by the capillary length ($l$):
The porosity ($f$) is the ratio of the empty section through which the liquid flows relative to the section Tube ($S$). The first is calculated by multiplying the section of each capillary, the tube radius ($R$), by the number of equal hydraulic resistances in parallel ($N_p$), so that:
The number of equal hydraulic resistances in parallel ($N_p$) is calculated as the fraction given by the porosity ($f$) of the section Tube ($S$), divided by the section of a capillary, the tube radius ($R$):
The total hydraulic resistance in parallel ($R_{pt}$) is the result for the hydraulic resistance ($R_h$) identical values, obtained by dividing this by the number of equal hydraulic resistances in parallel ($N_p$):
The total hydraulic resistance in series ($R_{st}$) wird berechnet, indem the hydraulic resistance ($R_h$) mit the number of identical hydraulic resistors in series ($N_s$) multipliziert wird:
Since the hydraulic resistance ($R_h$) is equal to the inverse of the hydraulic conductance ($G_h$), it can be calculated from the expression of the latter. In this way, we can identify parameters related to geometry (the tube length ($\Delta L$) and the tube radius ($R$)) and the type of liquid (the viscosity ($\eta$)), which can be collectively referred to as a hydraulic resistance ($R_h$):
The total hydraulic resistance ($R_t$) is calculated from a type of hydraulic resistance density, which depends on the viscosity ($\eta$), the porosity ($f$), and the tube radius ($R$), as well as the geometric factors the tube length ($\Delta L$) and the section Tube ($S$):
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:
If we have a tube with a the section Tube ($S$) moving a distance the tube element ($\Delta s$) along its axis, having displaced the volume element ($\Delta V$), then it is equal to:
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:
Darcy rewrites the Hagen Poiseuille equation so that the pressure difference ($\Delta p$) is equal to the hydraulic resistance ($R_h$) times the volume flow ($J_V$):
The porosity ($f$) expresses the relationship between the pore volume ($V_p$) and the total volume ($V_t$), allowing us to define the equation as follows:
The surface of a disk ($S$) of ERROR:5275.1 is calculated as follows:
The mean Speed ($\bar{v}$) can be calculated from the distance traveled in a time ($\Delta s$) and the time elapsed ($\Delta t$) using:
The total volume ($V_t$) is the sum of the pore volume ($V_p$), which includes both the micropores and the macropores in the soil, and the total Dry Mass of Sample ($M_s$), so that:
Since we already know the total Dry Mass of Sample ($M_s$) and the solid volume ($V_s$) from the sample, we can introduce the solid Density ($\rho_s$) and calculate it using the following equation:
ID:(2071, 0)