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Porous medium hydraulic resistance
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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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Hydrodynamic networks in porous media
Concept
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
| $\displaystyle\frac{1}{ R_{pt} }=\sum_k\displaystyle\frac{1}{ R_{hk} }$ |
,
is transformed into
| $ R_{pt} = \displaystyle\frac{ R_h }{ N_p }$ |
.
Similarly, the general series sum, which yields the total hydraulic resistance in series ($R_{st}$) according to
| $ R_{st} =\displaystyle\sum_k R_{hk} $ |
,
is transformed into
| $ R_t = N_s R_{pt} $ |
.
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Hydrodynamic resistance of a porous medium
Concept
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
| $ R_h =\displaystyle\frac{8 \eta | l | }{ \pi r ^4}$ |
,
and calculating the number of equal hydraulic resistances in parallel ($N_p$) from the porosity ($f$) and the section Tube ($S$) using
| $ N_p = \displaystyle\frac{ f S }{ \pi r ^2}$ |
,
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
| $ N_s = \displaystyle\frac{ \Delta L }{ l }$ |
,
the final result is obtained as
| $ R_t = \displaystyle\frac{ 8 \eta }{ f r ^2 }\displaystyle\frac{ \Delta L }{ S }$ |
.
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Model
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Target : Equation To be used 
Equations
$ \Delta p = \rho_w g \Delta h $
Dp = rho_w * g * Dh
$ \Delta p = R_t J_V $
Dp = R_h * J_V
$ V_t = S \Delta L $
DV = S * Ds
$ f = \displaystyle\frac{ N_p \pi r ^2 }{ S }$
f = N_p * pi * R ^2/ S
$ f =\displaystyle\frac{ V_p }{ V_t }$
f = V_p / V_t
$ j_s = \displaystyle\frac{ J_V }{ S }$
j_s = J_V / S
$ N_p = \displaystyle\frac{ f S }{ \pi r ^2}$
N_p = f * S /( pi * R ^2)
$ N_s = \displaystyle\frac{ \Delta L }{ l }$
N_s = DL / l
$ \rho_s = \displaystyle\frac{ M_s }{ V_s }$
rho_s = M_s / V_s
$ R_h =\displaystyle\frac{8 \eta | l | }{ \pi r ^4}$
R_h =8* eta * abs( DL )/( pi * R ^4)
$ R_{pt} = \displaystyle\frac{ R_h }{ N_p }$
R_pt = R_h / N_p
$ R_t = N_s R_{pt} $
R_st = N_s * R_h
$ R_t = \displaystyle\frac{ 8 \eta }{ f r ^2 }\displaystyle\frac{ \Delta L }{ S }$
R_t = 8* eta * DL /( f * R ^2* S )
$ S = \pi R ^2$
S = pi * r ^2
$ j_s \equiv\displaystyle\frac{ \Delta s }{ \Delta t }$
v_m = Ds / Dt
$ V_t = V_s + V_p $
V_t = V_s + V_p
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Number of hydraulic resistors in series
Equation
The number of identical hydraulic resistors in series ($N_s$) can be obtained by dividing the tube length ($\Delta L$) by the capillary length ($l$):
| $ N_s = \displaystyle\frac{ \Delta L }{ l }$ |
ID:(15904, 0)
Porosity via surface
Equation
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:
| $ f = \displaystyle\frac{ N_p \pi r ^2 }{ S }$ |
| $ f = \displaystyle\frac{ N_p \pi R ^2 }{ S }$ |
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Number of hydraulic resistances in parallel
Equation
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 ($R$):
| $ N_p = \displaystyle\frac{ f S }{ \pi r ^2}$ |
| $ N_p = \displaystyle\frac{ f S }{ \pi R ^2}$ |
ID:(15905, 0)
Sum of identical hydraulic resistances in parallel
Equation
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 ($N_p$):
| $ R_{pt} = \displaystyle\frac{ R_h }{ N_p }$ |
ID:(15902, 0)
Sum of identical hydraulic resistances in series
Equation
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:
| $ R_t = N_s R_{pt} $ |
| $ R_{st} = N_s R_h $ |
ID:(15903, 0)
Hydraulic resistance of a tube
Equation
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 ($R_h$):
| $ R_h =\displaystyle\frac{8 \eta | l | }{ \pi r ^4}$ |
| $ R_h =\displaystyle\frac{8 \eta | \Delta L | }{ \pi R ^4}$ |
Since the hydraulic resistance ($R_h$) is equal to the hydraulic conductance ($G_h$) as per the following equation:
| $ R_h = \displaystyle\frac{1}{ G_h }$ |
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:
| $ G_h =\displaystyle\frac{ \pi R ^4}{8 \eta | \Delta L | }$ |
we can conclude that:
| $ R_h =\displaystyle\frac{8 \eta | \Delta L | }{ \pi R ^4}$ |
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Hydraulic resistance of porous material
Equation
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 ($S$):
| $ R_t = \displaystyle\frac{ 8 \eta }{ f r ^2 }\displaystyle\frac{ \Delta L }{ S }$ |
| $ R_t = \displaystyle\frac{ 8 \eta }{ f R ^2 }\displaystyle\frac{ \Delta L }{ S }$ |
ID:(15907, 0)
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 $ |
ID:(4345, 0)
Volume element
Equation
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:
| $ V_t = S \Delta L $ |
| $ \Delta V = S \Delta s $ |
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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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Darcy's law and hydraulic resistance
Equation
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 ($J_V$):
| $ \Delta p = R_t J_V $ |
| $ \Delta p = R_h J_V $ |
The volume flow ($J_V$) can be calculated from the hydraulic conductance ($G_h$) and the pressure difference ($\Delta p$) using the following equation:
| $ J_V = G_h \Delta p $ |
Furthermore, using the relationship for the hydraulic resistance ($R_h$):
| $ R_h = \displaystyle\frac{1}{ G_h }$ |
results in:
| $ \Delta p = R_h J_V $ |
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Porosity
Equation
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:
| $ f =\displaystyle\frac{ V_p }{ V_t }$ |
ID:(4245, 0)
Surface of a disk
Equation
The surface of a disk ($S$) of ($$) is calculated as follows:
| $ S = \pi R ^2$ |
| $ S = \pi r ^2$ |
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Average Speed
Equation
The mean Speed ($\bar{v}$) can be calculated from the distance traveled in a time ($\Delta s$) and the time elapsed ($\Delta t$) using:
| $ j_s \equiv\displaystyle\frac{ \Delta s }{ \Delta t }$ |
| $ \bar{v} \equiv\displaystyle\frac{ \Delta s }{ \Delta t }$ |
ID:(3152, 0)
Total volume with general porosity
Equation
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:
| $ V_t = V_s + V_p $ |
ID:(4726, 0)
Solid density
Equation
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:
| $ \rho_s = \displaystyle\frac{ M_s }{ V_s }$ |
ID:(15073, 0)