Storyboard

Once the hydraulic resistance and conductivity have been calculated, it becomes possible to model a multi-layer soil system. To achieve this, it is essential to compute the total resistance and conductivity and, after establishing the overall flow, to determine the partial flows (in the case of parallel layers) or the pressure drop in each layer (in the case of series layers).

>Model

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Concept

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Concept

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In the case of a sum where the elements are connected in series, the total hydraulic resistance of the system is calculated by summing the individual resistances of each element.

Since the elements are connected in series, the pressure drop occurs in each of the elements while the flow remains constant. Therefore, the total pressure difference ($\Delta p_t$) will be equal to the sum of the pressure difference in a network ($\Delta p_k$). Each of these elements, according to Darcy's law, is equal to the hydraulic resistance in a network ($R_{hk}$) multiplied by the volume flow in a network ($J_{Vk}$):

$\Delta p_k = R_{hk} J_{Vk}$

Thus, the sum of the hydraulic resistance in a network ($R_{hk}$) will be equal to the total hydraulic resistance in series ($R_{st}$).

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Concept

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In the case of a sum where the elements are connected in series, the total hydraulic conductance of the system is calculated by adding the individual hydraulic conductances of each element.

Since the elements are connected in series, the pressure drop occurs in each of the elements while the flow remains constant. Therefore, the total pressure difference ($\Delta p_t$) will be equal to the sum of the pressure difference in a network ($\Delta p_k$). Each of these elements, according to Darcy's law, is equal to the volume flow in a network ($J_{Vk}$) divided by the hydraulic conductance in a network ($G_{hk}$):

$\Delta p_k = \displaystyle\frac{J_{Vk}}{K_{hk}}$

So, the sum of the inverse of the hydraulic conductance in a network ($G_{hk}$) will be equal to the inverse of the total Series Hydraulic Conductance ($K_{st}$).

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Concept

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A situation in the soil where the elements are connected in series occurs when water infiltrates vertically through several layers, eventually ending up in the water table. In this case, the column Section ($S$) remains constant, while each layer has a different width that acts as the width of the kth layer ($L_k$).

In this scenario, hydraulic resistances are directly summed, and their values depend on the type of soil, and therefore, on the hydraulic conductivity in the kth layer ($K_{sk}$) and the width of the kth layer ($L_k$).

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Concept

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In the case of a sum where the elements are connected in parallel, the total hydraulic resistance of the system is calculated by adding the individual resistances of each element.

Since the elements are connected in parallel, the pressure drop is the same for all elements, while the flow varies from one to another. The value of the total flow ($J_{Vt}$) will be equal to the sum of the volume flow in a network ($J_{Vk}$). Each of these elements, according to Darcy's law, is equal to the pressure difference ($\Delta p$) divided by the hydraulic resistance in a network ($R_{hk}$):

$J_{Vk} = \displaystyle\frac{\Delta p}{R_{hk}}$

Therefore, the sum of the inverses of the hydraulic resistance in a network ($R_{hk}$) will be equal to the inverse of the total hydraulic resistance in parallel ($R_{pt}$).

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Concept

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En el caso de una suma en la que los elementos están conectados en paralelo, la conductancia hidráulica total del sistema se calcula sumando las conductancias individuales de cada elemento.



Dado que los elementos están conectados en paralelo, la caída de presión es la misma para todos los elementos, mientras que el flujo varía de uno a otro. El valor de the total flow ($J_{Vt}$) será igual a la suma de the volume flow in a network ($J_{Vk}$). Cada uno de estos elementos, de acuerdo con la ley de Darcy, es igual a the pressure difference ($\Delta p$) multiplicado por the hydraulic conductance in a network ($G_{hk}$):

$J_{Vk} =G_{hk} \Delta p$

Por lo tanto, la suma de the hydraulic conductance in a network ($G_{hk}$) será igual al inverso de the parallel total hydraulic conductance ($G_{pt}$).

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Concept

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A situation in the soil where the elements are connected in parallel occurs when water flows through different layers in parallel. If the layers have a slope, a pressure difference is generated. If the layers have a similar thickness, the pressure difference will be the same in all layers. In this case, the sample length ($\Delta L$) is constant, while each layer has a different the section of the kth layer ($S_k$).

In this situation, hydraulic conductivities are directly summed, and their values depend on the type of soil, and therefore, on the hydraulic conductivity in the kth layer ($K_{sk}$) and the section of the kth layer ($S_k$).

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Concept

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Selected parameter

Calculations

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Equation

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The flow of liquid in a porous medium such as soil is measured using the variable the flux density ($j_s$), which represents the average velocity at which the liquid moves through it. When modeling the soil and how the liquid passes through it, it is found that this process is influenced by factors such as the porosity ($f$) and the radius of a generic grain ($r_0$), which, when greater, facilitate the flow, whereas the viscosity ($\eta$) hinders passage through capillaries, reducing the flow velocity.

The modeling eventually incorporates what we will call the hydraulic conductivity ($K_s$), a variable that depends on the interactions between the radius of a generic grain ($r_0$), the porosity ($f$), the liquid density ($\rho_w$), the gravitational Acceleration ($g$), the viscosity ($\eta$), and the generic own porosity ($q_0$):

$ K_s \equiv \displaystyle\frac{ r_0 ^2 }{8 q_0 }\displaystyle\frac{ f ^3 }{(1- f )^2}\displaystyle\frac{ \rho_w g }{ \eta }$

Conditions

Variables

$J_t$

Flujo Total 2 Capas

$m^3/s$

$J_1$

Own volume factor of sand

$-$

$J_2$

Silt own volume factor

$-$

Relation

Development

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:

$ j_s =-\displaystyle\frac{ r_0 ^2 }{8 q_0 }\displaystyle\frac{ f ^3 }{(1- f )^2}\displaystyle\frac{ \rho_w g }{ \eta }\displaystyle\frac{ \Delta h }{ \Delta L }$

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

$ K_s \equiv \displaystyle\frac{ r_0 ^2 }{8 q_0 }\displaystyle\frac{ f ^3 }{(1- f )^2}\displaystyle\frac{ \rho_w g }{ \eta }$

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

the hydraulic conductivity ($K_s$) expresses how easily the liquid is conducted through the porous medium. In fact, the hydraulic conductivity ($K_s$) increases with the porosity ($f$) and the radius of a generic grain ($r_0$), and decreases with the generic own porosity ($q_0$) and the viscosity ($\eta$).

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Equation

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As the total flow ($J_{Vt}$) 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 column Section ($S$), and the sample length ($\Delta L$), it is equal to:

Therefore, the hydraulic conductance ($G_h$) is equal to:

$ G_h = \displaystyle\frac{ r_0 ^2}{8 \eta q_0 }\displaystyle\frac{f ^3}{(1- f )^2 }\displaystyle\frac{ S }{ \Delta L }$

Conditions

Variables

Relation

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Equation

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In the context of electrical resistance, there exists its inverse, known as electrical conductance. Similarly, what would be the hydraulic conductance ($G_h$) can be defined in terms of the hydraulic resistance ($R_h$) through the expression:

$ R_h = \displaystyle\frac{1}{G_h }$

Conditions

Variables

$G_h$

Hydraulic conductance

$m^4/kg s$

$R_h$

Hydraulic resistance

$kg/m^4s$

Relation

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Equation

>Top, >Model

$ R_h = \displaystyle\frac{8 \eta q_0 }{ r_0 ^2}\displaystyle\frac{(1- f )^2 }{f ^3}\displaystyle\frac{ \Delta L }{ S }$

Conditions

Variables

$S_1$

Section in point 1

$m^2$

$S_2$

Section in Point 2

$m^2$

$S$

Section or Area

$m^2$

Relation

Development

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

$ \Delta p = R_h J_V $

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 ($\Delta L$):

$ J_{Vt} =-\displaystyle\frac{ r_0 ^2}{8 \eta q_0 }\displaystyle\frac{ f ^3}{(1- f )^2}\displaystyle\frac{ S }{ \Delta L } \Delta p $

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

$ R_h = \displaystyle\frac{8 \eta q_0 }{ r_0 ^2}\displaystyle\frac{(1- f )^2 }{f ^3}\displaystyle\frac{ \Delta L }{ S }$

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Equation

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Calculating the hydraulic resistance ($R_h$) with the viscosity ($\eta$), the generic own porosity ($q_0$), the radius of a generic grain ($r_0$), the porosity ($f$), the sample length ($\Delta L$), and the column Section ($S$) using

which can be rewritten using the expression for the hydraulic conductivity ($K_s$) with the liquid density ($\rho_w$) and the gravitational Acceleration ($g$), resulting in

$ R_h = \displaystyle\frac{ \rho_w g }{ K_s }\displaystyle\frac{ \Delta L }{ S }$

Conditions

Variables

$\rho_b$

Dry bulk density

$kg/m^3$

$f_k$

Mass fraction of sand in the sample

$-$

$f$

Porosity

$-$

$\rho_s$

Solid Density

$kg/m^3$

Relation

Development

Calculating the hydraulic resistance ($R_h$) using the viscosity ($\eta$), the generic own porosity ($q_0$), the radius of a generic grain ($r_0$), the porosity ($f$), the sample length ($\Delta L$), and the column Section ($S$) through

$ R_h = \displaystyle\frac{8 \eta q_0 }{ r_0 ^2}\displaystyle\frac{(1- f )^2 }{f ^3}\displaystyle\frac{ \Delta L }{ S }$

and utilizing the expression for the hydraulic conductivity ($K_s$)

$ K_s \equiv \displaystyle\frac{ r_0 ^2 }{8 q_0 }\displaystyle\frac{ f ^3 }{(1- f )^2}\displaystyle\frac{ \rho_w g }{ \eta }$

is obtained after replacing the common factors

$ R_h = \displaystyle\frac{ \rho_w g }{ K_s }\displaystyle\frac{ \Delta L }{ S }$

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Equation

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As the hydraulic resistance ($R_h$) is related to the hydraulic conductivity ($K_s$), the liquid density ($\rho_w$), the gravitational Acceleration ($g$), the column Section ($S$), and the sample length ($\Delta L$), it is expressed as

Since the hydraulic conductance ($G_h$) is the inverse of the hydraulic resistance ($R_h$), we can conclude that

$ G_h = \displaystyle\frac{ K_s }{ \rho_w g }\displaystyle\frac{ S }{ \Delta L }$

Conditions

Variables

$V_p$

Pore volume

$m^3$

$V_c$

Solid volume of clay

$m^3$

Relation

Development

As the hydraulic resistance ($R_h$) is associated with the hydraulic conductivity ($K_s$), the liquid density ($\rho_w$), the gravitational Acceleration ($g$), the column Section ($S$), and the sample length ($\Delta L$), it is expressed as

$ R_h = \displaystyle\frac{ \rho_w g }{ K_s }\displaystyle\frac{ \Delta L }{ S }$

And the relationship for the hydraulic conductance ($G_h$)

$ R_h = \displaystyle\frac{1}{G_h }$

leads to

$ G_h = \displaystyle\frac{ K_s }{ \rho_w g }\displaystyle\frac{ S }{ \Delta L }$

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Equation

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The total pressure difference ($\Delta p_t$) in relation to the various pressure difference in a network ($\Delta p_k$), leading us to the following conclusion:

$ \Delta p_t =\displaystyle\sum_k \Delta p_k $

Conditions

Variables

$\Delta p_k$

Pressure Drop in Each Layer

$Pa$

$\Delta p_t$

Total Pressure Difference Parallel Layers

$Pa$

Relation

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Equation

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Since the volume flow ($J_V$) can be calculated from the hydraulic conductance ($G_h$) and the pressure difference ($\Delta p$) using the following equation:

it can be expressed in terms of the pressure difference ($\Delta p$). Considering that the inverse of the hydraulic resistance ($R_h$) is the hydraulic conductance ($G_h$), we arrive at the following expression:

$ \Delta p = R_h J_V $

Conditions

Variables

$R_h$

Hydraulic resistance

$kg/m^4s$

$\Delta p$

Variación de la Presión

$Pa$

$J_V$

Volume flow

$m^3/s$

Relation

Development

In the case of a single cylinder the hydraulic resistance ($R_h$), which depends on the viscosity ($\eta$), the tube length ($\Delta L$), and the cylinder radio ($R$), it is calculated using the following equation:

$ R_h =\displaystyle\frac{8 \eta | \Delta L | }{ \pi R ^4}$

On the other hand, Hagen-Poiseuille's law allows us to calculate the volume flow ($J_V$) generated by the pressure difference ($\Delta p$) according to the equation:

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

Combining both equations, we obtain Darcy's law:

$ \Delta p = R_h J_V $

which Henry Darcy formulated to model the general behavior of more complex porous media through which a liquid flows.

The genius of this way of rewriting the Hagen-Poiseuille law is that it demonstrates the analogy between the flow of electric current and the flow of liquid. In this sense, Hagen-Poiseuille's law corresponds to Ohm's law. This opens up the possibility of applying the concepts of electrical networks to systems of pipes through which a liquid flows.

This law, also known as the Darcy-Weisbach Law, was first published in Darcy's work:

• "Les fontaines publiques de la ville de Dijon" ("The Public Fountains of the City of Dijon"), Henry Darcy, Victor Dalmont Editeur, Paris (1856).

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Equation

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In the case of ($$), its value is calculated using the viscosity ($\eta$), the cylinder radio ($R$), and the tube length ($\Delta L$) through the equation:

When there are multiple hydraulic resistances connected in series, we can calculate the total hydraulic resistance in series ($R_{st}$) by summing the hydraulic resistance in a network ($R_{hk}$), as expressed in the following formula:

$ R_{st} =\displaystyle\sum_k R_{hk} $

Conditions

Variables

$R_h$

Hydraulic resistance

$kg/m^4s$

$R_{ss}$

Hydraulic resistance in a network

$kg/m^4s$

Relation

Development

One way to model a tube with varying cross-section is to divide it into sections with constant radii and then sum the hydraulic resistances in series. Let's assume that we have a series of sections with radii R_{hk} and lengths L_k. The corresponding hydraulic resistances would be

$ R_h =\displaystyle\frac{8 \eta | \Delta L | }{ \pi R ^4}$

In each element, there will be an equal pressure drop where the flow is the same, and Darcy's law applies:

$ \Delta p = R_h J_V $

The total pressure difference will be equal to the sum of individual pressure drops

$ \Delta p_t =\displaystyle\sum_k \Delta p_k $

so

$\Delta p=\displaystyle\sum_k \Delta p_k=\displaystyle\sum_k (R_{hk}J_V)=\left(\displaystyle\sum_k R_{hk}\right)J_V\equiv R_{st}J_V$

Therefore, the system can be modeled as a single conduit with hydraulic resistance calculated as the sum of individual components:

$ R_{st} =\displaystyle\sum_k R_{hk} $

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Equation

>Top, >Model

In the case of the sum of elements in series, the total hydraulic resistance in series ($R_{st}$) is equal to the sum of the hydraulic resistance in a network ($R_{hk}$):

Since the hydraulic resistance in a network ($R_{hk}$) is the inverse of the hydraulic conductance in a network ($G_{hk}$), we have:

$\displaystyle\frac{1}{ G_{st} }=\displaystyle\sum_k\displaystyle\frac{1}{ G_{hk} }$

Conditions

Variables

$R_{h1}$

Hydraulic Resistance 1

$kg/m^4s$

$R_{h2}$

Hydraulic Resistance 2

$kg/m^4s$

$R_{pt}$

Total hydraulic resistance in parallel

$kg/m^4s$

Relation

Development

The total hydraulic resistance in series ($R_{st}$), along with the hydraulic resistance in a network ($R_{hk}$), in

$ R_{st} =\displaystyle\sum_k R_{hk} $

and along with the hydraulic conductance in a network ($G_{hk}$) and the equation

$ R_h = \displaystyle\frac{1}{G_h }$

leads to

$\displaystyle\frac{1}{ G_{st} }=\displaystyle\sum_k\displaystyle\frac{1}{ G_{hk} }$

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Equation

>Top, >Model

Since each the hydraulic resistance of the kth layer ($R_{sk}$), which is a function of the liquid density ($\rho_w$), the gravitational Acceleration ($g$), ($$), the width of the kth layer ($L_k$), and the hydraulic conductivity in the kth layer ($K_{sk}$), is

it follows that the total hydraulic resistance in series ($R_{st}$) is

$ R_{st} = \displaystyle\sum_k\displaystyle\frac{ \rho_w g }{ K_{sk} }\displaystyle\frac{ L_k }{ S }$

Conditions

Variables

Relation

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Equation

>Top, >Model

The sum of soil layers in parallel, denoted as the total flow ($J_{Vt}$), is equal to the sum of the volume flow in a network ($J_{Vk}$):

$ J_{Vt} =\displaystyle\sum_k J_{Vk} $

Conditions

Variables

$J_{Vt}$

Flow in System of Parallel Layers

$m^3/s$

$J_{Vk}$

Microporosity own volume factor

$-$

Relation

.

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Equation

>Top, >Model

With the introduction of the hydraulic conductance ($G_h$), we can rewrite the Hagen-Poiseuille equation with the pressure difference ($\Delta p$) and the volume flow ($J_V$) using the following equation:

$ J_V = G_h \Delta p $

Conditions

Variables

$G_h$

Hydraulic conductance

$m^4/kg s$

$\Delta p$

Pressure difference

$Pa$

$J_V$

Volume flow

$m^3/s$

Relation

Development

If we examine the Hagen-Poiseuille law, which allows us to calculate the volume flow ($J_V$) from the cylinder radio ($R$), the viscosity ($\eta$), the tube length ($\Delta L$), and the pressure difference ($\Delta p$):

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

we can introduce the hydraulic conductance ($G_h$), defined in terms of the tube length ($\Delta L$), the cylinder radio ($R$), and the viscosity ($\eta$), as follows:

$ G_h =\displaystyle\frac{ \pi R ^4}{8 \eta | \Delta L | }$

to arrive at:

$ J_V = G_h \Delta p $

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Equation

>Top, >Model

In the case of parallel elements, the pressure drop is equal across all of them. The total flow ($J_{Vt}$) is the sum of the volume flow in a network ($J_{Vk}$):

And since the volume flow in a network ($J_{Vk}$) is proportional to the hydraulic conductance in a network ($G_{hk}$), we can conclude that

$ G_{pt} =\displaystyle\sum_k G_{hk} $

Conditions

Variables

$R_{h1}$

Hydraulic Resistance 1

$kg/m^4s$

$R_{h2}$

Hydraulic Resistance 2

$kg/m^4s$

$R_{h3}$

Hydraulic Resistance 3

$kg/m^4s$

$R_{pt}$

Total hydraulic resistance in parallel

$kg/m^4s$

Relation

Development

With the total flow ($J_{Vt}$) being equal to the volume flow in a network ($J_{Vk}$):

$ J_{Vt} =\displaystyle\sum_k J_{Vk} $

and with the pressure difference ($\Delta p$) and the hydraulic conductance in a network ($G_{hk}$), along with the equation

$ J_V = G_h \Delta p $

for each element, it leads us to the conclusion that with the parallel total hydraulic conductance ($G_{pt}$),

$J_{Vt}=\displaystyle\sum_k J_{Vk} = \displaystyle\sum_k K_{hk}\Delta p = K_{pt}\Delta p$

we have

$ G_{pt} =\displaystyle\sum_k G_{hk} $

.

.

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Equation

>Top, >Model

In the case of a hydraulic resistance, its value is calculated using the equation:

When there are multiple hydraulic resistances connected in parallel, the hydraulic resistance of the entire system can be calculated using the following formula, specifically for parallel connections

$\displaystyle\frac{1}{ R_{pt} }=\sum_k\displaystyle\frac{1}{ R_{hk} }$

Conditions

Variables

$R_h$

Hydraulic resistance

$kg/m^4s$

$R_{sp}$

Resistencia hidráulica sumada en paralelo (multiple)

$kg/m^4s$

Relation

Development

The parallel total hydraulic conductance ($G_{pt}$) combined with the hydraulic conductance in a network ($G_{hk}$) in

$ G_{pt} =\displaystyle\sum_k G_{hk} $

and along with the hydraulic resistance in a network ($R_{hk}$) and the equation

$ R_h = \displaystyle\frac{1}{G_h }$

leads to

$\displaystyle\frac{1}{ R_{pt} }=\sum_k\displaystyle\frac{1}{ R_{hk} }$

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Equation

>Top, >Model

Considering that each the hydraulic conductance in a network ($G_{hk}$), dependent on the liquid density ($\rho_w$), the gravitational Acceleration ($g$), the soil layer length ($L$), the section of the kth layer ($S_k$), and the hydraulic conductivity in the kth layer ($K_{sk}$), is equal to:

Therefore, the parallel total hydraulic conductance ($G_{pt}$) is calculated as:

$ G_{pt} = \displaystyle\sum_k\displaystyle\frac{ K_{sk} }{ \rho_w g }\displaystyle\frac{ S_k }{ L }$

Conditions

Variables

$d$

Horizontal Distance

$m$

Relation

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