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The porosity of the soil allows rain or irrigation water to penetrate the soil and reach the napa. Therefore we must study how it can be modeled based on our geometric model as the water moves.
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Flow density and hydraulic conductivity
Concept
The flux density ($j_s$) can be expressed in terms of the hydraulic conductivity ($K_s$), in the infinitesimal limit with the column height differential ($dh$) and the distance differential ($dx$), as follows:
| $ j_s = - K_s \displaystyle\frac{ dh }{ dx }$ |
This means that the steeper the gradient or the steeper the terrain, the larger the flux density ($j_s$) will be, as illustrated in the graph:
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The graph shows how bars with equal distance differential ($dx$) values have progressively smaller column height differential ($dh$) values, resulting in a decreasing flux density ($j_s$). Since the volume of the liquid is conserved, this can only be possible if there is another flow that compensates for this reduction in flux density ($j_s$). This could be a flow perpendicular to the one shown, for example, if the shorter bars are wider in a direction perpendicular to the graph.
This issue leads to the following:
The height $h$ of the liquid can only be calculated as a result of solving a differential equation, as it must meet the requirement that volume is conserved throughout the entire area where flow occurs.
Additionally, it is important to keep in mind that:
The negative sign reflects the fact that flow always goes from the higher to the lower elevation zone. If the slope is negative, the negative sign results in positive flow (from left to right), and conversely, if the slope is positive, the flow is negative (from right to left).
ID:(930, 0)
Flow equation in one dimension
Concept
If we study the one-dimensional case, describing the process along the $x$-axis, we can observe how the height of the column $\Delta h$ varies over a time interval $\Delta t$. In this case, a column with width $\Delta x$ will change its volume per unit length over time as $\Delta x \Delta h/\Delta t$. On the other hand, the amount of liquid entering along the column at $x$ is $h(x) j_s(x)$, while at $x+\Delta x$ it exits as $h(x+\Delta x) j_s(x+\Delta x)$:
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Therefore, the variation of the height of the water column on the ground ($h$) over time is equal to the variation of the product of the height of the water column on the ground ($h$) and the flux density ($j_s$) at position:
| $\displaystyle\frac{\partial h}{\partial t} = - \displaystyle\frac{\partial}{\partial x}( h j_s )$ |
Partial derivatives are similar to ordinary derivatives, with the difference that they are applied to functions that depend on more than one variable. In these cases, the partial derivative, denoted by the symbol $\partial$, reminds us of the typical derivative denoted by the letter $d$, but with the peculiarity that the variables not mentioned in the denominator are held constant.
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Flow into a channel
Concept
In the case of flow towards a channel, the system can be modeled in a one-dimensional manner, where the height of the water column on the ground ($h$) is a function of the position of the water column on the ground ($x$) representing the flux density ($j_s$), and it satisfies the condition
| $ h j_s = h_0 j_{s0} $ |
with the flow at a reference point ($j_{s0}$) and the reference height of the water column ($h_0$) defining the water profile in the soil:
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The key to this equation is that the product of the height of the water column on the ground ($h$) and the flux density ($j_s$) must always remain constant. In this sense, if the height of the water column on the ground ($h$) increases, the flux density ($j_s$) decreases, and vice versa. Moreover, the sign remains the same; hence, flow towards the channel, i.e., negative flow, will occur only when the groundwater level is higher than that of the channel. As the liquid approaches the channel, the groundwater level decreases, leading to an increase in flow density.
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Flow from a channel
Concept
In the case where the flow emerges from the channel, a situation arises where the level of the height of the water column on the ground ($h$) must decrease as we move away from the channel, ensuring the existence of the pressure gradient that drives the flow. The problem is that if the flow rapidly moves within the medium, the height will tend to zero, and as a result, the flow will approach infinity, which doesn't make sense.
This means that there is no stationary solution in such a scenario, and the only solution is for the medium to fill up until it reaches the height of the channel, effectively becoming constant.
The question is whether there exists a non-trivial stationary situation that represents a real and interesting scenario. One possible case is when the level of the medium decreases to the point where it becomes lower than the column before the solution diverges. This case corresponds to the situation where the flow emerges at the surface, and there is no divergence in the solution. This would imply that a flow is generated that exits to the exterior at a certain point, with the risk of weakening the foundation and thereby destabilizing the medium, which acts as a dam.
ID:(4746, 0)
Flow into a well
Concept
In the case of groundwater flow towards a well, the height of the water column on the ground ($h$) as a function of the radius from center of well ($r$) with the water well radius ($r_0$), the characteristic length of the flow in the ground ($s_0$), and the reference height of the water column ($h_0$) is represented by
| $ r \displaystyle\frac{ dh^2 }{ dr } = 2 h_0 ^2\displaystyle\frac{ r_0 }{ s_0 } $ |
which defines the water profile in the ground:
ID:(4371, 0)
Model
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Calculations
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Target : Equation To be used 
Equations
$ \vec{j}_s = - K_s \nabla h $
&j_s = - K_s * @GRAD( h , x )
$ h \nabla^2 h + \nabla h \cdot \nabla h = 0 $
h * &D^2 h + &D h * &D h = 0
$ h j_s = h_0 j_{s0} $
h * j_s = h_0 * j_{s0}
$ j_s = -\displaystyle\frac{ k_s }{ \eta }\displaystyle\frac{ dp }{ dx }$
j_s = - k_s * dp /( eta * dx )
$ k_s = \displaystyle\frac{ \eta }{ \rho_w g } K_s $
k_s = eta * K_s /( rho_w * g )
$ K_s \equiv \displaystyle\frac{ r_0 ^2 }{8 q_0 }\displaystyle\frac{ f ^3 }{(1- f )^2}\displaystyle\frac{ \rho_w g }{ \eta }$
K_s = r_0 ^2 * f ^3 * rho_w * g /(8* q_0 *(1- f )^2* eta )
$ k_s \equiv \displaystyle\frac{ r_0 ^2 }{8 q_0 }\displaystyle\frac{ f ^3 }{(1- f )^2}$
k_s = r_0 ^2 * f ^3/(8* q_0 *(1 - f )^2)
$\displaystyle\frac{\partial h}{\partial t} = - \vec{\nabla} \cdot ( h \vec{j}_s )$
D_t h = - &D * (h &j_s)
ID:(15224, 0)
Soil hydraulic conductivity
Equation
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 }$ |
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 ($\eta$).
ID:(4739, 0)
Soil permeability
Equation
The hydraulic conductivity ($K_s$) represents how the liquid behaves within the medium. Part of the hydraulic conductivity ($K_s$) is inherent to the properties of the medium itself, while another part contains constants that describe the behavior of the liquid. Therefore, it makes sense to introduce a new constant that is specific to the medium and not to the flowing liquid.
As a result, the soil permeability ($k_s$) is related to the radius of a generic grain ($r_0$), the porosity ($f$), and the generic own porosity ($q_0$) through the following definition:
| $ k_s \equiv \displaystyle\frac{ r_0 ^2 }{8 q_0 }\displaystyle\frac{ f ^3 }{(1- f )^2}$ |
Since the hydraulic conductivity ($K_s$) 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 ($\eta$), and the generic own porosity ($q_0$) through
| $ K_s \equiv \displaystyle\frac{ r_0 ^2 }{8 q_0 }\displaystyle\frac{ f ^3 }{(1- f )^2}\displaystyle\frac{ \rho_w g }{ \eta }$ |
we can define the part that depends solely on the soil as the soil permeability ($k_s$), expressing it as follows:
| $ k_s \equiv \displaystyle\frac{ r_0 ^2 }{8 q_0 }\displaystyle\frac{ f ^3 }{(1- f )^2}$ |
ID:(10595, 0)
Permeability and hydraulic conductivity
Equation
The soil permeability ($k_s$) can be calculated from the hydraulic conductivity ($K_s$), the liquid density ($\rho_w$), the gravitational Acceleration ($g$), and the viscosity ($\eta$) using the following expression:
| $ k_s = \displaystyle\frac{ \eta }{ \rho_w g } K_s $ |
The soil permeability ($k_s$) is related to the radius of a generic grain ($r_0$), the porosity ($f$), and the generic own porosity ($q_0$), it is equal to
| $ k_s \equiv \displaystyle\frac{ r_0 ^2 }{8 q_0 }\displaystyle\frac{ f ^3 }{(1- f )^2}$ |
Therefore, with the equation for the hydraulic conductivity ($K_s$), along with the liquid density ($\rho_w$), the gravitational Acceleration ($g$), and the viscosity ($\eta$),
| $ K_s \equiv \displaystyle\frac{ r_0 ^2 }{8 q_0 }\displaystyle\frac{ f ^3 }{(1- f )^2}\displaystyle\frac{ \rho_w g }{ \eta }$ |
it results in the relationship between the soil permeability ($k_s$) and the hydraulic conductivity ($K_s$) as
| $ k_s = \displaystyle\frac{ \eta }{ \rho_w g } K_s $ |
Typically, soil characterization measurements are performed using a specific liquid, resulting in a value of a hydraulic conductivity ($K_s$). With this value, you can calculate the soil permeability ($k_s$) using the equation mentioned above.
ID:(34, 0)
Flow density and pressure gradient
Equation
The flux density ($j_s$) can be expressed in terms of the height of the water column on the ground ($h$) or, based on the water column pressure ($p$) generated by the liquid column. Using the definition of the soil permeability ($k_s$), we obtain the following expression for the viscosity ($\eta$) and the position of the water column on the ground ($x$):
| $ j_s = -\displaystyle\frac{ k_s }{ \eta }\displaystyle\frac{ dp }{ dx }$ |
The pressure difference ($\Delta p$) in relation to the liquid density ($\rho_w$), the gravitational Acceleration ($g$), and the height difference ($\Delta h$) is calculated as per the following equation:
| $ \Delta p = \rho_w g \Delta h $ |
in the infinitesimal limit where the pressure difference ($\Delta p$) equals the pressure differential ($dp$), denoted as:
$\Delta p \rightarrow dp$
and where the height difference ($\Delta h$) equals the column height differential ($dh$), denoted as:
$\Delta h \rightarrow dh$
Using the relationship of the flux density ($j_s$) with the hydraulic conductivity ($K_s$), the column height differential ($dh$), and the distance differential ($dx$), which is expressed as:
| $$ |
and the relationship for the soil permeability ($k_s$) with the viscosity ($\eta$), which is expressed as:
| $ k_s = \displaystyle\frac{ \eta }{ \rho_w g } K_s $ |
We can derive the following equation:
| $ j_s = -\displaystyle\frac{ k_s }{ \eta }\displaystyle\frac{ dp }{ dx }$ |
ID:(45, 0)
Flow density and height gradient in more dimensions
Equation
Given that the one-dimensional equation for the flux density ($j_s$) is expressed as the hydraulic conductivity ($K_s$), the height of the water column on the ground ($h$), and the position of the water column on the ground ($x$) as follows:
| $ j_s = - K_s \displaystyle\frac{ dh }{ dx }$ |
It is possible to generalize this equation for the case of a homogeneous medium, resulting in an equation for the flow density in more than one dimension ($\vec{j}_s$) where the hydraulic conductivity ($K_s$) remains constant, as follows:
| $ \vec{j}_s = - K_s \nabla h $ |
ID:(999, 0)
Flow equation in more than one dimension
Equation
If we generalize the equation in one dimension for the height of the water column on the ground ($h$) as a function of the time ($t$) and the position of the water column on the ground ($x$) with the flux density ($j_s$):
| $\displaystyle\frac{\partial h}{\partial t} = - \displaystyle\frac{\partial}{\partial x}( h j_s )$ |
and replace the partial derivative with a divergence, we obtain with the flow density in more than one dimension ($\vec{j}_s$):
| $\displaystyle\frac{\partial h}{\partial t} = - \vec{\nabla} \cdot ( h \vec{j}_s )$ |
ID:(15111, 0)
Static solution in one dimension
Equation
We can study the stationary case, which implies that the height of the water column on the ground ($h$) divided by the flux density ($j_s$) must be constant and, in particular, can take values at a specific point denoted by the reference height of the water column ($h_0$) and the flow at a reference point ($j_{s0}$):
| $ h j_s = h_0 j_{s0} $ |
If, for the height of the water column on the ground ($h$) divided by the flux density ($j_s$), the equation
| $\displaystyle\frac{\partial h}{\partial t} = - \displaystyle\frac{\partial}{\partial x}( h j_s )$ |
in the stationary case reduces to
$\displaystyle\frac{d}{dx} (h j_s) = 0$
which corresponds to the product of the height of the water column on the ground ($h$) and the flux density ($j_s$) being constant. If you have values for a specific point defined by the reference height of the water column ($h_0$) and the flow at a reference point ($j_{s0}$), then you have:
| $ h j_s = h_0 j_{s0} $ |
Note: The differential equation is an ordinary differential equation because it depends solely on the position $x$ and no longer on time $t$.
ID:(15107, 0)
Static solution in more than one dimension
Equation
The equation for the height of the water column on the ground ($h$) with the time ($t$) and the flow density in more than one dimension ($\vec{j}_s$) is:
| $\displaystyle\frac{\partial h}{\partial t} = - \vec{\nabla} \cdot ( h \vec{j}_s )$ |
In the stationary case and using the equation for the flow density in more than one dimension ($\vec{j}_s$), when we expand the derivatives, we obtain:
| $ h \nabla^2 h + \nabla h \cdot \nabla h = 0 $ |
The equation for the height of the water column on the ground ($h$) with the time ($t$) and the flow density in more than one dimension ($\vec{j}_s$) is:
| $\displaystyle\frac{\partial h}{\partial t} = - \vec{\nabla} \cdot ( h \vec{j}_s )$ |
in relation to
| $ \vec{j}_s = - K_s \nabla h $ |
results after replacing and developing the derivative
$\displaystyle\frac{\partial h}{\partial t}=-\vec{\nabla}\cdot(h\vec{j}_s)= K_s\vec{\nabla}\cdot(h\nabla h)=K_s(\vec{\nabla} h\cdot\vec{\nabla} h + h \nabla^2 h)$
which, in the stationary case, reduces to
| $ h \nabla^2 h + \nabla h \cdot \nabla h = 0 $ |
ID:(4375, 0)