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Flow into a channel
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If there is a water table at a given depth and a trench in the form of a channel, water will begin to flow into it. As the water flows at a rate equal to the inflow within the water table that replenishes it, the channel will reach a depth that depends on this flow.
ID:(2080, 0)
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.
ID:(15104, 0)
Flow height solution towards a channel
Concept
The solution to the one-dimensional flow equation towards a channel, where the height of the water column on the ground ($h$) is calculated as a function of the reference height of the water column ($h_0$) and the position of the water column on the ground ($x$) at the channel's edge, along with the characteristic length of the flow in the ground ($s_0$), takes the following form:
| $ \displaystyle\frac{ h }{ h_0 } = \sqrt{1 + \displaystyle\frac{ 2 x }{ s_0 }} $ |
This solution is graphically represented in terms of the additional factors $h/h_0$ and $x/s_0$ as follows:
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The profile reveals that, away from the channel, the height of the water column is significantly high. However, due to water extraction by the channel, this height begins to decrease until it reaches the channel's edge. Dynamically, the flux density ($j_s$) determines the amount of water flowing into the channel, while the reference height of the water column ($h_0$) gradually adjusts until it reaches an equilibrium state. In other words, if the value of the reference height of the water column ($h_0$) is too low relative to the total amount of water arriving at the channel, it increases; and if it is too high, it decreases. In this way, the reference height of the water column ($h_0$) acquires the value that balances the amount of incoming water with the amount of water flowing through the channel.
ID:(15109, 0)
Flux density solution towards a channel
Concept
The solution obtained for the height and the parameters the flow at a reference point ($j_{s0}$) and the reference height of the water column ($h_0$) reveals that the flux density ($j_s$) is given by:
| $ \displaystyle\frac{ j_s }{ j_{s0} } = \displaystyle\frac{1}{\sqrt{1 + \displaystyle\frac{ 2 x }{ s_0 }}} $ |
We can graphically represent the flux density ($j_s$) as a function of the additional factors $j_s/j_{s0}$ and $x/s_0$ as follows:
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It is noticeable that the flux density ($j_s$) continues to increase as we approach the channel, as the height of the water column on the ground ($h$) decreases. This increase is necessary to maintain the flow velocity in the flux density ($j_s$) or, alternatively, to increase it.
ID:(15110, 0)
Model
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Equations
$ h \displaystyle\frac{ dh }{ dx } = \displaystyle\frac{ h_0 ^2 }{ s_0 } $
h * @DIFF( h , x, 1) = h_0 ^2/ s_0
$ h j_s = h_0 j_{s0} $
h * j_s = h_0 * j_{s0}
$ \displaystyle\frac{ h }{ h_0 } = \sqrt{1 + \displaystyle\frac{ 2 x }{ s_0 }} $
h / h_0 = sqrt(1 + 2* x / s_0 )
$ \displaystyle\frac{ j_s }{ j_{s0} } = \displaystyle\frac{1}{\sqrt{1 + \displaystyle\frac{ 2 x }{ s_0 }}} $
j / j_s0 = 1/sqrt(1 + 2* x / s_0 )
$ s_0 \equiv \displaystyle\frac{| j_{s0} |}{ K_s h_0 }$
s_0 = abs( j_s0 )/( K_s * h_0 )
ID:(15819, 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)
Characteristic length of the flow in the ground
Equation
With the hydraulic conductivity ($K_s$), the flow at a reference point ($j_{s0}$), and the reference height of the water column ($h_0$), a characteristic length of the flow in the ground ($s_0$) can be defined as follows:
| $ s_0 \equiv \displaystyle\frac{| j_{s0} |}{ K_s h_0 }$ |
To avoid complicating the analysis, we have defined the expression while considering the absolute value of the flow at a reference point ($j_{s0}$), thus preventing situations where it might be negative. This means that, depending on the sign of the flow at a reference point ($j_{s0}$), we must express the relationship assuming a positive or negative value for the derivative, thereby defining the flow direction.
ID:(4747, 0)
Equation of flow into a channel
Equation
The differential equation for calculating the height of the water column on the ground ($h$) as a function of the position of the water column on the ground ($x$), the reference height of the water column ($h_0$), and the characteristic length of the flow in the ground ($s_0$) is:
| $ h \displaystyle\frac{ dh }{ dx } = \displaystyle\frac{ h_0 ^2 }{ s_0 } $ |
The equation for the product of the height of the water column on the ground ($h$) and the flux density ($j_s$) as a function of the reference height of the water column ($h_0$) and the flow at a reference point ($j_{s0}$) is:
| $ h j_s = h_0 j_{s0} $ |
And with the equation that describes the flux density ($j_s$) in terms of the hydraulic conductivity ($K_s$) and the position of the water column on the ground ($x$):
| $ j_s = - K_s \displaystyle\frac{ dh }{ dx }$ |
And with the expression for the characteristic length of the flow in the ground ($s_0$):
| $ s_0 \equiv \displaystyle\frac{| j_{s0} |}{ K_s h_0 }$ |
We can derive the resulting equation as follows:
| $ h \displaystyle\frac{ dh }{ dx } = \displaystyle\frac{ h_0 ^2 }{ s_0 } $ |
ID:(15108, 0)
Height of flow into a channel
Equation
The equation for the height of the water column on the ground ($h$) as a function of the characteristic length of the flow in the ground ($s_0$) and dependent on the reference height of the water column ($h_0$) and the characteristic length of the flow in the ground ($s_0$) is as follows:
| $ \displaystyle\frac{ h }{ h_0 } = \sqrt{1 + \displaystyle\frac{ 2 x }{ s_0 }} $ |
The equation for the height of the water column on the ground ($h$) as a function of the characteristic length of the flow in the ground ($s_0$) and dependent on the reference height of the water column ($h_0$) and the characteristic length of the flow in the ground ($s_0$) is as follows:
| $ h \displaystyle\frac{ dh }{ dx } = \displaystyle\frac{ h_0 ^2 }{ s_0 } $ |
We can rearrange it to facilitate integration as follows:
$h dh = \displaystyle\frac{h_0^2}{x_0}dx$
Then, integrating with respect to $h_0$, the height at the origin, we get:
$\displaystyle\frac{1}{2}(h^2 - h_0^2) =\displaystyle\frac{h_0^2}{s_0}x$
This leads us to the following expression:
| $ \displaystyle\frac{ h }{ h_0 } = \sqrt{1 + \displaystyle\frac{ 2 x }{ s_0 }} $ |
ID:(15105, 0)
Flow density into a channel
Equation
The solution for the flux density ($j_s$) and the flow at a reference point ($j_{s0}$) given the position of the water column on the ground ($x$) and the characteristic length of the flow in the ground ($s_0$), we obtain:
| $ \displaystyle\frac{ j_s }{ j_{s0} } = \displaystyle\frac{1}{\sqrt{1 + \displaystyle\frac{ 2 x }{ s_0 }}} $ |
With the solution for the flux density ($j_s$) and the flow at a reference point ($j_{s0}$) given the position of the water column on the ground ($x$) and the characteristic length of the flow in the ground ($s_0$), we have:
| $ \displaystyle\frac{ h }{ h_0 } = \sqrt{1 + \displaystyle\frac{ 2 x }{ s_0 }} $ |
We can calculate the flux density ($j_s$) with the hydraulic conductivity ($K_s$) using:
| $ j_s = - K_s \displaystyle\frac{ dh }{ dx }$ |
and employing
| $ s_0 \equiv \displaystyle\frac{| j_{s0} |}{ K_s h_0 }$ |
this way, we obtain:
| $ \displaystyle\frac{ j_s }{ j_{s0} } = \displaystyle\frac{1}{\sqrt{1 + \displaystyle\frac{ 2 x }{ s_0 }}} $ |
ID:(15106, 0)