User:
Flow into a well
Storyboard 
If there is a water table at a given depth and a trench deeper than the water table, water will start to flow into the trench. As water is extracted at a rate equal to the inflow from the water table replenishing it, a water mirror will form at a depth that depends on this flow.
ID:(2082, 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)
Flow height solution towards a well
Concept
The solution to the one-dimensional flow equation towards a well, in which the value of the height of the water column on the ground ($h$) is calculated as a function of the radius from center of well ($r$), the reference height of the water column ($h_0$), and the water well radius ($r_0$) at the well'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 r_0 }{ s_0 }\ln\left(\displaystyle\frac{ r }{ r_0 }\right)} $ |
This solution is graphically represented in terms of the additional factors $h/h_0$ and $r/r_0$ for various $r_0/s_0$ as follows:
The profile reveals that, far from the well, the height of the water column is significantly high. However, due to water extraction by the well, this height begins to decrease until it reaches the well's edge. Dynamically, the flux density ($j_s$) determines the amount of water flowing towards the well, while the reference height of the water column ($h_0$) gradually adjusts itself to reach 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 reaching the well, 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 water arriving with the amount of water being extracted through the well.
ID:(10591, 0)
Flux density solution towards a well
Concept
The solution obtained for the height and parameters the flow at a reference point ($j_{s0}$) and the radius from center of well ($r$), the water well radius ($r_0$), the characteristic length of the flow in the ground ($s_0$) shows us that the flux density ($j_s$) is equal to:
| $ \displaystyle\frac{ j_s }{ j_{s0} } = \displaystyle\frac{1}{\displaystyle\frac{ r }{ r_0 }\sqrt{1 + \displaystyle\frac{ 2 r_0 }{ s_0 }\ln\left(\displaystyle\frac{r}{r_0}\right)}}$ |
This solution is graphically represented in terms of the additional factors $j_s/j_{s0}$ and $r/r_0$ for various values of $r_0/s_0$ as follows:
None
the flux density ($j_s$) continues to increase as we approach the channel, while 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:(2209, 0)
Model
Top
Parameters
Variables
Calculations
to 
, then, select the variable: 
to 
Calculations
Calculations
Variable 
Given Calculate 
Target : Equation To be used 
Equations
$ h j_s = h_0 j_{s0} $
h * j_s = h_0 * j_{s0}
$ \displaystyle\frac{ h }{ h_0 } = \sqrt{1 + \displaystyle\frac{ 2 r_0 }{ s_0 }\ln\left(\displaystyle\frac{ r }{ r_0 }\right)} $
h / h_0 = sqrt(1 + 2* r_0 *log( r / r_0 )/ s_0 )
$ \displaystyle\frac{ j_s }{ j_{s0} } = \displaystyle\frac{1}{\displaystyle\frac{ r }{ r_0 }\sqrt{1 + \displaystyle\frac{ 2 r_0 }{ s_0 }\ln\left(\displaystyle\frac{r}{r_0}\right)}}$
j = j_s0 /(( r / r_0 )*sqrt(1 + 2* r_0 *log( r / r_0 )/ s_0 ))
$ r \displaystyle\frac{ dh^2 }{ dr } = 2 h_0 ^2\displaystyle\frac{ r_0 }{ s_0 } $
r * @DIFF( h ^2, r, 1) = h_0 ^2* r_0 / s_0
ID:(15821, 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)
Equation of flow into a well
Equation
In the case of the well, we can work with a polar coordinate system and assume angular symmetry, which means that the height of the water column on the ground ($h$) depends only on the radius from center of well ($r$) and satisfies the equation
| $ r \displaystyle\frac{ dh^2 }{ dr } = 2 h_0 ^2\displaystyle\frac{ r_0 }{ s_0 } $ |
As the height of the water column on the ground ($h$) satisfies
| $ h \nabla^2 h + \nabla h \cdot \nabla h = 0 $ |
and in polar coordinates with the radius from center of well ($r$) for the case of angular symmetry, we have
$\vec{\nabla}h = \displaystyle\frac{du}{dr}\hat{r}$
and
$\nabla ^2h = \displaystyle\frac{1}{r}\displaystyle\frac{d}{dr}\left(r\displaystyle\frac{dh}{dr}\right)$
we obtain
$\displaystyle\frac{1}{r}\displaystyle\frac{d}{dr}\left(r\displaystyle\frac{dh^2}{dr}\right)=0$
or
$r\displaystyle\frac{dh^2}{dr}=C$
with $C$ as a constant. On the other hand, the equation with the hydraulic conductivity ($K_s$) and the flow density in more than one dimension ($\vec{j}_s$)
| $ \vec{j}_s = - K_s \nabla h $ |
in polar coordinates with rotational symmetry reduces to
$j_s = - K_s \displaystyle\frac{dh}{dr}$
which at the well's surface with the water well radius ($r_0$), the characteristic length of the flow in the ground ($s_0$), the flow at a reference point ($j_{s0}$), and the reference height of the water column ($h_0$) leads to the conclusion that with
| $ s_0 \equiv \displaystyle\frac{| j_{s0} |}{ K_s h_0 }$ |
we have
$C=r_0\displaystyle\frac{dh^2}{dr}=r_02h_0\displaystyle\frac{dh}{dr}=2r_0h_0\displaystyle\frac{|j_{s0}|}{K_sh_0}=2h_0^2\displaystyle\frac{r_0}{s_0}$
resulting in
| $ r \displaystyle\frac{ dh^2 }{ dr } = 2 h_0 ^2\displaystyle\frac{ r_0 }{ s_0 } $ |
ID:(4430, 0)
Height of flow into a well
Equation
In the case of 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:
| $ \displaystyle\frac{ h }{ h_0 } = \sqrt{1 + \displaystyle\frac{ 2 r_0 }{ s_0 }\ln\left(\displaystyle\frac{ r }{ r_0 }\right)} $ |
The equation for 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 as follows:
| $ r \displaystyle\frac{ dh^2 }{ dr } = 2 h_0 ^2\displaystyle\frac{ r_0 }{ s_0 } $ |
This equation can be rearranged to facilitate integration as follows:
$dh^2 = 2h_0^2\displaystyle\frac{r_0}{s_0}\displaystyle\frac{dr}{r}$
Next, by integrating both sides, we obtain the height at the well's wall with the reference height of the water column ($h_0$) and the water well radius ($r_0$):
$h^2 - h_0^2 = 2h_0^2\displaystyle\frac{r_0}{s_0}\ln\left(\displaystyle\frac{r}{r_0}\right)$
Finally, by rearranging the height of the water column on the ground ($h$), we get:
| $ \displaystyle\frac{ h }{ h_0 } = \sqrt{1 + \displaystyle\frac{ 2 r_0 }{ s_0 }\ln\left(\displaystyle\frac{ r }{ r_0 }\right)} $ |
ID:(10593, 0)
Flow density into a well
Equation
The solution for the flux density ($j_s$) and the flow at a reference point ($j_{s0}$) given the radius from center of well ($r$), the water well radius ($r_0$), and the characteristic length of the flow in the ground ($s_0$), we obtain:
| $ \displaystyle\frac{ j_s }{ j_{s0} } = \displaystyle\frac{1}{\displaystyle\frac{ r }{ r_0 }\sqrt{1 + \displaystyle\frac{ 2 r_0 }{ s_0 }\ln\left(\displaystyle\frac{r}{r_0}\right)}}$ |
With the solution for the height of the water column on the ground ($h$) and the reference height of the water column ($h_0$) given the radius from center of well ($r$), the water well radius ($r_0$), and the characteristic length of the flow in the ground ($s_0$), we obtain:
| $ \displaystyle\frac{ h }{ h_0 } = \sqrt{1 + \displaystyle\frac{ 2 r_0 }{ s_0 }\ln\left(\displaystyle\frac{ r }{ r_0 }\right)} $ |
We can calculate the flow density in more than one dimension ($\vec{j}_s$) from the hydraulic conductivity ($K_s$) as follows:
| $ \vec{j}_s = - K_s \nabla h $ |
And with the flux density ($j_s$) and the flow at a reference point ($j_{s0}$) using
| $ s_0 \equiv \displaystyle\frac{| j_{s0} |}{ K_s h_0 }$ |
in this way, we obtain:
| $ \displaystyle\frac{ j_s }{ j_{s0} } = \displaystyle\frac{1}{\displaystyle\frac{ r }{ r_0 }\sqrt{1 + \displaystyle\frac{ 2 r_0 }{ s_0 }\ln\left(\displaystyle\frac{r}{r_0}\right)}}$ |
ID:(4368, 0)