Drainage formula (Water Overlay): Difference between revisions

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Depending on whether the [[Drainage (Water Overlay)|drainage]] is [[Drainage active (Water Overlay)|passive or active]], the formula for either passive or active drainage is used.
==Drainage passive==
==Drainage passive==
First the flow capacity is calculated
First the flow capacity is calculated


<math>Q_{p,t}=\Delta t \cdot q_t</math>
<math>Q_p=\Delta t \cdot q_t</math>


<math>Q_{b} = A_d \cdot f_s \cdot  \frac{(w_g - w_w) \cdot A_w}{A_d \cdot f_s + A_w}</math>
<math>Q_b = A_d \cdot f_s \cdot  \frac{||w_d - w_w|| \cdot A_w}{A_d \cdot f_s + A_w}</math>


if <math>w_w > w_g</math> then:
if <math>w_d > w_w</math> then:


<math>Q_{max} = A_w \cdot (w_w - z_w) </math>
<math>Q_d = A_d \cdot (w_d - z_d) \cdot f_s</math>
 
<math>Q_t = min ( Q_p, Q_b , Q_d )</math>


else:
else:


<math>Q_{max} = A_d \cdot (w_d - z_d) \cdot f_s</math>
<math>Q_w = A_w \cdot (w_w - z_w) </math>
 
<math>Q_{d} = min ( Q_{p,t}, Q_{b} , Q_{max} )</math>


<math>Q_t = min ( Q_p, Q_b , Q_w )</math>


Where:
Where:
: <math>w_g</math> is the ground water level in meters
: <math>q_t</math> is the [[Drainage q (Water Overlay)|DRAINAGE_Q]] of the {{Drainage}} at time t.
: <math>w_w</math> is the water level in the waterway
: <math>\Delta t</math> is the computational [[Timestep formula (Water Overlay)|timestep]].
: <math>w_d</math> is the ground water level, above the drainage, in meters.
: <math>w_w</math> is the water level in the waterway.
: <math>z_d</math> is the drainage datum height in meters.
: <math>z_d</math> is the drainage datum height in meters.
: <math>z_w</math> is the max waterway height.
: <math>A_d</math> is the drainage area size in square meters.
: <math>A_d</math> is the drainage area size in square meters.
: <math>A_w</math> is the waterway area size in square meters.
: <math>A_w</math> is the waterway area size in square meters.
: <math>z_w</math> is the max waterway height.
: <math>f_s</math> is the average storage percentage of the ground above the drainage.
: <math>f_s</math> is the average storage percentage of the ground above the drainage.
: <math>Q_{b}</math> is the amount transferred for a balanced ground and surface water level
: <math>Q_p</math> is the possible amount of water drained, or pumped back when negative, based on the [[Drainage q (Water Overlay)|DRAINAGE_Q]] of the {{Drainage}} and timestep <math>\Delta t</math>.
: <math>Q_{max}</math> is the amount available in the ground above the drainage.
: <math>Q_b</math> is the amount transferred for a balanced ground and surface water level.
: <math>Q_{p_t}</math> is the drainage capacity at time t
: <math>Q_d</math> is the amount of water available in the ground above the drainage.
: <math>Q_{d}</math> is the actual drained volume at time t
: <math>Q_w</math> is the amount of water available in the waterway.
: <math>Q_t</math> is the actual drained, or infiltrated backwards, volume at time t.


==Drainage active==
==Drainage active==
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''Case 1: '''Active Draining''':''
''Case 1: '''Active Draining''':''


If a positive Pump q is defined:
If a positive [[Drainage q (Water Overlay)|DRAINAGE_Q]] is defined:
: <math>Q_{p,t} = \Delta t \cdot q_t </math>
: <math>Q_p = \Delta t \cdot q_t </math>
: <math>Q_{g,t} = (w_{g,t} - z_d)\cdot f_s</math>
: <math>Q_d = (w_d - z_d)\cdot f_s</math>


If an overflow threshold <math>T_{o,t}</math> is defined as well:
If an overflow threshold <math>T_{o,t}</math> is defined as well:
: <math>Q_{o,t} =  max ( 0, T_{o,t} - w_{t,w} )</math>
: <math>Q_o =  max ( 0, T_{o,t} - w_w )</math>


The actual water pumped out of the drainage system is calculated. If any of the terms are undefined, they are not included.
The actual water pumped out of the drainage system is calculated. If any of the terms are undefined, they are not included.
: <math>Q_t = max( 0 , min( Q_{g,t} , Q_{o,t} , Q_{p,t}))</math>
: <math>Q_t = max( 0 , min( Q_{d} , Q_{o} , Q_{p}))</math>


''Case 2: '''Active Reverse Draining''':''<br>
''Case 2: '''Active Reverse Draining''':''<br>


If a negative Pump q is defined:
If a negative [[Drainage q (Water Overlay)|DRAINAGE_Q]] is defined:
: <math>Q_{p,t} = \Delta t \cdot q_t </math>
: <math>Q_p = \Delta t \cdot q_t </math>
: <math>Q_{w,t} =  min ( 0, z_w - w_{t,w}  )</math>
: <math>Q_w =  min ( 0, z_w - w_{t,w}  )</math>


If an overflow threshold  <math>T_{o,t}</math> is defined as well:
If an overflow threshold  <math>T_{o,t}</math> is defined as well:
: <math>Q_{o,t} =  min ( 0, T_{o,t} - w_{t,w}  )</math>
: <math>Q_o =  min ( 0, T_{o,t} - w_{t,w}  )</math>


The actual water pumped into the drainage system is calculated. If any of the terms are undefined, they are not included.
The actual water pumped into the drainage system is calculated. If any of the terms are undefined, they are not included.
: <math>Q_t =  max( 0 , max( Q_{p,t} , Q_{w,t} , Q_{o,t}) )  </math>
: <math>Q_t =  max( 0 , max( Q_p, Q_w , Q_o) )  </math>




Where:
Where:
: <math>w_g</math> is the ground water level in meters
: <math>q_t</math> is the [[Drainage q (Water Overlay)|DRAINAGE_Q]] of the {{Drainage}} at time t.
: <math>w_w</math> is the water level in the waterway
: <math>\Delta t</math> is the computational [[Timestep formula (Water Overlay)|timestep]].
: <math>Q_p</math> is the possible amount of water drained, or pumped back when negative, based on the [[Drainage q (Water Overlay)|DRAINAGE_Q]] of the {{Drainage}} and timestep <math<\Delta t</math>.
: <math>w_d</math> is the ground water level, above the drainage, in meters.
: <math>w_w</math> is the water level in the waterway.
: <math>z_d</math> is the drainage datum height in meters.
: <math>z_d</math> is the drainage datum height in meters.
: <math>A_d</math> is the drainage area size in square meters.
: <math>z_w</math> is the waterway's datum height.
: <math>A_w</math> is the waterway area size in square meters.
: <math>z_w</math> is the max waterway height.
: <math>f_s</math> is the average storage percentage of the ground above the drainage.
: <math>f_s</math> is the average storage percentage of the ground above the drainage.
: <math>Q_{b}</math> is the amount transferred for a balanced ground and surface water level
: <math>Q_d</math> is the amount of water available in the ground above the drainage.
: <math>Q_{max}</math> is the amount available in the ground above the drainage.
: <math>T_{o,t}</math> is the overflow threshold in meters at time t.
: <math>Q_{p_t}</math> is the drainage capacity at time t
: <math>Q_o</math> is the possible amount of water at time t, that can be added to the waterway until the overflow threshold is reached.
: <math>Q_{d}</math> is the actual drained volume at time t
: <math>Q_w</math> is the amount of water available in the waterway.
: <math>Q_{t}</math> is the actual drained (or pumped) volume at time t.


==Notes==
{{article end
A negative pump Q can potentially raise the ground water level at the drainage that it reaches the surface above the drainage.
|notes=
 
* A negative [[Drainage q (Water Overlay)|DRAINAGE_Q]] can potentially raise the ground water level at the drainage so much that it reaches the surface above the drainage.
==Related==
|related=
The following topics are related to this formula.
The following topics are related to this formula.
; Structures
; Structures
: [[Drainage (Water Overlay)|Drainage]]
: [[Drainage (Water Overlay)]]
; Models
; Models
: [[Surface model (Water Overlay)|Surface model]]
: [[Surface model (Water Overlay)]]
: [[Underground model (Water Overlay)|Underground model]]
: [[Ground model (Water Overlay)]]
 
}}
{{WaterOverlay formula nav}}
{{WaterOverlay formula nav}}

Latest revision as of 13:08, 5 March 2024

Depending on whether the drainage is passive or active, the formula for either passive or active drainage is used.

Drainage passive

First the flow capacity is calculated

if then:

else:

Where:

is the DRAINAGE_Q of the Drainage at time t.
is the computational timestep.
is the ground water level, above the drainage, in meters.
is the water level in the waterway.
is the drainage datum height in meters.
is the max waterway height.
is the drainage area size in square meters.
is the waterway area size in square meters.
is the average storage percentage of the ground above the drainage.
is the possible amount of water drained, or pumped back when negative, based on the DRAINAGE_Q of the Drainage and timestep .
is the amount transferred for a balanced ground and surface water level.
is the amount of water available in the ground above the drainage.
is the amount of water available in the waterway.
is the actual drained, or infiltrated backwards, volume at time t.

Drainage active

Case 1: Active Draining:

If a positive DRAINAGE_Q is defined:

If an overflow threshold is defined as well:

The actual water pumped out of the drainage system is calculated. If any of the terms are undefined, they are not included.

Case 2: Active Reverse Draining:

If a negative DRAINAGE_Q is defined:

If an overflow threshold is defined as well:

The actual water pumped into the drainage system is calculated. If any of the terms are undefined, they are not included.


Where:

is the DRAINAGE_Q of the Drainage at time t.
is the computational timestep.
is the possible amount of water drained, or pumped back when negative, based on the DRAINAGE_Q of the Drainage and timestep <math<\Delta t</math>.
is the ground water level, above the drainage, in meters.
is the water level in the waterway.
is the drainage datum height in meters.
is the waterway's datum height.
is the average storage percentage of the ground above the drainage.
is the amount of water available in the ground above the drainage.
is the overflow threshold in meters at time t.
is the possible amount of water at time t, that can be added to the waterway until the overflow threshold is reached.
is the amount of water available in the waterway.
is the actual drained (or pumped) volume at time t.

Notes

  • A negative DRAINAGE_Q can potentially raise the ground water level at the drainage so much that it reaches the surface above the drainage.

Related

The following topics are related to this formula.

Structures
Drainage (Water Overlay)
Models
Surface model (Water Overlay)
Ground model (Water Overlay)