Ground flow formula (Water Overlay): Difference between revisions

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Underground flow is calculated differently from surface flow, to account for the slowdown and porousness of the medium.
Underground flow is calculated differently from surface flow, to account for the slowdown and porousness of the medium.


In general, Darcy's law is used. When an [[Hydraulic features (Water Overlay)#Aquifer|aquifer]] is present a variant is applied.  
In general, The underground flow is implemented using formulas described in Harbaugh 2005<ref name="Harbaugh"/>. However, when an [[Hydraulic features (Water Overlay)#Aquifer|aquifer]] is present, the aquifer variant is applied.


=====Darcy's law=====
==Default horizontal flow==
Underground flow between cells is calculated using Darcy's law<ref name="Modflow"/>.
[[File:Undergroundflow.png|right|400px|thumb|Two adjacent cells, where underground water level of cell 1 is larger than cell 2.]]
The flow between the two cells is calculated as:
: Δw = w<sub>1</sub> - w<sub>2</sub>
: B<sub>ex</sub> = max ( B<sub>1</sub> , B<sub>2</sub> ) - D<sub>ground</sub>
: A<sub>c</sub> = Δx * ( w<sub>avg</sub> - B<sub>ex</sub> )
: C = min(C<sub>1</sub> , C<sub>2</sub> )
: q = Δw * C * A<sub>c</sub> / Δx * Δt


: ''C<sub>d</sub> = Δt * I<sub>und</sub> * width * A * ( (WL<sub>source</sub> - WL<sub>target</sub>) / distance )
where:
: w<sub>n</sub> = The underground water level of cell n.
: B<sub>n</sub> = The surface elevation of cell n.
: C<sub>n</sub> = The ground conductance of cell n. For this we use the vertical infiltration speed.
: D<sub>ground</sub> = The configured [[Ground_bottom_distance_m_(Water_Overlay)|ground bottom distance]].
: A<sub>c</sub> = Area of conductance.
: Δw = Underground water level difference.
: Δt = Computational timestep.
: Δx = Size of grid cell.
: w<sub>avg</sub> = Averaged underground water level, based on water levels in underground, underground storage fraction and potentially the surface water level, when the underground is filled to the top.
<br style="clear:both;">


Since both ''width'' and ''distance'' are directly related to the cell size, and the result should be in water height rather than volume, the formula can be rewritten as follows:
: ''C<sub>d</sub> = Δt * I<sub>und</sub> * A * (W<sub>source</sub> - W<sub>target</sub>) / cell
Because the underground may have a different porousness, the maximum amount of water that can flow to another cell has to take into account the relative water storage capacities of the underground.
: ''C<sub>wsp</sub> = (WL<sub>source</sub> - WL<sub>target</sub>) * (WSP<sub>source</sub> / (WSP<sub>source</sub> + WSP<sub>target</sub>) )
The amount of water which flows from the source to the target cell is calculated as follows:
: ''Δw = max( 0 , min( C<sub>d</sub> , C<sub>wsp</sub> ) )
Where:
* Δw = The underground flow which takes place.
* Δt = Computational timestep.
* cell = Cell size.
* C<sub>d</sub> = The capacity for water flow possible based on the relative water heights.
* C<sub>wsp</sub> = The capacity for water flow possible based on the relative water storage percentages.
* WL<sub>source</sub> = The amount of water in the saturated zone of the source cell. The height of the water column if the equivalent amount of water was placed on the surface.
* WL<sub>target</sub> = The amount of water in the saturated zone of the target cell. The height of the water column if the equivalent amount of water was placed on the surface.
* A = Contact area of the underground cells
* I<sub>und</sub> = The [[Ground infiltration md (Terrain) (Water Overlay)|GROUND_INFILTRATION_MD]] of the underground terrain type of the origin cell.
* WSP<sub>source</sub> = The [[Water storage percentage (Terrain) (Water Overlay)|WATER_STORAGE_PERCENTAGE]] of the underground terrain type of the origin cell.
* WSP<sub>target</sub> = The WATER_STORAGE_PERCENTAGE of the underground terrain type of the target cell.


=====Aquifer formula=====
=====Aquifer formula=====

Revision as of 14:28, 25 April 2019

Underground flow is calculated differently from surface flow, to account for the slowdown and porousness of the medium.

In general, The underground flow is implemented using formulas described in Harbaugh 2005[1]. However, when an aquifer is present, the aquifer variant is applied.

Default horizontal flow

Two adjacent cells, where underground water level of cell 1 is larger than cell 2.

The flow between the two cells is calculated as:

Δw = w1 - w2
Bex = max ( B1 , B2 ) - Dground
Ac = Δx * ( wavg - Bex )
C = min(C1 , C2 )
q = Δw * C * Ac / Δx * Δt

where:

wn = The underground water level of cell n.
Bn = The surface elevation of cell n.
Cn = The ground conductance of cell n. For this we use the vertical infiltration speed.
Dground = The configured ground bottom distance.
Ac = Area of conductance.
Δw = Underground water level difference.
Δt = Computational timestep.
Δx = Size of grid cell.
wavg = Averaged underground water level, based on water levels in underground, underground storage fraction and potentially the surface water level, when the underground is filled to the top.



Aquifer formula

When an aquifer is present, its hydraulic diffusivity is used to calculate the water flow.

First, the hydraulic diffusivity dictates the fraction of the water height difference which will flow.

F = 2 * sqrt( KD / WSPsource ) * sqrt( Δt ) * ( 1 / cell )

Based on this fraction, the actual amount of water flow is calculated.

Δw = ( (WLsource/WSPsource) - (WLtarget/WSPsource) ) * F

Where:

  • Δw = The underground flow which takes place.
  • Δt = Computational timestep.
  • cell = Cell size.
  • F = Fraction of water which flows between cells
  • KD = The AQUIFER_KD attribute of aquifer.
  • WSPsource = The WATER_STORAGE_PERCENTAGE attribute of the underground terrain type of the origin cell.
  • WSPtarget = The WATER_STORAGE_PERCENTAGE attribute of the underground terrain type of the target cell.
  • WLsource = The amount of water in the saturated zone of the source cell. The height of the water column if the equivalent amount of water was placed on the surface.
  • WLtarget = The amount of water in the saturated zone of the target cell. The height of the water column if the equivalent amount of water was placed on the surface.

References

  1. ↑ Cite error: Invalid <ref> tag; no text was provided for refs named Harbaugh

Cite error: <ref> tag with name "Modflow" defined in <references> is not used in prior text. Template:WaterOverlay nav

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