Ground flow formula (Water Overlay): Difference between revisions

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Underground flow is different from surface flow, since it has to account for the slowdown and porousness of the medium. In general, horizontal underground flow is calculated using formulas described in Harbaugh 2005<ref name="Harbaugh"/><ref name="Modflow"/>. However, when an [[Aquifer (Water_Overlay)|aquifer]] is present, the [[Underground flow formula (Water Overlay)#Aquifer formula|Aquifer formula]] is applied.
Ground flow is different from surface flow, since it has to account for the slowdown and porousness of the medium. In general, horizontal ground flow is calculated using formulas described in Harbaugh 2005<ref name="Harbaugh"/><ref name="Modflow"/>. However, when an [[Aquifer (Water_Overlay)|aquifer]] is present, the [[ground flow formula (Water Overlay)#Aquifer formula|Aquifer formula]] is applied.


It depends on the configuration of the [[Hydraulic conductivity with thickness (Water Overlay)|HYDRAULIC_CONDUCTIVITY_WITH_THICKNESS]] attribute value in the [[Water Overlay]] what Hydraulic Conductivity formula is used:  
It depends on the configuration of the [[Hydraulic conductivity with thickness (Water Overlay)|HYDRAULIC_CONDUCTIVITY_WITH_THICKNESS]] attribute value in the [[Water Overlay]] what Hydraulic Conductivity formula is used:  
* A value <math><= 0</math> means [[Underground flow formula (Water Overlay)#Hydraulic Conductivity without Thickness|Hydraulic Conductivity without Thickness]] formula will be used.
* A value <math><= 0</math> means [[Ground flow formula (Water Overlay)#Hydraulic Conductivity without Thickness|Hydraulic Conductivity without Thickness]] formula will be used.
* A value <math>> 0</math> means [[Underground flow formula (Water Overlay)#Hydraulic Conductivity with Thickness|Hydraulic Conductivity with Thickness]] formula will be used.
* A value <math>> 0</math> means [[Ground flow formula (Water Overlay)#Hydraulic Conductivity with Thickness|Hydraulic Conductivity with Thickness]] formula will be used.
===Hydraulic Conductivity without Thickness===
===Hydraulic Conductivity without Thickness===
[[File:Undergroundflow2.png|right|400px|thumb|Two adjacent cells, where underground water level of cell 1 is larger than cell 2.]]
[[File:Undergroundflow2.png|right|400px|thumb|Two adjacent cells, where ground water level of cell 1 is larger than cell 2.]]
The flow between the two cells is calculated as:
The flow between the two cells is calculated as:


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where:
where:
: <math>w_{n,t}</math> = The [[Groundwater level formula (Water Overlay)|underground water level]] of cell <math>n</math> at time <math>t</math>.
: <math>w_{n,t}</math> = The [[Groundwater level formula (Water Overlay)|ground water level]] of cell <math>n</math> at time <math>t</math>.
: <math>B_c</math> = the datum height of the surface of cell c, set by the [[Elevation (Water Overlay)|elevation]] or a [[Terrain elevation prequel (Water Overlay)|Terrain elevation prequel]].
: <math>B_c</math> = the datum height of the surface of cell c, set by the [[Elevation (Water Overlay)|elevation]] or a [[Terrain elevation prequel (Water Overlay)|Terrain elevation prequel]].
: <math>B_{im}</math> = the calculated datum height of the impenetrable layer.
: <math>B_{im}</math> = the calculated datum height of the impenetrable layer.
: <math>K_n</math> = The hydraulic conductivity of the cell, defined in [[Terrain hydraulic conductivity md (Water Overlay)|HYDRAULIC_CONDUCTIVITY_MD]] of the underground terrain.
: <math>K_n</math> = The hydraulic conductivity of the cell, defined in [[Terrain hydraulic conductivity md (Water Overlay)|HYDRAULIC_CONDUCTIVITY_MD]] of the ground terrain.
: <math>d_{b,c}</math> = The ground bottom distance of the cell c, defined by a [[Bottom distance prequel (Water Overlay)|Bottom distance prequel]] or a general [[Ground_bottom_distance_m_(Water_Overlay)|GROUND_BOTTOM_DISTANCE_M]] of the Water Overlay.  
: <math>d_{b,c}</math> = The ground bottom distance of the cell c, defined by a [[Bottom distance prequel (Water Overlay)|Bottom distance prequel]] or a general [[Ground_bottom_distance_m_(Water_Overlay)|GROUND_BOTTOM_DISTANCE_M]] of the Water Overlay.  
: <math>A_{c,t}</math> = Area of conductance at time <math>t</math>.
: <math>A_{c,t}</math> = Area of conductance at time <math>t</math>.
: <math>\Delta w</math> = Underground water level difference at time <math>t</math>.
: <math>\Delta w</math> = Ground water level difference at time <math>t</math>.
: <math>\Delta t</math> = Computational timestep.
: <math>\Delta t</math> = Computational timestep.
: <math>\Delta x</math> = Size of grid cell.
: <math>\Delta x</math> = Size of grid cell.
: <math>\bar{w_{t}}</math> = Averaged underground water level at time <math>t</math>, based on water levels in underground, [[Terrain water storage percentage (Water Overlay)|WATER_STORAGE_PERCENTAGE]] and potentially the [[Surface water level formula (Water Overlay)|surface water level]], when the underground is filled to the top.
: <math>\bar{w_{t}}</math> = Averaged ground water level at time <math>t</math>, based on water levels in ground, [[Terrain water storage percentage (Water Overlay)|WATER_STORAGE_PERCENTAGE]] and potentially the [[Surface water level formula (Water Overlay)|surface water level]], when the ground is filled to the top.
: <math>V_{K,t}</math> = The amount of water to be transported at time <math>t</math> between one cell and the other.
: <math>V_{K,t}</math> = The amount of water to be transported at time <math>t</math> between one cell and the other.
<br clear=both>
<br clear=both>
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where:
where:
: <math>w_{n,t}</math> = The [[Groundwater level formula (Water Overlay)|underground water level]] of cell <math>n</math> at time <math>t</math>.
: <math>w_{n,t}</math> = The [[Groundwater level formula (Water Overlay)|ground water level]] of cell <math>n</math> at time <math>t</math>.
: <math>KD_n</math> = The hydraulic conductivity of the cell, multiplied with the thickness of the layer, defined in [[Terrain hydraulic conductivity with thickness md (Water Overlay)|HYDRAULIC_CONDUCTIVITY_WITH_THICKNESS_MD]] of the underground terrain.
: <math>KD_n</math> = The hydraulic conductivity of the cell, multiplied with the thickness of the layer, defined in [[Terrain hydraulic conductivity with thickness md (Water Overlay)|HYDRAULIC_CONDUCTIVITY_WITH_THICKNESS_MD]] of the ground terrain.
: <math>\Delta w</math> = Underground water level difference at time <math>t</math>.
: <math>\Delta w</math> = Ground water level difference at time <math>t</math>.
: <math>\Delta t</math> = Computational timestep.
: <math>\Delta t</math> = Computational timestep.
: <math>\Delta x</math> = Size of grid cell.
: <math>\Delta x</math> = Size of grid cell.
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; Formulas
; Formulas
: [[Groundwater level formula (Water Overlay)|Groundwater level formula]]
: [[Groundwater level formula (Water Overlay)|Groundwater level formula]]
: [[Underground infiltration formula (Water Overlay)|Underground infiltration formula]]
: [[Ground infiltration formula (Water Overlay)|Ground infiltration formula]]
; Models
; Models
: [[Underground model (Water Overlay)|Underground model]]
: [[Ground model (Water Overlay)|Ground model]]
: [[Infiltration model (Water Overlay)|Infiltration model]]
: [[Infiltration model (Water Overlay)|Infiltration model]]
: [[Tracer flow model (Water Overlay)|Tracer flow model]]
: [[Tracer flow model (Water Overlay)|Tracer flow model]]
|seealso=
|seealso=
* [[Underground model (Water Overlay)#Horizontal flow and aquifers| Underground model]]
* [[Ground model (Water Overlay)#Horizontal flow and aquifers| Ground model]]
|references=
|references=
<references>
<references>

Revision as of 15:23, 26 January 2024

Ground flow is different from surface flow, since it has to account for the slowdown and porousness of the medium. In general, horizontal ground flow is calculated using formulas described in Harbaugh 2005[1][2]. However, when an aquifer is present, the Aquifer formula is applied.

It depends on the configuration of the HYDRAULIC_CONDUCTIVITY_WITH_THICKNESS attribute value in the Water Overlay what Hydraulic Conductivity formula is used:

  • A value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <= 0} means Hydraulic Conductivity without Thickness formula will be used.
  • A value Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle > 0} means Hydraulic Conductivity with Thickness formula will be used.

Hydraulic Conductivity without Thickness

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

The flow between the two cells is calculated as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta w_t = w_{1,t} - w_{2,t}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_{im} = max ( B_{1}- d_{b,1} , B_{2}- d_{b,2} ) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{c,t} = \Delta x \cdot ( \bar{w_{t}} - B_{im} )}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K = min(K_{1} , K_{2} )}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{K,t} = \frac{\Delta w \cdot K \cdot A_{c,t}}{ \Delta x} \cdot \Delta t}

where:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{n,t}} = The ground water level of cell Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} .
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_c} = the datum height of the surface of cell c, set by the elevation or a Terrain elevation prequel.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_{im}} = the calculated datum height of the impenetrable layer.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K_n} = The hydraulic conductivity of the cell, defined in HYDRAULIC_CONDUCTIVITY_MD of the ground terrain.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_{b,c}} = The ground bottom distance of the cell c, defined by a Bottom distance prequel or a general GROUND_BOTTOM_DISTANCE_M of the Water Overlay.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_{c,t}} = Area of conductance at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} .
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta w} = Ground water level difference at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} .
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta t} = Computational timestep.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x} = Size of grid cell.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{w_{t}}} = Averaged ground water level at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} , based on water levels in ground, WATER_STORAGE_PERCENTAGE and potentially the surface water level, when the ground is filled to the top.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{K,t}} = The amount of water to be transported at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} between one cell and the other.


Hydraulic Conductivity with Thickness

The flow between the two cells is calculated as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta w_t = w_{1,t} - w_{2,t}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle KD = min(KD_{1} , KD_{2} )}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{KD,t} = \frac{\Delta w \cdot KD \cdot \Delta x }{ \Delta x} \cdot \Delta t}

where:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{n,t}} = The ground water level of cell Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} .
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle KD_n} = The hydraulic conductivity of the cell, multiplied with the thickness of the layer, defined in HYDRAULIC_CONDUCTIVITY_WITH_THICKNESS_MD of the ground terrain.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta w} = Ground water level difference at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} .
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta t} = Computational timestep.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta x} = Size of grid cell.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{KD,t}} = The amount of water to be transported at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} between one cell and the other.

Aquifer formula

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

Based on conditions being true, the calculated volume of water that is transported through the aquifer is calculated as:

Where:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w_{n,t}} = Ground water level in cell Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} ;
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_a} = the datum height of the aquifer at the cell.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{a,t}} = Volume in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^{3}} that flows between the two adjacent cells due to the aquifer at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} .
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta w_{t}} = Ground water level difference between the two adjacent cells at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} ;
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta t} = Computational timestep.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {KD}_a} = The AQUIFER_KD attribute of aquifer.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{K(D),t}} = The calculated amount of water to be transported at time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} between one cell and the other.

Related

The following topics are related to this formula.

Features
Aquifer
Formulas
Groundwater level formula
Ground infiltration formula
Models
Ground model
Infiltration model
Tracer flow model

See also

References

  1. ↑ Harbaugh, A.W., 2005, MODFLOW-2005, the U.S. Geological Survey modular ground-water model-the Ground-Water Flow Process: U.S. Geological Survey Techniques and Methods 6-A16, variously paginated.
  2. ↑ Langevin, C.D., Hughes, J.D., Banta, E.R., Niswonger, R.G., Panday, Sorab, and Provost, A.M. (2017) ∙ Documentation for the MODFLOW 6 Groundwater Flow Model: U.S. Geological Survey Techniques and Methods, book 6, chap. A55 ∙ p 31 ∙ found at: https://doi.org/10.3133/tm6A55 (last visited 2019-02-04)
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