Ground flow formula (Water Overlay): Difference between revisions

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<math>\Delta w_t = w_{1,t} - w_{2,t}</math>
<math>\Delta w_t = w_{1,t} - w_{2,t}</math>


<math>B_{im} = max ( B_{1}- d_{b,1} , B_{2}- d_{b,2} ) </math>
<math>B_{bb} = max ( B_{1}- d_{b,1} , B_{2}- d_{b,2} ) </math>


<math>A_{c,t} = \Delta x \cdot ( \bar{w_{t}} - B_{im} )</math>
<math>A_{c,t} = \Delta x \cdot ( \bar{w_{t}} - B_{bb} )</math>


<math>K = min(K_{1} , K_{2} )</math>
<math>K = min(K_{1} , K_{2} )</math>
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: <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>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>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>B_{bb}</math> = the datum height of the bottom boundary at the edge of interaction between the two cells.
: <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>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>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> = Ground 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 formula (Water Overlay)|timestep]] in seconds.
: <math>\Delta x</math> = Size of grid cell.
: <math>\Delta x</math> = Size of grid cell.
: <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>\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.
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: <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>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> = Ground 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 formula (Water Overlay)|timestep]] in seconds.
: <math>\Delta x</math> = Size of grid cell.
: <math>\Delta x</math> = Size of grid cell.
: <math>V_{KD,t}</math> = The amount of water to be transported at time <math>t</math> between one cell and the other.
: <math>V_{KD,t}</math> = The amount of water to be transported at time <math>t</math> between one cell and the other.
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: <math>V_{a,t} =
: <math>V_{a,t} =
\begin{cases}
\begin{cases}
  \dfrac{ \Delta w_{n,t} \cdot {KD}_a \cdot \Delta x}{\Delta x} \cdot \Delta t & \text{if } w_{n,t} > z_a  \text{ and } KD_a > 0 \\
  \dfrac{ \Delta w_{t} \cdot {KD}_a \cdot \Delta x}{\Delta x} \cdot \Delta t & \text{if } w_{n,t} > z_a  \text{ and } KD_a > 0 \\
  V_{K(D),t}  & \text{otherwise}  
  V_{K(D),t}  & \text{otherwise}  
\end{cases}
\end{cases}
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Where:
Where:
: <math>\Delta w_{t}</math> = Ground water level difference between the two adjacent cells at time <math>t</math>;
: <math>{KD}_a</math> = The [[Aquifer kd (Water Overlay)|AQUIFER_KD]] attribute of aquifer.
: <math>\Delta x</math> = Size of grid cell.
: <math>\Delta t</math> = Computational [[Timestep formula (Water Overlay)|timestep]] in seconds.
: <math>w_{n,t}</math> = Ground water level in cell <math>n</math> at time <math>t</math>;
: <math>w_{n,t}</math> = Ground water level in cell <math>n</math> at time <math>t</math>;
: <math>z_a</math> = [[Aquifer datum (Water Overlay)|the datum height of the aquifer]] at the cell.
: <math>z_a</math> = [[Aquifer datum (Water Overlay)|the datum height of the aquifer]] at the cell.
: <math>V_{K(D),t}</math> = The calculated amount of water to be transported at time <math>t</math> between one cell and the other.
: <math>V_{a,t}</math> = Volume in <math>m^{3}</math> that flows between the two adjacent cells due to the aquifer at time <math>t</math>.
: <math>V_{a,t}</math> = Volume in <math>m^{3}</math> that flows between the two adjacent cells due to the aquifer at time <math>t</math>.
: <math>\Delta w_{t}</math> = Ground water level difference between the two adjacent cells at time <math>t</math>;
: <math>\Delta t</math> = Computational timestep.
: <math>{KD}_a</math> = The [[Aquifer kd (Water Overlay)|AQUIFER_KD]] attribute of aquifer.
: <math>V_{K(D),t}</math> = The calculated amount of water to be transported at time <math>t</math> between one cell and the other.


{{article end
{{article end
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The following topics are related to this formula.
The following topics are related to this formula.
; Features
; Features
: [[Aquifer (Water Overlay)|Aquifer]]
: [[Aquifer (Water Overlay)]]
; Formulas
; Formulas
: [[Groundwater level formula (Water Overlay)|Groundwater level formula]]
: [[Groundwater level formula (Water Overlay)]]
: [[Ground infiltration formula (Water Overlay)|Ground infiltration formula]]
: [[Ground infiltration formula (Water Overlay)]]
; Models
; Models
: [[Ground model (Water Overlay)|Ground model]]
: [[Ground model (Water Overlay)]]
: [[Infiltration model (Water Overlay)|Infiltration model]]
: [[Infiltration model (Water Overlay)]]
: [[Tracer flow model (Water Overlay)|Tracer flow model]]
: [[Tracer flow model (Water Overlay)]]
|seealso=
|seealso=
* [[Ground model (Water Overlay)#Horizontal flow and aquifers| Ground model]]
* [[Ground model (Water Overlay)#Horizontal flow and aquifers| Ground model]]

Latest revision as of 13:29, 5 March 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:

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:

where:

= The ground water level of cell at time .
= the datum height of the surface of cell c, set by the elevation or a Terrain elevation prequel.
= 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.
= the datum height of the bottom boundary at the edge of interaction between the two cells.
= The hydraulic conductivity of the cell, defined in HYDRAULIC_CONDUCTIVITY_MD of the ground terrain.
= Area of conductance at time .
= Ground water level difference at time .
= Computational timestep in seconds.
= Size of grid cell.
= Averaged ground water level at time , based on water levels in ground, WATER_STORAGE_PERCENTAGE and potentially the surface water level, when the ground is filled to the top.
= The amount of water to be transported at time between one cell and the other.


Hydraulic Conductivity with Thickness

The flow between the two cells is calculated as:

where:

= The ground water level of cell at time .
= The hydraulic conductivity of the cell, multiplied with the thickness of the layer, defined in HYDRAULIC_CONDUCTIVITY_WITH_THICKNESS_MD of the ground terrain.
= Ground water level difference at time .
= Computational timestep in seconds.
= Size of grid cell.
= The amount of water to be transported at time 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:

= Ground water level difference between the two adjacent cells at time ;
= The AQUIFER_KD attribute of aquifer.
= Size of grid cell.
= Computational timestep in seconds.
= Ground water level in cell at time ;
= the datum height of the aquifer at the cell.
= The calculated amount of water to be transported at time between one cell and the other.
= Volume in that flows between the two adjacent cells due to the aquifer at time .

Related

The following topics are related to this formula.

Features
Aquifer (Water Overlay)
Formulas
Groundwater level formula (Water Overlay)
Ground infiltration formula (Water Overlay)
Models
Ground model (Water Overlay)
Infiltration model (Water Overlay)
Tracer flow model (Water Overlay)

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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