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_{ | <math>B_{im} = max ( B_{1}- d_{b,1} , B_{2}- d_{b,2} ) </math> | ||
<math>A_{c,t} = \Delta x \cdot ( \bar{w_{t}} - B_{ | <math>A_{c,t} = \Delta x \cdot ( \bar{w_{t}} - B_{im} )</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)|underground water level]] of cell <math>n</math> at time <math>t</math>. | : <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>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>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 underground 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. | ||
Revision as of 12:35, 25 January 2024
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[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
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 underground 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 underground 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} = Underground 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 underground 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 underground, WATER_STORAGE_PERCENTAGE and potentially the surface water level, when the underground 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 underground 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 underground 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} = Underground 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:
- 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} = \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 \\ V_{K(D),t} & \text{otherwise} \end{cases} }
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
- Underground infiltration formula
- Models
- Underground model
- Infiltration model
- Tracer flow model
See also
References
- ↑ 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.
- ↑ 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)
- Models
- Surface • Ground • Rain • Evaporation • Infiltration • Elevation • Breach • Sewer • Storage • Tracer flow • Border • Order
- Model formulas
- Timestep • Season • Surface water level • Groundwater level
- Flow formulas
- Surface flow • Surface infiltration • Ground flow • Ground infiltration • Surface evaporation • Ground evaporation • Ground bottom flow
- Hydraulic structure formulas
- Culvert • Weir • Breach growth • Breach flow • Pump • Sewer Overflow • Inlet • Drainage
- Related