Ground transpiration formula (Water Overlay): Difference between revisions

Latest revision as of 11:17, 15 July 2026

Conceptually, water can transpire from the ground via crops, tree and plant foliage. This is called evapotranspiration. The amount a plant can evaporate depends on the saturation and ground water level, as well as the root structure. The factor by which roots can extract water from the ground is modeled using the Feddes trapezium, based on the root depths (Building or Terrain).

Feddes trapezium indicating for what water levels and by what factor roots of a plant can extract water from the ground.

The amount a plant, crop or tree evaporates also depends on its growth season. Therefore, the species-specific transpiration factor can be set as four seasonal values.

Last but not least, the calculated evapotranspiration factor is multiplied with the evaporation amount at time $t$ , which is configured for a Weather.

The transpired amount is first drawn from the unsaturated zone and then, when the expected amount is not yet reached, from the saturated zone.

Formulas

This transpiration is calculated per cell. For all ground transpiration calculation, the ground water depth at time t is used.

$d_{g w, t} = B_{c} - w_{g, t}$

Feddes oxygen stress factor:

$f_{o} = {\begin{matrix} 1.0 & if d_{g w, t} is \geq d_{r, 3} \\ \frac{d_{g w, t} - d_{r, 4}}{d_{r, 3} - d_{r, 4}} & if d_{g w, t} is \geq d_{r, 4} \\ 0.0 & otherwise \end{matrix}$

Feddes saturated zone root update factor:

$f_{s} = {\begin{matrix} 0.0 & if d_{g w, t} is \geq d_{r, 1} \\ 1.0 - \frac{d_{g w, t} - d_{r, 2}}{d_{r, 1} - d_{r, 2}} & if d_{g w, t} is \geq d_{r, 2} \\ 1.0 & otherwise \end{matrix}$

Seasonal transpiration factor:

$f_{t s, t} = (1.0 - f_{s e, t}) \cdot f_{t s, c} + f_{s e, t} \cdot f_{t s, n}$

Max potential transpiration:

$E_{m a x} = f_{t s, t} \cdot e_{w, t} \cdot f_{o} \cdot Δ t$

Transpiration unsaturated:

$E_{u} = {\begin{matrix} 0.0 & if d_{g w, t} is \leq 0.0 \\ m i n (E_{m a x}, h_{u} \cdot \frac{d_{r, 1}}{d_{g w, t}}) & otherwise \end{matrix}$

Potential transpiration saturated:

$d_{s} = w_{g, t} - (B_{c} - d_{b, c})$

$d_{r} = m a x (0.0, w_{g, t} - (B_{c} - d_{r, 1}))$

$E_{p, s} = (E_{m a x} - E_{u}) \cdot f_{s}$

Transpiration saturated zone:

$E_{s} = m a x (0.0, m i n (E_{p, s}, d_{s} \cdot f_{w s}, d_{r} \cdot f_{w s}))$

Transpiration total is the sum of transpiration in the saturated and unsaturated zone:

$E_{g} = E_{u} + E_{s}$

where

B_{c}

= the datum height of the surface of cell c, set by the elevation or a Terrain elevation prequel.

w_{g, t}

= the ground water level (datum height) at time t.

d_{g w, t}

= the depth of the ground water level, which is equal to the column height of the unsaturated zone at time t.

f_{t s, c}

= the transpiration factor at current season c, of the buildings (plants) on the surface, or the surface terrain's transpiration factor if none present.

f_{t s, n}

= the transpiration factor at coming season n, of the buildings (plants) on the surface, or the surface terrain's transpiration factor if none present.

f_{s e, t}

= Season fraction. Used for interpolating the transpiration factor.

f_{t s, t}

= the interpolated seasonal transpiration factor.

e_{w, t}

= the evaporation rate of the weather at time t.

f_{o}

= the calculated Feddes oxygen stress factor, based on the 3rd and 4th root depths (Building or Terrain), the Feddes trapezium interpolation model, and the ground water depth.

f_{s}

= the calculated Feddes saturated zone root update factor, based on the 1st and 2nd root depths (Building or Terrain), the Feddes trapezium interpolation model, and the ground water depth.

Δ t

= the computational timestep in seconds.

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.

d_{s}

= the total thickness (column height) of the saturated zone.

d_{r, n}

= the nth root depth value, defined by building attributes or terrain attributes.

h_{u}

= the amount of water in the unsaturated zone.

E_{m a x}

= the potential max transpiration.

E_{u}

= the transpired amount in the unsaturated zone.

E_{p, s}

= the potential transpiration in the saturated zone.

f_{w s}

= The WATER_STORAGE_PERCENTAGE of the ground terrain type, used here as a fraction.

E_{s}

= the transpired in the saturated zone.

E_{g}

= the total transpired from the ground.

The following topics are related to this formula.

Formulas: Groundwater level formula (Water Overlay)
Models: Ground model (Water Overlay); Evapotranspiration model (Water Overlay)

Notes

The ground water level cannot be lowered further than the defined root depth by evapotranspiration.

Models

Surface • Ground • Rain • Evapotranspiration • 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

Building attributes • Hydrological features • Hydraulic structures • Terrain attributes • Model attributes • Prequels • Results • Simulation data

@@ Line 1: / Line 1: @@
-Underground evaporation is calculated per cell.
+Conceptually, water can transpire from the ground via crops, tree and plant foliage. This is called evapotranspiration. The amount a plant can evaporate depends on the saturation and ground water level, as well as the root structure.
+The factor by which roots can extract water from the ground is modeled using the Feddes trapezium, based on the root depths ([[Root depth m (Water Overlay)|Building]] or [[Terrain root depth m (Water Overlay)|Terrain]]).
+[[File:Feddes_trapezium.jpg|thumb|right|400px|Feddes trapezium indicating for what water levels and by what factor roots of a plant can extract water from the ground.]]
-For all underground evaporation, the height of the unsaturated zone is used.
+The amount a plant, crop or tree evaporates also depends on its growth season. Therefore, the species-specific [[Water transpiration (Function Value)|transpiration factor]] can be set as four [[Season formula (Water Overlay)|seasonal values]].
-: ''H<sub>unsat</sub> = H<sub>surface</sub> - WL<sub>underground</sub>''
-First the capacity for saturated evaporation is calculated, based on how much of the saturated area is in contact with the roots.
+Last but not least, the calculated evapotranspiration factor is multiplied with the [[Evaporation m (Water Overlay)|evaporation amount]] at time <math>t</math>, which is configured for a [[Weather (Water Overlay)|Weather]].
-: ''C<sub>sat</sub> = max( 0 , min( RD , GBDM ) - H<sub>unsat</sub> ) * WSP''
-Next the height of the unsaturated zone, and based on that the capacity for unsaturated evaporation is calculated.
+The transpired amount is first drawn from the unsaturated zone and then, when the expected amount is not yet reached, from the saturated zone.
-: ''C<sub>unsat</sub> = max( 0 , min( RD , H<sub>unsat</sub> ) ) * ( W<sub>unsat</sub> / H<sub>unsat</sub> )''
-Finally, the actual evaporation is calculated:
+==Formulas==
-: ''Δw<sub>unsat</sub> = min( C<sub>unsat</sub> , Δt * E<sub>weather</sub> * E<sub>top</sub> )''
-: ''Δw<sub>sat</sub> = min( C<sub>sat</sub> , (Δt * E<sub>weather</sub> * E<sub>top</sub>) - Δw<sub>unsat</sub> )''
-: ''Δw = Δw<sub>unsat</sub> + Δw<sub>sat</sub>''
-Where:
+This transpiration is calculated per cell. For all ground transpiration calculation, the ground water depth at time t is used.
-* Δw = The total amount of evaporation which takes place.
-* Δt = Computational timestep.
-* Δw<sub>unsat</sub> = The amount of evaporation which takes place from the unsaturated zone.
+<math>d_{gw,t} = B_{c} - w_{g,t}</math>
-* Δw<sub>sat</sub> = The amount of evaporation which takes place from the saturated zone.
-* C<sub>sat</sub> = The amount of evaporation that can take place from the saturated zone.
-* C<sub>unsat</sub> = The amount of evaporation that can take place from the unsaturated zone.
-* H<sub>unsat</sub> = The height (column) of the unsaturated zone.
-* W<sub>unsat</sub> = The amount of water in the saturated zone. The height of the water column if the equivalent amount of water was placed on the surface.
+Feddes oxygen stress factor:
-* WL<sub>underground</sub> = The [[Groundwater level formula (Water Overlay)|groundwater level]], relative to {{datum}}.
-* H<sub>surface</sub> = The [[Terrain height (Water Overlay)|terrain height]] in the cell, relative to {{datum}}.
+<math>f_o =
-* RD = The [[Root depth m (Water Overlay)|ROOT_DEPTH_M]] of the construction if present, the [[Terrain root depth m (Water Overlay)|ROOT_DEPTH_M]] of the surface terrain otherwise.
+\begin{cases}
-* GBDM = The [[Ground bottom distance m model attribute (Water Overlay)|GROUND_BOTTOM_DISTANCE_M]] of the Water Overlay. (effectively available height in the underground).
+.0 & \text{if } d_{gw,t} \text{ is } \geq d_{r,3} \\
-* E<sub>weather</sub> = The [[Evaporation_m_(Water_Overlay)|evaporation rate]] of the [[weather]].
+\frac{d_{gw,t} -  d_{r,4}}{d_{r,3} -  d_{r,4}} & \text{if } d_{gw,t} \text{ is } \geq d_{r,4} \\
-* E<sub>top</sub> = The [[Water evaporation factor (Water Overlay)|WATER_EVAPORATION_FACTOR]] of the construction if present, the [[Terrain water evaporation factor (Water Overlay)|WATER_EVAPORATION_FACTOR]] of the surface terrain otherwise.
+.0 & \text{otherwise}
+\end{cases}
+</math>
+Feddes saturated zone root update factor:
+<math>f_s =
+\begin{cases}
+.0 & \text{if } d_{gw,t} \text{ is } \geq d_{r,1} \\
+.0 - \frac{d_{gw,t} -  d_{r,2}}{d_{r,1} -  d_{r,2}} & \text{if } d_{gw,t} \text{ is } \geq d_{r,2} \\
+.0 & \text{otherwise}
+\end{cases}
+</math>
+Seasonal transpiration factor:
+<math> f_{ts,t} = (1.0 - f_{se,t}) \cdot f_{ts,c} + f_{se,t} \cdot f_{ts,n}</math>
+Max potential transpiration:
+<math>E_{max} = f_{ts,t} \cdot e_{w,t} \cdot f_o \cdot \Delta t</math>
+Transpiration unsaturated:
+<math>E_{u} =
+\begin{cases}
+.0 & \text{if } d_{gw,t} \text{ is } \leq 0.0 \\
+min(E_{max}, h_u \cdot \frac{d_{r,1}}{d_{gw,t}}) & \text{otherwise} \\
+\end{cases}
+</math>
+Potential transpiration saturated:
+<math>d_{s} = w_{g,t} - ( B_{c} - d_{b,c} )</math>
+<math>d_{r} = max( 0.0 ,  w_{g,t} - ( B_{c} - d_{r,1} ))</math>
+<math>E_{p,s} = (E_{max} - E_{u}) \cdot f_s</math>
+Transpiration saturated zone:
+<math>E_{s} = max (0.0 , min ( E_{p,s}, d_{s}\cdot f_{ws}, d_r \cdot f_{ws}) ) </math>
+Transpiration total is the sum of transpiration in the saturated and unsaturated zone:
+<math>E_g = E_u + E_s</math>
+where
+: <math>B_{c}</math> = the datum height of the surface of cell c, set by the [[Elevation model (Water Overlay)|elevation]] or a [[Terrain elevation prequel (Water Overlay)|Terrain elevation prequel]].
+: <math>w_{g,t}</math> = the ground water level (datum height) at time t.
+: <math>d_{gw,t}</math> = the depth of the ground water level, which is equal to the column height of the unsaturated zone at time t.
+: <math>f_{ts,c}</math> = the [[Water transpiration factor (Water Overlay)|transpiration factor]] at current season c, of the buildings (plants) on the surface, or the surface terrain's [[Terrain water transpiration factor (Water Overlay)|transpiration factor]] if none present.
+: <math>f_{ts,n}</math> = the [[Water transpiration factor (Water Overlay)|transpiration factor]] at coming season n, of the buildings (plants) on the surface, or the surface terrain's [[Terrain water transpiration factor (Water Overlay)|transpiration factor]] if none present.
+: <math>f_{se,t}</math> = [[Season formula (Water Overlay)|Season fraction]]. Used for interpolating the transpiration factor.
+: <math>f_{ts,t}</math> = the interpolated seasonal transpiration factor.
+: <math>e_{w,t}</math> = the [[Evaporation m (Water_Overlay)|evaporation rate]] of the [[weather]] at time t.
+: <math>f_o</math> = the calculated Feddes oxygen stress factor, based on the 3rd and 4th root depths ([[Root depth m (Water Overlay)|Building]] or [[Terrain root depth m (Water Overlay)|Terrain]]), the Feddes trapezium interpolation model, and the ground water depth.
+: <math>f_s</math> = the calculated Feddes saturated zone root update factor, based on the 1st and 2nd root depths ([[Root depth m (Water Overlay)|Building]] or [[Terrain root depth m (Water Overlay)|Terrain]]), the Feddes trapezium interpolation model, and the ground water depth.
+: <math>\Delta t</math> = the computational [[Timestep formula (Water Overlay)|timestep]] in seconds.
+: <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_{s}</math> = the total thickness (column height) of the saturated zone.
+: <math>d_{r,n}</math> =  the nth root depth value, defined by [[Root depth m (Water Overlay)|building attributes]] or [[Terrain root depth m (Water Overlay)|terrain attributes]].
+: <math>h_u</math> = the amount of water in the unsaturated zone.
+: <math>E_{max}</math> = the potential max transpiration.
+: <math>E_{u}</math> = the transpired amount in the unsaturated zone.
+: <math>E_{p,s}</math> = the potential transpiration in the saturated zone.
+: <math>f_{ws}</math> = The [[Terrain water storage percentage (Water Overlay)|WATER_STORAGE_PERCENTAGE]] of the ground terrain type, used here as a fraction.
+: <math>E_{s}</math> = the transpired in the saturated zone.
+: <math>E_{g}</math> = the total transpired from the ground.
 ==Related==
 The following topics are related to this formula.
 ; Formulas
-: [[Groundwater level formula (Water Overlay)|Groundwater level formula]]
+: [[Groundwater level formula (Water Overlay)]]
 ; Models
-: [[Underground model (Water Overlay)|Underground model]]
+: [[Ground model (Water Overlay)]]
-: [[Evaporation model (Water Overlay)|Evaporation model]]
+: [[Evapotranspiration model (Water Overlay)]]
-{{Template:WaterOverlay_nav}}
+{{article end
-{{Water Module buttons}}
+|notes=
+* The ground water level cannot be lowered further than the defined root depth by evapotranspiration.
+}}
+{{WaterOverlay formula nav}}

Ground transpiration formula (Water Overlay): Difference between revisions

Latest revision as of 11:17, 15 July 2026

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