Ground transpiration formula (Water Overlay)

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.

$d_{gw,t}=B_c-w_{g,t}$

Feddes oxygen stress factor:

$f_o=\{\begin{array}{cc}1.0& \text{if }{d}_{gw,t}\text{ is }\ge d_{r,3}\\ \frac{d_{gw,t}-d_{r,4}}{d_{r,3}-d_{r,4}}& \text{if }{d}_{gw,t}\text{ is }\ge d_{r,4}\\ 0.0& \text{otherwise}\end{array}$

Feddes saturated zone root update factor:

$f_s=\{\begin{array}{cc}0.0& \text{if }{d}_{gw,t}\text{ is }\ge d_{r,1}\\ 1.0-\frac{d_{gw,t}-d_{r,2}}{d_{r,1}-d_{r,2}}& \text{if }{d}_{gw,t}\text{ is }\ge d_{r,2}\\ 1.0& \text{otherwise}\end{array}$

Seasonal transpiration factor:

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

Max potential transpiration:

$E_{max}=f_{ts,t}\cdot e_{w,t}\cdot f_o\cdot \Delta t$

Transpiration unsaturated:

$E_u=\{\begin{array}{cc}0.0& \text{if }{d}_{gw,t}\text{ is }\le 0.0\\ min(E_{max},h_u\cdot \frac{d_{r,1}}{d_{gw,t}})& \text{otherwise}\end{array}$

Potential transpiration saturated:

$d_s=w_{g,t}-(B_c-d_{b,c})$

$d_r=max(0.0,w_{g,t}-(B_c-d_{r,1}))$

$E_{p,s}=(E_{max}-E_u)\cdot f_s$

Transpiration saturated zone:

$E_s=max(0.0,min(E_{p,s},d_s\cdot f_{ws},d_r\cdot f_{ws}))$

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.

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

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

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

$f_{se,t}$ = Season fraction. Used for interpolating the transpiration factor.

$f_{ts,t}$ = 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.

$\Delta t$ = 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.

$E_{max}$ = the potential max transpiration.

$E_u$ = the transpired amount in the unsaturated zone.

$E_{p,s}$ = the potential transpiration in the saturated zone.

$f_{ws}$ = 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.

Related

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.

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