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.

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

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}

= Can be supplied as four values; it is an interpolated value based on the current and next season.

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.