Soil reservoir

The soil reservoir takes as input rainfall \(R\) (mm) and PET \(E_P\) (mm) to calculate the effective rainfall \(R_E\) (mm), the AET \(E_A\) (mm) and the UPET \(E_U\) (mm) of the watershed. \(H_s\) (mm) is the water level reservoir (soil moisture storage).

In Rameau, there are actually two methods to compute \(H_s\): the Thornthwaite method [Thornthwaite, 1948] and the GR3 method [Edijatno and Michel, 1989].

Physical parameters

Soil capacity of the Thornthwaite reservoir

The soil water holding capacity \(C_t\) (mm) is the maximum water level allowed in the soil reservoir.

Should be optimised. Default value is 70 mm.

[watershed.1]
thornthwaite.capacity = { value = 70.0, lower = 0, upper = 500, opti = true, sameas = 0 }

Soil capacity of the progressive reservoir

The soil water holding capacity \(C_g\) (mm) is the maximum water level allowed in the soil reservoir.

Should be optimised. Default value is 70 mm.

[watershed.1]
progressive.capacity = { value = 70.0, lower = 0, upper = 500, opti = true, sameas = 0 }

Exponential decrease of PET

If true, the GR3 formulation to compute AET is triggered only if the reservoir saturation is lower than 50% of its capacity.

Default value is false.

[watershed.1]
progressive.pet_decrease = false

Calculating effective rainfall

Thornthwaite model

If \(R\) > \(E_P\), then

\[\begin{split}E_A & = E_P \\ E_U & = 0 \\ H_s & = H_s + R - E_P\end{split}\]

Then, if \(H_s\) > \(C_t\) then

\[\begin{split}R_E & = H_s - C_t \\ H_s & = C_t\end{split}\]

else

\[R_E = 0\]

Conversely, if \(E_P\) > \(R\) then

\[\begin{split}E_A & = \min(R + H_s, E_P) \\ H_s & = \max(0, H_s - R + E_P) \\ E_U & = \max(0, E_P - E_A) \\ R_E & = 0.\end{split}\]

GR3 model

If \(R\) > \(E_P\), the amount of water that feeds into the soil reservoir over a time step is equal to \(R'= R - E_P\). Then, \(R_E\) (mm) is computed as follows:

\[R_E = \left ( \frac{H_s}{C_g} \right )^2 R'\]

The complement to 1 of \((1 - \left ( \frac{H_s}{C_g})^2 \right )R'\) will contribute to varying \(H_s\). For an elementary amount of rainfall \(dR'\) [Makhlouf et al., 1995]:

\[dS = \left ( 1 - \left ( \frac{H_s}{C_g} \right )^2 \right )dR'.\]

By integrating this equation over the time step:

\[H_s = \frac{ H_s + C_g\tanh \left ( \frac{R'}{C_g} \right ) }{ 1 + \frac{H_s}{C_g}\tanh \left (\frac{R'}{C_g} \right ) }.\]

Then,

\[\begin{split}E_A & = E_P \\ E_U & = 0.\end{split}\]

Conversely, if \(R\) < \(E_P\), the soil reservoir is subject to the PET \(E'= R - E_P\) and

\[E_A = \frac{H_s}{C_g} \left (2 - \frac{H_s}{C_g} \right )E'.\]

For an elementary change of PET \(dE\), the reduction of \(H_s\) is expressed as follows [Makhlouf et al., 1995]:

\[dH_s = -\frac{H_s}{C_g} \left (2 - \frac{H_s}{C_g} \right )dE.\]

By integrating this equation over the time step \(\Delta t\) (s)

(1)\[H_s(t + \Delta t) = H_s(t)\frac{ 1 - \tanh \left ( \frac{E'}{C_g} \right ) } { 1 + \left (1 - \frac{H_s(t)}{C_g} \right ) \tanh \left (\frac{E'}{C_g} \right ) }\]

and then, to make sure that \(H_s\) is always positive:

\[\begin{split}H_s(t + \Delta t) & = max(0, H_s(t + \Delta t)) \\ E_A & = R - (H_s(t + \Delta t)- H_s(t)) \\ E_U & = E_P - E_A.\end{split}\]

In Rameau, it is possible to trigger the AET decrease depending on the reservoir saturation only when it is below 50% of its capacity. In this case, we have:

\[C^{50}_g = 0.5C_g\]

If \(H_s(t) > C^{50}_g\), PET is fully satisfied similarly to the Thornthwaite approach:

\[H_s(t + \Delta t)= H_s(t) - E'\]

Else, \(H_s(t + \Delta t)\) is computed using Equation (1) replacing \(C_g\) with \(C^{50}_g\).