Transfer reservoir

The transfer reservoir computes surface runoff \(q_r\) (mm) and seepage \(q_s\) (mm) from the effective rainfall \(R_E\) (mm). \(H_t\) (mm) is the transfer reservoir water level.

Physical parameters

Runoff/seepage ratio height

The runoff/seepage ratio height \(H_r\) (mm) is the water level for which surface runoff and infiltration are equal. For example, if \(H_t\) is equal to \(n\) times \(H_r\), surface runoff is equal to \(n\) times infiltration.

Should be optimised. Default value is 70 mm.

[watershed.1]
transfer.runsee = { value = 70.0, lower = 0.1, upper = 5000., opti = true, sameas = 0 }

Transfer halflife time

Transfer reservoir halflife time \(t_{1/2}^t\) (months). This parameter characterizes the response time between an effective rainfall and an increase in the river flow slow component. Common values are:

• 0.5 months for rapid river flows

• 1 to 5 months for sources or deep groundwater level evolutions.

Should be optimised. Default value is 1 month.

[watershed.1]
transfer.halflife = { value = 0.5, lower = 0.01, upper = 10.0, opti = true, sameas = 0 }

Overflow fate

Fate of overflow.

loss	description
'no'	Overflow is directly added to the river flow.
'loss'	Overflow leaves the system.
'groundwater'	Overflow is added to the baseflow component of the river flow.

Default value is “no”.

[watershed.1]
transfer.overflow.loss = "no"

Overflow threshold

Overflow threshold \(H_o\) (mm). If \(H_t\) > \(H_o\) threshold, an overflow occurs. If equal to zero, overflow is deactivated.

Default value is 0.

[watershed.1]
transfer.overflow.threshold = { value = 0.0, lower = 0.0, upper = 10.0, opti = false, sameas = 0 }

Overflow halflife time

Overflow half-life time \(t_{1/2}^o\) (unit of time step). It controls the response time of the reservoir level decrease due to overflow. A value of zero means instantaneous overflow.

Default value is 0.

[watershed.1]
transfer.overflow.halflife = { value = 0.0, lower = 0.001, upper = 10.0, opti = false, sameas = 0 }

Calculation of surface runoff and seepage

Instantaneous runoff \(u_r\) (\(\mathrm{mm.s^{-1}}\)) and seepage \(u_s\) (\(\mathrm{mm.s^{-1}}\)) take place simultaneously according to two types of reservoir draining:

\[\begin{split}u_r &= \frac{H_t^2}{\tau H_r} \\ u_s &= \frac{H_t}{\tau}\end{split}\]

\(\tau\) (s) is a time constant defining the exponential decay of \(H_t\). \(\tau\) is related to \(t_{1/2}^t\) through the relationship:

\[\tau = (30.41 \cdot 86400)\frac{t_{1/2}^t}{\ln(2)}.\]

\(H_t\) decreases over a time step according to the following equation:

(1)\[\frac{dH_t}{dt} = -u_r -u_s\]

In Rameau, the transfer reservoir is fed at the beginning of each time step of duration \(\Delta t\) by a sudden inflow of effective rainfall \(R_E\) coming from the soil reservoir in the form of a Dirac. \(H_0\) is the value of \(H_t\) at the beginning of the time step (\(t = 0\)). Then,

\[H_0 = H_0 + R_E\]

Overflow \(q_o\) (mm) may occur if \(H_o\) > 0. In this case, if \(H_0 > H_o\), overflow occurs according to a linear exponential draining of half-life time \(t_{1/2}^o\) (unit of time step) and \(H_0\) is updated consequently:

\[\begin{split}q_o & = \frac{H_0 - H_o}{T_o} \\ H_0 & = \max \left ( H_0 - q_o, 0. \right )\end{split}\]

with

\[T_o = \frac{1}{1 - \exp \left(-\frac{\ln2}{t_{1/2}^o} \right)}.\]

The analytical solution of the equation (1) is:

\[\begin{split}H_t(t) & = \frac{CH_r\exp \left (\frac{-t}{\tau} \right )}{1 - C\exp \left (\frac{-t}{\tau} \right )} \\ C & = \frac{1}{1 + \frac{H_r}{H_0}}.\end{split}\]

Integrating \(u_s\) between \(t\) and \(t + \Delta t\) gives the seepage flowing during the time step \(q_s\) (mm):

\[\begin{split}q_s & = H_r\ln \left ( \frac{1 - C \exp \left (-\frac{\Delta t}{\tau} \right )}{1 - C} \right ) \\ & = H_r\ln \left ( 1 + \frac{H_0)}{H_r} \left (1 - C\exp \left(-\frac{\Delta t}{\tau} \right ) \right ) \right ) \\ & = H_r\ln \left ( 1 + \frac{H_0}{T_hH_b} \right )\end{split}\]

with:

\[T_h = \frac{1}{1 - \exp \left(-\frac{\Delta t}{\tau} \right)}.\]

Finally, mass conservation gives:

\[q_r = (H_0 - H_t(\Delta t)) - q_s\]