Simulation of water losses on routed flows based on GR groundwater exchange function

GR models have a groundwater exchange function calculating a groundwater exchange flow in mm/time step based on the level of the routing reservoir. GR5J and GR6J have a different formulation than GR4J (See Le Moine, 2008).

In order to simulate supposed leakage at low-flows on karstic intermediate river basins, we need to find a way to remove water volume from the routed upstream flows on intermediate river basins.

I propose the following formalism:

• To reuse the groundwater exchange function result which is available in GR outputs as base of leak calculation
• To apply this leak in mm/time step to the upstream flow in m3/s taking into account the area of the downstream intermediate basin (positive exchange are not taken into account)
• To differentiate the leaks by taking into account the distance travelled by each upstream flow in the downstream intermediate basin

Le Moine, Nicolas. 2008. « Le bassin versant de surface vu par le souterrain : une voie d’amélioration des performances et du réalisme des modèles pluie-débit? » These de doctorat, Paris 6. https://www.theses.fr/2008PA066468.

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changed title from Simulation water losses on routed flows based on GR groundwater exchange function to Simulation of water losses on routed flows based on GR groundwater exchange function

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Specification:

• E OutputsModel$Exch
• Qup_i i^th upstream flow in m³/s
• PT_i the delay of the upstream flow i in the downstream intermediate water basin (i.e. the time during the leak occurs on the flow)
• A_d the area of the downstream intermediate water basin

Qup_i(t_{Qup_i positive} + PT_i) = max(0, Qup_i(t_{Qup_i positive} + PT_i) + E(t_{Qup_i positive}) \times PT_i \times 1000 A_d)

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I think there's still an issue on the repartition of the water leaks. The current formula applied the leaks on the whole area of the downstream intermediate water basin. In the case of a tributary connected near the downstream outlet, that will oversize the leaks. A weight related to the distance to the upstream inlets must be applied.

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Current implemented equation for Calibration of the Talanoa model.

Water losses Eup_{i} (m3/time step) applied on the routed upstream flows Qup_i are computed as below:

Eup_i(t) = \min(E(t),0) PT_i \dfrac{L_i}{\sum_{1 \leq j \leq N} L_j} A_d \times 1000

With:

• E OutputsModel$Exch of GR model in mm/time step
• PT_i the delay of the upstream flow i in the downstream intermediate water basin (i.e. the time during the leak occurs on the flow in number of time steps)
• L_i the length of the water course between upstream inlet and downstream outlet
• A_d the area of the downstream intermediate water basin in km²

Hypotheses behind the calculation of the water losses are:

• inflows from underground water exchanges does not affect routed upstream flows, that's why only negative exchange flows are considered here.
• water leaks are applied on the volume of water upstream flows stored in the intermediate basin, that's why the leaks are proportional to the delay time or travel time of the routed upstream flow in the intermediate basin.
• to pass from a height of water losses in mm per time step to a volume, we consider that each upstream water course are distinct and are influenced by an area proportional to their respective lengths.

Then leaks are only applied on positive upstream flows. During the calculation of the routed upstream flow affected by the leak (Qup(t) + Eup(t)), we check that the leaks that the leaks don't imply negative flows (cap to 0) and consequently modify the variable Eup.