The GR4J model (modèle du Génie Rural
à 4 paramètres Journalier) is a lumped four-parameter model.
Its four parameters to be
: capacity of the production store
: groundwater exchange coefficient
: one day ahead capacity of the routing store (mm)
: time base of unit hydrograph HU1 (d)
In the following, raw rainfall is
denoted by P (mm) and potential
evapotranspiration (PE) is denoted by E (mm).
P is an
estimate of catchment areal rainfall and E can
be simply a regime curve of PE repeated every year. The following equation are
the results of integrations over the time step.
first operation is the subtraction of E from P to determine either
a net rainfall Pn or a net evapotranspiration capacity En. In
GR4J, this operation is computed as if there were an interception storage of
zero capacity. Pn and En are computed with the following equations:
Pn = P - E
En = 0
P < E, alors
Pn = 0
En = E – P
case Pn is not zero, a part Ps of Pn fills the production
store. It is determined as a function of the level S in the store by:
X1 (mm) is the maximum capacity of the SMA store.
the other case, when En is not zero, an actual evaporation rate is
determined as a function of the level in the production store to calculate the
quantity Es of water that will evaporate from the store. It is obtained
water content in the production store is then updated with:
= S - Es + Ps
percolation leakage Perc from the production store is then calculated as
a power function of the reservoir content:
is always lower than S. The reservoir content becomes:
total quantity Pr of water that reaches the routing functions is given
Pr = Perc +
is divided into two flow components according to a fixed split: 90 % of Pr
is routed by a unit hydrograph UH1 and then a non linear routing store,
and the remaining 10 % of Pr are routed by a single unit hydrograph UH2.
unit hydrographs depend on the same time parameter X4
expressed in days.
their discrete form, unit hydrographs UH1 and UH2 have n
and m ordinates respectively, where n and m are the
smallest integers exceeding x4 and 2.x4
respectively. This means that the water is staggered into n unit
hydrograph inputs for UH1 and m inputs for UH2. The
ordinates of both unit hydrographs are derived from the corresponding S-curves
(cumulative proportion of the input with time) denoted by SH1 and SH2
is defined along time t by:
For 0 <
is defined along time t by:
X4 < t < 2X4
and UH2 ordinates are then calculated by:
j is an integer.
At each time step i, the outputs Q9 and Q1 from the unit hydrographs are
calculated by :
where l = int(X4)+1 and m = int(2.X4)+1, with int(.) the integer part.
groundwater exchange term F that acts on both flow components, is then
R is the level in the routing store, x3 its
“reference” capacity and x2 the water exchange
coefficient. x2 can be either positive in case of water
imports, negative for water exports or zero when there is no water exchange.
level in the routing store is updated by adding the output Q9 of UH1
and F as follows:
R = max (0 ; R + Q9 + F)
outflow Qr of the reservoir is then calculated as:
level in the reservoir becomes:
= R – Qr
the content of the routing store, the output Q1 of UH2 is subject
to the same water exchange F to give the flow component Qd as
Qd = max (0 ;
streamflow Q is finally obtained by:
Q = Qr + Qd
Nascimento, N.O., Yang, X., Makhlouf, Z. et Michel, C. (1999). GR3J : a daily watershed model with three free parameters. Hydrological
Sciences Journal, 44(2), 263-278.
Perrin, C., 2002. Vers une amélioration d'un modèle
global pluie-débit au travers d'une approche comparative. La Houille Blanche,
n°6/7 : 84-91.
Perrin, C., Michel, C. and Andréassian, V., 2003.
Improvement of a parsimonious model for streamflow simulation. Journal of
Hydrology, 279 : 275-289.