Ordinary differential equations 15
stage in the reservoir during the filling–emptying process, the maximum
value of the outflow discharge, the remaining water storage volume and
other critical quantities can be easily estimated so that management decisions can be reached.
An interpolation process may be required for estimation of Q in (t n+1 ) if
the inflow hydrograph is provided in discrete time steps δt, different from
the computational time step Δt (Figure 2.3). Then, the value of Q in (t n+1 ) is
estimated in terms of the measured values Q in (t m+1 ) and Q in (t m ):
Q t
Q t
t
Q t
Q t
in n
i n m
in m
i n m
( )
( )
[ (
)
( )]
+
+
=
+
−
1
1
1
λ
∆
(2.13)
where λ 1 = t n+1 – t m .
Summarizing, the mathematical model consists of the ODE, the input
parameters and the initial condition value for z. The solution algorithm
consists of the following steps:
1. Provide input values for the
a. Initial depth z 0
b. Discharge coefficient C o (or C w )
c. Inflow hydrograph Q in (t m ), for t m = m·δt
d. Stage–reservoir surface relationship S(z i ), for z i = n·Δz
e. Computational time step Δt
S(z n )
S i
S i+1
z i+1
Water
elevation
λ 2
S
Δz
Δz
z i
z n+1
λ 1
Δz
z
Reservoir
surface area
Figure 2.2 Interpolation from the stage–surface area curve.