Chapter 3 : Reservoir models - KU Leuven

If the concentration time is not taken into account as a variable parameter with
the runoff, the extra variation in smoothing and peak shift can be modelled using
the dynamic storage. In that way the smoothing module and the dynamic storage
module overlap to some extent, although they cannot replace each other completely.
The differential equation that describes a linear static reservoir model is :
dV
dt
= Q − Q
(3.8)
in
through
with :
Q
through
V
=
V
Q m
(3.9)
m
In this set of equations, V is the instantaneous storage volume in the reservoir,
V m the maximum storage, Q in the inflow, Q m the maximum throughflow and
Q out the instantaneous throughflow. The analytical solution for this set of equations
at time t+dt as a function of the instantaneous storage V t in the previous time step t is :
V Q V ⎛
m
Qm
Q
V dt V Qm
t+ dt = in − ⎛ ⎞⎞
⎜−
⎟
⎜
t
V
dt
m ⎝ ⎝ m ⎠
⎟
+ ⎛ ⎞
1 exp exp ⎜−
⎟ (3.10)
⎠
⎝ m ⎠
When the maximum storage V m is reached, the overflow starts spilling and the overflow
discharge is equal to the difference between inflow Q in and maximum throughflow Q m .
The reservoir modelling system Remuli however uses piecewise linear relationships for
the storage/throughflow relationships, which also makes the inclusion of extra storage
during the overflow event possible (see paragraph 3.3.2).
The dynamic storage and the static storage are fitted together as shown in figure 3.15
using the following equations :
Vm
V k Q k Q
Q Q Vdm
= stat through + dyn in = through +
Q
Q in
(3.11)
in which V dm is the maximum dynamic storage corresponding with the maximum
inflow Q dm .
Combined with the continuity equation this leads to :
Or solved :
dV
dt
⎛ Vdm
V
Vt+ dt = Qin
⎜ +
⎝ Q Q
dm
⎛ Vdm
Qm
⎞ Qm
= Qin
⎜1 + ⎟ −
⎝ Q V ⎠ V
V
(3.12)
m
m
dm
m
m
m
⎞ ⎛ Qm
V dt V Qm
⎟ − ⎛ ⎞⎞
⎜−
⎟
t
⎠
⎜
⎝
V
dt
⎝
m ⎠
⎟ + ⎛ − ⎞
1 exp exp ⎜ ⎟ (3.13)
⎠ ⎝ m ⎠
dm
Chapter 3 : Reservoir models 3.29