3.1.2 Definition of an appropriate time to concentration
During the initial model runs, times to concentration derived from
Fathom’s global flood model (Sampson et al. 2015) were used. The model
uses the velocity method (United States National Resources Conservation
Service, National Engineering Handbook. Section 630, Hydrology. Chapter
15, Time of Concentration) to calculate the time to concentration as a
sum of the travel time in shallow concentrated flow and the travel time
in open channel flow. The travel time is derived using the longest flow
path from the point of interest and an average velocity derived using
Manning’s coefficient. This produced a substantial underestimation of
the observed water levels. To understand the reason of this behaviour,
several experiments were undertaken to help understand model parameter
sensitivity. All the following tests were performed using the discharge
estimated for a return period of one hundred years.
Initially, the model was run for each lake keeping the same weir value
(best estimate from GIS and remotely sensed data) and varying the time
to concentration from 1 hour up to 600 hours (24 days), in order to
evaluate model sensitivity to this variable. In some cases, the time to
concentration had a big influence on the modelled water level, while in
others it seemed to be relatively insensitive. In all cases the time to
concentration showed an asymptotic trend. The asymptotic behaviour
indicates that it is essential not to underestimate time to
concentration, while overestimation will be less harshly penalised in
terms of model performance. This is intuitively correct as water levels
in lakes are naturally self-regulating, with outflow increasing as lake
level increases until an equilibrium level is reached. The weir equation
represents this, with discharge being proportional toh 3/2, where h is water height above the
weir crest.
The other variable shown to have a strong influence on model behaviour
is the weir width. To evaluate model sensitivity to this variable, the
time to concentration value was held constant while weir width was
varied across a wide range of values. Again, some test cases proved to
be very sensitive to this variable while others exhibited minimal
sensitivity, with water level increases remaining almost constant
regardless of weir width.
Following the univariate analysis of time to concentration and weir
width, the next step was to try and delineate the behaviour of these
lakes and reservoirs using a bivariate analysis. A range of different
simulations were therefore run for each lake, varying both the weir
width and the time to concentration. Figure 2 represents an example of
the results obtained for the Lake Massawippi (Quebec) station,
representing the absolute error between the peak water level increase
produced by the model and the maximum recorded water level increase
(difference between annual maximum and annual mean).
[Figure 2]
From these results, it is possible to identify some general patterns
across a subset of 31 water level measuring gauging stations with time
records longer than 25 years, known lake area and synthetic discharge in
Quebec. Overestimation typically occurs when the weir is narrow or when
the time to concentration drastically increases, whilst it appears more
difficult to provoke underestimation from the model. In most cases it is
also possible to note that time to concentration maintains its
asymptotic trend: once the inflow hydrograph has a long enough duration,
the water level fluctuation stabilises and grows very slowly. Bigger
lakes generally appear to be more sensitive to the time to
concentration, and less to the weir width, while for smaller lakes the
best estimation of the water level seems to be very dependent on a good
estimate of the weir width whilst still requiring a long enough
hydrograph. Unfortunately, it doesn’t seem possible to generalise
overall behaviour in water levels as even lakes that seem to be similar
in size and with a comparable inflow show different values of recorded
water lake fluctuation. Since these analyses highlighted how an
overestimation of the inflow duration shouldn’t heavily penalize the
model performance, a fixed value of 200 hours was chosen for the time to
concentration to use hereafter.