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.