THE WATERSHED AND ITS CHARACTERISTICS
3.2 HYDROLOGICAL BEHAVIOUR
3.2.1 Hydrological Response of a Watershed
The analysis of the hydrological behavior of a watershed is generally carried out by studying the hydrological response of this hydrological system to an impulse (precipitation). This response is measured by observing the quantity of water that exits the outlet of the system. The reaction of the discharge Q with respect to time t is represented graphically by a runoff hydrograph. Watershed response can also be represented with a limnigraph, which basically shows the depth of water measured with respect to time. Figures 3.4, 3.5 and 3.6 show, respectively, the principle for analyzing hydrological behavior, the hydrological response for a given precipitation event (the hyetograph is the curve representing the intensity of the rain as a function of time), and an example of a measured hydrological response of a watershed (e.g., the Bois-Vuacoz watersheed.
The hydrological response of a watershed to a particular event is characterized by, among other things, its velocity (time to peak tp which is the time between the beginning of the water flow and the peak of the hydrograph), and its intensity (peak flow Qmax, maximum volume Vmax…). However, understanding the hydrological response cannot be reduced to these two parameters alone. The analysis is actually more delicate than it seems because the flow measured at the outlet is related to the watershed’s scale. A number of factors influence the hydrological response to a particular precipitation event and it is hard to isolate any particular factor1. There are other characteristics that allow us to analyze the hydrological response of a watershed, as well, and in particular those resulting from studying the hyetograph and the resulting hydrograph (Section 11.8). One particularly useful parameter is time of concentration.
1. These factors will be discussed further in later chapters, including Chapter 6 which deals with the study of infiltration and flow, and Chapter 11 which discusses the analysis of hydrological processes.
Q H
t
Q Qmax
Qmoy Qmoy
Fig. 3.4 : Principles of the analysis of watershed response.
time[h]
intensity [mm / h]discharge [m3/s]
tm
Qmax
Vmax
time[h]
Fig. 3.5 : Hyetograph and hydrograph.
3.2.2 Time of Concentration and Isochrones
Definition of the Time of Concentration
The time of concentration tc of waters in a watershed is defined as the maximum time needed for a drop of water to flow from a particular point in the watershed to the watershed’s outlet.
It is composed of three different terms:
• th - runoff initiation time - the time necessary for the soil to absorb the water before surface runoff begins;
• tr - runoff time - the time corresponding to water flowing over the surface or in the top soil layers into a collection system (natural waterway, collectors or pipes…);
• ta – routing time - the time necessary for the water in the waterway to reach the outlet.
The time of concentration is thus equal to the maximum of the sum of these three times:
(3.1) Theoretically, tc is estimated to be the time duration between the end of the rain event and the end of direct surface runoff (Chapter 11). In practice, the time of concen-tration can be deduced from field measurements or estimated using empirical formulas.
Fig. 3.6 : Example of hydrological response for the Bois-Vuacoz sub-catchment (Haute-Mentue, Switzerland).
Isochrone curves
Isochrone curves are lines connecting the points in a watershed that have equal times of concentration. The isochrone farthest from the outlet represents the time required for the water from a uniform rainfall to reach the outlet from the entire water-shed surface. The design of the isochrone pattern makes it possible to visualize the entire hydrological behavior of a watershed and the relative importance of each of the surfaces contained between two isochrone curves (Figure 3.7). As we will see, these isochrones curves make it possible, by making certain assumptions, to create a flood hydrograph of a rainfall event over the watershed.
The Rational Method
The rational method was developped by the Irish engineer Mulvanay in about 1850, and could be considered as the first hydrological model. It is undoubtedly the most widely known and applied method, essentially because of its simplicity. The basic idea underlying this method is that when a rainfall of intensity i with an infinite duration starts at the same instant over an entire watershed, the outflow at the watershed outlet will grow until the entire watershed surface contributes to the surface runoff. The time that elapses between the beginning of runoff and this peak flow is obviously equal to the time of concentration. Therefore, peak flow can be determined using the following equation:
(3.2) where Qmax is the maximum discharge [l/s/ha], i is the intensity of the rain for a duration equal to the time of concentration [l/s/ha], and A is the surface area of the watershed [ha]. Cr is a coefficient called the runoff coefficient.2
2. The units of measurement in this relationship (3.2) can be expressed in m3/s for the flow (Qmax), in ha for the area (A), and in mm/h for the rainfall intensity (i) by applying a conver-sion factor of u = 0.0028 to the right side of the equation.
isochrones
1
A2 A4 A5 A6
area [km2]
time of concentration 2. t 3. t 4. t 5. t 6. t 1. t
A3
A
Fig. 3.7 : Representation of watershed with isochrones, and area-time of concentration diagram.
Note the curvatures of the isochrones near the drainage network.
The runoff coefficient (which we will discuss further in section 3.4) is defined as the ratio between the depth of runoff and the depth of precipitation. This coefficient conveys the idea that not all the water landing on a watershed necessarily ends up at its outlet, because there may be losses in the water budget equation due to interception, evaporation or infiltration.
In theory, the value of this coefficient can lie anywhere between zero and one.
However, it is entirely possible in practice to obtain a runoff coefficient of greater than one. This can be due to measurement errors or an error in the delineation of the watershed boundaries, where the effective limits of the watershed do not coincide with its topographical limits.
Calculating Discharge Using Isochrones
The isochrone method, which might be regarded as an extension of the rational method, is quite simple. It consists of estimating the discharge after sub-dividing the watershed into sectors divided by isochrones lines.
Before evaluating a flood hydrograph resulting from precipitation over a water-shed, which is not necessarily constant over time but is uniform over the area, we first assess the effect of precipitation of duration 't falling over sector Ai. The time it takes for the water to arrive at the outlet is between (i1)x't and i x't. If the time interval is relatively small, we can state the time for the water to reach the outlet is equal to (i1)x't.
Thus, precipitation of intensity Ii falling over sector Ai between t and t +'t will cause a flow of Qi = Crix i x Ai between time t + (i1)x't and t + i x't. As a result, the flow at the outlet of the watershed can be determined as the total of the flows from the precipitation on each sector:
• for area A1 between t and t't,
• for area A2 between t't and t,
• for area A3 between t2't and t't,
• ...
• for area Ai between t i1 't and t i2 't,
• ...
• for area An between t n1 't and t n2 't, By adding up all the partial flows, the following is obtained:
(3.3) where, Ii>t i1 't@ represents the intensity of precipitation on sector Ai for time
>t i1 't@. Assuming that the precipitation is homogenous over the watershed and
that the runoff coefficient is the same for all sectors, the final equation is:
(3.4) For the purpose of illustration, Figure 3.8 shows the case of a watershed divided into 7 sectors, as well as the hyetograph and the hydrological response obtained using the isochrone method.