zones, models are used to know the relationship which can result from the comparison between rainfall and runoff.
These models are statistical and/or conceptual, depending on the case study and the hydrological regime of the basin.
But, they differ in the length and the nature of available data which can be hourly, daily, monthly, or yearly events.
The following models are used in Morocco.
3.1 Case of gauged basins 3.1.1 Statistical models
Because of the availability of computers and the easy use of statistical models, the correlation techniques are widely used between rainfall and runoff. The results, however, vary upon the time interval of data.
Concerning yearly data, these models fit the data, with reasonable correlation coefficients, where only the length of data influences the accuracy. For monthly data, the same 52 Promdings of the UNESCO /NWRC/ACSAD Workshops on “Wadi Hydrology “and “‘Groundwater Pmtection ”
Methodology for Assessment, Operation, Management and Development of Water Resources in Morocco
relationships are observed, but sometimes when the phenomenon occurs between months the relationship can not be justified. Data from events are, generally, the most suitable data for best correlation. The application of simple regression to Taghjijt wadi, in the South West of Morocco, shows that the correlation coefficient -between rainfall storm events and corresponding runoff in mm- is always the highest. The formula applicable to this wadi is presented below, not necessarily as the best one, but to give an idea about the values of a (regression coefficient) and b (constant).
and runoff can be written as follows:
L = 0.40 * (P - 15) the regression formula, which can have a high correlation coefficient, extracted from data of the observed period of any gauged basin. The recession formula is also adjusted on available data for isolated events. This method, mentioned here, allows the values for runoff of any gauged basin to extend to the historical unknown period.
As it is expressed above, this formula is valid for certain values of rainfall (above 30 mm). The equation is not usually linear for the whole range of rainfalls, and other exponentials in forms can be written as follows:
L=a Pb
The same method is proposed in the hydrological study of arid zones in Morocco. The idea is to calculate the maximum instantaneous discharge q0 from the daily rainfall, and to apply the recession formula for the following days. The instantaneous discharge can be calculated from the following equation:
where: 40 = c* Pd
L = runoff in mm, P = rainfall in mm and a, b = coefficients. and the recession formula is written as:
The above formula was applied to two basins: Taghjijt (1400 km*) and Assaka (2450 km*); as given in Table 1. threshold P, can take high or small values upon the value of the coefficient a, according to small or high floods. The coefficient a determines the fraction, of either the threshold P, or rainfall P, to produce runoff. The threshold P, limits, effective rainfall from the raw rainfall.
q,, = maximum instantaneous discharge in m3/s, P, = daily rainfall in mm, c,, = coefficient, k = recession coefficient and t = time in days.
As examples, for the same wadi of Taghjijt, we find (Master Plan for south basins) that the relationship between rainfall
3.1.2 Approximating runoff by numbering floods and their characteristics
Arid regions are generally subject to a small number of floods (which do not exceed ten, except in wet years). Most of the surface water comes from floods only, as the local people say, ‘you are passing the water of the year’. This means that all the water of the year is passing through their location in one voluminous and high flood. The expression includes some regret, because the people are incapable of doing anything to gain profit from the flooding water. In contrast to what we imagine as scarcity of water in such zones, considerable quantities of water characterize them.
Table 1. Deduced Formula and Coefficients for Taghjijt and Assaka.
Wadi
If people can regulate these quantities, they will satisfy their IPi = (Si’TRi - STM) /a3
needs and make some additional profit. STSi = STTRi - IPi
If we have the number of floods for any previous year, and the mean volume of the flood, we can estimate the annual runoff of any wadi (even for the ungauged one). These statistics are easy to learn from rainfall gauges; yet but the volume of floods is quite difficult to determine.
where:
IPi = deep infiltration, STTRi = part of rainfall which percolates the soil, STSi = part of rainfall which supplies the storage of the upper reservoir, a3 = parameter and STM threshold.
In the case ofAssaka wadi, (hydrological studies of Assaka and Assayad Dams), the relationships between annual runoff, and the total volume of floods for two wadis, is as follows:
The deep reservoir is supposed to release water following the rule:
It is important to underline that, sometimes, it is difficult to know the volume of the floods. For this purpose, we need to search for additional methods.
Qi = Rdi + Rli
3.1.3 Water balance with multiple reservoirs methods This kind of model is used when the rainfall is more distributed along the year, with an annual rainfall of more than 300 mm, which is true in the mountain parts of wadis in Morocco. The model used is based on monthly water balance, with 9 parameters which were reduced to 5. The full description is given hereafter:
This model needs to be fitted for gauged stations by using any criteria function. In this case, we use the sum of squares function of: the difference between calculated and observed runoffs. The task is to minimize the following formula:
(Qi - Di)2 where:
Qi = calculated runoff and Di = observed runoff.
- The first fraction of water is the direct runoff which is expressed by the following equation:
Rdi = al (Pi - SRD) where:
Pi = rainfall of the i month, SRD = the threshold of direct runoff and al = runoff coefficient.
The model was developed and computerized for the purpose of the Souss basin Master Plan, and was used in many other locations. In other semi arid zones, for example, the model
-The second fraction of rainfall is divided in two parts again as follows:
1) Evapotranspiration which is based on the Turc formula:
ETRi = TOTi/(l +(TOTi/a2 Tu)~) Tu = 300 + 25 Ti + 0.05 Ti3
It is important to underline that, even with these models, it is difficult to reproduce the extreme values, minimum and maximum. However, these models allow the extrapolation of results by direct calculation for any ungauged basin, or for unknown periods of runoff for any gauged station.
where:
ETRi = monthly evapotranspiration in mm, TOTi = monthly rainfall in mm, Ti = monthly mean temperature in “C and a2 = parameter.
3.2 Case of ungauged wadi
2) Total infiltration amount of water which, at certain threshold of STM, supplies both of the upper reservoir (affected by evapotranspiration through the following months) and the groundwater reservoir. The equations related to this description are as follows:
For this purpose, we need many gauged stations, or at least two in a homogenous region. The simplest model used for the estimation of runoff for an ungauged basin is the one based on a correction factor taking into account the surface areas of two basins and their mean annual rainfalls, as in the following equation:
K = (SUS2) (PI/P2)
54 Pro&dings
of
the UNESCO ! NWRC f ACSAD Wohhopr on “Wadi Hydrology” and “Groundwater Protection ”---
Methodology for Assessment, Operation, Management and Development of Water Resources in Morocco
where:
Sl = surface area of the known basin in km*, S2 = surface area of the unknown basin in km2, Pl = annual mean rainfall for the known basin in mm, P2 = annual mean rainfall for the unknown basin in mm and K = correction factor. This kind of correction must be verified between gauged basins.
Sometimes, when several gauged wadis are available in homogenous regions, (even in arid regions), empirical formulae can be extracted from the analysis of data in which annual runoff is expressed versus surface area, and/or mean annual rainfall. In the case of Morocco, many attempts were made to find suitable regressions. The variability of the annual mean rainfall (ranging from less than 100 mm in the plain near the mouths of basins, to more than 400 mm in the crest of mountains) makes this analysis difficult. Perhaps, in the future, when the data of the stations which were installed from the beginning of the eighties will be available (long periods of data), this analysis will be possible.
Another simple method, which will give an idea about runoff of ungauged basins, is the use of the runoff coefficient. But, for best approximations, the difference between surface areas of the basins (unknown and known) must be small.
Otherwise, if more than one gauged basin is available, the best approximation is obtained when the correlation between surface areas of basins is used to calculate the runoff coefficient.