Issue 
La Houille Blanche
Number 5, Août 1965



Page(s)  445  456  
DOI  https://doi.org/10.1051/lhb/1965034  
Published online  24 March 2010 
L'application du calcul des probabilités aux problèmes d'explication des réservoirs
Relationships between reservoir capacity and use application to algarian wadis
Ingénieur des Ponts et Chaussées, Service des Etudes Générales et Grands Travaux hydrauliques, Alger.
An interannual reservoir with a capacity C designed to meet the overall annual demand D of a number of users is considered, and the object is to show how, by Markoff's simple chain process theory, a relationship f (C, D, N) = 0 can be found between capacity C, demand D ant the average number of years N between two successive occasions on which demand could not be guaranteed. This relationship enables "guaranteed flow" to be calculated in terms of reservoir capacity. The method is used in designing dams on Algerian wadis for irrigation and town, water supply. Annual wadi flows allowed for are analysed statistically and their distribution is found to be as represented by the GaltonGibrat law. Reservoir outflows are then analysed (leakage, evaporation, various uses of water, including hydropower production and town water supplies). The last two take priority over all others, and require that a certain safety margin be guaranteed, being a certain percentage CI. of demand D. Mathematical formulation of the problem is first done on a " continuous " basis, the function for the distribution of reservoir filling states then being given by a certain Fredholm integral equation of the second kind [sec (3) and (3')]. As there are no analytical solutions to these equations, a "discontinuous" study is done, which assumes that only a fixed number of reservoir water levels are considered between the empty and full states. If these various intermediate states are Ej ... Er, the matrix T for the probability Pji that a state Ei in a year n 1 will go over to astate Ei during the following year n is determined from the inflow probability law and the fixed reservoir management rule. Probable reservoir contents Qni (water levels) in year n can be considered to be the elements of a matrix Pn1 with k lines and one column, Pn being connected to Pn  1 by the following matricial operation : At the probability limit, Pn and Pn  1 tend towards a stationary distribution P, which is the solution of the fol1owing equation : This matricial equation represents a linear system of k equations with k unknowns, the solution of which gives the probable reservoir water levels, and hence the average recurrence frequency of periods during which demand D cannot be guaranteed. A detailed numerical application is given for the case of Wadi Fodda, with remarks on how evaporation and seasonal variations from year to year were allowed for. Numerical calculation gave a very simple f (C, D, N) = 0 relationship for the various cases considered, as follows : log N / N0 = a (C  Ca) (1 / D  / /D0) Constants N0, a, C0, D0 vary in cach typical case.
© Société Hydrotechnique de France, 1965