Volume 1, numéro 1-2, 1988

FR :

Un modèle physique est constitué de quatre éprouvettes de sable percolées à des vitesses de flux différentes par L'eau d'un forage colmaté.

On a pu ainsi obtenir, en quelques semaines, un colmatage différentiel de deux éprouvettes percolées à des vitesses supérieures à 1 cm/s, et confirmer les hypothèses émises sur l'influence de La vitesse réelle des filets liquides dans L'apparition du colmatage, et sur ta nature bactérienne du processus colmatant.

Ce modèle constitue un véritable "colmatomètre" qui a donné lieu au dépôt d'un brevet (BOURGUET et al., 1985). Avant La réalisation d'un champ captant nouveau, L'emploi d'un tel colmatomètre devrait permettre d'apprécier les risques de colmatage spécifiques au site, et de définir Les normes de crépinage et d'exploitation permettant de Les éviter, ou du moins de les minimiser.

EN :

INTRODUCTION

A study on 300 wells drilled in the Ypresian aquifer, over 60 % of which were clogged, revealed that there were significant statistic relationships between the frequency of ctogging, the high velocities of the water close to the well screens and the presence of hydrogen sulfide in the water; the latter means the existence of sulfato-reducing bacterial activity.

The experimental model described here was designed to check the effect of the velocity of the water in the aquifer on the creation of clogging, as well as its relationship with the development of the bacterial population induced by a sufficient nutrient flow rate.

This model, comprising sterilized sand, through wich water from a clogged well percolated, was set up prior to a model which will soon be built, consisting of cores of Ypresian sand containing its own microorganisms.

APPARATUS AND METHODS

Four stainless-steel tubes (length 14 cm, inner diameter 26 mm) were filled with sterilized siliceous sand, the initial porosity of which was 35 %. The particle size distribution (40 to 140 µm) and the permeability were similar to those of the Ypresian sand. Tube seeding was carried out for a week by percolation with water from a clogged well; no clogging in the tubes was observed. This water was then percotated simultaneously through the tubes (figure 2) for 58 days at rates, maintained by weekly corrections, close to 1, 3, 12 and 30 mm/s respectively (figure 3).

The microbiological analyses consisted in : direct numeration of bacterial bodies by epifluorescence microscopy, indirect numeration by counting revivifiable heterotrophic aerobic bacteria after 15 days incubation within a medium chosen to be the least selective possible and, lastly, the numeration of both permanent and casual anaerobia, as well as sulfatoreducing bacteria following Hungate's method.

HYDRODYNAMIC RESULTS

(figure 4)The reduction of permeability, which results from clogging, after 58 days of percolation was the following :

- tube nb 1          0 % for an average percolation rate of 0.8 mm/s,

- tube nb 2         2 % for an average percolation rate of 2.5 mm/s,

- tube nb 3         29,5 % for an average percolation rate of 11.5 mm/s,

- tube nb 4         59 % for an average percolation rate of 30.8 mm/s.

Tubes nb 1, 2 and 4 were destroyed in order to carry out microbiological analyses. Tube nb 3, maintained in percolation, reached 99,4 % clogging after 142 days.

MICROBIOLOGICAL RESULTS AND DISCUSSION

Sampling procedures for analyses were set up by using tube nb 2. Three different samples of sand were used for analyses : from the inlet, from the middle and from the outlet of tubes nb 1 et 4. Moreover analyses were made on the percolation water upstream and downstream of tube nb 4.

The revivifiable heterotrophic aerobic bacteria are more numerous in the water at the outlet of tube nb 4 than at the inlet, white direct counting shows a slight decrease.

There are greater quantities of microorganisms in the sand than in the percolation water. The presence of sulfato-reducing bacteria can even be observed whereas they are too few to be counted in the percolation water. The filtering effect of the sand is not the only, element responsible for the increase amount of bacteria. Indeed the ratio between the number of revivifiable heterotrophic bacteria in the clogged tube and the non clogged tube was about 3 times greater than the ratio between their flow rates.

The higher velocity in tube nb 4 induced favorable conditions for the development of certain species of bacteria, some of which generate hydrogen producing organic molecules, thus allowing the development of sulfato-reducing bacteria.

Clogging itself should depend either on the increase of the biomass or on the development of microorganisms producing a significant quantity of organic molecules outside the cells. By taking into account the amount of bacteria and the volumes of percolated water, it can be seen that 2.01 x 10¹¹ cells, i.e. 0.2 cm³ , were retained by filtration in tube nb 1, and 0.7 cm³ in tube nb 4. Basing the estimations on bacterial counts in the sand, biovolumes of 0.2 mm³ are obtained for tube nb 1 (non clogged), and 8 mm³ for tube nb 4 (clogged). Consequently the clogging substances consist mainly of products situated outside the cells.

The tube is the center of chemolithotrophic organisms (made evident by C14 tracers), thus the aerobic heterotrophic microflora and the sulfato-reducing species are no doubt linked to the use of the metabolism products of these organisms.

After 142 days percolation, the sand in tube nb 3 (clogged at 99,4 %) had a cell content of 10¹¹ cells/cm³ of sand at the inlet (filamentous aggregates) and 10⁷ cells/cm³ at the outlet (no aggregates). On an average, roughly a third of the pore volume is filled with cells, which is in agreement with a decrease in porosity from 35 to 27 % and therefore with the clogging in the tubes.

CONCLUSION

Tubes of sand could be clogged experimentalty and rapidly. The most significant clogging was obtained for the fastest percolation rate.

It seems as though clogging is due to a proliferation of bacteria, essentially oligotrophic, and to the creation of a nutrient-chain rising to the sulfato-reducing bacteria (which are heterotrophic).

The clogging process described in the case of the Ypresian sand aquifer is certainly similar to the processes occurring in other granular aquifers.

The model " clog-meter" could be used by water supptiers exploiting aquifers fiable to clogging. Indeed it can determine the best flow rate, to avoid clogging in newly exploited pumping fields and decide on the most adequate well equipment, the number of wells needed in newly exploited areas and their exploitation yield in order to obtain the desired flow rates.

FR :

La production primaire phytoplanctenique et l'activité hétérotrophe bactérienne ont été étudiées parallèlement sur le lac d'AYDAT en 1985. L'activité photosynthétique, mesurée au alentours du midi solaire évolue entre 10 et 141 mg C m^-2h^-1, le maximum étant atteint en juillet. La vitesse maximale, V max, d'assimilation de glucose par les bactéries, (en présence de concentrations de traceur non limitantes) se situe entre 0,06 - 1,26 mg C m^-2h^-1. Le compartiment phytoplanctonique, de par sa production, semble être l'un des éléments régulateurs de l'activité hétérotrophe bactérienne en zone épilimnique.

EN :

The seasonal distributions of phytoptanktonic primary production and bacterial heterotrophicpotentials (using respectivey ¹⁴C and ¹⁴Glucose incorporations) were studied in a eutrophic lake (Lake Aydat, Massif Central, France).

Primary production and bacterial heterotrophic potentials vary respectively from 10 to 141 mg.C.m^-2h^-1 and from 0,06 to 1,26 mg.C.m^-2h^-1.

The comparative evolution of these two activities in relation to the temperature shows that the general idea about "the temperature - bacterial development" model seems to be insufficient when the investigation takes place in a complex ecosystem such as Lake Aydat. Indeed, the phytoplanktonic compartment, apart from its production, seems to be one of the regulating elements of the heterotrophic bacterial activity whose variations cannot be explained by changes in temperature alone.

In spite of the significant correlation obtained between bacterial activity and temperature, we should consider that the influence of this abiotic parameter on the bacterial development is an indirect one.

FR :

Le travail a eu pour but d'étudier l'efficacité de la photolyse du peroxyde d'hydrogène sur la dégradation de quelques composés organochlorés aliphatiques saturés (chiorométhanes et chloroéthanes) en milieu aqueux (pH 7,5). Les expériences ont été réalisées en réacteur statique, avec une Lampe basse pression à vapeur de mercure et avec des concentrations initiales en produit chloré de l'ordre de 10^-6 mol l^-1 et en H₂0₂ comprises entre 10^-5 et 10^-3 mol L^-.

Les résultats montrent que le système H₂0₂/UV peut oxyder les composés organochlorés étudiés à l'exception des composés ne possédant pas d'atome d'hydrogène (CCL₄ et C₂ CL₆). Les rendements d'oxydation obtenus avec Le réacteur utilisé dépendent du temps de réaction, de la concentration initiale en H₂0₂, du flux photonique et peuvent être nettement diminués par la présence de pièges à radicaux (ions bicarbonates) dans le milieu réactionnel.

Par ailleurs, une étude cinétique de la photolyse du peroxyde d'hydrogène en absence de matière organique est également présentée.

EN :

The aim of this work was to study oxidation of certain volatile polychlorinated hydrocarbons, using hydrogen peroxide photoactivated by UV. This research was carried out with different mixtures of diluted aqueous solutions of chloromethanes (CHCl₃, CCl₄) and chloroethanes (C₂H₃Cl₃, C₂H₂Cl₄, C₂HCl₅, CCl₆), which are typical halogenated compounds most frequently found in contaminated groundwater. The effect of the hydrogen peroxide concentration, the light intensity and the bicarbonate concentration on the rate of 1,1,2-trichloroethane (TCE) oxidation was determined. A kinetic study on hydrogen peroxide photolysis in a solution free of organic compounds was also carried out.

EXPERIMENTATION

Experiments were conducted in a batch reactor (V = 4 l), equipped with an immersed mercury low-pressure lamp. The intensity emitted at 253.7 nm was roughly 2 10¹⁹ photons s^-1. The temperature of the reaction mixture was maintained with a regulation system at 16 ± 0.5 °C (figure 1).

The solutions were prepared in a phosphate buffer µ = 2 10^-2 M, pH = 7.5). The outer surface of the lamp was masked with strips of aluminium, so as to obtain various percentages of initial energy (20 to 100 %).

The concentration of the hydrogen peroxide of the samples was determined by spectrophotometry and the chlorinated compounds were analysed by electron capture gas chromatography.

RESULT

Kinetics of hydrogen peroxide photolysis : H₂0₂ was decomposed by UV tb produce two hydroxyl radicals. In diluted solutions ([H₂0₂] < 10^-3 M), the concentration decreases in accordante with a first order law. The rate constant depends on the initial light intensity (I_o), on the characteristics of the reactor (volume and distance between the lamp and the watt. of the reactor) and on the motar extinction coefficient of the irradiated solution (equation C). The decomposition rate appears to be dependent on pH, the rate of constant rire has been found to be proportional to the dissociation of hydrogen peroxide into its basic form (E_H2O2 = 20 mol^-1 cm^-1,E_H2O2_ = 240 mol^-1 cm^-1) (figure 2 and 3).

Oxidation of the chlorinated compounds : H₂0₂/UV is very efficient for the removal of organic compounds. Preliminary experiments showed that both UV and H₂O₂ treatments do not decompose halogenated compounds. Hydroxyl radicals are extremly reactive and attack organic compounds preferentialty by abs-tracting a hydrogen atom from an organic molecule. This is confirmed by the results which show that chloromethanes and chloroethanes with an H atom are eliminated, but net compounds such as tetrachloride and hexachloroethane (figure 4 to 7).

The effectiveness of an H₂0₂/UV system depends on various parameters. Studies on the TCE elimination show that the oxidation yields an increase when the reaction time, the UV irradiation dose (figure 9b) and the hydrogen peroxide concentration (figure 8) increase. However, the efficiency decreases in the presence of radical traps such as bicarbonate and carbonate ions (figure 10).

FR :

L'étude concerne le chrome (VI) et son élimination dans l'eau par les celluloses greffées (échangeuse d'anions) AE₂ sous forme OH¯, Cl¯, S0₄⁼.

La capacité d'échange correspond à 0,5-0,6 meq g^-1 de cellulose. Les cinétiques d'échanges du Cr (VI) avec les ions CL¯ OH¯ et SO₄ ont été déterminées en fonction du pH du milieu, de la température, de ta force ionique, de La quantité de détergents et des formes de celluloses AE₂. Les résultats expérimentaux montrent que les formes de celluloses sont efficaces dans l'ordre OH¯ > Cl¯ > SO₄⁼ vis-à-vis du Cr (VI) et que l'échange décroît pour les ions étudiés dans l'ordre suivant : CrO₄⁼ > Cr₂0₇⁼ > HCrO₄¯.

EN :

Ion exchange is used extensively for water and waste water treatment for the removal of a variety of ion species. It is known that at acidic pH nome anion-exchange resins can selectively remove and recover chromate from cooling water (SENGUPTA, 1986) where chromate is used as a corrosion inhibitor. This method is employed for plating waste treatment and to recover hexavalent chromium. The waste stream is first passed through a cation exchanger to remove metal ions such as iron, zinc, nickel and trivalent chromium. The hexavalent chromium passes through as CrO₄⁼, and is subsequently removed in a anion exchanger.

Hexavalent chromium may exist in the aqueous phase in different ionic forms with total chromate concentration and pH dictating which particular chromate species will predominate. The equilibrium reactions for different Cr (VI) species are :

H₂CrO₄ ⇌ HCrO^-₄ + H⁺

HCrO^-₄ ⇌ CrO^-₄^- + H⁺

Cr₂O^-₇^- + H₂O ⇌ 2HCrO^-₄

Cr₂O^-7- + H₂O ⇌ 2CrO^-₄^- + 2H⁺

In the present study a description is given of the extraction of small quantities of Cr (VI) ion in water by modified cellulose (anionic form, (AE₂) counterions OH¯, Cl¯, S0₄⁼ ; 0,5-0,6 meq l^-1 ion exchange capacity) which is represented by the following chemical formula : (CHATELIN, 1982 ; WATTIEZ, CHATELIN, 1981).

Cell - O - CH₂ - CH - O - CH₂ - CH -

                          |                         |

                        CH₂                    CH₂

                          |                         |

                         N⁺    CL^-             N⁺    CL^-

                       /  |  \                   /  |  \

             CH₃  CH₃  CH₃         CH₃  CH₃  CH₃

Kinetics of exchange and effects of pH, competiting ions, concentration, temperature, ionic strength, surfactants and types of cellulose (Cl¯, OH¯, SO₄⁼) have been studied.Chromate only, chromate-pH, chromate-chloride, chromate-sulfate and chromate-surfactant kinetics (20 ± 1 °C) were determined for different anionic forms of modified cellulose. Kinetics data were generated by a batch equilibration technique where a weighed amount (5 g) of modified cellulose was shaken gently for 0.5-1 h with a fixed volume of solution (500 ml) containing Cr (VI) with successively H⁺, Cl¯, SO₄⁼, a surfactant of known initial composition. A the end of the equilibration solution, the composition was determined again to calculate the Cr (VI) uptake. Equilibrium was achieved in 0.5 h, but 75 % of Cr (VI) was removed in 5 min.

RESULTSSome specific aspects of the ion-exchange process for chromate recovery from industriel waste have been reported.It was observed that the more efficient types of cellulose for chromium removal where OH¯ > Cl¯ > SO₄⁼ respectively and that ion exchange decreased as follows

CrO₄⁼ > Cr₂0₇⁼ > HCrO₄¯.

FR :

L'exploitation des systèmes aquifères karstiques est toujours effectuée sous des contraintes liées à leur forte vulnérabilité. Une étude complète d'un épisode de crue de la source du Lez a été réalisée grâce à une coopération entre les Services de Santé et l'Université. Cette étude comporte une observation des paramètres physico-chimiques et bactériologiques sur une période de 15 jours, correspondant à une réponse hydrodynamique impulsionnelle sur la totalité du bassin. Les pas de temps d'échantillonnage varient de 4 à 24 heures.

L'interprétation des résultats physico-chimiques met en évidence une disjonction nette entre les variations piézométriques et le passage de différents volumes d'eau. L'écoulement des eaux plus chaudes (16,5 °C) observé lors de l'étiage, eaux d'origine profonde plus chargées chimiquement se poursuit jusque vers le milieu de la décrue pour faire place à l'arrivée d'eaux plus froides correspondant à des infiltrations rapides puis retardées. Les eaux les plus chaudes sont caractérisées par des variations des teneurs en magnésium, les teneurs en calcium restant à peu prés constantes. Inversement; les eaux froides de l'infiltration retardée ont de faibles variations des teneurs en magnésium et sont tracées par une forte augmentation des valeurs en calcium.

Le risque sanitaire maximal est lié au début de l'arrivée des eaux froides. Les eaux d'étiage et les eaux profondes sont légèrement contaminées. Les eaux d'infiltration retardée sont peu contaminées. En dehors des zones de perte du réseau hydrographique et des axes d'infiltration rapide, la vulnérabilité de l'aquifère est faible à l'échelle du bassin.

EN :

The exploitation of karstic aquifer systems for the supply of potable water raises the problem of the evaluation of sanitary risks, in view of their great vulnerability. To supply a great urban tenter, a yield as high as several cubic metres per hour is required, and this magnitude corresponds to that of an underground basin of several km², so that it is difficult to monitor and protect.

Analytical observations from sanitary control platforms on potable water catchments have shown the existence of periods of high bacteriological pollution synchronizing systematically with periods of a rise of water.

These bacteriological pollution transfers depend on the general conditions in which the aquifer functions. It seemed interesting, for the prevention of contamination, to examine all these hydrogeologic functions observed during a water rise, using bacteriological parameters.

This study was based on observations made during an overall flood period following a prolonged drought throughout the whole basin supplying the city of Montpellier (France). The aim was to carry out a correlation test on the functioning of a karstic aquifer, with regard to :

- the variation of the main physico-chemical parameters,

- the principal pathogenic germs,

- the commonly observed indicators of faecal contamination.

This operation, undertaken in close collaboration with the sanitation board and the university authorities, required considerable investment in human, materiat and financial resources.

Montpellier, a city with a population of 250 000 inhabitants, is supplied with potable water from the source of the River Lez. The water catchment is-situated upstream of a. major resurgence (12 m³/s during the water rise), tapping water from a principal drainage channel at a depth of 40 metres.

The area of the basin feeding the spring was evaluated to be 150 km² by adjustment to the 450 km² of Jurassic and Cretaceous outcropping certified limestone, north of Montpellier. Even if the limits of the basin are not precisely defined, some of its parameters are well known. The aquifer consists of different structural zones, almost like independent sub-systems, having in common a rapid drainage network. The response observed at the exurgence is compared globally with the pluviometric signs, as well as with the effects produced in each structural zone, the transfer time, chemistry, temperature, recession, etc.

The whole aquifer must be considered under pressure and it is this state of pressure which is probably responsible for the directional flow of the water from the rock matrix and of that circulating in the major karstification areas. The aquifer may be either in a state of injection or drainage and this notion is supported by the chemical and thermal fluctuations at the exurgence point.

A one-year experimental programme, based on our knowledge of this aquifer, has been set up in collaboration with forecasters at the National Meteorological Office. All have been on the alert and ready to intervene, equipped with all the necessary staff and apparatus to ensure reliable sampling and analyses. The following procedures were performed :

- sampling every 4 hours during the first 48 hours ;

- sampling every 12 hours during the following 48 hours.

Then until the end of the study, sampling took place every 24 hours with :

- 1 bacteriological sample in sterile condition,

- 1 sample for physico-chemical analyses in the laboratory,

- temperature measurements with a 1/10° mercury thermometer,

- resistivity measurements,

- pH measurements with two standards,

- a limigraphic reading.

This period of water rise, hydrologically homogeneous throughout the whote basin in the form of a single impulsive response tasting about 10 days, was characterised by the circulation of two types of water masses : from the rock matrix and resutting from rapid infiltration. The mixture of these two bodies, each with a distinct chemical character, is a function of the state of pressure in the aquifer. At the end of the rise, a third type of water intervenes, that of retarded infiltration.

The best criteria for identifying these volumes of water are thermal ones. The most variable elements are magnesium and calcium. Deep waters with a temperature of 16 °5 are characterised by a variation in magnesium, and cold water, by a variation in calcium. The highest value of calcium is related to the passage of water bodies attributed to the influence of retarded infiltration.

The geological observations and the processing of results by factorial analyses show a clear distinction between the chemical rise and the piezometric rise. The hydraulic rise point is constituted by water with the saure chemical characteristics as that of the lowest water level. The most important dilution due to rapid transfer in the basin is observed in the middle of the water fait. These synchronized with the highest batteriological pollution rate for most of the elements.

This study has provided a global appreciation of the sanitary risks and shows how pollution mechanisms function. Risks are permanent, even during the passage of the deepest water with a long residence time. This phenomenon raises the question of the duration and mode of survive of non encystable bacteria indicators in the karst system. The survival time appeared to be about a year or more.

There is always a covariation between the physico-chemical elements and bacteriological pollution where the aquifer is under high pressure and when the water rises or has reached its maximum level, or when the water starts to fall. It is when the level of the water starts to drop that the main peaks of bacteriological pollution are observed. From this moment onwards, the whole basin functions like a drained system and the absence of covariation in the physico-chemical and bacteriological parameters are signs that the flow has become heterogeneous.

For sanitation purposes it may be sufficient to monitor temperature and other chemical elements, in order to determine when an additional treatment of the water is required to maintain its potability. An appropriate apparatus is being devised to this end.

FR :

Dans les régions de plaine, la présence d'un cours d'eau s'accompagne généralement de celle d'une nappe alluviale. Des échanges entre nappe et rivière ont lieu à travers le fond du lit et les berges. Ces écoulements ont une influence sur leur régime thermique respectif. Dans un premier temps, afin de préciser ce terme souvent négligé du bilan thermique des cours d'eau, nous avons calculé, à l'aide d'un modèle numérique, le débit et les apports thermiques correspondants en provenance de la nappe lorsque la nappe alimente la rivière. Les résultats sont fonction de paramètres décrivant la forme du système, de la perméabilité de la couche aquifère et de la pente de la surface piézométrique. A l'inverse, lorsque le cours d'eau alimente la nappe de façon permanente, le régime thermique du cours d'eau, dont les fluctuations annuelles sont plus marquées que celles de la nappe, est susceptible d'influencer celui de la nappe. Nous avons également simulé cette influence. De plus, on a envisagé l'effet sur la nappe d'un échauffement du cours d'eau au-dessus de sa température naturelle. Des abaques ont été tracés pour regrouper les résultats. Ils permettent de connaître pour un grand nombre de situations, la distance depuis la rive à laquelle subsiste un échauffement dans l'aquifère égal à la moitié de l'échauffement de la rivière.

EN :

In the plain, rivers are generally connected to an aquifer, with water exchanges occurring between the two in both directions. In order to quantify the water and heat exchanges a model based on a cross-section of the alluvial aquifer was designed. A parameter study was conducted with this model to draw nomograms pointing to a broad range of application of the results. These results are expressed in terms of the following parameters :

- hydraulic conductivity of the aquifer,

- slope of the hydraulic head in the aquifer,

- geometric ratios describing the river-aquifer contact and thicknesses of the saturated and unsaturated parts of the aquifer.

First the water discharges and corresponding heat transfers from the aquifer toward the river are estimated. This factor of the thermal balance of the river, usually neglected, is important in order to avoid systematic errors when modeling the thermal. behaviour of the river. Restricting the energy balance of a river to the exchanges through the water-atmosphere interface, thus neglecting the inputs of water from the aquifer which is colder than the river in summer and warmer in winter, lead to a biased calibration of the river thermal modes.

On the other hand, the river affects the thermal behaviour of the aquifer, since the annual fluctuations of this temperature are greater important than the temperature fluctuations in the aquifer. When river and aquifer only exchange heat by conduction, the river influences the aquifer in a very restricted zone, only a few metres away from the bank. As soon as the river water enters the aquifer however, the thermal influence of the river may extend to several hundreds metres from the bank.

The effect on the aquifer of an artificial increase in the river temperature was then simulated to produce situation at steady state for one or two years. The resulting increase in the aquifer temperature is calculated for a cross section. Results are gathered on monograms giving the distance between the river bank and the site where the temperature increase in the aquifer equals 50 % of the temperature increase of the river. This distance depends on the following main variables : water velocity in the aquifer, thickness of the saturated and unsaturated parts of the aquifer.

Finally we have shown that the thermal balance of a river shoutd include heat exchanges with the alluvial aquifer, when noticeable inputs from the aquifer exist. Thermal modifications in the alluvial aquifer, due to water advection from the river are also reported.

The search for wide applications of the results led us to simplify the actual configurations of the shape of the system and its hydraulic and thermal boundary conditions. At a given site the direction of the water fluxes between river and aquifer may change several times a year and transient hydraulic conditions will have to be considered in some real cases.

Applications of these findings may be made, for example, in the fields of water supply or for a heat pump using water from this type of river-aquifer system.

FR :

Le modèle mathématique décrit dans cet article a pour tâche de choisir les sites sur une rivière où des installations hydro-électriques seront aménagées, puis de trouver la taille optimale de ces installations. La solution de ce problème dépend naturellement du montant d'argent que la compagnie est prête à investir sur la rivière. Toutefois, ce montant n'est pas connu au départ puisqu'il est lui-même fonction de ce que les installations pourront produire. Il est donc nécessaire de résoudre le problème pour tous les niveaux possibles de production étant donné qu'on ne connaît pas le niveau qui sera choisi. Ce problème est résolu dans cet article par une méthode très efficace qui regroupe l'énumération implicite, la programmation linéaire successive et l'analyse paramétrique. De façon succincte, l'énumération implicite fait le choix des sites qui seront aménagés pour un niveau de production donné. La programmation linéaire successive, quant à elle, se charge de déterminer la taille optimale des installations. Enfin, l'analyse paramétrique montre comment la taille des installations varie avec le niveau de production. L'efficacité de cette méthode vient du fait que l'algorithme d'énumération implicite, qui consomme beaucoup de temps de calcul, est appelé un nombre minimal de fois.

EN :

The purpose of the work described in this paper was to find a method for identifying the development scheme of a valley that will allow a utility to meet its electrical energy need at minimum cost. Suppose, for instance, that the utility wishes to build hydroelectric power plants in a virgin river valley so as to produce D gigawatthours of firm energy each year, and that preliminary surveys of the river show n possible sites. The first step is to select the sites and then to determine, for each, the dam height and hence size of the reservoir, elevation of the water inlet, elevation of the water outlet, and capacity of the power plant. The selection has to be clone in terms of minimizing the investment cost.

One of the utility's major difficulties with the above problem is to determine the value of D and hence the amount of money to invest in the valley. The solution depends on the alternatives to such an investment : a nuclear or thermal power plant or development of another valley, for example. In fact, the only completety reliable way to determine the value of D is to consider all the investment possibilities. The question is how do it without overly magnifying the problem ? One way, and incidentally the easiest, is to use a two-step solution : first, determine the optimal development scheme for each river for ail possible values of D, then find the optimal investment policy among the valleys and hence the optimal value of D. In this paper, only the first step is solved.

Determination of the optimal development scheme of a valley for ail possible values of D is formulated here as a parametric mixed-integer linear programming problem, which takes the form:

minimize cx

subject to

ax ≽ b + Ɵh

x ≽ 0

x^-= 0 or 1

where c and x are n vectors, A is an m by n matrix, b is an m vector, h is a change vector of dimension m, and 0 is a scalar that is varied continuously. x is a subvector of x containing those elements that must be 0 or 1. This formulation was chosen despite the nonlinearity of the real problem, in order to take advantage of the relative computational efficiency of mixed-integer linear programming as offered by IBM's MPSX/370 package. The nonlinear functions, like the costs, the production, and the relation between reservoir level and content, were linearized using separable programming in some cases and successive linear approximation in others. Since MPSX/370 inhibits the simultaneous use of separable and mixed-integer programming, the author wrote his on branch-and-bound algorithm. The parametric analyses are clone in such a way that the branch-and-bound algorithm, which consumes most of the computer time, is called a minimum number of times.