A MULTIVARIATE STATISTICAL APPROACH OF OVERTOPPING PROBABILITY FOR RISK ANALYSIS AT DAMS

H. Bergmann, H. Breinhälter and O. Hable

Graz University of Technology, Institute for Hydraulics and Hydrology

A-8010 Graz, Mandellstraße 9, Austria

Tel: ++43-316-873-6260, Fax: ++43-316-873-6264

E-mail: hable@hydro.tu-graz.ac.at

Abstract: flood and wind-induced high water behind a dam are two of the possible geophysical events causing its overtopping. in this paper a procedure is laid out for the evaluation of the overtopping probability due to flood and wind events. currently in austria spillway flood capacity is designed to accommodate some fixed flood, such as the once in 5000 year event for dams. The design flood itself is defined as the peak runoff. No other variables, such as the reservoir storage volume or wave heights are considered. Additionally other parameters of the design event (e.g. Time to peak, total flood volume) are assumed to be suitable. Dissatisfaction with this approach has prompted an investigation of a probabilistic approach taking account of all the other important variables, such as initial reservoir storage, wind induced wave height, outlet valve and gate openings. A research project has the aim to develop a new probabilistic design concept for spillways of dams with consideration of the reservoir management. The proposed design concept is able to optimise both safety and economy. A practicable design software has been developed to estimate hydraulic-hydrologic failure situations by means of the monte carlo method of risk analysis. The development and the practical application of the proposed method is carried out in the context of a case study concerning a reservoir of a hydro power plant in austria. 

Keywords: overtopping probability, probabilistic design concept, dams, bivariate flood model, risk analysis, monte carlo method

1   INTRODUCTION

Failure of a dam occurs when one or more of the failure events occurs, such as overtopping, seepage, piping, instability, etc. Numerous studies of dam failures have indicated that overtopping is a major failure mode of earth and rockfill dams (Cheng, S., et al., 1993). The average dam failure probabilities for (all kind of) dams was found to be approximately 10^-3 per year per dam (Cheng, S. 1993). In thirty percent of all cases overtopping is the cause of earth dam failure (Vrijling, J. 1993). Overtopping occurs when the water level of the reservoir behind the dam rises above the dam crest. It may result from failure to make timely and adequate releases of floods through the spillway and flood release outlets, wave action induced by wind, landslides, earthquakes, and other geophysical forces, and the combination of these effects. In the context of this paper only flood and wind events are considered for the evaluation of the overtopping probability of the dam. Hydrological phenomena like precipitation or runoff always appear as multivariate events. For many problems in water management it is not satisfying to know only the frequency of single characteristics. It will be more important to know the probability of the whole event. Therefore they should be characterized by numerous parameters. Dam failure is not a determinable event and hence quantitative evaluation of dam safety requires probability theory. The proposed probabilistic design concept for spillways of dams includes a multivariate statistical model for the simulation of incoming flood events, which is able to describe the hydrological process entirely.

2    CURRENT DESIGN PRACTICE

The actual hydraulic design practice in Austria is to select a definite load case (or several cases, if necessary) for which the calculation is made. This design concept deals with only one random variable, named the "peak discharge" of the flood (e.g. HQ₅₀₀₀ for dams). The design flood itself is defined as the peak runoff (an univariate value), which can be discharged by the spillway outlets under special (mostly fixed) conditions, without endangering the dam or important parts of the construction. Other parameters of the design event (e.g. time to peak, total flood volume, etc.) are more or less, as suitable assumed. Currently operating states due to reservoir management, do not find any consideration in the official water law procedure (e.g. the initial water level in the reservoir at the beginning of a flood event or the condition of the crest and spillway gates and the bottom outlet or the wind depending freeboard height) and are specified as fixed values. The uncertainty in the estimation of design floods induced the water authorities to set up safety principles for the dimensioning of flood spillways, which seem to be relatively highly compared to other endangering potentials. Mostly it has not been considered, that there exists a spectrum of hydrographs (different combinations of peak and volume, Fig. 8) for each probability, from which the decisive event and the resulting peak flow has to be found out, while considering the retention.

3    PROPOSED SOLUTION

These are reasons to develop a probabilistic design concept for spillways, which is taking account of the demands of safety (against uncontrolled overtopping of dams) as well as of economy due to reservoir management. Nevertheless it has to be considered, that the safety of hydraulic structures does not only depend on the chosen computing method and the choice of the design probability, but also on the quality of the available hydrological data. Especially the choice of the design probability varies in the countries of the world. The return period of the design events for large storage reservoirs vary from a 100 years up to 10.000 years and is limited in most countries by the probable maximum flood (DVWK, 1999).

4    MAIN LOAD VALUES

The proposed method, originally developed by Pohl, R. (1997) at the Dresden University of Technology, is standing out by the fact, that the computed water level in the reservoir is consequently the only design value for evaluating the overtopping safety of the dam. This design value is resulting of three main load values (Fig. 1), namely the 

l         random initial reservoir water level at the beginning of a flood event...h₀

l         additional elevation height due to a random flood event (inflow)...h_r

l         required freeboard height due to a random wind event...h_f. 

According to this, the design value (=water level) itself is also a random variable, which can be estimated by means of the method of statistical trials (Monte Carlo Method). An exact solution is difficult or impossible, due to flood routing and freeboard calculations. Within this design concept (Fig. 1), the "failure" is defined as a non-desired hydraulic state, independently of damages to property or personal injuries.

h₀ + h_r + h_f > H_C = Crest level                     hydraulic-hydrological “failure”

Fig. 1  Definition sketch, approach of the overtopping probability

4.1    Initial reservoir water level (reservoir management)

At the beginning of a flood event an initial reservoir water level h₀ (Fig. 1) is to be found, which is mathematically considered to be random. The separate entry of the initial reservoir water level guarantees the economy of the design concept and is representative for previous floods or droughts, the reservoir operation and the flood management, the availability of crest and spillway gates and the availability of the bottom outlet and the penstocks.

4.2    Additional elevation height (flood event)

Of large influence for the safety against overtopping of the dam is the incoming flood event. The retention of the flood hydrograph in the uncontrolled storage normally rises the level in the reservoir. For this reason the flood event itself has a special meaning within this design concept. A bivariate flood model has been developed at the Institute for Hydraulics and Hydrology at the Graz University of Technology (Sackl & Bergmann,1986) and should become a part of the design concept.

Characteristics of flood events

The peak Q_D and volume V_D of direct runoff are the best describing characteristics of a flood event (Fig. 2, Table 1). With a sufficiently large sample size it is possible to evaluate design events (V_D and Q_D) of a certain return period T_n using a bivariate statistical analysis. In this paper no detailed description of the used bivariate statistical model is given but further information are available by Sackl, B. (1987).

The method is divided into the following steps (Sackl & Bergmann,1986):

Bivariate sample (V_D and Q _D)

The part of direct runoff of a flood event is estimated by the separation of the base flow, which is done by an increasing straight line. For this quantitative method the actual course is of no interest, it is still important to objectify the separation. For instance this can be done by a constant separation criterion (V_D/V_B,1 = const.).

Fig. 2  Schematic representation of a flood event

Fig. 3  Bi-normal density function

Table 1  Flood characteristics

Bivariate distribution function

To prevent an escalation of variety in methods, the bivariate normal distribution function (Fig. 3) is used exclusively. An excellent fit can be achieved by a transformation of single samples into normal distributed samples. The easiest way to get an approximate “normalization” is to transform the single samples in a way that the coefficients of skewness become zero.

Lines of equal probability density function (PDF)

First of all it is possible to compute the ISO-PDF-lines (lines of equal bivariate PDF), which are limiting certain random areas. They are horizontal cuts through the bi-normal PDF, which are ellipsis in the normalized V_D, Q_D – system. With the help of these lines is possible to make a semi-graphical test of fit.

“Design curves” – lines of equal return period

The probability of an event occurring in a certain V_D, Q_D – area is equivalent to the volume above this area up to the bivariate PDF. There are theoretically infinite possibilities to delimit such an area for integration. The only efficient definition of a bivariate flood probability is the so called “upper right probability” p_ur which is the probability of the occurrence of a pair of values V_D, Q_D in the upper right quadrant (Fig. 4). An ISO-p_ur-line is a horizontal cut through the bivariate distribution function relating to the “upper right quadrant”. It is also possible to define an ISO-p_ur-line as a line of equal return period (Fig. 6) or as a “flood design-curve” (Sackl & Bergmann, 1987). For different return periods a spectrum of design-curves is obtained. The intersection points of an ISO-p_ur-line with the V_D– and Q_D-axes corresponding to a certain return period T_n are the respective values V_n and Q_n. The course within these points respectively within the event-range has nearly the shape of a quarter-ellipse (regarding only bivariate flood analysis). So design-curves can be approximated as quarter ellipses (Fig. 5) after the collection of flood-peaks for certain return periods Q_D [m³/s], volumes of direct runoff for certain return periods V_D [m³], a typical local standard hydrograph or standard hydrograph-range, typical local range of the time index t_m = V_D/Q_D [hours], and mean base flow or initial base flow Q_B,0 [m³/s] corresponding with the increase of base flow V_D/V_B,1 [-].

Fig. 4  ISO-PDF-LINES

Fig. 5  Ellipse as an approximation for an ISO-p_ur-line (“design-curve”), event range from t_m,min to t_m,ma

Fig. 6  Curves of flood events of equal return period 

        Fig. 7  Standard hydrograph

Standard hydrograph – event range

Fig. 8  Spectrum of design flood hydrographs 　

A direct runoff hydrograph standardized to the peak u = 1 and the time to peak t = 1 is called standard hydrograph. It is necessary for the model, to reduce the multivariate “flood” process to two dimensions. The chosen analytical 2-parametric function makes it possible to choose the type of standard hydrograph (Fig. 7) in a wide range according to the basins properties. To obtain the standard parameters (a, m), direct runoff is separated from measured flood events and then standardized, by the determination of the standard volumes VS and the standard times t_max= T/t_a (Table 1). There are no floods with an extreme peak runoff and a volume equal to zero. For this reason, there always must be a certain “event range”, which is possible to delimit approximately with straight lines through the origin. This straight lines are defined by the time index t_m. The time to peak t_a of an event point lying on a design curve results directly from the value of the time index t_m and the assumed standard hydrograph. Therefore the boundaries t_m,min and t_m,max (Fig. 6) can be estimated by the probable minimal and maximal values for t_a for a certain watershed. The course of the “design-curves” is similar to a quarter ellipse only within the range of t_m,min and t_m,max (Fig. 5).

Design flood hydrographs

After construction of the design curves for different return periods (Fig. 5), the events along a “T_n-years” line are converted into a “spectrum of design flood hydrographs” (Fig. 8) with the constant return period T_n.

4.3    Freeboard height

The calculation of the freeboard height as shown in the 3^rd column of Fig. 9 widely corresponds to the guideline 246/1997 of the German Association of Water Resources Management and Land Improvement (DVWK, 1997). Both wind set-up in the reservoir and wave run-up on the slope of the dam are considered. They are functions of the wind direction, fetch, water depth, and wind velocity.  

5    FLOW CHART OF THE CALCULATION PROCEDURE

The flow chart of the calculation procedure is schematically shown in Fig. 9. In this procedure the flood event with the maximum discharge peak of n arbitrary years is simulated n-times using random numbers, whereby the overtopping probability is distinguished, concerning the direct probability of overtopping p₂ (only due to flood meeting the initial water level) and the indirect probability of overtopping p₃ (due to flood and wind effect). A practicable design software has been developed to estimate hydraulic-hydrologic failure situations by means of the method of statistical testing and includes the multivariate flood model described before.

6    CASE STUDY AND RESULTS

Salza dam impounding the river Salza is an annual storage in Austria has been chosen to explain the results of the proposed design method. The 53 [m] high constant-angle arch dam was built between 1947 and 1949 and covers a catchment area of A_E = 150 [km²]. Salza dam was the first arch dam of medium size in Austria. This is the reason why an excellent flood data base (observation period 1954 – 1999) is available for any kind of investigations. For more details concerning technical data of the Salza dam see Table 4. The calculation of the exceedence probability was carried out under the conditions shown in Table 2. The results of this case study are collected in Table 3. Both heights, the normal top water level and the overflow crest are in any case overtopped, even in an arbitrary year. The overtopping probability of the dam crest in an arbitrary year is between 0,7 and 15 [%] due to flood event, respectively flood and (the assumed) wind event together (85 and 87 [%] in the design case). A specific feature of the Salza dam is its ungated spillway crest over the whole crest length (121 [m]) which, rising towards the flanks from elevation 771,5 to 772,5 [m a.s.l.], directs overflowing water into the middle portion irrespective of the flow rate. So the Salza dam is rather insensitively against overtopping and this is the reason why such high overtopping probabilities are acceptable. Actually, priming of the spillway occurs several times a year at the Salza dam.

Table 2  Calculation assumptions to the case study

l         The initial water level was uniformly distributed between normal top water level and 8 [m] below normal top water level

l         The bottom outlet is closed during a flood event

l         Because of missing wind data at the Salza dam site,wind velocity was assumed with 15 [m/s]

Table 3  Results of the overtopping probability of the Salza dam in Austria

Further results of another case study concerning a combined concrete gravity and rockfill dam with an controlled spillway using the proposed method are reported by Hable, O. et al., (2000).

7  CONCLUSIONS

By applying the proposed method, it is possible to estimate the correlation between the overtopping probability and a freely selectable "critical water level" (e.g. the maximum storage level in the reservoir, the crest level or the overflow crest of the spillway). Therefore prognoses about the different probabilities (e.g. of keeping of the storage level, of overtopping of the dam or of priming of the spillway) are possible. Beyond this it is possible to dimension spillways of hydraulic structures (not only dams) by matching the demands of safety as well as of economy. For this reason the main advantage of this probabilistic design method including the proposed multivariate flood model are an almost complete and realistic processing of the available information, an excellent simulation of the incoming flood event, a risk estimation tool as a base for taking measures if necessary and the comparability of dams with different spillways, storages, types of constructions and purposes concerning their security. A research project together with the association of the power plants of Austria (VEÖ) has the aim to develop and improve the proposed method. It should be possible to open up new paths in the field of design works, if a reorientation in finding of design values is made possible in the official water law procedure.

References

Cheng, S. 1993: Statistics on dam failures, In: Reliability and uncertainty analysis in hydraulic design, edited by Ben Chie Yen and Yeou-Koung Tung, 1993.

Cheng, S., Yen, B., Tang, W. 1993: Stochastic risk modeling of dam overtopping, In: Reliability and uncertainty analysis in hydraulic design, edited by Ben Chie Yen and Yeou-Koung Tung, 1993.

DVWK, 1997: Freibordbemessung an Stauanlagen - DVWK - Merkblatt 246/1997, Hrsg. Deutscher Verband für Wasserwirtschaft und Kulturbau (in German).

DVWK, 1999: Hochwasserabflüsse - DVWK - Schriften 124/1999, Hrsg. Deutscher Verband für Wasserwirtschaft und Kulturbau (in German).

Hable, O., Bergmann, H., Breinhälter, H., Krainer, R. 2000: Flood management at dams with consideration of the reservoir operation.- In: Proceedings of the International Symposium on Flood Defence, University of Kassel, Germany.

Pohl, R. 1997: Überflutungssicherheit von Talsperren - In: Mitteilungen des Institutes für Wasserbau und Technische Hydromechanik der TU Dresden, Heft Nr. 11, 1997 (in German).

Sackl, B.; Bergmann, H. 1986: A Bivariate Statistical Model and its Application to Flood Protection Projects – In: Proceedings of the International Symposium on Flood Frequency and Risk Analysis, Louisiana State University, Baton Rouge, USA 1986.

Sackl, B. 1987: Ermittlung von Hochwasser-Bemessungsganglinien in beobachteten und unbeobachteten Einzugsgebieten, Graz University of Technology, Dissertation 1987 (in German).

Simmler, H. 1977: Large dams in Austria, Revised and amended edition of the publication of 1964, Eigenverlag des Österreichischen Wasserwirtschaftsverbandes, Wien, in Kommission bei Springer-Verlag Wien – New York, Vienna 1977.

Vrijling, J. 1993: Development in probabilistic design of flood defenses in the Netherlands, In: Reliability and uncertainty analysis in hydraulic design, edited by Ben Chie Yen and Yeou-Koung Tung, 1993.

Fig. 9  Flow chart of the calculation procedure,Pohl, R. (1997), changed and completed  

Table 4.1  Main technical data of Salza dam, Simmler, H. (1977)

Table 4.2  Main technical data of Salza dam in Austria Simmler, H. (1977), Changed And Completed