Coincidence of Flood Flow of the and its Tributaries

Abstract

Within the regional co-operation of the Danubian countries related to hydrologic issues, and in accordance with the UNESCO IHP program - phase IV and V, Yugoslavia was put in charge to realise Project No. 4 entitled "Coincidence of Flood Flow of the Danube River and its Tributaries". This project was completely realised by the Yugoslav experts. The experts from other Danubian countries provided assistance regarding data collection and the project review. The results of the study is presented in this publication, which also represents fourth – supplementary part of the monograph "The River Danube and Its Basin - A Hydrological Monograph", which was published in 1986 within the regional co-operation of the Danubian countries. The starting point for the analyses is that the classical approach to the risk assessment aims at determining the probability that a pre-selected value of the flood characteristic will be exceeded by the flood event. This is actually equivalent to determining of the return period of that event. The applied procedure consists of statistical analyses of hydrological data on nearby gauging stations. This approach, from the point of view of engineering, gives satisfactory results in accomplishing a large number of tasks. In particular, it appears to be the most frequently applied approach when dealing with reasonably simple river systems, i.e. when the river does not receive large tributaries within the examined reach. However, when the system encompasses of large , the described approach cannot give a reliable flood characteristics estimate. Namely, for various reasons flood wave geneses in two catchments are different, so that flood peaks do not occur simultaneously at both the main river and its tributary. Yet, the flood wave of one river may greatly affect water level of the other stream. In addition, hydrological data are usually collected at gauging stations located outside of the reach where influence can be realised, so that the interaction of one river upon another is not accounted for by the gathered data. If this is the case, reliable estimation of flood events coincidence on both the main river and its tributary becomes a vital issue. To that end, a methodology to be presented in the publication pertains to the estimation of the coincidence of flood flows at the main river and its tributary. The term "coincidence" is used to denote the probability of simultaneous occurrence of the two random variables, X and Y, which denote random events at the main stream and its tributary. According to the theory of statistics, a two-dimensional probability distribution function of the normally distributed bi-variate random process, X and Y is defined as follows:

 2  1  1 ()x −µ 2ρ(x −µx )(y −µy ) ()y −µy  f ()x, y = ⋅exp−  x − +  (1) 2 21− ρ 2 σ 2 σ σ σ 2 2π ⋅σ x ⋅σ y ⋅ 1− ρ  () x x y y 

The symbols in expression (1) stand for the following:

X and Y random variables (flood flow characteristics at the main river and its tributary or at two adjacent profiles of the main river); x and y simultaneous realisation of the random variables X and Y respectively; µx and µy the expected values of X and Y variables; σx and σy the standard deviations of X and Y variables; ρ the coefficient of correlation between X and Y

In this publication the procedure to assess the coincidence of flood flows i.e. to obtain the probability, P(X ≥ x I Y ≥ y), that two flood events will simultaneously be exceeded was outlined in details. The subsequent analysis of the flood wave coincidence on the main river and its tributary is based upon the bi-variate statistical distribution of the following variable combination: a) Maximum annual flood on the main river, X, and maximum annual flood on the tributary,Y; b) Maximum annual flood of the main river, X, and the corresponding (simultaneous) flood of the tributary, Ycor; c) Maximum annual flood on the tributary, Y, and the corresponding (simultaneous) flood on the main river, Xcor.

It should be noted that in the three preceding paragraphs the term flood is used to denote various flood wave characteristics, such as: flood peak discharge, flood wave volume, flood wave duration, and lag-time between the occurrence of two flood waves. In a similar manner instead of consideration of the main river and tributary, the flood characteristics at two profiles on the main river - (upstream and downstream from the tributary) can be investigated. The lines of equal probabilities (bi-variate joint probability density function), as well as the lines which define the exceedance probability (cumulative distribution function) of the mentioned variables/characteristics of floods are the result of a coincidence calculation of the variable combinations in question. The computing scheme for the cumulative exceedance probabilities is given by the following equations:

∞∞ P[]X > X 1;Y > Y1 = ∫∫g()X ,Y, R dxdy XY11 ∞∞ P[]X > X 1;Ycor > Y1 = ∫∫g(X ,Ycor , R)dxdycor (2) XY11 ∞∞ P[]X cor > X 1;Y > Y1 = ∫∫g()X cor ,Y, R dxcor dy XY11

Flood events are considered to be the flood waves in which the peak flow exceeds pre-defined magnitude. That magnitude can be taken from the flow duration curve or selected in some other way. As already stated, the following characteristics of floods can be analysed:

1. Extreme value (flood peak) – Qmax; 2. Flood hydrograph volume above the pre-defined flow – W; 3. Duration of flood wave above the pre-defined flow – T; 4. Time lag between the occurence of flood events - τmax

Depending upon the overall hydrological regime, the pre-defined value can be exceeded more than once during a single calendar year. It implies that the number of previously defined events change from year to year. Hence, the annual frequency of floods appears as an important flood characteristic to be analysed in due course. In addition, the period of year when the flood occurs is of interest as well. When analysing the flood coincidence of the main stream and its tributary or of the main river upstream and downstream from the tributary, attention should be paid to flood wave genesis. In connection with this, the following characteristics are important: (snow melt, heavy rainfall, concentration time etc.). With regard to flood coincidence analyses, it is necessary to consider gauging stations immediately downstream and upstream from the tributary. The most interesting characteristics to be included into coincidence analyses are described in Fig. 2.1-2.4. The ensuing text gives a rather extensive list of flood variables to be discussed. The most interesting points where flood coincidence should be assessed along the Danube river are those where some significant tributaries merge the main watercourse. To that end, the following nine tributaries were chosen: the Inn, Enns, , , Tisa, , Velika Morava, and . The analyses are applied to those sections of the Danube River that are bounded by the adjacent gauging stations where long-term observations are available. In another words, the sections consist of the river reach from the nearest upstream gauge to the first downstream gauging station, as well as the first gauging station at each tributary. In accordance with the available data, the following sections on the Danube River have been examined:

1. The Danube river (Hofkirchen - Achleiten) - the Inn river (-ingling) 2. The Danube river ( - Krems) - the Enns river (Steyr) 3. The Danube river ( - ) the Morava river (Moravsky Jan) 4. The Danube river (Bezdan - Bogojevo) - the Drava river (Donji Miholjac) 5. The Danube river (Bogojevo - Slankamen) - the Tisa river (Senta) 6. The Danube river (Slankamen - Pancevo) - the Sava river (Sremska Mitrovica) 7. The Danube river (Pancevo - Veliko Gradiste - the Velika Morava river (Ljubicevski Most) 8. The Danube river (Vadu Oii - Ceatal Izmail) - the Siret river (Lungoci) 9. The Danube river (Vadu Oii - Ceatal Izmail) - the Prut river (Tchernovtsy)

Coincidence analysis of the most important flood hydrograph features of the Danube river and its tributaries confirmed that flood wave genesis is very complex within the Danube basin. In particular, this complexity was emphasised at the points. Evaluation of characteristics of flood hydrographs, their interaction and simultaneousness, plays an important part in the design of hydraulic structures within the flood defence system from the point of view of both the system development cost and a provided degree of protection. The results obtained trough the coincidence analyses may be of multifold importance. Firstly, they can be used for evaluation of statistical significance of coincidence of various flood wave characteristics, and consequently for comprehensive assessment of the flood situation on the Danube river and its tributaries. Practical significance of these results is that, in the case when coincidence of floods in the main river and its tributaries is not significant, a less expensive flood protection system can be developed based on the classical uni-variate approach to flood analyses. Naturally, that system must provide the required degree of safety. Secondly, the above proposed approach gives an opportunity to determine designed parameters of optimal combinations of the examined variables with respect to both the system development cost and safety of hydraulic structures. Also, the methodology can be used to calculate the designed water stage of the confluence where hydrologic records are missing for one gauging station on the main river (either upstream or downstream from the tributary). In addition, the above-presented analyses can prove advantageous under the circumstances when a flat area should be protected from two adjacent streams. The protection system, based on the classical flood estimate procedure of each river (uni-variate analyses), is often expensive and difficult to develop since the spatial constraints are usually very rigid and the required degree of protection is high. The proposed methodology produces results, which allow for integral examination of flood protection from both rivers taking into account the coincidence of flood waves. The basic assumption associated with this solution is that there is a possibility to construct (one or more) connecting channels between two adjacent streams with or without gates. If coincidence is significant, a part of flood flow can then be directed from one river to the other, and vice versa. The approach can be used for overall evaluation of flood flow characteristics. In addition, upon comprehensive examination of various flood hydrograph parameters (peak discharge, flood volume and flood duration of either the same or different probability), the "designed hydrograph" can be defined at each input/output profile (gauging stations). In that way the most "unfavourable" shape of the designed flood can be defined for the purpose of developing a flood protection system within the confluence. The outlined results imply that the most significant coincidence on the Danube river and its tributaries can be expected when flood events are examined on the upstream and downstream profiles of the main river. Somewhat less significant coincidence is associated with the flood event realised on the main river downstream profile and the tributary. Coincidence of flood waves occurring at the main river upstream profile and the tributary in most cases is statistically insignificant. Taking into account the above it can be concluded that the results obtained in this study have a large practical value. They can be used to determine input parameters for the design of flood protection structures on key sectors along the Danube rivers, i.e. in the vicinity of tributaries. The implementation of these results leads to development of the optimal size of flood protection system, which in addition assures higher safety for the Danube's lower reaches. Finally, it should be emphasised once again, this study should give an incentive to professionals and government authorities dealing with flood protection to pay appropriate attention to flood coincidence along the Danube river. The use of the described approach can greatly help in avoiding the waste of precious flood protection funds, while providing at the same time the required degree of safety to the flood-prone areas.