Evaluated data fro» basins in North-Rhine Westphalia. type

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1 213 REGIONAL ANALYSIS OF FLOW DURATION CURVES Case Studies on River Basins in North- West Germany Hartmut Wittenberg Fachhochschule Nordostniedersachsen Department of Civil Engineering D 3113 Suderburg, Federal Republic of Germany ABSTRACT Flow duration curves or nuabers indicate on how «any days of an average year discharges through a cross section of a river are reached or exeeded. They are an important information for the planning of water resources projects where storage is not substantial, e.g. for the assessment of run off river potential, availability of water and the concentration of pollution. In «any countries flow duration curves are therefore derived and published for gauging stations with sufficient records. Planning engineers however feel often difficulty to derive duration curves for ungauged river basins. A research study had the following objectives! Parameterization of duration curves, i.e. choice of a suitable mathematical equation and determination of parameter values for gauged river basins. regionalization, correlation between shape parameters and basin properties to establish regional regression equations for the synthesis of duration curves for ungauged basins. analysis af the influence of urbanization on shape and parameters of flow duration curves. Table 1 Evaluated data fro» basins in North-Rhine Westphalia basi n no. o-f stations type area km 2 Upper Ruhr Upper Si eg Emscher hills, -forests hills, -forests urban Good results were obtained by using the three parateter gasma function to model density functions of the duration curves. As a siaplification, duration curves of smaller non urbanized basins in the region may be described by the «ass curve of the linear reservoir, thus depending on the parameter K only. This parameter is related to basin properties. Its value corresponds to mean flow of the considered basin. THE DATA BASE In order to allow a regional analysis of duration curves, data of 24 basins in the regions were collected and evaluated. Mean flow QM, mean annual basin precipitation and the duration curves of diurnal flows were determined when ever possible for the common time intervall For some gauging stations this data period is shorter. Morphometric data were derived from topographical maps 1: Average annual basin precipitation was determined from 64 stations by the «ethod of Thiessen polygones.

2 214 Some physical and morphoaetrical properties are given in Table 2, where A = basin area in km 2 Sw = mean (weighted) slope of the «ain channel in X 1 = length o-f the main channel in km hn = mean geodetic altitude o-f the main channel in m msl hp = average annual basin precipitation in mm The river basins controlled by gauging stations 1 to 16 belong to the upper Ruhr or to the tributary rivers Lenne and Volme. Basins 17 to 23 are situated in the hydro- clieatologically very similar basin o-f the upper Sieg river. Both regions are hilly and mainly covered by forests and agricultural areas. Moderate influences «ay occur due to soae smaller reservoirs and urban areas. As characteristics are similar, data o-f stations 1 to 23 were used for regional analysis. The Emscher basin with gauging station 24 has quite different characteristics. Situated in the center of the industrial Ruhr district it is essentially built up and urbanized. The portion of impervious areas with its iapact on flood runoff increased from 8 X in 1952 to about 20 X in the present (Wittenberg, 1980). The influence of this develop«ent on water balance and duration curves is overlapping with changing contribution of sewage water and water punped fro» mines. The identification of this influence will be the aim of further research work. THE MODEL EQUATION The density function of diurnal average flows 0 through the cross section of a river is «iathe«atically the derivative function of the duration curve. It has a curved curved shape with an origin at Bo > 0 and a theoretically open end, as shown in Fig. 1 by an example. After trying some other approaches the three parameter ga««a distribution was chosen as the Model equation. This function corresponds to the Pearson III distribution or the cascade of linear reservoirs used for hydrograph synthesis, shifted by a fix «iniaun value. The duration curve is obtained as the integral of the density function, thus T = Q(T) (Q(t)-Qo)' s (n-l) Kf r <n) Q=0 * exp(-<q<t)-qo)/k) ( 1 > wi th T time of non exeedence as fraction of the average year Q(T) discharge not exeeded during this time in «3/s Qo minimum flow in «3/s, shifting constant K flow parameter in «3/s (for the cascade of reservoirsi retention constant) n shape parameter (number of linear reservoirs of the cascade) without dimension r gamma function

3 215 Table 2 Physical properties of studied basins No QM A Sw hw hp GAUGING STATION MOHNESEE-NEUHAUS NICHTINGHAUSEN OEVENTROP MESCHEDE AMECKE ECKESEY AMBROCK STEFANSOHL KIERSPE BORLINGHAUSEN BAMENOHL HOHENLIMBURG MENKHAUSEN HÛPPCHERHAMMER KRAGHAMMER OBERKIRCHEN ALSDORF BETZDORF KAAN-MARIENBORN NIEDERDIELFEEN KREUZTAL WEIDENAU NAUHOLZBACH OBERHAUSEN

4 216 ^Q/m 3 / 50 H 40 A Emscher A 10 DURATION CURVE 0 DAYS Fig. 1 Flow duration curve -for the gauging station Oberhausen/ Enscher river and density function Duration curves are thus deternined by three parameter», i.e. Qo, K and n. As the integral of equation 1 cannot be solved «atheaatically for arbitrary values of n, this can be done numerically by sunning up for stall increments of Û with the help of a computer. As an alternative the procedure comuonly used to compute values Q(T! for the Pearson type III distribution can be applied, i.e. the general frequency equation Q(T) QM k * SQ Qo < 2 ) with SQ k CS TR K t n in n3/s («ean value or first nouent) 4 (K t QH) in «3/s (standard deviation) frequency factor taken fro» tables for the Pearson type III distribution as a function of the skew coefficient CS and the recurrence interval TR 2 / 4n 365 / (366 - T) with T = number of days of non eneedence in the average year

5 217 Values of the model parameters K, n and Qo were computed -for the duration curves o-f all gauging stations using a nonlinear leastsquares curve fitting procedure. The computer program written by the author has been applied for hydrograph analysis and is described in earlier publications (Wittenberg, 1980). Fig. 2 shows as an example a duration curve derived -from recorded discharge data and the calibrated model function. Fig. 2 Flow duration curve, gauging station Qberhausen/ Emscher river, and model function Parameter values K and n computed for the duration curves of the 23 gauging stations are listed in the third and fourth column of Table 3. It shows that, especially for the smaller basins, values n are very close to 1. Furthermore the minimum value Qo for these basins revealed to become practically 0. As the parameters K and n are correlated through the statistical moments, Equ. 1 could be simplified setting n = 1 and Qo = 0! T = 1 exp(-q<t)/ki> ( 3 ) This equation corresponds to the integral functon of the linear reservoir. The duration curve is thus related to one parameter Kl only, with the dimension m3/s. Optimal values «1 were calibrated for the 23 gauging stations. They are given in the fifth column of Table 3.

6 218 Table 3 Parameter values determined for discharge duration curves No QM "K n Kl Klcoml Klcom GAUGING STATION MOHNESEE-NEUHAUS NICHTINGHAUSEN OEVENTROP MESCHEDE AMECKE ECKESEY AMBROCK STEPHANSOHL KIERSPE BORLINGHAUSEN BAMENOHL HOHENLIMBURG MENKHIUSEN HÔPPCHERHÂMMER KRAGHAMMER OBERKIRCHEN ALSDORF BETZDORF KAAN-MARIENBORN NIEDERDIELFEN KREUZTAL WEIDENAU NAUHOLZBACH

7 219 Differences between original duration curves and those recalculated by equation 3 are rather small. The variation coefficient computed as an index of deviation has an average value of 5 X. An example of an original and recomputed duration curve is shown in Figure 3. REGIONAL ANALYSIS Parameter Kl is mainly related and nearly proportional to the size of the basin area A. To detect the dependences to the other basin parameters (Table 1), regression analysis was carried out with the specific values Kl / A, giving thus equal «eights to all basins. Multiple linear regression analysis between Kl / A and basin area A, length o-f main channel 1 and mean annual basin precipitation hn yielded the following equationst Klcoml = A/ t < S A % 1 ) ( 4! variations coefficient 18.5 X between original values Kl correlations coefficient and recomputed values Klgerl and Klcom2 = A/100000*< * A t *hN) ( 5 ) variations coefficient 14.5 X between original values Kl correlations coefficient and recomputed values Kiger2 Computed values Klcoml and Klco«2 are listed in Table 3. It is obvious that the values Kl whether original or recalculated are very close to mean discharge QH. This is plausible as Kl in Equ. 3 is the first statistical moment of the density function, i.e. the mean value. Generally it can be stated that Kl ~ MQ < 6 ) and at least for river basins of comparable size in the studied region («responding to Equ. 3) results in T s 3è5 t (i - exp(-q<t)/qm>) days ( 7 ) This equation allows the estimation of duration numbers or curves for a basin in the region of which only «orphoaetric properties or «tean flow are known, as demonstrated by the following examples Given! A basin with a mean discharge QH of 1.3 mz/% or a value Kl of i # 3 {1)3/5, computed by regional equation 4 or 5. How nanny days of the average year is discharge less than 2 «3/s? Solutions T = 365 * ( 1 - exp( -2 / 1.5 )) = 269 days

8 220 Fig. 3 Flow duration curve for the station Nichtinghausen/ Henne river, original and computed with Equation 3 ACKNOWLEDGEMENTS Data used for this investigation were proportioned by the Staatliches Amt fiir Wasser- und Ab-fallwirtschaft (StAWA) Hagen and the Enschergenoesenschaft (Emscher river authority) in Essen. The work was sponsored by the Deutsche Forschungsgemeinschaft (German Research Association). REFERENCE Wittenberg, H. (1980) A parallel cascades model to predict the effects of urbanization on watershed response. Studies and reports in hydrology 28, , IHP, UNESCO, Paris.