Nonlinear analysis of flow recession curves

Transcription

1 FRIEND: Flow Regimes from International Experimental and Network Data (Proceedings of the Braunschweig Conference, October 1993). IAHS Publ. no. 221, Nonlinear analysis of flow recession curves H. WITTENBERG Fachhochschule Nordostniedersachsen, Department of Civil Engineering (Water Resources arid Environmental Management), D Suderburg, Germany Abstract Flow recession curves of rivers are commonly simulated by the equation of a single linear reservoir which is determined by one parameter, the retention constant. In reality, however, retention-discharge characteristics are hardly linear and recession curves can be approximated only sectionwise by linear reservoirs with retention "constant" values which increase while discharge decreases. In the present study a nonlinear reservoir algorithm with an exponential relationship between storage and discharge of the type S = a-çt is applied on given recession curves of gauging stations in different regions. The coefficients are determined by iterative curve fitting or a non linear regression procedure. Resulting computed recession curves show an excellent fit to the given data. In most cases the obtained power coefficient b is about 0.4, thus the storage-discharge relationship is far from being linear. The model parameters are related to catchment properties for regional analysis and physical interpretation. INTRODUCTION The flow recession curve at a river cross section is the discharge hydrograph of the basin during a rainless or dry period. Its analysis yields information on the retention characteristics of the basin and of groundwater storage and depletion. Commonly the algorithm of the linear reservoir is applied, e.g. Barnes (1939), Bako & Hunt (1988), Demuth (1989), as it is a simple model using only one parameter. The storage volume S of the reservoir is then assumed to be proportional to the discharge Q: S = k-q (!) with the retention constant k, which represents the retarding time of the system. The depletion of such a linear reservoir can be described by an exponential recession: Q t = <2 0 -exp(-f/ ) (2) where: Q t is the outflow at any time t in m 3 s" 1, Q 0 is the outflow at time t 0 in m 3 s" 1, and k is the retention constant with the dimension of time. As this recession curve becomes a straight line for the logarithms of Q values, the retention constant k can be determined easily from the slope of the plot of In Q versus t. Retention-discharge characteristics in nature, however, are hardly linear and recession curves can be approximated only sectionwise by linear reservoirs with different retention constant values, as demonstrated in Fig. 1. The value of the retention

2 62 H. Wittenberg LENNE RIVER AT KICKENBACH M d ^ DATA LINEAR RESERVOIRS S k - Q 10 k 19 d Fig. 1 Sectionwise approximation of a recession curve by linear reservoir. \ "constant" is smaller for the upper part of a recession hydrograph and increases continuously with the recession of the discharge. Thus, there would not only be one k- value to describe the recession, but an arbitrary number of different ^-values. In hydrological practice it is often assumed that recession flow is the superposition of different flow components, such as groundwater discharge and direct runoff, each being the outflow of a linear reservoir (Schwarze et al., 1989). A model of two parallel linear reservoirs was fitted to the data of Fig. 1. As there are now three parameters instead of one, i.e. the retention constants k x and k 2 and a partition factor for the two components, there is a much better fit to the given data as shown in Fig. 2. The variation coefficient between the data and the computed recession curve is only 9.4%. The assumption of linearity of reservoirs is a tool to facilitate separation of flow components. Nevertheless it is fiction. As in most reservoirs, storage and retention effects are in reality nonlinear, this must also be assumed for the groundwater and its discharge. GROUNDWATER AS A NONLINEAR RESERVOIR In a current investigation the aptitude of a nonlinear reservoir algorithm for recession curve modelling is studied. In this, the relationship between storage and discharge can be expressed by the equation: S = a-q b (3) with the coefficients a and b to be calibrated from observed flow recession curves. COMPUTATION OF THE FLOW RECESSION CURVE When the coefficients a and b are known, the recession hydrograph of the nonlinear

3 Nonlinear analysis of flow recession curves 63 LENNE RIVER AT K1CKENBACH \ - D DATA! COMPUTED! S 51 CT0.4 ; Fig. 2 Recession curve approximation by nonlinear reservoir. reservoir is computed by the same procedure as used for reservoir routing, e.g. Linsley et al. (1975). The change in storage AS in a time interval At is determined by the discharge during this time. At a time step i it can be approximated by the following equation: ASjAt = (S i -S i _ l )/At = -G2, + e M )/2 (4) or, rearranged so that all known terms are on the right-hand side: S t IAt + Q,l2 = S^/At-Q^/2 (5) As there is a fixed relationship between S and Q as given in equation (3), both values at the time i can be deduced from S J At + Q/2. Figure 3 shows the recession curve computed by the nonlinear reservoir model for the same discharge data used in Figs 1 and 2. It becomes evident that the adaptation of this curve is much better than that of any linear reservoir and that one curve fits the entire range of discharge data. This is not only because there is one more parameter in the storage-discharge relationship yielding more flexibility. Even compared with the model concept of two parallel linear reservoirs with 3 parameters as shown in Fig. 2, the present nonlinear model showed a better fit with a variation coefficient of only 4.5%, probably because it is a more realistic approach to the complex hydraulic processes. More examples of recession curves simulated by the nonlinear reservoir model are shown in Figs 4 and 5. CALIBRATION OF PARAMETERS OF THE NONLINEAR MODEL The coefficients a and b for recession curves of gauging stations can be calibrated by an iterative fitting procedure. Using a computer program the value for b is varied systematically, while the value for a is adjusted so that the computed discharge volume

4 64 H. Wittenberg FERNDORF RIVER AT KREUZTAL " DATA COMPUTED S 21.7 Q Fig. 3 Recession curve approximation by nonlinear reservoir. '- o 3 a n is equal to that of the given recession curve: a = E^-x + C,)-At/2i:(Qli-QD (6) For every set of the parameters a and b, the recession curve is computed by nonlinear reservoir routing as described above. The values of a and b for which the variation coefficient between the given and the computed curve is a minimum, are considered as representative. Another approach similar to that used for the linear reservoir, which can even be achieved graphically, may be applied as follows. Differentiation of equation (3) gives: PEGEL 22, 1984 SHANDONG a DATA COMPUTED S 23 Q a n -i w- -B D 8 O 0 Q G Fig. 4 Recession curve approximation by nonlinear reservoir.

5 Nonlinear analysis of flow recession curves 65 PEGEL 72, 1984, SHANDONG DATA 20 COMPUTED S Q a-3-a-a eg D a Fig. 5 Recession curve approximation by nonlinear reservoir. ds/dq = a-b-q b ' 1 ( 7 ) Considering the time interval At between i 1 and i and substituting AS by the mean discharge {Q t + Q^) At/2 yields: or: {Q^Q^)-Atl2{Q r Q^) =a-i-((g i + GU)/2) 6-1 (Q, + Qi-Ù 2 tolmq, ~ Qt-i) =a-b- (( + Qi.0/2)» (9) Since on a given flow recession curve, all values Q are known, equation (9) is of the type: 7 = c-x b ( 10 ) where Y is the value of the left side of the equation, while X = {Q t + Q ia )/2 and c = a b. The coefficients b and c with respect to a can be easily determined from given data by nonlinear (logarithmic) regression. The logarithmic transformation, however, causes a slightly sub-optimal approximation. Therefore, the iterative approach described above is recommended and was used for the analyses. PRELIMINARY RESULTS The coefficients a and b were calibrated using recession curve data from gauging stations in northwest Germany and in the Ping Fei Meng region, Shandong Province, China. Some resulting computed recession curves are shown in the Figs 3 to 5. They show an excellent fit to the given data, which cannot be achieved by the single linear reservoir model as it is demonstrated in Fig. 1. In many cases it shows a better fit than the model of two parallel linear reservoirs (compare Figs 2 and 3), though the latter one has three free parameters.

6 66 H. Wittenberg The obtained parameter values a and b of the nonlinear reservoir are listed in Table 1. The power coefficient b, would be 1 for a linear reservoir, but in most cases here it is about 0.4, indicating that the storage-discharge relationship is far from being linear. The value of b «0.4 found from present data appears to be typical. Therefore it was fixed for a second calibration of the values of the factor a u giving the relationship: The obtained values of a { are also given in Table 1. They are related to the basin area Table 1 Parameter values determined for recession curves, nonlinear reservoir S = a ÇJ?. No Gauging station A a b «l 1 Môhnesee-neuhaus Nichtinghausen Oeventrop Meschede Amecke Ambrock Kierspe Bôrlinghausen Bamenohl Hohenlimburg Menkhausen Hiippcherhammer Oberkirchen Kickenbach Betzdorf Kreuztal Weidenau Shandong Shandong Shandong Shandong A basin area in km 2 a,b parameters of storage-discharge relationship S = a g* a l factor of relationship with fixed power S = a r Q A Q discharge in m 3 s" 1 S storage volume in m 3

7 Nonlinear analysis of flow recession curves 67 and the regional basin characteristics. In Fig. 6 the average linear relationships of a t to basin area are shown for both German and Chinese data sets. The significantly lower a { values found for basins in China's Shandong province indicate that basin storage in that region is much lower than in the German basins analysed. As this is a current research project, only preliminary results on the recession curve analysis based on nonlinear storage-discharge relationships are presented here. Further work will be done for the hydraulic and hydrological interpretation of the results, their generalization and regionalization. FACTOR a RECESSION CURVE ANALYSIS, N.L.RESERVOIR S a * CT BASIN AREA in km2 Fig. 6 Factor a versus basin area. Acknowledgements The research work is sponsored by the Deutsche Forschungsgemeinschaft (German Research Association). REFERENCES Bako, M. D. & Hunt, D. N. (1988) Derivation of basefiow recession constant using computer and numerical analysis. Hydrol. Set J. 33(4), Barnes, B. S. (1939) The structure of discharge-recession curves. Trans. Am. Geophys. Un., S. Demuth, S. (1989) The application of the West German IHP recommendations for the analysis of data from small research basins. In: FRIENDS in Hydrology (Proc. Bolkesjo Symp., April 1989), IAHS Publ. no Linsley, R. K., Kohler, M. A. & Paulhus, J. L. (1975) Hydrology for Engineers. McGraw-Hill, New York, USA. Schwarze, R., Griinewald, U., Becker, A. & Frôhlich, W. (1989) Computer-aided analyses of flow recession and coupled basin water balance investigations.in: FRIENDS in Hydrology (Proc. BolkesjeSymp., April 1989), IAHS Publ. no. 187.

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