Low Flow Estimation for Several Durations and Return Periods Using Mean Annual Flow Distribution

Low Flow Estimation for Several Durations and Return Periods Using Mean Annual Flow Distribution

Rodica MIC1, Monica GHIOCA1, Gilles GALÉA2

1 National Institute of Hydrology and Water Management 97 Şos. Bucureşti – Ploieşti, Sector 1, Bucharest cod 013686, Romania,

E-mail: [email protected] : [email protected]; 2 Cemagref-Lyon, Unité de Recherche Hydrologie Hydraulique,

3bis quai Chauveau 69336 Lyon, Cedex 09 France, E-mail: [email protected]

Abstract: The characteristics of low flow, in Timis-Bega hydrographical area, are presented in this paper. The low flow analysis will be done using the flow-duration-frequency (QdF) modelling, establishing for each analyzed sub-basin a local modelling associated to a Weibull Law with two parameters. Local modelling requires the knowledge of two local descriptors: the characteristic times of the low flow ∆e and the median value of annual daily minima quantils. Based on these models, a dimensionless regional model for low flow was established. On other hand, the theoretical distribution of mean annual discharges was drawn for each analysed sub-basin using also Weibull Law with two parameters. Theoretical distribution comparison of dimensionless regional models corresponding to low and mean flow shows that these are almost identical. Thus a low flow estimation for several durations and mean return periods was tested, based on mean flow characteristics. The utilization of this method to the ungauged sites supposes the knowledge of two parameters of Weibull regional distribution corresponding to annual mean discharges, as well as an estimation of two local descriptors of low flow based on the river basin characteristics.

Key words: low flow, QdF modelling, mean low, regionalisation.

Introduction Precursory studies in the Timis-Bega River Basin had shown the adequacy of the law of Weibull in 2 parameters to represent the temporal and space changeability of the low flow and mean annual discharge of the sub-basins. On the other hand we had observed a weak distinction between the local dimensionless distributions established for the normal annual discharges and low flows. Our intention is to see, if the use of distribution established for the normal annual discharge of one of sub-basin can be used for the estimation of low flow with different durations.

General context and available input data series of Timis-Bega River Basin. Located in the Oust of Romania the Timis and Bega Rivers drain respectively 7489 km2 and 5248 km2. The mean altitude (Zm) is of 415m for the basin of Timis and of 236m for the basin of Bega, it culminates in 2000m about in the southeast of the drainage basin. Of the network observations we used in this study only 19 gauging stations (14 sub-basins in the Timis River Basin and 5 sub-basins in the Bega River Basin, figure 1) among which low flow and normal annual discharge are not influenced (table 1). The normal annual discharge vary from 7 l/s/km² at 41 l/s/km² and the daily mean low discharge vary from 2 l/s/km² at 10 l/s/km². The diversified flows are explained, between others, by the size of sub-basins, the orographical context, the nature and occupation of the soil of these 19 sub-basins.

Figure 1. Location of the gauging stations of the analysed sub-basins

Table 1 Presentation of the 19 discreet sub-basins of Timis-Bega

Observations S Zm Sus-basins Code

period years (km2) (m) Bega - LUNCANI 50105 1976-2001 26 73,5 739 Bega - FAGET 50110 1974-2001 28 474 470 Bega - BALINT 50115 1970-2001 32 1064 329 Sasa - POIENI 50205 1966-2001 36 80 763 Gladna - FIRDEA 50301 1980-2001 22 57 456 Timis - TEREGOVA 53105 1953-2001 49 167 906 Timis - SADOVA 53110 1971-2001 31 560 936 Timis - CARANSEBES 53115 1974-2001 28 1072 769 Timis - LUGOJ 53125 1970-2001 32 2706 665 Raul Rece - RUSCA 53205 1953-2001 49 163 1157 Fenes - FENES 53305 1964-2001 38 125 973 Sebes - TURNU RUENI 53405 1964-1998 35 122 819 Bistra - VOISLOVA BUCOVEI 53505 1951-2001 51 232 886 Bistra - VOISLOVA 53510 1963-2001 39 404 827 Bistra -OBREJA 53515 1982-2001 20 863 880 Bistra Marului - POIANA MARULUI 53605 1953-2001 49 79 1442 Sucu - POIANA MARULUI 53705 1953-2001 49 77 1434 Nadrag - NADRAG 53805 1963-2001 39 35 742 Golet - GOLET 55105 1982-2001 20 41 751

Data extraction and treatment For the chosen basins, the frequency study of the low flow was performed from series of daily discharge. The variable of different durations d used in the hydrologic analysis of low flow is VCNd - average discharge (volume) on a continued duration d, minimal in season (figure 2) (Mic, 1998; Galéa & al., 1999). This variable associated of the duration d, notated vcnd, is extracted by the daily discharges series for each analysed sub-basins. In the case of our study we shall be studying more precisely for d = 1, 3, 6, 10, 30 and 90 days.

time

d

Q (m3/s)

Figure 2. Characteristic discharges of low flow

The adjustment of statistical laws on these series requires the attribution of an experimental frequency in every value of the series. These statistical laws represent the theoretical frequency evolution of studied events. After sampling, for a pouring basin given with a number n of years by observations they acquire stocks of vcnd discharges is appointed an experimental frequency or a mean return period. The samples of vcnd are sorted by descending order and an indication of rank i is appointed to every value. So, the weakest value sees itself allocating the rank i=1 and the strongest value the rank i=n (where n represent the number of years with observations). Theoretical distribution is represented by a law of Weibull with 2 parameters (1). This is a simple law, adapted to the study of low and mean discharges (Galéa & Canali, 2005).

L/1FLFL Uβ)U(Q α= (1)

Where: UF = - ln (1-F), with F : not exceeding frequency; QL(UF) : theoretical quantil deducted from the observed discharges (m3/s) This law may the computation for a duration d, of quantils (Mic, 1998; Galéa & al., 1999; Galéa & al. 2000):

- dry (when F<0.5, T=1/F) ; - humid (when F>0.5, T=1/1-F) ; - année moyenne (F=0.5, T=2)

On the other hand we used the values of mean annual discharge for the observed period at the analysed gauging stations.

Local modelling A local QdF model of low flow and a local model for mean annual discharge have been calibrated for each of the analysed sub-basin. Concerning the low flow local model, this has obtained using all the dimensionless series, for the analysed durations (1, 3, 6, 10, 30 and 90 days). The dimensionless values are computed by division with the median values of each series. Parameters α and β expressing function (1) of the dimensionless curve, can be computed using the series of mean dimensionless values using an optimisation of error method. An example with the dimensionless values for different duration and the theoretical curve of the local model is presented in the figure 3.

Bega - LUNCANI (S = 73.5 km²)

0

0.5

1

1.5

2

2.5

0 1 2 3 4Up

Q (m

3 /s)

1 3 6 10 30 90days W2 Loc.

Dimensionless local law calibration di i ll lid

Figure 3. Calibration of dimensionless local law for Bega-Luncani River Basin

On the other hand the parameters α and β of function (1) are computed for the dimensionless values of series of mean annual discharges (Galea & al., 2002). Then, we compare the functions calibrate for low flow and for mean annual discharge (figure 4) we can observe that these functions are very close, from all the analysed sub-basin.

Bega - LUNCANI (S = 73.5 km²)

0

0.5

1

1.5

2

0 1 2 3 4Up

Q /Q

med

ian

Local Model of Low Flow Local Model of Annual Discharge

Dimensionless local laws

Figure 4. Bega-Luncani River Basin - Comparison between

dimensionless local laws of low flow and normal annual discharge distribution

Therefore, we can use the parameters of α and β of function (1) computed for the dimensionless values of series of mean annual discharges ( QA

Lα and QALβ ) to estimate the low flow, with different duration and

frequency (2).

+⋅= = 1∆e

1dvcnUβ)U,d(V 1d/1

FQALFL

QAL

-α (2)

with: QAL/1

FQAL

αUβ Theoretical distribution of local dimensionless of mean annual discharge; ∆e - Characteristic times of low flow;

1dvcn = - Daily discharge of low flow.

The characteristic times of low flow (∆e) is a parameter linked up with the dynamics of low flow. This allows deducting, the theoretical distribution consolidated by the daily discharge of low-flow ( 1dvcn = ), with the durations d=3 days, 6 days, 10 days, 30 days and 90 days (Galea & al., 2002, Galéa & Canali, 2005). In figure 5 are presented, for example, the distribution of low flow with the duration d=10 days and the observed values with the same duration.

Bega - LUNCANI (S = 73.5 km²)

0

0.5

1

1.5

0 1 2 3 4Up

Q (m

3 /s)

vcn 10 day obs vcn 10 day local mod QA

Figure 5. Local modelling of low flow using the local dimensioneless mean annual discharge

distribution

In order to validate the local modelling we have computed the relative root mean square error between the observed values and the estimated local values. The relation after which the rRMSE is computed is (3):

∑

∑∑s

1mm

n

1i

2

Fobsi

Felmod

iFobsi

s

1i

n

)U(Q)U(Q)U(Q

100)%(RMSERE i

ii

=

==

−

= (3)

where: s and n are respectively the number of sub-basins and the number of durations taken into account at each site. The error, for each duration and for both dry and humid domain, are presented in figure 6.

05

101520

2530

d=1 d=3 d=6 d=10 d=30 d=90days

RR

MS

E (%

)

Dry domain Humid domain

Figure 6. Relative root mean square error for local modelling

Regional modelling The regional model is obtained using the dimensionless distributions of local models

(Mic & al., 2002; Galéa &al., 2006). In addition, two sub-sets have been constituted: a calibration set (12 sub-basins) and a validation set (7 sub-basins).

The regional parameters of Weibull distribution αR and βR are computed by the mean of the values corresponding of the calibration set.

Figure 7 presents obtained curve for the regional distribution and it can be seen that the dispersion between the local curves is small, indicating a good homogeneity of the region.

Low Flow - Regional Modellisation

0.00.51.01.52.02.53.0

0 1 2 3 4 5U F

Q/Q

med

ian

Low Flow Local Model Low Floow Regional Model

QA - Regional Modellisation

0.0

0.51.0

1.5

2.02.5

3.0

0 1 2 3 4 5U F

QA

/QA

med

ian

QA - Local Model QA - Regional Model

Figure 7. Low flow and mean annual discharge regional modelling of Timis–Bega River Basin

Then, we compared the regional distribution of low-flow and mean annual discharge (figure 8) these distribution are very close. As a consequence, it was considered the regional model of mean annual discharge could be used to estimate the low-flow for different duration and frequency.

Dimensionless Regional distribution

0.00.5

1.01.52.0

2.53.0

0.0 1.0 2.0 3.0 4.0 5.0

U F =-LN(1-F)

Q/Q

med

Low Flow QA

Figure 8. Low flow and mean annual discharge regional modelling of Timis–Bega River Basin

To reveal the fact that in the ensemble of the analysed sub-basins the regional model is representative, the relative root mean square errors (3) between the quantiles of the local and regional model were computed. The vcnd values, for 5 and 10 years return period in dry and humid domain are used.

Figure 9 presents the relative root mean square errors in the case of dry and humid domain for both calibration and validation sub-basin. It can be observed that, this error is not more then 12 %.

Regional modelling

8.87.8

11.8

4.9

02468

101214

Calibration Validation

RR

MS

E (%

)

Dry domain Humid domain

Figure 9. Relative root mean square error for regional modelling

Estimation of normalising parameters at ungauged basins For the estimation of low-flow at ungauged basins, the knowledge of normalising variables of

the model, daily discharge of low flow ( 1dvcn = ) and characteristic times of low flow (∆e) is necessary. Relations involving physiographic and climatic characteristics of the basins were established

(Galéa et al., 2006). The tested characteristics were: river basin area (S), the river length (L), the river

slope (Ir), the mean basin altitude (Hm), the river basin slope (Ib).

Conclusions The results of the modelling, using the flow-duration-frequency associated to a Weibull Law

with two parameters, of the mean annual discharge and the low-flow of the basin Timis- Bega show a great similarity on the whole. Theoretical distribution of dimensionless regional models corresponding to low and mean annual flow are almost identical. The low-flow for several durations and mean return periods was estimated, based on mean annual flow characteristics. Local modelling requires the knowledge of two local descriptors: the characteristic times of the low flow ∆e and the median value of annual daily minima quantils. Based on these models, a dimensionless regional model for low flow was established. The utilization of this method to the ungauged sites supposes the knowledge of two parameters of Weibull regional distribution corresponding to annual mean discharges, as well as an estimation of two local descriptors of low flow based on the river basin characteristics.

References

Galéa, G., Mic, R., Chaput, N., 1999 : Prise en compte d'observations locales épisodiques pour un meilleur usage opérationnel des modèles débit-durée-fréquence d'étiage au sein d'un réseau hydrométrique. 5ième rencontre Hydrologique Franco-Roumaine "Suivi intégré des eaux continentales". Lyon, 6 - 8 septembre 1999

Galéa, G., Javelle, P., Chaput, N., 2000 : Un modèle débit-durée-fréquence pour caractériser le régime d’étiage d’un bassin versant. Revue des Sciences de l’Eau, 13(4) : 421-440.

Galéa, G., Canali, S., 2005 : Régionalisation des modules annuels et des régimes d’étiage du bassin hydrographique de la Moselle française : lien entre modèles régionaux. Revue des Sciences de l’Eau, 18(3) : 331-352.

Galéa G., Mic R., Ghioca M., 2006 : Modélisation statistique des modules annuels et des étiages du bassin roumain du Timis-Bega, similitudes régionales avec la Moselle française. Soumis à la Revue des Sciences de l’Eau.

Mic, R., 1998 : Régimes d’étiage Bassins de la Moselle (France) et de l’Arges (Roumanie), Modèles de synthèse QdF, l’usage opérationnel sur bassin versant non observe. CEMAGREF – Lyon.

Mic R., Galéa, G., Javelle, P., 2002 : Modélisation régionale des débits de crue du bassin hydrographique du Cris (Roumanie) : approche régionale classique et par modèles de référence. Revue des Sciences de l’Eau 15(3) : 677-700.