Debi Süreklililik Çizgisinin Matematik Modelleri

Abstract

Bu çalışma akarsulardaki akımların süreklilik çizgileri ile ilgilidir. Debi süreklilik çizgisi su kaynakları mühendisliğinin birçok dalında uygulama alanı bulmasına karşın bu konu üzerinde yapılmış çalışmalar konunun önemine oranla oldukça düşük sayıdadır. Debi süreklilik çizgisinin matematik modelleri ile ilgili bir çalışmaya ise literatürde rastlanmamıştır. Bu çalışmada öncelikle debi süreklilik çizgisinin su kaynakları mühendisliğindeki kullanımı ile ilgili ayrıntılı bilgili verilmiştir. Debi süreklilik çizgisini etkileyen bileşenler belirlenmiş ve bu bileşenlerin debi süreklilik çizgisi üzerindeki etkisi incelenmiştir. Çalışmada debi süreklilik çizgileri, stasyoner (yıllık) ve periyodik (aylık ve günlük) akımların debi süreklilik çizgileri olarak ayrı ayrı incelenmiştir. Stasyoner akımlarda otokorelasyon katsayısının, periyodik akımlarda ise farklı sayıda harmonik kullanılması durumlarında 9 faz açısının debi süreklilik çizgisi üzerindeki etkisi incelenmiştir. Debi süreklilik çizgisinin akım modelleri ile ilgisi incelenmiş ve bir akım modeli verildiğinde süreklilik çizgisini elde etmek için algoritmalar geliştirilmiştir. Daha sonra süreklilik çizgisinin analitik yoldan elde edilmesi ile ilgili yöntemler geliştirilmiştir. Bu yöntemler, olasılıkların ortalaması ile debi süreklilik çizgisinin elde edilmesi ve Z=XY değişkeninin dağılımını kullanılması ile debi süreklilik çizgisinin elde edilmesidir. Türkiye'deki bazı nehir akım ölçüm istasyonları için, gerek akım modelleri kullanılarak simülasyon yolu ile gerekse analitik metodlarla elde edilen debi süreklilik çizgileri, tarihi debi süreklilik çizgileri ile karşılaştırılmıştır. Sonuçta elinde tarihi nehir akımları ile ilgili bazı temel parametre değerleri bulunan su kaynakları mühendislerinin debi süreklilik çizgisi değerlerini rahatlıkla elde edebilmeleri için bazı grafik ve tablolar verilmiştir. Yapılan çalışmanın, akım gözlemleri yetersiz olan veya bulunmayan kesitlerde süreklilik çizgilerinin elde edilmesi gibi bir çok su kaynakları mühendisliği alanında faydalı olacağı umulmaktadır.

This study is related to the duration curves of river flows. The flow duration curve plots cumulative frequency of discharge, that is, discharge as a function of time that the discharge is exceeded. Though flow duration curve is widely used in water resources engineering the number of the studies related to this subject are low compared with the importance of the subject. The majority of the realized studies are about the utilization domains of duration curves in water resources engineering and some of them cover the determination of the regional flow duration curves. A study related to the mathematical models of the flow duration curve is not found in the literature. In this study firstly detailed information is presented about the utilization areas of the flow duration curves. The components of the flow duration curve are determined and their effects on the duation curve are examined. The relation between flow duration curve and flow models is investigated and algorithms are developed to obtain a duration curve as a flow model is present. Following it methods are developed to obtain flow duration curve analytically. Then for some gauged Turkish river flow stations duration curves obtained analytically are compared with the original observed duration curves and the duration curves based on the simulation using the flow models. FLOW DURATION CURVE OF THE STATIONARY SERIES Annual mean flows are stationary and they do not contain periodic component. Therefore they have only the stochastic component. Annual flows can be presented with the 1. order Markov model as follows: X| = n X,.i + E where ri is the 1. order auto correlation coefficient while xt.i and xt are the annual flow values succeeding each other, e is the stochastic component with mean "0" and variance "1- ri2 ". To investigate the effect of the auto correlation coefficient, flow duration curves are obtained for the simulated series based on the 1. order Markov model for various auto correlation coefficient values. It is seen that there is no difference between the shapes of the curves. As a case study flow duration curve of the annual flows in Missouri river in U.S.A. having a high auto correlation coefficient value is compared with the flow duration curve of the standard normal variables obtained from the Normal Distribution table. Again here both curves agree well with each other. It can be concluded that the auto correlation coefficient has no effect on flow duration curve. For the stationary flows the exceedance probability of a flow value x can be found by subtracting the corresponding cumulative distribution value from 1. Accordingly it can be written as P(x < xo) = 1 - F(xo) where P(x < xo) and F(xo) represent the probability that x exceeds xo and the cumulative frequency value of x0 respectively. FLOW DURATION CURVE OF THE PERIODIC SERIES CONSISTING OF ONLY PERIODIC COMPONENT Periodic series (monthly and daily series) are not stationary and have periodic components. If the flow value is represented only with one harmonic, the exceedance At probability -can be defined as: At, 1 fx-|i - = 1 - - arccos, T % K C where T, C and u. represent the period (year), the amplitude and the annual mean value respectively. It is clear from the above formula that the phase angle 0 has no effect on the flow duration curve in the case of 1 harmonic. In the case of two harmonics the periodic component is defined in the dimensionless form as given below: f- 2tü\ C 2 = cos + - - COS \T J C, 47rt V T + 9 where 9, Cj and Cz are the phase angle and the amplitudes of the 1. and 2. harmonics C respectively. Simulations are realized for various -~r and 9 values. It is seen that the C shape of the flow duration curve is sensitive to the changes in -~ and 0 valaues FLOW DURATION CURVE OF THE PERIODIC SERIES HAVING BOTH PERIODIC AND STOCHASTIC COMPONENTS 1. Determination of the flow duration curve using the average of the probabilities If the number of the time intervals considered through a T period is represented with At N and - is the exceedance probability (At = 1 for a month N = 12, At = 1 for a day N = 365), it can be assumed that the flow values at different time intervals are xvi homogeneous, that is their distributions and statistical parameters are constant. In that case for each Atj time interval (i=l,2, N) the probability that the flow x; exceeds a xo value, P(xj > Xo), is determined by considering the distribution function and parameters of the flow in that time interval. The average of these probabilities gives the probability, that the x flow value exceeds the mentioned Xo through a year: ZWx, >x0) P (x> Xo)= - n For the monthly flows this equation is defined as: 12 P (x> Xo)= SP(x, >x0) i=l 12 and for the daily flows: 365 IP(Xj >x0) '<*>?*>-" 365 Using the above mentioned formulas flow duration curves are obtaianed for different gauged flow stations in Turkey for different probability distributions. Analytically obtained flow duration curves for daily flows of Karahacih station for Normal distribution and skewed distribution are compared with the original flow duration curve and the duration curves obtained by simulating daily flows based on the Thomas Fiering model (Figure 1). It is seen that the flow duration curves based on the analytical method agree well with the original duration curve and the duration curves based on the simulation. 2. Determination of the flow duration curve using the distribution of the Z = X Y variable The periodic flows can be expressed in the following form: x=xt + sxte where xt, Sxt and e are periodic mean, periodic standard deviation and gamma distributed stochastic component (mean 0, standard deviation 1, skewness coefficient Cst) respectively. Since s* is proportional to xt the above equation becomes x=xt + kxjE 1600 1200 800 - 400 Excedance probability Figure 1. Analytically obtained flow duration curves for Normal Distribution, (1), constant skewness coefficient, (2), and periodic skewness coefficient, (3), and the flow duration curves of the simulated daily synthetic series for a duration of 30 years based on the Thomas Fiering model using periodic skewness coefficient and auto correlation coefficient, (4), using periodic skewness coefficient but without considering auto correlation coefficient, (5), and the original flow duration curve, (6), for the Karahacih station. S- t=l_ where k = '. The equation can be factorized as N x=xt(l + ke) If the elements in the above equation are represented as Z=x, R=xt and 'Y=l+ks\ the probability density function of the Z variable is as follows; f,W= J y=~oo.y; fv(y)dy After some substitutions and changes the cumulative distribution function is obtained by integrating the probability density function as presented below: xvm *v Fz(u)= J FR - fY(y)dy y=- For discrete variables the above equation is defined in the following form: F^z^EF^^Jf^yOAy, R is not a random variale and represents the periodic mean values of the periodic series (daily and monthly flows). Since the stochastic component e has a gamma distributed marginal distribution also 'yi = 1 + ke' variable has the same distribution. Therefore the mean and the standard deviation of 'yj = 1 + ks' variable are E(l + ke) = 1 ; Var(l+ke) = k2Var(e) = k2. The three paramater gamma distribution has the following form: The parameters of the distribution are found in terms of k and c^: kc 2k a = sy sy Substituting these values into the probability density function we obtain, My)= kc sy k 2 VCsy / y-l + \ - j-i,(y-yo 'sy/ Considering this formula flow duration curve can be obtained using the Fz^zjj = ZFRl - jfY(yi)Ayi equation. As a case study for the daily flows of Karahacılı station analytic flow duration curve is compared with original duration curve and the duration curve based on the simulation of the daily flows using x =x,(l + k8). It is seen clearly that the three curves agree well with each other (Figure 2). xrx If xt is represented with two harmonics the flow value is expressed in the dimensionless form as: 1+- -cos + - cosl +6 x V x x (1 + ke) where x represent the overall mean of the daily flows. Accordingly the cumulative distribution function of - is expressed in terms of 5 x parameters. f \ x VxV = f(^-> -f-,e,k,c.x) X X 1200 -, BOO X 400 0.00 i ? r 0.40 080 Exceedance probability 120 Figure 2. The flow duration curves obtained by simulation, (1), and analytically, (2), together with the original flow duration curve, (3). Flow duration curves are obtained using the mentioned analytic method for various k and Ck values keeping the other three parameters constant. It is seen that for k<0.5, k has no effect on the flow duration curve if cra<1.25, which is the case for the majority of the Turkish rivers. Then flow duration curves are obtained using the analytic method for various 0 phase angle values keeping the other 4 parameter values constant. It is seen that 9 has no effect on the flow duration curve. In the light of these results the cumulative distribution function is expressed in terms of three variables: ( \ x X X Accordingly the exceedance probability values for various - values are tabulated for x C C different - -, -^- and k values. x x Table 1. The exceedance probabilities for different - =- values (0.1;0.2;0.3) x Q corresponding to various k values (0. 1 ;0.2;0.3;0.4;0.5) as - = 0.5 found analytically for - values. x It is expected that the realized study will be useful in the determination of the flow duration curves in river sections where the flow data is missing and in many other water resources problems.