SEES 503 SUSTAINABLE WATER RESOURCES. Floods. Instructor. Assist. Prof. Dr. Bertuğ Akıntuğ

Transcription

1 SEES 503 SUSTAINABLE WATER RESOURCES Floods Instructor Assist. Prof. Dr. Bertuğ Akıntuğ Civil Engineering Program Middle East Technical University Northern Cyprus Campus SEES 503 Sustainable Water Resources 1/69

2 Introduction FLOODS EXTREME EVENTS DROUGHTS A natural disaster. An overflow of water from a lake or other body of water due to excessive rainfall or other input of water. Associated with sustained periods of significantly lower rainfall, soil moisture, surface-water storage, streamflow, and groundwater levels. Source: Source: SEES 503 Sustainable Water Resources 2/69

3 Outline What is Flood? Management of Flood (Flood Control) Flood Modeling Case study: Modeling of Guzelyurt-Bostanci Flood (Calculation of Design Flow) Flood Routing Flood Forecasting SEES 503 Sustainable Water Resources 3/69

4 What is Flood? Flood: A high flow that exceeds the capacity of a stream or drainage channel. Flood Stage: The elevation at which the flood overflows the embankments. Floodplain: The normally dry land adjoining rivers, streams, lakes, bays, or oceans that is inundated during flood events. SEES 503 Sustainable Water Resources 4/69

5 What is Flood? According to Federal Emergency Management Agency (FEMA), USA: Definition of Floodplain: Any land area susceptible to being inundated by flood waters from any source Source: HECPrograms.html SEES 503 Sustainable Water Resources 5/69

6 What is flood? Regulatory mechanisms used to control development in floodplains Flood insurance requirements. To minimize the future flood loss To allow floodplain occupants to be mostly responsible for flood-damage costs instead of taxpayers. Building restrictions 100-yr flood has been adopted as the base flood for delineating flood plains in USA. Cross-section of a typical floodplain SEES 503 Sustainable Water Resources 6/69

7 Outline What is Flood? Management of Flood (Flood Control) Flood Modeling Separate file Case study: Modeling of Guzelyurt-Bostanci Flood (Calculation of Design Flow) Flood Routing Flood Forecasting SEES 503 Sustainable Water Resources 7/69

8 Outline What is Flood? Management of Flood (Flood Control) Flood Modeling Case study: Modeling of Guzelyurt-Bostanci Flood (Calculation of Design Flow) Flood Routing Flood Forecasting SEES 503 Sustainable Water Resources 8/69

9 Introduction Water Resources Projects require the matching of water demand and existing water supply for the future. Absolute magnitude and exact occurring times of flows cannot be forecasted without probability theory. It is almost impossible to solve a hydrologic problem using deterministic methods. Therefore, probability and statistical methods must be employed. SEES 503 Sustainable Water Resources 9/69

10 Introduction Hydrologic events have variable nature and uncertainties. Therefore, hydrologic variables have both deterministic and random (stochastic) components. Prediction of hydrologic events for the future are very important for the design of hydraulic structures. Statistical and stochastic methods are used for the prediction of hydrologic events. In order to estimate future values of hydrologic variables past records of their variables are used. SEES 503 Sustainable Water Resources 10/69

11 Selection of Data All the hydrologic variables are continuous. Statistical analysis require that the sample consist of the recorded data in a series have the following properties: Random each observation should have the same probability of occurrence. Independent each observation should be independent of the other observations. Homogeneous each observation should be obtained under similar conditions. Relevant the property of the should be related to the problem. The mean annual, mean monthly, instantaneous hourly discharge values are all flow data. Adequate the length of the record period. SEES 503 Sustainable Water Resources 11/69

12 Types of Data Series If a river has been gauged daily for 10 years, there will be 3650 observations. However, these are not independent random events. The array of these observations is termed a full series or a complete series. From the 10 year complete series if we take the maximum events of each year we can call it an annual series. In this case annual flows will be independent. This is why we define the water year. SEES 503 Sustainable Water Resources 12/69

13 Types of Data Series Water years clearly separate high and low flow periods. Therefore, it is necessary that water years be used in defining hydrologic events. Types of data series SEES 503 Sustainable Water Resources 13/69

14 Types of Data Series Some of the annual peaks are smaller than the secondary peaks of other years. This problem may be overcome by listing a partial duration series. Partial duration series all peaks above a certain value are included, provided that they are independent events. The choice of the series to be used depends on the purpose of the analysis. Partial duration series is required during the construction period of a large dam (i.e. 4 years). Annual series is used for the design flood of a dam s spillway. Full series are used to determine the statistical characteristics of the related variable such as mean and standard deviation. SEES 503 Sustainable Water Resources 14/69

15 Frequency Histogram Why use a Frequency Distribution? It is a way to summarize numerical data It condenses the raw data into a more useful form... It allows for a quick visual interpretation of the data SEES 503 Sustainable Water Resources 15/69

16 Frequency Histogram Class Intervals and Class Boundaries Each class grouping has the same width Determine the width of each interval by Width of interval number of range desired class groupings Usually at least 5 but no more than 15 groupings Class boundaries never overlap Round up the interval width to get desirable endpoints SEES 503 Sustainable Water Resources 16/69

17 Frequency Histogram Example: A manufacturer of insulation randomly selects 20 winter days and records the daily high temperature 24, 35, 17, 21, 24, 37, 26, 46, 58, 30, 32, 13, 12, 38, 41, 43, 44, 27, 53, 27 SEES 503 Sustainable Water Resources 17/69

18 (continued) Sort raw data in ascending order: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Find range: = 46 Select number of classes: 5 (usually between 5 and 15) Compute class interval (width): 10 (46/5 then round up) Determine class boundaries (limits): 10, 20, 30, 40, 50, 60 Compute class midpoints: 15, 25, 35, 45, 55 Count observations & assign to classes SEES 503 Sustainable Water Resources 18/69

19 Data in ordered array: (continued) 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Class Frequency Relative Frequency Percentage 10 but less than but less than but less than but less than but less than Total SEES 503 Sustainable Water Resources 19/69

20 Data in ordered array: 12, 13, 17, 21, 24, 24, 26, 27, 27, 30, 32, 35, 37, 38, 41, 43, 44, 46, 53, 58 Class Frequency Percentage Cumulative Frequency Cumulative Percentage 10 but less than but less than but less than but less than but less than Total SEES 503 Sustainable Water Resources 20/69

21 A graph of the data in a frequency distribution is called a histogram The class boundaries (or class midpoints) are shown on the horizontal axis the vertical axis is either frequency, relative frequency, or percentage Bars of the appropriate heights are used to represent the number of observations within each class SEES 503 Sustainable Water Resources 21/69

22 Class (No gaps between bars) Class Midpoint Frequency 10 but less than but less than but less than but less than but less than Frequency Histogram: Daily High Temperature Class Midpoints SEES 503 Sustainable Water Resources 22/69

23 Class Lower class boundary Cumulative Percentage Less than but less than but less than but less than but less than but less than Cumulative Percentage Daily High Temperature Class Boundaries (Not Midpoints) SEES 503 Sustainable Water Resources 23/69

24 Frequency Histogram In hydrologic design problems, frequency histogram is used to obtain the relationship between the magnitudes of hydrologic events and the corresponding recurrence intervals of the events. To conduct a frequency analysis Independent data series should be selected. Suitable class interval is selected. The number of events in each class interval is plotted with respect to the interval to obtain the frequency histogram. Frequency Histogram SEES 503 Sustainable Water Resources 24/69

25 Frequency Histogram The histogram gives a good prediction of the distribution of the flood magnitudes. However engineers are more interested in the number of times a flood of a given magnitude is equaled or exceeded in a stated period. Cumulative frequency histogram (plot of the total number of floods above the lower limit of any class interval) is obtained. Frequency histogram is not recommended when record period is less than 10 years. Cumulative Frequency Histogram SEES 503 Sustainable Water Resources 25/69

26 Return Period The recurrence interval (return period, T r ) is the average time that elapses between two events, which is equal or exceed a particular level. T r =N N year event, the event which is expected on the average to be equaled or exceeded once every N years. The probability of occurrence or the probability of exceedence in one year for a T r year flood is denoted by p and gives as follows. p = 1/T r The probability of non-occurrence or probability of non-exceedence, q, will then be q = 1 p = 1 SEES 503 Sustainable Water Resources 26/69 1 T r

27 Probability Distribution Functions Statistical functions are used to obtain the maximum information from short duration hydrologic observations and to evaluate the most probable nature of the corresponding populations. The probability distribution functions used most often in hydrology: Normal (Gaussian) Distribution, Log-normal Distribution, Pearson Type III Distribution, Log-Pearson Type III Distribution, and Extreme Value Type I (Gumbel) Distribution. Solution is possible by analytical and graphical methods. SEES 503 Sustainable Water Resources 27/69

28 Probability Distribution Functions Analytical Solutions Many probability distribution functions for continuous random variables can be expressed as either: X = X + K S where K : frequency factor. Read from the published tables previously developed by integrating the appropriate probability density function (pdf). X : sample mean X = 1 n n i = 1 X i S : sample standard deviation 1 n 1 SEES 503 Sustainable Water Resources 28/69 S = n i = 1 ( X i X 2 )

29 Probability Distribution Functions Analytical Solutions or log X = log X + K S log X where K : frequency factor. Read from the published tables previously developed by integrating the appropriate pdf. log X : sample mean of log transformed variable X. S log X : sample standard deviation of log transformed variable X. SEES 503 Sustainable Water Resources 29/69

30 Probability Distribution Functions Analytical Solutions Model parameters are computed from sample observation. Sample mean: 1 X = n n i = 1 X i Sample standard deviation: S = 1 n 1 n i = 1 ( X i X 2 ) Skew coefficient: G = SEES 503 Sustainable Water Resources 30/69 n ( n n i = 1 ( X i 1)( n X ) 2) S 3 3

31 Probability Distribution Functions Analytical Solutions Normal Distribution Parameters: µ (mean) and σ (standard deviation) unknown They are substituted by their estimates obtained form sample µ = X (sample mean) σ = s (sample standard deviation) Probability Density Function (PDF) f ( x) = 1 e σ 2π ( x µ ) 2 / 2 σ 2 PDF of Normal Distribution SEES 503 Sustainable Water Resources 31/69

32 Probability Distribution Functions Analytical Solutions Normal Distribution (con t) Cumulative Density Function (CDF) F( x) = 1 σ 2π + e ( x µ ) 2 / 2 σ 2 dx CDF of Normal Distribution SEES 503 Sustainable Water Resources 32/69

33 Probability Distribution Functions Analytical Solutions Normal Distribution (con t) Standard Unit z = x µ σ or z = x s X When z is found from the table, the value of x can be obtained by the inverse transformation X = µ + σz or X = X + sz SEES 503 Sustainable Water Resources 33/69

34 Probability Distribution Functions Analytical Solutions Normal Distribution (con t) X T-yr value: X T = X + yr K T yr S where K T-yr : frequency factor for T-yr return period. Read from the published tables previously developed by integrating the Normal pdf. X : sample mean X = 1 n n i = 1 X i S : sample standard deviation 1 n 1 SEES 503 Sustainable Water Resources 34/69 S = n i = 1 ( X i X 2 )

35 Probability Distribution Functions Analytical Solutions Log-Normal Distribution Hydrologic variables do not fit normal distribution perfectly. They are mostly skewed to the right, since they generally have positive values or values larger than a certain lower limit. Most of the time their logarithms fit the normal distribution. Therefore, the variables x are transformed to their lognormals and then the normal distribution is applied. y=log(x) or y=ln(x) Probability density function of the log-normal distribution is given as: ( y µ y ) / 2σ y f ( x) = e σ 2π y SEES 503 Sustainable Water Resources 35/69

36 Probability Distribution Functions Analytical Solutions Log-Normal Distribution (con t) X T-yr value: log X T yr = log X + K T yr S log X where K T-yr : frequency factor for T-yr return period. Read from the published tables previously developed by integrating the Normal pdf. log X : sample mean of log transformed data S log X : sample standard deviation of log transformed data SEES 503 Sustainable Water Resources 36/69

37 Probability Distribution Functions Analytical Solutions Pearson Type III and Log- Pearson Type III Distribution Pearson (1930) proposed a general formula that fits many probability distribution, including normal, beta and gamma distribution. Pearson Type III parameters: mean standard deviation skew coefficient. K values determined from the following table. X = X + K S log X = log X + K Slog X The Log-Pearson Type III distribution of X is equivalent to applying the Pearson Type III distribution to the transformed random variables log X. SEES 503 Sustainable Water Resources 37/69

38 Probability Distribution Functions Analytical Solutions Pearson Type III and Log- Pearson Type III Distribution The values for K for G=0 are the same as the values in the normal probability table. SEES 503 Sustainable Water Resources 38/69

39 Probability Distribution Functions Analytical Solutions Extreme-Value (Gumbel) Distribution Gumbel in 1958 introduced the theory of extremes considering the distribution of the largest or the smallest values observed. The annual extreme values are the largest of smallest observed values among the 365 daily values in a year. Gumbel used this theory for the analysis of floods. In this distribution, the probability of the occurrence of a magnitude equal to or greater than any value x (exceedence probability) is p b e 1 = 1 e where b = ( X X S) S, e= Can be applied directly without needing a table. SEES 503 Sustainable Water Resources 39/69

40 Probability Distribution Functions Graphical Solutions Plotting the frequency distributions provides a visual inspection of the fit. It gives an idea about how well the distribution represents the data set. Different graph papers are used for different probability distributions in such a way that the plot should be a straight line. SEES 503 Sustainable Water Resources 40/69

41 Probability Distribution Functions Graphical Solutions Data plotting position SEES 503 Sustainable Water Resources 41/69

42 Probability Distribution Functions Graphical Solutions There are different definitions of plotting position. q: prob. of non-occurrence Weibull will be used in this course SEES 503 Sustainable Water Resources 42/69

43 Probability Distribution Functions Graphical Solutions Weibull plotting positions for graphical solution SEES 503 Sustainable Water Resources 43/69

44 Probability Distribution Functions Graphical Solutions Since plotting positions are distribution free, they can be used for any distribution. Plotted points show, in general, a straight-line trend. A best-fit straight line is decided either by visual inspection or by another suitable methods. Using the best-fit straight line, the return period corresponding to a magnitude or a magnitude for a desired return period can be obtained. SEES 503 Sustainable Water Resources 44/69

45 Probability Distribution Functions Graphical Solutions For T r =100, p=1/100=0.01=1% SEES 503 Sustainable Water Resources 45/69

46 Probability Distribution Functions Graphical Solutions SEES 503 Sustainable Water Resources 46/69

47 Probability Distribution Functions Graphical Solutions SEES 503 Sustainable Water Resources 47/69

48 Probability Distribution Functions Distribution of Hydrologic Phenomena Experiences show that: Mean annual flows Normal Distribution Maximum annual flows Log-Normal Distribution Extreme Value Distribution Mean monthly flows Log-Normal Distribution Extreme Value Distributions Monthly or annual volumes of runoff Normal Distribution Log-Normal Distribution Gamma Distribution SEES 503 Sustainable Water Resources 48/69

49 Probability Distribution Functions Distribution of Hydrologic Phenomena Experiences show that: Annual total precipitation Cubic or Square root Normal Dist. Annual max. hourly or daily precipitation Gumbel Dist. Log-Pearson Dist. Log-Normal Dist. Annual summer maxima and winter minima temp. Normal Dist. SEES 503 Sustainable Water Resources 49/69

50 Outline What is Flood? Management of Flood (Flood Control) Flood Modeling Case study: Modeling of Guzelyurt-Bostanci Flood (Calculation of Design Flow) Flood Routing Flood Forecasting SEES 503 Sustainable Water Resources 50/69

51 Flood Routing Flood routing is a procedure through which the temporal variation of discharge at a point on a stream channel or on a reservoir may be determined by consideration of similar data from a point upstream. Change in flood hydrograph due to storage Flood routing is a process, which shows how a flood wave is reduced in magnitude, and lengthened in time by the storage in a reach or reservoir between the two points. SEES 503 Sustainable Water Resources 51/69

52 Flood Routing - Overview Introduction Storage Equation Reservoir Routing Routing in Natural Channels SEES 503 Sustainable Water Resources 52/69

53 Storage (Continuity) Equation SEES 503 Sustainable Water Resources 53/69 Change in flood hydrograph due to storage S = ( I Q) t Q or I Q = I : inflow Q: outflow ds/dt: change in storage I I ds dt S S2 S1 Q = = t t : the average inflow (I 1 +I 2 )/2 during t. : the average outflow (Q 1 +Q 2 )/2 during t. S 1 and S 2 : storage at the beginning and at the end of t.

54 Storage (Continuity) Equation Although the storage equation is correct and simple, the movement of a flood wave in a river channel or in a reservoir is a condition of unsteady flow. Therefore following assumptions and approximations are required. The stream should be divided into parts (reaches) where channel characteristics are constant or assumed to be constant. There should be a stream gauge at each end of the reach. The time interval, t, should be as large as possible so as to have less number of computation steps, but at the same time it should be small enough to observe the passage of the flood peak. SEES 503 Sustainable Water Resources 54/69

55 Storage (Continuity) Equation Assumptions and approximations are required. (con t) If there are tributaries entering the reach whose discharges are small compared to the inflow, they may be ignored. SEES 503 Sustainable Water Resources 55/69

56 Flood Routing - Overview Introduction Storage Equation Reservoir Routing Routing in Natural Channels SEES 503 Sustainable Water Resources 56/69

57 Reservoir Routing Reservoir routing provides methods for evaluating the effects of a reservoir on a flood wave passing through it. For the design and planning of hydraulic structures, it applies to the determination of the location and the capacity of reservoir, of the size of outlets or spillways, etc. In reservoir routing, it is assumed that the water surface in the reservoir is horizontal at all times and the relationship between the storage and discharge is constant. Change in flood hydrograph due to storage in the reservoir SEES 503 Sustainable Water Resources 57/69

58 Reservoir Routing 2S t 2S t 1 2 ( I 1 + I 2) + Q1 = + Q2 Known Unknown Known Obtained SEES 503 Sustainable Water Resources 58/69

59 Reservoir Routing 2S 2S I + I + Q = + t t 1 2 ( 1 2) 1 Q2 If not known Q 0 =I 0 Unknown SEES 503 Sustainable Water Resources 59/69

60 Reservoir Routing 2S 2S I + I + Q = + t t 1 2 ( 1 2) 1 Q2 If not known Q 0 =I 0 If Q n > I n : to complete the outflow hydrograph after the end of the inflow hydrograph, the values of inflow are taken to be constant and equal to the last inflow value I n and routing process is continued till an outflow value equal to or closer to I n value is obtained. This way the lengthening on the base time can be determined. SEES 503 Sustainable Water Resources 60/69

61 Overview Introduction Storage Equation Reservoir Routing Routing in Natural Channels SEES 503 Sustainable Water Resources 61/69

62 Routing in Natural Channels In natural channels, storage is not only a function of outflow but also inflow. Wedge storage may be positive or negative depending upon whether the food in coming toward the reach (+) or going away from the reach (-). Storage in natural channels SEES 503 Sustainable Water Resources 62/69

63 Routing in Natural Channels (gain to storage during rising) > (loss from storage during falling) The routing procedure in the channels is obtained by expressing storage as a function of both outflow and inflow. Change in flood hydrograph due to storage in the channel Outflow vs storage SEES 503 Sustainable Water Resources 63/69

64 Routing in Natural Channels Muskingum Method is generally used in the channel routing procedure. S = KQ + Kx( I Q) ( xi + ( 1 x Q) S = K ) TRIAL & ERROR METHOD Assume x plot S vs [xi+(1-x)q] The correct x is the one that gives closest to the straight line. K: slope of this line. K: travel time of the center of mass of flood wave from the upstream end of the reach to the downstream end. x: dimensionless constant. It indicates the effective weights of inflow and outflow. Determination of x by trial and error SEES 503 Sustainable Water Resources 64/69

65 Routing in Natural Channels Muskingum Method S ( xi + ( 1 x Q ) = S = K( xi + ( 1 x Q ) 1 K 1 ) ) 2 2S t 2S t 1 2 ( I 1 + I 2) + Q1 = + Q2 2K ( xi + (1 x) Q ) 2K( xi + (1 x) Q ) ( I 1 + I 2) + Q1 = + Q2 t t SEES 503 Sustainable Water Resources 65/69

66 Routing in Natural Channels Muskingum Method + 2K + Rearranging above equation for inflow and outflow terms 2K(1 x) + t Q t ( xi + (1 x) Q ) 2K( xi + (1 x) Q ) ( I 1 I 2) Q1 = + Q2 2 t = t 2Kx I t 2 t + 2Kx + I t t 1 2K(1 x) t + t Q 1 Q 2 = t 2K(1 2Kx x) + t I 2 + t 2K(1 + 2Kx x) + t I 1 + 2K(1 2K(1 x) t x) + t Q 1 Q = + C + C + C 1 2 C0I 2 + C1I1 C2Q = SEES 503 Sustainable Water Resources 66/69

67 Routing in Natural Channels Muskingum Method Channel Routing SEES 503 Sustainable Water Resources 67/69

68 Outline What is Flood? Management of Flood (Flood Control) Flood Modeling Case study: Modeling of Guzelyurt-Bostanci Flood (Calculation of Design Flow) Flood Routing Flood Forecasting SEES 503 Sustainable Water Resources 68/69

69 Flood Forecasting Flood forecasting: to forecast flow rates and water levels for periods ranging from a few hours to days ahead using the real-time precipitation and streamflow data in rainfall-runoff models. Models: Rainfall-runoff models Snowmelt models Streamflow models SEES 503 Sustainable Water Resources 69/69