Report

Applied Hydrology Rainfall-Runoff Modeling (2) Professor Ke-Sheng Cheng Dept. of Bioenvironmental Systems Engineering National Taiwan University RSLAB-NTU Lab for Remote Sensing Hydrology and Spatial Modeling 1 Peak flow estimation Rational method Empirical method Unit hydrograph technique RSLAB-NTU Lab for Remote Sensing Hydrology and Spatial Modeling 2 The concept of Isochrones Let’s assume that raindrops fall on a spatial point x within the watershed require an amount of travel time t(x) to move from point x to the basin outlet. RSLAB-NTU Lab for Remote Sensing Hydrology and Spatial Modeling 3 Contour lines of the travel time are termed as isochrones (or runoff isochrones) of the watershed. The highest value of isochrones represents the time of concentration of the watershed. Assume the effective rainfall has a constant intensity (in time) and is uniformly distributed over the whole watershed, i.e., ie ( x, t ) ie constant RSLAB-NTU Lab for Remote Sensing Hydrology and Spatial Modeling 4 If the effective rainfall duration tr = 2 hours, what is the peak direct runoff at the basin outlet? When will the peak flow occur? When will the direct runoff end? RSLAB-NTU Lab for Remote Sensing Hydrology and Spatial Modeling 5 The time of peak flow occurrence is dependent on the relative magnitude of Ai’s. RSLAB-NTU Lab for Remote Sensing Hydrology and Spatial Modeling 6 If the effective rainfall duration tr = 10 hours, what is the peak direct runoff at the basin outlet? When will the peak flow occur? When will the direct runoff end? RSLAB-NTU Lab for Remote Sensing Hydrology and Spatial Modeling 7 The watershed storage effect is neglected. RSLAB-NTU Lab for Remote Sensing Hydrology and Spatial Modeling 8 Under the assumptions of constant intensity and uniform spatial distribution for effective rainfall, the peak direct runoff occurs at the time t = tc, if the duration of the effective rainfall is longer than or equal to tc. RSLAB-NTU Lab for Remote Sensing Hydrology and Spatial Modeling 9 tr tc RSLAB-NTU Lab for Remote Sensing Hydrology and Spatial Modeling 10 tr tc However, for storms with duration shorter than tc, the peak flow and its time of occurrence depend on relative magnitude of contributing areas (Ai’s). RSLAB-NTU Lab for Remote Sensing Hydrology and Spatial Modeling 11 Rational method for peak flow estimation Consider a rainfall of constant intensity and very long duration occurring uniformly over a basin. The runoff gradually increases from zero to a constant value as shown in the following figure. RSLAB-NTU Lab for Remote Sensing Hydrology and Spatial Modeling Qp ie A ciA c runoff coefficient 12 Practical application of the rational method for peak flow estimation Calculating the time of concentration, tc. Determining the average rainfall intensity i from the IDF curve using tc as the storm duration and a predetermined return period T. Calculating the runoff coefficient, c. Calculating the peak direct runoff Q p ie A ciA c runoff coefficient RSLAB-NTU Lab for Remote Sensing Hydrology and Spatial Modeling In engineering practice, it is normally restricted to catchments with basin area not exceeding 15 km2. 13 Unit hydrograph of the modified rational method Original rational method only estimates the peak flow. It does not yield the complete runoff hydrograph. tr tc tr tc tr tc 2tc tc+tr Neglecting the watershed storage effect. RSLAB-NTU Lab for Remote Sensing Hydrology and Spatial Modeling 14 Synthetic unit hydrographs For areas where rainfall and runoff data are not available, unit hydrograph can be developed based on physical characteristics of the watershed. Clark’s IUH (time-area method) SCS unit hydrograph Linear reservoir model (Nash model) 15 Clark’s IUH (time-area method) The concept of isochrones Isochrones are lines of equal travel time. Any point on a given isochrone takes the same time to reach the basin outlet. Therefore, for the following basin isochrone map and assuming constant and uniform effective rainfall, discharge at the basin outlet can be decomposed into individual contributing areas and rainfalls. 16 Contributing area and contributing rainfall 17 18 Using the basin isochrone map, the cumulative contributing area curve can be developed. The derivatives or differences of this curve constitute the instantaneous unit hydrograph IUH(t). Time base of the IUH is the time of concentration of the watershed. 19 If the effect of watershed storage is to be considered, the unit hydrograph described above is routed through a hypothetical linear reservoir with a storage coefficient k located at the watershed outlet. For a linear reservoir with storage coefficient k, we have 20 Consider the continuity of the hypothetical reservoir during a time interval t. 21 Example. A watershed of 1000-acre drainage area has the following 15-minute time-area curve. The storage coefficient k of the watershed is 30 minutes. Determine the 15-minute unit hydrograph UH(15,t). 22 23 Area between isochrones t = 0 and t = 0.25 hr is 100 acres. Since we are interested in the unit hydrograph UH(15-min, t), the rainfall intensity in the time period (0.25 hr) should be 4 inch/hr. Therefore, the ordinate of UH(15-min, t) at time t = 0.25 hr is: 4 inch/hr (1/12 ft/inch) (1/3600 hr/sec) 100 acres (43560 ft2/acre) = 403.33 cfs 24 Synthetic unit hydrographs Clark’s IUH (time-area method) SCS unit hydrograph Linear reservoir model (Nash model) 25 SCS unit hydrograph Note: tb > tr+tc. The SCS UH takes into account the storage effect. 26 Misuse of SCS UH – an example tc=0.5 27 Synthetic unit hydrographs Clark’s IUH (time-area method) SCS unit hydrograph Linear reservoir model (Nash model) 28 Nash’s linear reservoir model Consider a linear reservoir which has the following characteristics: S (t ) KQ(t ) 29 30 31 Q(t ) I (1 e t / k ) 32 33 34 35 Summary of the Conceptual IUH model Assumption: Linear Reservoir S(t)=kQ(t) 36 Relationship between IUH and the S curve RSLAB-NTU Lab for Remote Sensing Hydrology and Spatial Modeling 37 38 39 40 41 42 [ A(1 e t t K ) A(1 e ( t t ) K 1 )] t (t t ) ( t t ) t 1 1 t t K K K K K A[e e ] A[e e e ] t t t 1 A t K t K t K K Ae (e 1) e (e 1) t t 43 44 The Nash Linear Reservoir Model 45 46 The above equation represents a gamma density function, and thus the integration over 0 to infinity yields 1 (one unit of rainfall excess). 47 Parameter estimation for n and k The IUH of the n-LR system is characterized by a function which is identical to the gamma density. =M1 =M2 48 The direct runoff can be expressed as a convolution integral of the rainfall excess and IUH, i.e. t Q(t ) i( )u(t )d 0 If the direct runoff hydrograph (DRH) is divided by the total volume of direct runoff, it can be viewed as a density function. Thus, Q(t ) / VDRH q(t ) denotes a density function of a random variable t. The first moment of this rescaled DRH is M Q1 tq(t )dt 0 1 VDRH 0 0 t i( )u (t )ddt 49 The second moment of the rescaled DRH is M Q 2 t q(t )dt 0 1 2 VDRH t 0 2 0 i( )u(t )ddt Similarly, the first and second moments of the rescaled effective rainfall can be respectively expressed as M I1 MI2 1 VERH 1 VERH 0 ti(t )dt 0 2 t i(t )dt 50 It can be shown that M Q1 M I 1 nk MQ2 M I 2 n(n 1)k 2 2nkMI1 Thus, given the ERH and DRH, the parameters n and k can obtained from the above equations. 51 Calculation of M I 1 and M I 2 52 Calculation of M Q1 and M Q 2 53 坡地開發滯留池之水文設計 Hydrological Analysis for Detention Pond Design in Hillslope Development 54 坡地開發水文設計相關規範 55 56 57 58 59 60 61 62 63 64 65 66 現行設計規範之檢討 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 5.75 (60 / 60) / 2 10350 159.14 (15 / 60) 103 (20104 ) 7957 91 92 93 94 95 96 97 98 99 100 101 102 103 逕流量分析 (水土保持技術規範第 十七條 ) 洪峰流量之估算，有實測資料時，得採用單位歷 線分析；面積在一千公頃以內者，無實測資料時， 得採用合理化公式(Rational Formula)計算。合理 化公式如下： 1 Qp CIA 360 式中， Q ：洪峰流量 (立方公尺/ 秒), p C：逕流係數(無單位), I：降雨強度(公釐/ 小時), A：集水區面積(公頃。 ) 104 滯洪設施規劃設計原則 (水土保持技術規範第 九十五 條 ) 105 滯洪量之估算 (水土保持技術規範第 九十六 條 ) 106