1. Rainfall-Runoff Modeling (2)
Professor Ke-Sheng Cheng
Dept. of Bioenvironmental Systems Engineering
National Taiwan University

2. Peak flow estimation Rational method Empirical method
Unit hydrograph technique

3. 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.

4. 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.,

5. 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?

6. The time of peak flow occurrence is dependent on the relative magnitude of Ai’s.

7. 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?

8. The watershed storage effect is neglected.

9. 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.

11. 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).

12. 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.

13. 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
In engineering practice, it is normally restricted to catchments with basin area not exceeding 15 km2.
For areas with complicated landuse/landcover condition, applications of the rational method should be restricted to even smaller catchments.

14. Unit hydrograph of the modified rational method
Original rational method only estimates the peak flow. It does not yield a complete runoff hydrograph.
Neglecting the watershed storage effect.
2tc
tc+tr

15. 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)

16. 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.

17. Contributing area and contributing rainfall

19. 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.

20. 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

21. Consider the continuity of the hypothetical reservoir during a time interval t.

22. 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).

24. Area between isochrones t = 0 and t = 0. 25 hr is 100 acres
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) = cfs

25. Synthetic unit hydrographs
Clark’s IUH (time-area method)
SCS unit hydrograph
Linear reservoir model (Nash model)

26. SCS unit hydrograph
Note: tb > tr+tc. The SCS UH takes into account the storage effect.

27. Misuse of SCS UH – an example
tc=0.5

28. Synthetic unit hydrographs
Clark’s IUH (time-area method)
SCS unit hydrograph
Linear reservoir model (Nash model)

29. Nash’s linear reservoir model
Consider a linear reservoir which has the following characteristics:

36. Summary of the Conceptual IUH model Assumption: Linear Reservoir S(t)=kQ(t)

37. Relationship between IUH and the S curve

45. The Nash Linear Reservoir Model

47. The above equation represents a gamma density function, and thus the integration over 0 to infinity yields 1 (one unit of rainfall excess).

48. 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

49. The direct runoff can be expressed as a convolution integral of the rainfall excess and IUH, i.e.
If the direct runoff hydrograph (DRH) is divided by the total volume of direct runoff, it can be viewed as a density function. Thus, denotes a density function of a random variable t.
The first moment of this rescaled DRH is

50. The second moment of the rescaled DRH is
Similarly, the first and second moments of the rescaled effective rainfall can be respectively expressed as

51. It can be shown that
Thus, given the ERH and DRH, the parameters n and k can obtained from the above equations.

52. Calculation of

53. Calculation of

56. 坡地開發滯留池之水文設計
Hydrological Analysis for Detention Pond Design in Hillslope Development

57. 坡地開發水文設計相關規範

69. 現行設計規範之檢討

106. 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：

107. 滯洪設施規劃設計原則 (水土保持技術規範第 九十五 條 )
滯洪設施規劃設計原則 (水土保持技術規範第 九十五 條 )

108. 滯洪量之估算 (水土保持技術規範第 九十六 條 )