1. Modified Rational Method for Texas Watersheds
Nirajan Dhakal and Xing Fang
Auburn University
May 11-13, 2009

2. Runoff coefficients estimated from land use
90 watersheds in Texas were used to estimate standard (table) rational runoff coefficient using ArcGIS - Cvlit
From a previous TxDOT project, we had a geospatial database containing watershed boundaries for 90 Texas watersheds, which were delineated using 30-meter Digital Elevation Model (DEM).
National Land Cover Data (NLCD) for 2001 were obtained for the Texas from the USGS website
The first task was to cut out the NLCD layer using watershed boundary for a particular watershed and find out areas of different classes of land cover within that watershed.
Clipping polygon method was used using original NAD_1983_Albers projected coordinate system.

3. Runoff coefficient estimation ….
For clipping polygon method Raster NLCD layer was converted into the polygon feature layer.
The “Clip” function of the Arc Toolbox was used by selecting “Analysis Tools”, then “Extract”.
“Input Features” should be 2001 NLCD panel containing selected watershed, “Clip Features” should be the layer containing selected watershed boundary, and “Output Feature Class” is to provide a shape file name for storing clipped NLCD area.
The attribute table was opened, “Area” field was added and then Calculate Geometry function was used to determine the areas for different grid (land cover) codes.
Using the Statistics and Summarize functions, the total area as well as the individual area of each land cover class for the watershed was obtained.

4. Runoff coefficient estimation...
Each different watershed has different land cover classes distributed inside its watershed boundary.
It was found that there were total 15 land cover classes involved for the 90 watersheds studied.
Our next task was to assign runoff coefficient for particular land cover class.
From different literature sources studied, typically we do not find runoff coefficients for most of 15 NLCD land-cover classes, but we identified similar land use types to match them.
Assuming that all of the rainfall is converted into runoff for open water and wetlands, the value of C assigned to these land-cover classes is 1.
For the other land cover classes a range of C values are available in the mentioned sources under similar land use types.
Average values were assigned for them after determining under which land use type the particular class falls or closely matches.

6. Kuichling’s Original Paper
Q = m A r
where Q = maximum discharge of the outlet sewer (cfs), A = magnitude of the drainage area in acres (42.8 – acres), r = maximum rate of rainfall (in/hour), and m is proportion of impervious surface on said area = the proportion of the rainfall discharged during the period of greatest flow.
The use of average rates of rainfall for an unduly long period of time creates the error
r is computed over a short period within which the surface drainage waters from all portions of the area may be concentrated at the outfall.

7. Kuichling’s Original Paper
“.. For a uniform rate of precipitation, the concentrated discharge from a given surface will become a maximum for the condition when the duration of such rate is equal to the time required for the water which falls upon the most distant points to reach the place of observation; or, in other words, that the entire area is contributing to said discharge; …”
“…the rates of rainfall adopted in computing the dimensions of a main sewer must correspond to the time required for the concentration of the drainage waters from the whole tributary area…”

8. Runoff coefficient estimation..
A weighted C value was calculated in excel spreadsheet for each watershed using the following equation:
Weighted C value compared with “Table C” values and observed runoff coefficient values from Kirt Harle (2003) and from another TxDOT project supervised by Dr. David B. Thompson for 36 watersheds.

10. “C varies from zero to 1.0 as T ranges from zero to infinity” by Stanley S. Butler,
Discussion on “Experimental examination of the rational method”. T is return period.

11. Calomino et al. (1997) showed that C = 0. 57 R0. 042 (R2 = 0
Calomino et al. (1997) showed that C = 0.57 R0.042 (R2 = 0.96) for a 1.89 Ha urban watershed (91.5% impervious area) with C ranging from 0.31 to 0.81.

14. 14

15. Future work for runoff coefficient
Rainfall and runoff events used for the study were from 1960 to 1980.
Land cover conditions could be more reliable or representative if NLCD data more close to that period is used.
Reliable NLCD as old as 1992 is found.
Repetition of the work for runoff coefficient calculation using the NLCD 1992 has started.

16. Modified Rational Method - MRM
For MRM three different possible types of hydrograph can be developed for the given sub-basin.
When rainfall duration = tc (from Wanielista, Kersten and Eaglin)

17. When rainfall duration > tc
Assumptions of rainfall distribution and rainfall loss: (1). Uniform rainfall distribution (2). No initial loss (3). Constant ratio for rainfall loss over the duration
When rainfall duration > tc

18. Kuichling’s Original Paper
“…the period of maximum discharge is usually equal to the duration of the corresponding maximum intensity of the rain, and that the flow is not in the form of a short, sharp-crested wave of momentary duration like that produced from a flushing tank. Thus, in a heavy shower lasting twenty minutes at uniform intensity, the maximum sewer discharge will continue for about the same length of time, …”

19. When rainfall duration < tc
MRM can be extended to applications for non- uniform rainfall distribution.
The runoff hydrograph from the MRM for the rainfall event with the duration less than the time concentration can be converted to a unit hydrograph (UH).
UH generated can be used to compute runoff hydrographs for any nonuniform rainfall events using unit hydrograph theory (convolution).
(“Urban Surface Water Management” by Walesh, 1989)

20. Rational Hydrograph Method (RHM) proposed by Guo (2000, 2001) for continuous nonuniform rainfall events.
RHM was used to extract runoff coefficient and time of concentration from observed rainfall and runoff data through optimization.
Considered the time of concentration as the system memory (Singh 1982) and used a moving average window from (T- tc) and T to estimate uniform rainfall intensity for the application of the rational method to determine hydrograph ordinates.
For 0 ≤ T < tc [(T-t c)<0]
For tc ≤ T ≤ Td (the rainfall duration)
For Td ≤ T ≤ (Td+Tc), Guo (2001) adopted linear approximation for a small catchment

21. A hypothetical non-uniform rainfall event tested with 5-min MRM unit hydrograph and then with Guo’s RHM .
DRH predicted by the two methods show some differences after the rainfall ceases.
Guo’s RHM (2002, 2001) used a linear approximation from the discharge Q(Td) at the end of the rainfall event to zero at the time Td + Tc.
For nonuniform rainfall events this approximation is not correct because this will result the violation of the conservation of volume for the rainfall excess and the runoff hydrograph.

22. V. P. Singh and J. F. Cruise (1982) used a systems approach for the analysis of rational formula.
They approved that the rational method was based on one fundamental assumption, i.e., watershed is represented as a linear, time-invariant system.
Nash (1958) equation was used to obtain instantaneous unit hydrograph (IUH) – h(t):
for 0 ≤ t ≤ Tc .
for t ≥ Tc
The convolution method was used to derive the S-hydrograph and then D-hr unit hydrograph for the rational method.

23. IUH
S-hydrograph
Rational UH
Three key parameters:
Watershed area
Time of concentration
Duration of UH (D)

24. A symmetric trapezoidal shape unit hydrograph was developed for D < Tc .
for 0 ≤ t ≤ D
for D ≤ t ≤Tc
for Tc ≤ t ≤ (Tc + D)
They concluded probability density function (PDF) of the rational method is a uniform distribution with entropy increasing with the increasing value of Tc.

25. Nonlinear Systems
The HEC-1/HMS programs use a synthetic time-area curve derived from a generalized basin shape:
AI = TI1.5,
(0 ≤ TI ≤ 0.5)
AI = (1-TI) (0.5 < TI <1)
where AI is the cumulative area as a fraction of watershed area and TI is fraction of time of concentration.

26. MRM UH development and convolution
For all the 90 watersheds the unit hydrograph duration (D) used is 5 minutes.
Time of concentration obtained from several methods (1) the square root of area of the watersheds, and (2) Kirpich method with watershed parameters developed by LU, UH and USGS.
Runoff coefficients tested were obtained from two methods: Cvlit and Cvbc.
Fortran code was developed to calculate the rainfall excess and finally perform convolution to get DRH.
Different combinations of runoff coefficient and time of concentration used to get different simulated results of DRH.

27. Seventeen error parameters between observed and predicted DRHs were calculated after applying MRM UH to each event – Have not analyzed.
Average and standard deviation of the error parameters were determined for all events in the same station.
Several events were selected to plot the observed and modeled DRH.
Summarizing plot was made between the peak value of the modeled and observed discharges.
Summarizing plot was made between the time to peak between the modeled and observed data.

28. Statistical error parameters Nash - Sutcliffe efficiency (ME) and the coefficient of determination (R2) were calculated for each run.
Gamma UH was also developed using the regression equations with MRNG Optimized Qp and Tp (Pradhan 2007) of UH.
Mean Gamma UH developed were applied to all the events corresponding to the same station to generate DRHs by convolution.
Two runs were made one with Cvbc and then with Cvlit .

29. Results using Tc from square root of area and Cvbc

31. Results using Tc computed by Kirpich method (USGS data) and Cvbc

37. Statistical summary of peak discharge results using Cvbc and different Tc

38. Statistical summary of time to peak results using Cvbc and different Tc

39. Results using Tc from square root of area and Cvlit

42. Statistical summary of peak discharge results using Cvlit and different Tc

43. Statistical summary of time to peak results using Cvlit and different Tc

46. Results using Gamma UH and Cvbc

53. Results using Gamma UH and Cvlit

57. Statistical summary of peak discharge results using Gamma UH and different C

58. Statistical summary of time to peak results using Gamma UH and different C

59. Statistical summary of peak discharge results using Cvbc

60. Statistical summary of peak discharge results using Cvlit

61. Peak Discharge Results

62. Statistical summary of time to peak results using Cvbc

63. Statistical summary of time to peak results using Cvlit

64. Time to peak results