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Limits of the Unified Rational Method for Texas There are several major conceptual limits to URAT. First, as identified in the previous paragraph, T ⋆ < 10 minutes is to be truncated at 10 minutes; the equivalent drainage area for such a time is difficult to express because of interplay between predictive variables on T ⋆ . Second, an upper limit of T ⋆ also is difficult to identify. Third, from Appendix C, the author interprets the general time-area linearity in figure C.5 and the runoff coefficient-area relation in figure C.6 to indicate that the rational method might be suitable for larger upper limits of drainage area than often used. The hydrologic-engineering community often uses upper limits for the rational method at a quarter to half a square mile. The computational elements producing the URAT provide no additional insight into what the upper drainage area limit should be. In practice, the steady-state assumption of the watershed for the rational method and anticipated importance of routing and storage are keys to guiding the engineer in judging applicability of URAT (and other forms of the rational method). 3.2.2. Spatial Influence on 10-year T-star The statewide equations in eq. 3.6 for Texas do not accommodate the substantial changes in flood hydrology across Texas. In an effort to accommodate spatial differences in flood hydrology across Texas, URAT takes into account the influence of spatial location by county. The influence of spatial location manifests in the development of T ⋆ through P , OmegaEM, and county-specific IDF curves. This spatial influence is represented in URAT by the regression coefficients by county in tables 3.2–3.4. To provide example computations concerning influence of location in Texas, suppose that six 640-acre watersheds with half-percent slopes exist in Brewster (far-west Texas), Brooks (south Texas), Borden (southern Texas Panhandle), Baylor (north-central Texas), Blanco (central Texas), and Bowie (northeast Texas) counties. What are the estimated values of watershed time or storm duration using T ⋆ TEXAS and T ⋆ county for the 10-year annual recurrence interval for each of these counties? The statewide estimated value is and the individual county values are T ⋆[10] TEXAS = 0.099 × 6400.315 × 0.005 −0.693 = 30 minutes, (3.7) T ⋆[10] Brewster = 2.822 × 6400.218 × 0.005 −0.488 = 153 minutes, T ⋆[10] Brooks = 0.496 × 6400.308 × 0.005 −0.655 = 117 minutes, T ⋆[10] Borden = 0.384 × 6400.287 × 0.005 −0.657 = 80 minutes, T ⋆[10] Baylor = 0.084 × 6400.336 × 0.005 −0.751 = 39 minutes, T ⋆[10] Blanco = 0.070 × 6400.318 × 0.005 −0.694 = 22 minutes, and T ⋆[10] Bowie = 0.052 × 6400.317 × 0.005 −0.700 = 16 minutes. 24
These times vary greatly; relative to the the statewide estimate, the time for Bowie county is about half and the time for Brewster county is about 5 times longer. How can these times be interpreted? As the general aridity of the county increases (consider arid Brewster county compared to humid Bowie county), the effective rainfall duration increases substantially. The author suggests that this is attributed in part to the generally lower antecedent moisture conditions and higher initial abstraction potential in the semi-arid to arid counties of central to western Texas. Longer duration rainfall (and hence more total rainfall depth for flood-producing rainfall) is generally required to initially saturate soil and fill depressions of the surface of watersheds in western Texas compared to watersheds in other parts of the State. General hydrologic knowledge of Texas indicates that the thin soils and limestone geologic setting of Blanco county would result in some of the smallest watershed times in Texas—the previous computations show this to be the case. 3.2.3. Relative Influence of Recurrence Interval on T-star for Two Low-Slope Settings An S of 0.002 is representative of a low-slope setting. A comparison of T ⋆ results for the essentially flat (low-slope topography) of Harris and Lubbock counties in Texas is made. For Harris county, the time ensemble of URAT for a 200-acre watershed is T ⋆[2] Harris = 0.135 × 2000.542 × 0.002 −0.445 = 38 minutes, T ⋆[5] Harris = 0.068 × 2000.405 × 0.002 −0.665 = 36 minutes, T ⋆[10] Harris = 0.074 × 2000.326 × 0.002 −0.715 = 35 minutes, T ⋆[25] Harris = 0.075 × 2000.229 × 0.002 −0.807 = 38 minutes, T ⋆[50] Harris = 0.068 × 2000.169 × 0.002 −0.882 = 40 minutes, and T ⋆[100] Harris = 0.090 × 2000.102 × 0.002 −0.907 = 43 minutes. and for Lubbock county, the time ensemble is T ⋆[2] Lubbock = 0.885 × 2000.449 × 0.002 −0.400 = 115 minutes, T ⋆[5] Lubbock = 0.505 × 2000.347 × 0.002 −0.571 = 110 minutes, T ⋆[10] Lubbock = 0.378 × 2000.292 × 0.002 −0.658 = 106 minutes, T ⋆[25] Lubbock = 0.402 × 2000.203 × 0.002 −0.737 = 115 minutes, T ⋆[50] Lubbock = 0.411 × 2000.148 × 0.002 −0.789 = 121 minutes, and T ⋆[100] Lubbock = 0.395 × 2000.092 × 0.002 −0.853 = 129 minutes. For the time ensembles T ⋆ for Harris and Lubbock counties, the computations show a non-monotonic pattern of time. This non-monotonicity is important and is extensively considered in later discussion. For now, it is observed that watershed times of equivalence T ⋆ in Harris (Houston area) are systematically about 33 percent less than the times for Lubbock county. Although direct interpretation of 25
Page 2 and 3: NOTICE The United States Government
Page 4 and 5: This page intentionally left blank.
Page 6 and 7: AUTHOR’S DISCLAIMER The contents
Page 8 and 9: Acknowledgements A project of the m
Page 10 and 11: 4.2 Application of the Modified Rat
Page 12 and 13: List of Figures 2.1 The MRM hydrogr
Page 14 and 15: D.5 Modeled and observed peak disch
Page 16 and 17: The modified rational method extend
Page 18 and 19: 2. Technical Aspects of the Rationa
Page 20 and 21: unoff, with a negligible amount of
Page 22 and 23: 2.1.4. Summary and Important Implic
Page 24 and 25: If the storm duration is equal to t
Page 26 and 27: Figure 2.4: The instantaneous unit
Page 28 and 29: the ratio of peak rate of runoff to
Page 30 and 31: in these cases strictly converts th
Page 32 and 33: 3. A Proposed Unified Rational Meth
Page 34 and 35: with the middle of the C-value rang
Page 36 and 37: 10. The intercepts of β and expone
Page 40 and 41: T ⋆ as a “travel time” is ten
Page 42 and 43: 5. Determine IU ⋆ using T L ⋆ b
Page 44 and 45: Table 3.1: Countywide values of mea
Page 46 and 47: Table 3.1: Countywide values of mea
Page 48 and 49: Table 3.2: Countywide values of sca
Page 56 and 57: Table 3.3: Countywide values of exp
Page 64 and 65: Table 3.4: Countywide values expone
Page 66 and 67: Table 3.4: Countywide values expone
Page 68 and 69: Table 3.5: Comparison of annual pea
Page 70 and 71: LATITUDE, IN DEGRESS 26 30 34 2−y
Page 72 and 73: PEAK DISCHARGE, IN CFS 0 5 10 15 20
Page 74 and 75: Figure 4.1: Subdivision X Drainage
Page 76 and 77: Figure 4.2: Depth-Duration diagram
Page 78 and 79: coefficient C as in Equation 4.1. W
Page 80 and 81: (Row 27) and the excess rainfall in
Page 82 and 83: Figure 4.6: Modified rational metho
Page 84 and 85: Figure 4.8: Modified rational metho
Page 86 and 87: this report to estimate T ∗ . Wit
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5. Summary and Conclusions This sec
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coefficients entirely). The method
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Bibliography Akan, A. O. (2002). Mo
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FHWA (1983). Urban Highway Storm Dr
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Roussel, M. C., D. B. Thompson, X.
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A. Literature Review The rational m
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Intensity (in/hr) 0 1 2 3 4 5 0 20
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storms with durations greater than
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The intent is to estimate the disch
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Figure A.3: Modified Rational Formu
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where L is a characteristic length
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Table A.1: Rational method runoff c
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Table A.1: Rational method runoff c
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Figure A.4: Runoff Coefficients for
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B. Basin Development Factor The bas
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3. Storm drains, or storm sewers
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C. The iPar Program and Parsing of
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The values for C can not be decoupl
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Figure C.2: Screenshot of custom-bu
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Figure C.3: Screenshot of output fo
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logSrhs 0.26969 0.04056 6.650 3.44e
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hook operator provides a compact no
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20 MEDIAN TIME-R (LAG TIME), IN HOU
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1 0.9 0.8 0.7 MEDIAN CR 0.6 0.5 0.4
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# print ( DDF .at.F); # print ( IDF
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the .I
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eturn ( get (x, COUNTY_NAMES ))), M
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mycol25
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D. Modified Rational Method The pur
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100.0 Cumulative Distribution (%) 8
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10000 Modeled Q p (cfs) 1000 100 1:
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10000 Modeled Q p (cfs) 1000 100 1:
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where A is the drainage area in squ
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500 Station: 08048550 DryBranch For
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10000 MRM Gamma 1:1 line Modeled Q
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