DISCUSSIONS AND CLOSURES

Transcription

1 DISCUSSIONS AND CLOSURES Discussion of Rational Hydrograph Method for Small Urban Watersheds by James C. Y. Guo July/August 2001, Vol. 6, No. 4, pp Sushil K. Singh 1 1 Scientist El, National Inst. of Hydrology, G.P.S. Regional Center, Sagar M.P., India. sukusi@hotmail.com The author has presented an application of rational method to obtain a hydrograph for small urban catchments considering the temporal variation of fractional catchment area contributing to runoff at the outlet. A system memory for computing the average rainfall intensity while applying Eq. 1 is assumed equal to time of concentration, T c. The author expresses the contributing catchment area by Eq. 5 without any condition on T. While a condition T c T 0 is put for Eq. 6, Eq. 5 is not applicable for all values of T. Hence, Eq. 5 needs some correction to maintain consistency and to avoid the error of A e A for T T c. The following equations should be substituted for Eq. 5 : A e A T T c for T c T 0 (1a) A e A for T d T T c (1b) Eq. 1 of this discussion shows that the contributing catchment area varies linearly with T for T T c and is equal to the catchment area for T T c. The runoff for the respective cases is given by Eqs. 6 and 3. For unit rainfall intensity continuous in time is assumed to occur; the resulting hydrograph will be an S-hydrograph. Eqs. 6 and 3 lead to an S-hydrograph, which is a straight line between T 0 and T T c. For T T c, the ordinate of the S-hydrograph is constant, the value of which is given by CA. Hence, Eq. 5 may not be justified or appropriate for all catchments, irrespective of their shape. HEC-1 recommends a different form in place of Eq. 1 of this discussion for representing temporal variation in the contributing catchment area. In view of this, further elaboration is needed. The author states p. 353 that the parameter, T c, was optimized from the observed hydrograph using the least-squares method. But he uses Eq. 10, which is not in a form of leastsquares. Does the use of Eq. 10 in place of minimization of integral squared error have any specific advantage? On pages , the author states that the discrepancy on the rising hydrograph Fig. 2 is caused mainly by surface detention. It is better to express in a different way that the cause of this discrepancy is the lack of representation of actual conditions by Eq. 1a of this discussion. The region of discrepancy is dominated by T T c. Since small catchments are being considered, the storage effects may be neglected. As evident from the Introduction and Conclusion, the application of the proposed method was compared to that of HEC-1. Since the details of such comparisons are not available in the paper, this point appropriate for discussion could not be taken up. However, any comparison with application of HEC-1 should be a cautious one, as it uses a different form of Eq. 5. Closure to Rational Hydrograph Method for Small Urban Watersheds by James C. Y. Guo July/August 2001, Vol. 6, No. 4, pp James C. Y. Guo 1 1 Professor, Dept. of Civil Engineering, Univ. of Colorado at Denver, Denver, CO jguo@carbon.cudenver.edu The writer is thankful to the discusser, Sushil K. Singh, for his interest in the paper. The replies to the questions are given as follows: 1. The discusser s suggestion on the applicability limits for Eq. 5 is correct. The discusser s Eq. 1 agrees with the writer s statements in the paper. The paper provides three sets of equations that apply to three distinctive segments on the hydrograph. Eq. 5 is only applicable to the rising hydrograph, as stated in the paragraph immediately before Eq. 5 in the paper, and Eqs. 6 and 7 further define that the rising hydrograph is the segment between 0 T T c, in which T elapsed time and T c time of concentration. 2. The discusser raises an interesting question on the S-hydrograph. The ordinates of a S-hydrograph are defined as t T S T t 0 q t (1) in which S(T) ordinate on S-hydrograph at time T; q(t)-runoff rate on direct runoff hydrograph; and t time variable. Generated from a small watershed, the S-hydrograph reaches its constant as S T B t 0 t T B q t CAP in which T B base time of direct runoff hydrograph; C runoff coefficient; A tributary area; P rainfall depth; and t incremental time interval. The discusser stated that the S-hydrograph can reach its constant given by the product of CA after T T c. This observation is correct only if the rainfall excess is uniform and long Guo In this paper, Eqs. 3 and 6 were developed to predict the runoff rates under a nonuniform rainfall distribution. The resultant runoff hydrograph fluctuates according to the temporal variations on the rainfall distribution, and its S-hydrograph increases with respect to the incremental rainfall depths. The constant of the S-hydrograph is not only determined by C and A, but also by P and t. 3. Eq. 10 in the paper was intended for the least-square method. Taking the square root on the accumulative squared error was to reduce the data scattering from all tests used in the paper. 4. During an event, both surface and depression detention vol- t (2) JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MAY/JUNE 2003 / 165

2 Therefore, the writer suggests that the rational hydrograph method be applicable to urban catchments up to 150 acres. Fig. 1. Comparison between rational hydrograph method and SCS unitgraph method using HEC-1 model umes impose storage effects on the runoff process. The rational method is essentially a special case of the kinematic wave method using the runoff coefficient to model the hydrologic losses Guo The tests represented in the paper indicate that the runoff coefficient can model the soil infiltration loss during the event, but not the initial and depression losses occurred at the beginning of the event. This conclusion agrees with the concept of incipient runoff depth investigated by EPA Guo and Urbonas 1996, Guo and Hughes 2001, and Guo and Urbonas Not every rainfall event can produce runoff until the rainfall depth exceeds the incipient runoff depth. The linearity nature of the rational hydrograph method does not abstract the initial losses separately. As a result, the method tends to overestimate the runoff rates at the beginning of the event when the initial losses are significant. However, such deficiency can be quickly recovered as time goes on. 5. The discusser s comment on the catchment shape effect on Eq. 5 is acceptable. As stated in the paper, without knowing the time-area relationship, the linear assumption using Eq. 5 can be otherwise used. The tests presented in the paper gave good agreements with the laboratory data and with the field measurements. Although Eq. 5 is simple, the linear time-area relationship seems to be applicable to small urban catchments. This observation may be attributed to the cascading overland flows through urban settings. 6. Often the size of the watershed dictates the selection of hydrologic methods. The application limit of the rational hydrograph method was investigated in the paper by comparing the linear predictions to the nonlinear predictions using the Soil Conservation Service s SCS unit hydrograph method. Since the default time-area curve in HEC-1 had never been used in the study, the consistence between methods was not a concern. A set of hypothetical square watersheds was studied under the SCS 6 h rainfall distribution. The flow path through a hypothetical watershed was assumed to be diagonal on a ground slope of The hydrologic losses were modeled by the rational hydrograph method with a runoff coefficient of 0.75 or the HEC-1 procedure using a curve number of 85. As shown in Fig. 1, the predicted peak flow rates by both the linear and the nonlinear methods closely agree with each other until the tributary area becomes greater than 150 acres. References Environmental Protection Agency EPA Methodology for analysis of detention basins for control of urban runoff quality. EPA440/ , Washington, D.C. Guo, J. C. Y Overland flow on a pervious surface. International Water Resources Association, Int. J. Water, Guo, J. C. Y Storm hydrographs for small catchments. American Water Resources Association, Int. J. Water, Guo, J. C. Y., and Hughes, W Storage volume and overflow risk for infiltration basin design. J. Irrig. Drain. Eng., 127 3, Guo, J. C. Y., and Urbonas, B Maximized detention volume determined by runoff capture rate. J. Water Resour. Plan. Manage., 122 1, Guo, J. C. Y., and Urbonas, B Runoff capture and delivery curves for storm-water quality control designs. J. Water Resour. Plan. Manage , Discussion of Explicit Solution to Green and Ampt Infiltration Equation by Sergio E. Serrano July/August 2001, Vol. 6, No. 4, pp D. A. Barry 1 ; J.-Y. Parlange 2 ; G. C. Sander 3 ;L.Li 4 ; D.-S. Jeng 5 ; and W. L. Hogarth 6 1 Professor, Contaminated Land Assessment and Remediation Research Centre, Institute for Infrastructure and Environment, School of Engineering and Electronics, The Univ. of Edinburgh, Crew Building, King s Buildings, Edinburgh EH9 3JN, U.K. 2 Professor, Dept. of Biological and Environmental Engineering, Cornell Univ., Ithaca, NY Reader, Dept. of Civil and Building Engineering, Loughborough Univ., Loughborough, Leicestershire LE11 3TU, U.K. 4 Senior Lecturer, Dept. of Civil Engineering, The Univ. of Queensland, Brisbane, Queensland 4072, Australia. 5 Senior Lecturer, School of Engineering, Griffith Univ., PMB 50 Gold Coast Mail Centre, Gold Coast, Queensland 9726, Australia. 6 Pro Vice Chancellor, Faculty of Science and Information Technology, Univ. of Newcastle, University Drive, Callaghan, New South Wales 2308, Australia. The author presents approximations for cumulative infiltration, F(t), modeled by the classical Green and Ampt formula, as well as the Green Ampt surface flux, f (t). For the sake of brevity, we focus on f (t), and so consider Eq. 17 of the original paper first. The author notes that Barry et al showed that the Green Ampt infiltration formula can be expressed explicitly in terms of the Lambert W function as shown in Eq. 7 of the original paper. That is, approximations to the Green Ampt infiltration formula amount to approximations for W and vice versa. The discussers recall, see Eq. 6, that W(x) is the function that satisfies W exp(w) x. Barry et al provide some simple approximations to the Lambert W function, which can be used to approximate Green and Ampt infiltration. Rather than proceed down this obvious path, we note that Barry et al. 1993, pg. 46 mention the similarity of their approximations on the even simpler approximation of Brutsaert 1977, pg The Brutsaert approximation for W(x) can be written in the form 18 W x 7 3 /2 1/2, exp1x 0 (1) 166 / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MAY/JUNE 2003

3 Fig. 1. Flux according to different solutions using parameters as in Fig. 4 of the original paper. Fig. 2. Comparison of approximations for f (t) with exact Green and Ampt formula, Eq. 10. where 1 ln( x). In the author s notation, Brutsaert s approximation for F(t) is, from Eqs. 7 and 1 above, simply where 36a F ta 6 G 62G 1/2, t t p (2a) G t B t ln a (2b) a Differentiating Eq. 2a above with respect to t gives f t K 1 1/2 1 2G 1 2G 1/2 2, t t 6 p (3) Both Eqs. 2 and 3 presented here are remarkably simple formulas. The accuracy of Eq. 3 above was checked by plotting it for the case considered in Fig. 4. In Fig. 1, we give the exact solution, as well as Eqs. 17 and 3. Obviously, Eq. 17 is a worse approximation for Green and Ampt infiltration than Eq. 3. The maximum relative error of Eq. 3 is just 0.035%. In contrast to the case shown in Fig. 1, Green Ampt infiltration is more generally applied for infiltration into a dry soil, a case we consider here also, i.e., we take t p F p 0. It is clear that the final term on the right-hand side of Eq. 17 is dimensionally incorrect. Because of the ambiguous dimensions in Eq. 17, we shall consider a 1 in a set of consistent units this procedure circumvents the dimensionality error in Eq. 17 and so permits a check of its predictive capability and plot f /K versus Kt. In Fig. 2 then, the predictions of Eq. 17 are again compared with the exact solution Eq. 10. Clearly, the accuracy of Eq. 17 is poor. On the other hand, Eq. 3 of this discussion is a very satisfactory approximation for all Kt. The maximum relative error for Eq. 3 in Fig. 2 is less than 0.5%. This error is likely acceptable for most practical applications. More accuracy is, in any case, easily obtained using the analytical approximation for W given by Barry et al ; their formula is still very simple, yet has a maximum relative error of 0.025%. Arbitrary precision estimates of W are also available Barry et al. 1995a,b. The infiltration rate, f (t), in Eq. 17 was obtained after differentiating Eq. 16 with respect to t. We note that Eq. 16 is also dimensionally incorrect. Direct differentiation of Eq. 16 yields a very complicated expression, quite different to Eq. 17, which we do not write here. However, Fig. 2 includes its predictions; clearly, the results are poor. In contrast to Eqs. 16 and 17, Eq. 15 is a dimensionally correct approximation for F(t), although, undoubtedly, it is unwieldy. This expression was differentiated the resulting expression is, again, too lengthy to report to derive another expression for f (t), with the results plotted in Fig. 2. In this case, again, the loss of accuracy for small Kt is apparent. Fig. 2 shows that Eqs. 15, 16, and 17 of the original paper all give very different behaviors for f (t) as t 0. Series expansions for these equations give, for Eq. 15, f /K 5, for Eq. 16, f /Kt 3, and for Eq. 17, f /Kt 2. Besides being internally inconsistent Eqs. 16 and 17 were both derived directly from Eqs. 15 none of these limiting results is consistent with the t 1/2 small-time behavior for Green Ampt infiltration, a behavior which is reproduced by Eq. 3. In summary, we have shown that Eq. 17 is a poor predictor of Green Ampt surface flux, and clearly offers nothing over the existing approximation of Brutsaert 1977 and, by extension, those of Barry et al In any case, the expressions for infiltration and flux, i.e., Eqs. 16 and 17, are dimensionally incorrect, a defect that makes them wholly unsuitable for practical application. Although dimensionally correct, the derivative of Eq. 15 is also a poor predictor of Green Ampt flux, with the incorrect small t behavior. It is somewhat perplexing that the flux expressions calculated from Eqs. 15 and 16 do not reduce to Eq. 17, and in particular give inconsistent estimates of f (t) for small t, although they are all versions of the same formula. In contrast, the formula Eq. 3 is simple, reduces to the correct limiting behavior for small and large t, with an accuracy sufficient for most routine use. Finally, although we have not presented results for the cumulative infiltration, the significant flaws that are evident in the author s approximations for f (t) not surprisingly carry over to F(t). JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MAY/JUNE 2003 / 167

4 References Barry, D. A., Barry, S. J., and Culligan-Hensley, P. J. 1995a. Algorithm 743: WAPR: A FORTRAN routine for calculating real values of the W-function. ACM Trans. Math. Softw., 21 2, Barry, D. A., Culligan-Hensley, P. J., and Barry, S. J. 1995b. Real values of the W-function. ACM Trans. Math. Softw., 21 2, Barry, D. A., Parlange, J.-Y., Sander, G. C., and Sivaplan, M A class of exact solutions for Richards equation. J. Hydrol., , Brutsaert, W Vertical infiltration in a dry soil. Water Resour. Res., 13 2, Closure to Explicit Solution to Green and Ampt Infiltration Equation by Sergio E. Serrano July/August 2001, Vol. 6, No. 4, pp Sergio E. Serrano 1 1 Professor of Hydrologic Engineering, Dept. of Civil Engineering, Univ. of Kentucky, Lexington, KY sergio@engr.uky.edu In this closure, the writer conclusively restates and demonstrates the features of the result presented in Serrano 2001a it is the first explicit solution of the Green Ampt infiltration equation. It is neither a difficult-to-calculate, mostly unavailable Lambert W series approximation nor an empirical approximation to the Lambert W series approximation. The results in Serrano 2001a were obtained using a decomposition series, which is a systematic procedure to obtain analytical solutions to nonlinear equations without linearization, perturbation, or discretization. In this case, decomposition yields a remarkably accurate result. However, there are several characteristics that are obvious to anyone familiar with asymptotic series, as we shall see when we analyze the discussion by Barry et al. I invite the open-minded hydrologists and engineers to find new hydrologic solutions to many unresolved problems with decomposition. For an introduction to the method, the reader may consult Adomian For linear and nonlinear examples in hydrology, the reader may consult Serrano 1997, 2001b. In their discussion, Barry et al. conveniently avoid for the sake of brevity the main result presented in Serrano 2001a that is the infinite series represented by Eq. 15. Barry et al. fail to acknowledge the remarkable accuracy achieved with just a few terms of this equation. Figs. 1, 2, and 3 in Serrano 2001a prove this point. However, an infinite series, made of infinite terms, may in principle produce an infinite set of curves depending on which terms of the series, and in which arrangement, the user applies them. Thus, if we use the first three terms of the series, we obtain one curve. If we use five terms, we obtain another similar curve, etc. In a convergent series, however, all those possible curves approach a unique shape when many terms are used. Eq. 16 in Serrano 2001a is simply a manipulation of Eq. 15. Granted, there are some transcription errors, but even if there were not, Eqs. 15 and 16 may contain different terms at the time of calculation and thus produce slightly different curves. This does not prove anything against the results. It only illustrates some obvious features of asymptotic series. Barry et al. seem amazed to discover this and spend a good part of their discussion on this. Their Fig. 6 and most of this discussion are better suited for an algebra textbook. Fig. 1. Comparison between decomposition, exact, Lambert W series, and Barry et al. solutions for cumulative infiltration. To further prove my point, let us rewrite Eq. 15 with the same notation but in shorter form. F 1 a ln F 0 ta, F 0 t B tf p, B tk t t p F 2 a ln F 0 ta a F 0 ta (1) F 3 a ln F 0 ta a 2 a ln F 0 ta 2 F 0 ta a F 0 ta F 4 a ln F 0 ta a 3 F 0 ta ln2 3 2 a F 0 ta a 3 a ln F 0 ta 3 F 3 0 ta a F 0 ta ] We immediately see a pattern that allows us to use more terms. For simplicity set m 1 (t)f 0 (t) a /(), and m 2 (t) a/ F 0 (t) a. Thus, from Eq. 1, or from Eq. 15 in Serrano 2001a, F F 0 F 1 F 2 or F F 0 ta ln m 1 t a ln m 1 tm 2 t 1 m 2 tm 2 2 tm 2 3 t a ln 2 m 1 t m 2 2 t1 m 2 tm 2 2 t a ln 3 m 1 t 1 m 2 tm 2 2 t By identifying partial series and simplifying, we arrive at a simple expression very close to the exact solution: 2 (2) 168 / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MAY/JUNE 2003

5 Fig. 2. Comparison between decomposition, exact, Lambert W series, and Barry et al. solutions for infiltration rate. Fig. 3. Behavior of decomposition and Lambert W series solution as t 0 F tf 0 ta ln m 1 t m 2 t 1 (3) 1 m 2 t 1 m 2 t ln m 1 t Eq. 15 in Serrano 2001a is sufficiently accurate for most practical applications. However, Eq. 3 contains more terms of the decomposition series than Eq. 15 in Serrano 2001a and improves accuracy slightly. Using the same data set and Eq. 3 in this closure, we reproduce Fig. 3 in Serrano 2001a as Fig. 1 in this closure. The exact decomposition series and Lambert W approximation are virtually indistinguishable from one another. This result corroborates those obtained in Serrano 2001a and completely disproves all claims in Barry et al. Clearly, the accuracy of decomposition is remarkable. Fig. 1 in this closure also shows the form of Eqs. 2a and 2b in Barry et al. allegedly offered as a most accurate one. It is obvious that such approximation yields erroneous results. In fact, I cannot replicate any of the figures in Barry et al. and since no explanation is given as to their calculations we can only conclude they are incorrect. Barry et al. argue that the Lambert W series approximation is simpler and more accurate than decomposition. The writer included the Lambert W series in Serrano 2001a analysis for historical completeness. However, the Lambert W does not constitute a standard function. It is an expression of the series that results when attempting to solve the Lambert W equation. These series are not simple to compute. There are several approximations available in the literature with various degrees of accuracy. By far the most important limitation lies in the fact that the Lambert W series is unavailable in nearly all calculators and spreadsheets software. In fact, the writer is only aware of its availability in Maple software algebra. As a result, the user who wants to use the Lambert W series must program one of their equations in the computer and deal with the usual issues of convergence and numerical approximation. This contradicts the aim of simplicity in infiltration calculations. Barry et al. argue that Eqs. 2a, 2b, and 3 are better approximations. However, as we have seen, this approximation of approximations gives erroneous results. Thus, I understand the desire by Barry et al. to resurrect their old theories, but the new results are clearly superior. Differentiation of Eqs. 15 or 16 in Serrano 2001a, orof Eq. 3 in this closure, yields the series Eq. 17 in Serrano 2001a for the infiltration rate f (t). As we stated earlier, there are some transcription errors in Eq. 16 which were carried into Eq. 17 in Serrano 2001a. Even with these errors, the accuracy of Eq. 17 gives relative errors of less than 0.5%. This error is likely acceptable in most practical applications to quote Barry et al. statement for one of their calculations with a 0.5% relative error. Nonetheless, let us correct the errors and further prove the accuracy of decomposition. Barry et al. state that direct differentiation of Eq. 16 is a very complicated expression.... To prove them wrong again, let us differentiate Eq. 3 in this closure, which contains many more terms in the decomposition series than Eq. 16 in Serrano 2001a. It is simply m f K 2 2 t 1 m 2 t 1 m 2 t 1 m 2 t ln m 1 t 1 ln m 1 t m 2 t lnm 1 t 1 m 2 t m 2 t ln m 1 t 1 ln m 1 1 m 2 t ln m 1 t (4) Fig. 2 in this closure shows the infiltration rate from Eq. 4 in this closure using the same data set. Clearly, the accuracy of decomposition is remarkable. Fig. 2 also shows Eq. 2b and 3 in Barry et al. Once again, their empirical approximation gives erroneous results. Barry et al. further attempt to show the behavior of these solutions when t 0. Without any manipulation of the data, let us observe the small-time behavior of these solutions. Thus, by setting t p 0 and plotting f /K Fig. 3 in this closure is produced. Note again the vast superiority of the decomposition solution and how it approaches to the physically based theoretical curve of t 1/2, that is, the small-time behavior for Green and Ampt infiltration. In contrast to that, observe the behavior of the Lambert W series, which collapses at around t 0.7 min, when it enters the complex plain and no real solutions are possible. In conclusion, we have demonstrated that the arguments in Barry et al. discussion are false, and that the results contained in Serrano 2001a constitute a new contribution to the modeling of infiltration and the first explicit solution to the Green and Ampt infiltration equation. This solution exhibits remarkable accuracy that converges fast to the exact solution, and simplicity far superior to the old Lambert W series approximations, and their empirical approximations. JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MAY/JUNE 2003 / 169

6 References Adomian, G Solving frontier problems in physics The decomposition method, Kluwer Academic, Boston. Serrano, S. E Hydrology for engineers, geologists, and environmental professionals. An integrated treatment of surface, subsurface, and contaminant hydrology, HydroScience Inc., Lexington, Ky. Serrano, S. E. 2001a. Explicit solution to Green and Ampt infiltration equation. J. Hydrologic Eng. 6 4, Serrano, S. E. 2001b. Engineering uncertainty and risk analysis. A balanced approach to probability, statistics, stochastic modeling, and stochastic differential equations, HydroScience Inc., Lexington, Ky. 170 / JOURNAL OF HYDROLOGIC ENGINEERING ASCE / MAY/JUNE 2003