Comparison of constitutive flow resistance equations based on the Manning and Chezy equations applied to natural rivers

WATER RESOURCES RESEARCH, VOL. 41,, doi:10.1029/2004wr003776, 2005 Comparison of constitutive flow resistance equations based on the Manning and Chezy equations applied to natural rivers David M. Bjerklie U.S. Geological Survey, East Hartford, Connecticut, USA S. Lawrence Dingman Department of Earth Sciences, University of New Hampshire, Durham, New Hampshire, USA Carl H. Bolster U.S. Department of Agriculture, Agricultural Research Service, Bowling Green, Kentucky, USA Received 1 November 2004; revised 18 July 2005; accepted 26 July 2005; published 10 November 2005. [1] A set of conceptually derived in-bank river discharge estimating equations (models), based on the Manning and Chezy equations, are calibrated and validated using a database of 1037 discharge measurements in 103 rivers in the United States and New Zealand. The models are compared to a multiple regression model derived from the same data. The comparison demonstrates that in natural rivers, using an exponent on the slope variable of 0.33 rather than the traditional value of 0.5 reduces the variance associated with estimating flow resistance. Mean model uncertainty, assuming a constant value for the conductance coefficient, is less than 5% for a large number of estimates, and 67% of the estimates would be accurate within 50%. The models have potential application where site-specific flow resistance information is not available and can be the basis for (1) a general approach to estimating discharge from remotely sensed hydraulic data, (2) comparison to slope-area discharge estimates, and (3) large-scale river modeling. Citation: Bjerklie, D. M., S. L. Dingman, and C. H. Bolster (2005), Comparison of constitutive flow resistance equations based on the Manning and Chezy equations applied to natural rivers, Water Resour. Res., 41,, doi:10.1029/2004wr003776. 1. Introduction [2] Modeling of gradually varied or unsteady open-channel flows relies on the use of a general flow resistance or constitutive equation that characterizes the relation between energy gradient and flow rate. The constitutive relation is generally described as Q ¼ K A g 1=2 R p S q e ; where Q is discharge (L 3 T 1 ), A is cross-sectional area (L 2 ), R is hydraulic radius (L), S e is the energy slope, K is the channel conductance (inverse of resistance) and is dimensionless, g is the acceleration due to gravity (L T 2 ), and p and q are exponents. Specification of the constitutive relation is not straightforward because there is uncertainty about the values of p and q [Manning, 1889; Henderson, 1966; Leopold et al., 1960; Golubstov, 1969] and the way in which the coefficient, K, varies with flow and boundary characteristics. [3] The most widely used constitutive relation is the Manning equation: ð1þ Q ¼ ðu m =nþa R 2=3 S 1=2 ; ð2þ where u m is a unit-dependent proportionality constant and n is the Manning resistance coefficient (TL 1/3 ). The Copyright 2005 by the American Geophysical Union. 0043-1397/05/2004WR003776 Manning equation is an empirical modification of the Chezy equation: Q ¼ K 0 A ðg R SÞ 1=2 ; ð3þ where K 0 is a dimensionless conductance coefficient. The Chezy equation can be derived from force-balance relations, is dimensionally homogeneous, and is based on the assumption that resistance is proportional to velocity squared (V 2 ). As pointed out by Leopold et al. [1960], that assumption may only be true if the flow boundary does not change as velocity is varied. This condition is generally true for pipe flow but not for open channels where the boundary changes substantially with discharge. Flow resistance, particularly when considering laminar flow, is a function of the Reynolds number. However, when considering fully turbulent flow (when the shear Reynolds number is greater than about 400), and for relatively small-scale roughness (when the water depth is five to ten times the roughness height [Katul et al., 2002]), the Reynolds number is less important, and flow resistance can be defined as the ratio between stream velocity and shear velocity [Smart et al., 2002]. [4] Several studies, including Golubstov [1969], Riggs [1976], Jarrett [1984], and Dingman and Sharma [1997], suggest that a wide range of flows can be successfully modeled using a universal value for the conductance coefficient when nontraditional values of p and q are specified. Henderson [1966] also suggests that it may be possible to model open-channel flows using a constant conductance 1of7

BJERKLIE ET AL.: TECHNICAL NOTE Table 1. Flow Measurement Database: Range of Hydraulic Variables Parameter Symbol Units Mean Standard Deviation Coefficient Variation Maximum Minimum Calibration Data N = 680 Discharge Q m 3 /s 860 2434 2.83 27576 0.01 Top width W m 128 159 1.24 1009 2.9 Mean depth (hydraulic radius) Y m 2.38 2.24 0.94 12.39 0.1 Mean velocity V m/s 1.15 0.62 0.54 5.1 0.02 Water surface slope (average) S m/m 0.0029 0.0056 1.93 0.04 0.000043 Validation Data N = 357 Discharge Q m 3 /s 717 1960 2.73 17837 0.02 Top width W m 126 146 1.16 765 3.1 Mean depth (hydraulic radius) Y m 2.33 2 0.86 12.7 0.18 Mean velocity V m/s 1.11 0.59 0.53 3.53 0.02 Water surface slope (average) S m/m 0.0021 0.0042 2.00 0.04 0.000043 coefficient for stable channels at bank-full stage, at least to the accuracy obtainable by the usual subjective methods. This is especially important, because confirmation of this finding would free the modeler from the inherently subjective and highly uncertain [Hydrologic Engineering Center, 1986] process of estimating the value of the conductance coefficient. [5] In this paper, generally appropriate constant values of K and q are developed for use in natural rivers. The models are derived for application under the same conditions that the Manning or Chezy equation would be used, fully rough turbulent flow and low relative roughness (shear Reynolds number greater than about 400 and water depth five to ten times the roughness height). Additionally, the models are intended for use with hydraulic information that can be directly measured and interpreted in the field including width, depth, and channel slope, and where site-specific flow resistance information is lacking. For this reason, we have based our analysis on data that have been obtained entirely from measurements made in actual rivers. 2. Calibration and Validation Data [6] A large discharge measurement database was compiled for calibration and comparison of the discharge-estimating models. The database includes 1037 flow measurements from 103 river sites in the United States and New Zealand. The data were obtained from Barnes [1967] (N = 22); Hicks and Mason [1991] (N = 330); Coon [1998] (N = 215); and from the U.S. Geological Survey s (USGS) online National Water Information System (NWIS) database (see http:// nwis.waterdata.usgs.gov/usa/nwis/measurements) (N = 470). The data do not include rivers with maximum top widths of less than about 10 m, and with the exception of a few extreme low flows, the data used for the analysis are considered to be associated with hydraulically wide flow conditions [Chow, 1959]. Thus throughout this paper the hydraulic radius and the mean depth of flow are considered equivalent, and the mean depth was used to represent the characteristic flow depth. [7] We excluded reaches that were highly braided, that exhibited control on the slope due to backwater effects, and that had large expansion or contraction of flow (evaluated based on a subjective inspection of available maps and documentation). These channel selection criteria were implemented so that the depth and velocity could be considered a reflection of channel resistance. Thus rivers contracted by a bridge, a natural feature such as a canyon or narrows, and rivers whose slope is controlled by a dam are not included in this database. Information about bed load and sediment transport conditions was not available or considered in the channel selection criteria. Additionally, the data are not homogeneous with respect to channel form other than the exclusion of braided channels. [8] The channel characteristics of each river in the database were evaluated on the basis of information available from the data sources or from inspection of topographic maps of the channel at each station. For comparative purposes, the database was randomly divided into a calibration data set (N = 680) and a validation data set (N = 357) (Table 1). With the exception of the data from Barnes [1967], which includes only one discharge measurement per river station, each river station includes 5 to 20 in-bank discharge measurements, across a wide range of flow conditions. The data set includes a measured discharge, width and/or cross-sectional area, mean depth and/or hydraulic radius, mean velocity, and channel or average water surface slope. [9] Bjerklie et al. [2003] have shown that when analyzing a diverse data set of river discharge measurements, the channel slope (or average water surface slope) can be used in lieu of a measured water surface slope without significantly changing the predictive qualities of the relations derived from the data. The data used for this study were obtained from reaches that do not exhibit a break in slope or backwater control on the slope, determined as a criterion of the data collection or from inspection of topographic maps. Thus either an average water surface slope (averaged from five or more measurements) or a topographic channel slope for the reach is used instead of a measured water surface slope. This practice is further justified because water surface is often difficult to measure, especially remotely [Bjerklie et al., 2003]. 3. Comparative Models and Statistical Measures [10] Multiple regression analysis of discharge measurements from a wide range of river size and morphology [Dingman and Sharma, 1997; Bjerklie et al., 2003] suggests that, in natural rivers, the exponent on the slope term, q, ina general constitutive equation is closer to a value of 0.33 instead of 0.5, as proposed by Manning. A slope exponent 2of7

BJERKLIE ET AL.: TECHNICAL NOTE Table 2. Multiple Regression Model Statistics Coefficient Value t Statistic p-value Lower 95% Upper 95% Intercept (log k) 0.68 14.51 <0.0001 0.59 0.78 X 1 (W X ) 1.10 50.02 <0.0001 1.06 1.15 X 2 (Y X ) 1.63 55.57 <0.0001 1.57 1.68 X 3 (S X ) 0.33 19.03 <0.0001 0.30 0.37 of 0.33 can also be derived from analysis of flume data presented by Leopold et al. [1960]. On the basis of these studies, it is proposed that a modified form of the Manning and Chezy equations that use a value of 0.33 for the slope exponent rather than 0.5 be tested for specific applications in natural rivers. Assuming that the mean flow depth is equivalent to the hydraulic radius (Y = R), the Manning equation is modified to Q ¼ k m W Y 1:67 S 0:33 ; and the modified Chezy equation is Q ¼ k c g 0:5 W Y 1:5 S 0:33 ; where k 1 and k 2 are coefficients of conductance and the cross-sectional area, A, is replaced by the mean depth times the top width (W Y). [11] Four conceptual discharge estimation models are compared with a model derived using multiple regression. Each model was calibrated using the same data. The models are based on the constitutive elements of flow (width, depth, and slope) with four different combinations of p and q. These comparative models include the Manning equation (6), the modified Manning equation (7), the Chezy equation (8), the modified Chezy equation (9), and the regression model (10): Q ¼ k 1 W Y 1:67 S 0:5 ; Q ¼ k 2 W Y 1:67 S 0:33 ; ð4þ ð5þ ð6þ ð7þ (9) were derived from the Chezy p ffiffiffi equation, the values for k 3 and k 4 include the constant g (see equation (3)) and thus these coefficients could p ffiffiffi be derived in their dimensionless form by dividing by g. An optimized conductance coefficient was determined for each model by finding the constant value that reduced the mean of the log-residuals of the predicted discharge and observed discharge for the calibration data set to approximately zero. The coefficients were also derived by minimizing the sum of squares; however, the resulting values for the coefficients tended to increase overprediction at low discharges. For this reason and because the log-residual distribution was the most random measure of error, the zero mean of the log-residual was retained as the best approach to determining the values for the conductance coefficients. [13] The predictive qualities of the models given by discharge estimation models (6) through (10) were compared by examining the mean and standard deviation of the log-residual, the relative residual, and the actual residual for each observation in the calibration data. These residual measures are defined for each observation as Log-residual ¼ log Q p log ð Qi Þ; ð11þ Relative residual ¼ Q p Q i Q i ; ð12þ Actual residualðr i Þ ¼ Q p Q i ; ð13þ where Q p is the predicted discharge and Q i is the observed discharge (in cubic meters per second). The residuals were calculated for each pair of predicted and observed values from the calibration (N = 680) and validation data sets (N = 357). The validation data indicate model performance for data not used in the calibration. The mean and standard deviation of the residuals generated by each model provide a measure of the average and expected variability of model uncertainty. [14] The performance of the proposed models was also evaluated using a general measure of total uncertainty, the root mean square error (RMSE) defined as Q ¼ k 3 W Y 1:5 S 0:5 ; Q ¼ k 4 W Y 1:5 S 0:33 ; ð8þ ð9þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P u N t ðr i Þ 2 i¼1 RMSE ¼ N ð14þ Q ¼ 4:24 W 1:10 Y 1:63 S 0:33 : ð10þ The coefficient of determination (r 2 ) for the regression model is 0.97 with a standard error of 0.19 (log 10 (Q[m 3 /s])). The regression statistics indicate that all of the coefficients are highly significant (Table 2) at the 95% confidence level. It is of interest to note that within the 95% confidence interval, the exponents on width, W, and depth, Y, are nearly equal to the values used in the modified Manning equation, and the exponent on slope, S, is equal to 1/3, similar to previous studies. [12] The values for k 1 through k 4 are unit-dependent (meters and seconds); however, because equations (8) and 3of7 and a model selection criterion (MSC) described by Koeppenkastrop and DeCarlo [1993] that compares the information content of each model and is defined as 0 MSC ¼ lnb @ 2 Q i Q avg P N i¼1 P N i¼1 ð Þ 2 r i 1 2 m C N A ; ð15þ where r i is the ith actual residual between model prediction and observation, Q i is the ith observed discharge, Q avg is the average of the observed values, N is the number of observations, and m is the number of fitting parameters.

BJERKLIE ET AL.: TECHNICAL NOTE Figure 1. Distribution of the discharge coefficient determined from the quasi-uniform flow data for (a) Manning, (b) modified Manning, (c) Chezy, and (d) modified Chezy. The coefficient of variation (CV) is indicated for each distribution. The model with the smallest RMSE has the smallest total uncertainty, and the model with the largest MSC contains the highest information content and is generally considered the most appropriate model for describing the observed data. 4. Results of the Model Comparisons [15] Histograms of the fitted values for the conductance coefficients k 1 through k 4 (Figure 1) show that the conductance coefficient tends to have a smaller range and simpler distribution when an exponent of 0.33 is used (k 2 and k 4 ). In addition, the lowest coefficient of variation (CV) values are associated with the models that use a slope exponent of 0.33 (equations (7) and (9)). This indicates that more of the variance normally associated with the conductance coefficients is explained using the modified value for q of 0.33 compared to the traditional value of 0.5. Thus in general it can be said that there is less potential for estimation error and therefore greater prediction accuracy when using constitutive equations that assume q = 0.33 for natural rivers. [16] The regression model (equation (10)) shows the least estimate uncertainty for the calibration and validation data set (Tables 3 and 4) and provides only slightly better prediction accuracy than the modified Manning equation (7). However, the modified Manning equation performed somewhat better than the regression model with regard to the RMSE and the MSC for the validation data set. These results suggest that the modified Manning equation can provide similar or even slightly better estimates of discharge compared to a statistically derived model, with the advantage that the model coefficients can be calibrated to specific reaches if information is available. The models with a slope exponent of 0.5 (equations (6) and (8)) performed the worst, indicating that when evaluating a diverse set of rivers over a wide range of flow conditions, the slope exponent is better approximated by a value of 0.33 rather than 0.5. On the basis of the statistics of the log and Table 3. Model Calibration Statistics Model N = 680 Log Residual Relative Residual Actual Residual, m 3 /s RMSE, m 3 /s MSC Q = 23.3 W Y 1.67 S 0.5 mean 0.0004 0.23 227 969 1.84 (Manning equation (6)) stdev 0.24 1.36 943 Q = 7.14 W Y 1.67 S 0.33 mean 0.0002 0.15 45 474 3.27 (Modified Manning equation (7)) stdev 0.19 1.04 473 Q = 25.2 W Y 1.5 S 0.5 mean 0.0005 0.34 353 1372 1.14 (Chezy equation (8)) stdev 0.27 1.92 1327 Q = 7.73 W Y 1.5 S 0.33 mean 0.0001 0.20 213 932 1.92 (Modified Chezy equation (9)) stdev 0.21 1.39 909 Q = 4.84 W 1.10 Y 1.63 S 0.33 mean 0.0023 0.14 71 469 3.28 (Regression equation (10)) stdev 0.19 0.89 464 4of7

BJERKLIE ET AL.: TECHNICAL NOTE Table 4. Model Validation Statistics Model N = 680 Log Residual Relative Residual Actual Residual, m 3 /s RMSE, m 3 /s MSC Q = 23.3 W Y 1.67 S 0.5 mean 0.0216 0.14 173 726 1.98 (Manning equation (6)) stdev 0.22 1.22 706 Q = 7.14 W Y 1.67 S 0.33 mean 0.0031 0.15 24 436 2.99 (Modified Manning equation (7)) stdev 0.20 1.04 438 Q = 25.2 W Y 1.5 S 0.5 mean 0.0256 0.2 275 1036 1.27 (Chezy equation (8)) stdev 0.25 1.64 1000 Q = 7.73 W Y 1.5 S 0.33 mean 0.0066 0.17 159 690 2.08 (Modified Chezy equation (9)) stdev 0.20 1.35 673 Q = 4.84 W 1.10 Y 1.63 S 0.33 mean 0.0014 0.14 72 566 2.46 (Regression equation (10)) stdev 0.19 0.92 562 relative residuals for the validation data, 67% of the estimates derived from equation (7) and (10) would be expected to be within plus or minus 50 to 60% of the actual value, with the mean accuracy for a large number of estimates within 5% of the actual value. [17] For the validation data, the modified Manning equation (7) and the regression equation (10) both overpredict at low discharge (Figure 2). Additionally, the models show a significant negative trend slope between the log residual and the log of discharge (at the 95% confidence level), with the regression model showing the flattest slope (Figure 2). This suggests that the models are generally incomplete and that additional information would improve their accuracy. Figure 3 shows that the log-residuals for the modified Manning and regression equations are strongly correlated with the Froude number with all of the models underpredicting discharge at high Froude numbers, and greatly overpredicting discharge at low Froude numbers. These results indicate that the models will be less applicable where flow is very rapid relative to depth (high Froude number) Figure 2. Observed versus estimated discharge and log residual distribution as a function of observed discharge for equations (7) and (10). Trend lines are shown on the residual distribution plots. 5of7

BJERKLIE ET AL.: TECHNICAL NOTE Figure 3. Log residual distribution as a function of width and Froude number for equations (7) and (10). Trend lines are shown on the plots. and where flow is very slow relative to depth (low Froude number). The results indicate that all of the models could be improved by developing general calibration functions for the conductance coefficient that are based on flow conditions. 5. Discussion and Conclusions [18] The need for general discharge-estimating equations is important where there is little or no reach-specific information available for calibration. This is particularly the case with respect to remote sensing of land surface hydrologic processes [Vörösmarty et al., 1999]. Attempts to quantify river discharge over large areas, especially on a near-real-time basis, will necessitate the use of prediction models that use remotely measured or estimated hydraulic variables and observable channel hydraulic properties. In this capacity, statistically based relations have been developed that use hydraulic information including width, channel slope, and either mean depth or mean velocity to estimate river discharge [Dingman and Sharma, 1997; Bjerklie et al., 2003]. The conceptually based models described or developed in this paper are generally preferable to multiple-regression-based models because they are not restricted by the data used in model development, provide similar accuracy, and are based on physical concepts that reflect or can be correlated with observed conditions. [19] Because of the large variance in the estimate uncertainty, application of these models may not be sufficient for individual rivers or small-scale studies. However, they may be appropriate for use (1) in regional applications where an estimate of the mean discharge over spatially diverse areas is needed, (2) in cases where no data exist, and (3) where the necessary accuracy is similar to slope-area measurements, which typically have a much wider range of uncertainty compared to standard discharge measurements [Kirby, 1987]. As indicated by the trends between the logresiduals and the hydraulic variables (discharge, width, depth, slope), the various models exhibit different error characteristics depending on the flow conditions. The residual trends indicate that prediction error would be reduced by additional explanatory information such as specification of K as a variable function of stream power [van den Berg, 1995], width, or the ratio of flow cross-sectional area to the bank-full channel cross-sectional area (A/A b ). Additionally, a priori classification of river types by general energy regime, which would account for some of the variance 6of7

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