COMPARISON OF FRACTIONAL SEDIMENT TRANSPORT METHODS

Baosheng Wu

Associate Professor, Department of Hydraulic Engineering,

Tsinghua University, Beijing 100084, China. Tel: 86-10-62772097,

E-mail: baosheng@tsinghua.edu.cn

Abstract: A comparison of fractional sediment transport methods is presented in this paper. The performance and variations in different methods are evaluatd based on flume and field data in the sand size range.  Results may be used as guidelines for the selection of fractional transport methods in engineering practice.

Keywords: sediment transport, fractional transport, fractional load, sediment mixtures

1  INTRODUCTION

The fractional transport of nonuniform sediment mixtures is a very challenging subject in the hydraulics of river flow.  Because of the complexity of the sediment transport processes, prediction of sediment transport rates by size fractions has not been accomplished following purely analytical methods.  All existing fractional sediment transport methods have been established relying on calibration using limited flume and field data collected under so-called steady uniform flow conditions.  When different methods are applied to a given river, computed results of fractional transport rates can vary drastically from each other and from actual measurements.

In the literature, comparisons of sediment transport functions are generally limited to the total load or bed-material load.  The performance and accuracy of fractional load computations using different methods are not known.  As a result, there are no guidelines available for the selection of fractional transport methods in engineering practice.  In this paper, the performance and variations in different fractional bed-material transport methods are evaluated based on flume and field data in the sand size range.

2  FRACTIONAL SEDIMENT TRANSPORT METHODS

Fractional transport of nonuniform sediment mixtures has intrigued scientists for decades.  Starting from Einstein (1950), many attempts including field measurements, laboratory studies, empirical and theoretical analysis, and numerical simulations have been made in the past to understand the mechanisms of nonuniform sediment transport and to predict the transport rates by size fractions.  Based on the treatment in formulations and the physical considerations in the development, the existing fractional sediment transport methods can be classified into four categories (Wu and Molinas, 1996; Wu, 1999): i) direct computation by the size fraction approach; ii) shear stress correction approach; iii) bed material fraction approach (BMF); and iv) transport capacity fraction approach (TCF).

The direct computation by the size fraction approach aims at computing sediment transport rates for each size fraction present in nonuniform mixtures.  After the computation of transport capacities corresponding to each size group, the bed-material load is calculated by the summation of fractional sediment transport rates.  The shear stress correction approach focuses on extending a uniform sediment transport formula to compute fractional transport rates for nonuniform sediment mixtures.  In doing so, the actual shear stresses acting on each size fraction are corrected by introducing a correction factor.  The BMF approach relates the fractional transport rate directly to the size distribution of bed material.  The fractional transport rate for a give size group is determined combining by its potential transport capacity and its availability on the channel bed.  The TCF approach computes the fractional transport rates by distributing the bed-material transport rate into size groups through a transport capacity distribution function.  First, the bed-material sediment concentration is computed by the use of a bed-material load equation.  Then, the computed bed-material concentration is broken into fractional concentrations by a transport capacity distribution function.

Fractional bed-material load methods selected for comparisons are: (i) the direct computation by size fraction methods of Einstein(1950), Laursen(1958), and Toffaleti (1969);( ii) the BMF approach using Engelund and Hansen=s (1967),  Ackers and White=s (1973), and Yang=s equations (1973), and the Karim=s modified BMF method (1998); and (iii) the TCF approach using the Yang (1973) equation and transport capacity distribution functions of Karim and Kennedy (1981), Li (1988), and Wu and Molinas (1996).  Detailed procedures for using these methods can be found in many sediment transport text books.  Since most of the methods derived following the shear stress correction approach are only applicable to the computations of fractional bedload for gravel-bed materials, methods in this category are excluded from this comparison.

3  COMPARISON OF COMPUTED RESULTS

For the comparison of fractional sediment transport methods, it is required that the data include measurements of size distributions for both bed material and the sediments in transport.  The sediment transprt data selected for this study is listed in Table 1.  It incorporates 154 data sets containing a total of 1007 data points.  These data are limited to sand sizes with median diameter in the range of 0.10 to 0.90 mm.  The lower and upper bounds of various size fractions used in different data sources vary.  The size fractions and sediment loads falling within each size group used in this study follow the reported size groups and fractional loads by each of the data sources.       First, the fractional transport rates are computed using SedWin (Wu and Molinas, 1998), a program for sediment transport computations.  Then the computed results are evaluated to demonstrate the performance and varistions of different methods.  In the statistical analysis, three different statistical methods are adopted to indicate the goodness of fit between the computed and measured results.  They are the Mean Normalized Error, MNE, the Average Geometric Deviation, AGD, and the correlation coefficient, R.  Each parameter provides a measure of the goodness of fit between the computed and measured results from a diferent perspective.

Statistical results of the computed results are given in Table 2. All the three statistical parameters, including MNE, AGD and R, indicate the best predictions of fractional bed-material loads are given by the TCF approach using Wu and Molinas= function.  Reasonable predictions are also given by the TCF approach using Karim and Kennedy=s function.  Overall, large errors in predictions are produced by methods in the direct computation by the size fraction category, and slitely better results are given by the BMF approache using Englund and Hansen’s and Yang’s equations. Using the MNE, AGD, and R as the criteria seperatly, the BMF approache using Ackers and White’s equation, the TCF approache using Li’s function, and the direct computation by size fraction method of Einstein reult in the largest errors, respectively.

                           Table 1  A list of sediment transport data

                             Table 2   Comparison between computed and measured fractional bed-material concentrations

(1)       (2)

  Fig. 1 shows the discrepancy ratio distributions of computed fractional bed-material concentrations for 5 selected methods, where the discrepancy ratio is defined as R_i = C_tci/C_tmi.  The unit value of R_i indicates the perfect agreement between the computed and measured results. As seen in this figure, for the Einstein method 15.3% of data lie in the range of discrepancy ratios of 0<R_i<1/7.5. This indicates high degree of bias or greatly underestimation of the fractional transport rates.  For the Karim method, this bias is even higher, there is 30.8% of data lie in the range of discrepancy ratios of 0<R_i <1/7.5.  For the BMF approach using the Engelund and Hansen equation, the discrepancy ratio distributions are close to normal distribution, but the predictions are not concentrated around perfect agreement. Overall, the discrepancy ratios resulting from Karim and Kennedy’s and Wu and Molinas’s methods are normally distributed and are more concentrated around perfect agreement.

Fig. 2 shows the variations of C_tci/C_tmi with the relative diameter D_i/D₅₀. Generally, large scatters of computed results can be observed for finer and coarser size fractions. However, the discrepancy ratios between the computed and measured fractional transport rates become relatively smaller for size fractions around D₅₀.

4  CONCLUSIONS

A comparison of different fractional bed-material load computation methods in sand-bed channels was conducted.  Conclusions from this comparative study are:

(1) Overall, the discrepancy ratios between the computed and measured fractional transport rates are small for size fractions around D₅₀.  The discrepancy ratios become larger for finer and coarser size fractions.

(2) Even though methods in the direct computation by the size fraction category were derived from sound theories, none of them can give reliable predictions.  The BMF approach using Yang=s and Engelund and Hansen=s equations produce slitly better results.

(3) Methods based on the TCF approach, with the exception of Li=s function, result in smallest mean normalized errors and provide reasonable predictions.

References

Ackers, P. and White, W. R. (1973). Sediment Transport: New Approach and Analysis, Journal of Hydraulic Division, ASCE, 99(HY11), 2041-2060.

Colby, B. R. and Hembree, C. H. (1955). Computations of Sediment Discharge, Niobrara River near Cody, Nebraska, U.S. Geological Survey Water Supply Paper 1357.

Culbertson, J. K., Scott, C. H., and Bennett, J. P. (1972). Summary of Alluvial-Channel Data From Rio Grande Conveyance Channel, New Mexico, 1965-69, U. S. Geological Survey Professional Paper 562-J.

Einstein, H. A. (1950). The Bed Load Function in Open Channel Flows, Technical Bulletin No.1026, United States Department of Agriculture, Soil Conservation Service.

Einstein, H. A. and Chien, N. (1953). Transport of Sediment Mixtures with Large Ranges of Grain Sizes, MRD Sediment Series No.2, U.S. Army Engineer Division, Missouri River, Corps of Engineers, Omaha, Neb.

Einstein, H. A. (1978). Sediment Transport Data in Laboratory Flumes, International Research and Training Center on Erosion and Sedimentation, Publication Circular No.2, Beijing, China.

Engelund, F. and Hansen, E. (1967). A Monograph on Sediment Transport in Alluvial Streams, Danish Technical Press (Teknish Forlag).

Hsu, S. M. and Holly, F. M. (1992). Conceptual Bed-Load Transport Model and Verification for Sediment Mixtures, Journal of Hydraulic Engineering, ASCE, 118(8), 1135-1152.

Hubbel, D. W. and Matejka, D. Q. (1959). Investigation of Sediment Transportation, Middle Loup River at Dunning, Nebraska, U.S. Geological Survey Water Supply Paper 1476.

Karim, M. F. and Kennedy, J. F. (1981). Computer-Based Predictors for Sediment Discharge and Friction Factor of Alluvial Streams, Iowa Institute of Hydraulic Research Report No. 242, University of Iowa, Iowa City, Iowa.

Karim, M. F. (1998). Bed Material Discharge Prediction for Nonuniform Bed Sediments, Journal of Hydraulic Engineering, ASCE, 124(6), 597-604.

Laursen, E. M. (1958). The Total Sediment Load of Streams, Journal of Hydraulic Division, ASCE, 84(HY1), 1-36.

Li, Y., (1988). A Study on the Computation of Two Dimensional Deformations in Streams, Sediment Research (in Chinese), No. 1.

Samaga, B. R., Ranga Raju, K. G., and Garde, R. J. (1986a). Bed Load Transport of Sediment Mixtures, Journal of Hydraulic Engineering, ASCE, 112(11), 1003-1018.

Samaga, B. R., Ranga Raju, K. G., and Garde, R. J. (1986b). Suspended Load Transport of Sediment Mixtures, Journal of Hydraulic Engineering, ASCE, 112(11), 1019-1035.

Toffaleti, F. B. (1969). Definitive Computations of Sand Discharge in River. Journal of Hydraulic Division, ASCE, 95(HY1), 225-246.

Vanoni, V. A. and Brooks, N. H. (1957). Laboratory Studies of the Roughness and Suspended Load of Alluvial Streams, Report No. E-68, Sedimentation Laboratory, California Institute of Technology, Pasadena, California.

Wu, B. and Molinas, A. (1996). Modeling of Alluvial River Sediment Transport, Proceedings of the International Conference on Reservoir Sedimentation, Vol. I, Edited by M. L. Albertson,  A. Molinas, and R. Hotchkiss, Fort Collins, Colorado, USA, pp. 281-325.

Wu, B. and Molinas, A. (1998). SedWin (Visually Interactive Sediment Transport Model for Windows 95/98), Hydrau-Tech, Inc., Fort Collins, Colorado.

Wu, B. (1999). Fractional Transport of Bed-Material Load in Sand-Bed Channels, Ph.D. Dissertation, Department of Civil Engineering, Colorado State University, Fort Collins, Colorado, USA, 193pp.

Yang, C. T. (1973). Incipient Motion and Sediment Transport, Journal of Hydraulic Division, ASCE, 99(HY10), 1679-1704.

Fig. 1  Discrepancy ratio distributions of fractional bed-material concentrations:             (a) einstein (1950) method;             (b) BMF approach using the engelund and hansen (1967) equation;             (c) karim (1998) method;             (d) TCF approach using the karim and kennedy (1981) function;             (e) TCF approach using the Wu and molinas (1996) function

Fig. 2    Variation of CCCI /CCMI versus DI /D50:
(a) einstein (1950) method;
(b) BMF  approach using the engelund and hansen (1967) equation;
(c) karim (1998) method;
(d) TCF approach using the karim and kennedy (1981) function;
(e) TCF approach using the wu and molinas (1996) function
[*data of middle loup river and niobrara river]