REASSESSMENT OF SELECTED SEDIMENT DISCHARGE FORMULAS

Hyoseop Woo and Kwonkyu Yu

     Senior Research Fellow, Korea Inst. of Construction Tech., 2311 Daehwa-dong, Ilsan-gu, Koyang-si, Kyeonggi-do 411-410, Korea. Tel: 82-31-9100-545, fax: 82-31-9100-251,

E-mail: hswoo@kict.re.kr and Graduate Student, Dept. of Civil Eng., Univ. of Iowa, Iowa, USA

Abstract: Existing sediment discharge formulas are reassessed by using carefully selected stream sediment data that are found in the literature. In order to clarify the result of this comparative study, only the total sediment discharge formulas and the data collected from the streams are considered, excluding suspended load or bed load formulas and flume data. Direct comparison was made between the measured and calculated sediment discharges for the selected sediment discharge formulas by using the data showing a well-defined, a single relation between water discharge and sediment discharge at each gauging station. The result of this assessment is disappointing with the best accuracy of less than 70%. Among the eight sediment discharge formulas tested in this study, Einstein’s and Toffaleti’s formulas appear least reliable, while the European formulas appear relatively reliable for determining the sediment discharge in the alluvial channel. This result generally agrees with the previous ones.

Keywords: total sediment discharge formulas, sediment–rating curve, field data, comparative study

1    INTRODUCTION

Determining sediment discharge in the alluvial channel with a good reliability is as important as determining the water discharge in the channel. It can be calculated by using the sediment discharge formulas and the flow and channel variables such as water velocity, depth, slope, and the bed sediment variables of sediment size and gradation. The problems embedded in the presently used sediment discharge formulas are the fact that their reliabilities are sometimes disappointing with the discrepancy ratios between the calculated values by the formulas and the measured ones exceeding more than the order of two.  Because of this, it is very essential for the river engineers to select proper sediment discharge formulas that are suitable for the channel they concern.

The purpose of this study is to reassess the existing sediment discharge formulas. For this, this study uses the data of the stream-sediment discharges that are selected carefully in the literature with a unique criterion, that is, showing a single relation between water discharge and sediment discharge at each gauging station. In addition, this study only uses the field (stream) data excluding flume data. It is because there are some inconsistencies on the results of the assessments of the sediment discharge formulas between the stream and flume data.

The sediment discharge formulas assessed in this study include eight ones developed mostly in Western Europe and USA. All are the total sediment discharge formulas excluding the bed load or suspended load ones. The data used in this study includes a total of 722 points collected from the fifteen river stations throughout the world. These are excerpted from Brownlie's compendium of sediment discharge data (1981a).

2    REVIEW OF ASSESSMENT OF SEDIMENT DISCHARGE FORMULAS

Since 1960's, numerous assessments of sediment discharge formulas, for example, the early work of Vanoni(1960; ASCE 1975) and the recent work of Yang and Wan(1991), have been introduced in the literature. Table 1 shows a summary of those assessments.

This table shows contradicting results. For example, the Yang formula (1973) is ranked “poor” by Brownlie(1981b) and van Rijn(1984), while it is ranked “average”by Shen(1982), and “good” by Yang and other evaluators. Another example can be found from Ackers and White’s formula. This formula is considered “poor” by Shen(1982), while it is considered “average” by Yang and others, and “good” by White, et al(1973) and Brownlie(1981b). It is certain that these discrepancies are due mainly to the fact that each evaluator used different data, although other possible causes, such as the improper application of the sediment discharge formulas, could not be excluded.

The reliability of the measured data used for the assessment also would play an important role in the reliability of the assessment result. Fig. 1 shows water discharge versus sediment discharge, that is, the sediment-rating curve in the log-log plot at the Niobrara River, which shows a well-defined relation between the two variables. On the other hand, Fig. 2 shows the sediment-rating at the Mississippi River. It is seen in Fig. 2 that the relation between water discharge and sediment discharge is not so singular as that in Fig. 1, which implies that the flows in the Mississippi River might not be in equilibrium at the time of data collection. Or, some of the data points in Fig. 2 might be less reliable. Any plural values of sediment discharge corresponding to a single value of water discharge in the sediment rating curve implies that the sediment equilibrium, which is a fundamental basis in constructing the sediment discharge formulas, is not achieved. In the alluvial channel with the flow in equilibrium, the sediment discharge at a channel is uniquely determined with the water discharge unless it crosses the flow regime (upper and lower). Therefore, any plural values with large difference between them for the sediment discharge corresponding to a water discharge in a station may be less adequate to be used for the objective assessment of the sediment discharge formulas.

Because of the uncertainty in evaluating the sediment discharge formulas by using less reliable data, therefore, the contradicting result in Table 1 appears inferential.  Nevertheless, it can be concluded from Table 1 that, in general, Engelund & Hansen’s, Ackers & White's, Yang's, and van Rijn’s formulas are relatively more reliable than the other formulas.

3    REASSESSMENT OF SEDIMENT DISCHARGE FORMULAS

3.1    Selection of sediment discharge formulas compared

In this study, the following eight sediment discharge formulas are selected for the comparison: They are the ones that have been frequently compared with measured data in the past.

l         Formulas appearing more reliable as shown in Table 1: Ackers & White(AW, 1973), Engelund & Hansen(EH, 1967), van Rijn(RN, 1984), Yang(YN, 1973)

l         Formulas developed earlier and known widely: Einstein(EN, 1950), Laursen(LS, 1958), Toffaleti(TF, 1968)

l         Formula developed using most extensive data: Brownlie(BR, 1981)

3.2    Review of measured field data of sediment discharge

Brownlie's compendium of sediment discharge data is used for the selection of data to be used in this reassessment. From the compendium, the 44 stream data with total data points of 1,601 are available. From these data sets, only the data sets for the sand range with their relations

(Legend: G Good, A Average, P Poor ) Evaluations are based on the following criteria; 1) Evaluator's own evaluation. 2) When no written evaluation is available, 'good' is ranked for more than 70 % of total data points being within the error range of 50 %, 'average' for 50∼70 %, and 'poor' for less than 50 %. 3) In case of the graphical comparison or other means, 'good' for high-graded 3-4 formulas, 'average' for average-graded 3-4 ones, 'poor' for the rest. between water discharge and sediment discharge, expressed by the equation, Q_s = a Q^b, being well-defined with the coefficients of determination exceeding 0.81 are selected. By this criterion, a total of 15 data sets with 722 data points as shown in Table 2 are used for the reassessment.

Table 2    Field data used for reassessment

3.3    Comparison with measured field data

Sediment discharge for each data point is calculated by each sediment discharge formula selected in this study. The computer program named GUIDE, which were developed by the authors in 1989, was used for this calculation. The accuracy of the program GUIDE, which calculates bed-material discharge in the channel by the sediment discharge formula selected by users, has been verified by another computer program calculating sediment discharge, SEDTRAN, which was developed by Stevens and Yang (1989).

The calculated sediment discharge is compared with measured one and the difference between them is expressed by the discrepancy ratio r as the ratio of the calculated value over the measured one. The geometric mean (m_r) and the geometric standard deviation (S_r) of the discrepancy ratio are expressed, respectively, as：

log m_r  = log r_i                                             (1)

(log S_r)² = _r)²                   (2)

where n is the number of data point.

4    DISCUSSION

Table 3 shows the result of calculation. In this table, "accuracy" indicates the percentage of the data points the discrepancy ratio of which is between 0.5 and 2.0. The reasons for the numbers of the data points in Table 3 being a little different for each formula is the fact that some formulas can not be applicable to a few data points with extreme values. Fig. 3 shows the geometric mean of the discrepancy ratio and the accuracy for each formula.

Table 3    Results of calculation for discrepancy ratio

Generally speaking, more reliable formula is the one having the mean of the discrepancy ratio nearer to the unity, the accuracy higher, and the standard deviation smaller. In the average sense, EH and RJ formulas are more reliable among them tested, while AW and BR formulas are next. It should be mentioned at this point that the BR formula was developed through the multiple regression analysis by using the basically same data sets that are used in this study. The AW formula is not applicable to the streams with their bed-sediment size less than 0.25 mm (Woo et al. 1990).

5    CONCLUSIONS

There are several causes that would make the existing assessments of sediment discharge formulas to lack consistency. This study focuses the possible problem of using the less reliable data. This study used the field data showing a well-defined, single relation between the water discharge and sediment discharge at each gauging station, which implies that they were collected when the flows were in the equilibrium.

The result of this assessment is also disappointing with the best accuracy of less than 70%. Among the eight sediment discharge formulas tested in this study, Einstein’s and Toffaleti’s formulas appear least reliable, while Engelund and Hansen's, van Rijn's, and Ackers and White's formulas appear, in general, relatively reliable for determining the sediment discharge in the alluvial stream. However, it is the best practice to calibrate, prior to use, with the data collected from the stream investigated, the sediment discharge formula chosen for that stream.

References

Alonso, C.V. (1980), Selecting a Formula to Estimate Sediment Transport Capacity in Nonvegetated Channels, Chapter 5 in CREAMS, W. G. Knisel ed., U.S. Dept. of Agriculture, Conservation Research Report 26, 426-439.

ASCE (1975), Sedimentation Engineering, Vanoni, V. A. edited, 220-223.

Brownlie, W.R. (1981a), Compilation of Alluvial Channel Data: Laboratory and Field, W.M. Keck Laboratory of Hydraulics and Water Resource, Report No. KH-R-43B, California Inst. of Technology, Pasadena, California, Nov.

Brownlie, W.R. (1981b), Prediction of Flow Depth and Sediment Discharge in Open Channels, W.M. Keck Laboratory of Hydraulics and Water Resource, Report No. KH-R-43A, California Inst. of Technology, Pasadena, California, Nov.

Nakado, T. (1990), Test of Selected Sediment Transport Formulas, J. of Hydraulic Engineering, ASCE, 116(3), 362-379.

Ranga Raju, K.G., Garde, R.J., and Bhardwaj, R.C. (1981), Total Load Transport in Alluvial Channels, J. of Hydraulic Engineering, ASCE, 107(2), 179-191.

Shen, H.W. and Hung, C.S. (1971), An Engineering Approach to Total Bed Material Load by Regression Analysis, Proc. of Sedimentation Symposium, Berkeley, California.

Stevens, H. H., Jr., and Yang, C. T. (1989), Summary and Use of Selected Fluvial Sediment Discharge Formulas, U.S. Geological Survey, Water Resources Investigation Report 89-4026.

van Rijn, L.C. (1984), Sediment Transport, Part II: Suspended Load Transport, J. of Hydraulic Engineering, ASCE, 110(1), 1613-1641.

Vetter, D.M. (1989), Gesamttransport von Sediment in offenen Gerinnen, The Institute of Hydrology, Report No.26., Univ. of the German Federal Army, 1988, translated into English Total Sediment Transport in Open Channels, by I.C. Hollingsworth, U.S. States Department of the Interior, Bureau of Reclamation.

White, W.R., Milli, H. and Crabbe, A.D. (1973), Sediment Transport: An Appraisal Methods, Vol.2, Performance of Theoretical Methods, when Applied to Flume and Field Data, Hydraulics Research Station, Report No.IT 119., Wallingford, U.K., Nov.

Woo, H., and Yu, K. (1989), Development of a Guideline for the Selection of Sediment Discharge Formulas, Korea Institute of Construction Technology, Seoul , Korea, p. 69 (in Korean).

Woo, H., Yoo, K., and Seoh, B. (1990), Performance Test of Some Selected Sediment Transport Formulas, Proc. of the 1990 National Hydraulic Engineering Conference, Volume I, edited by H.H. Chang and J.C. Hill, ASCE, San Diego, 694-699.

Yang, C.T. and Molinas, A. (1982), Sediment Transport and Unit Stream Power Function, J. of Hydraulic Engineering, ASCE, 108(6), 774-793.

Yang, C.T., and Wan, S. (1991), Comparison of Selected Bed-Material Load Formulas, J. of Hydraulic Engineering, ASCE, 117(8), 973-988.

Fig. 1    Sediment-water relation at the Niobrara river

Fig. 2    Sediment-water relation at the Mississippi river

Fig. 3    Comparison of performance of sediment discharge formulas

Table 1    Summary of assessment of sediment discharge formulas

( Legend: G Good, A Average, P Poor ) Evaluations are based on the following criteria; 1) Evaluator's own evaluation. 2) When no written evaluation is available, 'good' is ranked for more than 70 % of total data points being within the error range of 50 %, 'average' for 50∼70 %, and 'poor' for less than 50 %. 3) In case of the graphical comparison or other means, 'good' for high-graded 3-4 formulas, 'average' for average-graded 3-4 ones, 'poor' for the rest.