Sediment transport capacity in rivers Capacité de transport de sédiment dans les fleuves

Transcription

1 Journal of Hydraulic Research Vol. 42, No. 3 (2005), pp International Association of Hydraulic Engineering and Research Sediment transport capacity in rivers Capacité de transport de sédiment dans les fleuves SHU-QING YANG, Principal Research Scientist, Maritime Research Center, School of Civil and Environmental Engineering, Nangyang Technological University, Singapore csqyang@nus.edu.sg ABSTRACT This paper investigates the sediment transport capacity in rivers, the correlations between the total sediment discharges and various hydraulic parameters are examined by using 1593 records in the database compiled by Brownlie, in which the particle sizes are in the range of mm, the water depth ranges from m, the energy slope is in the range of and the maximum concentration is 8.47 kg/m 3. Among the existing hydraulic parameters including the widely cited parameters, such as VS/, V 3 /(gh) and T, the highest correlation coefficient is achieved by the new parameter T T when the database is used for comparison. This investigation also shows that the measured sediment discharge g t is linearly correlated with the parameter T T and the proportional factor (=12.5) obtained from flume data is still valid for expressing sediment discharge in rivers and canals. The measured and calculated total loads from various rivers are in a good agreement. RÉSUMÉ Cet article étudie la capacité de transport de sédiment dans les fleuves, les corrélations entre les décharges totales de sédiment et divers paramètres hydrauliques sont examinées à l aide des 1593 enregistrements de la base de données compilée par Brownlie, dans laquelle les tailles de particules sont dans la gamme mm, la profondeur d eau s étend de à m, la pente d énergie varie de à , et la concentration maximum est de 8.47 kg/m 3. Parmi les paramètres hydrauliques existants comprenant les paramètres largement cités, comme VS/, V 3 /(gh) ett, le coefficient de corrélation le plus élevé est réalisé par le nouveau paramètre T T quand la base de données est employée pour comparaison. Cette recherche prouve également que la décharge mesurée de sédiment g t est corrélée linéairement avec le paramètre T T et que le facteur de proportionnalité (=12.5) obtenu à partir des données en canalisation est encore valide pour exprimer la décharge de sédiment dans les fleuves et les canaux. Les charges totales mesurées et calculées de divers fleuves sont dans un bon accord. Keywords: River flow, sediment discharge, particle size, correlation coefficient. 1 Introduction and literature review The transport of granular material, such as silt, sand and gravel by the flow of water has been a subject of study for decades due to its importance as it determines the evolution of river beds and banks, estuaries and coastal-lines, and consequently the sediment transport exerts a considerable influence on the formation of the topography and stratification of the earth s surface. Hence, the sediment transport is of interest to a wide circle of professionals including hydraulic engineers, coastal engineers, geologists, hydrologists, geographers and so on. The mechanism of sediment transport has been investigated intensively due to its complexity and importance in determination of river revolution, pollution dispersion, boundary current interactions, etc. The investigations of the problem have resulted in numerous studies on calculating the sediment discharge. To date, many sophisticated equations have been proposed to calculate the total-load rate or bed material load that excludes the wash load and is made up of only those solid particles represented in the bed. Most of the existing models involve factors that are summarized by Chien and Wan (1998) as: C t = f(v, g, h, S, ρ, ρ s,ν,,d 50,B,σ g ) (1) where C t = total sediment transport concentration in kg/m 3 ; V = mean velocity; g = gravitational acceleration; h = water depth; S = energy slope; ρ = water density; ρ s = sand density; ν = fluid viscosity; = particle fall velocity; d 50 = particle median diameter; B = width of channel; σ g = gradation of sediment size. Since the phenomenon of sediment transport is related to so many variables shown in Eq. (1), and some of these variables are interrelated and dependent on each other, researchers generally select some relatively important variables to characterize the sediment transport processes, thus, dimensionless parameters formed by part dominant factors are proposed by previous investigators. Chien and Wan (1998) summarized that well cited equations, including those proposed by Einstein (1942), Meyer-peter and Muller (1948), Yalin (1972), Bagnold (1973), Engelune and Revision received July 29, 2004 / Open for discussion until November 30,

2 132 Yang Fredsøe (1976) and Ackers and White (1973) can be expressed by two dimensionless parameters = g ( ) t γ 1/2 ( ) 1 1/2 (2) γ s γ s γ gd 3 and = γ s γ d γ R b S (3) where γ s and γ are the specific weight of sand and water, respectively, R b = the hydraulic radius corresponding to grain friction on the bed; g t = total load transport per width in kg/m/s, and g t can be converted from the total sediment concentration C t using the relationship g t = qc t (4) where q = water discharge per unit width (m 2 /s). Some researchers believe that the sediment concentration C t is only governed by gravitational acceleration g; particle fall velocity ; mean velocity V and hydraulic radius R or water depth h, a dimensionless parameter V 3 /(gh) was first proposed by Velikanov (1954), and further developed by Zhang (1959) with the following form: ( V 3 ) m C t = k 1 (5) gh where k 1 and m are empirical factors to be determined using measured data. Dou (1974) assumed that the energy loss of flowing water is used to overcome the bed resistance, and to keep sediment particles in suspension as well as to transport the bed load. The total sediment discharge proposed by Dou has the following form: g t = ρ s ρ s ρ ρu 2 (0.123 V 2 0.1VV c where V c is the critical velocity at an incipient motion, the shear velocity is defined as /( u = V 2.5ln 11h ) (7) d 50 Equations (5) and (6) suggest that the energy slope S has no direct influence on the sediment transport. Similarly, van Rijn (1984a) believed that, without energy slope, two dimensionless parameters [ ] γs γ g 1/3 D = d (8) γ ν 2 and T = u 2 u2 c (9) u 2 c can fully express the process of sediment transport near a bed, in which u = (g0.5 /C )V = bed-shear velocity related to grains; C = Chezy-coefficient related to grains and u c = critical bedshear velocity according to Shields. To describe the suspended load transport, van Rijn (1984b) defined one more parameter Z ) (6) to express the influence of the upward turbulent fluid forces and the downward gravitational forces, and Z is defined as Z = (10) βκu in which β = coefficient related to diffusion of sediment particles; κ = constant of Von Karman; u = overall bed-shear velocity. Different from the aforementioned researchers, Yang (1996) is one of the first scholars to include the energy slope S and the mean velocity V simultaneously in his equations of sediment transport. The dimensionless parameter (VS)/ is termed as unit stream power. He expresses the equation of sediment concentration in the form of C t = f(vs/). Recently, a new parameter T T in kg/m/s is proposed by Yang and Lim (2003), Yang (2005). He found that, based on laboratory data, the parameter T T provides a good and linear relationship with the sediment discharge, g t, i.e. g t = kt T (11) T T = γ s γ s γ τ u 2 u2 c 0 (12) where k = a constant, τ 0 is the overall bed shear stress, T T is the total load transport parameter defined by Yang (2004). The objectives of this paper are: (1) to investigate the physical relationship between T T and other hydraulic parameters; (2) to investigate the correlation of sediment total load with the mentioned hydraulic parameters based on field data; and (3) to investigate the validity of Eq. (11). 2 Remarks on various correlation parameters The following discussion concerns the inter-relationship among various hydraulic parameters that play important roles for sediment transport, i.e., the unit stream power, (VS)/ proposed by Yang; the Velikanov s parameter, V 3 /(gh); van Rijn s transport stage parameter, T ; and the new total-load transport parameter, T T. If u 2 u 2 c, then from Eq. (11), g t is proportional to γ s τ 0 u 2 /(γ s γ)/ which can be rewritten in terms of the sediment concentration as follows g t q = C t γ s u 2 γ s γ τ 0 Vh (13) Equation (13) can be written as follows after τ 0 = ρgrs is introduced. ( ) γs γ u 2 C t = k S R (14) γ s γ V h Substituting the expression of u into Eq. (14) yields ( ) VS C t = k 0 (15) where ( ) /( γs γ k 0 = kr 2.5h ln 11R ) (16) γ s γ 2d 50 Equation (16) shows that sediment concentration is a function of unit stream power, VS/, which is proposed by Yang (1996).

3 Sediment transport capacity in rivers 133 However, k 0 is not a constant, it is affected by the sediment size, d 50, and the water depth, this may be the reason why Yang s (1996) formulas consist of two parts, one for fine sediment and another for gravel bed material. The Velikanov parameter can be obtained from Eq. (15) by substituting the Chezy s equation, S = V 2 /(C 2 R) where C = Chezy s coefficient, C t = k 3 ( V 3 gr ) (17) where k 3 = gk 0 /C 2 and V 3 /(gr) = Velikanov parameter. A more clear relationship between T T and others could be provided in the following form: ( ) ( ) ( g t γ g VS γ s Vh = k u 2 )( ) u2 c u 2 c γ s γ C 2 u 2 c u 2 (a) (b) (c) (d) (e) (f) (18) It is seen from Eq. (18) that the non-dimensional concentration (a) appears to be proportional to five familiar, non-dimensional parameters: inverse of the relative density of the material (b); Chézy overall stream resistance (c); Yang s (1996) unit stream power (d); van Rijn s (1984a) grain mobility (e) and critical Shields parameter (f). Therefore, it can be concluded that Eq. (11) combines the parameters developed by Yang (1996) and van Rijn (1984a,b). It is also interesting to note that the stream power τ 0 V used by Bagnold (1973) and Dou (1974) is quite similar to T T, instead of the mean velocity, the grain-shear velocity is used in Eq. (11), the grain-shear velocity plays more important roles in sediment transport than the mean velocity does, the fact is known by van Rijn (1984a,b). 3 Computation of total sediment discharge Equation (11) has provided good prediction for sediment transport in laboratory conditions (Yang, 2004), in that case, coefficient k is found to be a constant and equal to In this paper, Eq. (11) will be extended to compute the total load in rivers using hydraulic variables: hydraulic radius, R; mean velocity, V ; energy slope S; grain diameter d σ (= d 50 /σ g ); water and sediment densities ρ and ρ s ; gravitational acceleration g; cross-sectional averaged bed shear stress τ 0 ; and particle fall velocity,. It can be seen that Eq. (11) includes all parameters shown in Eq. (1). The complete method to compute the total sediment transport rate is shown as follows: 1. Compute the critical bed-shear velocity u 2 c, using the Shields curve, or using an empirical relationship proposed by Guo (1997). u 2 c (s 1)gd σ = 0.095S 2/ [1 exp( S 3/4 /20)] (19) where S = d σ (s 1)gdσ (20) 4ν An equation proposed by Van Rijn (1984a) is also recommended to estimate u 2 c. 2. Compute the shear velocity u from following equation: ( ) V 11R = 2.5ln (21) u 2d σ 3. Compute the fall velocity based on d σ. 4. Compute the bed-shear stress τ 0 = γrs. 5. Compute the total load transport parameter T T using Eq. (12). 6. Estimate the sediment discharge using Eq. (11). 4 Data sources The field database used for the verification of Eq. (11) was compiled by Brownlie (1981a,b); he stated that he eliminated all errors in his database with total satisfaction, therefore the data sets in Brownlie s report are used in the present analysis. The data base includes Acop canal (ACP) data that were recorded on five canals in Pakistan in American canal data (AMC) were obtained by Simons in Atchafalaya (ATC), Rio Grande (RIO), Mississipi (MIS) and Red river (RED) data were obtained by Toffaleti in India canal data of Chaudry et al. (1970) were observed in West Pakistan (CHP). Colorado River data (COL) were measured by the US Bureau of Reclamation in Hii River data (HII) were obtained by Shinohara and Tsubaki in 1959 in Japan. Middle Loup River data (MID) were observed by Hubbell and Matejka in Einstein observed sediment discharge on Mountain Greek (MOU) in Rio Magdalena and Canal del Dique data were collected in Columbia by Nedeco in 1973 (NED). Colby and Hembree measured sediment transport on the Niobrara River (NIO) in Sediment concentrations in North Saskatchewan River (NSR) were measured by Samide in Oak Creek data (OAK) were obtained by Milhous in Portugal River data (POR) were measured by Da Cunha in Rio Grande Conveyance data (RGC) were measured by Culbertson et al. in Rio Grande river data (RGR) were obtained by Nordin and Beverage in New Mexico in The range of field data is shown in Table 1, in which Brownlie (1981b) calculated σ g using the following equation σ g = 1 2 ( d84 + d ) 50 d 50 d Correlation coefficients (22) The first step in testing the validity of Eq. (11) is to check the correlation coefficients between total sediment discharge g t and proposed hydraulic parameter T T. Table 2 shows the correlation coefficients of measured sediment discharge g t and proposed parameter T T. For comparison, Table 2 also shows the

4 134 Yang Table 1 Summary of basic variable and parameter range in rivers River Particle size Water depth Gradation Energy slope No. of data Concentration d 50 (mm) h (m) σ g S 1000 ppm (max.) ACP RED MIS AMC ATC CHO CHP COL HII LED MID MOU NED NIO POR RGC RGR RIO OAK NSR Total Table 2 Correlation coefficients of total load transport and hydraulic parameters based on field data River (no. of runs) C t and (VS)/ C t and V 3 /(gr) and C t and D C t and T g t and T T k S k ACP (142) RED (30) MIS (165) AMC (11) ATC (68) CHO (33) CHP (33) COL (104) HII (38) LED (55) MID (38) MOU (100) NED (74) NIO (40) POR (219) RGC (8) RGR (293) RIO (38) OAK (17) NSR (55) Mean correlation coefficients between sediment concentration C t and hydraulic parameters: (VS)/, V 3 /(gr), D and T proposed by Yang (1996), Velikanov (1954) and van Rijn (1984a), respectively. The correlation coefficients between and are also calculated. The correlation coefficients in the present analysis are calculated by the following equation Correlation coefficient = (X X)(Y Ȳ) (X X) 2 (Y Ȳ) 2 (23)

5 Sediment transport capacity in rivers 135 in which the over bar denotes the mean value, Y is the measured sediment transport load (C t, g t or ) and X is the hydraulic parameter, and Y = f(x) has been expressed by previous researchers in different ways. The last row of Table 2 shows the overall results and it can be seen that in the view of correlation coefficients, the parameter T T yields the best result. 4.2 Determination of factor k The mean value of factor k can be determined by following equation: k = 1 n g t,measured = 1 n ( ) (24) n n i=1 γ s γ s γ τ 0 u 2 u 2 c i The standard deviation is a measure of how widely values are dispersed from the average value (the mean), the standard deviation of factor k can be assessed using Sk 2 = 1 n (k i k) 2 (25) (n 1) i=1 where n is total run number of measurements in a data set. The last two columns of Table 2 show the mean value of k and its deviation i=1 T T i for each data set determined by Eqs (24) and (25), respectively. Although the mean value of k for each river varies from 3.73 to 26.9, the over-all averaged value (=14.9) is very close to the standard value (=12.5) obtained from laboratory data. In order to show the validity of Eq. (11) in the assessment of sediment transport in rivers, the comparison of measured sediment discharge with Eq. (11) is shown in Figs (1 4). In these figures, the abscissa is the hydraulic parameter kt T in unit of kg/m/s, the ordinate is the product of measured values from Eq. (11) with , 0.01, 1, 100 and The solid lines represent perfect agreements in which the dimensionless coefficient k is shown in the legends and the dotted lines represent the ±100% error band. Figure 1 shows the data of ACP, RED, MIS, AMC and ATC. Figure 1 clearly indicates that k = 12.5 is acceptable to express the total sediment discharge in these rivers and canals. The total sediment discharges measured in CHO, COL, HII, LED and MID are shown in Fig. 2. The measurement methods shown in Fig. 2 differ from each other, however, Fig. 2 shows that k = 12.5 is acceptable for the total load prediction. Figure 3 shows the measured total load in MOU, NED, NIO, OAK and NSR. In MOU, the velocity was measured by means of floats. Sediment discharge was measured by trapping sediment Figure 1 1.E+06 1.E+05 1.E+04 1.E+03 1.E+02 1.E+01 1.E+00 1.E-01 (kg/m/s) 1.E-02 Acop canal (k=12.5) 1.E-03 Red River (k=12.5) 1.E-04 Mississippi (k=12.5) 1.E-05 American canal(k=12.5) 1.E-06 1.E-07 ktt (kg/m/s) Atchafalaya River(k=12.5) 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 Relationship of kt T and sediment total load g t measured from ACP, RED, MIS, AMC and ATC rivers. Figure 2 (kg/m/s 1.E+06 1.E+05 1.E+04 1.E+03 1.E+02 1.E+01 1.E+00 1.E-01 1.E-02 Chop Canal (k=12.5) 1.E-03 Colorado(k=12.5) 1.E-04 HII river (k=12.5) 1.E-05 River by Leopold (k=12.5) Middle Loup River (k=12.5) 1.E-06 1.E-07 1.E-03 1.E-02 1.E-01 ktt (kg/m/s) 1.E+00 1.E+01 Relationship of kt T and sediment total load g t measured from CHO, COL, HII, LED and MID rivers.

6 136 Yang (kg/m/s) 1.E+06 1.E+05 1.E+04 1.E+03 1.E+02 1.E+01 1.E+00 1.E-01 1.E-02 Moutain Creek (k=3.75) South American River,k= E-03 1.E-04 Niobrara River (1955, k=6.15) 1.E-05 N.Saskatchewan River(k=4.5) 1.E-06 Oak Creek(k=12.5) 1.E ktt (kg/m/s) Figure 3 (kg/m/s) Relationship of kt T and g t measured from MOU, NED, NIO, NSR and OAK rivers. Portugal River(k=12.5) Rio Grande 1976 (k=12.5) Rio Grande 1965 (k=12.5) Rio Grande 1968 (K=12.5) ktt (kg/m/s) 10 Figure 4 1.E+06 1.E+05 1.E+04 1.E+03 1.E+02 1.E+01 1.E+00 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 1.E-07 Relationship of kt T and sediment total load g t measured from POR, RGC, RGR and RIO rivers. in a mesh covered hopper and pumping it into a weight tank. Because only the bed load was measured, the coefficient k is less than the standard value (=12.5). Similarly, in NIO stream flow observations were made at the gaging-station section 580 m upstream from the cross section where the concentration was measured. Slope S was not measured on every day, the slope values were the observations made on other days, this is probably why the coefficient k is less than the standard value in this case. In NSR, only velocity and water depth were observed, the discharges shown in Brownlie s report were computed by multiplying the local velocities and depth and the assumed widths. The transport rates were determined with basket-type bed-load samplers, which only measured the gravel bed load. According to Brownlie s comments, at least 10% sand was not included. This is why k is below the standard value. However, Fig. 3 shows strong relationship between the measured g t and hydraulic parameter T T, which implies that Eq. (11) is also valid for prediction of bed load transport if the coefficient k is calibrated. Figure 4 shows the measured total load in POR, RIO, RGR and RIO. It can be seen that k = 12.5 is acceptable. From Table 2 and Figs 1 4, it can be concluded that there is a linear relationship between measured sediment discharge g t and proposed hydraulic parameter T T. 4.3 Comparison with other formulas It is useful to provide a comparison on the predictability of the widely cited formulas for sediment transport including Eq. (6) which is very popular in China, Yang s (1996) equations based on unit stream power, van Rijn s (1984a,b), Karim s (1998) and Engelund and Hansen s (1972) formulas and Eq. (11). The results are shown in Table 3, tabulated in terms of a discrepancy ratio, r defined as r = g t(measured) g t(computed) (26) The first column of Table 3 shows the score of Eq. (11) when k = 12.5 is applied; the second column also shows the score of Eq. (11) when k in last column of Table 2 is applied. The overall results in the last row of Table 3 shows that Eq. (11) yields the best results; it also indicates that there is little difference for Eq. (11) when k = 12.5 or the mean value from Eq. (24) is used. This

7 Sediment transport capacity in rivers 137 Table 3 Comparisons on predictability of various formulas for field sediment discharge River % Score of predicted sediment discharge and range of discrepancy r 0.75 <r< <r< <r<3 Y Y D K CTY VR EH Y Y D K CTY VR EH Y Y D K CTY VR EH (k = 12.5) (k from (k = 12.5) (k from (k = 12.5) (k from Eq. (24) Eq. (24) Eq. (24) ACP RED MIS AMC ATC CHO CHP COL HII LED MID NED POR RGC RGR RIO OAK Mean Note on formulas: Y = Eq. (11), D = Eq. (6), K = Karim, CTY = C.T. Yang, VR = van Rijn, EH = Engelund and Hansen. interesting outcome suggests that k = 12.5 could be used for estimating the total sediment discharge in rivers. 5 Conclusions The correlations between the total sediment discharge and various hydraulic parameters are examined by using 1593 records in the database compiled by Brownlie, in which the particle sizes are in the range of mm, water depth from to m, energy slope in the range and the maximum concentration is 8.47 kg/m 3. The following conclusions can be drawn from this study: 1. Among the existing hydraulic parameters including the widely cited parameters, such as Einstein s flow intensity parameter, the dimensionless unit stream power (VS)/, Velikanov s parameter V 3 /(gh) and van Rijn s transportstage parameter T, the highest correlation coefficient is achieved by the new parameter T T when the Brownlie s database is used for comparison. 2. This investigation shows that the measured sediment discharge g t in rivers is linearly correlated with the parameter T T and the proportional factor (=12.5) obtained from flume data is still valid for expressing sediment discharge in rivers and canals and it yields best results among the well cited formulas. This indicates that k = 12.5 is acceptable in the prediction of sediment discharge in rivers. 3. Although the database shows that the factor k is insensitive to other variables, the theoretical results indicate that k is a function of the relevant variables such as aspect ratio, relative roughness, etc., thus more research is necessary to investigate the expression of factor k. 4. To yield better agreement with measurements, it is recommended that prior to predicting the sediment discharge in a particular river reach, the factor k be calibrated using some available measurement data. Notation C t = Mean sediment concentration C, C = Chezy s coefficient d 50 = Median sediment size d σ = d 50 /σ g D = Particle parameter g t = Total sediment discharge g = Gravitational acceleration h = Water depth q = Discharge per unit width k, k 1 = Coefficient m = Coefficient R b = Hydraulic radius corresponding to bed R = Hydraulic radius S = Energy slope T = Transport stage parameter u = Shear velocity = (grs) 0.5 u = Shear velocity derived from mean velocity u c = Critical shear velocity corresponding to Shields curve V = Mean velocity

8 138 Yang V c = Critical velocity for sediment transport Z = Coefficient β = Coefficient related to diffusion of sediment particles κ = Constant of Von Karman = Sediment transport parameter = Hydraulic parameter = Fall velocity ρ = Fluid density ν = Kinematic viscosity γ s = Specific weight of sand γ = Specific weight of water τ 0 = Shear stress = ρgrs σ g = Sand gradation References 1. Ackers, P. and White, W.R. (1973). Sediment Transport: New Approach and Analysis. J. Hydraul. Div., ASCE 99(11). 2. Bagnold, R.A. (1973). The Nature of Saltation and of Bed Load Transport in Water. Proc. R. Soc. London Ser. A, Brownlie, W.R. (1981a). Discussion of Total Load Transport in Alluvial Channels. J. Hydraul. Div., ASCE, 107(12). 4. Brownlie, W.R. (1981b). Compilation of Alluvial Channel Data: Laboratory and Field. California Institute of Technology, California. Report KH-R-43B. 5. Chien and Wan (1998). Mechanics of Sediment Transport. ASCE, Reston, VA. 6. Dou, G.R. (1974). Similarity Theory and its Application to the Design of Total Sediment Transport Model. Research Bulletin of Nanjing Hydraulic Research Institute, Nanjing, China (in Chinese). 7. Engelune, F. and Fredsøe J. (1976). A Sediment Transport Model for Straight Alluvial Channels. Nordic Hydrol. 7, Engelund, F. and Hansen, E. (1972). A Monograph on Sediment Transport in Alluvial Streams, Teknisk Forlag, Copenhagen, Denmark. 9. Einstein, H.A. (1942). Formulas for the Transportation of Bed Load. Trans. Soc. Civil Eng Guo, J. (1997). Discussion of The Albert Shields Story. J. Hydraul. Eng., ASCE 121(7). 11. Karim, F. (1998). Bed Material Discharge Prediction for Nonuniform Bed Sediment. J. Hydraul. Eng., ASCE 124(6), Meyer-Peter, E. and Muller, R. (1948). Formulas for Bed Load Transport. Proceedings of the 3rd Meeting of IAHR, Stockholm, Switzerland, Vol. 6, pp van Rijn, L.C. (1984a). Sediment Transport Part I: Bed Load Transport. J. Hydraul. Eng., ASCE 110(10), van Rijn, L.C. (1984b). Sediment Transport Part II: Suspended Load Transport J. Hydraul. Eng., ASCE 110(11), Velikanov, M.A. (1954). Gravitational Theory for Sediment Transport. J. Sci. Soviet Union Geophys. 4 (in Russian). 16. Yalin, M.S. (1972). Mechanics of Sediment Transport. Pergamon Press, Oxford. 17. Yang, C.T. (1996). Sediment Transport: Theory and Practice. McGraw-Hill International Editions. 18. Yang, S.Q. and Lim, S.-Y. (2003). Total Load Transport Formula for Flow in Alluvial Channels. J. Hydraul. Eng. ASCE 129(1), Yang, S.Q. (2005). Prediction of Total Bed Material Discharge. J. Hydraul. Res. IAHR 43(1), Zhang, R. (1959). A study of sediment transport capacity of middle and lower Yanze River. J. Sediment Res. 4(2), Beijing, China (in Chinese).