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Assessment of total sediment load in rivers using lateral distribution method
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ScienceDirect Journal of Hydro-environment Research xx (2014) 1e7 www.elsevier.com/locate/jher
Research paper
Assessment of total sediment load in rivers using lateral distribution method Sung-Uk Choi a,*, Jinhwi Lee b,1 a
School of Civil and Environmental Engineering, Yonsei University, Seoul 120-749, Republic of Korea b ISAN Research Institute, ISAN Corporation, Gyeonggi-do 431-060, Republic of Korea Received 26 March 2013; revised 12 May 2014; accepted 3 June 2014
Abstract A new numerical model for predicting the total sediment load in the river is presented. The model is comprised of two parts, namely flow and sediment transport parts. The flow analysis is carried out using the lateral distribution method, which distributes the flow and sediment load across the width, based on channel geometry and flow dynamics. To obtain the total sediment load, the bedload and suspended load are predicted separately. Conventional formulas are used for the bedload, and Rousean distribution with an entrainment function is used for the suspended load. Einstein partition is used to separate the form drag component from the total shear stress. The model is applied to the Danube River, a gravel-bed river in Slovakia and the Han River, a sand-bed river in Korea. Predicted distributions of sediment load in the lateral direction are given. The predicted results are compared with those obtained using the 1D approach as well as from measured data. Total sediment load rating curves are presented and the mixture effect of bed-materials is discussed. © 2014 International Association for Hydro-environment Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved.
Keywords: Bedload; Lateral distribution method; Sediment transport; Suspended load; Total sediment load
1. Introduction Sediment transport in rivers is a major issue in many countries. Sediment particles that are transported by the river induces morphodynamic changes. An over-supply of sediment results in aggradation of the river bed that can raise the flood level. It may also fill the habitats of aquatic animals and cause a decrease in the function of stream installations. A sediment under-supply results in the degradation of the river bed, which can endanger stream installations such as banks and bridge piers by undermining their foundations. Moreover, it can have a serious impact on the physical habitats of aquatic biota. Instream vegetation can be eradicated by the flow, and the
* Corresponding author. E-mail addresses: [email protected] (S.-U. Choi), [email protected] (J. Lee). 1 Formerly, Graduate student at Department of Civil & Environmental Engineering, Yonsei University, Seoul 120-749, Republic of Korea.
spawning ground for various types of fish can deteriorate by flushing fine sediment particles from the substrate. In spite of its importance, predicting sediment transport in a river has not been so successful, from the engineering point of view. A variety of different formulas have been proposed, but none of them have found to be universally applicable. A formula, that accurately predicts sediment transport in one river, is not applicable to another river. The difficulty in making such predictions arises from inherently complicated nature of the physics of sediment transport in a river. The conventional approach is the 1D method, in which the total sediment load is computed by multiplying the sediment load per unit width to the channel width, with the assumption that the channel cross section is rectangular in shape. This approach is valid for the case of a wide channel whose flow depth is relatively constant. However, for rivers whose flow depth varies to a large extent, the 1D method may underpredict the bedload seriously (Bradely et al., 1999). This is why Camenen et al. (2011) and de Almeida and Rodriguez
http://dx.doi.org/10.1016/j.jher.2014.06.002 1570-6443/© 2014 International Association for Hydro-environment Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved.
Please cite this article in press as: Choi, S.-U., Lee, J., Assessment of total sediment load in rivers using lateral distribution method, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.06.002
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(2011) computed the total sediment load in a large river by distributing the shear stress laterally based on the channel geometry. Previous studies, however, did not take into account the impact of flow dynamics on the lateral distribution of the discharge. This motivated the present study. The objective of this study was to develop a new, more accurate and reliable method, for numerically predicting the total sediment load in a river. The method predicts the total sediment load with information on discharge, channel geometry and slope, and the size of bed materials. The bedload and suspended load are separately computed, based on a flow analysis by the lateral distribution method. The lateral distribution method provides the lateral profile of the flow variables, including the depth-averaged velocity and boundary shear stress, based on the channel geometry and flow dynamics. The method is applied to the bedload transport in the Danube River, in Slovakia and to the total sediment load transport in the Han River, in Korea. The predicted results are compared with results computed by the conventional 1D approach, as well as with measured data. 2. Proposed method The proposed method consists of two parts, namely the flow analysis and the computation of sediment transport. The lateral distribution method is used for the flow analysis. Separating the impact of bedform from the total shear stress is then achieved by means of the Einstein partition. Finally, the bedload and suspended load are estimated separately using the formula and Rousean distribution, respectively. The total sediment load is then obtained by adding the bedload and the suspended load. 2.1. Flow analysis Shiono and Knight (1990) developed the following equation by integrating Reynolds equations over depth for a uniform open-channel flow: rgHSx þ
v vU rH 3 Bg tb ¼ G vy vy
ð1Þ
where y denotes the lateral direction, r is the water density, g is the gravitational acceleration, H is the flow depth, Sx is the bed slope in the streamwise (x) direction, 3 is the y-component eddy viscosity, U is the depth-averaged streamwise velocity, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bg is the geometric factor defined by 1 þ S2x þ S2y (here, Sy is the bed slope in the lateral direction), tb is the bed shear stress, and G is the term related with secondary currents. In Eq. (1), the bed shear stress and y-component eddy viscosity can be represented by, respectively, tb ¼ 3
MODEL
rgn2 2 U H 1=3
¼ lu* H
ð2Þ ð3Þ
where n is the roughness coefficient, l is the non-dimensional eddy viscosity, and u* is the local shear velocity. The roughness coefficient can be varied, for example, to reflect higher roughness near the bank, but a constant value of roughness coefficient is used herein. In addition, the term related with secondary currents in Eq. (1), is given by 0 H 1 Z v@ G¼ ru v dzA ð4Þ vy 0
where u and v represent time-averaged streamwise and lateral velocity components, respectively. In the present study, the value of non-dimensional eddy viscosity of l ¼ 0.067 is used and the impact of secondary currents is ignored, i.e., G ¼ 0. Before the computations, Manning's roughness coefficient is calibrated using measured stageedischarge relationships. The calibrated values for the roughness coefficient are normally larger than the value estimated by formula such as ManningeStrickler's relationship. This is due to the presence of a bedform on the river bed, channel meandering, changes in channel width, and so on. Then, for a given discharge, the finite difference form of Eq. (1) is solved numerically. Freeslip boundary conditions are imposed at the sides of the channel. The computations are repeated by changing the stage until the computed discharge becomes identical with the given discharge. Solving the difference equation provides the lateral distributions of depth-averaged velocity U and local shear velocity u*. 2.2. Computation of total sediment load In general, two approaches are available for predicting the total bed-material load in a river. One is to estimate the bedload and suspended load separately. This approach is founded on the fact that the hydrodynamics of each mode of sediment transport is different. Thus, models of the bedload and suspended load should be based on accurate mechanics. The methods proposed by Einstein (1950), van Rijn (1984a,b), and Toffaleti (1969) utilize this approach. The other approach is to estimate the total bed-material load directly without dividing the transport mode into two parts. This approach is empirical and mostly relates the dimensionless rate of sediment transport to a dimensionless flow parameter. This approach is simpler, but ignores the details of the physics of particle transport by a flow. This approach can only be applied in the strict sense to equilibrium or quasi-equilibrium flows (Garcia, 2008). However, models based on this approach have advantages in that they have been calibrated with laboratory or field data. 2.3. Computation of bedload In the present study, Meyer-Peter and Muller's (1948) formula and Camenen and Larson's (2005) formula are used to predict bedload transport (hereafter they are denoted as MPM and CL formulas, respectively). They are given by, respectively,
Please cite this article in press as: Choi, S.-U., Lee, J., Assessment of total sediment load in rivers using lateral distribution method, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.06.002
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S.-U. Choi, J. Lee / Journal of Hydro-environment Research xx (2014) 1e7
3=2 q*sb ¼ 8 t* t*c with t*c ¼ 0:0470
ð5Þ
3=2 t* q*sb ¼ 12 t* exp 4:5 c* with t*c ¼ 0:055 t
ð6Þ
where q*sb is Einstein bedload number defined by pffiffiffiffiffiffiffiffiffi qsb =ð RgDDÞ with qsb the volume rate of bedload transport per unit width [L2/T] (here, R and D are the submerged specific gravity and diameter of the sediment particles, respectively), and t* and t*c are Shields stress and its critical value, respectively. MPM formula is a typical DuBoys type relationship for bedload transport in that the formula yields the amount of bedload transport only if the Shields stress exceeds its critical value. CL formula is a modification of MPM formula based on the probabilistic approach for the incipient motion of particles on the bed (Camenen and Larson, 2005). CL formula predicts an exponentially-decaying bedload even if the Shields stress is less than its critical value.
The volume rate of the suspended sediment load per unit width is obtained by qss ¼
ZH u c dzz
0
u c dz
ð7Þ
b
where c is the time-averaged concentration of suspended sediment and b is the height close to bed. Normally, b is taken to be 0.05H. In order to estimate suspended load, Rousean distribution of suspended sediment is assumed. That is, Z ð1 2Þ=2 cð2Þ ¼ cb ð8Þ ð1 2b Þ=2b where cb is the near-bed sediment concentration, 2 ¼ z=H, 2b ¼ b=H, and Z is Rouse number defined by vs =ðku* Þ (here, vs is the settling velocity of the particle, and k is von Karman constant). For equilibrium suspension, the near-bed concentration of suspended sediment is the same as dimensionless entrainment rate E, i.e., cb ¼ E. In the present study, the following dimensionless entrainment rate by Wright and Parker (2004) is used: E¼
AZu5 A 5 1 þ 0:3 Zu
2.5. Mixture effect For a sediment mixture on the channel bed, fine particles tend to be located behind coarser particles. As a result, larger particles are more exposed to the flow, allowing them to move more readily. This is called the hiding effect. Normally, after sediment particles are divided into finite classes, the hiding effect is considered by changing the critical Shields stress for jth particle, i.e., B Dj * * ð10Þ tc;j ¼ tc D50 where the exponent B is a constant. For B, Ferguson et al. (1989) proposed a constant value of 0.12 and Wilcock and Crowe (2003) proposed the following relationship: B¼
0:67 1 þ exp 1:5 Dj D50
ð11Þ
In the present study, the relationship by Wilcock and Crowe (2003) is used.
2.4. Computation of suspended load
ZH
3
3. Application to the gravel-bed river The proposed method was initially applied to bedload transport in a Slovak reach of the Danube River. Fig. 1 shows the location of the study reach. According to Camenen et al. (2011), the Danube River has experienced a dramatic change due to flood protection projects, the construction of navigation channels, and instream structures in the upstream reach of the river. As a result, the downstream reach of the SlovakHungarian Danube River has become degraded. To study the impact of the hydropower plant, Camenen et al. (2011) made extensive measurements of the bedload with a basket-type bedload sampler. The sampling point of the bedload is located about 11 km downstream from the gauging station. Sediment particles range from fine to coarse gravel with
ð9Þ
where A is a constant equal to 5:7 107 and Zu is the similarity variable defined by Zu ¼
u*s 0:6 0:07 Re S vs p
with u*s is the shear velocity partition due to skin friction. Eq. (9) is a modification of Garcia and Parker's (1991) relationship for applications to large and low slope streams.
Fig. 1. Location of the study reach in the Danube River (modified from Camenen et al. (2011)).
Please cite this article in press as: Choi, S.-U., Lee, J., Assessment of total sediment load in rivers using lateral distribution method, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.06.002
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D50 ¼ 9 mm and D90 ¼ 22 mm, corresponding to medium and coarse gravels, respectively. The grain size distribution of the bed sediment particles can be found in Camenen et al. (2011). By fitting to measured datasets for average velocity versus discharge, a value of n ¼ 0.026 was obtained. This value of the roughness coefficient is slightly larger than n ¼ 0.022 estimated by Strickler's formula using D50. This may be due to the presence of bedforms and changes in channel width. Fig. 2 shows the lateral distributions of unit discharge and depth-averaged velocity for Q ¼ 4640 m3/s and the channel geometry of the sampling point. The cross section in the figure is a view facing downstream and the flow direction is normal to the cross section. The maximum unit discharge on the left hand side of the channel is about 2.5 times than that on the right hand side, however, the velocity does not vary to this extent over the channel width. In general, the lateral location of maximum velocity changes with discharge. However, although not shown here, this trend was not observed because the effect of secondary currents in Eq. (1) is ignored in the present study. Fig. 3 shows the lateral distribution of the bedload per unit width over the channel width. Plots of both predicted and measured data are shown for comparison. MPM and CL formulas are used to predict the bedload in Fig. 3(a) and (b), respectively. In the figure, various symbols do not denote the raw measured data, but rather, the best fit of the measured data. Otherwise no trend for the measured data is apparent. In the prediction, the roughness height is estimated by ks ¼ 2:0D90 , as proposed by Camenen et al. (2011). It can be seen in Fig. 3(a) that MPM formula predicts bedload only on the left hand side of the channel where the velocity is higher, which becomes serious as the discharge decreases. This indicates that the Shields stress on the right hand side of the channel does not exceed its critical value. Although the critical Shields stress is only a matter of convenience for correlating experimental data as indicated by Parker (2004), the resulting pattern of the bedload transport
Fig. 2. Lateral distribution of unit discharge and depth-averaged velocity (up) and the geometry of channel section (down) in the Danube River.
Fig. 3. Lateral distribution of bedload per unit width in the Danube River.
predicted by MPM formula is completely different from that obtained by CL formula. That is, CL formula yields the bedload over the entire section, once the discharge is greater than 1750 m3/s. This is consistent with the feature observed in the measured data by Camenen et al. (2011). In general, it appears that CL formula is a better predictor of bedload. However, this formula still under-predicts the bedload per unit width, which is more serious on the right hand side of the channel. It can also be noted in the figure that the fitted curve from measured data shows the peak value of bedload at about y ¼ 80 m with a smooth increase from the left bank. However, the raw data in Camenen et al. (2011) do not show this tendency clearly. If the fact that the peak value occurs at about y ¼ 80 m is true, then the present model predicts the location of the peak of the bedload incorrectly. This is directly related to the velocity distribution in Fig. 2, showing the maximum at the deepest point in the cross section, predicted without consideration of the secondary flows. Fig. 4 shows the change in the total bedload with discharge. Measured data are compared with bedloads computed by the two formulas. The total bedloads predicted using ks ¼ 2:5D50 and ks ¼ 2:0D90 are provided for comparisons. The value of
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prediction by the 1D model is due to the fact that the shear stress averaged over the width yields a smaller sediment load than that estimated by the lateral distribution method, which becomes serious particularly for the case of a low discharge. According to Camenen et al. (2011), the total bedload tends to decrease if the discharge increases continuously, i.e., Q > 3500 m3/s. This occurs because the mean velocity and bed shear stress rarely change or even decrease for such large discharges. However, in the present application, this is not observed because the roughness coefficient is kept constant throughout the computations. Fig. 5 presents the change in total bedload with discharge with and without the effect of a mixture of sediment particles. If MPM formula is used, consideration of the mixture effect leads to a slight over-prediction of the total bedload for Q < 1300 m3/s. For Q > 1300 m3/s, the prediction using the mixture effect consistently under-estimates the bedload by about 20%. When CL formula is used, the prediction using the mixture effect consistently under-estimates the bedload by about 16%, compared with the total bedload obtained with uniform sediment assumed. 4. Application to the San-Bed River
Fig. 4. Total bedload versus discharge in the Danube River.
the effective roughness height estimated by the latter is about twice that estimated by the former. It can be seen in the figure that a higher effective roughness height yields a larger total bedload. This is simply because the bed shear stress increases with the effective roughness height. For both formulas, the prediction using ks ¼ 2:0D90 appears to be better than that using ks ¼ 2:5D50 . In a similar study for bedload prediction in the same river by Camenen et al. (2011), the former relation for effective roughness height was proposed. When ks ¼ 2:0D90 is used, MPM formula under-predicts the total bedload for a low discharge such as Q < 3000 m3/s and over-predicts for a high discharge such as Q > 3000 m3/s. However, CL formula successfully predicts the total bedload compared with measured data. In Fig. 4, the total bedload predicted by the 1D approach is also given for comparison. It can be seen that the total bedload computed by the 1D model is significantly smaller than the value obtained using the proposed method for a small discharge. This is consistent with findings reported by Seed (1996) and Bradely et al. (1999). The under-prediction by MPM formula appears to be more serious. However, as the discharge increases, the difference becomes smaller. Under-
The Han River is located at the middle part of the Korean peninsula. The river, which flows from east to west, runs through the metropolitan area of the city of Seoul and flows toward the Yellow Sea (see Fig. 6). The river is 459.3 km long, and the watershed area is 34,674 km2. The Yeojoo Station is located 174.3 km upstream from the mouth of the river. The average slope of the water surface is 0.000175. The cross section of the station is given in Fig. 7, showing a compound channel with a floodplain on the right hand side. The bed materials at this station consist mostly of sand, with a median diameter of 0.67 mm. Fig. 7 shows the lateral distributions of unit discharge and the depth-averaged velocity for the total discharge of 5827 m3/ s at Yeojoo Station in the Han River. It can also be seen that a thalweg has been developed at the left hand side of the cross section, the ratio of the maximum depth-averaged velocity to the section-averaged velocity is 1.32. The bed elevation profile is also given together with flow depth partition due to the presence of bedforms. The flow direction is normal to the cross section, which is a view facing downstream in the figure. The vertical distance between the channel bottom and the dotted line is the flow depth contributed by the resistance due to sediment particles, and the vertical distance between the dotted line and the free surface is due to the form drag. As can be seen in the figure, about 37% of the total flow depth is due to skin friction, which contributes to sediment transport. The lateral distributions of sediment loads are given in Fig. 8. Herein, CL formula is used to predict the bedload. Solid and dotted lines represent the total and suspended sediment loads, respectively, as computed by the proposed method. The vertical distance between the solid and dotted lines denotes the amount of bedload. In the computations, the mixture effect is taken into account. From the figure, the
Please cite this article in press as: Choi, S.-U., Lee, J., Assessment of total sediment load in rivers using lateral distribution method, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.06.002
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Fig. 7. Lateral distribution of unit discharge and depth-averaged velocity (up) and the geometry of channel with flow depth partition due to bedforms (down) in the Han River.
Fig. 5. Total bedload versus discharge in the Danube River.
amount of bedload is estimated to be about 7.5% of the total sediment load. If the sediment particles are assumed to be uniform, then the computed portion of the bedload in the total sediment load decreases slightly to 7%.
Fig. 9 presents the change in the total sediment load with discharge at the Yeojoo Station on the Han River, Korea. Predicted results with and without sediment mixture being considered are plotted together with measured data and their best fit. Here, the measured data mean the total sediment load estimated by the modified Einstein procedure using sampled suspended sediment concentrations. Detailed information on the sampling method and devices used are given in the Hydrological Survey Center, Korea (HSC, 2008). In general, the proposed method successfully predicts the total sediment load. Specifically, the total sediment load predicted by the proposed method with the mixture effect is slightly larger than that without the mixture effect for Q < 2500 m3/s and vice versa for Q > 2500 m3/s. However, the difference is negligible. It should also be noted that the 1D approach under-predicts the total sediment load significantly.
Fig. 6. Location of the study reach in the Han River, Korea and grain size distribution of bed materials. Please cite this article in press as: Choi, S.-U., Lee, J., Assessment of total sediment load in rivers using lateral distribution method, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.06.002
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Fig. 8. Lateral distribution of total sediment load in the Han River.
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discharges. In addition, consideration of the mixture effect led to a slight under-prediction of the total bedload. Finally, the model was applied to the Han River, a large sand-bed river in Korea. Einstein partition estimated that about 37% of the total shear stress was due to skin friction. The lateral distribution of bedload predicted by Camenen and Larson's formula was presented, revealing that about 7% of the total sediment load was made up of bedload. The predicted total sediment load with discharge compared favorably to that estimated from sampled data for the concentration of suspended sediment. In this case, consideration of mixture effect under- and over-estimated slightly the total sediment load for low and large discharges, respectively.
5. Conclusions
Acknowledgments
A new method for numerically predicting the total sediment load in a river is described. The method predicts the total sediment load using information on channel geometry and slope, discharge, and the size of bed materials. For a given discharge, the lateral profile for the depth-averaged velocity was obtained using the lateral distribution method. The portion of skin friction in the total shear stress was estimated by means of Einstein partition. The bedload and suspended load were then estimated separately. Formulas such as MyerePeter and Muller's and CameneneLarson's relationships were used to predict bedload transport and Rouse distribution with Wright and Parker's entrainment function was used for suspended sediment transport. The proposed method was applied to a station in the Danube River, in Slovakia. The study reach has a gravel bed, whose median size of bed materials can be classified as medium gravel. A DuBoys type formula such as MyereMuller and Peter's relationship predicted no bedload if the shear stress is less than its critical value although a significant bedload is transported in reality. However, CameneneLarson's formula appropriately predicted the lateral distribution of the bedload. Comparisons with measured data revealed that the proposed method successfully predicted the total bedload, while the 1D approach resulted in a significant under-prediction. This under-prediction by the 1D approach became serious for low
This study is supported by the Korea Agency for Infrastructure Technology Advancement (KAIA) under grant contract number #11 CTIP-C04.
Fig. 9. Total sediment load versus discharge in the Han River.
References Bradely, J.B., Williams, D.T., Walton, R., 1999. Applicability and limitations of sediment transport modeling in gravel-bed rivers. In: Klingeman, P.C., et al. (Eds.), Gravel-Bed Rivers in the Environment. Water Resources Publications, Littleton, CO. Camenen, B., Larson, M., 2005. A bed-load sediment transport formula for the nearshore. Estuar. Coast. Shelf Sci. 63, 249e260. Camenen, B., Holubova, K., Lukac, M., Le Coz, J., Paquier, A., 2011. Assessment of methods used in 1D models for computing bed-load transport in a large river: the Danube River in Slovakia. J. Hydraul. Eng., ASCE 137 (10), 1190e1199. de Almeida, G.A.M., Rodriguez, J.F., 2011. Understanding pool-riffle dynamics through morphological simulations. Water Resour. Res., AGU 47, W01502. Einstein, H.A., 1950. The Bed Load Function of Sediment Transportation in Open Channel Flows (Technical Bulletin 1026). USDA, Soil Conservation Service. Ferguson, R., Prestegaard, K., Ashworth, P., 1989. Influence of sand on hydraulics and gravel transport in a braided gravel bed river. Water Resour. Res., AGU 25 (4), 635e643. Garcia, M., 2008. Sediment transport and morphodynamics. In: Garcia, M.H. (Ed.), Sedimentation Engineering. ASCE, Reston, VI (Chapter 2). Garcia, M.H., Parker, G., 1991. Entrainment of bed sediment into suspension. J. Hydraul. Eng., ASCE 117 (4), 414e435. HSC, 2008. Measurement of Sediment Discharge (2007 Hydrological Survey Center Report). Ministry of Land, Transport, and Maritime Affairs, Korea (in Korean). Meyer-Peter, E., Muller, R., 1948. Formulas for bed-load transport. In: Proceedings of 2nd IAHR Congress, IAHR, 39e64, Madrid, Spain. Parker, G., 2004. 1d Sediment Transport Morphodynamics with Applications to Rivers and Turbidity Currents (E-book). Seed, D., 1996. Lateral Variation of Sediment Transport in Rivers (SR454). HR Wallingford, Wallingford, UK. Shiono, K., Knight, D.W., 1990. Mathematical models of flow in two or multi stage straight channels. In: White, W.R. (Ed.), Proceedings of International Conference on River Flood Hydraulics, pp. 229e238 (Wallingford Paper G1). Toffaleti, F.B., 1969. Definitive computations of sand discharge in rivers. J. Hydraul. Div., ASCE 95 (1), 1831e1842. van Rijn, L.C., 1984a. Sediment transport, part I: bed load transport. J. Hydraul. Eng., ASCE 110 (10), 1431e1456. van Rijn, L.C., 1984b. Sediment transport, part II: suspended load transport. J. Hydraul. Eng., ASCE 110 (11), 1613e1641. Wilcock, P., Crowe, J., 2003. Surface-based transport model for mixed-size sediment. J. Hydraul. Eng., ASCE 129 (2), 120e128. Wright, S., Parker, G., 2004. Flow resistance and suspended load in sand-bed rivers: simplified stratification model. J. Hydraul. Eng. 130 (8), 796e805.
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Available online at www.sciencedirect.com
ScienceDirect Journal of Hydro-environment Research xx (2014) 1e7 www.elsevier.com/locate/jher
Research paper
Assessment of total sediment load in rivers using lateral distribution method Sung-Uk Choi a,*, Jinhwi Lee b,1 a
School of Civil and Environmental Engineering, Yonsei University, Seoul 120-749, Republic of Korea b ISAN Research Institute, ISAN Corporation, Gyeonggi-do 431-060, Republic of Korea Received 26 March 2013; revised 12 May 2014; accepted 3 June 2014
Abstract A new numerical model for predicting the total sediment load in the river is presented. The model is comprised of two parts, namely flow and sediment transport parts. The flow analysis is carried out using the lateral distribution method, which distributes the flow and sediment load across the width, based on channel geometry and flow dynamics. To obtain the total sediment load, the bedload and suspended load are predicted separately. Conventional formulas are used for the bedload, and Rousean distribution with an entrainment function is used for the suspended load. Einstein partition is used to separate the form drag component from the total shear stress. The model is applied to the Danube River, a gravel-bed river in Slovakia and the Han River, a sand-bed river in Korea. Predicted distributions of sediment load in the lateral direction are given. The predicted results are compared with those obtained using the 1D approach as well as from measured data. Total sediment load rating curves are presented and the mixture effect of bed-materials is discussed. © 2014 International Association for Hydro-environment Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved.
Keywords: Bedload; Lateral distribution method; Sediment transport; Suspended load; Total sediment load
1. Introduction Sediment transport in rivers is a major issue in many countries. Sediment particles that are transported by the river induces morphodynamic changes. An over-supply of sediment results in aggradation of the river bed that can raise the flood level. It may also fill the habitats of aquatic animals and cause a decrease in the function of stream installations. A sediment under-supply results in the degradation of the river bed, which can endanger stream installations such as banks and bridge piers by undermining their foundations. Moreover, it can have a serious impact on the physical habitats of aquatic biota. Instream vegetation can be eradicated by the flow, and the
* Corresponding author. E-mail addresses: [email protected] (S.-U. Choi), [email protected] (J. Lee). 1 Formerly, Graduate student at Department of Civil & Environmental Engineering, Yonsei University, Seoul 120-749, Republic of Korea.
spawning ground for various types of fish can deteriorate by flushing fine sediment particles from the substrate. In spite of its importance, predicting sediment transport in a river has not been so successful, from the engineering point of view. A variety of different formulas have been proposed, but none of them have found to be universally applicable. A formula, that accurately predicts sediment transport in one river, is not applicable to another river. The difficulty in making such predictions arises from inherently complicated nature of the physics of sediment transport in a river. The conventional approach is the 1D method, in which the total sediment load is computed by multiplying the sediment load per unit width to the channel width, with the assumption that the channel cross section is rectangular in shape. This approach is valid for the case of a wide channel whose flow depth is relatively constant. However, for rivers whose flow depth varies to a large extent, the 1D method may underpredict the bedload seriously (Bradely et al., 1999). This is why Camenen et al. (2011) and de Almeida and Rodriguez
http://dx.doi.org/10.1016/j.jher.2014.06.002 1570-6443/© 2014 International Association for Hydro-environment Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved.
Please cite this article in press as: Choi, S.-U., Lee, J., Assessment of total sediment load in rivers using lateral distribution method, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.06.002
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(2011) computed the total sediment load in a large river by distributing the shear stress laterally based on the channel geometry. Previous studies, however, did not take into account the impact of flow dynamics on the lateral distribution of the discharge. This motivated the present study. The objective of this study was to develop a new, more accurate and reliable method, for numerically predicting the total sediment load in a river. The method predicts the total sediment load with information on discharge, channel geometry and slope, and the size of bed materials. The bedload and suspended load are separately computed, based on a flow analysis by the lateral distribution method. The lateral distribution method provides the lateral profile of the flow variables, including the depth-averaged velocity and boundary shear stress, based on the channel geometry and flow dynamics. The method is applied to the bedload transport in the Danube River, in Slovakia and to the total sediment load transport in the Han River, in Korea. The predicted results are compared with results computed by the conventional 1D approach, as well as with measured data. 2. Proposed method The proposed method consists of two parts, namely the flow analysis and the computation of sediment transport. The lateral distribution method is used for the flow analysis. Separating the impact of bedform from the total shear stress is then achieved by means of the Einstein partition. Finally, the bedload and suspended load are estimated separately using the formula and Rousean distribution, respectively. The total sediment load is then obtained by adding the bedload and the suspended load. 2.1. Flow analysis Shiono and Knight (1990) developed the following equation by integrating Reynolds equations over depth for a uniform open-channel flow: rgHSx þ
v vU rH 3 Bg tb ¼ G vy vy
ð1Þ
where y denotes the lateral direction, r is the water density, g is the gravitational acceleration, H is the flow depth, Sx is the bed slope in the streamwise (x) direction, 3 is the y-component eddy viscosity, U is the depth-averaged streamwise velocity, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Bg is the geometric factor defined by 1 þ S2x þ S2y (here, Sy is the bed slope in the lateral direction), tb is the bed shear stress, and G is the term related with secondary currents. In Eq. (1), the bed shear stress and y-component eddy viscosity can be represented by, respectively, tb ¼ 3
MODEL
rgn2 2 U H 1=3
¼ lu* H
ð2Þ ð3Þ
where n is the roughness coefficient, l is the non-dimensional eddy viscosity, and u* is the local shear velocity. The roughness coefficient can be varied, for example, to reflect higher roughness near the bank, but a constant value of roughness coefficient is used herein. In addition, the term related with secondary currents in Eq. (1), is given by 0 H 1 Z v@ G¼ ru v dzA ð4Þ vy 0
where u and v represent time-averaged streamwise and lateral velocity components, respectively. In the present study, the value of non-dimensional eddy viscosity of l ¼ 0.067 is used and the impact of secondary currents is ignored, i.e., G ¼ 0. Before the computations, Manning's roughness coefficient is calibrated using measured stageedischarge relationships. The calibrated values for the roughness coefficient are normally larger than the value estimated by formula such as ManningeStrickler's relationship. This is due to the presence of a bedform on the river bed, channel meandering, changes in channel width, and so on. Then, for a given discharge, the finite difference form of Eq. (1) is solved numerically. Freeslip boundary conditions are imposed at the sides of the channel. The computations are repeated by changing the stage until the computed discharge becomes identical with the given discharge. Solving the difference equation provides the lateral distributions of depth-averaged velocity U and local shear velocity u*. 2.2. Computation of total sediment load In general, two approaches are available for predicting the total bed-material load in a river. One is to estimate the bedload and suspended load separately. This approach is founded on the fact that the hydrodynamics of each mode of sediment transport is different. Thus, models of the bedload and suspended load should be based on accurate mechanics. The methods proposed by Einstein (1950), van Rijn (1984a,b), and Toffaleti (1969) utilize this approach. The other approach is to estimate the total bed-material load directly without dividing the transport mode into two parts. This approach is empirical and mostly relates the dimensionless rate of sediment transport to a dimensionless flow parameter. This approach is simpler, but ignores the details of the physics of particle transport by a flow. This approach can only be applied in the strict sense to equilibrium or quasi-equilibrium flows (Garcia, 2008). However, models based on this approach have advantages in that they have been calibrated with laboratory or field data. 2.3. Computation of bedload In the present study, Meyer-Peter and Muller's (1948) formula and Camenen and Larson's (2005) formula are used to predict bedload transport (hereafter they are denoted as MPM and CL formulas, respectively). They are given by, respectively,
Please cite this article in press as: Choi, S.-U., Lee, J., Assessment of total sediment load in rivers using lateral distribution method, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.06.002
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3=2 q*sb ¼ 8 t* t*c with t*c ¼ 0:0470
ð5Þ
3=2 t* q*sb ¼ 12 t* exp 4:5 c* with t*c ¼ 0:055 t
ð6Þ
where q*sb is Einstein bedload number defined by pffiffiffiffiffiffiffiffiffi qsb =ð RgDDÞ with qsb the volume rate of bedload transport per unit width [L2/T] (here, R and D are the submerged specific gravity and diameter of the sediment particles, respectively), and t* and t*c are Shields stress and its critical value, respectively. MPM formula is a typical DuBoys type relationship for bedload transport in that the formula yields the amount of bedload transport only if the Shields stress exceeds its critical value. CL formula is a modification of MPM formula based on the probabilistic approach for the incipient motion of particles on the bed (Camenen and Larson, 2005). CL formula predicts an exponentially-decaying bedload even if the Shields stress is less than its critical value.
The volume rate of the suspended sediment load per unit width is obtained by qss ¼
ZH u c dzz
0
u c dz
ð7Þ
b
where c is the time-averaged concentration of suspended sediment and b is the height close to bed. Normally, b is taken to be 0.05H. In order to estimate suspended load, Rousean distribution of suspended sediment is assumed. That is, Z ð1 2Þ=2 cð2Þ ¼ cb ð8Þ ð1 2b Þ=2b where cb is the near-bed sediment concentration, 2 ¼ z=H, 2b ¼ b=H, and Z is Rouse number defined by vs =ðku* Þ (here, vs is the settling velocity of the particle, and k is von Karman constant). For equilibrium suspension, the near-bed concentration of suspended sediment is the same as dimensionless entrainment rate E, i.e., cb ¼ E. In the present study, the following dimensionless entrainment rate by Wright and Parker (2004) is used: E¼
AZu5 A 5 1 þ 0:3 Zu
2.5. Mixture effect For a sediment mixture on the channel bed, fine particles tend to be located behind coarser particles. As a result, larger particles are more exposed to the flow, allowing them to move more readily. This is called the hiding effect. Normally, after sediment particles are divided into finite classes, the hiding effect is considered by changing the critical Shields stress for jth particle, i.e., B Dj * * ð10Þ tc;j ¼ tc D50 where the exponent B is a constant. For B, Ferguson et al. (1989) proposed a constant value of 0.12 and Wilcock and Crowe (2003) proposed the following relationship: B¼
0:67 1 þ exp 1:5 Dj D50
ð11Þ
In the present study, the relationship by Wilcock and Crowe (2003) is used.
2.4. Computation of suspended load
ZH
3
3. Application to the gravel-bed river The proposed method was initially applied to bedload transport in a Slovak reach of the Danube River. Fig. 1 shows the location of the study reach. According to Camenen et al. (2011), the Danube River has experienced a dramatic change due to flood protection projects, the construction of navigation channels, and instream structures in the upstream reach of the river. As a result, the downstream reach of the SlovakHungarian Danube River has become degraded. To study the impact of the hydropower plant, Camenen et al. (2011) made extensive measurements of the bedload with a basket-type bedload sampler. The sampling point of the bedload is located about 11 km downstream from the gauging station. Sediment particles range from fine to coarse gravel with
ð9Þ
where A is a constant equal to 5:7 107 and Zu is the similarity variable defined by Zu ¼
u*s 0:6 0:07 Re S vs p
with u*s is the shear velocity partition due to skin friction. Eq. (9) is a modification of Garcia and Parker's (1991) relationship for applications to large and low slope streams.
Fig. 1. Location of the study reach in the Danube River (modified from Camenen et al. (2011)).
Please cite this article in press as: Choi, S.-U., Lee, J., Assessment of total sediment load in rivers using lateral distribution method, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.06.002
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D50 ¼ 9 mm and D90 ¼ 22 mm, corresponding to medium and coarse gravels, respectively. The grain size distribution of the bed sediment particles can be found in Camenen et al. (2011). By fitting to measured datasets for average velocity versus discharge, a value of n ¼ 0.026 was obtained. This value of the roughness coefficient is slightly larger than n ¼ 0.022 estimated by Strickler's formula using D50. This may be due to the presence of bedforms and changes in channel width. Fig. 2 shows the lateral distributions of unit discharge and depth-averaged velocity for Q ¼ 4640 m3/s and the channel geometry of the sampling point. The cross section in the figure is a view facing downstream and the flow direction is normal to the cross section. The maximum unit discharge on the left hand side of the channel is about 2.5 times than that on the right hand side, however, the velocity does not vary to this extent over the channel width. In general, the lateral location of maximum velocity changes with discharge. However, although not shown here, this trend was not observed because the effect of secondary currents in Eq. (1) is ignored in the present study. Fig. 3 shows the lateral distribution of the bedload per unit width over the channel width. Plots of both predicted and measured data are shown for comparison. MPM and CL formulas are used to predict the bedload in Fig. 3(a) and (b), respectively. In the figure, various symbols do not denote the raw measured data, but rather, the best fit of the measured data. Otherwise no trend for the measured data is apparent. In the prediction, the roughness height is estimated by ks ¼ 2:0D90 , as proposed by Camenen et al. (2011). It can be seen in Fig. 3(a) that MPM formula predicts bedload only on the left hand side of the channel where the velocity is higher, which becomes serious as the discharge decreases. This indicates that the Shields stress on the right hand side of the channel does not exceed its critical value. Although the critical Shields stress is only a matter of convenience for correlating experimental data as indicated by Parker (2004), the resulting pattern of the bedload transport
Fig. 2. Lateral distribution of unit discharge and depth-averaged velocity (up) and the geometry of channel section (down) in the Danube River.
Fig. 3. Lateral distribution of bedload per unit width in the Danube River.
predicted by MPM formula is completely different from that obtained by CL formula. That is, CL formula yields the bedload over the entire section, once the discharge is greater than 1750 m3/s. This is consistent with the feature observed in the measured data by Camenen et al. (2011). In general, it appears that CL formula is a better predictor of bedload. However, this formula still under-predicts the bedload per unit width, which is more serious on the right hand side of the channel. It can also be noted in the figure that the fitted curve from measured data shows the peak value of bedload at about y ¼ 80 m with a smooth increase from the left bank. However, the raw data in Camenen et al. (2011) do not show this tendency clearly. If the fact that the peak value occurs at about y ¼ 80 m is true, then the present model predicts the location of the peak of the bedload incorrectly. This is directly related to the velocity distribution in Fig. 2, showing the maximum at the deepest point in the cross section, predicted without consideration of the secondary flows. Fig. 4 shows the change in the total bedload with discharge. Measured data are compared with bedloads computed by the two formulas. The total bedloads predicted using ks ¼ 2:5D50 and ks ¼ 2:0D90 are provided for comparisons. The value of
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prediction by the 1D model is due to the fact that the shear stress averaged over the width yields a smaller sediment load than that estimated by the lateral distribution method, which becomes serious particularly for the case of a low discharge. According to Camenen et al. (2011), the total bedload tends to decrease if the discharge increases continuously, i.e., Q > 3500 m3/s. This occurs because the mean velocity and bed shear stress rarely change or even decrease for such large discharges. However, in the present application, this is not observed because the roughness coefficient is kept constant throughout the computations. Fig. 5 presents the change in total bedload with discharge with and without the effect of a mixture of sediment particles. If MPM formula is used, consideration of the mixture effect leads to a slight over-prediction of the total bedload for Q < 1300 m3/s. For Q > 1300 m3/s, the prediction using the mixture effect consistently under-estimates the bedload by about 20%. When CL formula is used, the prediction using the mixture effect consistently under-estimates the bedload by about 16%, compared with the total bedload obtained with uniform sediment assumed. 4. Application to the San-Bed River
Fig. 4. Total bedload versus discharge in the Danube River.
the effective roughness height estimated by the latter is about twice that estimated by the former. It can be seen in the figure that a higher effective roughness height yields a larger total bedload. This is simply because the bed shear stress increases with the effective roughness height. For both formulas, the prediction using ks ¼ 2:0D90 appears to be better than that using ks ¼ 2:5D50 . In a similar study for bedload prediction in the same river by Camenen et al. (2011), the former relation for effective roughness height was proposed. When ks ¼ 2:0D90 is used, MPM formula under-predicts the total bedload for a low discharge such as Q < 3000 m3/s and over-predicts for a high discharge such as Q > 3000 m3/s. However, CL formula successfully predicts the total bedload compared with measured data. In Fig. 4, the total bedload predicted by the 1D approach is also given for comparison. It can be seen that the total bedload computed by the 1D model is significantly smaller than the value obtained using the proposed method for a small discharge. This is consistent with findings reported by Seed (1996) and Bradely et al. (1999). The under-prediction by MPM formula appears to be more serious. However, as the discharge increases, the difference becomes smaller. Under-
The Han River is located at the middle part of the Korean peninsula. The river, which flows from east to west, runs through the metropolitan area of the city of Seoul and flows toward the Yellow Sea (see Fig. 6). The river is 459.3 km long, and the watershed area is 34,674 km2. The Yeojoo Station is located 174.3 km upstream from the mouth of the river. The average slope of the water surface is 0.000175. The cross section of the station is given in Fig. 7, showing a compound channel with a floodplain on the right hand side. The bed materials at this station consist mostly of sand, with a median diameter of 0.67 mm. Fig. 7 shows the lateral distributions of unit discharge and the depth-averaged velocity for the total discharge of 5827 m3/ s at Yeojoo Station in the Han River. It can also be seen that a thalweg has been developed at the left hand side of the cross section, the ratio of the maximum depth-averaged velocity to the section-averaged velocity is 1.32. The bed elevation profile is also given together with flow depth partition due to the presence of bedforms. The flow direction is normal to the cross section, which is a view facing downstream in the figure. The vertical distance between the channel bottom and the dotted line is the flow depth contributed by the resistance due to sediment particles, and the vertical distance between the dotted line and the free surface is due to the form drag. As can be seen in the figure, about 37% of the total flow depth is due to skin friction, which contributes to sediment transport. The lateral distributions of sediment loads are given in Fig. 8. Herein, CL formula is used to predict the bedload. Solid and dotted lines represent the total and suspended sediment loads, respectively, as computed by the proposed method. The vertical distance between the solid and dotted lines denotes the amount of bedload. In the computations, the mixture effect is taken into account. From the figure, the
Please cite this article in press as: Choi, S.-U., Lee, J., Assessment of total sediment load in rivers using lateral distribution method, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.06.002
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Fig. 7. Lateral distribution of unit discharge and depth-averaged velocity (up) and the geometry of channel with flow depth partition due to bedforms (down) in the Han River.
Fig. 5. Total bedload versus discharge in the Danube River.
amount of bedload is estimated to be about 7.5% of the total sediment load. If the sediment particles are assumed to be uniform, then the computed portion of the bedload in the total sediment load decreases slightly to 7%.
Fig. 9 presents the change in the total sediment load with discharge at the Yeojoo Station on the Han River, Korea. Predicted results with and without sediment mixture being considered are plotted together with measured data and their best fit. Here, the measured data mean the total sediment load estimated by the modified Einstein procedure using sampled suspended sediment concentrations. Detailed information on the sampling method and devices used are given in the Hydrological Survey Center, Korea (HSC, 2008). In general, the proposed method successfully predicts the total sediment load. Specifically, the total sediment load predicted by the proposed method with the mixture effect is slightly larger than that without the mixture effect for Q < 2500 m3/s and vice versa for Q > 2500 m3/s. However, the difference is negligible. It should also be noted that the 1D approach under-predicts the total sediment load significantly.
Fig. 6. Location of the study reach in the Han River, Korea and grain size distribution of bed materials. Please cite this article in press as: Choi, S.-U., Lee, J., Assessment of total sediment load in rivers using lateral distribution method, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.06.002
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Fig. 8. Lateral distribution of total sediment load in the Han River.
7
discharges. In addition, consideration of the mixture effect led to a slight under-prediction of the total bedload. Finally, the model was applied to the Han River, a large sand-bed river in Korea. Einstein partition estimated that about 37% of the total shear stress was due to skin friction. The lateral distribution of bedload predicted by Camenen and Larson's formula was presented, revealing that about 7% of the total sediment load was made up of bedload. The predicted total sediment load with discharge compared favorably to that estimated from sampled data for the concentration of suspended sediment. In this case, consideration of mixture effect under- and over-estimated slightly the total sediment load for low and large discharges, respectively.
5. Conclusions
Acknowledgments
A new method for numerically predicting the total sediment load in a river is described. The method predicts the total sediment load using information on channel geometry and slope, discharge, and the size of bed materials. For a given discharge, the lateral profile for the depth-averaged velocity was obtained using the lateral distribution method. The portion of skin friction in the total shear stress was estimated by means of Einstein partition. The bedload and suspended load were then estimated separately. Formulas such as MyerePeter and Muller's and CameneneLarson's relationships were used to predict bedload transport and Rouse distribution with Wright and Parker's entrainment function was used for suspended sediment transport. The proposed method was applied to a station in the Danube River, in Slovakia. The study reach has a gravel bed, whose median size of bed materials can be classified as medium gravel. A DuBoys type formula such as MyereMuller and Peter's relationship predicted no bedload if the shear stress is less than its critical value although a significant bedload is transported in reality. However, CameneneLarson's formula appropriately predicted the lateral distribution of the bedload. Comparisons with measured data revealed that the proposed method successfully predicted the total bedload, while the 1D approach resulted in a significant under-prediction. This under-prediction by the 1D approach became serious for low
This study is supported by the Korea Agency for Infrastructure Technology Advancement (KAIA) under grant contract number #11 CTIP-C04.
Fig. 9. Total sediment load versus discharge in the Han River.
References Bradely, J.B., Williams, D.T., Walton, R., 1999. Applicability and limitations of sediment transport modeling in gravel-bed rivers. In: Klingeman, P.C., et al. (Eds.), Gravel-Bed Rivers in the Environment. Water Resources Publications, Littleton, CO. Camenen, B., Larson, M., 2005. A bed-load sediment transport formula for the nearshore. Estuar. Coast. Shelf Sci. 63, 249e260. Camenen, B., Holubova, K., Lukac, M., Le Coz, J., Paquier, A., 2011. Assessment of methods used in 1D models for computing bed-load transport in a large river: the Danube River in Slovakia. J. Hydraul. Eng., ASCE 137 (10), 1190e1199. de Almeida, G.A.M., Rodriguez, J.F., 2011. Understanding pool-riffle dynamics through morphological simulations. Water Resour. Res., AGU 47, W01502. Einstein, H.A., 1950. The Bed Load Function of Sediment Transportation in Open Channel Flows (Technical Bulletin 1026). USDA, Soil Conservation Service. Ferguson, R., Prestegaard, K., Ashworth, P., 1989. Influence of sand on hydraulics and gravel transport in a braided gravel bed river. Water Resour. Res., AGU 25 (4), 635e643. Garcia, M., 2008. Sediment transport and morphodynamics. In: Garcia, M.H. (Ed.), Sedimentation Engineering. ASCE, Reston, VI (Chapter 2). Garcia, M.H., Parker, G., 1991. Entrainment of bed sediment into suspension. J. Hydraul. Eng., ASCE 117 (4), 414e435. HSC, 2008. Measurement of Sediment Discharge (2007 Hydrological Survey Center Report). Ministry of Land, Transport, and Maritime Affairs, Korea (in Korean). Meyer-Peter, E., Muller, R., 1948. Formulas for bed-load transport. In: Proceedings of 2nd IAHR Congress, IAHR, 39e64, Madrid, Spain. Parker, G., 2004. 1d Sediment Transport Morphodynamics with Applications to Rivers and Turbidity Currents (E-book). Seed, D., 1996. Lateral Variation of Sediment Transport in Rivers (SR454). HR Wallingford, Wallingford, UK. Shiono, K., Knight, D.W., 1990. Mathematical models of flow in two or multi stage straight channels. In: White, W.R. (Ed.), Proceedings of International Conference on River Flood Hydraulics, pp. 229e238 (Wallingford Paper G1). Toffaleti, F.B., 1969. Definitive computations of sand discharge in rivers. J. Hydraul. Div., ASCE 95 (1), 1831e1842. van Rijn, L.C., 1984a. Sediment transport, part I: bed load transport. J. Hydraul. Eng., ASCE 110 (10), 1431e1456. van Rijn, L.C., 1984b. Sediment transport, part II: suspended load transport. J. Hydraul. Eng., ASCE 110 (11), 1613e1641. Wilcock, P., Crowe, J., 2003. Surface-based transport model for mixed-size sediment. J. Hydraul. Eng., ASCE 129 (2), 120e128. Wright, S., Parker, G., 2004. Flow resistance and suspended load in sand-bed rivers: simplified stratification model. J. Hydraul. Eng. 130 (8), 796e805.
Please cite this article in press as: Choi, S.-U., Lee, J., Assessment of total sediment load in rivers using lateral distribution method, Journal of Hydro-environment Research (2014), http://dx.doi.org/10.1016/j.jher.2014.06.002