Approach to Separate Sand from Gravel for Bed-Load Transport Calculations in Streams with Bimodal Sediment

Transcription

1 Approach to Separate Sand from Gravel for Bed-Load Transport Calculations in Streams with Bimodal Sediment Jaber H. Almedeij 1 ; Panayiotis Diplas, M.ASCE 2 ; and Fawzia Al-Ruwaih 3 Abstract: The bed material found in gravel-bed streams is nonuniform in terms of grain size and can typically be classified as unimodal or bimodal. The latter type of sediment distribution is usually represented by two modes, one of sand size and another of gravel. For this case, the movement of one mode becomes nonlinearly influenced by the other. As a result, the presence of the two modes in a bimodal material complicates the calculation of bed-load transport rates. The present study proposes an approach to separate the calculation of bed-load transport rates for bimodal materials into two independent fractions of sand and gravel, thereby rendering the bed sediment into two unimodal components. This approach is accomplished by decoupling the two fractions through scaling the reference Shields stresses of the sand and gravel modes to match the value of the mode of unimodal materials. Consequently, the contribution of each fraction to bed load can be estimated using a suitable relation derived for unimodal materials. Laboratory and field bed-load data available in the literature are used to examine the validity of the overall approach. DOI: / ASCE : CE Database subject headings: Sand; Gravel; Calculation; Streams; Sediment; Bed load. Introduction The bed material in gravel-bed streams is usually well represented by either a unimodal or a bimodal grain-size distribution e.g., Kondolf In the latter case, the presence of two modes, typically one of sand size and another of gravel, complicates the bed-load transport rate predictions e.g., Klingeman and Emmett 1982; Ferguson et al. 1989; Kuhnle 1993a; Wilcock 1993; Sambrook Smith et al. 1997; Powell 1998; Wilcock The traditional approach of calculating bed load using a single characteristic grain size such as the median is better suited for unimodal materials and may not be appropriate in this case. This is especially evident when the median falls in the gap between the two modes and, therefore, represents a size class containing a small percentage of the overall sediment. One approach to estimate bed-load transport rates for bimodal sediments consists of dividing the bed material into two unimodal fractions, each with a representative particle diameter e.g., Bagnold 1980; Kuhnle 1992; Wilcock Bagnold 1980 suggested using the corresponding mode grain size to represent each of the two unimodal components. The mode is a suitable 1 Assistant Professor, Civil Engineering Dept., Kuwait Univ., P.O. Box 5969, Safat 13060, Kuwait. almedeij@civil.kuniv.edu.kw 2 Professor, Dept. of Civil and Environmental Engineering, Virginia Polytechnic Institute and State Univ., Blacksburg, VA pdiplas@vt.edu 3 Professor, Dept. of Earth and Environmental Sciences, Kuwait Univ., P.O. Box 5969, Safat 13060, Kuwait. Fawzia@ kuc01.kuniv.edu.kw Note. Discussion open until April 1, Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on May 19, 2005; approved on November 30, This paper is part of the Journal of Hydraulic Engineering, Vol. 132, No. 11, November 1, ASCE, ISSN /2006/ / $ choice for the unimodal component, because among the other sizes available its motion is the most stable and it has the advantage of always having the highest percentage by weight Almedeij and Diplas Under these circumstances, the simplest possible approach would be to carry out two separate computations of bed load, one for each mode, to determine the contributions of the respective components. However, such an approach to calculate bed-load transport rates treats the mobility of sand and gravel as completely independent of each other, which is contrary to what one might expect and has been reported in the literature for most cases of bimodal mixtures e.g., Bagnold For some cases, the material represented by the sand mode tends to hide in the crevices created by the coarser gravel and requires a higher shear stress to be entrained as compared to that necessary for the same size when available as unimodal. In contrast, the coarser gravel becomes more exposed to the hydrodynamic forces and can be moved by shear stresses lower than that necessary for the same size when available as unimodal. The objective of this study is to present an approach to separate the calculation of bed-load transport rates for bimodal materials into two independent unimodal components, sand and gravel. First, the degree of influence that the presence of one sediment fraction exerts on the mobility of the other is examined using laboratory bed-load data collected by Kuhnle 1993b from experiments having three different bimodal materials, with 90, 75, and 55% sand. Then, an approach to transform the bimodal material into two independent unimodal fractions is proposed and verified. Furthermore, field bed-load data collected by Kuhnle 1992 from Goodwin Creek with 25% of surface sand is used to examine the validity of the approach for a more complicated condition, when the bimodal bed material becomes stratified in the vertical direction into surface and subsurface layers / JOURNAL OF HYDRAULIC ENGINEERING ASCE / NOVEMBER 2006

2 in the original bed material. For example, the original mixture of SG45 has sand and gravel modes, respectively, of D s =0.42 mm and D g =5.657 mm, as shown in Fig. 1, which are similar to the corresponding sizes in the transported materials SG45-4, -5, and -6 Table 1. This finding supports the use of the mode values to represent the calculation of bed-load transport rates for a bimodal bed material. It should be noted, however, that for low and intermediate shear stresses the contribution of each sediment component to bed load is different from that present in the bed material. Degree of Interference between Sand and Gravel An effort is made in this section to determine the degree of interference between sand and gravel during bed-load motion as follows. The bimodal bed material is assumed to consist of two independent unimodal components. Then, the total bed-load transport rate contributed from both components is calculated and compared to that measured under the same flow condition. The reference Shields stresses of the sand and gravel modes are then estimated to obtain the degree of interference resulting from this assumption. Initially, by assuming two independent unimodal components of sand and gravel, the total bed-load transport rate, q B, can be expressed as q B = f s q s + f g q g 1 Fig. 1. Grain size distribution of bed materials: data from Kuhnle 1993b Laboratory Bed-Load Data The laboratory bed-load data considered here were collected by Kuhnle 1993b in a flume 15.2 m long, m wide, and m deep. Three bed materials were employed: SG10, SG25, and SG45 Fig. 1, where the letters S and G refer to sand and gravel fractions, respectively, and the two-digit number indicates the percentage of gravel in the original bed material that is, the original mixture designated as SG45 consists of 45% gravel and 55% sand. For each bed material, six runs were performed, and bed-load transport rates were measured Table 1. As reported by Kuhnle 1993b, during all the experimental runs, no vertical stratification in terms of grain size developed for the bed materials, similar to that found in natural streams with a surface layer overlying a finer subsurface. Fig. 2 presents the varying composition of the transported bedload material, measured by Kuhnle 1993b, in response to the changing boundary shear stress,. Each series of plots deals with a single type of bed material composition and is arranged in terms of the increasing values of. As can be seen for relatively low values, the transported bed-load material consists exclusively of sand. However, as increases, part of the gravel component is entrained by the flow. As continues to increase, the contribution of gravel to bed load becomes progressively greater. For the highest available values, the proportions of gravel and sand appear to be similar to those found in the original bed materials see SG10-6, SG25-6, and SG45-5. It is interesting to note that, once the whole range of particle sizes for one component is set in motion, the corresponding mode becomes similar to that available where q s and q g =calculated bed-load transport rates of sand and gravel, respectively; and f s and f g =sand and gravel fractions % in the original bed material, respectively. This expression in a dimensionless form is W = f s W s + f g W g 2 where W =R s q B / g ds 1.5 =total dimensionless bed-load transport rate; R s =submerged specific gravity of sediment; g=gravitational acceleration; d=flow depth; S=energy slope; and W s =R s q s / g ds 1.5 and W g =R s q g / g ds 1.5 are dimensionless bed-load transport rates of sand and gravel, respectively. To find W from Eq. 2, W s and W g need to be calculated separately using any bed-load transport formula suitable for unimodal sediments. The formula proposed by Almedeij and Diplas 2003, which is an empirical fit to bed-load data considered to represent a wide range of dimensionless Shields stresses, will be used here. This formula can be written for sand and gravel in the forms W 1 s = s s and W 1 g = g g where s and g =Shields stresses based on the mode of sand and gravel, respectively. The Shields stress parameter is expressed as x =ds/r s D x, corresponding to the mode of x=s and g. It should be noted that the form of Eq. 4 Eq. 4 considered here assumes the sand gravel mode of the bed surface to be similar to that for the subsurface material, which is the case for Kuhnle s 1993b laboratory data. Fig. 3 a provides a comparison between the total bed-load transport rates measured by Kuhnle 1993b during the flume experiments and those calculated using the preceding formulation. JOURNAL OF HYDRAULIC ENGINEERING ASCE / NOVEMBER 2006 / 1177

3 Fig. 2. Grain size distribution of bed-load materials arranged from left to right based on Nm 2, from lowest to highest values: data from Kuhnle 1993b Table 1. Bed-Load Material, Original Bed Mixture, and Exposed Shear Stress Conditions for Experiments of Kuhnle 1993b Bed-load material Original bed material Measured Nonscaled calculations Scaled calculations Boundary shear stress Run D s D g D s D g q s q g q s q g q s q g s b number mm mm mm mm kg m 1 s 1 kg m 1 s 1 kg m 1 s 1 kg m 1 s 1 kg m 1 s 1 kg m 1 s 1 Nm 2 SG N a 8.60E E E E E E SG N 2.19E E E E E E SG N 2.82E E E E E E SG N 4.50E E E E E E SG E E E E E E SG E E E E E E SG N 5.71E E E E E E SG N 2.08E E E E E E SG N 2.06E E E E E E SG N 4.00E E E E E E SG E E E E E E SG E E E E E E g b SG N 2.30E E E E E E SG N 5.72E E E E E E SG N 7.59E E E E E E SG E E E E E E SG E E E E E E SG E E E E E E a N=not enough material in motion to identify clear mode grain size. b Calculated using mode grain size of corresponding fraction in original bed material / JOURNAL OF HYDRAULIC ENGINEERING ASCE / NOVEMBER 2006

4 Fig. 3. Measured bed-load transport versus bed shear stress for SG data adapted from Kuhnle 1993b, with solid line representing calculated bed-load transport rates: a bed-load calculation based on nonscaled approach; b bed-load calculation based on scaled approach From a visual inspection, it is apparent that the agreement for cases SG10 and SG25 is closer than that for SG45. A more quantitative criterion for determining the overall fitting accuracy for each case is obtained by calculating the mean absolute standard error MASE proposed by Almedeij and Diplas 2003 : where q icalculated q ri = qimeasured q icalculated q imeasured MASE = n q ri i=1 n if q imeasured q icalculated if q imeasured q icalculated 5 and n=total number of bed-load samples. Eq. 5 indicates that MASE 1; the closer the MASE value is to one, the better the accuracy of an equation, with MASE= 1 representing the condition of perfect agreement. The MASE values calculated for SG10 and SG25 are equal to 2.2 and 2.8, respectively, while for SG45 it is The results obtained for the available bed conditions suggest that decreasing the amount of sand in bimodal materials increases the error in the total bed-load transport rate calculated using this approach, which ignores the interference between sand and gravel components. More insight with regard to the degree of interference between sand and gravel components can be obtained from estimating the reference values of the Shields stress for the corresponding modes. The reference Shields stresses can be estimated using the technique of fractional transport analysis proposed by Parker et al and Diplas Initially, each of the two bimodal components is divided into N grain size ranges, each having a representative grain-size diameter, D i, and fraction, f i. The volumetric bed-load transport rate of each size range, q Bi, is then calculated from the total volumetric bed-load discharge, q B,byq Bi = p i q B, where p i =fraction of bed-load material having diameter D i. Then, dimensionless plots of bed-load transport against boundary shear stress are prepared for each size fraction using the parameters JOURNAL OF HYDRAULIC ENGINEERING ASCE / NOVEMBER 2006 / 1179

5 Fig. 4. Plot of W i versus i for SG data with D i sizes indicated and W R s q Bi i = f i g ds 1.5 i = ds 7 R s D i where W i =dimensionless bed-load transport rate for the ith grain size; and i =Shields stress for the ith grain size. The reference Shields stress for the mode size of sand, rs, and gravel, rg, can be defined from these plots such that W i equals a low reference value, W r. This value can be considered as W r =0.002 Parker et al To interpolate or extrapolate from the values of W i to W r =0.002, the technique of Wilcock and Southard 1988 can be used. The points nearest the value of W r =0.002 were given the most weight. For the SG data, the plots of W i against i are shown in Fig. 4, with the references rs and rg presented in Table 2. 6 As can be seen, the estimated reference Shields stresses capture the degree of interference between sand and gravel, especially near the threshold condition. When the amount of sand in the channel bed decreases from 90 to 55%, then rs increases from to and rg from to For comparison, the reference Shields stress for unimodal sediments is about 0.03, estimated at W r =0.002 e.g., Paintal 1971; Parker et al. 1982; Diplas 1987; Almedeij and Diplas The present results indicate that, for materials such as SG10 and SG25, the mobility of the major component of sand is not influenced in a significant Table 2. Bed Material Characteristics for Kuhnle 1993b Data Bed material Sand amount % rs rg 0.03/ rs 0.03/ rg SG SG SG / JOURNAL OF HYDRAULIC ENGINEERING ASCE / NOVEMBER 2006

6 Fig. 5. Similarity collapse for SG data with D i sizes indicated degree by the presence of gravel, because the corresponding rs values do not differ much from 0.03 Table 2. Hiding effects for the sand component become pronounced when the gravel constitutes a sizable percentage of the bed material, as in SG45. These results are in agreement with prior observations made by Iseya and Ikeda 1987 and Wilcock Transformation Approach Similarity Hypothesis For each SG material in Fig. 4, the trends exhibited by the measured bed-load transport rate plotted against shear stress are similar in shape for all grain size fractions. The obvious difference is that the trends for the finer grains are shifted laterally to the right. It is also apparent that the shift for the trends of the sand component is more pronounced than that of the gravel. Then, the trends can be collapsed onto a single one with the aid of an appropriate transformation similar to that used by Parker et al and Diplas The transformation is based on similarity and stands on plotting W i in terms of the normalized Shields stress parameter, i, where i = i / ri, and ri =reference Shields stress for the ith grain size determined for W i =W ri = Fig. 5 shows the similarity collapse for the three bimodal materials using the same individual grain sizes employed in Fig. 4. The solid line represents the unimodal trend of bed-load transport proposed by Almedeij and Diplas 2003 in terms of i, obtained using a reference Shields stress equal to As can be seen, the fractional bed-load transport rates plotted in this form nearly collapse onto the unimodal trend. The reason is that, for a specific W i value, the similarity renders the normalized Shields stresses of the individual grain sizes equivalent, 1 / r1 = 2 / r2 =...= i / ri. Another technique to demonstrate this similarity is by plotting for the entire sand component the parameter ri / rs against D i /D s, as well for the gravel, as presented in Fig. 6. Here, the dashed line corresponds to the condition of equal mobility if perfect similarity were satisfied Parker et al As is seen, this is probably true to a firstdegree approximation. Owing to the similarity, it is possible to use the transformation to collapse the two sediment components of sand and gravel on one trend using their corresponding representative mode grain sizes, i.e., s / rs = g / rg. This expression can be written once in terms of sand and another in terms of gravel, each involving the normalized Shields stress of unimodal sediment: and Fig. 6. Test for perfect similarity = s 0.03 Þ s = 0.03 s 8 s rs g = g rg 0.03 Þ g = 0.03 g 9 rg where s and g =scaled Shields stress parameters based on the sand and gravel modes, respectively. The parameters s and g, which are scaled with reference to the value of 0.03, are expected to render the mobility characteristics of the corresponding modes equivalent to that of unimodal sediments, that is, sand and gravel components independent of each other. As an example, for the sand fraction of SG45, if s is equal to 0.08, then Eq. 8 suggests s to be The implication is that equations for unimodal materials such as Eqs. 3 and 4 can be employed by using the scaled Shields stress parameters to calculate the contributions of the two bimodal components to total bed load. Fig. 7 a shows the similarity collapse obtained for the sand and gravel components using the scaled parameters. Here, in the expression of W i, f p represents the percentage of the entire sand or gravel component in the bed bed load material. As can be seen in this way, the W i values for the two components nearly collapse onto the unimodal trend. It is interesting to compare this analysis with that performed using the nonscaled parameters of Shields stress. Fig. 7 b shows that most of the W i values determined from the nonscaled parameters do not follow the unimodal trend; the W i values for all gravel components fall well above the trend, whereas for the sand component of SG45, they are, in general, below. However, it is worth seeing that, for the sand components of SG10 and SG25, having rs close to 0.03, the W i values follow the unimodal trend properly. rs JOURNAL OF HYDRAULIC ENGINEERING ASCE / NOVEMBER 2006 / 1181

7 Fig. 8. Measured versus calculated bed-load transport rate W = f s W s + f g W g 12 where W =R s q B / g ds 1.5 =total transformed dimensionless bed-load transport rate. Results Fig. 7. Dimensionless bed-load transport versus Shields stress for sand and gravel fractions of SG materials, with solid line representing unimodal trend of Almedeij and Diplas 2003 : a analysis based on scaled approach; b analysis based on nonscaled approach Transformed Bed-Load Parameters To calculate the bed-load transport rate of sand and gravel, the scaled parameters of Shields stress can be used in Eqs. 3 and 4 to obtain the forms and W s = W g = s / rs s / rs = s s g / rg g / rg = g g where W s =R s q s / g ds 1.5 and W g =R s q g / g ds 1.5 are the transformed dimensionless bed-load transport of sand and gravel, respectively. Obviously, the only difference between the transformed bed-load parameters, W s and W g, and the corresponding nontransformed ones, W s and W g, is that the former are calculated using the scaled Shields stresses, which decouple sand from gravel and thus render the bimodal material into two independent unimodal components. The total bed-load transport rate is The validity of the concept and the proposed methodology are examined by comparing the total bed-load transport rates obtained from the scaled and nonscaled approaches against the values measured by Kuhnle 1993b during his experiments Fig. 3. About 83% of the total bed-load transport rates calculated by the scaled approach fall within a range of discrepancy ratio, q calculated /q measured, between 0.5 and 2, as compared to 55% from the nonscaled approach Fig. 8. The calculated MASE values indicate improvement with respect to the overall fitting accuracy, especially for the bed-load transport data of SG45. The new MASE value for SG45 is equal to 1.78, which is about 6.6 times smaller than that obtained from the nonscaled approach. For SG10 and SG25, the calculated MASE values are, respectively, 1.1 and 2 times smaller than that from the nonscaled approach. Application to Goodwin Creek A set of bed-load data collected by Kuhnle 1992 from Goodwin Creek, a gravel-bed stream located in north central Mississippi, will be employed to examine further the validity of the methods developed in this study. The channel, which is about 25 m wide and 3 m deep, has a weakly bimodal bed material with the sand component comprising about 25% of the surface and 32% of the subsurface layers Kuhnle Apparently, the material of the surface layer is coarser than that of the subsurface, with corresponding median grain sizes equal to 11.7 and 8.3 mm. Nevertheless, both the surface and subsurface materials have similar mode grain sizes of D g =26.9 mm and D s =0.5 mm Fig. 9. Owing to the vertical segregation of the bed material, the approach proposed here to separate the calculation of bed-load transport rates for bimodal sediments becomes more complicated. The reason is that, for low shear stresses, the surface material is the main source of bed load, while for sufficiently high values the 1182 / JOURNAL OF HYDRAULIC ENGINEERING ASCE / NOVEMBER 2006

8 Fig. 9. Grain size distribution for surface and subsurface materials of Goodwin Creek data adapted from Kuhnle 1992 subsurface dominates e.g., Almedeij and Diplas The variation of the makeup of the bimodal surface fractions during bed-load motion is not considered in this study. A more comprehensive analysis for bimodal sediment requires further bed-load data covering a wide range of Shields stresses. For a first-order approximation, however, only the sand and gravel fractions of the original surface material will be used to handle the calculations herein. To determine the reference Shields stresses of sand and gravel modes for Goodwin Creek, the technique of fractional transport analysis can be used in a similar manner to that performed earlier. By plotting W i against i Fig. 10, the references for the surface material are found to be rs =1.79 and rg It is interesting to mention that those references are similar to the corresponding values obtained for the subsurface material. Based on the references determined for Goodwin Creek, as well as for SG10 and SG25, two conclusions can be drawn. For bimodal sediments, if one of the two components is dominant, then it would not be affected in a major way by the presence of the other and, thus, would behave like an equivalent unimodal material e.g., rs and for SG10 and SG25, respectively, and rg for Goodwin Creek. Conversely, the minor component would experience the greatest interference, resulting in the most pronounced hiding or exposure effects for the cases of sand e.g., rs 1.79 for Goodwin Creek and gravel e.g., rg and for SG10 and SG25, respectively. Furthermore, Rakoczi 1975 and Wilcock 1998 suggested that, when the mobility of the sand depends completely on gravel, both bimodal components would be entrained at roughly the same dimensional shear stress value; that is, rs rg, where rs and rg =dimensional reference shear stresses of the sand and gravel modes, respectively. It follows that the reference Shields stresses of the sand and gravel modes will be related to each other by the form rs rg D g /D s 13 The reference values of Shields stress obtained from Goodwin Creek satisfy this relation. Fig. 11 presents the measured bed-load transport rates of Goodwin Creek and a comparison between the trends obtained from the scaled and nonscaled approaches. It can be seen that, Fig. 10. Plot of W i versus i for surface and subsurface materials of Goodwin Creek: data from Kuhnle 1992 JOURNAL OF HYDRAULIC ENGINEERING ASCE / NOVEMBER 2006 / 1183

9 mobility, such as cohesion for the case of clay particles. This approach could also be generalized for bed materials with grainsize distributions having more than two modes. It should also be mentioned that the approach does not consider variation of the makeup of the surface components during bed-load motion. A more comprehensive analysis would require further bed-load data covering a wide range of Shields stresses. Moreover, additional future work would be necessary to further improve the collapse of the bed-load data by accounting for hiding and exposure effects of sediment existing in each unimodal component. Notation Fig. 11. Comparison between measured and calculated bed-load transport rates for Goodwin Creek; only bed-load data points with 40 Nm 2 are used in this comparison, as recommended by Kuhnle 1992 although the measured bed-load data exhibit some scatter, the scaled approach yields better estimations. The MASE value calculated for the nonscaled approach is 450, while for the scaled it is considerably reduced, to a value of 11. The much higher MASE for the nonscaled approach resulted from the sufficiently higher rs 1.79 from This interference of sand, though comprising only 25% from the surface, yielded the largely overestimated calculations of bed-load transport rates shown in Fig. 11. It is also seen that the scaled approach, although it has a better performance, tends to overestimate, to some extent, the calculation of bed-load transport rates for the upper range of shear stresses. This deficiency likely results from excluding the variation of the makeup of the surface material during the calculation of bed load. Conclusion The interference between sand and gravel components in bimodal bed materials complicates the calculation of bed-load transport rates. For this case, an approach to separate the bed-load transport calculations into two independent unimodal fractions of sand and gravel is proposed. The approach relies on scaling the reference Shields stresses of the sand and gravel modes to match the value of the mode of unimodal sediment, assumed to be equal to This approach allows the use of a suitable relation developed for unimodal sediment to calculate the bed-load transport rate separately for each of the two bimodal components. The validity of the approach was examined using available bed-load data obtained by Kuhnle 1992, 1993b from laboratory and field studies. Though the initial results are favorable, additional testing would be necessary to examine the approach further. The results obtained in this study indicate that, for bimodal materials, if one component is dominant, then its behavior during bed-load motion resembles that of unimodal, while the minor component becomes highly influenced, exhibiting the highest degree of hiding or exposure effects for the cases of sand and gravel, respectively. When both bimodal components are represented with similar percentages, mutual interference is evident. Though a sand-gravel combination was assumed throughout this work, the proposed approach should be valid for any type of bimodal material unless some other factors influence bed-load The following symbols are used in this paper: D g mode grain size of gravel; D i mean grain size of ith grain size range; D s mode grain size of sand; d flow depth; f g fraction of gravel in original bed material; f i fraction of material in ith grain size range; f s fraction of sand in original bed material; g gravitational acceleration; p i fraction of bed load in ith grain size range; q B volumetric bed-load transport rate per unit channel width; q Bi volumetric bed load per unit width in ith grain size range; q g volumetric bed-load transport rate of gravel; q s volumetric bed-load transport rate of sand; R s submerged specific gravity of sediment; S energy slope; W dimensionless bed-load parameter; W transformed dimensionless bed-load transport; W g dimensionless bed-load transport of gravel; W g transformed dimensionless bed-load transport of gravel; W i dimensionless bed-load transport of ith grain size range; reference dimensionless bed-load transport; W r W s dimensionless bed-load transport of sand; W s transformed dimensionless bed-load transport of sand; bed shear stress; g Shields stress of gravel mode; g i rg rs s s scaled Shields stress of gravel mode; Shields stress of ith grain size range; reference Shields stress of gravel mode; reference Shields stress of sand mode; Shields stress of sand mode; and scaled Shields stress of sand mode. References Almedeij, J., and Diplas, P Bedload transport in gravel-bed streams with unimodal sediment. J. Hydraul. Eng., , Bagnold, R. A An empirical correlation of bed load transport rates in flumes and natural rivers. Proc. R. Soc. London, Ser. A, 372, Diplas, P Bedload transport in gravel-bed streams. J. Hydraul. Eng., 113 3, / JOURNAL OF HYDRAULIC ENGINEERING ASCE / NOVEMBER 2006

10 Ferguson, R. I., Prestegaard, K. L., and Ashworth, P. J Influence of sand on hydraulics and gravel transport in a braided gravel bed river. Water Resour., 25 4, Iseya, F., and Ikeda, H Pulsations in bedload transport rates induced by a longitudinal sediment sorting: A flume study using sand and gravel mixtures. Geogr. Ann., 69, Klingeman, P. C., and Emmett, W. W Gravel bedload transport processes. Gravel-bed rivers, R. D. Hey, J. C. Bathurst, and C. R. Thorne, eds., Wiley, New York, Kondolf, G. M Salmonid spawning gravels: A geomorphic perspective on their size distribution, modification by spawning fish, and criteria for gravel. Ph.D. thesis, Johns Hopkins Univ., Baltimore. Kuhnle, R. A Fractional transport rates of bed load on Goodwin Creek. Dynamics of gravel-bed rivers, P. Billi, R. D. Hey, C. R. Thorne, and P. Tacconi, eds., Wiley, Chichester, U.K., Kuhnle, R. A. 1993a. Fluvial transport of sand and gravel mixtures with bimodal size distributions. Sediment. Geol., 85, Kuhnle, R. A. 1993b. Incipient motion of sand-gravel sediment mixtures. J. Hydraul. Eng., , Paintal, A. S Concept of critical shear stress in loose boundary open channels. J. Hydraul. Res., 9 1, Parker, G., Klingeman, P. C., and McLean, D. G Bedload and size distribution in paved gravel-bed streams. J. Hydr. Div., 108 4, Powell, D. M Patterns and processes of sediment sorting in gravel-bed rivers. Prog. Phys. Geogr., 22 1, Rakoczi, L Influence of grain-size composition on the incipient motion and self-pavement of bed materials. Proc., 16th Congress International Association for Hydraulic Research, Delft, The Netherlands, 2, Sambrook Smith, G. H., Nicholas, A. P., and Ferguson, R. I Measuring and defining bimodal sediments: Problems and implications. Water Resour. Res., 33 5, Wilcock, P. R Bed-load transport of mixed-size sediment. Dynamics of gravel-bed rivers, P. Billi, R. D. Hey, C. R. Thorne, and P. Tacconi, eds., Wiley, Chichester, U.K., Wilcock, P. R Critical shear stress of natural sediments. J. Hydraul. Eng., 119 4, Wilcock, P. R Two-fraction model of initial sediment motion in gravel-bed rivers. Science, 280, Wilcock, P. R Toward a practical method for estimating sediment-transport rates in gravel-bed rivers. Earth Surf. Processes Landforms, 26, Wilcock, P. R., and Southard, J. B Experimental study of incipient motion in mixed-size sediment. Water Resour. Res., 24, JOURNAL OF HYDRAULIC ENGINEERING ASCE / NOVEMBER 2006 / 1185