Grain-size distributions of bed load: Inferences from flume experiments using heterogeneous sediment beds

Sedimentary Geology 223 (2010) 1–14

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Sedimentary Geology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / s e d g e o

Grain-size distributions of bed load: Inferences from flume experiments using heterogeneous sediment beds K. Ghoshal b, B.S. Mazumder a,⁎, Barendra Purkait c a b c

Fluvial Mechanics Laboratory, Physics and Earth Sciences Division, Indian Statistical Institute, Kolkata-700 108, India Department of Mathematics, Indian Institute of Technology, Kharagpur-721 302, India Environmental Geology Division, E.R, Geological Survey of India, Kolkata-700091, India

a r t i c l e

i n f o

Article history: Received 27 June 2008 Received in revised form 10 August 2009 Accepted 9 September 2009 Keywords: Flume experiments Bedload Active layer Critical shear stress Sand/gravel mixtures Bed roughness Statistical distributions

a b s t r a c t The grain-size fractions in the bedload transported over the five heterogeneous sediment beds of different values of bed roughness were computed from the flume experiments. The existence of an entrapment factor associated with the sorting observed from the bed to active layer was modeled based on the modified critical shear stress to estimate the grain-size fractions in the transport layer under given hydraulic conditions. The efficiency of these models was tested with the observed data. Subsequently, the patterns of observed grainsize distributions in the transport layer were tested to identify the distributions developed in the active layer due to sorting using three probability density functions (pdf), such as, log-normal, log-hyperbolic and logskew-Laplace. Tests indicated that a log-skew-Laplace distribution fitted best for 49%, a log-hyperbolic for 31%, and a log-normal for 20% out of forty-five bedload samples collected under unidirectional flow with changes in flow discharge and bed roughness. The results of this study would be useful to specify the grainsize distributions in the bedload formed under different hydrodynamic conditions in various sedimentary environments. © 2009 Published by Elsevier B.V.

1. Introduction Transport of non-cohesive sediments such as silt, sand or gravel under hydraulic conditions has received much attention from geologists and engineers over the years with the objective to study the size sorting process and to interpret the grain-size distributions found in sedimentary deposits (Niekerk et al., 1992). During transportation, the movement of non-cohesive sediment is commonly classified into two categories: bedload and suspended load. In channels, these two types of transportation have the largest influence on grain-size distributions. Bedload is transported in three ways: by sliding, rolling and saltation. A grain at the bed surface will begin to move when the boundary shear stress due to water current just exceeds the critical shear stress. The physics underlying the bedload transport of sediment mixtures has received attention only very recently, though the mixed character of sediment is a significant characteristic of present-day river sediments, as well as of other recent and ancient sedimentary deposits. The main objective of the present contribution is twofold: first, to determine the active layer (near-bed) concentration of each size fraction that is transported as bedload over sand/sand–gravel mixture beds, incorporating the modified critical shear stress due to sediment heterogeneity; and second, to determine the probability density function (pdf) which fits the patterns of grain-size distributions at the ⁎ Corresponding author. Tel.: +91 033 25753033; fax: +91 033 25773026. E-mail address: [email protected] (B.S. Mazumder). 0037-0738/$ – see front matter © 2009 Published by Elsevier B.V. doi:10.1016/j.sedgeo.2009.09.008

active layer (exchange of sediment between bed sub-surface and transport layer) formed from the variations of bed roughness (due to addition of coarse materials to the reference sand-size distribution) and flow discharge. It hitherto remains unclear whether the patterns of size distributions in the active layer during transportation depend on the bed roughness produced by the sand/sand–gravel mixture, or it merely reflects part of the bed's size distribution. The recognition of pattern might provide a clue for understanding the transport process in a particular environment where the sedimentary particles were sorted. Laboratory experiments were performed with sedimentary beds of different values of bed roughness under different flow conditions. The question was whether any special grain-size distribution develops in the active layer as an effect of entrapment of finer grains in the interstices of gravel-sized particles, and/or as a result of variations in discharge. 1.1. Previous work Flume experiments indicated that the grain-size distributions in both the active layer (which was defined here as the current-affected top part of the bed surface) and the suspension under unidirectional current conditions were related to the current velocity, the characteristics of the bed materials, and the suspension height (Sengupta, 1979; Ghosh et al., 1981; Sengupta et al., 1991, 1999; Mazumder, 1994; Mazumder et al., 2005a,b). These studies showed that size sorting process took place immediately above the surface layer

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(referred to bed surface) leading to a specific size distribution in the active layer, and that the grain sizes of the active layer effected the size distribution in suspension. Several investigations were also carried out to compute the active layer concentration of each individual size fraction of sediments using the incipient motion of a sediment particle as determined from the mixture of non-uniform sizes (Gessler, 1967; Smith and McLean, 1977; Ghosh et al., 1981; Van Rijn, 1984; Garde and Rangaraju, 1985; Wilcock and Southard, 1988; McLean, 1992; Wilcock, 1993; Zyserman and Fredsoe, 1994; Patel and Rangaraju, 1999 and others). Ghoshal and Mazumder (2005) calculated the active layer concentrations at the 2D50 level (where D50 is that sieve size of which 50% of the mixture by weight is finer, and D is the grain diameter in mm), modifying the Smith and McLean (1977)’s active layer model. However, the bed materials of all studies mentioned above consisted only of sand sizes. A field study by Kuhnle (1992) identified a stream with a bimodal distribution (sand/gravel mixture) in which the sand fraction was entrained at lower velocities than the gravel fraction. Later, Kuhnle (1993) performed a series of experiments to determine the critical shear stress of each size fraction from five different sediment beds built of sand, gravel and sand–gravel mixture. The experiments showed that in 100% of both the sand and the gravel beds, all grain sizes of sand and gravel beds began to move at nearly the identical shear stress (mean shear stress for sand=0.236 N m− 2; for gravel=3.560 N m− 2). It was observed that most sand sizes showed nearly equal entrainment mobility in both laboratory and field studies (Parker et al., 1982; Wilcock and Southard, 1988; Church et al., 1991). For the beds composed of sand/gravel mixture, however, all sand sizes showed essentially a constant relationship between the critical shear stress and the grain sizes, whereas for the gravel fraction the critical shear stress increased with increase in size. Lanzoni and Tubino (1999) noticed in an experimental flume that the different mobilities of the various grain-size fractions in the mixture not only modified the sediment-transport capacity, but also induced a longitudinal and a transverse pattern in sorting. On transport of mixed size sediments, the effect of changing gravel and sand contents on overall transport rate was studied in the laboratory flume (Wilcock et al., 2001; Wilcock and Crowe, 2003; Curran and Wilcock, 2005). An increase in sediment supply could increase the mobility of gravel fractions in stream bed, which could lead to bed degradation and preferential evacuation of this sediment from the river bed (Curran, 2007). Mazumder et al. (2005b) investigated the effect of bed roughness on suspension above the sediment beds consisting of heterogeneous mixtures (sand/sand–gravel) in an experimental channel. However, they did not study the effects of bed roughness and flow discharge on sorting and on the pattern of grain-size distribution in the active layer. The pattern of size distribution might provide a clue for understanding the transport process in a particular environment. Several investigators studied the grain-size distributions in different sedimentary deposits. It was recognized that granulometry seemed to follow a log-normal distribution (Krumbein, 1938; Blench, 1952; Kennedy and Koh, 1961). However, most of the sediments did actually not follow so. Several researchers (Bagnold and BarndorffNielsen, 1980; Ghosh and Mazumder, 1981; Barndorff-Nielsen et al., 1982; Christiansen et al., 1984; Wyrwoll and Smyth, 1985; Sengupta et al., 1991) suggested that grain-size distributions be represented as log-hyperbolic rather than log-normal; on the other hand, Fieller et al. (1984) and Fieller and Flenley (1992) suggested a simplified version, known as the log-skew-Laplace distribution. Moreover, the lognormal and the log-skew-Laplace distributions were the most extreme cases of a hyperbolic family (Christiansen and Hartmann, 1991). Kothyari (1995) used the method of power transformation to study the grain-size distribution of a river bed. Sengupta et al. (1991), Purkait and Mazumder (2000) and Purkait (2002) observed that, after a considerable transport distance of sediments in the Usri River (Jharkhand, Eastern India), the conditions of a log-normal distribution were satisfied. Purkait (2006) further observed that, irrespective of

bed-form size, the bedload sediments of the Usri River were also lognormally distributed. 2. Experimental techniques and methodologies 2.1. Experimental Channel Experiments were conducted in a re-circulating ‘close-circuit’ flume (Mazumder et al., 2005a; Ghoshal, 2005) specially designed at the Fluvial Mechanics Laboratory (FML) of the Indian Statistical Institute in Kolkata, India. Both the experimental and the re-circulating channels of the flume have the identical dimensions (10 m long×50 cm wide×50 cm high). To get a clear view of the movements of the sedimentary particles, the walls of the flume were made of perspex windows over a distance of 8 m. long. Two non-clogging types of centrifugal pumps and valves were arranged in such a way that the flow could be set at any desired speed up to 1.30 m/s for a water depth of up to 35 cm. The inlet and outlet pipes were freely suspended from an overhead structure to allow tilting of the flume. The upstream end of the channel was divided into three sub-channels of equal dimensions, and a honeycomb cage placed at each end of the subchannels in order to ensure a smooth, vortex-free, uniform flow of water through the experimental channel. 2.2. Experiments A series of experiments was conducted over five different sediment beds (Bed Nos. 10C1–10C5) with known grain-size distributions; these beds consisted of heterogeneous mixtures of particles ranging from 0.032 to 8 mm (Mazumder et al., 2005b), with different values of bed roughness. The roughness values of the various sediment beds were computed for the experiments based on the measure of D65 (the sieve size for which 65% of the mixture by weight is finer). All five sediment beds' size distributions had the identical modal value with a peak at 2.0ϕ (where ϕ = −log2D; D is particle diameter in mm). The roughness of the different sediment beds was gradually increased with respect to the roughness of the reference grain-size distribution (unimodal, No. 10C1) by adding ever coarser particles successively to the beds 10C1–10C4. Beds 10C1–10C3 consisted of 100% sand (range 0.03–2 mm), whereas the other two beds (Nos. 10C4 and 10C5) consisted of sand–gravel mixture (range 0.03–8 mm) with 14% gravel and 86% sand, and with 25% gravel and 75% sand, respectively. The mean specific gravity of the sedimentary particles used for the experiments was 2.65. Cumulativepercentage plots of the grain-size distributions of all five sediment beds used in the experiments are shown in Fig. 1, and their characteristics are summarized in Table 1. All experiments, each with a specific mixture of grain sizes spreading over the flume base, were started with initially smooth and plane sediment bed of 3–4 cm thick. The water depth was kept at a constant height of 35 cm above the flume base for each of the experiments. Unidirectional flows over the sediment beds were kept at a specific, desired velocity. Bedforms were developed when sediment transport started. During fluid flows over sediment beds in recirculating flumes, different types of bedforms were seemed to be generated among which asymmetric, symmetric, nearly flat bed and a series of asymmetric waves were observed. The dimensions of bedforms, such as amplitudes and wavelengths, depended on the flow velocity and the sand bed materials. With the development of bedforms, sediment jets were ejected from the ripple crests and dispersed into the main flow. This process seemed to reach a steady state in suspension over a period of time and at each maximum velocity, the ‘fullydeveloped’ bedforms (equilibrium category) seemed to be evolved from an approximately steady uniform flow. 2.3. Velocity measurements and bedload sampling Water velocity was measured at the mid-section of the channel by inserting a propeller (2.5 cm in diameter) of an Ott Laboratory

K. Ghoshal et al. / Sedimentary Geology 223 (2010) 1–14

Fig. 1. Size distributions of five sediment beds in weight (kg), percentage (%) and cumulativepercentage plots.

current meter to the desired sampling heights above the beds. In order to assure the establishment of a fully-developed flow, the measuring section was chosen at a distance of 7.5 m downstream from the upstream honeycomb cage. The mean values of at least five velocity measurements, made over a period of 3–4 min at the same height above the bed, were recorded at the matured bedforms. The streamwise mean velocities (u) measured over all five sediment beds under controlled conditions are plotted against a dimensionless vertical coordinate ξ (=y/d) in Fig. 2(a–c) for three different maximum velocities (umax = 68, 101 and 116 cm/s). The controlled velocity data were so close that it followed a single line to fit usual log-law, even for different values of bed roughness. The velocity data had been fitted to the log-wake law by the least square method, and the calculated velocities were consistent with the observed data for more than 85% of the flow depth (Mazumder et al., 2005b). The friction velocities

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(u⁎ = 4.94, 6.45 and 7.35 cm/s) were determined by means of the fitted log-wake law for all three maximum velocities. All together fifteen experiments were conducted in three runs or discharges (Q = 0.087, 0.148 and 0.175 m3/s) for each sediment bed. A rack and pinion arrangement of siphon tubes of 10 mm in diameter mounted on a trolley was set up on the flume for the sampling of bed load and suspended particles at a distance of 7.5 m from the source. After the sufficient time at each maximum velocity or discharge, the bedforms were considered as having equilibrium when it hardly changed anymore in size and shape, or when it formed a nearly flat bed during transportation. The bedforms had wave lengths varying from 80 to 100 cm at a maximum velocity of 68–101 cm/s above the beds. When the velocity was about 116 cm/s, the bedforms were nearly a flat. The suspension was considered steady when repeated sampling of the suspended sediments from different heights showed only very small changes in the proportion of grain sizes. When the bedforms and water surface slope seemed to be steady and equilibrium with the materials in suspension, three repeated samples per run with a volume of approx. 0.2 dm3 were collected through siphoning from the top of the newly formed sedimentary active layer (that actively exchanges sediment between bed sub-surface and the transport layer). The samples were collected at three locations in the transverse section of the channel, viz. in the middle of the section (M. S.) and at approximately 5 cm of both the inner wall (I. W.) and the outer wall (O. W.) of the flume. An attempt was made to collect the samples from the stoss side of the migrating bedforms with a mild slope at the upstream face, and gradually it went to the trough regions. In this situation, the locations of samples were not considered on the bedforms that created during the runs. It was obvious that the bedloads from the beds (Nos. 10C1–10C3), consisting of particles ranging from 0.032 to 2 mm, were easily siphoned, whereas for the case of beds (Nos. 10C4–10C5) ranging from 0.032 to 8 mm, the siphoning was somewhat hampered due to the presence of gravel of sizes >2 to 8 mm in the mixtures. The coarser grains at the bed surface resisted the erosion/bedload transport due to their weight. The smaller the grain with low settling velocity, the easier it was to move as bedload. Since the present study dealt with the hindrance effects on the sorting due to the presence of gravel in the mixed beds, the identical method was used for sampling the active layer from all the five sediment beds. It was noticed that bedload-transported materials consisted mainly of sand particles, whereas the movements of gravel particles were minimal, which were ascribed to their weight. Thus, resistance to sorting was observed. Though the velocity at the near-wall regions was less than that at the mid-section of the channel due to the side-wall frictional effects and the secondary circulations, bedload samples from the near-wall region were collected and presented for comparative study. The correction factor for the side-wall effects was not considered while studying the theoretical models. However, the frictional effect and the secondary circulation in the theoretical models due to side-walls will be considered in a subsequent paper. Each sample was oven-dried and sieved at ½ ϕ intervals by means of a Ro-Tap sieve shaker for 15 min. The amount of each sieve fraction was weighed by an electronic balance. The mean values of three repeated active layer samples collected at all three locations (M. S., I. W. and O. W.) of the flume above five different sediment beds were analyzed and plotted (next section).

Table 1 Characteristics of all five sediment beds used for the experiments. Sediment beds

Bed No. 10C1

Bed No. 10C2

Bed No. 10C3

Bed No. 10C4

Bed No. 10C5

Mean grain size Bed roughness (D65) SD, sorting (σ) Skewness (Sk1) Kurtosis (KG) Bed conc. (Cb) (dimless) Distributions

1.703(ϕ) (0.307 mm) 1.250(ϕ), 0.42 mm 0.660(ϕ) (0.632 mm) − 0.171 3.883 0.570 Log-hyperbolic

1.462(ϕ) (0.363 mm) 0.950(ϕ), 0.51 mm 0.780(ϕ) (0.58 mm) − 0.109 2.706 0.580 Log-hyperbolic

1.10(ϕ) (0.462 mm) 0.550(ϕ), 0.68 mm 1.090(ϕ) (0.47 mm) − 0.449 2.325 0.620 Log-hyperbolic

0.706(ϕ) (0.613 mm) 0.030(ϕ), 0.98 mm 1.410(ϕ) (0.38 mm) − 0.655 2.107 0.650 Log-hyperbolic

0.287(ϕ) (0.82 mm) − 0.610(ϕ), 1.53 mm 1.740(ϕ) (0.30 mm) − 0.773 1.962 0.700 Log-hyperbolic

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3.1. Gessler (1967)'s model In order to deal in a practical and manageable way with bedload transport, Gessler (1967) assumed the followings: 1) the turbulent fluctuations of the bed shear stress are statistically distributed according to the normal-error law, and 2) a grain starts to move when the effective bottom shear stress τ0 exceeds a critical value (τc) which is a function of the grain size and the grain Reynolds number. On the basis of these assumptions the probability p of a grain being eroded from the bed under given hydraulic conditions is given by p = 1−prob½τ0 < τc  τc ð1Þ −1  1 2 2 expð−t = 2σ Þdt = 1− pffiffiffiffiffiffi τ0 σ 2π −∞ τ0 − τ0 follows a Gaussian distribution with mean zero and since  τ

∫

0

variance σ 2, and τ̅0 is the mean bottom shear stress. The value of σ (=0.57) was estimated by Gessler from his experimental data. The relative active layer concentration Cbli of each size fraction i was obtained using the bed concentration Cbi and the probability (pi) being eroded from the bed as: Cbli =

Cbi pi ∑Cbi pi

ð2Þ

where Cbi = wi/w with wi being the weight for ith size class and w = ∑wi in the total weight of the sediment in the bed. 3.2. Van Rijn (1984)'s model Van Rijn (1984) proposed a formula for estimation of the active layer (near-bed) concentration Cbli of ith grain-size fraction at 2D level as: Cbli = 0:18Cbi

Si D⁎

ð3Þ

where Cbi is the maximum value of bed concentration (Table 1) for firmly packed grains, Si = θ0/θci − 1 is the normalized excess shear stress of ith grain size, θ0 and θci are the dimensionless shear stress and critical shear stress respectively and D⁎ is the non-dimensional particle diameter, defined as  D⁎ = Dm

Fig. 2. Vertical velocity distribution above all five sediment beds for three different maximum velocities (umax = 68, 101, 116 cm/s).

gA ν2 f

1 = 3

ð4Þ

where Dm is the median grain size of sediment, A is the relative density of sediment, νf is the kinematic viscosity of water and g is the acceleration due to gravity. 3.3. Zyserman and Fredsoe (1994)'s model

The observed grain-size fractions in the active layer above the five sediment beds are provided in Tables 2(a) and 2(b) for umax = 116 cm/s (Q = 0.175 m3/s). Details of velocity, slopes, temperature and sediment data collected at different discharges (Q = 0.087, 0.148 and 0.175 m3/s) for all five sediment beds (Nos. 10C1–10C5) can be found in Mazumder et al. (2001).

Zyserman and Fredsoe (1994) suggested a formula for determination of the active layer (near-bed) concentration Cbli of ith grain size at 2D level, which is independent of bed materials, as:

3. Estimation of active layer concentrations

where θ0 and θci are the dimensionless shear stress and critical shear stress respectively related to skin friction. In their studies, the value of θci equal to 0.045 was adopted for simplicity, as there was no significant influence of critical value of the Shields parameter on the bed concentration used. In fact, the constant value 0.045 as used by Zyserman and Fredsoe (1994) is the asymptotic value of the

An attempt was made to study the different models to estimate size fractions of the active layer concentration (near-bed), when the bed materials with different roughnesses and flow parameters were given; but none of them were sufficient to describe it properly.

Cbli =

0:331ðθ0 −θci Þ1:75 1:75 1 + 0:331 0:46 ðθ0 −θci Þ

ð5Þ

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Table 2a Observed active layer (near-bed) concentration (Cbl) above the three sediment beds (Nos. 10C1–10C3) for umax = 116 cm/s (Q = 0.175 m3/s). Size (phi)

− 0.1 − 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Size (mm)

2.00 1.40 1.00 0.701 0.495 0.351 0.246 0.175 0.124 0.088 0.061 0.043 0.032

Bed 10C1

Bed 10C2

Bed 10C3

I. W. Cbl × 10− 3

O. W. Cbl × 10− 3

I. W. Cbl × 10− 3

O. W. Cbl × 10− 3

M. S. Cbl × 10− 3

I. W. Cbl × 10− 3

O. W. Cbl × 10− 3

M. S. Cbl × 10− 3

– – 0.01139 0.02007 0.17111 2.72289 5.74338 2.78271 0.47802 0.02712 0.04015 0.01293 0.00888

– – 0.5347 0.1417 0.6263 3.2353 3.3740 1.2367 0.1880 0.0183 0.0203 0.0075 0.0040

– – 0.7306 1.1427 2.8666 4.7253 3.5960 1.1170 0.1350 0.0208 0.0740 0.0035 0.0035

– – 1.8560 2.0360 3.1920 3.3370 1.6970 0.4600 0.1880 0.0678 0.0750 0.0037 0.0026

– – 0.0000 0.0000 0.0133 0.0630 0.1700 0.1680 0.0700 0.0100 0.0190 0.0120 0.0160

– 0.2300 0.2300 0.2830 0.6950 1.7060 3.1540 1.5760 0.2142 0.0150 0.0018 0.0068 0.0041

– 0.1600 0.1800 0.2550 1.0100 1.9300 3.1860 0.9200 0.1470 0.0168 0.0326 0.0152 0.0160

– – 0.5800 0.5900 1.2000 2.3300 2.4000 0.7900 0.1300 0.0120 0.0130 0.0054 0.0046

dimensionless critical shear stress for relatively coarse, unisize sediment in hydraulically rough bed. Here the value of θci will be used for heterogeneous sediment mixtures. 3.4. McLean (1992)'s model According to McLean (1992), the active layer concentration Cbli of each size fraction at the active layer level is given by Cbli =

γ0 Cbi Si 1 + γ0 Si

ð6Þ

where Si = θ0/θci − 1 is the normalized excess shear stress for each grain size and Cbi is the maximum value of bed concentration (Table 1) for each size fraction. The value of constant γ0 was taken to be 0.004 by McLean (1992) for a limited set of data. 4. Verification of theoretical models In most of sediment-transport studies, the critical shear stress for each grain size is found from the Shields' curve, which is the relation between the dimensionless critical shear stress and the grain Reynolds number. In fact, the Shields' curve was determined from the bed of uniform sand grain size. However, in order to compute the Gessler (1967)’s model in the present study the probability of a grain being eroded from a sand/sand–gravel mixture bed was determined using the values of critical shear stress (τc) for different grain sizes. The values of critical shear stress were determined from Kuhnle (1993), who obtained the incipient motion following the technique of Parker et al. (1982) for ith grain-size fraction in sand, gravel and sand–gravel mixtures, which were rather different from those

obtained from the Shields curve. According to Kuhnle (1993) the transport function depends on grain-size and it is strongly non-linear in nature. For the sand–gravel mixtures, the value of modified critical shear stress (τc) for each size fraction is higher than that observed for 100% sand beds and lower than that found for 100% gravel beds. Modified critical shear stress for each grain-size is expected to provide better results in the present case, because some sort of ‘hiding factor’ affects the finer particles significantly during sorting. As such, a portion of the sand was entrapped into the interstices of the coarse particles and thus removed from the surface (Parker and Klingeman, 1982). The value of σ (=0.57) estimated by Gessler from his experimental data was used for the present computation, although the actual value of σ might be different. Fig. 3 shows the plots of measured and computed relative active layer concentrations for three locations (M. S., I. W. and O. W.) for all five sediment beds of different values of roughness and three different maximum velocities (umax = 68, 101 and 116 cm/s.). For sand beds (Nos. 10C1–10C3), the computations were made for grain-size fractions up to the size 2.5 phi and for sand–gravel mixtures (Nos. 10C4–10C5) they were made up to 4.0 phi according to the available values of modified critical shear stress. The Gessler's method predicted better for sand beds rather than the sand–gravel mixture beds, if the modified critical shear stress was used. Computations using Gessler's equation provided the relative active layer concentration and it reflected the bed distribution only, which was a drawback of this method. Models of Van Rijn (1984) and Zyserman and Fredsoe (1994) given by the Eqs. (3) and (5) respectively were computed to estimate the active layer (near-bed) concentration Cbli of ith grain-size fraction in the sediment mixtures at 2D level, using the modified dimensionless critical shear stress (θci) for different grain sizes. Computations of active layer concentration Cbli using these two models showed highly

Table 2b Observed active layer (near-bed) concentration (Cbl) above the two sediment beds (Nos. 10C4 and 10C5) for umax = 116 cm/s (Q = 0.175 m3/s). Size (phi)

− 0.1 − 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Size (mm)

2.00 1.40 1.00 0.701 0.495 0.351 0.246 0.175 0.124 0.088 0.061 0.043 0.032

Bed 10C4

Bed 10C5

I. W. Cbl × 10− 3

O. W. Cbl × 10− 3

M. S. Cbl × 10− 3

I. W. Cbl × 10− 3

O. W. Cbl × 10− 3

M. S. Cbl × 10− 3

– – 0.0000 0.0000 0.0200 0.0120 0.0640 0.1120 0.0530 0.0020 0.0100 0.0010 0.0120

– – 0.0000 0.0000 0.0002 0.0300 0.0310 0.0250 0.0390 0.0350 0.0220 0.0100 0.0002

– – 0.0000 0.0100 0.0600 0.0200 0.0500 0.0580 0.0460 0.0390 0.0520 0.0270 0.0370

0.0030 0.3000 0.0070 0.0200 0.0430 0.1100 0.1200 0.1115 0.0600 0.0400 0.0030 0.0000 0.0100

0.0000 0.0660 0.0250 0.0300 0.0900 0.2000 0.2700 0.0800 0.0230 0.0120 0.0040 0.0000 0.0000

– – 0.1000 0.0050 0.0000 0.0800 0.1900 0.0520 0.0460 0.0000 0.0000 0.0000 0.0000

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Fig. 3. Relative active layer concentration above all five sediment beds for three different runs: continuous line stands for Gessler (1967), solid triangle for inner wall (I. W.); dotted line for mid-section (M. S.) and solid circle for outer wall (O. W.) of the channel.

overestimation and are provided in Table 3 for Van Rijn (1984) and in Table 4 for Zyserman and Fredsoe (1994). Finally, it became possible to estimate the active layer concentration modifying Eq. (6). In this study Eq. (6) was used to estimate the active layer concentrations of all size fractions in the bed load samples for given flow discharges over all five different sediment beds. It was more reasonable to use the modified critical shear stress because the

movement of each grain-size was different when it occurrred in a heterogeneous sand/sandy-gravel mixture beds than in a sand bed of equal grain-size. While computing the bed load, the value of the parameter γ0 of Eq. (6) was adjusted for each size class to obtain the best-fit of the desired active layer concentrations with the actual observed values. The value of the parameter γ0 was determined as 0.004 by McLean (1992) for a limited set of data contrary to the

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Table 3 Computed active layer concentration (Cbl) using Van Rijn's model above the five sediment beds (Nos. 10C1–10C5) for umax = 116 cm/s (Q = 0.175 m3/s). Grain size (φ)

Grain size (mm)

Bed 10C1 Active layer conc. (Cbl) computed by Eq. (3)

Bed 10C2 Active layer conc. (Cbl) computed by Eq. (3)

Bed 10C3 Active layer conc. (Cbl) computed by Eq. (3)

Bed 10C4 Active layer conc. (Cbl) computed by Eq. (3)

Bed 10C5 Active layer conc. (Cbl) computed by Eq. (3)

− 3.0 − 2.5 − 2.0 − 1.5 − 1.0 − 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

8.000 5.657 4.000 2.828 2.000 1.400 1.000 0.701 0.495 0.351 0.246 0.175 0.124 0.088 0.061

– –

– – – – – – 0.03069 0.03896 0.05549 0.07762 0.09167 0.03701 – – –

– – – – 0.02928 0.02928 0.02975 0.03459 0.04958 0.06912 0.08170 0.03453 – – –

– – 0.01255 0.01255 0.01848 0.02176 0.02176 0.02562 0.03674 0.04955 0.05139 0.01995 0.00373 0.00032 0.00038

0.00560 0.00642 0.00800 0.00957 0.01321 0.01588 0.01626 0.01891 0.02719 0.03789 0.04473 0.01884 0.00337 0.00029 0.00034

– – – 0.01062 0.01747 0.04132 0.09533 0.11274 0.04781 – – –

present experimental data, where parameter γ0 was a function of grain size. The fitted values of log γ0 are plotted against the grain size (ϕ) for all values of bed roughness in Fig. 4(a, b) for the data collected near the inner and outer walls (I. W. and O. W.) of the flume and in Fig. 4(c, d) for the mid-section (M. S.) of the flume during three different runs (nos. 1, 2 and 3). Though the values of log γ0 were scattered, an empirical relationship between γ0 and ϕ was established in Fig. 4(a, b); and hence γ0 was estimated for a given grain-size. Two relationships between the parameter γ0 and the size (ϕ) had been established by simple regression for the side-wall boundaries: one for bed roughness (D65) ranging from 0.042 cm to 0.068 cm (beds 10C1–10C3) and the other for D65 ranging from 0.099 cm to 1.53 cm (beds 10C4 and 10C5). The relationships between γ0 and ϕ near the side-walls (inner and outer walls) are log γ0 = 0:3626ϕ−3:9867

ð7Þ

for beds 10C1–10C3 and log γ0 = 0:42ϕ−4:5

ð8Þ

for beds 10C4 and 10C5. Similarly, empirical relations for the mid-section (M. S.) of the channel (see Fig. 4(c, d)) are established by multiple regression, where the parameter γ0 is a function of bed roughness D65 (in ϕ) and grain size ϕ; and are given by log γ0 = − 4:07D65 + 0:4628ϕ−1:611

ð9Þ

for beds 10C1–10C3 and log γ0 = 0:131D65 + 0:446ϕ−4:23

ð10Þ

for beds 10C4 and 10C5. From the above relations (Eqs. (7)–(10)) the value of γ0 was estimated for each grain-size at fixed u⁎ or umax above the five sediment beds. The computed active layer concentration of each size fraction using the modified McLean's formula (Eq. (6)) was plotted together with the observed data in Fig. 5(a) for both side-walls (inner and outer) and in Fig. 5(b) for the mid-section of the flume at maximum velocity (umax = 116 cm/s, Run 3). When a particular umax or run was considered (say, umax = 116 cm/s, Run 3), the amount of each size fraction of active layer samples per unit volume (as mentioned) increased with an increase of bed roughness (D65) and subsequently decreased with further increase of bed roughness (Fig. 5). A diminishing concentration at the active layer was most conspicuous, when the bed roughness was increased >0.68 mm. Increasing the proportion of gravel of the sandy-gravel mixture induced a higher bed roughness and led to the entrapment of fine sand particles to the interstices of the gravel and removed from the bed surface. A higher shear stress was needed to entrain a comparable amount of sediment in the active layer when the bed roughness was increased by adding gravel. The comparison of observed and computed active layer concentrations showed that though the model overestimated or underestimated the amount; the observed and calculated modes matched well. The estimated results for the sand

Table 4 Computed active layer concentration (Cbl) using Zyserman and ’Fredsoe's model above the five sediment beds (Nos. 10C1–10C5) for umax = 116 cm/s (Q = 0.175 m3/s). Grain size (φ)

Grain size (mm)

Bed 10C1 Active layer conc. (Cbl) computed by Eq. (5)

Bed 10C2 Active layer conc. (Cbl) computed by Eq. (5)

Bed 10C3 Active layer conc. (Cbl) computed by Eq. (5)

Bed 10C4 Active layer conc. (Cbl) computed by Eq. (5)

Bed 10C5 Active layer conc. (Cbl) computed by Eq. (5)

− 3.0 − 2.5 − 2.0 − 1.5 − 1.0 − 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

8.000 5.657 4.000 2.828 2.000 1.400 1.000 0.701 0.495 0.351 0.246 0.175 0.124 0.088 0.061

– –

– – – – – – 0.03651 0.06540 0.11176 0.17373 0.24702 0.31357 – – –

– – – – 0.01033 0.02080 0.03833 0.06923 0.11567 0.17714 0.24943 0.31498 – – –

– – 0.00187 0.00453 0.01099 0.02218 0.04000 0.07114 0.11761 0.17872 0.25020 0.31528 0.36812 0.40481 0.42929

0.00002 0.00051 0.00207 0.00520 0.01149 0.02277 0.04072 0.07195 0.11843 0.17957 0.25120 0.31603 0.36850 0.40499 0.42936

– – – 0.03498 0.06715 0.11354 0.17525 0.24806 0.31413 – – –

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Fig. 4. (a, b). Variation of γ0 against grain size (ϕ) near side-walls (inner and outer) of the channel: (a) for beds 10C1–10C3, and (b) for beds 10C4 and 10C5. (c, d). Variation of γ0 against grain size (ϕ) at the mid-section of the channel: (c) for beds 10C1–10C3, and (d) for beds 10C4 and 10C5.

beds (Nos. 10C1–10C3) were closer to the observed values than those for the sand–gravel mixtures (beds 10C4 and 10C5) (Fig. 5(a, b)). 5. Patterns of grain-size distributions in the transport layer The patterns of the grain-size distributions in the active layer (nearbed) had been established for various discharges (Q = 0.087, 0.148 and 0.175 m3/s) over all five different sediment beds. The identification of size distribution provided a clue for understanding the transport process in a particular sedimentary environment. The question was whether any special grain-size distribution developed in the active layer as an effect of entrapment of finer grains in the interstices of gravel-sized particles, and/or as a result of variations in discharge. Among the family of size-frequency distributions, the density function of a normal (Gaussian) distribution is given by: 1 1 ðϕ−μÞ2 gðϕ; μ; σÞ = pffiffiffiffiffiffi exp − 2 σ2 σ 2π

ð11Þ

pffiffiffiffiffiffiffiffiffiffiffi γ1 γ2 pffiffiffiffiffiffiffiffiffiffiffi δðγ1 + γ2 ÞK1 ðδ γ1 γ2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 × exp − ðγ1 + γ2 Þ × δ2 + ðϕ−μ1 Þ2 2 1 + ðγ1 −γ2 Þðφ−μ1 Þ 2

½

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Ln pðϕ; γ1 ; γ2 ; μ 1 ; δÞ = ν− ðγ1 + γ2 Þ δ2 + ðϕ−μ1 Þ2 2 1 + ðγ1 −γ2 Þðφ−μ 1 Þ 2

ð13Þ

where

!

where the grain size (ϕ) is the random variable, μ is the mean of the distribution, and σ > 0 is the standard deviation. In a normal distribution, the mean, median and mode values coincide, the skewness is 0, and the kurtosis is 3.0. Barndorff-Nielsen (1977) first introduced the hyperbolic distribution to describe the mass-size distribution of aeolian sand. A plot of a probability density function of the grain-size distribution with a logarithmic scale on the ordinate axis is referred to as a hyperbola. Denoting the size variable by ϕ and the probability density function p(ϕ), the four-parameter equation of the hyperbolic curve is: pðϕ; γ1; γ2; μ1 ; δÞ =

where γ1 and γ2 represent the slopes of two linear asymptotes for the graph of Ln p(ϕ; γ1, γ1, μ1, δ), μ1 is the abscissa of the intersecting point of two asymptotes, K1(.) is the modified Bessel function of the third kind, and pthe scale parameter δ (>0) can be expressed as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi δ = 2ðμ1 ⁎ − μ1 Þ ðγ1 γ2 Þ = ðγ1 −γ2 Þ, with μ1⁎ being the observed mode of the distribution. The slopes γ1 and γ2 correspond to the fine-grain and the coarse-grain coefficient, respectively. The scale parameter δ represents the difference in ordinates of the point of intersection of the asymptotes and the maximum of the parabola. When the variable ϕ follows a log-hyperbolic distribution, Eq. (12) can be written as:



ð12Þ

ν = Ln

pffiffiffiffiffiffiffiffiffiffiffi γ1 γ2 pffiffiffiffiffiffiffiffiffiffiffi δðγ1 + γ2 ÞK1 ðδ γ1 γ2 Þ

ð14Þ

is the norming constant, which is adjusted to agree with the observed frequency. For strongly skewed distributions, μ1⁎ may differ considerably from μ1. Clearly, μ1⁎=μ1 if and only if γ1 =γ2. In this case, the hyperbolic distribution is symmetric around μ1⁎. Moreover, the normal distribution is the limiting form of the hyperbolic family if the slope of the left asymptote (γ1) equals the slope of ffi the2 right asymptote (γ2) of the pffiffiffiffiffiffiffiffiffiffiffiffiffiffi hyperbola, and δ→∝, while δ ðγ1 γ2 Þ→σ (σ is the standard deviation, δ is a scale parameter that is a measure of the spread in sorting). Fieller et al. (1984) introduced the ‘log-skew-Laplace’ distribution, which is essentially described by two straight lines instead of a hyperbola. The log-skew-Laplace distribution of particle size is regarded as the limiting form of the hyperbolic family obtained by letting the scale parameter δ→0, and can be written as: 1

expfðϕ−μ2 Þ = αg for ϕ≤μ2

ð15aÞ

−1

expfðϕ−μ2 Þ = βg for ϕ≥μ2

ð15bÞ

gðϕ; α; β; μ2 Þ = ðα + βÞ

= ðα + βÞ

K. Ghoshal et al. / Sedimentary Geology 223 (2010) 1–14

9

Fig. 5. (a). Active layer concentration near the side-walls of the channel as computed by modified McLean (1992) above all five sediment beds for Run 3 (umax = 116 cm/s). (b). Active layer concentration at the mid-section (M. S.) of the channel as computed by modified McLean (1992) above all five sediment beds for Run 3 (umax = 116 cm/s) (in the first figure the measured data is not available).

where ϕ indicates the observed size variable, and α, β, μ2 are distribution parameters that are defined, respectively, as the reciprocals of the arc tangents of the acute angle made by the two lines with the horizontal axis and the abscissa of their point of intersection. Clearly, α is the slope of the left asymptote (i.e. for the coarser fractions), whereas β is the slope of the right asymptote (i.e. for the finer fractions). The values of α and β are estimated from Eqs. (15a) and (15b).

6. Verification of statistical distributions The weight frequency of the various grain-size fractions of the five sediment beds (Nos. 10C1–10C5) before the experiments has been plotted in the probability scale in Fig. 6 (thick continuous lines in the first column). All five sediment beds show a three-segment shape on probability paper. Only in bed 10C1 that the classical three-segment

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K. Ghoshal et al. / Sedimentary Geology 223 (2010) 1–14

shape is not very clear. All the three models mentioned above are fitted for all five sediment beds, and it is found that the grain-size distributions of all five beds follow a log-hyperbolic form (Eq. (13)). The estimated parameters and relative errors of the three distribution models fitted for the experimental beds are shown in Tables 5a and 5b respectively. The formula for calculating the relative error (Eq. (16)) is discussed underneath.

The grain-size distributions of transport layer samples collected across the width of the channel at three different locations (M. S., I. W. and O. W.) for three values of maximum velocity (umax)—ranging from 68 to 116 cm/s (Runs 1, 2 and 3)—have been plotted in Fig. 6 for all five sediment beds (Nos. 10C1–10C5). Columns in Fig. 6 represent the variation of the distributions with umax (Runs 1, 2 and 3), whereas the rows represent the variation with sediment beds (Nos. 10C1–

Fig. 6. Probability plots of the observed bed loads' grain-size distributions above the five sediment beds for all runs: continuous thick line in the first column represents the size distributions of five sediment beds (10C1–10C5) and other symbols are same as Fig. 3.

K. Ghoshal et al. / Sedimentary Geology 223 (2010) 1–14

11

Table 5a Estimated parameters of the three hypothesized distributions for sediment beds. Bed nos.

Log-normal

Log-hyperbolic

μ(ϕ)

σ(ϕ)

γ1

γ2

μ1(ϕ)

δ(ϕ)

α

Log-skew-Laplace β

μ2(ϕ)

10C1 10C2 10C3 10C4 10C5

1.467 1.193 0.807 0.423 0.027

0.626 0.799 1.131 1.458 1.776

2.14 0.908 0.497 0.315 0.208

3.242 3.234 3.232 3.229 3.248

1.761 1.819 1.913 1.978 2.057

0.746 0.315 0.285 0.258 0.244

0.493 0.497 1.117 1.51 1.928

0.512 0.863 0.574 0.624 0.668

1.664 1.097 1.641 1.591 1.547

Table 5b Computed relative error (E) for the three hypothesized distributions fitted for sediment beds. Bed no

Log-normal

Log-hyperbolic

Log-skew-Laplace

10C1 10C2 10C3 10C4 10C5

0.480 0.394 0.537 0.675 0.889

0.351⁎ 0.328⁎ 0.341⁎ 0.342⁎ 0.331⁎

0.364 0.817 0.466 0.516 0.560

⁎ Indicates lowest error value.

10C5) or bed roughness (D65). It is distinctly observed from the figures that the size distributions in the bedload for the sand beds (Nos. 10C1 and 10C2, first two rows) at the mid-section (M. S.) and the side-walls (I. W. and O. W.) differ from those of the sandy-gravel mixtures (Nos. 10C4 and 10C5, fourth and fifth rows), particularly for the velocity range of 68–101 cm/s (Runs 1 and 2). At the mid-section of the channel, the size of the bedload tends to be relatively fine on active layer of the sandy beds (Nos. 10C1 and 10C2), and more irregular on the top layer of the sandy/gravel mixtures (Nos. 10C4 and 10C5). This relatively irregular grain-size distribution of the bedload on the mixed beds indicates at some kind of ‘critical bed roughness’ together with sediment entrapment on the beds if the bed roughness is increased; this results in a relatively scattered pattern of the plots. It is also interesting to note that the grain-size distributions in the bedload samples collected for bed 10C3 (third row) across the flume width (M. S., I. W. and O. W.) are almost the identical for all three discharges (Q = 0.087, 0.148 and 0.175 m3/s). A coarse bed surface is formed over the beds (Nos. 10C3–10C5), trapping the finer grains in the interstices of the coarse particles, which increases the proportion of coarse grains exposed to the surface and diminishes the mobility of finer particles. These characteristics suggest that relatively fine particles occur for the sandy beds (Nos. 10C1, 10C2) at the midsection of the channel for almost all discharges, and that the resulting increase of coarser materials in the sand beds leads to diminish the mobility of the finer particles. A total of forty-five samples of bedload collected across the width of the channel for the five sediment beds and three different discharges (Q = 0.087, 0.148 and 0.175 m3/s) are fitted using all three statistical models. The estimated parameters are plotted against bed roughness (D65) in Fig. 7 for a normal distribution; in Fig. 8 for a log-hyperbolic distribution; and in Fig. 9 for a log-skew-Laplace distribution. To obtain a quantitative measure of the discrepancies, the weighted relative errors between the calculated and the observed values have been determined for these three distributions using the formula (Ghosh et al., 1986; Mazumder, 1994; Purkait, 2002): Relative Error ðEÞ =

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ∑½ðCC −CO Þ = CO T

ð16Þ

where CC is the calculated value (i.e., the weight percent of the calculated amount of bedload per grain-size fraction) as obtained from the different distribution models, CO is the observed value (i.e., the weight percent of the measured amount of bedload per sieve fraction),

Fig. 7. Two estimated parameters of log-normal distribution of all runs above five sediment beds: (a, c) for side-walls (inner and outer) and (b, d) for mid-section of the channel.

and T is the total observed value (T equals to 100, as the frequencies are calculated in percentages). The relative errors between the observed and calculated values against bed roughness (D65) for three runs (maximum velocities) and three distributions are plotted in Fig. 10(a–c) for the bedload collected near the inner and outer walls, and in Fig. 10 (d–f) for the mid-section samples of the channel. The effects of the velocity on the grain-size distribution patterns of all bedload samples are presented in Table 6 for three maximum velocities (umax) applied: low (68 cm/s), medium (101 cm/s) and high (116 cm/s). For each umax, fifteen bedload samples were collected. It was found that (1) at a low velocity, the log-normal distribution fitted best in six cases, (2) at a medium velocity, the log-skew-Laplace distribution fitted best in ten cases, and (3) at a high velocity, log-skewLaplace distribution fitted very well for seven cases. It was therefore concluded that, at medium and higher velocity, the log-skew-Laplace distribution was the best-fit distribution model among the three models, irrespective of the bed roughness. With reference to the medium velocity it should be noted that, when the velocity was decreased to the low range, the log-normal distribution yielded the best-fit, and that, when the velocity was increased to the high range, it might be true that the log-skew-Laplace distribution fitted well, but that the bedload particles had a tendency to attain a log-hyperbolic distribution. Particularly near the inner and outer walls (I. W. and O. W.) of the channel the log-skew-Laplace distribution fitted well at medium and high velocities, whereas no particular distribution was found for low velocities. At the mid-section of the channel, the distribution followed the log-skew-Laplace with velocity. An observation by Sengupta (1979) was interesting in this context. He observed that, at a particular water depth, a hyperbolic distribution yielded lognormal with increasing velocity in the case of suspended load, whereas the hyperbolically distributed bed attained a log-skew-Laplace distribution in the present study where a particular velocity range and water depth were maintained. Table 7 shows the effect of bed roughness on the grain-size distribution of the bedload. A critical examination of this effect indicated that, out of five experiments, a log-skew-Laplace distribution gave the best-fit for three experiments, and that one log-normal and one log-hyperbolic distribution did so. The log-skew-Laplace distribution, that fitted best among the three distribution models, did

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K. Ghoshal et al. / Sedimentary Geology 223 (2010) 1–14

Fig. 9. Three estimated parameters of log-skew-Laplace distribution: (a, c, e) for sidewalls and (b, d, f) for mid-section of the channel.

Fig. 8. Four estimated parameters of log-hyperbolic distribution: (a, c, e, g) for sidewalls and (b, d, f, h) for mid-section of the channel.

so irrespective of a change in bed roughness. It must therefore be deduced that the bed roughness did neither have a vital nor a consistent role in modifying the grain-size distribution pattern, provided that the bed roughness was of the same order as those used in our experiments. A comparative study of the three distributions out of 45 cases indicated that a log-skew-Laplace distribution fitted best for 22 cases (49%), whereas a log-hyperbolic distribution for 14 cases (31%), and a log-normal distribution for 9 cases (20%). 7. Conclusions The purpose of this study was to ascertain the relative effect of bed roughness and discharge of water flows on the active layer concentration of each size fraction brought into motion for the five sediment beds of different bed roughness; and to identify the effective grain-size distribution pattern of resulting bed load sediments. Existing well-known equations (Gessler, 1967; Van Rijn, 1984; McLean, 1992; Zyserman and Fredsoe, 1994) for estimating the active layer concentration of each size fraction above the sediment beds had

been tested with the observed data, but none of them was capable of doing so for heterogeneous sediment mixtures (sand/sand–gravel). Gessler (1967)’s equation for estimating the near-bed concentration, utilizing the concept of entrapment of fine particles into the interstices of coarse particles, was unable to predict the concentration of each size fraction, whereas the results were qualitatively reasonably well comparable (for mixed sand/sand–gravel beds) with the computation using the modified critical shear stress. The formulae proposed by Van Rijn (1984) and Zyserman and Fredsoe (1994) overestimated substantially the active layer concentration of each size fraction. The modification of McLean (1992)’s equation was proposed to estimate the active layer concentration above the five sediment beds. The proposed modifications essentially concerned with an adaptation of the functional form of γ0 in McLean (1992)’s formula, and introduced the non-linear effect of the transport stage function (S) for entrapment. New empirical relationships between the parameter (γ0), the grain size (ϕ) and the bed roughness (D65) were suggested for active layer concentration, based on the samples collected near the side-walls (inner and outer) and mid-section of the channel. The suggested relationships between γ0 and the modified critical shear stress were used for computation of active layer (near-bed) concentration of each size fraction. The accuracy of the above formulae was tested with the observations. The deviations from the actual values obtained by the Gessler (1967) and McLean (1992) formulae were generally of the same order. The modified McLean's active layer formula seems to be more realistic because it utilizes the parameter γ0, as a function of grain-size and bed roughness.

K. Ghoshal et al. / Sedimentary Geology 223 (2010) 1–14

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Table 7 Effect of bed roughness on bed loads’ grain-size distribution patterns above all five sediment beds for all runs. Bed characteristics

Distribution pattern out of 9 samples in each bed i.e., total of 45 samples Distribution

Bed No. 10C1, Runs 1, 2, 3 Bed roughness (D65) = 1.25ϕ (0.42 mm)

Log-normal Log-hyperbolic Log-skewLaplace Bed No. 10C2, Runs 1, 2, 3 Log-normal Bed roughness (D65) = 0.95ϕ (0.51 mm) Log-hyperbolic Log-skewLaplace Bed No. 10C3, Runs 1, 2, 3 Log-normal Bed roughness (D65) = 0.55ϕ (0.68 mm) Log-hyperbolic Log-skewLaplace Bed No. 10C4, Run 1, 2, 3 Log-normal Bed roughness (D65) = 0.03ϕ (0.98 mm) Log-hyperbolic Log-skewLaplace Bed No. 10C5, Runs 1, 2, 3 Log-normal Bed roughness (D65) = −0.61ϕ (1.53 mm) Log-hyperbolic Log-skewLaplace

Fig. 10. Computation of relative errors for all three different distributions against bed roughness (D65 in mm): (a, b, c) for side-walls (inner and outer) for all runs and (d, e, f) for mid-section of the channel for different distributions.

The grain-size distributions of forty-five bedload samples collected across the width of the channel over five sediment beds were compared with the outcomes of three statistical distributions. Increase of the flow velocity led to a change in the grain-size distribution at the sedimentary surface. At medium velocity (101 cm/s), the log-skew-Laplace distribution appears to give the best-fit for bedload sediments among the three distributions. When the flow velocity was reduced to 68 cm/s, the grain size showed a log-normal distribution, whereas at an increased velocity (116 cm/s) the bedload sediments approached to reflect the identical distribution as the beds (Mazumder et al., 2005b) although the log-skewLaplace distribution was the best-fit one. It could therefore be deduced Table 6 Effect of velocity on bed loads' grain-size distribution patterns above all five sediment beds. Run no

Maximum velocity (umax)

Distribution pattern out of 15 samples in each run i.e., total of 45 samples

1

68 cm/s

2

101 cm/s

3

116 cm/s

Distribution Log-normal Log-hyperbolic Log-skew-Laplace Log-normal Log-hyperbolic Log-skew-Laplace Log-normal Log-hyperbolic Log-skew Laplace

No. of times wins 6 4 5 1 4 10 2 6 7

No. of times wins 2 2 5 1 4 4 0 4 5 4 2 3 2 2 5

that, at both an intermediate and a relatively high velocity, the log-skewLaplace was the best-fit irrespective of the bed roughness. The effect of the bed roughness on the bedload grain-size distribution indicated that, out of five experiments with different values of bed roughness, a log-skewLaplace distribution showed the best-fit for three experiments, whereas other two distributions each showed one best-fit. The effect of the bed roughness had no consistent role in the modification of the grain-size distribution, provided that the bed roughness was of same order as used during the experiments. A comparative study indicated that a log-skewLaplace distribution fitted best for 49% of all cases, a log-hyperbolic for 31%, and a log-normal for 20%, under unidirectional flow with changes in flow velocity and bed roughness. The results of the present study may help to distinguish between the various sedimentary environments under different hydrodynamic conditions of deposition and bed roughness. The pattern of size distribution might provide a clue for understanding the transport process. The motivation was to determine if the transport process could be deduced from a sedimentary record where grain-size distribution sorting was observed. Acknowledgements The authors would like to express their sincere thanks to Prof. J. K. Ghosh, Prof. S. R. McLean, Prof. P. L. de Boer, Prof. A. J. Van Loon, Dr. Rajat Mazumder and two anonymous reviewers for their constructive comments and suggestions for improvement of the paper. Appendix A. Notations A Cb, Cbl, Co, Cc D D⁎ Dm D65 d g P

relative density of sediments bed and active layer concentrations observed and calculated concentrations grain diameter (mm) dimensionless grain diameter median grain diameter bed roughness grain size for which 65% of the mixture is finer depth of water acceleration due to gravity probability of a grain being eroded

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Q flow discharge S normalized excess shear stress of each grain size T total observed concentration t dummy index u stream-wise mean velocity u⁎, umax shear and maximum velocities w total weight of sediment in the bed α, β, μ2 three parameters of log-skew-Laplace distribution ξ(=y/d) dimensionless vertical height ϕ grain size (ϕ = −log2(D)) [D in mm] γ1, γ2, μ1, δ four parameters of log-hyperbolic distribution γ0 a parameter related to grain size (D) ν normalizing constant to log-hyperbolic distribution μ, σ mean and standard deviation of normal distribution νf kinematic viscosity of fluid θ dimensionless critical shear stress τ0 bottom shear stress τ̅0 mean bottom shear stress τc critical shear stress Index i represents ith size fraction. References Bagnold, R.A., Barndorff-Nielsen, O., 1980. The pattern of natural size distributions. Sedimentology 27, 199–207. Barndorff-Nielsen, O., 1977. Exponentially decreasing distributions for the logarithm of particle size. Proceedings of Royal Society of London 353 (A), 401–419. Barndorff-Nielsen, O., Dalsgaard, K., Halgreen, C., Kuhlman, M., Mller, J.T., Shou, G., 1982. Variations in particle size distribution over a small dune. Sedimentology 29, 53–65. Blench, T., 1952. Normal size distributions found in samples of river bed sand. Civil Engineering (USA) 22, 147. Christiansen, C., Hartmann, D., 1991. The hyperbolic distribution. In: Syvitski, J.P.M. (Ed.), Principles, Methods and Application of Particle Size Analysis. Cambridge University Press, pp. 237–248. Christiansen, C., Blaesild, F., Dalsgaard, K., 1984. Re-interpreting “segmented” grain size curves. Geological Magazine 121, 47–51. Church, M., Wolcott, J.F., Fletcher, W.K., 1991. A test of equal mobility in fluvial sediment transport: behavior of sand fraction. Water Resources Research 27, 2941–2951. Curran, J.C., 2007. The decrease in shear stress and increase in transport rates subsequent to an increase in sand supply to a gravel-bed channel. Sedimentary Geology 202, 572–580. Curran, J.C., Wilcock, P.R., 2005. Effect of sand supply on transport rates in a gravel-bed channel. Journal of Hydraulic Engineering, ASCE 131 (11), 961–967. Fieller, N.R.J., Flenley, E.C., 1992. Statistics of particle size data. Journal of Applied Statistics 41, 127–146. Fieller, N.R.J., Gilbertson, D.D., Olbricht, W., 1984. A new method for environmental analysis of particle size distribution data from shoreline sediments. Nature 311, 648–651. Garde, R.J., Rangaraju, K.G., 1985. Mechanics of Sediment Transportation and Alluvial Stream Problem. Wiley Eastern Limited, New Delhi. Gessler, J., 1967. The beginning of bed load movement of mixtures investigated as natural armouring in channels. Rep. No. 69. Laboratory of hydraulic research and soil mechanics of the Swiss Federal Institute of Technology. Ghosh, J.K., Mazumder, B.S., 1981. Size distribution of suspended particles — unimodality, symmetry,and log-normality. In: Taillie, C., et al. (Ed.), Statistical Distribution in Scientific Work, vol. 6. D. Reidel Publishing Comp, The Netherlands, pp. 21–32. Ghosh, J.K., Mazumder, B.S., Sengupta, S., 1981. Methods of computation of suspended load from bed materials and flow parameters. Sedimentology 28, 781–791. Ghosh, J.K., Mazumder, B.S., Saha, M.R., Sengupta, S., 1986. Deposition of sand by suspended currents: Experimental and theoretical studies. Journal of Sedimentary Petrology 56, 57–66. Ghoshal, K., 2005. On velocity and suspension concentration in a sediment-laden flow. Experimental and theoretical studies: Ph.D. Thesis, Jadavpur University, Calcutta.

Ghoshal, K., Mazumder, B.S., 2005. Sediment-induced stratification in turbulent openchannel flow. Envirometrics 16 (7), 673–686. Kennedy, J.F., Koh, R.C.Y., 1961. The relation between the frequency distributions of sieve diameters and fall velocities of sediment particles. Journal of Geophysical Research 66, 4233–4246. Kothyari, U.C., 1995. Frequency distribution of river materials. Sedimentology 42, 283–291. Krumbein, W.C., 1938. Size frequency distributions of sediments and the normal phi curve. Journal of Sedimentary Petrology 8, 84–90. Kuhnle, R.A., 1992. Frictional transport rates of bed load on Goodwin Creek. In: Bill, P., Hey, R.D., Throne, C.R., Tacconi, P. (Eds.), Dynamics of Gravel Bed Rivers. John Wiley and Sons Ltd., Chichester, UK, pp. 141–155. Kuhnle, R.A., 1993. Incipient motion of sand–gravel sediment mixtures. Journal of Hydraulic Engineering (ASCE) 119 (12), 1400–1415. Lanzoni, S., Tubino, M., 1999. Grain sorting and bar instability. Journal of Fluid Mechanics 393, 149–174. Mazumder, B.S., 1994. Grain-size distribution in suspension from bed materials. Sedimentology 41, 271–277. Mazumder, B.S., Ghoshal, K., Dalal, D.C., 2001. Influence of bed roughness on sediment suspension: ISI Flume Laboratory Data. Technical Report No. PESD/Flume/2/2001, pp. 1–65. Mazumder, B.S., Ray, R.N., Dalal, D.C., 2005a. Size distributions of suspended particles in open channel flow over sediment beds. Environmetrics 16 (2), 149–165. Mazumder, B.S., Ghoshal, K., Dalal, D.C., 2005b. Influence of bed roughness on sediment suspension: experimental and theoretical studies. Journal of Hydraulic Research, IAHR 43 (3), 245–257. McLean, S.R., 1992. On the calculation of suspended load for non-cohesive sediments. Journal of Geophysical Research 97 (C4), 5759–5770. Niekerk, A.V., Vogel, K.R., Slingerland, R.L., Bridge, J.S., 1992. Routing of heterogeneous sediments over movable bed model development. Journal of Hydraulic Engineering, ASCE 118, 246–262. Parker, G., Klingeman, P.C., 1982. On why gravel bed streams are paved. Water Resources Research 18, 1409–1423. Parker, G., Klingeman, P.C., McLean, D.G., 1982. Bedload and size distribution in paved gravel-bed streams. Journal of Hydraulic Division, ASCE 108 (4), 544–571. Patel, P.L., Rangaraju, K.G., 1999. Critical tractive stress of non-uniform sediment. Journal of Hydraulic Research, IAHR 37 (1), 39–58. Purkait, B., 2002. Patterns of grain size distribution in some point bars of the Usri River, India. Journal of Sedimentary Research 72, 367–375. Purkait, B., 2006. Grain-size distribution patterns of a point bar system in the Usri River, India. Earth Surface Processes and Landforms 31, 682–702. Purkait, B., Mazumder, B.S., 2000. Grain size distribution — a probabilistic model for Usri river sediments in India. In: Wang, Zhao-Yin, Hu, Shi-Xiong (Eds.), Stochastic Hydraulics 2000. Balkema, A.A, pp. 291–297. Sengupta, S., 1979. Grain size distribution of suspended load in relation to bed materials and flow velocity. Sedimentology 26, 63–82. Sengupta, S., Ghosh, J.K., Mazumder, B.S., 1991. Experimental — theoretical approach to interpretation of grain size frequency distributions. In: Syvitski, J.P.M. (Ed.), Principles, Methods and Applications of Particle Size Analyses. Cambridge University Press, Cambridge, UK, pp. 264–279. Sengupta, S., Das, S.S., Maji, A.K., 1999. Sediment transportation and sorting processes in streams. Proceedings of Indian National Science Academy 65A (2), 167–206. Smith, J.D., McLean, S.R., 1977. Spatially averaged flow over a wavy surface. Journal of Geophisical Research 82 (12), 1735–1746. Van Rijn, L.C., 1984. Sediment transport, part II: suspended load transport. Journal of Hydraulic Engineering, ASCE 110 (11), 1613–1641. Wilcock, P.R., 1993. Critical shear stress of natural sediments. Journal of Hydraulic Engineering ASCE 119 (4), 491–505. Wilcock, P.R., Crowe, J.C., 2003. Surface-based transport model for mixed-size sediment. Journal of Hydraulic Engineering, ASCE 129 (2), 120–128. Wilcock, P.R., Southard, J.B., 1988. Experimental study of incipient motion in mixed-size sediment. Water Resources Research 24, 1137–1151. Wilcock, P.R., Kenworthy, S.T., Crowe, J.C., 2001. Experimental study of the transport of mixed sand and gravel. Water Resources Research 37 (12), 3349–3358. Wyrwoll, K.H., Smyth, G.K., 1985. On using the log-hyperbolic distribution to describe the textural characteristics of eolian sediments. Journal of Sedimentary Petrology 55, 471–478. Zyserman, J.A., Fredsoe, J., 1994. Data analysis of bed concentration of suspended sediment. Journal of Hydraulic Engineering, ASCE 120 (9), 1021–1042.