Research Articles

Assessment of the fraction of bed load concentration towards the sediment transport of a monsoon-dominated river basin of Eastern India

  • KAR Rohan ,
  • SARKAR Arindam , *
  • School of Infrastructure, Indian Institute of Technology Bhubaneswar, Argul, Khordha, Odisha 752050, India
*Arindam Sarkar, PhD, E-mail:

Received date: 2022-06-13

  Accepted date: 2023-01-03

  Online published: 2023-05-11

Supported by

Ministry of Water Resources, Government of India(28/1/2016-R&D/228-245)


Given the challenges of re-creating complex bed load (BL) transport processes in rivers, models are preferred over gathering and examining field data. The highlight of the present research is to develop an approach to determine the ungauged bed load concentration (BLCu) utilizing the measured suspended sediment concentration (SSC) and hydraulic variables of the last four decades for the Mahanadi River Basin. This technique employs shear stress and SSC equations for turbulent open channel flow. Besides, the predicted BLCu is correlated with SSC using a power relation to estimate BLCu on the river and tributaries. Eventually, different BL functions (BLF) efficiency is assessed across stations. The model predicted BLCu is comparable with the published data for sandy rivers and falls within ± 20%. Outliers in hydraulic and sedimentological statistics significantly influence estimating the BL fraction apart from higher relative ratios and catchment geology. The constants of power functions are physically linked to sediment transport configuration, mechanism, and inflow to the stream. The stream power-based BLF best predicts the BL transport, followed by shear stress and unit discharge approaches. The disparity in the estimation of BLCu results from station-specific physical factors, sampling data dispersion, and associated uncertainties.

Cite this article

KAR Rohan , SARKAR Arindam . Assessment of the fraction of bed load concentration towards the sediment transport of a monsoon-dominated river basin of Eastern India[J]. Journal of Geographical Sciences, 2023 , 33(5) : 1023 -1054 . DOI: 10.1007/s11442-023-2118-6

1 Introduction

Numerous studies have enhanced our understanding of the trend of riverine suspended sediments transport across the world, but our knowledge of bed sediment transport in such rivers is quite restricted (Wang et al., 2007; Azamathulla et al., 2013; Armijos et al., 2017; Montanher et al., 2018; Kar and Sarkar, 2021; Luo et al., 2022). Quantitatively estimating alluvial bed transport processes is crucial in fluvial engineering, managing water resources, and accurately predicting flood damage (Gordon et al., 2004; Sinnakaudan et al., 2006; MacArthur et al., 2008; Joshi and Xu, 2017; An et al., 2021). The total sediment load (TL) carried by a river is composed of the bed load (BL), suspended load (SL) and wash load (sediment size < 0.062 mm) (Vanoni, 2006; Knighton, 2014). However, the particles of the wash load were considered independent of the river discharge (Bettes, 2008). Also, few researchers have found that sediments of sizes between 0.063 to 1.2 mm were complex to classify as BL or SL. They regularly alternate between rolling, sliding, and suspension phases (Gomez, 1991; Church, 2006; Nittrouer et al., 2008). In general, the TL of a stream is, therefore, effectively defined as the sum of bed load and suspended load, which varies spatiotemporally and majorly assists in the morphological evolution of alluvial channels.
Bed load transport rate is widely used in various practical applications, including the design of channels, estimation of the life of reservoirs, and effective design of bridge scour control mechanisms. However, this data is often not readily accessible when it is desired. Although the Helley-Smith bed load sampler was considered efficient for alluvial rivers (Emmett, 1980; Lemma et al., 2019); however, Leopold and Emmett (1997) found no sampling tool has computed consistent bed load along the channel bed. Therefore, BL is usually expressed as bed load transport rates (BLT). The BLT is a function of depth, discharge and sediment particle size; accordingly, it varies for different rivers over a smaller time interval. Besides, BLT depends on basin geology, hydrology, anthropogenic activities and transport power of an alluvial channel. The observed BLT fluctuates with sampling duration for coarse-bed rivers (Bunte and Steven, 2005); however, similar findings for sand-bed rivers are difficult to present due to a lack of observed BL data. Due to cross-sectional and temporal fluctuation and other issues with field techniques and equipment, acquiring field measurements of bed load rates is challenging, tedious, and time-consuming. BL information for rivers worldwide is rarely found (Emmett, 1980; Erskine and Saynor, 2015; Frings and Vollmer, 2017). Also, in India, the measurements of BL are laborious and uneconomical due to various operational constraints compared to cost-effective acoustic or physical sampling methods of capturing SL.
Because of the lack of BL data for Indian alluvial rivers, precise assessment, validation, and segmentation of the total load transport rate of these rivers remain an issue. BLT models are developed for calibration and evaluation using computational techniques, theoretical formulation, and flume data sets. Based on measurable data, one can divide these methods into two categories. In the first category, formulations estimate sediment transport rate based on river geometry and bed composition (Einstein, 1950; Schoklitsch, 1962; Engelund and Hansen, 1967; Karim, 1998; Cheng, 2002; Recking, 2013). Due to their different theoretical basis, these models give different results. Several laboratory-based models do not effectively predict sediment transport in river channels or canals. In the second category, formulations require quantifiable load data (such as SL) to estimate bed load or to compute total sediment transport (Gray et al., 2010; Ashley et al., 2020). The second-category procedures only apply to rivers with sand beds and suspended sediments. There is no unique method for all river channels. Therefore, it is essential to thoroughly evaluate the theoretical background of a model and the availability of data before making a model selection.
Measuring bed load data over a mobile bed channel is extremely infrequent or challenging. In most cases worldwide, the bed load component was either neglected or taken as a certain proportion of SL or TL or calculated from the sediment rating curve (Maddock and Borland, 1950; Holeman, 1968; Milliman and Meade, 1983; Turowski et al., 2010; Ziegler et al., 2014; Ellison et al., 2016). This is crucial since Turowski et al. (2010) found BL flux might account for a significant portion of TL in suspension-dominated streams at low discharges. However, the proportion of the BL decreased with an increase in discharge, and also BLT was seen as more prominent in coarser sediment-bed rivers than sand-bed rivers (Topping et al., 2000; Yasi and Hamzepouri, 2008). In general, the BL was established to be present in the ratio of 1 to 75% of the SL (Gomez, 1991; Babinski, 2005; Church, 2006; Cantalice et al., 2013, 2015; Joshi and Xu, 2017; Lemma et al., 2019; Ashley et al., 2020). However, the low range of BL elucidates its absence from entering the coastal zone, which is most certainly one of the causes of coastal erosion (Cantalice et al., 2015).
Bed shear stress, critical velocity, stream power, and stochastic modelling are methods developed to estimate BLT from hydraulic and sediment data (Yang, 2003). However, most BLT formulae were developed from laboratory flume data under ideal conditions. This contrasts the varying hydraulic and sediment environments in natural streams. Consequently, the estimated BLT varies across equations and from observed data. Many previous studies on BL prediction found that stream power-based equations were efficient and comparable with observed BL data for sandy rivers (Molinas and Wu, 2010; Hassanzadeh et al., 2011; Cantalice et al., 2013; Lemma et al., 2019; Armijos et al., 2021). The efficacy of bed load formulae has been extensively established compared to observed data collected through flumes and natural streams (Yang and Huang, 2001; Bravo-Espinosa et al., 2003; Martin and Ham, 2005; Barry et al., 2008; Wu et al., 2008; Recking et al., 2012). The BLT equations developed by Meyer-Peter and Müller (1948), Schoklitsch (1962) and Parker et al. (1982) provide acceptable performance for transport limited sediment environment, while that of Bagnold (1980) and Schoklitsch (1962) testified the best performance for supply limited sediment environment. Secondly, for field measurements carried out for larger durations, the effectiveness of BLT equations was stated to be enhanced. However, field data quality plays a vital role while validating BLT formulae (Molinas and Wu, 2001; Frings and Vollmer, 2017).
Recent studies on BL estimation were mainly based on approaches like artificial neural network techniques, bayesian hierarchical modelling, gene expression programming, bedform velocimetry method, HEC-RAS modelling and regression analysis using observed hydraulic and sediment data (Sinnakaudan et al., 2006; Yasi and Hamzepouri, 2008; Holmes, 2010; Sasal et al., 2010; Zakaria et al., 2010; Kazemi et al., 2011; Cantalice et al., 2013; Ghani and Azamathulla, 2014; Asheghi and Hosseini, 2020; Ashley et al., 2020; Armijos et al., 2021). However, it was understood that such approaches require an extensive amount of in-situ BL measurements, which are otherwise rarely measured in India. Studies on BLT for Indian alluvial rivers are limited. Few have evaluated the BLT of different rivers using effective shear stress, total load and sediment rating curves (Singh et al., 2007; Yadav and Samtani, 2008; Waikhom and Yadav, 2017, 2018). While others have used computational models to study the velocity distribution or the SSC profile for sediment-loaded flow (Tsai and Tsai, 2000; Jung et al., 2004; Tsai et al., 2010). Pektas and Dogan (2015) proposed the incorporation of the SL in predicting the BL of a channel wherever possible to improve the model accuracy. However, the techniques that estimate the BL using observed load data exclusively apply to sandy rivers with SL.
To the author’s knowledge, currently, there are no previous studies conducted in the Mahanadi River Basin (MRB) that scientifically approximate the contribution of BL towards the total sediment yield of the basin. However, some recent studies have focused on modelling suspended sediment transport of the basin (Yadav and Satyanarayana, 2020; Yadav et al., 2021; Kar and Sarkar, 2022). At the same time, the declining trend of suspended sediment load across the basin and its related influences was reported (Bastia and Equeenuddin, 2016; Kar and Sarkar, 2021). In this regard, evaluating BL is considered significant to assess the effects of dams on sediment transport and basin yield. Dams in the MRB played a critical role in regulating sediment transport from upstream of the catchment to the coastal zones (Kar and Sarkar, 2021). The BLT can only be estimated if the hydraulic and sediment characteristics of a river channel are understood, although, there is not a single approach to BL estimation that works for every type of river channel.
Consequently, the theoretical basis of the model and the availability of data must be taken into account while picking an appropriate model for the prediction of BL. These variables are the longitudinal bed slope, flow velocity, bed composition, and water temperature. As a result, an attempt has been made in this study to devise a plausible approach for estimating the ungauged bed load concentration (BLCu) of the MRB when only the gauged suspended sediment concentration (SSCg) and related hydraulic variables for a hydrological station are accessible. Instead, this study will provide a scientific substitute for the constant percentages of bed load that were usually considered for the studies related to basin sediment transport. The justification for adopting this approach is that BL flux per unit width depends on unit water discharge, suspended sediment concentration, and particle sizes (Rubin and Topping, 2001). Also, including suspended sediment concentration in the model formulation improves the performance and contributes substantially to model accuracy (Pektas and Dogan, 2015). It is believed that identical causative conditions at the river reach scale drive both the bed and suspended sediments. In addition, for stations lacking gauged BL data, relevant bed load functions (BLF) were tested by evaluating their outcomes with the model-predicted BLCu of the river basin.
The initial goal of the present study is to build a model of BLCu for the major sub-basins of the MRB based on suspended sediment concentrations and other hydraulic parameters measured at the outlet hydrological stations of the sub-basins. The method uses vertical profiles of flow velocity and sediment concentration determined from the cross-sectional mean SSCg and average flow velocity to calculate the BLCu at a cross-section. The calculated BLCu as a fraction of total sediment concentration or SSCg of sub-basins is validated using previously published methods for sand-bedded streams, as the in-situ bed load observations are unavailable. This verification would instead justify the certainty of the computed bed load concentration for the channel section. The association between predicted BLCu and SSCg is assessed, and a functional relationship is proposed for all the major sub-basins involved. These proposed functional equations can eventually be used to quantify the ungauged bed load concentration for the only known suspended sediment concentration at a station. In addition, multiple established BLFs are assessed and compared with the predicted BLCu to determine the suitable BLF for the individual sub-basins of the MRB. The performance of these BLFs is statistically established for its use as an alternative to the earlier proposed equation. This approach will objectively quantify the fraction of BL transported through a river basin, previously ignored in basin sediment transport studies or taken as a set percentage of the measured suspended sediment or total load. In addition, the specific BLF that may be directly employed for any sub-basin to achieve an acceptable degree of performance in estimating BLCu will be summarized. The developed model will therefore act as an indirect tool for assessing bed load concentration for hydrological studies in the river basin, which is otherwise not directly gauged.

2 Study area

With a catchment area of 139,681.5 km2, the Mahanadi River Basin (MRB) is among the key tropical river basins of the Indian Peninsula (CWC, 2014), as shown in Figure 1. It travels over 851 kilometres before emptying into the Bay of Bengal. Longitude 80°28°E to 86°43°E and latitude 19°08°N to 23°32°N make up the catchment terrain. It rises about 442 metres above sea level from the Sihawa mountains in Chhattisgarh’s Dhamtari district. The basin spans many Indian states, with Chattisgarh accounting for 75,136 km2, Odisha for 65,580 km2, Bihar for 635 km2, and Maharashtra for 238 km2. Six main tributaries flow into the Mahanadi River (MR) in the upstream portion of the watershed. The four of these, namely Seonath, Hasdeo, Mand, and Ib, join on the left bank of the mainstream in terms of flow route, whereas the Pairi and Jonk connect on the right bank of the river. Additionally, the Tel and Ong tributaries join the main river downstream of the Hirakud reservoir (Figure 1). The MRB can thus be subdivided into 8 major sub-basins corresponding to the 8 tributaries mentioned. The last gauging station Tikarpara caters as the outlet of the entire MRB. The outlet gauging stations of the 8 sub-basins are mentioned in Figure 1.
Figure 1 A map of the Mahanadi River Basin showing hydrological stations
The basin receives an average of 1350 mm of precipitation per year, most of which occurs during the monsoon season when more than 80% of the yearly flow occurs. The runoff of the MR is widely heterogeneous throughout the year since it is essentially a rain-fed river. Furthermore, the average maximum temperature in the watershed is 39.56℃, with a mean minimum temperature of 20.01℃. The river maintains a constricted and shallow channel for eight months. It becomes enraged during the monsoon season, destroying its banks and flood plains. According to the CWC (2014), the average yearly discharge of the basin is 66.9 BCM (billion cubic metres). The river branches into numerous distributaries before discharging into the Bay of Bengal.
Furthermore, agricultural land makes up 54.3% of the basin’s size; the forest makes up 32%, and wastelands, water bodies, and built-up areas make up 14%. Fine-textured soil comprises around 42% of the watershed, while medium-textured soil covers 51.3% (CWC, 2014). The upstream geology of the watershed is occupied by the heavy rock of the Eastern Ghats, which dates back to Precambrian times. On the other hand, deltaic alluvium from the river prevails downstream of the watershed. Furthermore, compared to suspended sediments, bed sediments are often coarse-grained (Chakrapani and Subramanian, 1990). Yet, the basin produces more silt than all remaining national rivers (Mahalik, 2000).

3 Data sources

The Central Water Commission (CWC), Government of India, provided the long-term stage (d, in m), discharge (Q, in m3/s), suspended sediment concentration (SSCg, in g/l), and transverse bed-surface profiles for 9 hydro-observation stations for this study. The data covers 1973 to 2017, but the 9 hydrological stations were installed at different times across the basin by CWC. As a result, the length of the data interval varies for each station due to data availability (Table 1). Those mentioned hydrological stations correspond to each of the major sub-basins of the MR and one ultimate gauging station. Because monsoon rain (June to September) carries a more significant proportion of the basin’s annual sediment load than non-monsoon rain (October to May), sediment observations were done daily during the monsoons and irregularly throughout the non-monsoons. However, daily observations were still intermittent for some years, while no data were recorded for a few years at some stations of the MRB. Hence, the target was to analyze the existing recorded data from 1973-2017 for the mentioned gauging stations of the MRB (Table 1). As a result, the data extending from June to November, averaging on a monthly scale across available years, were considered for further analysis to have a majority of the data as continuous as possible. Also, the mean monthly temperature data across the MRB were collected from CWC.
Table 1 Summary of hydrological stations considered in the present study
Gauging stations Latitude (o) Longitude (o) Data interval Altitude (m.a.s.l) Drainage area (km2) Tributary/Sub-tributary
Baronda 20.91 81.89 1980-2017 283 3225 Pairi
Jondhra 21.73 82.35 1981-2017 219 29,645 Seonath
Rampur 21.65 82.52 1977-2017 219 2920 Jonk
Bamnidih 21.90 82.72 1973-2017 223 9730 Hasdeo
Kurubhata 21.99 83.20 1980-2017 215 4625 Mand
Sundargarh 22.12 84.01 1980-2017 214 5870 Ib
Salebhata 20.98 83.54 1973-2017 130 4650 Ong
Kantamal 20.65 83.73 1977-2017 118 19,600 Tel
Tikarpara 22.63 84.62 1973-2017 50 124,450 Mahanadi

Note: m.a.s.l = m above sea level

In addition, topographic data with a spatial resolution of 30 m in the form of Cartosat-1 digital elevation models (Carto DEM) was used in this study to calculate the average longitudinal bed slope of the channel (Sa), which was recorded by the Indian Space Research Organisation (ISRO). The primary soil information was gathered from the National Bureau of Soil Survey and Land Use Planning (NBSS and LUP, Nagpur, India) soil map (1999) on a scale of 1:500,000. The Water Technology Centre for Eastern Region, on the other hand, provided information on particle size for each dominating soil classification (Singh et al., 2009). Table 1 includes extensive information on watershed features and hydrological (observation/gauging) stations. In particular, Table 2 indicates the proportion of particle sizes in each dominating soil taxonomic class.
Table 2 Particulars of percentage classification of dominant soil classes across stations of the study
Hydrological stations Percentage sand, Psand (%) Percentage silt, Psilt (%) Percentage clay, Pclay (%)
Baronda 32.400 19.900 47.700
Jondhra 64.400 13.600 22.100
Rampur 64.400 13.600 22.100
Bamnidih 64.400 13.600 22.100
Kurubhata 23.300 36.700 40.000
Sundargarh 61.100 15.300 23.700
Salebhata 59.150 20.300 20.550
Kantamal 32.000 25.100 43.000
Tikarpara 56.400 15.300 28.400

4 Methodology

4.1 Estimation of hydraulic parameters

The daily average depth of flow at all sub-basin outflow stations was calculated by subtracting a station’s zero-gauge level (determined by CWC) from its daily mean water stage for the whole study period. Subsequently, the monthly average flow depth (ha, m) was calculated. The monthly average cross-sectional area (A, m2) and average flow width (FW, m) of a station were deduced using the transverse bed surface profiles and the monthly average flow depth. Further, the average monthly flow velocity (va, m/s) was computed as discharge divided by the daily cross-sectional area at any station. The representative sediment size at n per cent finer (dn, mm) was estimated as a weighted average of the standard sediment sizes of sand (0.075 to 4.75 mm), silt (0.002 to 0.075 mm), and clay (0 to 0.002 mm) provided by the Indian Standard Soil Classification System (ISSCS) at n%, adopting the following equation:
${{d}_{50}}=\left( {{P}_{sand}}\times {{S}_{avg}} \right)+\left( {{P}_{silt}}\times {{M}_{avg}} \right)+\left( {{P}_{clay}}\times {{C}_{avg}} \right)$
where Psand = percentage sand (%), Savg = weighted average sand size diameter at n%, Psilt = percentage silt (%), Mavg = weighted average silt size diameter at n%, Pclay = percentage clay (%), Cavg = weighted average clay size diameter at n%, and n = 50, 65, 84 and 90. The kinematic viscosity of water (ν) at different monthly mean temperatures was deduced using Julien (1998).

4.2 Formulation of physical equations for sediment-laden flow

The shear stress equation for a 2-D turbulent flow in an open channel is defined by applying Prandtl’s mixing length concept as follows:
$\tau =\mu \frac{\partial u}{\partial z}-\rho \overline{{u}'{w}'}={{\mu }_{sw}}\frac{\partial u}{\partial z}+{{\rho }_{sw}}l_{m}^{2}{{\left( \frac{\partial u}{\partial z} \right)}^{2}}$
where τ is the shear stress, μ and μsw are the dynamic viscosity of water and sediment-laden flow, respectively, ρ and ρsw are the density of water and sediment-laden flow, u is the average velocity of flow, z is the distance of the vertical axis with reference to the channel bed, $\overline{{{u}'}}$ and $\overline{{{w}'}}$ are the mean variation of velocity in horizontal and vertical directions respectively, lm is the mixing length. Figure 2 shows the schematic of the velocity and suspended sediment concentration.
Figure 2 Schematic of the distribution of the vertical velocity and suspended sediment concentration
The dynamic viscosity of sediment-laden flow was stated by Graf (1984) as follows:
${{\mu }_{sw}}=\mu \sum\limits_{a=0}^{\infty }{{{\left( k{{C}_{v}} \right)}^{a}}}$
where k is Einstein’s viscosity constant (≈ 2.5), Cv is the volumetric suspended sediment concentration at any depth from the channel bed, and a is considered up to 3 as recommended by Tsai et al. (2010). The density of sediment-laden flow is determined from the equation below:
${{\rho }_{sw}}=\rho \left( 1-{{C}_{v}} \right)+{{\rho }_{s}}{{C}_{v}}$
where ρs is the density of suspended sediment. The mixing length expression was given by Tsai and Tsai (2000) as follows:
${{l}_{m}}=\kappa z{{\left( 1-\frac{z}{{{h}_{a}}} \right)}^{0.5\left( 1+\frac{\alpha {{C}_{v}}}{{{C}_{r}}} \right)}}$
where κ is the Von Karman coefficient (≈ 0.4), α is the turbulent Schmidt number, Cr is the reference concentration at depth zr, and zr is the thickness of the bed layer (Figure 2). Yalin (1972) provided a 2D shear stress equation for sediment-laden flow in open channels as below.
$\tau ={{\left( {{\rho }_{sw}} \right)}_{z}}u_{*}^{2}\left( 1-\frac{z}{{{h}_{a}}} \right)$
where u* is the shear velocity =$\sqrt{gS_ah_a}$, g is the acceleration due to gravity, (ρsw)z is the density of sediment-laden flow from z to the free water surface$=\frac{1}{h_a-z}\int\limits_{z}^{h_a}\rho_{sw}dz$. Combining Eqs. (2) and (6) leads to
${{\mu }_{sw}}\frac{\partial u}{\partial z}+{{\rho }_{sw}}l_{m}^{2}{{\left( \frac{\partial u}{\partial z} \right)}^{2}}={{\left( {{\rho }_{sw}} \right)}_{z}}u_{*}^{2}\left( 1-\frac{z}{{{h}_{a}}} \right)$
Using Eqs. (3), (4) and (5) in Eq. (7) results in the following principal equation of flow velocity distribution of river water containing suspended sediments.
$\begin{align} & \left[ \mu \sum\limits_{i=0}^{\infty }{{{\left( k{{C}_{v}} \right)}^{i}}}+\left\{ \rho \left( 1-{{C}_{v}} \right)+{{\rho }_{s}}{{C}_{v}} \right\}{{\kappa }^{2}}{{z}^{2}}{{\left( 1-\frac{z}{{{h}_{a}}} \right)}^{\left( 1+\alpha \frac{{{C}_{v}}}{{{C}_{r}}} \right)}}\frac{\partial u}{\partial z} \right]\frac{\partial u}{\partial z}= \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \rho u_{*}^{2}\left( 1-\frac{z}{{{h}_{a}}} \right)+\left( {{\rho }_{s}}-\rho \right)\frac{u_{*}^{2}}{{{h}_{a}}}\int\limits_{z}^{{{h}_{a}}}{{{C}_{v}}dz} \\\end{align}$
The distribution of suspended sediment concentration (SSC) in a steady turbulent flow following the advection-diffusion concept is given by
${{\varepsilon }_{sw}}\frac{\partial {{C}_{v}}}{\partial z}=-{{\omega }_{sw}}{{C}_{v}}\left( 1-{{C}_{v}} \right)$
where ωsw is the terminal velocity of sediment-water and εsw is the suspended sediment diffusivity, which is given below
${{\varepsilon }_{sw}}={\alpha }'l_{m}^{2}\frac{\partial u}{\partial z}$
The constant α', as given by Coleman (1970) and Van Rijn (1984), is specified as
${\alpha }'=1+2{{\left( \frac{\omega }{{{u}_{*}}} \right)}^{2}}\text{ for }0.1<\frac{\omega }{{{u}_{*}}}<1$
where ω is the terminal velocity of transparent water, which is defined by the following equations:
$\omega =\frac{1}{25.6}\left( S-1 \right)\frac{gd_{50}^{2}}{\nu },\text{ }{{d}_{50}}<0.1\text{ mm}$
$\omega =\sqrt{{{\left( \frac{13.95\nu }{{{d}_{50}}} \right)}^{2}}+1.09g{{d}_{50}}\left( S-1 \right)}-\frac{13.95\nu }{{{d}_{50}}}\text{, 0}\text{.1 mm }{{d}_{50}}<4\text{ mm}$
$\omega =1.044\sqrt{g{{d}_{50}}\left( S-1 \right)}\text{, }{{d}_{50}}>4\text{ mm}$
where S is the relative density of sediments. The terminal velocity of sediment-laden flow, as established by Chien and Wan (1999), is found by using
$\frac{{{\omega }_{sw}}}{\omega }={{\left( 1-{{C}_{v}} \right)}^{{\hat{p}}}}$
where $\hat{p}$ is a parameter given as a function of Reynolds number of sediment particles $\left( {{\operatorname{R}}_{e}}=\frac{\omega {{h}_{a}}}{\nu } \right)$ (Table 3).
Table 3 Correlation between Re and$\hat{p}$
Re ≤0.1 0.2 0.5 1.0 2.0 5.0 10.0 20.0 50.0 100.0 200.0 ≥500.0
$\hat{p}$ 4.91 4.89 4.83 4.78 4.69 4.51 4.25 3.89 3.33 2.92 2.58 2.25
The principal equation of SSC distribution of sediment-water is derived using Eqs. (9-15) and is indicated as follows:
$\left\{ 1+2{{\left( \frac{\omega }{{{u}_{*}}} \right)}^{2}} \right\}{{\kappa }^{2}}{{z}^{2}}{{\left( 1-\frac{z}{{{h}_{a}}} \right)}^{\left( 1+\frac{\alpha {{C}_{v}}}{{{C}_{r}}} \right)}}\frac{\partial u}{\partial z}\frac{\partial {{C}_{v}}}{\partial z}+\omega {{C}_{v}}{{\left( 1-{{C}_{v}} \right)}^{\hat{p}+1}}=0$
To solve Eqs. (8) and (16), boundary conditions comprising the velocity of flow, ur, and SSC, Cr, at the top of the bed layer are required. The ur can be calculated using
${{u}_{r}}=5.75{{u}_{*}}{{\log }_{10}}\left( \frac{30.2{{z}_{r}}\lambda }{{{k}_{b}}} \right)$
where kb is the roughness of channel bed (≈ d65), λ is the correction factor and is correlated with d65° (Table 4), δʹ is the height of the viscous sub-layer and is given by
${\delta }'=\frac{11.6\nu }{{{{{u}'}}_{*}}}$
where ${{{u}'}_{*}}$ is the particle shear velocity determined using Einstein (1950).
Table 4 Correlation between d65/δ° and λ
d65° 0.2 0.3 0.5 0.7 1.0 2.0 4.0 6.0 10.0
λ 0.7 1.0 1.38 1.56 1.61 1.38 1.10 1.03 1.0
On the other hand, SSC at the top of the bed layer is seldom measured; instead, the average SSC is recorded. As a result, va and SSCg were used to calculate Cr adopting the trial run process.

4.3 Steps involved in the development of the bed load transport model

The principal equations (8) and (16) were solved using the numerical method. It discretizes the space between the free surface of the water and the bed layer surface using p nodes. It uses the forward finite difference approach for solving the resulting equations. The differential equations involved in solving it are as follows:
$\frac{\partial u}{\partial z}=\frac{{{u}_{k+1}}-{{u}_{k}}}{{{z}_{k+1}}-{{z}_{k}}},\frac{\partial {{C}_{v}}}{\partial z}=\frac{{{\left( {{C}_{v}} \right)}_{k+1}}-{{\left( {{C}_{v}} \right)}_{k}}}{{{z}_{k+1}}-{{z}_{k}}}$
$u_{k+1}^{i+1}=u_{k}^{i}+\frac{\left\{ \rho u_{*}^{2}\left( 1-{{z}_{m}}/{{h}_{a}} \right)+C \right\}}{D}$
$C=\left( \frac{\mu \sum\limits_{a=0}^{\infty }{{{\left( k\left( {{C}_{v}} \right)_{m}^{i} \right)}^{a}}}}{{{z}_{k+1}}-{{z}_{k}}} \right)+\left( \frac{\left( \rho +\left( {{\rho }_{s}}-\rho \right)\left( {{C}_{v}} \right)_{m}^{i} \right)\left( {{\kappa }^{2}}z_{m}^{2}{{\left( 1-\frac{{{z}_{m}}}{{{h}_{a}}} \right)}^{\left( 1+\frac{\alpha \left( {{C}_{v}} \right)_{m}^{i}}{{{C}_{r}}} \right)}} \right)\left( u_{k+1}^{i}-u_{k}^{i} \right)}{{{\left( {{z}_{k+1}}-{{z}_{k}} \right)}^{2}}} \right)$
$D=\left( {{\rho }_{s}}-\rho \right)\frac{u_{*}^{2}}{{{h}_{a}}}\int\limits_{z}^{{{h}_{a}}}{C_{v}^{i}}dz$
$\left( {{C}_{v}} \right)_{k+1}^{i+1}=\left( {{C}_{v}} \right)_{k}^{i}-\left( \frac{\omega \left( {{C}_{v}} \right)_{m}^{i}{{\left( 1-\left( {{C}_{v}} \right)_{m}^{i} \right)}^{\widehat{p}+1}}}{\left( \frac{{{E}^{i}}}{{{z}_{k+1}}-{{z}_{k}}} \right)} \right)$
${{E}^{i}}=\left\{ 1+2{{\left( \frac{\omega }{{{u}_{*}}} \right)}^{2}} \right\}{{\kappa }^{2}}z_{m}^{2}{{\left( 1-\frac{{{z}_{m}}}{{{h}_{a}}} \right)}^{\left( 1+\frac{\alpha \left( {{C}_{v}} \right)_{m}^{i}}{{{C}_{r}}} \right)}}\left( \frac{u_{k+1}^{i}-u_{k}^{i}}{{{z}_{k+1}}-{{z}_{i}}} \right)$
where k = 1, 2, …, p-1, m shows the mean value of nodes k and k+1, i indicates the value of the former iteration. The following procedure was followed in determining the ungauged bed load concentration on a monthly scale at all sub-basin stations:
a) The hydraulic variables involved in the development of the model as input were determined from sections 3 and 4.1.
b) The turbulent Schmidt number α was set with an initial guess of zero. The α for the subsequent iterations was determined using the following equation as developed by Tsai and Tsai (2000)
$\log \alpha =3.87\log \left( \frac{{{C}_{r}}}{{{C}_{m}}} \right)-2.6\log \left( \frac{\omega }{{{u}_{*}}} \right)+1.89\log \left( \frac{{{z}_{r}}}{{{h}_{a}}} \right)-1.2$
where Cm is the mean sediment concentration along the entire vertical z.
c) The flow velocity at the bed layer surface was computed using Eq. (17).
d) The sediment concentration at the bed layer surface was specified with a first value assumed ten times the observed volumetric concentration.
e) The distribution of velocity and SSC along the vertical z was computed using Eq. (20) and (23), adopting the trial run process.
f) The mean velocity and concentration were evaluated using the findings of the preceding step.
g) The mean concentration calculated in the previous step was compared with SSCg with an acceptable tolerance limit$\left(\left|\frac{\left(S S C_{g}-C_{m}\right)}{S S C_{g}}\right| \leq 10^{-4}\right)$. The procedure starting from (e) was repeated till the tolerance limit set for each iteration was acknowledged.
h) The BLT per unit width, qb (in kg/s/m), was calculated using the following equation
${{q}_{b}}=\beta {{\rho }_{s}}{{u}_{r}}{{C}_{r}}{{z}_{r}}$
where β is the coefficient of correction, considered as one here, it was assumed that the concentration of BL does not vary across the depth of the bed layer. The depth of the bed layer was considered as 0.05ha in this study based on published literature. According to Tsai et al. (2010), field-measured data yields a reasonable bed load estimation at a bed layer thickness of 0.05 times the flow depth. Additionally, it was discovered that the given bed layer causes increased sediment concentration and velocity, typical for monsoon flow conditions in a river. The suspended sediment concentration at a few particle sizes above the bed cannot be empirically quantified, according to Dey (2014). The reference level proportionate to the flow depth, or the bed layer thickness equal to 0.05 times the flow depth, is therefore regarded as satisfactory for verifying the concentration distribution. As a result, based on the. above rationality, the bed layer thickness of 0.05 times the flow depth is assumed in the present study. The BLT (or BLCu) in g/l is finalized by dividing Eq. (26) with unit flow discharge (m3/s/m).

4.4 Evaluation of BLCu using established approaches

The bed load transport for an outlet sub-basin station determined from the mathematical model was validated using some recognized approaches. The mean BLCu for all stations were initially compared and analyzed with the bed load classification of Maddock and Borland (1950) and Lane and Borland (1951), as tabulated in Table 5. The researchers classified the rivers as sand-bed or gravel-bed based on the prevailing particle sizes and attributed a proportion of bed load based on the measured SSC. The SSC less than 1000 ppm was classified as low, 1000 to 7500 ppm as medium and greater than 7500 as high concentration values. However, it should be noted that the methodologies given by these authors provide the most relevant results for sand-bed rivers, except for the most significant concentrations, but under-estimate the bed load fraction for gravel-bed rivers. As a result, these approaches were primarily considered a benchmark for validating the model outputs in the present study.
Table 5 Fraction of the ungauged bed load concentration as per Maddock and Borlanda (1950) and Lane and Borlandb (1951)
Suspended sediment concentration (ppm) Bed material class Texture of suspended sediment % BL relative
to TLa
% BL relative
to SLb
< 1000 Sand Similar to bed material Up to 50 25-150
< 1000 Gravel, rock or consolidated clay Small amount of sand 5 5-12
1000-7500 Sand Similar to bed material 10-20 10-35
1000-7500 Gravel, rock or consolidated clay 25% sand or less 5-10 5-12
> 7500 Sand Similar to bed material 10-20 5-15
> 7500 Gravel, rock or consolidated clay 25% sand or less 2-8 2-8

Note: BL: Bed load, SL: Suspended load, TL: Total load

Based on field data, Williams and Rosgen (1989) revised the methodologies used by the preceding authors (Table 6) and are therefore called ‘reference values’ hereafter. In this modified one, the author has split the data for gravel and sand-bed streams based on 15 categories of SSC, which makes it more precise to be mentioned. The result of the modelled BLT was eventually examined with this modified technique to make the findings of the current study more rational. The precision of the predicted mean proportion of BL for all stations was evaluated using the relative error (RE) with respect to the reference values. The RE (in %) is defined as the ratio of a difference between the model-predicted and reference value to the reference value. The positive and negative values of RE signify the under and over-estimating fraction of TL transported as BL.
Table 6 Fraction of the ungauged bed load concentration as per Williams and Rosgen (1989) in sandy bed rivers
Average concentration of SSC (g/l) (× 10-3) 1 3 5.2 10.9 20.7 40.8 80.4 160.6
(Average ± SD) of BL
relative to TL (%)
99.7±0 99.2±0.80 84±19 80±21 70±16 57±26 52±30 36±31
Average concentration of SSC (g/l) (× 10-3) 317.1 631.1 1131 2395 4918 8006 23450
(Average ± SD) of BL
relative to TL (%)
27±25 11±9 9±7 10±11 13±8.20 8±6.30 1±0.04

Note: BL: Bed load, TL: Total load, SD: Standard deviation

4.5 Correlation between observed SSC and the estimated bed load concentration

The relationship between the gauged SSC and the estimated ungauged bed load concentration is typically required to calculate bed load from suspended load and vice versa. Past literature established that bed load data could be described as a power function of suspended load data (Métivier et al., 2004; Meunier et al., 2006; Turowski et al., 2010). The predicted BLCu in the present study was also correlated with the SSCg. A power curve was fitted, and a functional relationship (BLCu = a×SSCgb) was reported for all sub-basin outlet stations, where a and b are the constants of the power regression. The performance of the power regression was evaluated using the error statistics recommended by Moriasi et al. (2015), namely the coefficient of determination (R2), Nash Sutcliffe efficiency (NSE), observed standard deviation ratio (RSR), and per cent bias (PBIAS). R2 is given by the equation (27). where it ranges from 0 to 1, it determines the degree of variance in the explanatory variable described by the proposed model, with higher values indicating less error variance. The NSE measures residual variance relative to observed data variance and is given by Eq. (28). It measures how closely the observed versus predicted data plot fits the 1:1 line. Moriasi et al. (2007) presented another statistic, the observed standard deviation ratio (RSR), to measure the error index of the model. RSR is defined by Eq. (29) (Moriasi et al., 2015). Lower RSR means better model prediction. RSR was proposed above RMSE since it normalizes model performance across investigations (Moriasi et al., 2007). Per cent bias (PBIAS) assesses the average tendency of predicted data to be larger or smaller than observed data (Gupta et al., 1999). Equation (30) quantifies bias overestimation and underestimation (Moriasi et al., 2015). Positive values represent model underestimation bias, while negative values show overestimation. R2 and NSE have ideal values of 1.0; however, RSR and PBIAS have optimal values of 0.0.
Performance evaluation criteria for a catchment scale sediment model are provided by Moriasi et al. (2015), including R2, NSE, and PBIAS (%). R2 > 0.4, NSE > 0.45, and PBIAS ≤ ±20% are all deemed satisfactory for a sediment model on a monthly time scale. However, for models with a monthly time step, an RSR of 0.7 or less is considered acceptable (Moriasi et al., 2007).
${{R}^{2}}={{\left[ \frac{\sum\limits_{i=1}^{n}{({{O}_{i}}-{{O}_{m}})({{P}_{i}}-{{P}_{m}})}}{\sqrt{\sum\limits_{i=1}^{n}{{{({{O}_{i}}-{{O}_{m}})}^{2}}}}\sqrt{\sum\limits_{i=1}^{n}{{{({{P}_{i}}-{{P}_{m}})}^{2}}}}} \right]}^{2}},0\le {{R}^{2}}\le 1$
$NSE=\left[ 1-\frac{\sum\limits_{i=1}^{n}{{{({{O}_{i}}-{{P}_{i}})}^{2}}}}{\sum\limits_{i=1}^{n}{{{\left( {{O}_{i}}-{{O}_{m}} \right)}^{2}}}} \right],-\infty \le NSE\le 1$
$RSR=\left[ \frac{\sqrt{\sum\limits_{i=1}^{n}{{{({{O}_{i}}-{{P}_{i}})}^{2}}}}}{\sqrt{\sum\limits_{i=1}^{n}{({{O}_{i}}}-{{P}_{m}}{{)}^{2}}}} \right],0\le RSR\le \infty $
$\operatorname{PBIAS}(\%)=\left[\frac{\sum_{i=1}^n\left(O_i-P_i\right)}{\sum_{i=1}^n\left(O_i\right)} \times 100\right],-\infty \leqslant$ PBIAS $\leqslant+\infty$
where Oi = ith measured data, Om = mean of the measured data, Pi = ith estimated data, Pm = mean of the estimated data and n = the number of data included for the investigation.
Furthermore, the constants of the proposed relation were physically connected with the hydraulic variables under evaluation. Kendall’s tau (τk) was used to assess the strength of the monotonic (linear and nonlinear) relationship between constants and hydraulic variables (Helsel et al., 2020). The τk was chosen because it is a rank-based process and robust to outliers. For Correlations, τk is deemed extremely weak for values less than ± 0.25, weak for values between ± 0.25 and ± 0.34, moderate for values between ± 0.35 and ± 0.39, and strong for values greater than ± 0.40 (Botsch, 2011).

4.6 Estimation of bed load transport rate using bed load functions

It is pertinent to mention that the present research has been conducted considering only the monsoon flows (which is dominant) over the river basin. The equilibrium sediment transport method, which is the traditional approach to bed load transport, is a one-to-one function of water flow regardless of the flow regime (i.e., for steady and unsteady non-uniform flows). Assuming that the flow shear stress exceeds the initial motion, this condition is appropriate (Bohorquez and Ancey, 2016). In this regard, the bed load equations extensively used in the literature provide an acceptable bed evolution process.
Additionally, information on non-uniform sediments is required for non-equilibrium sediment transport, which is limited in our instance. According to Einstein (1937), defining a representative diameter for non-uniform sediments is challenging since the size distribution of conveyed sediments and the size distribution of non-uniform sediment bed material are very different. As a result, dividing the sediment mixture into different sizes was advised before adding the individual size rates to determine the overall transport rate. Soil particle gradation information is essential for effectively determining non-equilibrium sediment transport. However, the scope of this data is currently constrained because the representative sediment particle size was determined using the solely available information on the soil classification of the relevant study stations. The present research has therefore established the performance of various well-known equilibrium sediment transport equations considering the data availability and the analysis period.
Multiple bed load transport equations developed using various techniques were employed in this regard. The BLCu for the MRB was estimated by applying 7 established bed load functions developed by Shields (1936), Schoklitsch (1962), Bagnold (1980), Misri et al. (1984) and Samaga et al. (1986) (Roorkee approach), Julien (2002), Huang (2010) and Recking (2013). The details of the above formulae are given in Table 7. These models were decided based on bed shear stress, discharge, stream power, suitability in sand-bed streams, and performance for rivers with unlimited sediment supply.
Table 7 Bed load functions incorporated in the present study
Bed load functions (Approaches
Functional equations Parameters involved
Shields, 1936
(Shear stress)
& \frac{{{q}_{b}}{{\gamma }_{s}}}{q\gamma {{S}_{a}}}=10\frac{\tau -{{\tau }_{c}}}{\left( {{\gamma }_{s}}-\gamma \right){{d}_{50}}} \\
& \tau =\gamma {{h}_{a}}{{S}_{a}} \\
& {{\tau }_{c}}={{\theta }_{c}}\left( s-1 \right)\gamma {{d}_{50}} \\
& {{\theta }_{c}}=0.1414S_{*}^{-0.23},{{S}_{*}}\le 6.61 \\
& {{\theta }_{c}}=\frac{{{\left[ 1+{{\left( 0.0223{{S}_{*}} \right)}^{2.84}} \right]}^{0.35}}}{3.09S_{*}^{0.68}},6.61<{{S}_{*}}<282.84 \\
& {{\theta }_{c}}=0.045,{{S}_{*}}\ge 282.84 \\
& {{S}_{*}}=\frac{{{d}_{50}}\sqrt{\left( s-1 \right)g{{d}_{50}}}}{\upsilon } \\
qb = bedload transport rate [(m3/s)/m]
q = unit flow discharge [(m3/s)/m]
γs = specific weight of sediment [KN/m3]
γ = specific weight of water [KN/m3]
Sa = average longitudinal bed slope [m/m]
τ = bed shear stress [KN/m2]
τc = critical shear stress [KN/m2]
d50 = particle size [mm]
ha = mean flow depth [m]
θc = critical shields stress given by Cao et al. (2006)
s = specific gravity of sediment = 2.65
ν = kinematic viscosity of water [m2/s]
g = acceleration due to gravity = 9.81 [m/s2]
Schoklitsch, 1962
(Unit flow discharge)
& {{q}_{c}}=\frac{1.944\times {{10}^{-5}}}{{{S}_{a}}^{{}^{4}/{}_{3}}} \\
& {{q}_{b}}=\frac{7000\times S_{a}^{1.5}\times \left( {{q}_{w}}-{{q}_{c}} \right)}{d_{50}^{0.5}} \\
qb = bedload transport rate [(kg/s)/m]
qw = unit flow discharge [(m3/s)/m]
qc = unit critical flow discharge [(m3/s)/m]
d50 = particle size (mm)
Sa = average longitudinal bed slope (m/m)
Bagnold, 1980
(Stream power)
& \frac{{{i}_{b}}}{i_{b}^{*}}=\frac{{{\left( W-{{W}_{0}} \right)}^{1.5}}}{{{\left( W-{{W}_{0}} \right)}^{*}}}{{\left( \frac{Y}{{{Y}^{*}}} \right)}^{-2/3}}{{\left( \frac{D}{{{D}^{*}}} \right)}^{-0.5}} \\
& {{W}_{0}}=290\times {{D}^{1.5}}\log \left( \frac{12Y}{D} \right) \\
ib = bedload transport rate [(kg/s)/m]
ib* = 0.1 [(kg/s)/m]
W = stream power [(kg/s)/m]
W0 = threshold stream power [(kg/s)/m]
Y = mean flow depth (m)
D = d50 = particle size (m)
(W-W0) * = 0.5 [(kg/s)/m]
Y* = 0.1 (m)
D* = 1.1×10-3 (m)
Roorkee (Misri et al., 1984; Samaga et al., 1986)
(Shear stress)
& {{\phi }_{b}}=4.6\times {{10}^{7}}{{{{\tau }'}}_{*}}^{8},{{{{\tau }'}}_{*}}\le 0.065 \\
& {{\phi }_{b}}=\frac{8.5{{{{\tau }'}}_{*}}^{1.8}}{{{\left( 1+5.95\times {{10}^{-6}}{{{{\tau }'}}_{*}}^{-4.7} \right)}^{1.43}}},{{{{\tau }'}}_{*}}>0.065. \\
& {{{{\tau }'}}_{*}}=\frac{{{R}_{b}}{{S}_{a}}}{(s-1){{d}_{50}}},{{R}_{b}}={{\left( \frac{{{v}_{a}}}{{{K}_{b}}\sqrt{{{S}_{a}}}} \right)}^{1.5}} \\
& {{K}_{b}}=\frac{K{{K}_{w}}{{(FW)}^{2/3}}}{{{\left[ (FW)K_{w}^{1.5}+2{{h}_{a}}\left( K_{w}^{1.5}-{{K}^{1.5}} \right) \right]}^{2/3}}} \\
& K=\frac{25.8}{{{\left( 2{{d}_{90}} \right)}^{1/6}}} \\
& {{K}_{w}}=\frac{1}{0.009} \\
& {{\phi }_{b}}=\frac{{{q}_{b}}}{{{d}_{50}}\sqrt{\left( s-1 \right)g{{d}_{50}}}} \\
ϕb = non-dimensional bed load transport rate
${{{\tau }''}_{*}}$= non-dimensional grain shear stress
Rb = hydraulic radius of bed region (m)
Sa = average longitudinal bed slope (m/m)
s = specific gravity of sediment = 2.65
d50 = particle size (m)
va = average flow velocity (m/s)
Kb = resistance coefficient of bed
K = general resistance coefficient
Kw = resistance coefficient of channel walls
FW = flow width (m)
d90 = particle size at which 90% of the sediment is finer by weight (m)
qb = unit bedload transport [(m3/s)/m]
Julien, 2002
(Shear stress)
& {{q}_{b}}=18\sqrt{g}d_{50}^{1.5}\tau _{*}^{2} \\
& {{\tau }_{*}}=\frac{{{h}_{a}}{{S}_{a}}}{(s-1){{d}_{50}}} \\
qb = unit bedload transport [(m3/s)/m]
g = 9.81 m/s2
d50 = particle size (m)
τ* = Shields parameter
ha = mean flow depth (m)
Sa = average longitudinal bed slope (m/m)
s = specific gravity of sediment = 2.65
Huang, 2010
[Reformulated Meyer-Peter and Muller, 1948]
(Shear stress)
& \phi =6{{\left( \eta \tau _{b}^{*}-0.047 \right)}^{5/3}} \\
& \eta =0.9997{{\left( \frac{{{K}_{b}}}{{{K}_{s}}} \right)}^{1.1981}} \\
& {{K}_{b}}=\frac{K{{K}_{w}}{{(FW)}^{2/3}}}{{{\left[ (FW)K_{w}^{1.5}+2{{h}_{a}}\left( K_{w}^{1.5}-{{K}^{1.5}} \right) \right]}^{2/3}}} \\
& {{K}_{s}}=\frac{25.8}{{{\left( 2{{d}_{90}} \right)}^{1/6}}} \\
& {{K}_{w}}=\frac{1}{0.009} \\
& \tau _{b}^{*}=\frac{{{R}_{b}}{{S}_{a}}}{(s-1){{d}_{50}}},{{R}_{b}}={{\left( \frac{{{v}_{a}}}{{{K}_{b}}\sqrt{{{S}_{a}}}} \right)}^{1.5}} \\
ϕb = non-dimensional bed load transport rate
η = bedform correction
τb* = non-dimensional bed shear stress
Rb = hydraulic radius of bed region (m)
Sa = average longitudinal bed slope (m/m)
s = specific gravity of sediment = 2.65
d50 = particle size (m)
va = average flow velocity (m/s)
Kb = resistance coefficient of bed
Ks = general resistance coefficient for skin friction
Kw = resistance coefficient of channel walls
FW = flow width (m)
d90 = particle size at which 90% of the sediment is finer by weight (m)
qb = unit bedload transport [(m3/s)/m]
Recking, 2013
(Shear stress)
& \phi =\frac{14\tau _{84}^{{{*}^{2.5}}}}{1+{{\left( \tau _{m}^{*}/\tau _{84}^{*} \right)}^{4}}} \\
& \tau _{m}^{*}=0.045\text{ for sand} \\
& \tau _{84}^{*}=\frac{{{S}_{a}}}{(s-1){{d}_{84}}\times \left[ \left( 2/FW \right)+74{{p}^{2.6}}{{\left( g{{S}_{a}} \right)}^{p}}{{q}^{-2p}}d_{84}^{3p-1} \right]} \\
& \text{where }p=0.23\text{ when }\frac{q}{\sqrt{g{{S}_{a}}d_{84}^{3}}}<100\text{ and }p=0.3\text{ otherwise} \\
& q=Q/FW \\
ϕ = Einstein non-dimensional bed load transport rate $\phi =\frac{{{q}_{b}}}{\sqrt{\left( s-1 \right)gd_{84}^{3}}}$
qb = unit bedload transport [(m3/s)/m]
g = 9.81 m/s2
d84 = particle size at which 84% of the sediment is finer by weight (m)
τ84* = Shields stress for d84
τm* = mobility shear stress
Sa = average longitudinal bed slope (m/m)
s = specific gravity of sediment = 2.65
FW = flow width (m)
The appropriateness of the BLF for the estimation of BLCu was established by comparing it with the previous mathematical model predicted BL concentration, and the performance of the same was ascertained using multiple statistical indices. Eventually, the best-performed BLF was expressed for straightforward quantification of ungauged BL concentration across sub-basins.

4.7 Assessment statistics of bed load transport equations

Different statistical indices were used to evaluate the performance of the predicted BLCu from the BLF of the MRB. The quantitative evaluation of the BLF was carried out using an average discrepancy ratio (DRa), dispersion index (DI), and inequality coefficient (IC) (Bravo-Espinosa et al., 2003; Molinas and Wu, 2010; Armijos et al., 2021). The DRa is given by equation (31). The bed load equations underestimate observed data for DRa < 1. However, the bed load functions are supposed to replicate the individual observations for 0.5 ≤ DRa ≤ 2 (Waikhom and Yadav, 2018; Armijos et al., 2021).
$D{{R}_{a}}=\left( \frac{1}{n} \right)\sum\limits_{i=1}^{n}{\left( \frac{{{P}_{i}}}{{{O}_{i}}} \right)}$
where n is the number of observations, Pi is the predicted data, and Oi is the observed data. A score was computed to evaluate the percentage of data coverage between accepted lower and higher bounds of DRa, i.e., between 0.5 and 2. Equation (32) gives the value of DI, where ANE is the average normalized error, and APF is the average prediction factor. ANE is the normalized absolute difference between observed and predicted data (Eq. 33). In contrast, the maximum value between the observed to predicted data ratio and the ratio of predicted to observed ones is denoted by APF (Eq. 34). DI is inferred satisfactory for values less than 10, and prediction power enhances towards zero.
$DI=\left( \frac{ANE\times APF}{100} \right)$
$ANE=\left( \frac{1}{n}\sum\limits_{i=1}^{n}{\left| \frac{{{O}_{i}}-{{P}_{i}}}{{{O}_{i}}} \right|} \right)\times 100$
$APF=\left[ \frac{1}{n}\sum\limits_{i=1}^{n}{\max \left( \frac{{{O}_{i}}}{{{P}_{i}}},\frac{{{P}_{i}}}{{{O}_{i}}} \right)} \right]$
The IC is a goodness-of-fit criterion based on a modelling measure called the root mean square error (RMSE). The bed load equation is thought to describe the observed data accurately for IC ≤ 0.5. RMSE and IC are given by Eqs. (35) and (36), respectively.
$RMSE=\sqrt{\sum\limits_{i=1}^{n}{\left[ \frac{{{\left( {{O}_{i}}-{{P}_{i}} \right)}^{2}}}{n} \right]}}$
$IC=\frac{RMSE}{\sqrt{\left[ \frac{1}{n}\sum\limits_{i=1}^{n}{O_{i}^{2}} \right]}+\sqrt{\left[ \frac{1}{n}\sum\limits_{i=1}^{n}{P_{i}^{2}} \right]}}$

5 Results and discussion

5.1 Basic statistics of hydraulic parameters

The details of the hydraulic variables on a monthly average scale with regard to the hydrological stations employed in the present study are listed in Table 8. Among the sub-basin stations, the tributaries Mand, Ib, and Tel carried higher mean SSCg than the other MRB tributaries (Table 9). The unusually high SSCg at these stations was attributed to the average longitudinal bed slope, sub-basin geology and drainage area in contrast to other tributaries. The nearly identical maximum SSCg of 1.48 g/l at Tikarpara was because it is the penultimate gauging station of the MRB carrying the highest discharge. However, the lower SSCg at Baronda and Bamnidih can be attributed to the presence of hydraulic structures immediately upstream of these stations. It also increases the maximum flow width of the Pairi and Hasdeo rivers downstream of these structures. The representative particle size estimated from Eq. (1) demonstrates that, according to the ISSCS, the bed sediments over the MRB are predominantly medium sand. Chakrapani and Subramanian (1990) also determined the average grain size classification of bed sediments as medium sand. The detailed statistical summary of the hydraulic variables is tabulated in Table 9. Among the tributaries, Bamnidih holds the highest coefficient of variation (CV), while Rampur has the highest skewness (Sk) and kurtosis (Kurt) in SSCg. However, in the entire basin, the last gauging station holds the highest CV, Sk and Kurt in SSCg. The high Kurt and CV of the SSCg imply that it is more dispersed, extremely unpredictable, and complex, with a non-normal distribution. In addition, it indicates the data is more prone to outliers than a Gaussian distribution. Accordingly, the low performance of the model in sediment prediction is anticipated at these locations. The Q and SSCg data are positively skewed, while other variables have mixed skewness. It reveals the presence of exceptionally high Q and SSCg as the analysis exclusively focuses on monsoon months. The high Sk, Kurt, and CV of SSCg in the whole data set of the MRB suggest its complicated and highly irregular behaviours compared to other hydraulic variables.
Table 8 Characteristics of hydraulic variables at all major sub-basin stations of the Mahanadi River Basin
ha(m) Q(m3/s) va(m/s) FW(m) SSCg(g/l) Sa
min Max min Max min Max min Max min Max
Baronda 1.79 2.77 158.92 775.38 0.41 1.28 310.97 548.46 0.03 0.44 0.00175 0.789
Jondhra 1.69 6.87 77.73 3760.21 1.25 2.71 130.32 507.11 0.03 0.72 0.00092 1.559
Rampur 2.27 3.59 68.06 533.23 0.47 1.56 147.10 170.84 0.09 0.58 0.00771 1.559
Bamnidih 1.87 3.65 57.74 1373.84 0.42 1.72 250.12 603.47 0.01 0.85 0.00130 1.559
Kurubhata 1.30 3.03 60.36 720.71 0.37 1.33 183.44 240.36 0.05 1.56 0.00100 0.576
Sundargarh 3.45 5.11 66.09 1209.26 0.42 1.81 131.09 313.9 0.07 1.61 0.00460 1.480
Salebhata 1.24 2.27 118.07 538.69 0.61 1.79 218.19 322.24 0.03 0.55 0.00177 1.435
Kantamal 1.76 6.77 52.34 3783.76 0.26 2.15 225.86 340.18 0.02 1.33 0.00204 0.782
Tikarpara 5.26 15.12 549.76 13172.60 0.43 2.22 231.34 839.21 0.04 1.48 0.00141 1.366

Note: ha = average depth of flow, Q = discharge, va = average flow velocity, FW = flow width, SSCg = suspended sediment concentration, Sa = average longitudinal bed slope, d50 = representative mean particle diameter, min = minimum, Max = maximum

Table 9 Descriptive statistics of hydraulic variables at all major sub-basin stations of the Mahanadi River Basin
Hydrological stations Statistics ha (m) Q (m3/s) va (m/s) FW (m) SSCg (g/l)
Baronda Mean 2.278 380.328 0.896 455.476 0.234
CV 0.097 0.382 0.215 0.120 0.439
Sk (Kurt) -0.384 (0.104) 0.738 (0.227) -0.406 (0.851) -0.360 (0.191) 0.215 (-0.547)
Jondhra Mean 3.891 1188.591 1.913 354.204 0.334
CV 0.242 0.526 0.164 0.244 0.501
Sk (Kurt) 0.212 (0.663) 1.012 (3.115) 0.396 (-0.195) -0.399 (-0.108) 0.133 (-0.359)
Rampur Mean 2.849 249.702 0.866 161.392 0.212
CV 0.114 0.402 0.219 0.039 0.468
Sk (Kurt) 0.131 (-0.916) 0.670 (0.356) 1.044 (3.198) -0.588 (-0.287) 1.767 (4.517)
Bamnidih Mean 2.607 383.544 0.998 394.77 0.233
CV 0.143 0.685 0.281 0.245 0.842
Sk (Kurt) 0.299 (-0.526) 1.358 (1.932) 0.087 (-0.701) 0.442 (-0.798) 1.203 (0.747)
Kurubhata Mean 2.052 227.490 0.802 210.392 0.541
CV 0.192 0.515 0.266 0.053 0.636
Sk (Kurt) 0.373 (-0.337) 1.407 (3.209) 0.225 (-0.170) 0.090 (-0.193) 0.891 (0.391)
Sundargarh Mean 4.317 369.928 0.889 284.426 0.546
CV 0.084 0.506 0.246 0.115 0.649
Sk (Kurt) -0.420 (-0.168) 1.551 (4.729) 1.069 (3.661) -2.814 (9.197) 1.213 (1.353)
Salebhata Mean 1.813 284.328 0.916 269.380 0.161
CV 0.149 0.375 0.240 0.101 0.779
Sk (Kurt) -0.401 (-0.516) 0.483 (-0.327) 1.700 (5.386) -0.361 (-0.710) 1.728 (3.065)
Kantamal Mean 3.886 840.552 1.057 287.088 0.415
CV 0.271 0.897 0.350 0.083 0.728
Sk (Kurt) 0.560 (-0.271) 1.536 (2.294) 0.498 (0.125) -0.162 (-0.318) 0.769 (-0.037)
Tikarpara Mean 9.018 3966.988 1.178 575.287 0.228
CV 0.245 0.694 0.308 0.198 0.843
Sk (Kurt) 0.686 (-0.107) 1.179 (0.891) 0.321 (-0.225) -0.489 (0.461) 3.648 (18.619)

Note: CV: Coefficient of variation, Sk: Skewness, Kurt: Kurtosis, ha = average depth of flow, Q = discharge, va = average flow velocity, FW = flow width, SSCg = suspended sediment concentration

5.2 Evaluation of model-predicted BLCu

The BLCu calculated using the developed mathematical model varied monthly across the study period at all sub-basin stations based on the input mean SSCg and flow velocity fluctuations. The bed load categorization of Maddock and Borland (1950) and Lane and Borland (1951) were used to compare and analyze the mean monthly BLCu for all stations. Table 10 lists the percentage predicted monthly mean BLCu at all sub-basin stations. The measured monthly mean SSC at all sub-basin stations of the basin falls below 1000 ppm (Table 9), corresponding to a low concentration as per Table 5. Accordingly, as per Table 5, the BLCu should lie within (25-150)% of SSCg and up to 50% of TL for sand-bed rivers. It was verified that all the values conform to the range of per cent BLCu per SSCg as mentioned (Table 10). However, since the classification provided in Table 5 is coarser, as a result, the proportion of BLCu relative to TL was eventually compared with the reference values. It can be seen that, for a given mean SSCg, the fraction of BLCu lies within the reference values mentioned in Table 6 for all stations within the basin, except at Sundargarh and Kurubhata. The fractional BL percentage relative to SSCg and TL is tabulated using Mean ± Standard deviation (Table 10) and compared with Table 6.
Table 10 Model predicted per cent BLCu relative to SSCg and TL
Stations Mean SSCg(g/l) Mean fraction of BLCu (%) w.r.t SSCg Mean fraction of BLCu (%) w.r.t TL RE(%)
Baronda 0.234 60.25 ± 20.36 37.60 ± 7.10 -18.61
Jondhra 0.334 46.51 ± 16.74 30.89 ± 7.72 -18.35
Rampur 0.212 43.55 ± 15.02 29.72 ± 6.07 9.72
Bamnidih 0.233 64.13 ± 32.46 36.50 ± 13.54 -14.77
Kurubhata 0.541 28.27 ± 15.18 21.09 ± 8.21 -56.22
Sundargarh 0.546 74.88 ± 31.14 41.06 ± 10.24 -167.97
Salebhata 0.161 43.54 ± 17.91 29.21 ± 9.29 18.66
Kantamal 0.415 32.37 ± 22.16 22.91 ± 9.57 -4.09
Tikarpara 0.228 53.07 ± 32.50 34.67 ± 11.30 -7.78

Note: BLCu: Ungauged model predicted bed load concentration, SSCg: Gauged suspended sediment concentration, TL: Total load concentration, RE: Relative error in the fraction of average predicted BLCu w.r.t. Table 6, Negative and positive RE’s denotes over and under predictions.

The relative error (RE) in calculating the mean fraction of BL in TL using the data of Williams and Rosgen (1989) varies widely across stations (Table 10). Except for Sundargarh and Kurubhata, all stations were within ± 20% of their expected level. Besides, the best prediction was noticed at Kantamal, which may be attributed to the least outliers in SSCg as understood from the kurtosis score. Although 7 out of 9 stations overestimated the BLCu fraction, the overestimation limits were reasonable for all sub-basins except the Mand and the Ib. The high RE values at these sub-basins were due to the increased SSCg, high kurtosis value in mean water discharge, and higher relief ratio. In general, it was observed that the predicted fraction of the TL transported as BL reduces with an increase in SSCg. In this regard, a typical variation of the per cent BL in TL versus the SSCg at Kantamal is shown in Figure 3. The sandstone, shale, and limestone strata in the Mand and Ib sub-basins resulted in much higher total erosion rates of nearly 500 tons/km2/yr relative to 100 to 300 tons/km2/yr at other locations (Chakrapani and Subramanian, 1990). It leads to higher SSCg, thereby creating a significant mismatch in the prediction of the BL as a fraction of TL. Authors have also reported similar observations at higher SSCs (Maddock and Borland, 1950; Lane and Borland, 1951). Also, the distribution of mean water discharge at these stations comprises the highest outliers, which may cause a significant departure in the modelling of BLT.
Figure 3 A typical covariation of the proportion of total load conveyed as bed load with observed suspended sediment concentration at Kantamal
In addition, the maximum relief ratio of 0.005 at Sundargarh led to an exceptionally higher overestimation of the fraction of BL. Despite conveying roughly comparable amounts of SSCg at Kurubhata and Sundargarh, the proportion of BLCu transported was nearly double for Sundargarh. This is because of more anomalies in the observed mean flow discharge at Sundargarh relative to Kurubhata, which eventually leads to a more predicted discharge of BL concentration. Similar observations of higher BL fractions at larger watershed areas were observed for Lena, Ganga and Brahmaputra rivers (Galy and France-Lanord, 2001; Alekseevskiy et al., 2008). Also, other researchers have proposed that the fraction of sediment delivered as BL is more prominent at a given catchment area for sand-bed rivers (Vezzoli, 2004; Turowski et al., 2010). Altogether, it was realized that about a third of the total sediment concentration, on average, is transported as bed sediment across the sub-basins of the Mahanadi River.

5.3 Correlation of model predicted BLCu and SSCg

In this study, the estimated BLCu was associated with the SSCg, and a power relationship was reported for all subbasin outlet stations. The variation in predicted BLCu (in g/l) with corresponding SSCg is depicted in Figure 4 across the sub-basins of the Mahanadi River. The R2 indicates the proportion of variation that is explained by each equation. All equations show an excellent fit to the data, accounting for 45 to 75% of the variation with a probability of significance < 0.05. The remaining unexplained variance may be attributable to sampling problems, hydrological variability, and the influence of numerous man-made controls in the river basin that develop through time.
Figure 4 Variation of the predicted ungauged bed load concentration (BLCu) and gauged suspended sediment concentration (SSCg) across stations (Solid black lines denote the fitted power regression, and p defines the probability of significance)
Table 11 displays various error statistics obtained to evaluate the performance and robustness of the proposed equations. For all investigated stations of the MRB, the metrics NSE, RSR, and PBIAS (%) are in the ranges of 0.57-0.80, 0.43-0.64, and 0.44-15.21, respectively. The NSE is lowest at Tikarpara (= 0.57) and ranges from 0.63-0.80 for the remaining stations under consideration. Although significant, the lowest NSE at the last gauging station may be attributed to the cumulative impact of the non-linearity in the sediment transport mechanism prevailing in the river basin. The per cent bias indicates that the model is under-estimated at all sites; nonetheless, it is small compared to the requirements established by Moriasi et al. (2015). Furthermore, the RSR performs well across stations. The consistency of error metrics throughout the model evaluation demonstrates that the proposed power regressions can estimate the BLCu from SSCg and vice versa for the key sub-basins under consideration.
Table 11 Error statistics of the proposed power regression model at various stations
Stations NSE RSR PBIAS (%) Stations NSE RSR PBIAS (%)
Baronda 0.63 0.60 2.28 Sundargarh 0.68 0.56 5.82
Jondhra 0.68 0.55 4.60 Salebhata 0.80 0.43 10.59
Rampur 0.69 0.55 4.65 Kantamal 0.69 0.54 7.48
Bamnidih 0.72 0.52 15.21 Tikarpara 0.57 0.64 8.82
Kurubhata 0.58 0.64 0.44
The observed SSC varies up to less than 0.6 g/l for 33% of the stations under observation. In addition, SSC levels ≥ 0.6 g/l were typically reported at stations with drainage areas over 5000 km2. The only exception was at Kurubhata, which has the highest relief among the tributaries with a catchment size of fewer than 5000 km2. The relief of Kurubhata is akin to that of Jondhra, which has more than six times the drainage area of Kurubhata. The predicted BL concentration is relatively less scattered around the fitted regression line at SSCg, less than 0.4 g/l and 0.8 g/l, corresponding to lower (< 5000 km2) and higher (> 5000 km2) drainage areas, respectively. As a result, adopting these proposed regression equations in the sub- basins is expected to perform satisfactorily within those limits of SSCg. Comprehensively, the proposed functional relationships would be helpful in the estimation of the basin sediment budget as a whole.
Additionally, constants a and b of the power regression between SSCg and BLCu vary from 0.17 to 0.57 and 0.51 to 1.07, respectively. The Kendalls tau, τk = 0.69 at the probability of significance, p < 0.05, denotes the constants to be positively correlated (Figure 5). However, it also reveals the importance of numerous hydraulic influences in preserving the natural balance between the two coefficients across the watershed.
Figure 5 Association between the constants of the proposed relations between SSCg and BLCu (Solid black line indicates the nonlinear trend, and p defines the probability of significance)
The analysis shows that constant a is linked positively with the typical particle size (d50) of the stations (τk = 0.57, p < 0.05). However, coefficient b is negatively associated with SSCg of sub-basins (τk = -0.44, p < 0.05). Since these correlations were statistically significant, the coefficient a can be used to describe the mode of sediment transport and the associated mechanism. Similarly, b signifies the natural supply of sediments to the stream. Besides that, only Bamnidih and Salebhata have coefficients b greater than one. It results from big multipurpose hydraulic projects close upstream of these stations, about 50 km from Bamnidih and 4 km from Salebhata (CWC, 2014). As a result, recorded sediment concentrations were restricted, and coefficient b was eventually elevated.

5.4 Impact of the dam on the relationship between bed and suspended sediments

Wang et al. (2007) discovered that sediment load variation is a good indicator for assessing the effects of human activities on the river basin. A decadal pattern of large dam development spanning the whole MRB is seen between 1977 and 2017, according to the National Register of Large Dams (NRLD, 2018). A consistent growth measuring up to a 50% gain is recognized between the first and last decades of 1977 and 2017 (Figure 6). Figure 7 depicts the impact of the dam on the relation between the bed and suspended sediment concentration. However, of all stations under consideration, only one (Bamnidih) is located directly downstream of a large dam (The Minimato Bango, Hasdeo sub-basin) and has a sufficient length of data pre- and post-impoundment. As a result, the influence of dam building on suspended and bed sediment concentrations are only being examined at Bamnidih. Looking at Figure 7a, we see a reverse trend in sediment concentrations before and after 1990. After dam impoundment in 1990, the percentage drop in mean suspended and bed sediment concentrations was around 66% and 73%, respectively. It demonstrates the significant impact of dam construction on bed sediment transport and, ultimately, the lack of bed sediments reaching the coastal zone. The same is true for the relationship between suspended and bed sediment concentrations as understood by Kendall’s tau (τk), as shown in Figure 7b. The τk decreases from 0.64 before dam impoundment to 0.39 following dam impoundment. However, when the whole study period is included, there is an overall strong association (τk = 0.63, p < 0.01) identified between the variables.
Figure 6 Variation of the construction of large dams in the river basin in the ten-year interval from 1977-2017
Figure 7 Impact of the Bango dam impoundment (1990) on the (a) variation of bed and suspended sediment concentration and (b) correlation between the bed (BLCu) and suspended sediment concentration (SSCg), as observed at Bamnidih

5.5 Determination of suitable bed load functions (BLF)

The BLFs used in the present study are based on past literature that was found suitable for alluvial rivers. The 7 BLFs used in the present study are derived from different theoretical approaches. Their performance was measured relative to the model-predicted BLCu for all the stations involved in the study. The quantitative evaluation of the potential of the different BLFs to replicate the findings of the estimated mathematical model was performed using three statistical indicators, as stated in section 4.6. A proper selection and validation of BLFs are needed to develop the BLT of the MRB. Figure 8 shows a graphical illustration of the DRa for all sub-basin stations, with only the equations with the top performance highlighted for convenience. Also, Table 12 summarizes the efficacy of the 7-bed load discharge formulas. The top performer and accepted BLFs for a sub-basin station are marked in bold. The average DRa for all stations except Kurubhata, Sundargarh and Tikarpara is greater than 1, which denotes that the BLFs over-predict the mathematical model data.
Figure 8 Graphical summary of Discrepancy ratio (DRa) across all stations, where the observations between the solid black lines are considered within acceptable limits, and black dotted lines represent the line of perfect agreement
Table 12 Statistical performance of different bed load functions (BLFs) across stations of the basin
Stations BLFs DRa Score (%) DI IC
Baronda Shields 1936 3984609 0 1.59×1013 0.99
Schoklitsch 1962 4.88 12.50 18.94 0.52
Bagnold 1980 0.82 56.25 1.46 0.40
Roorkee 1.89 62.50 3.57 0.36
Julien 2002 214.88 0 45960.89 0.98
Huang 2010 1.12 56.25 2.59 0.35
Recking 2013 564.30 0 317907.74 0.99
Jondhra Shields 1936 2163377.34 0 4.68×1012 0.99
Schoklitsch 1962 1.73 77.94 2.13 0.22
Bagnold 1980 0.36 16.17 3.56 0.69
Roorkee 6.31 45.58 34.27 0.58
Julien 2002 68.20 0 4583.21 0.95
Huang 2010 4.01 64.70 14.12 0.47
Recking 2013 2.47 23.52 14.15 0.74
Rampur Shields, 1936 1.38×108 0 1.91×1016 1
Schoklitsch 1962 51.06 0 2556.37 0.984
Bagnold 1980 8.84 0 69.45 0.74
Roorkee 2.48 42.50 4.23 0.41
Julien 2002 6383.28 0 40739959.30 0.99
Huang 2010 1.34 65 1.32 0.32
Recking 2013 9285.78 0 86216436.35 0.99
Bamnidih Shields 1936 2287781.84 0 5.23×1012 0.99
Schoklitsch 1962 1.88 63.49 2.36 0.28
Bagnold 1980 0.28 19.04 3.98 0.76
Roorkee 1.61 57.14 2.57 0.28
Julien 2002 198.10 0 39047.07 0.97
Huang 2010 0.85 41.26 1.99 0.40
Recking 2013 152.44 1.58 23087.79 0.98
Kurubhata Shields 1936 3511056.36 0 1.23×1013 0.99
Schoklitsch 1962 2.42 50.63 3.45 0.33
Bagnold 1980 0.55 37.97 1.34 0.45
Roorkee 0.82 72.15 1.05 0.29
Julien 2002 109.90 0 11968.71 0.97
Huang 2010 0.47 32.91 2.24 0.47
Recking 2013 156.60 0 24367.13 0.99
Sundargarh Shields 1936 24937514.25 0 6.22×1014 1
Schoklitsch 1962 7.66 2.59 51.04 0.64
Bagnold 1980 0.79 64.93 0.86 0.39
Roorkee 0.86 48.05 1.63 0.43
Julien 2002 2451.75 0 6008655.03 0.99
Huang 2010 0.46 35.06 2.84 0.59
Recking 2013 2199.84 0 4837097.88 0.99
Salebhata Shields 1936 7741251.25 0 7.34×1013 1
Schoklitsch 1962 8.57 6.25 80.12 0.63
Bagnold 1980 1.76 53.12 3.24 0.36
Roorkee 3.44 46.87 27 0.55
Julien 2002 331.08 0 124870.37 0.98
Huang 2010 1.81 50 10.42 0.49
Recking 2013 464.49 0 214443.40 0.99
Kantamal Shields 1936 28790604.46 0 5.98×1014 1
Schoklitsch 1962 10.6 0 72.84 0.72
Bagnold 1980 1.94 70.54 1.32 0.25
Roorkee 1.66 59.60 2.39 0.36
Julien 2002 885.01 0 807412.07 0.99
Huang 2010 1.01 60.46 1.67 0.40
Recking 2013 374 8.52 754199.84 0.99
Tikarpara Shields 1936 14290344.01 0 2.04×1014 1
Schoklitsch 1962 3.72 22.64 10.20 0.44
Bagnold 1980 0.59 42.45 1.41 0.50
Roorkee 0.73 37.73 4.22 0.49
Julien 2002 513.68 0 263359.66 0.99
Huang 2010 0.43 22.64 16.27 0.60
Recking 2013 2.61 32.07 20.88 0.62

Note: DRa: Discrepancy ratio, DI: Dispersion index, IC: Inequality coefficient, Roorkee BLF is developed by Misri et al. (1984) and Samaga et al. (1986).

A DRa equal to 1 implies a complete agreement between calculated and model-predicted BLT. As depicted in Figure 8, the prediction pattern varies significantly between stations over the investigation period. It generally follows a trend of larger over-prediction to lesser over-prediction from upstream to downstream stations of the MRB. However, the data falling within acceptable limits of DRa ranges from 78% at the upstream Jondhra to 42% at the downstream station Tikarpara. It shows a high deviation, although different BLFs were valid across stations. This could be due to the vital assumption of BLFs, primarily designed in laboratories, considering indefinite sediment input presuming steady-state for the given hydraulic and sediment parameters (Gray and Simoes, 2008). This is not true with the natural rivers since BLT varies significantly in space and time due to fluctuations in bed-shear stress and sediment sizes. Other vital elements influencing the efficiency of BLFs include data dispersion and uncertainties in hydraulic parameters. The existence of bedforms can also explain the over-prediction of calculated results using BLFs. This is because a significant part of the flow energy gets expended to overcome the form resistance induced by these bedforms, and the remaining energy is available for the transport of the sediments (Julien et al., 2002). In addition, Armijos et al. (2021) also noted that BLFs might overpredict since mean hydraulic variables take the whole cross-section into account. In contrast, only a part of the bed may be dynamic at any particular time.
The scaling of the denominator in the inequality coefficient (IC) (Eq. 36) ensures that IC consistently lies between 0 and 1. When IC is equal to 0, Pi becomes equal to Oi, and the fit is ideal. However, if IC = 1, Pi is not the same as Oi, and the equation has no predictive ability. IC is approximately 1 for Shield’s (1936), Julien’s (2002), and Recking’s (2013) BLFs at most stations, indicating that these three BLFs are incapable of determining BLT. Similarly, the DI value tends to zero whenever the error is minimal but can potentially take tremendous values with the increase in mean discrepancy error (Aguirre-Pe et al., 2004).
Based on the obtained statistical indicators, Huang’s (2010) BLF performed well in one station, the BLFs of Roorkee and Schoklitsch (1962) each performed well in 2 stations, and Bagnold’s (1980) BLF performed the best in the remaining 4 stations under investigation. Huang’s (2010) BLF, a reformed equation of Meyer-Peter and Muller (1948), outperformed the other BLFs only at Rampur. This is because of using a correction factor for the impacts of bedforms while calculating the BLT using Huang (2010). The approach utilized the linearity theory of alluvial channel flow, which asserts that flow shear stress is linearly connected with channel geometry, i.e., width and depth of flow. As a result, the highest performance of Huang’s (2010) BLF at Rampur is attributed to the lowest relative dispersion of channel geometric data (mean depth and width of flow) across all stations, as inferred from Table 9. The Bagnold (1980) BLF also performed considerably well at Baronda according to DRa, DI and IC; however, the Roorkee BLF has the highest relative score for which it is selected at this station. Similarly, Roorkee BLF performed reasonably well at Bamnidih, Sundargarh, Kantamal and Tikarpara, following DRa, DI and IC values. The final BLFs at any station were selected based on the highest score apart from DRa, DI and IC values. The Bagnold (1980) equation produced the best results, followed by Schoklitsch (1962) and Roorkee. These BLFs were also considered suitable for estimating BLT based on a large field and laboratory data globally (Lane, 1982; Bravo-Espinosa et al., 2003).
Moreover, the sediment size was well within the limit employed to construct the formulae of Schoklitsch (1962), Bagnold (1980), and Roorkee. The poorest score of BLFs at Tikarpara relative to other tributary stations is attributed to the larger coefficient of variation in observed discharge and the maximum contributing drainage area. Lastly, it is worth noting that the hydraulic input variables, i.e., depth of flow, flow velocity, discharge, longitudinal slope, and sediment sizes, are also crucial sources of error that can result in poor BLT computations (Wilcock, 2009). These errors are similar to those caused by sampling ambiguity. Nevertheless, the marked approved BLF at various sub-basin outlets of the MRB may be more helpful in estimating bed load transport rates for preliminary research in the future.

6 Conclusions

The proportion of the ungauged monthly bed load concentration (BLCu) at crucial sub-basins of the Mahanadi River is quantified using a computational model from 1973 to 2017. The predicted ungauged bed load concentration is eventually validated with the methodology of Williams and Rosgen (1989) for sand-bedded rivers. A power relationship is proposed to determine the BLCu from the gauged suspended sediment concentration (SSCg) at all stations. Moreover, the applicability of different bed load functions at all hydrological stations is evaluated and recommended.
Using the data of Williams and Rosgen (1989), the relative error (RE) in determining the mean fraction of BL in TL among stations ranges within ±20%, except for Sundargarh and Kurubhata. Outliers in the SSCg and water outflow data significantly impact estimating the bed load fraction. Furthermore, catchment factors such as higher relative ratio and catchment geology all considerably affected the relative error of prediction. Again, the postulated power relationship between BLCu and SSCg is demonstrated to perform as expected within the limitations of SSCg and the catchment area. The physical interpretation of the power relation constants specifies the form of sediment transport, the related mechanism, and the natural input of sediments to the stream.
Bagnold’s (1980) bed load function, based on the stream power technique, has the most acceptable performance in predicting bed load transport, followed by Roorkee and Schoklitsch (1962). Apart from data dispersion and uncertainties in hydraulic parameters, which may be related to sampling inconsistency, the existence of bedforms explains some over-prediction of estimated outcomes using BLFs. According to the analytical results, in the absence of observed bed load data for the Mahanadi River, bed load can be anticipated using either the proposed power functions or the recommended bed load functions. This finding also demonstrates that around 33% of the total load is conveyed as bed load in the Mahanadi River Basin during monsoons. However, in the present study, the representative particle size of the sediment is calculated from the only available data on the soil classification of the respective stations under study. The knowledge of soil particle gradation is crucial for efficiently determining non-equilibrium sediment transport, which is limited in the present investigation.
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