Analysis of Bed Load Equations and River Bed Level Variations Using Basin-Scale Process-Based Modelling Approach

Abstract

Bed load transport is a key process in maintaining the dynamically stable channel geometry for restoring the form and function of river ecosystems. Bed load consists of relatively large sediment particles that are moved along the streambed by rolling, sliding or saltation. Currently, various empirical correlations are used to estimate bed load transport rates since no single procedure, whether theoretical or empirical, has yet to be universally accepted as completely satisfactory in this aspect. Bed load particles are primarily sourced from river bed materials or banks. The amount of bed load and its spatial distribution contributes significantly to river bed level changes. Hillslope sediment contribution, mostly available to the river in the form of suspended load, also plays an important role in river bed level changes. This study aims to analyse different bed load equations and the resultant computations of river bed level variations using a process-based sediment dynamic model. Analyses have revealed that different bed load equations were mainly deduced from the concept of relating bed shear stresses to their critical values which are highly factored by the slope gradient, water discharge and particle sizes. In this study, river bed level variations are calculated by estimating total surplus or deficit sediment loads (suspended loads and bed loads) in a channel section. This paper describes the application of different widely used bed load equations, and evaluation of their various parameters and relative performances for a case study area (Abukuma River Basin, Japan) using a basin-scale process-based modelling approach. Relative performances of river bed level simulations obtained by using different bed load equations are also presented. This paper elaborates on the modelling approaches for river bed load and bed level simulations. Although verifications were not done due to unavailability of field data for bed load, qualitative evaluations were conducted vis-à-vis field data on flow and suspended sediment loads as well as the bed loads presented in different past studies.

Notation

A list of symbols used in this paper

Symbol	Meaning	Unit
β s	Correction factor for cohesive soil erosion	–
ρ	Water density	kg m−3
σ	Soil density	kg m−3
λ	Porosity of the bed sediment	–
τ	Bed shear stress	kPa
τ ri	Reference shear stress	kPa
τ rm	Reference shear stress for the geometric mean particle size on the bed surface	kPa
${\tau_{{*j}}}$	Nondimensional Shields stress associated with j-th grain size	–
${\tau_{{*ej}}}$	Nondimensional effective stress associated with j-th grain size	–
${\tau_{{*cj}}}$	Nondimensional critical Shields stress associated with j-th grain size	–
ω	Straining function varies from 0.453 to 1.011	–
θ	Ratio of bed shear stress to reference shear stress in Wilcock and Crowe formula	–
θ m	Volumetric moisture content	–
ϕ	Dummy variable used in Parker formula	–
ϕ sg0	Surface-based dimensionless measure of bed shear stress (Parker formula)	–
Δt	Temporal interval	s
Δz	Elevation change in Δt time	m
A	Water flow cross-section	m2
A s	Cross-section of sediment flow	m2
c	Manning-Strickler coefficient of bed roughness associated with skin friction and bed forms	–
${c^{\prime }}$	Manning-Strickler coefficient of bed roughness associated with skin friction only (i.e., after form drag due to bed forms has been excluded)	–
C r	Nondimensional stone cover (assumed 0.7 for channels, 0 otherwise)
C s	Suspended sediment concentration	m3 m−3
C v	Vegetation type dependent coefficient	–
d i	Representative particle diameter for i-th size class	m
d j	Representative particle diameter for j-th size class	m
d 50	Median grain size	m
d 90	Particle diameter for 90% finer grain size	m
D r	Stone size (assumed 0.1)	m
d sg	Suface geometric mean diameter	m
d sm	Geometric mean particle diameter on the bed surface	m
DF	Flow detachment or deposition	m3 s−1 m−1
DR	Soil detachment by rainfall	m3 s−1 m−1
DQ s	Residual suspended sediment in control section	m3 s−1
e −Zh	Water ponding correction factor	–
ee	Flow detachment (+ve) / deposition (-ve)	m3 s−1 m−1
E p	Potential evaporation	mm
g	Acceleration due to gravity	m s−2
G(ϕ)	Bed load transport function by Parker formula	–
h	Depth of water	m
IN	Depth of interception	mm
k	Soil detachability index	g J−1
KE	Kinetic energy	J g−1
L	Channel length	m
LAI	Leaf Area Index	–
n	Manning’s roughness	–
n /	Manning’s coefficient corresponding to the grain roughness	–
N	No. of observations	–
P bj	Fraction of j-th sediment particle size class in hillslope surface/river bed materials.	–
P bs	Fraction of the sand on the channel bed surface	–
P hi , P ei	Total hidden and exposed probabilities of particles d i	–
q bj	Estimated amount of bed load per unit width by assuming that there exists only a single particle size class (j-th class).	m2 s−1
q l	Lateral water discharge	m2 s−1
q s	Lateral sediment flow	m2 s−1
q w	Water discharge per unit channel width (without any sidewall correction)	m2 s−1
$q_w^{\prime }$	Water discharge per unit channel width (after sidewall correction)	m2 s−1
Q	Channel water discharge	m3 s−1
Q bj	Estimated amount of bed load by assuming that there exists only a single particle size class (j-th class).	m3 s−1
Q b	Bed load flow	m3 s−1
Q s	Suspended sediment flow	m3 s−1
R	Hydraulic radius	m
RDF	Root distribution function	–
S	Slope gradient	–
S 0 , S f	Water surface and friction slope	–
TC cs	Transport capacity concentration for suspended load	m3 m−3
u*	Shear velocity	m s−1
u *cj	Critical shear velocity associated with j-th grain size	m s−1
u *ej	Effective shear velocity associated with j-th grain size	m s−1
V	Flow velocity	m s−1
v s	Particle settling velocity	m s−1
w	Water flow width	m
w i	Inertial fall velocity	m s−1
Y	Shields parameter (nondimensional shear stress)	–
Y cr	Critical Shields parameter	–
Z	Soil texture index	–