in rivers or water stream is an expediting estimation as mentioned above using, e.g.,
streamflow gauges or remote sensing which allow quantifying channel slope.
A traditional pathway for characterizing the potential for sediment transportation
in river basins is based on the abovementioned assumption that the amount of
flowing water energy needed for the onset of an incipient particle motion and
sediment transport should be higher than a critical threshold of shear stress. The
rationale is that solid particles remain without acceleration if the acting forces such
as gravity, buoyancy, drag, or friction are in equilibrium. Shear stresses produced
by vertical velocity gradient deliver a drag force against solid particles in bedrock
that, if higher than the balance of forces, induce the beginning of the acceleration of
these particles. The magnitude of shear stresses is dependent on the surface slope,
channel geometry, and flow dynamics. This critical shear stress can be expressed by
a dimensionless ratio which is the Shields parameter with its near-constant critical
value h c , expressed as
h c ¼
s c
q s À q
ð
ÞD s
¼
hS
s À 1
ð
ÞD s
ð6:152Þ
where s c is the critical shear stress (Pa), q s and q are sediment and fluid density (kg
m
−3 ), g is the gravitational acceleration (ms
−2 ), h, is the flow depth, D s is the
representative grain size (m) usually considered as the median grain size D 50 , and
s is the sediment specific gravity.
The application of this methodology is limited by the reliance on flow depth,
which can be calculated from discharge and flow resistance, under assumptions on
channel geometry not always applicable. Another limitation is that Eq. (6.152) is
based on mean values, averaged over the width of the channel, instead of the real
forces on individual solid particles due to practical difficulties in accurately estimating the real force acting on individual particles. This reduces the accuracy of
this shear stress-based methodology for evaluation of sediment transport contradicting the assumption that real shear stress should be more representative of
sediment transportation mechanisms than specific stream power. Another limitation
of the average formulation of shear stress in Eq. (6.153) may be the fact, reported
by meta-analysis works, that specific stream power was more correlated with
bedload transport rates than shear stress or velocity (e.g., Parker 2010; Parker et al.
2011; Martin and Church 2000).
Another handicap with the Shields parameter can be a bias with channel slope
which could deliver an over-prediction of sediment transport rates. Stream
power-based transport equations can be more accurate with higher channel slopes
due to the absence of correlations between stream power and channel slope
(Lammers and Bledsoe 2018).
The following empirical bedload equation presented by Bagnold (1980) was
pivotal for the discussion and estimation of the critical stream power, x c , expressed
in kg m
−1 s
−1 , for delivering incipient motion of sediments
6.5 Mass Transfer
225
streamflow gauges or remote sensing which allow quantifying channel slope.
A traditional pathway for characterizing the potential for sediment transportation
in river basins is based on the abovementioned assumption that the amount of
flowing water energy needed for the onset of an incipient particle motion and
sediment transport should be higher than a critical threshold of shear stress. The
rationale is that solid particles remain without acceleration if the acting forces such
as gravity, buoyancy, drag, or friction are in equilibrium. Shear stresses produced
by vertical velocity gradient deliver a drag force against solid particles in bedrock
that, if higher than the balance of forces, induce the beginning of the acceleration of
these particles. The magnitude of shear stresses is dependent on the surface slope,
channel geometry, and flow dynamics. This critical shear stress can be expressed by
a dimensionless ratio which is the Shields parameter with its near-constant critical
value h c , expressed as
h c ¼
s c
q s À q
ð
ÞD s
¼
hS
s À 1
ð
ÞD s
ð6:152Þ
where s c is the critical shear stress (Pa), q s and q are sediment and fluid density (kg
m
−3 ), g is the gravitational acceleration (ms
−2 ), h, is the flow depth, D s is the
representative grain size (m) usually considered as the median grain size D 50 , and
s is the sediment specific gravity.
The application of this methodology is limited by the reliance on flow depth,
which can be calculated from discharge and flow resistance, under assumptions on
channel geometry not always applicable. Another limitation is that Eq. (6.152) is
based on mean values, averaged over the width of the channel, instead of the real
forces on individual solid particles due to practical difficulties in accurately estimating the real force acting on individual particles. This reduces the accuracy of
this shear stress-based methodology for evaluation of sediment transport contradicting the assumption that real shear stress should be more representative of
sediment transportation mechanisms than specific stream power. Another limitation
of the average formulation of shear stress in Eq. (6.153) may be the fact, reported
by meta-analysis works, that specific stream power was more correlated with
bedload transport rates than shear stress or velocity (e.g., Parker 2010; Parker et al.
2011; Martin and Church 2000).
Another handicap with the Shields parameter can be a bias with channel slope
which could deliver an over-prediction of sediment transport rates. Stream
power-based transport equations can be more accurate with higher channel slopes
due to the absence of correlations between stream power and channel slope
(Lammers and Bledsoe 2018).
The following empirical bedload equation presented by Bagnold (1980) was
pivotal for the discussion and estimation of the critical stream power, x c , expressed
in kg m
−1 s
−1 , for delivering incipient motion of sediments
6.5 Mass Transfer
225