Morris & Fan. Reservoir Sedimentation Handbook
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HYDRAULICS OF SEDIMENT TRANSPORT 9.44
FIGURE 9.19 Variation in the consolidation rate of cohesive sediments (modified from
M
igniot, 1982).
9.11.3 Rheology of Cohesive Sediment Mixtures
In a moving fluid the change in velocity v as a function of depth y along a vertical profile,
is termed the shear rate. A graph which illustrates the relation between the shear rate
dv/dy and shear stress is called a rheogram, and the equation describing this relationship
is called a rheological equation.
Clear water is a Newtonian fluid because the shear rate is directly proportional to the
applied shear stress at every depth. The rheological equation for a Newtonian fluid is:
= dv/dy
(9.44)
where μ = dynamic viscosity (N•s/m
2
). The characteristic shape of the rheogram for a
Newtonian fluid, as shown in Fig. 9.20, applies to clear water and sediment-water
mixtures of low concentration. These fluids will start moving as soon as the applied shear
stress exceeds zero.
At some higher sediment concentration, the suspended particles begin to interact
significantly with one another, creating a weak structural lattice within the fluid which
resists deformation by an applied shear stress. When this occurs the fluid will remain
immobile until the applied shear exceeds a threshold value termed the initial rigidity (τ
B
).
Water-sediment mixtures of this nature may be represented using the Bingham fluid
model, the rheogram of which is also presented in Fig. 9.20. The rheological equation for
a Bingham fluid is:
=
B
+ dv/dy (9.45)
where
B
= Bingham yield stress (N/m
2
) or initial rigidity, and η = plastic viscosity or
coefficient of rigidity, and represents the ratio of stress to shear, dv/dy, which is constant
HYDRAULICS OF SEDIMENT TRANSPORT 9.45
FIGURE 9.20 Rheograms illustrating the patterns characteristic of (a) Newtonian fluid, such as
clear water, (b) Bingham fluid, such as hyperconcentrated sediment-water mixtures and muds,
and (c) fluid model used by Otsubo and Muraoka in Fig. 9.23.
i
n a Bingham fluid. The rheological equation for a Bingham fluid is an empirical formula
derived from experimental data and which has proven useful in the analysis of a variety
of water-sediment mixtures including hyperconcentrated flows, clay slurries, and debris
flows. If the stress/shear ratio is not constant the fluid behavior is termed plastic. A fluid
is termed thixotropic if shear stress declines with time for a given value of dv/dy, and
antithixotropic if shear stress increases over time.
As a practical matter, a slurry may be considered a Bingham fluid when the Bingham
yield stress τ
B
exceeds 0.5 N/m
2
(Wan and Wang, 1994). The initial rigidity is not
constant for a given sample of cohesive sediment, but will increase as a function of
sediment concentration due to compaction with time (Fig. 9.21). Several important
behavioral characteristics of cohesive sediment have been related to the initial rigidity.
9.11.4 Laboratory Testing of Cohesive Sediment
Several basic laboratory and field tests may be used to characterize the bulk
characteristics behavior of cohesive sediment. The liquid limit is the moisture content
expressed as the weight percentage of oven-dried soil at which the soil will just begin to
flow when jarred slightly. The plastic limit is the lowest moisture content, expressed as a
percentage by weight of the oven-dried soil, at which it can be rolled into threads 3 mm
(1/8 inch) in diameter without breaking into pieces. Soils which cannot be rolled into
threads at any moisture content are considered nonplastic. The plasticity index is the
difference between the liquid limit and the plastic limit. It is the range of moisture content
in which a soil is plastic. When the plastic limit is equal to or greater than the liquid limit,
the plasticity index is recorded as zero. In the vane shear test a vane-shear apparatus
containing several vanes on a central axis is inserted into either a laboratory or field
sample and rotated to test the cohesive strength of the bulk sample (Fig. 9.22a). Torque is
applied to the sample through a calibrated spring, and the torque at which sample failure
occurs provides a direct reading of the shear stress of the sample.
Several types of tests can be used to directly measure erodibility of cohesive sediment
samples. In the rotating cylinder test a sediment sample is transferred to a laboratory
where it is shaped into a cylinder and inserted into a clear cylindrical container (Fig.
9.22b). The space between the sample and the outer cylinder is filled with water, and the
outer cylinder is rotated about the stationary soil sample to produce shear stress across the
sample surface. Rotation speed (torque) is increased until the sample surface is observed
to fail through cracking, peeling, or sloughing. A change in torque is also observable at
this point.
HYDRAULICS OF SEDIMENT TRANSPORT 9.46
FIGURE 9.21 Increase in initial rigidity as a function of sediment concentration (afte
r
Migniot, 1968).
The er
osion rate for sediment samples may be measured in a laboratory flume test.
Samples from the field are molded into a container placed in the bottom of a hydraulic
flume so that the sample surface is flush with the bottom of the flume. The erosion rate at
a given flow velocity is measured by the rate of weight loss in the sample container, or
the increase in solids concentration in the circulating water. Testing of this type can also
be performed in a closed rectangular conduit.
The degree of gravity setting or compaction can be measured as a function of time
using a column settling test. The relationship between the sediment concentration at the
onset of hindered settling, and the final sediment concentration after a settling period, has
been found to be a useful parameter for estimating several characteristics of cohesive
sediment. The procedure described by Otsubo and Muraoka (1988) used a 60 mm i.d.
clear cylinder 230 mm high. Samples were added and agitated, allowed to settle for one
day, again fully agitated for 5 minutes, and then the test was initiated. The height of the
initial sediment interface H
i
was recorded, as well as the interface level H
f
after 144 hours
of quiescent settling. The settling ratio V
may be computed as
H
f
Hi
Ci
C
f
(9.46)
HYDRAULICS OF SEDIMENT TRANSPORT 9.47
FIGURE 9.22 Devices used for the (a) vane shear test and (b) rotating cylinder test.
whe
re H is the interface height, C is the mass sediment concentration (kg/m
3
), and
subscripts
i and f refer to the initial and final condition respectively. Otsubo and Muraoka
used pure water in their settling tests; for field applications it may be more appropriate to
use native water. They also tested each mud using several different initial concentrations
for column settling tests. Mingniot (1968) used 200 g/L as the initial sediment
concentration in column settling tests.
9.11.5 Erosion Thresholds for Cohesive Sediment
Although absolute thresholds for sediment transport do not actually exist (Lavelle and
Mofjeld, 1987), approximate threshold conditions may be defined when erosion rates
become readily observable or significant. In the case of cohesive sediment, two erosion
threshold values may be defined.
1. Surface erosion or particle erosion refers to the condition when individual particles
or aggregates visibly begin to be removed from the surface of a bed of cohesive
sediment. Bulk measures of cohesive sediment mechanical parameters (solids
concentration, plasticity, Atterberg limits, etc.) have not been found useful for
predicting the critical flow conditions for initiation of surface erosion (Partheniades,
1986).
2. Mass erosion occurs at higher values of shear stress (velocity) and is characterized
by the removal of large clumps of sediment from the bed.
The onset of mass would normally be of interest in reservoir studies.
Threshold values of bed shear stress for surface erosion
c1
, and mass erosion
c2
were
related to initial rigidity by Otsubo and Muraoka (1988), who performed a series of flume
tests using both lake and estuary sediments to develop the relationship shown in (Fig.
9.23). Critical values for initiation of surface or particle erosion
c1
are given by
HYDRAULICS OF SEDIMENT TRANSPORT 9.48
FIGURE 9.23 Critical values for surface and mass erosion, as a function of initial rigidity
asdefined in Fig. 9.20c (after Otsubo and Muraoka, 1988).
c1
= 0.27
1
0.6
(9.47)
Critical values for the initiation of mass erosion
c2
are given by
c2
. = 0.79
1
0.94
(9.48)
These relationships were developed for values of bed shear stress (
c1
and τ
c2
)
between
0.08 and 3 N/m
2
.
Cohesive sediments compact over time, and the rate and extent of compaction will
reflect the mechanical shear strength of the aggregates. Higher cohesive forces will form
stronger and denser aggregates, and will in turn exhibit greater resistance to erosion by
shear forces. These denser aggregates also resist compaction by gravity within a settling
column. Based on this logic, Otsubo and Muraoka (1988) suggested that cohesive settling
behavior in laboratory columns might be correlated to erosion characteristics. Use of the
settling parameter is advantageous because it is easier to measure than initial rigidity
using a viscometer.
HYDRAULICS OF SEDIMENT TRANSPORT 9.49
FIGU
RE 9.24. Erosion resistance of successive sediment layers used by Bouchard (1988)
to model cohesive sediments in the small Kembs Reservoir on the Rhine.
Th
eir experiments indicated that several parameters could be correlated to the settling
ratio given in Eq. (9.47). The relationship between the settling ratio and initial rigidity is
given by
log (
B
)
= 3.58(H
f
/H
i
)-3.17 (9.49)
valid over the range 0.65 < H
/H
i
< 0.95. The initiation of mass erosion may be directly
related to the settling ratio by first solving Eq. (9.50) and then Eq. (9.49).
Based on flume testing of recently deposited sediment, Bouchard used erosion
threshold values ranging from 2.75 to 30 N/m
2
for modeling erosion in Kembs reservoir,
using the bedding pattern shown in Fig. 9.24.
Bed shear threshold values for mass erosion can vary greatly as a function of the clay
characteristics, and sediments that have compacted for many decades and have been
subject to desiccation can be very resistant to erosion, as illustrated by flume data using
cohesive sediment from several rivers in China (Fig. 9.25).
9.11.6 Erosion Rate of Cohesive Sediment
The erosion rate of cohesive sediment has been described by Partheniades (1962).
q
e
cr
1
(9.50)
where q
e
= mass of sediment eroded per unit area of bed surface per unit of time
(kg/s•m
2
), = shear stress (N/m
2
),
cr
= critical stress at which erosion commences
(N/m
2
), and = coefficient of erodibility.
9.11.7 Deposition Rate of Cohesive Sediment
The deposition rate of cohesive sediment can be described using the Krone equation.
q
d
C
v
1
d
(9.51)
HYDRAULICS OF SEDIMENT TRANSPORT 9.50
FIGURE 9.25 Critical shear stress values for initiation of mass erosion of cohesive
sediment (after Du and Zhang, 1989)
whe
re C
v
= volumetric sediment concentration, = fall velocity, and
d
= critical stress
for sediment deposition.
9.11.8 Angle of Repose
The angle of repose of cohesive sediment increases as a function of concentration. At low
concentrations the angle of repose will be less than that of cohesionless sediment, but as
concentration increases the angle of repose will exceed that of cohesionless sediment.
HYDRAULICS OF SEDIMENT TRANSPORT 9.51
FIGURE 9.26 Angle of repose of cohesive sediment as a function of initial rigidity fo
r
emergent and submerged sediments (after Migniot, 1968).
Migniot (1968) has related the angle of repose of cohesive sediment to the initial rigidity
(
B
) for several sediment samples, as shown in Fig. 9.26 for submerged and emergent
conditions. The general equation is:
tan= C
B
(9.52)
where = angle of repose,
B
= initial rigidity (N/m
2
), and C = coefficient with a value of
0.007 for emergent sediment and 0.025 for submerged sediment.
9.11.9 Settling of Individual Coarse Particles
The flocculant interaction of fine particles creates the Bingham yield stress, and larger
particles having a certain diameter d can be supported and will not settle in a quiescent
HYDRAULICS OF SEDIMENT TRANSPORT 9.52
Bingham fluid. The diameter d of the nonsettling particles is proportional to the Bingham
yield stress. This flocculant structure can be disrupted by turbulence, giving rise to the
situation in which coarse sediment can be sustained in suspension within a quiescent
fluid, yet will settle out when the same fluid begins to move and the turbulence within the
fluid partially destroys the flocculant structures. As flow velocity and turbulence
increase, sedimentation of the coarse materials will also increase, until the turbulence due
to velocity is itself sufficient to entrain and suspend the coarse sediment (Wan and Wang,
1994, p. 59).
The critical condition when the concentration of a suspension is so high that the
submerged particle weight is balanced by the yield stress of the flocculant mass is given
by:
d K
1
6
B
s
m
(9.53)
where d = particle diameter (m), y
s
and y
m
are the specific weights of the sediment
particles and the mixture respectively (N/m
3
),
B
= initial rigidity (N/m
2
), and the K
1
coefficient value has been reported in the range of 0.875 to 0.95 by different researchers
(Wan and Wang, 1994).
9.11.10 Group Settling of Cohesionless Sediment
Due to particle interaction, the fall velocity of a group of cohesionless coarse particles
within a suspension will be smaller than the fall velocity of individual particles in clear
water. The gross fall velocity of a group of uniformly sized coarse noncohesive particles
is given by the previously introduced equation by Richardson and Zaki (1954):
m
= (1 — C
v
)
M
(9.40)
where (ω
m
= gross fall velocity of uniform discrete particles in a mixture, ω = fall
velocity of an individual particle in clear water, C
v
= volumetric concentration of
particles in the mixture, and M = exponent value which is a function of the particle
Reynolds number (Qian, 1980). Based on data graphed by Wan and Wang (1994),
approximately:
for Re ≤3 M = 4.65
for 3 < Re <300 M = 4.65 — 1.075 [log(Re) — 0.477]
for Re ≥ 300 M = 2.5
Some experimenters have reported values of M around 7 for sand with d
50
= 0.067 mm
and 0.15 mm. In a water-mixture having nonuniformly sized sediment particles, each
grain size will settle at the velocity appropriate to its respective diameter.
9.12 CLOSURE
Sediment transport analysis is not an exact science. Even under carefully controlled
laboratory conditions there is significant scatter in the data, and field conditions are far
more complex. Conditions in reservoirs are more complex than in rivers. Numerical
techniques can provide only very approximate estimates of field conditions unless good
calibration data are available, which is rarely the case. The quantitative analysis of
HYDRAULICS OF SEDIMENT TRANSPORT 9.53
cohesive sediments can be particularly challenging. The purpose of these remarks is not
to detract from the use of analytical techniques; they constitute the essential foundation of
sedimentation engineering. Rather, it is to emphasize the imprecise nature of the results
that are to be anticipated, and to emphasize the importance of engineering judgment in
the application of numerical methods and interpretation of results.