TRACER STONES MOVING AS BEDLOAD IN GRAVEL-BED STREAMS

1
CHAPTER 28 TRACER STONES MOVING AS BEDLOAD IN
GRAVEL-BED STREAMS
This chapter was written by Miguel Wong and Gary
Parker It is preliminary code will be added
later.
Tracer stones (painted particles) in motion
during a flume experimentat St. Anthony Falls
Laboratory (SAFL)
2
BEDLOAD TRANSPORT-DOMINATED STREAMS
Mountain streams not only convey water, but also
transport large amounts of bed sediment,
including sand and gravel, and in some cases
cobbles and boulders. The transport of gravel and
coarser material is primarily in the form of
bedload, with particles sliding, rolling or
saltating within a thin layer near the stream
bed. Another characteristic of these streams is
that bedload transport events are sporadic and
are associated with floods. Thus in a perennial
stream, significant bedload transport may occur
for e.g. only about 5 of the time that water is
flowing. When bedload transport occurs, or for
that matter when the combined processes of
particle entrainment, transport and deposition
take place, the morphology of the stream may
evolve toward a new channel shape or bed profile.
Reliable and accurate estimates of the bedload
transport rate are essential, therefore, to the
quantification of such morphodynamic evolution.
The common approach is to obtain these estimates
via empirically derived relations, which are
based on characteristic driving parameters (i.e.
those of the flowing water) and the corresponding
resistance properties of the bed material.
Examples of some of these bedload transport
relations are presented in Chapter 7.
3
CHANNEL-AVERAGED DETERMINISTIC APPROACH
One good example of a relation still extensively
used in basic research and engineering
applications is that of Meyer-Peter and Müller
(1948). All terms defined below are deterministic
and represent channel-averaged values.
qb 1 is the dimensionless volume bedload
transport rate per unit width of stream (or
Einstein number), t 1 is the dimensionless bed
shear stress (or Shields number), and tc 1 is
the critical Shields number for particle
incipient motion. These variables are defined in
Chapter 7. (The notation inside the brackets
denotes the dimensions of the parameter preceding
it.)
In a 1D model of channel morphodynamic evolution,
restricted to tracking the time variation of the
longitudinal profile (bed elevation) of a study
reach, one can simply make use of the Exner
equation derived in Chapter 4.
4
BUT THE STORY IS NOT ALWAYS THAT SIMPLE
There are two limitations in the use of the
channel-averaged deterministic approach. First,
it does not contain the mechanics necessary to
describe the displacement patterns of individual
particles, hence it lacks the option of
explicitly linking changes in the composition and
surface configuration of the bed deposit with the
overall evolution of the channel morphometry of
the river (Blom, 2003). Second, bedload transport
is intrinsically a stochastic process (Einstein,
1950).
One alternative is the use of passive tracer
stones (e.g., painted or magnetically tagged
particles). The working hypothesis is that their
(vertical and streamwise) displacement history
may serve as a good indicator of the bedload
transport response of a stream to given water
discharge and sediment supply conditions
(DeVries, 2000).
Use of tracer stones in Shafer Creek, WA. Image
courtesy P. DeVries and T. Brown.
5
SEDIMENT CONTINUITY
The Exner equation derived in Chapter 4 relates
the time evolution of the bed elevation h L at
a given streamwise location x L with the volume
bedload transport per unit stream width qb L2/T
where lp 1 denotes bed porosity, and t T
represents time.
6
SEDIMENT CONTINUITY
The Exner equation derived in Chapter 4 relates
the time evolution of the bed elevation h L at
a given streamwise location x L with the volume
bedload transport per unit stream width qb L2/T
where lp 1 denotes bed porosity, and t T
represents time.
A different way to present the continuity
equation for the bed sediment is in terms of the
rate of exchange of particles between the bedload
and the bed deposit. This is the entrainment
formulation
where Db(x) L/T is the volume rate of
deposition from bedload per unit bed area at
location x, and Eb(x) L/T is the volume rate of
entrainment into bedload per unit bed area at
location x.
7
ENTRAINMENT FORMULATIONS
The last equation of Slide 6 reduces in the limit
as Dx ? 0 to
where Db L/T and Eb L/T represent the spatial
averages of the entrainment and deposition rates,
respectively. This form of mass continuity for
the sediment in the bed deposit is completely
equivalent to the form of Slide 5, i.e.
8
ENTRAINMENT FORMULATIONS
The last equation of Slide 6 reduces in the limit
as Dx ? 0 to
where Db L/T and Eb L/T represent the spatial
averages of the entrainment and deposition rates,
respectively. This form of mass continuity for
the sediment in the bed deposit is completely
equivalent to the form of Slide 5, i.e.
In a form analogous to Slide 12 of Chapter 4 for
suspended sediment, it can be shown that the
entrainment formulation of mass continuity for
the sediment in the (moving) bedload layer is
where x L is the volume concentration of
bedload per unit bed area.
9
ENTRAINMENT FORMULATIONS
The last equation of Slide 6 reduces in the limit
as Dx ? 0 to
where Db L/T and Eb L/T represent the spatial
averages of the entrainment and deposition rates,
respectively. This form of mass continuity for
the sediment in the bed deposit is completely
equivalent to the form of Slide 5, i.e.
In a form analogous to Slide 12 of Chapter 4 for
suspended sediment, it can be shown that the
entrainment formulation of mass continuity for
the sediment in the (moving) bedload layer is
where x L is the volume concentration of
bedload per unit bed area.
The term ?x/?t can be neglected for most cases of
interest, as seen from dimensional analysis and
the observation that bedload particles
are typically at rest far longer than in motion.
10
THE ACTIVE LAYER
The conservation equations of mass continuity
presented in the previous slides are intended for
bed sediment of uniform size in a 1D
bedload-dominated stream. Hirano (1971) extended
their application to size mixtures by introducing
the concept of an active layer of thickness La
L, so that the time evolution of the bed
sediment composition could be tracked in response
to changes in the sediment supply, overall bed
aggradation / degradation or flood hydrographs.
The basic equations are presented in Chapter 4,
and some applications are given in Chapters 17
and 18.
The use of this concept has allowed the
successful modeling of various morphodynamic
situations. However, the basis for its
formulation has two drawbacks (Parker et al.,
2000). First, the exchange of particles between
the bed deposit and the bedload is limited to a
surface layer of well-mixed sediment and finite
thickness (Fi and La in the upper sketch,
respectively). Second, entrainment of bed
sediment is represented by a step function (blue
dashed region in lower sketch), with the sediment
below the active layer (i.e., the substrate)
participating only as the bed degrades.
11
THE ACTIVE LAYER
The conservation equations of mass continuity
presented in the previous slides are intended for
bed sediment of uniform size in a 1D
bedload-dominated stream. Hirano (1971) extended
their application to size mixtures by introducing
the concept of an active layer of thickness La
L, so that the time evolution of the bed
sediment composition could be tracked in response
to changes in the sediment supply, overall bed
aggradation / degradation or flood hydrographs.
The basic equations are presented in Chapter 4,
and some applications are given in Chapters 17
and 18.
The use of this concept has allowed the
successful modeling of various morphodynamic
situations. However, the basis for its
formulation has two drawbacks (Parker et al.,
2000). First, the exchange of particles between
the bed deposit and the bedload is limited to a
surface layer of well-mixed sediment and finite
thickness (Fi and La in the upper sketch,
respectively). Second, entrainment of bed
sediment is represented by a step function (blue
dashed region in lower sketch), with the sediment
below the active layer (i.e., the substrate)
participating only as the bed degrades.
Are these realistic assumptions? Is there any
coupling between the bed sediment composition and
bedload transport rate that is missed with this
formulation?
12
THE ACTIVE LAYER, VIRTUAL VELOCITY AND TRACERS
Even under steady, uniform transport conditions,
bedload particles constantly interchange with the
bed. A moving particle is eventually deposited on
the bed surface or buried, where it may remain
for a substantial amount of time. Fluctuations in
bed elevation may cause the grain to be exhumed
and re-entrained, however. Now let vb L/T
denote the mean velocity of a particle while it
is moving, and vv L/T denote its virtual
velocity averaged over both periods of motion and
periods of rest. For typical gravel-bed streams,
vv ltlt vb, implying that a particle spends most of
its time at rest (even during equilibrium
transport).
13
THE ACTIVE LAYER, VIRTUAL VELOCITY AND TRACERS
Even under steady, uniform transport conditions,
bedload particles constantly interchange with the
bed. A moving particle is eventually deposited on
the bed surface or buried, where it may remain
for a substantial amount of time. Fluctuations in
bed elevation may cause the grain to be exhumed
and re-entrained, however. Now let vb L/T
denote the mean velocity of a particle while it
is moving, and vv L/T denote its virtual
velocity averaged over both periods of motion and
periods of rest. For typical gravel-bed streams,
vv ltlt vb, implying that a particle spends most of
its time at rest (even during equilibrium
transport). Two alternative statements of
equilibrium sediment mass conservation can be
stated using these velocities. Let ? denote the
volume of bedload particles per unit bed area,
and let the active layer thickness La
specifically denote a characteristic thickness
within which buried bedload particles reside.
Then,
Here La and vv can be quantified in the field
from measurements of the depth of burial of
tracers and distance moved by tracers, both over
a flood of known hydrograph.
14
ONE FIRST STEP TOWARD A MORE GENERAL MODEL

• One ambitious goal is to establish a relation
  between the statistics of vertical and streamwise
  displacement of a group of identifiable particles
  (tracer stones), the channel hydraulics and the
  bedload transport rate of a stream (see e.g.,
  Hassan and Church, 2000 Ferguson and Hoey,
  2002). This could allow the investigation of, for
  instance, how the vertical structure of the bed
  deposit (i.e. its stratigraphy) influences the
  overall morphodynamic evolution of a mountain
  stream.
• The material presented here corresponds to one of
  the simplest morphodynamic scenarios. The
  theoretical framework is developed with the aid
  of results from flume experiments.
  Simplifications considered in the analysis
  include
• Straight channel of constant width.
• 1D normal flow approximation valid at geomorphic
  time scales.
• Lower-regime plane-bed equilibrium transport
  conditions.
• Bedload transport predominating.
• Bed sediment of uniform size and given density,
  with constant bed porosity.
• A particle located at a given elevation in the
  bed deposit can be entrained into transport only
  if the instantaneous bed surface is at that
  elevation.

15
THE BED ELEVATION FLUCTUATES!!!
A first important fact to recognize is that even
for the case of lower-regime plane-bed conditions
equilibrium bedload transport and normal
(uniform and steady) flow, the bed elevation
fluctuates in time t at any streamwise location x
(see e.g., Wong and Parker, 2005).
Double-click on the image to run the video.
Experiments at SAFL Tracer stones (painted
particles) are gradually entrained from
ever-deeper locations and replaced with
non-painted particles, resulting in an
approximately constant mean bed elevation.
rte-bookbedload.mpg to run without relinking,
download to same folder as PowerPoint
presentations.
16
HOW TO HANDLE THESE VARIATIONS?
The variation in time t of the instantaneous bed
elevation z' L at a given streamwise location
x, can be tracked in terms of the fluctuations
around its local mean value h. Thus, a new
vertical coordinate system (positive downward)
can be defined in terms of the variable y L,
which is boundary-attached to h
Lets trace now a line at relative level y,
parallel to the mean bed elevation h. The
fraction of sediment pores in this line
(depicted by the sum of the thick green strips in
the sketch above) is represented by PS(y) 1.
Hence, for time scales shorter than those
corresponding to overall net bed aggradation or
degradation, it can be intuitively argued that
(Parker et al., 2000)
All water All sediment pores
and that PS(y) is a monotonically increasing
function ranging from 0 to 1.
17
PROBABILISTIC CHARACTERIZATION
Finding the shape of PS(y) is not necessarily
important at this stage, but the assumption that
this function is monotonically increasing from 0
to 1 is key. Thus, PS(y) defines a cumulative
distribution function, in this case of the amount
of sediment pores at a relative level y. In a
more physical context, PS(y) can be interpreted
as the probability that the instantaneous
relative bed elevation is less than or equal than
y, with its associated probability density
function pe(y) 1/L computed from
which by definition must satisfy
18
MEASURING FLUCTUATIONS
Simultaneous measurements of bed elevation
fluctuations at 6 different streamwise locations
in a flume were conducted for 10 different
equilibrium bedload transport conditions (Wong
and Parker, 2005). A sonar-multiplexing system
was used for this purpose. Let em L denote the
measurement error, fm L denote the measuring
footprint, and D50 L represent the median size
of the bed material. The respective values of
em/D50 and fm/D50 were 0.020 and 1.00 for the
four 0.5 MHz type probes used, and 0.005 and 0.53
for the two 1.0 MHz type probes used. The
algorithm developed was successful in
discriminating between particles in bedload
motion and the actual bed elevation.
Experiments at SAFL foreground Pulser
computer used for data acquisition and
processing background cables and metal frames
for ultrasonic probes in flume.
19
MEASURING FLUCTUATIONS
Simultaneous measurements of bed elevation
fluctuations at 6 different streamwise locations
in a flume were conducted for 10 different
equilibrium bedload transport conditions (Wong
and Parker, 2005). A sonar-multiplexing system
was used for this purpose. Let em L denote the
measurement error, fm L denote the measuring
footprint, and D50 L represent the median size
of the bed material. The respective values of
em/D50 and fm/D50 were 0.020 and 1.00 for the
four 0.5 MHz type probes used, and 0.005 and 0.53
for the two 1.0 MHz type probes used. The
algorithm developed was successful in
discriminating between particles in bedload
motion and the actual bed elevation.
Close-up picture of ultrasonic transducer probe
for measuring bed elevation fluctuations. Major
concerns in setting the probes were to avoid the
entrainment of air bubbles below them and to keep
their bottom as far as possible from the gravel
bed.
20
TIME SERIES OF BED ELEVATION
Time series of bed elevation z', and thus of the
bed elevation fluctuations y around the
corresponding mean value h, were obtained for the
10 equilibrium test conditions. From the
analysis carried out, it was found that for any
given equilibrium state, the time series were
stationary in their first two statistical
moments. Aggregated time series were then formed
for each equilibrium state.
Time series of bed elevation at various points in
a flume. Case (a) is for a relatively low bedload
transport rate, and case (b) is for a relatively
high bedload transport rate. Note that
fluctuations in bed elevation increase from case
(a) to case (b).
21
NORMAL PROBABILITY MODEL
Working with the aggregated time series of bed
elevation fluctuations y, an empirical cumulative
distribution was constructed for each equilibrium
state. Based on Chi-square tests, it was found
that a normal distribution model gave a good fit
of the probability density function of bed
elevation fluctuations, pe(y)
Empirical vs. theoretical normal cumulative
distribution function for a sample equilibrium
test.
where sy L is the standard deviation of bed
elevation fluctuations, which was found to
correlate with D50 and excess Shields number (t
- 0.055) as follows
22
ELEVATION-SPECIFIC ENTRAINMENT AND DEPOSITION
The following probabilistic terms can be
defined pEnt(y) 1/L probability density
function that a particle entrained from the bed
deposit into bedload
transport comes from a depth y relative to the
mean bed elevation h pDep(y)
1/L probability density function that a
bedload particle is deposited onto
the bed deposit at a depth y relative to
the mean bed elevation h The following
properties must be satisfied
such that
entrainment rate from level y to level y
Dy
and,
such that
deposition rate at level y to level y Dy
23
ENTRAINMENT OF TRACER STONES
A total of 80 flume runs with tracer stones were
conducted under conditions of plane-bed
lower-regime equilibrium bedload transport. They
corresponded to 10 different equilibrium cases, 8
tests each, and durations ranging from 1 min to
120 min. The experimental procedure consisted of
running the system until equilibrium was reached
seeding tracer stones in 4 spots, 4 layers per
spot, about 200 particles per layer, with the
color of tracers used as a proxy for initial
vertical position re-running the system for a
predetermined duration and, counting the number
of particles displaced per color.
Layered placement of tracer stones
The main results can be summarized as follows
the longer the duration of competent flow and/or
the larger the driving force (excess Shields
number), the larger is the fraction of tracer
stones displaced, and the deeper is the
layer accessed.
24
THIS EXPERIMENT LASTED 1-min ONLY
flow direction
Top yellow tracers on LHS of channel are
quickly displaced (lower spot), while the same
does not happen with top orange tracers on RHS.
Then the situation is reversed, likely because on
the RHS there are more tracer stones exposed
than on the LHS (they are buried or
already gone!).
25
ENTRAINMENT PER LAYER
The uniqueness of the experimental runs conducted
at SAFL is that they allow a direct measurement
of elevation-specific particle entrainment. By
looking at the additional fraction of tracers
displaced when comparing two runs of different
duration but both corresponding to the same
equilibrium conditions, entrainment rates can be
computed.
The plots to the left show the percents of
tracers moved from each layer (top, second, third
and bottom) as a function of experiment duration.
Case (a) is for a relatively low bedload
transport rate and case (b) is for a relatively
high bedload transport rate. Note that particle
entrainment per layer increases with bedload
transport rate, i.e. from (a) to (b), as well as
with experiment duration.
26
VANILLA MASS BALANCE FOR TRACER STONES
In the SAFL experiments, tracers of a given color
are not replaced with stones of the same color
once they are displaced. This is because all
tracers that moved out of the system were
captured at a sediment trap and were not
permitted to re-enter the flume. Thus, in an
entrainment formulation of mass balance for
tracer stones in a control volume corresponding
to their seeding position, the deposition term
vanishes. Let Ltr L denote the thickness of a
layer of tracers. The conservation equation for
the fraction of tracers fbts(t) 1 per layer at
time t then takes the form
Solving this ODE for Eb results in
Time evolution of the fraction of non-displaced
tracers as a function of test duration and excess
Shields number (t - 0.055). All 4 layers are
aggregated in the plot.
27
VANILLA MASS BALANCE FOR TRACER STONES
In the SAFL experiments, tracers of a given color
are not replaced with stones of the same color
once they are displaced. This is because all
tracers that moved out of the system were
captured at a sediment trap and were not
permitted to re-enter the flume. Thus, in an
entrainment formulation of mass balance for
tracer stones in a control volume corresponding
to their seeding position, the deposition term
vanishes. Let Ltr L denote the thickness of a
layer of tracers. The conservation equation for
the fraction of tracers fbts(t) 1 per layer at
time t then takes the form
The values of Eb determined in this way were
found to correlate with D50 and excess Shields
number (t - 0.055) as follows
Solving this ODE for Eb results in
where R is the submerged specific gravity of the
bed sediment 1, and g is the acceleration of
gravity L/T2
28 different combinations of fbts(t)-pairs have
been used to estimate the value of Eb for each
experimental equilibrium state.
28
ENTRAINMENT AND DEPOSITION FUNCTIONS
The probability density function pEnt(y) that a
particle entrained into bedload transport is
removed from depth y, and the corresponding
probability density function pDep(y) that a
particle deposited from the bedload is emplaced
at depth y were introduced in Slide 22. Here the
following general forms for pEnt and pDep are
assumed where pB(y) is an appropriately
chosen probability density function, and ybe and
ybd represent offset distances from the mean bed
(at y 0) for the erosion and deposition
functions, respectively.
The experiments reported here allow for
quantification of only the offset ybe at
equilibrium conditions. It is possible, however,
to speculate about the general relation between
the offset ybe and the offset ybd under
conditions that may or not be at equilibrium.
Here it is assumed that
29
ENTRAINMENT AND DEPOSITION FUNCTIONS contd.
The assumptions of the previous slide thus give
the following relations
Here y0 is an offset common to both functions,
which could be greater than or less than 0. The
case y0 gt 0 biases both functions downward below
the mean bed elevation. The case y1 gt 0 biases
erosion upward in the bed, and deposition
downward in the bed (see sketch to the right).
The above forms are assumed to be valid for both
equilibrium and disequilibrium cases. At
equilibrium, however, erosion and deposition must
balance within every layer, i.e.
so that y1 ? 0 as equilibrium is approached. This
point is illustrated in more detail in subsequent
slides.
30
EXPONENTIAL MODEL
The data allow estimation of the probability
density function pEnt(y) at equilibrium
conditions. Specifically, it was found that
pEnt(y) could be fitted to an exponential
function of the form
Note that the entrainment function pEnt(y) is
continuous in y, thus overcoming the step
function approximation of the active layer
formulation (Slide 10 see also Chapter 4).
Moreover, according to the above equation pEnt(y)
depends on the standard deviation sy of bed
elevation fluctuations, and hence correlates with
excess Shields number (t - 0.055) (Slide 21).
Exponential fitting for a sample equilibrium test.
Setting ybd y0 y1, the corresponding form for
the deposition function pDep(y) is
31
THE STORY SO FAR
32
THE STORY SO FAR
Predictors to complete a modified version of the
Parker et al. (2000) formulation have now been
developed up to the specification of forms for
the parameters y0 and y1.
33
PROBABILISTIC FORMULATION OF MASS CONTINUITY FOR
SEDIMENT IN THE BED DEPOSIT
As indicated in Slide 16, the control volume
(strip with fill) is boundary-attached to the
mean bed elevation h, which is free to move up or
down in time. The conservation equation for the
sediment in the bed deposit within any layer from
y to y Dy can then be expressed as follows
Time rate of change of mass in control volume
Flux of mass going into the control volume
Flux of mass going out from the control volume


34
PROBABILISTIC FORMULATION OF MASS CONTINUITY FOR
SEDIMENT IN THE BED DEPOSIT
As indicated in Slide 16, the control volume
(strip with fill) is boundary-attached to the
mean bed elevation h, which is free to move up or
down in time. The conservation equation for the
sediment in the bed deposit within any layer from
y to y Dy can then be expressed as follows
Note how the stone (solid circle) is moved out of
the control volume as the control volume is
advected upward
Apparent convective transfer as a result of
moving from the green to the blue strip.
35
DOES PS(y) HAVE TO BE STATIONARY?
The conservation equation for sediment in the bed
deposit presented in the previous slide reduces
to
Further reducing with the relation between PS(y)
and pe(y) of Slide 17
36
DOES PS(y) HAVE TO BE STATIONARY?
The conservation equation for sediment in the bed
deposit presented in the previous slide reduces
to
Further reducing with the relation between PS(y)
and pe(y) of Slide 17
0 ???
The expression above could be simplified more if
PS(y) is assumed to be stationary (independent of
time), even under disequilibrium conditions. By
doing so, however, the term on the LHS of the
relation to the right becomes independent of y.
The only way that this can be true is if the
following condition is satisfied
37
DOES PS(y) HAVE TO BE STATIONARY?
The conservation equation for sediment in the bed
deposit presented in the previous slide reduces
to
Further reducing with the relation between PS(y)
and pe(y) of Slide 17
0 ???
This is not only a very restrictive assumption
for cases of non-equilibrium transport, but it
can be seen from Slides 21 and 30 that even at
equilibrium pe(y) differs in form from pEnt(y)!
Thus in general PS(y) should not be expected to
be stationary. The term ?PS/?t should be expected
to vanish only for equilibrium conditions.
38
THE ENTRAINMENT FORMULATION FOR SEDIMENT
CONTINUITY OF SLIDE 9 CAN BE RECOVERED FROM THE
PROBABILISTIC FORMULATION
Lets integrate the equation for conservation of
bed sediment presented in Slide 35 over the whole
range of possible relative levels y
39
THE ENTRAINMENT FORMULATION FOR SEDIMENT
CONTINUITY OF SLIDE 9 CAN BE RECOVERED FROM THE
PROBABILISTIC FORMULATION
Lets integrate the equation for conservation of
bed sediment presented in Slide 35 over the whole
range of possible relative levels y
1
1
1
This yields
Applying integration by parts to the term on the
LHS of the expression above, and using the
relation between PS(y) and pe(y) of Slide 17, it
is found that
40
THE ENTRAINMENT FORMULATION FOR SEDIMENT
CONTINUITY OF SLIDE 9 CAN BE RECOVERED FROM THE
PROBABILISTIC FORMULATION
Lets integrate the equation for conservation of
bed sediment presented in Slide 35 over the whole
range of possible relative levels y
1
1
1
This yields
Applying integration by parts to the term on the
LHS of the expression above, and using the
relation between PS(y) and pe(y) of Slide 17, it
is found that
0, because mean of y is 0 by definition
0, assuming thin tails for pe(y)
41
THE ENTRAINMENT FORMULATION FOR SEDIMENT
CONTINUITY OF SLIDE 9 CAN BE RECOVERED FROM THE
PROBABILISTIC FORMULATION
Lets integrate the equation for conservation of
bed sediment presented in Slide 35 over the whole
range of possible relative levels y
1
1
1
This yields
0
Applying integration by parts to the term on the
LHS of the expression above, and using the
relation between PS(y) and pe(y) of Slide 17, it
is found that
then,
0, because mean of y is 0 by definition
0, assuming thin tails for pe(y)
42
TIME EVOLUTION EQUATION FOR PS(y)
Substituting the entrainment formulation of bed
sediment conservation,
into the relation of Slide 35,
and reducing, an equation for the time evolution
of PS(y) is obtained
The first term in brackets on the RHS of the
equation above indicates that when net deposition
occurs at relative level y, the amount of
sediment pores at that level increases, a
physically reasonable result. The second term in
brackets on the RHS is less intuitive in its
interpretation. Recalling the relation between
pe(y) and PS(y) in Slide 17, it represents the
(vertical) advection of mass due to overall bed
aggradation or degradation.
43
CASE OF EQUILIBRIUM BEDLOAD TRANSPORT
The following conditions hold for equilibrium
bedload transport in which case reduces
to and reduces to This justifies the
statement made at the bottom of Slide 29.
44
PROBABILISTIC FORMULATION OF MASS CONTINUITY FOR
TRACER STONES IN BED DEPOSIT
Making use again of a boundary-attached control
volume, lets now derive the sediment continuity
equation for the fraction of tracer stones in the
bed deposit at vertical position y, fb
fb(x,y,t) 1 (depicted by the sum of the green ?
blue solid squares)
where ftr ftr(x,t) 1 denotes the fraction of
tracer stones in bedload transport (depicted by
the sum of the red solid squares). Reducing,
45
REDUCTION WITH THE HELP OF THE TIME EVOLUTION
EQUATION FOR PS(y)
Expanding the conservation equation for tracer
stones in the bed deposit presented in the
previous slide, and using the relation between
PS(y) and pe(y) of Slide 17
46
REDUCTION WITH THE HELP OF THE TIME EVOLUTION
EQUATION FOR PS(y)
Expanding the conservation equation for tracer
stones in the bed deposit presented in the
previous slide, and using the relation between
PS(y) and pe(y) of Slide 17
according to the last equation of Slide 35
47
REDUCTION WITH THE HELP OF THE TIME EVOLUTION
EQUATION FOR PS(y)
Expanding the conservation equation for tracer
stones in the bed deposit presented in the
previous slide, and using the relation between
PS(y) and pe(y) of Slide 17
according to the last equation of Slide 35
Making the substitution indicated and cancelling
out terms, it is found that
The expression above captures an interesting
physical process that is not possible to describe
with the channel-averaged formulation. It is
represented by the convective term on the LHS,
which implies that changes in the composition of
the bed deposit are not only due to the direct
effect of overall bed aggradation or degradation,
but also due to the interaction between bed level
change and the vertical variation of the
background stratigraphy of the deposit.
48
PROBABILISTIC FORMULATION OF MASS CONTINUITY FOR
TRACER STONES IN BEDLOAD TRANSPORT
In an analogous form, lets proceed with the
formulation of the sediment continuity equation
for the fraction of tracer stones in the bedload
layer, ftr
Note that the term in brackets on the RHS
accounts for the total rate of entrainment of bed
tracers into bedload transport from any relative
level y. Reducing,
Expanding the conservation equation above, and
assuming that the time variation of x can be
neglected
49
WITH THE HELP OF THE EXNER AND ENTRAINMENT
FORMULATIONS
The equivalent formulations of sediment
continuity for the bed deposit presented in
Slides 5 and 7 are
Replacing this in the last equation of the
previous slide and cancelling out terms on its
RHS, it is found that
Note that in this case the convective term on the
LHS accounts for the streamwise imbalance in the
amount of tracer stones transported.
50
WITH THE HELP OF THE EXNER AND ENTRAINMENT
FORMULATIONS
The equivalent formulations of sediment
continuity for the bed deposit presented in
Slides 5 and 7 are
Replacing this in the last equation of the
previous slide and cancelling out terms on its
RHS, it is found that
Note that in this case the convective term on the
LHS accounts for the streamwise imbalance in the
amount of tracer stones transported.
Predictors for two additional variables are
needed in order to compute the time evolution of
the tracer stones displacement patterns these
variables are the volume bedload transport rate
qb, and the volume concentration of bedload per
unit bed area x.
51
PREDICTOR FOR qb
The following relation for qb has been derived
empirically from the results of the 10
equilibrium states for which runs with tracer
stones were conducted, combined with an
additional dataset of 20 other experimentally
obtained equilibrium states
in which, for normal flow conditions in a
hydraulically wide open channel
where H is the water depth L, and S is the
streamwise bed slope 1. (See Slide 14 of
Chapter 5.)
Empirical bedload transport relation based on
experiments at SAFL
52
PREDICTOR FOR x
The setup of our experiments did not allow the
direct measurement of x. We can indirectly
derive, however, a predictor for this variable.
The two transport relations of Slide 13 can be
written in the dimensionless forms where,
The reader is reminded that in the above
relations vb mean velocity of moving bedload
particles, vv virtual velocity of particles
including periods in motion and periods at rest,
and La active layer thickness.
53
PREDICTOR FOR x contd.
Fernández Luque and van Beek (1976), who
performed experiments similar to ours, proposed
the following relation to estimate the mean
velocity vb of a moving bedload particle
As shown in the plot to the right, this relation
can be accurately approximated by the
form Between this equation, the bedload
transport equation of Slide 51 and the
conservation equation , it is found
that
54
STREAMWISE DISPLACEMENT AND VIRTUAL VELOCITY OF
TRACER STONES
The results of 60 out of the 80 flume runs with
tracer stones referred to in Slide 23 served an
additional purpose to derive a predictor for
virtual particle velocity. After completing an
experiment for a given equilibrium state and test
duration, four groups were identified (according
to number of particles) for every color of tracer
stones (i) particles that did not move, Nop
(ii) particles that moved all the way out of the
flume, Ntr (iii) displaced particles that were
found at the bed surface, Nfs and, (iv)
displaced particles that were found buried in the
bed deposit, Nfb. Recall from Slide 26 that the
tracer stones that moved down to the sediment
trap were captured there, so they were not
allowed to make more than one loop. In all
experiments travel distances were recorded for
all particles in groups Nfs and Nfb. An
elevation-specific average (truncated) particle
travel distance could then be computed.
Final position of displaced tracer stones
55
DISCRIMINATION BY VERTICAL LOCATION OF SEEDING
Estimated mean travel distances (including
particles that ran out of the flume) for test
durations of 120 minutes. Depicted information is
discriminated by the initial vertical location of
the tracer stones, as well as a function of
excess Shields number (t - 0.055).
Calculations can be greatly simplified, however,
by assuming that for a given equilibrium state
the estimates of virtual particle velocity are
independent of the vertical seeding of the tracer
stones. This is a reasonable consideration since
the influence of initial vertical location is
already accounted for in the calculation of
elevation-specific particle entrainment rates
into bedload motion.
56
TRUNCATED DISTRIBUTIONS OF TRAVEL DISTANCES
The immediate question that arises is how to
estimate the scaled-up mean travel distance for
those particles that ran out of the flume (i.e.
group Ntr). This problem can be approached by
means of the following assumption for a given
equilibrium state not only the mean virtual
particle velocity vv but also its probability
density function should be invariant to the
duration of the experiment (Stedinger and Cohn,
1986). In other words, the information recorded
in a short-duration run for which all particles
that moved were in groups Nfs and Nfb (no
truncated distribution) should allow the
extrapolation of the distribution of travel
distances for a long-duration run (in which some
of the particles moved were in group Ntr) with
water discharge and sediment supply conditions
equal to those used in the short-duration
run. Such an analysis is underway, but
preliminary results are promising. In the
succeeding slides, vv denotes mean virtual
velocity, vv' L/T denotes any given virtual
velocity, rv vv'/vv 1, and Pv(rv) 1 denotes
the probability density function of rv. The goal
here is to demonstrate that Pv(rv) is independent
of the duration of a run.
57
TRUNCATED DISTRIBUTIONS OF TRAVEL DISTANCES contd.
In any given run a distribution of virtual
velocities was obtained from measurements of
tracer displacement. The data allowed
construction of the probability density function
Pv(rv), which is defined so that Pv(rv) x ?rv
denotes the probability that the normalized
virtual velocity rv ( vv'/vv) falls in the range
from rv to rv ?rv. It is interesting to see in
the diagram to the right, that the function
Pv(rv) appears to be independent of the excess
Shields number (? - 0.055). This is found for
the experiments of shortest duration, in which
the number of particles that ran out of the flume
(Ntr) was no more than 10 percent that of the
total displaced (Nfs Nfb Ntr).
Probability density function Pv of normalized
virtual travel velocity rv for runs of shortest
duration. The values of excess Shields number (?
- 0.055) are given in the legend. Note that Pv
appears to be independent of (? - 0.055)!
58
TRUNCATED DISTRIBUTIONS OF TRAVEL DISTANCES contd.
Again denoting rv vv'/vv, the probability of
non-exceedance Pne of any normalized travel
distance rv is given as where trun
specifically denotes the duration of the run from
which the data were collected. If Pne and thus Pv
are invariant with respect to run time, then it
follows that the following should hold for any
two run times trun1 and trun2 The results to
date do indeed indicate this invariance, as
illustrated by the plot to the left.
The probability of non-exceedance Pne is plotted
against virtual velocity vv for two runs of
different duration but at the same Shields
number. The correspondence indicates invariance
to run time.
59
PREDICTOR FOR vv
Estimates of virtual velocity vv can be obtained
in the following simple form
where Ltd(t) L represents the mean travel
distance for a test duration of time t, including
not only the measured values for particles that
stayed in the flume (i.e. groups Nfs and Nfb),
but also scaled-up estimates for particle that
ran out of the flume (i.e. group Ntr). A
predictor for mean virtual velocity was developed
using short-duration runs satisfying the
criterion that no more than 10 percent of the
tracers that were displaced ran out of the flume.
The probability analysis presented in the
previous three slides allowed reasonably accurate
estimation of vv even though some particles ran
out of the flume. The preliminary result obtained
is given below
60
IS THIS REALLY A GOOD PREDICTOR?
From Slide 52
Furthermore, from Slides 51 and 21, respectively
and,
A reasonable assumption concerning the thickness
of the active layer La is that it should vary
linearly with the standard deviation sy of bed
elevation fluctuations. Thus where ? is a
constant, it is assumed that Substituting the
above relation for into the relation above it
for , and the predictor for presented in
the previous slide, a new predictor for q is
found
61
IS THIS REALLY A GOOD PREDICTOR? contd.
The two bedload equations and were derived
from very different considerations. The first
relation was derived from direct measurements of
bedload transport. The second relation was
derived from a) measurements of particle virtual
velocity, b) measurements of the standard
deviation of bed elevation fluctuations and c)
the assumption La ?sy. Up to the very small
difference in exponents, the equations are in
agreement for the evaluation The above equation
provides the first objective evaluation of active
layer thickness that specifically indicates that
it should increase with increasing Shields number!
62
TRACER DISPERSAL FOR EQUILIBRIUM BEDLOAD TRANSPORT
Under morphodynamic equilibrium transport
conditions
and
then,
The conservation equation from Slide 47 for the
tracer stones in the bed deposit, fb, becomes
And the conservation equation from Slide 49 for
the tracer stones in bedload transport, ftr,
reduces to the following
With the aid of the predictors developed for
pe(y), pEnt(y), sy, qb, Eb and ?, one can solve
numerically the two equations above to determine
the fraction of tracers in the bed fb(x, y, t)
and the fraction of tracers in the moving bedload
ftr(x, t).
63
TRACER DISPERSAL FOR EQUILIBRIUM BEDLOAD
TRANSPORT contd.
The initial conditions on fb and ftr are For
example, ftrI might be set equal to zero (no
tracers in the bedload initially) and fbI might
be set so that only a specific area of the bed is
seeded with tracers to a specific depth. The
upstream condition on ftr is where ftrF denotes
the fraction of tracers in the feed. Once fb and
ftr are solved, it is possible to determine the
statistics of longitudinal dispersion of tracers.
For example, the mean streamwise travel distance
and the standard deviation of travel
distance are given as,
64
REVISITING THE ACTIVE LAYER FORMULATION
The numerical model will be presented in a
revised version of this chapter. It is of value,
however, to rearrange the formulation presented
here to an equivalent of the active layer model
of Chapter 4. And more importantly, this can be
done without losing generality for the
equilibrium case when applied to bed sediment of
uniform size. Lets begin by considering that
for a given equilibrium state, there is a
maximum depth of scour Lna L, measured with
respect to the mean bed elevation h. Then,
or,
In an analogous way,
and
65
REVISITING THE ACTIVE LAYER FORMULATION contd.
Recalling the conservation equation for tracer
stones in the bed deposit presented in Slide 47
Integrating it in y, applying integration by
parts and using the relation between PS(y) and
pe(y) of Slide 17, it is found that
66
REVISITING THE ACTIVE LAYER FORMULATION contd.
Recalling the conservation equation for tracer
stones in the bed deposit presented in Slide 47
Integrating it in y, applying integration by
parts and using the relation between PS(y) and
pe(y) of Slide 17, it is found that
1
0
Then,
67
REVISITING THE ACTIVE LAYER FORMULATION contd.
The essence of the active layer formulation is
the supposition of a mixed layer near the surface
with no vertical structure. With this in mind,
the following approximation is introduced
Substituting this into
yields the form
68
REVISITING THE ACTIVE LAYER FORMULATION contd.
The essence of the active layer formulation is
the supposition of a mixed layer near the surface
with no vertical structure. With this in mind,
the following approximation is introduced
Substituting this into
yields the form
1
1
1
to finally obtain
69
HERE IS THE EQUIVALENT FORMULATION
Lets define the average thickness of sediment
above y Lna as La L, which becomes equivalent
to the active layer thickness. Note that La now
depends on PS(y), which in turn is a function of
the magnitude of the driving force (that is, of
excess Shields number)
Lets also define the variable fI 1 to
represent the fraction of tracer stones at the
maximum depth of scour Lna
Replacing these two definitions in the last
equation of the previous slide
This expression above accounts for two different,
not necessarily related factors controlling the
fraction of tracers in the active layer (i)
changes in the composition of the sediment supply
material, and (ii) overall bed aggradation or
degradation.
70
AND FOR THE TRACER STONES IN BEDLOAD TRANSPORT
Recalling now the conservation equation for
tracer stones in bedload transport presented in
Slide 49, and considering that integration is up
to y Lna only
1
Hence,
which can be solved together with a re-arranged
expression for the tracer stones in the bedload
deposit (after substituting the entrainment
formulation of sediment continuity from Slide 7
in the last equation of the previous slide)
While the above active layer formulation is more
primitive than the continuous structure of e.g.
Slide 62, the analysis illustrates that the two
formulations are closely related to each other.
71
REFERENCES FOR CHAPTER 28
Blom, A., 2003. A vertical sorting model for
rivers with non-uniform sediment and dunes. PhD
thesis, University of Twente, the Netherlands,
267 pp. DeVries, P., 2000. Scour in low gradient
gravel bed streams Patterns, processes, and
implications for the survival of salmonid
embryos. PhD thesis, University of Washington,
Seattle, 365 pp. Einstein, H.A., 1950. The
bed-load function for sediment transportation in
open channel flows. Technical Bulletin No. 1026,
U.S. Department of Agriculture, SCS, Washington,
D.C., 78 pp. Ferguson, R.I. Hoey, T.B., 2002.
Long-term slowdown of river tracer pebbles
Generic models and implications for interpreting
short-term tracer studies. Water Resources
Research, doi10.1029/2001WR000637. Hassan, M.A.
Church, M., 2000. Experiments on surface
structure and partial sediment transport on a
gravel bed. Water Resources Research, 36(7),
1885-1895. Hirano, M., 1971. River bed
degradation with armouring. Transactions Japan
Society of Civil Engineering, 195,
55-65. Meyer-Peter, E. Müller, R., 1948.
Formulas for bed-load transport. Proc. 2nd
Meeting IAHR, Stockholm, Sweden, 39-64. Parker,
G., Paola, C. Leclair, S., 2000. Probabilistic
Exner sediment continuity equation for mixtures
with no active layer. Journal Hydraulic
Engineering, 126(11), 818-826.
72
REFERENCES FOR CHAPTER 28 cntd.
Stedinger, J.R. Cohn, T.A., 1986. Flood
frequency analysis with historical and paleoflood
information. Water Resources Research, 22(5),
785-793. Wong, M. Parker, G., 2005. Flume
experiments with tracer stones under bedload
transport. Proc. River, Coastal and Estuarine
Morphodynamics, Urbana, Illinois, 131-139.