1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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 experiment
at St. Anthony Falls Laboratory (SAFL)
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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.
1.000
Original relation of MPM
0.100
*
qb
Amended version of MPM
bedload transport relation
ETH - Dm = 28.65 mm
ETH - Dm = 5.21 mm
GIL - Dm = 3.17 mm
GIL - Dm = 4.94 mm
GIL - Dm = 7.01 mm
Equation (2)
Equation (22)
0.010
0.001
0.01
0.10
1.00
q*b [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 t*c [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.)
t -tc
*
*
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.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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]:
h
qb
1  p   
t
x
where p [1] denotes bed porosity, and t [T] represents
time.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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]:
h
qb
1  p   
t
x
where p [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:

1  p  
t 

x h dx 

x  x
x  x
b
h
1
b
x
water
D(x)
bed sediment + pores
x  x
 D x  dx   E x  dx
x
E(x)
x
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.
x
x + x
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
ENTRAINMENT FORMULATIONS
The last equation of Slide 6 reduces in the limit as x → 0 to:
h
1  p   Db  Eb
t
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.
h
qb
1  p   
t
x
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
ENTRAINMENT FORMULATIONS
The last equation of Slide 6 reduces in the limit as x → 0 to:
h
1  p   Db  Eb
t
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.
h
qb
1  p   
t
x
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:
 qb

 Eb  Db
t x
where  [L] is the volume concentration of bedload
per unit bed area.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
ENTRAINMENT FORMULATIONS
The last equation of Slide 6 reduces in the limit as x → 0 to:
h
1  p   Db  Eb
t
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.
h
qb
1  p   
t
x
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:
 qb

 Eb  Db
t x
where  [L] is the volume concentration of bedload
per unit bed area.
The term ∂/∂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.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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.
Fi
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.
f Ii
La
h
'
f i (x, z', t)
z'
Datum
x
Eb(x, h-La < z' < h) = constant
La
h
Datum
x
z'
10
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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?
11
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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 << vb, implying that a particle spends most of its
time at rest (even during equilibrium transport).
mean bed
elevation
A particle can deposit in the
surface, stay there, and later
be re-entrained into motion
x
Or, a particle can deposit in the
surface, get buried, be reexposed at the surface, and later
be re-entrained into motion
La
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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 << 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,
qb  vb
, qb  Lav v
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.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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.
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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).
instantaneous bed elevation
mean bed elevation
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.
15
rte-bookbedload.mpg: to run without relinking, download to same folder as PowerPoint presentations.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
HOW TO HANDLE THESE VARIATIONS?
water
h
y
sediment + pores
z'
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:
y  h  z'
Let’s 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):
PS y    0
PS y    1
All water
All sediment + pores
and that PS(y) is a monotonically increasing function ranging from 0 to 1.
16
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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:
dPS y 
pe y  
dy
which by definition must satisfy:
0

pe(y)
y (+)
 p y  dy  1
Ps(y)
e

0
1
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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).
Experiments at SAFL:
foreground – Pulser + computer
used for data acquisition and
processing; background – cables
and metal frames for ultrasonic
probes in flume.
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.
18
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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).
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.
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.
19
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
TIME SERIES OF BED ELEVATION
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.
(a)
420
400
Bed elevation (mm)
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.
380
360
340
320
300
0
600
1200
1800
2400
3000
3600
Time (sec), with measurements once every 3 sec
Probe 1
(b)
Probe 3
Probe 4
Probe 5
Probe 6
420
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).
Bed elevation (mm)
400
380
360
340
320
300
0
600
1200
1800
2400
3000
3600
Time (sec), with measuremens once every 3 sec
Probe 1
Probe 3
Probe 4
Probe 5
Probe 6
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
NORMAL PROBABILITY MODEL
1.0
Working with the aggregated time series
of bed elevation fluctuations y, an
empirical cumulative distribution was
constructed for each equilibrium state.
Probability of non-exceedance
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
Empirical
0.1
0.0
-18.0
Theoretical Normal
-14.4
-10.8
-7.2
-3.6
0.0
3.6
7.2
10.8
14.4
18.0
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):
Level relative to mean bed elevation, y (mm)
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:
pe y  
sy
D50
2

1
1  y  

exp 
2s y
 2  s y  





 3.09 t  0.055
0.56
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1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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:

 p y dy  1
Ent
such that
Eb pEnt y y
such that
Db pDep y y

and,

 p y  dy  1
Dep

= entrainment rate
from level y to
level y + y
= deposition rate at
level y to level
22
y + y
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
ENTRAINMENT OF TRACER STONES
A total of 80 flume runs with tracer stones were
conducted under conditions of plane-bed lowerregime 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.
23
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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!).
24
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
ENTRAINMENT PER LAYER
(a)
100%
Percent moved
80%
60%
40%
20%
0%
0
20
40
60
80
100
120
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.
Duration of experiment (min)
top
(b)
second
third
bottom
100%
Percent moved
80%
60%
40%
20%
0%
0
20
40
60
80
100
Duration of experiment (min)
top
second
third
bottom
120
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
25
(a) to (b), as well as with experiment duration.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
“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:
 fbts t 2 
Ltr
Eb  
 ln 
t 2  t1  fbts t1 
t* - 0.055
0.90
Fraction of tracers after run completed
d
fbts t  L tr   Eb fbts t 
dt
1.00
0.0209
0.0211
0.0294
0.0343
0.0359
0.0366
0.0495
0.0497
0.0503
0.0644
0.80
0.70
0.60
0.50
0.40
0.30
0
20
40
60
80
100
120
Duration of experiment (min)
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.
26
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
“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:
d
fbts t  L tr   Eb fbts t 
dt
Solving this ODE for Eb results in:
 fbts t 2 
Ltr
Eb  
 ln 
t 2  t1  fbts t1 
28 different combinations of fbts(t)-pairs
have been used to estimate the value of
Eb for each experimental equilibrium state.
The values of Eb determined in this way
were found to correlate with D50 and
excess Shields number (t* - 0.055) as
follows:


1.85
Eb

 0.05 t  0.055
RgD50
where R is the submerged specific
gravity of the bed sediment [1], and g is
the acceleration of gravity [L/T2] 27
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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:
pEnt y   pB ( y  ybe )
pDep ( y)  pB ( y  ybd )
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
ybe  y0  y1
ybd  y0  y1
28
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
ENTRAINMENT AND DEPOSITION FUNCTIONS contd.
The assumptions of the previous slide thus give the following relations:
pEnt y   pB (y  y0  y1)
pDep (y)  pB (y  y0  y1)
Here y0 is an offset common to both functions, which
could be greater than or less than 0. The case y0 > 0
biases both functions downward below the mean bed
elevation. The case y1 > 0 biases erosion upward in
the bed, and deposition downward in the bed (see
sketch to the right).
pEnt(y) > pDep(y)
mean bed
elevation h
instantaneous
relative level y
pDep(y) > pEnt(y)
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.
pEnt y  pDep (y)
so that y1  0 as equilibrium is approached. This point is illustrated in more detail
in subsequent slides.
29
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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:
1.0E-04




predicted
Entrainment rate, Eb (m/s)
 y  y0  y1
1
pEnt y  
exp 

2s y
sy

measured
1.0E-05
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).
Relative level from mean bed elevation, y (mm)
Moreover, according to the above equation pEnt(y)
depends on the standard deviation sy of bed
Exponential fitting for a
elevation fluctuations, and hence correlates with sample equilibrium test.
excess Shields number (t* - 0.055) (Slide 21).
1.0E-06
1.0E-07
0
Setting ybd = y0 + y1, the corresponding
form for the deposition function pDep(y) is:
5
10
15
20
 y  y0  y1 
1

pDep y  
exp 


2s y
sy


25
30
30
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
Level relative to mean bed elevation
THE STORY SO FAR
pEnt(y)
pDep(y)
y - y0 - y1
y
y - y0 + y1
pe(y)
Probability density function
31
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
Level relative to mean bed elevation
THE STORY SO FAR
pEnt(y)
 y  y0  y1
1
pEnt y  
exp 

2s y
sy

pDep(y)
y - y0 - y1
y
 y  y0  y1 
1

pDep y  
exp 


2s y
sy


y - y0 + y1
pe y  
pe(y)




2



1
1 y  

exp 
2s y
 2  s y  


Probability density function
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 32
y0 and y1.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
PROBABILISTIC FORMULATION OF MASS CONTINUITY
FOR SEDIMENT IN THE BED DEPOSIT
As indicated in Slide 16, the control
volume (strip with fill) is boundaryattached to the mean bed elevation
h, which is free to move up or
down in time.
y
y + y
y
y + y
h(t + t)
The conservation equation for the
sediment in the bed deposit within
any layer from y to y + y can then
be expressed as follows:
Time rate of change of
Flux of mass going into
=
mass in control volume
the control volume
h(t)
–
Flux of mass going out
from the control volume
33
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
PROBABILISTIC FORMULATION OF MASS CONTINUITY
FOR SEDIMENT IN THE BED DEPOSIT
As indicated in Slide 16, the control
volume (strip with fill) is boundaryattached to the mean bed elevation
h, which is free to move up or
down in time.
y
y + y
y
y + y
h(t + t)
The conservation equation for the
sediment in the bed deposit within
any layer from y to y + y can then
be expressed as follows:
1     P y y  D
p
t
S
h(t)
Note how the stone (solid circle) is moved out
of the control volume as the control volume is
advected upward
b pDep y   Eb pEnt y  y  1   p 

h
PS y  y  PS y  y y
t

Apparent “convective” transfer as a result
of moving from the green to the blue strip.
34
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
DOES PS(y) HAVE TO BE STATIONARY?
The conservation equation for sediment in the bed deposit presented in the
previous slide reduces to:
 PS y h PS y
1 p  

  Db pDep y  Eb pEnt y
t
y 
 t
Further reducing with the relation between PS(y) and pe(y) of Slide 17:
1     P y  h p y  D
S
p

t
t
e

b
pDep y   Eb pEnt y 
35
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
DOES PS(y) HAVE TO BE STATIONARY?
The conservation equation for sediment in the bed deposit presented in the
previous slide reduces to:
1    P y  h P y  D
S
p

t
S
t
y
b

pDep y  Eb pEnt y
Further reducing with the relation between PS(y) and pe(y) of Slide 17:
1     P y  h p y  D
S
p

t
t
e

b
pDep y   Eb pEnt y 
0 ???
The expression above could be simplified
h Db pDep y  Eb pEnt y
more if PS(y) is assumed to be stationary
1  p

t
pe y
(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
36
condition is satisfied: pe y  pEnt y  pDep y


1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
DOES PS(y) HAVE TO BE STATIONARY?
The conservation equation for sediment in the bed deposit presented in the
previous slide reduces to:
1    P y  h P y  D
S
p

S
t
t
y
b

pDep y  Eb pEnt y
Further reducing with the relation between PS(y) and pe(y) of Slide 17:
1     P y  h p y  D
S
p

t
t
0 ???
e

b
pDep y   Eb pEnt y 
pe y  pEnt y  pDep y
This is not only a very restrictive assumption for cases of nonequilibrium 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.
37
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
THE ENTRAINMENT FORMULATION FOR SEDIMENT CONTINUITY OF SLIDE 9
CAN BE RECOVERED FROM THE PROBABILISTIC FORMULATION
Let’s integrate the equation for conservation of bed sediment presented in Slide 35
over the whole range of possible relative levels y :

 PS y 

h
1  p   
dy 
pe y  dy  Db

t 
 t



 p y dy  E  p y dy
Dep

b
Ent

38
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
THE ENTRAINMENT FORMULATION FOR SEDIMENT CONTINUITY OF SLIDE 9
CAN BE RECOVERED FROM THE PROBABILISTIC FORMULATION
Let’s integrate the equation for conservation of bed sediment presented in Slide 35
over the whole range of possible relative levels y :

 PS y 

h
1  p   
dy 
pe y  dy  Db

t 
 t


 p y dy  E  p y dy
Dep
b

1
This yields:

Ent

1
1
  PS y 
h 
1  p   
dy    Db  Eb
t 
  t
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:
PS y 

 PS y 
dy

y

y pe y  dy


 t

t   t 




39
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
THE ENTRAINMENT FORMULATION FOR SEDIMENT CONTINUITY OF SLIDE 9
CAN BE RECOVERED FROM THE PROBABILISTIC FORMULATION
Let’s integrate the equation for conservation of bed sediment presented in Slide 35
over the whole range of possible relative levels y :

 PS y 

h
1  p   
dy 
pe y  dy  Db

t 
 t


 p y dy  E  p y dy
Dep
b

1
This yields:

Ent

1
1
  PS y 
h 
1  p   
dy    Db  Eb
t 
  t
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:
PS y 

 PS y 
dy

y

y pe y  dy


 t

t   t 



0, assuming
“thin” tails for pe(y)

0, because mean
of y is 0 by definition
40
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
THE ENTRAINMENT FORMULATION FOR SEDIMENT CONTINUITY OF SLIDE 9
CAN BE RECOVERED FROM THE PROBABILISTIC FORMULATION
Let’s integrate the equation for conservation of bed sediment presented in Slide 35
over the whole range of possible relative levels y :

 PS y 

h
1  p   
dy 
pe y  dy  Db

t 
 t

This yields:

 p y dy  E  p y dy
Dep
  PS y 
h 
1  p   
dy    Db  Eb
t 
 t
b

1
0

Ent

1

1
1    h  D
p
t
b
 Eb
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:
PS y 

 PS y 
dy

y

y pe y  dy


 t

t   t 



0, assuming
“thin” tails for pe(y)

PS y 
 t dy  0

then,
0, because mean
of y is 0 by definition
41
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
TIME EVOLUTION EQUATION FOR PS(y)
Substituting the entrainment formulation of bed sediment conservation,
1    h  D
p
t
b
 Eb
into the relation of Slide 35,
1     P y  h p y  D
S
p

t
t
e

b
pDep y   Eb pEnt y 
and reducing, an equation for the time evolution of PS(y) is obtained:
1    P y  D
S
p
t
b
pDep y   Eb pEnt y   Db  Eb  pe y 
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
42
aggradation or degradation.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
CASE OF EQUILIBRIUM BEDLOAD TRANSPORT
The following conditions hold for equilibrium bedload transport:
h
0 ,
t
1    h  D
in which case
and
p
t
b
1    P y  D
S
p
t
b
 Eb
PS y 
0
t
reduces to
Db  Eb
pDep y   Eb pEnt y   Db  Eb  pe y 
reduces to
pDep y  pEnt y
This justifies the statement made at the bottom of Slide 29.
43
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
PROBABILISTIC FORMULATION OF MASS CONTINUITY
FOR TRACER STONES IN BED DEPOSIT
Making use again of a boundaryattached control volume, let’s 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):
1     P y f
p
t
S
b
y
y + y
y + y
h(t + t)
h(t)
y  Db ftr pDep y   Eb fb pEnt y  y
 1  p 

h
PS y  fb y  PS y  fb y  y
t

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,
1     P y f   h P y f   D
S
p

t
b
S
t
y
y
b

f pDep y   Eb fb pEnt y 
b tr
44
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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:
 fb h fb 
1  p  PS y   
  1   p  fb

t

t

y


 Db ftr pDep y   Eb fb pEnt y 
 PS y  h


pe y 

t
 t

45
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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:
 fb h fb 
1  p  PS y   
  1   p  fb

t

t

y


 Db ftr pDep y   Eb fb pEnt y 
 PS y  h


pe y 

t
 t

Db pDep y  Eb pEnt y
according to the last
equation of Slide 35
46
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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:
 fb h fb 
1  p  PS y   
  1   p  fb

t

t

y


 Db ftr pDep y   Eb fb pEnt y 
 PS y  h


pe y 

t
 t

Db pDep y  Eb pEnt y
according to the last
equation of Slide 35
Making the substitution indicated and cancelling out terms, it is found that:
 fb h fb 
1 p PS y  
  Db ftr  fb  pDep y
 t t y 
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 47
vertical variation of the “background” stratigraphy of the deposit.
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
PROBABILISTIC FORMULATION OF MASS CONTINUITY
FOR TRACER STONES IN BEDLOAD TRANSPORT
In an analogous form, let’s proceed with the formulation of the sediment continuity
equation for the fraction of tracer stones in the bedload layer, ftr:
 


 ftr x   qb ftr x  qb ftr x x    Eb fb pEnt y dy x  Db ftr x
t
 -

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,
  ftr   qb ftr 

  Eb fb pEnt y  dy  Db ftr
t
x
-

Expanding the conservation equation above, and assuming that the time variation of
 can be neglected:

f
f
q
 tr  qb tr  ftr b   Eb fb pEnt y  dy  Db ftr
t
x
x -
48
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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:
1    h   q
b
p
t
x
 Db  Eb
Replacing this in the last equation of the previous slide and cancelling out terms on
its RHS, it is found that:
 

ftr
ftr

 qb
 Eb   fb pEnt y  dy  ftr 
t
x
 -

Note that in this case the convective term on the LHS accounts for the streamwise
imbalance in the amount of tracer stones transported.
49
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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:
1    h   q
b
p
t
x
 Db  Eb
Replacing this in the last equation of the previous slide and cancelling out terms on
its RHS, it is found that:
 

ftr
ftr

 qb
 Eb   fb pEnt y  dy  ftr 
t
x
 -

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 .
50
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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:


1.50
qb

q 
 2.66 t  0.055
RgD50 D50

q*b
0.100
in which, for normal flow conditions in
a hydraulically wide open channel:
t 
0.010
HS
RD 50
where H is the water depth [L], and S
is the streamwise bed slope [1]. (See
Slide 14 of Chapter 5.)
0.001
0.01
0.10
t -tc
*
SAFL data
*
Line of best fit
Empirical bedload transport relation
based on experiments at SAFL
51
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
PREDICTOR FOR 
The setup of our experiments did not allow the direct measurement of . We can
indirectly derive, however, a predictor for this variable. The two transport relations
of Slide 13 can be written in the dimensionless forms:
q  ˆ vˆ b
where,
q 
qb
RgD 50 D50
L
Lˆ a  a
D50
,
vˆ v 
, q  Lˆ avˆ v

, ˆ 
D50
,
vˆ b 
vb
RgD 50
vv
RgD 50
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.
52
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
PREDICTOR FOR 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:
vˆ b 
vb
 11.5
RgD50
 t  0.7 t 


cr
10
As shown in the plot to the right, this relation
can be accurately approximated by the form


Fernandez Luque &
van Beek
0.45
vˆ b1
v
vˆ b  8.00 t  0.055
, tcr  0.0455
Between this equation, the bedload
transport equation of Slide 51 and the
conservation equation q  ˆ vˆ b, it is
found that


ˆ  0.33 t  0.055
1.05
0.1
0.01
Approximation
0.1

t .0455
t 0
53
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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
Final position of displaced
moved all the way out of the flume, Ntr; (iii) displaced
tracer stones
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.
54
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
DISCRIMINATION BY VERTICAL LOCATION OF SEEDING
16.0
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).
Travel distance (m)
12.0
8.0
4.0
0.0
0.01
0.10
excess Shields number
top
second
third
bottom
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.
55
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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 shortduration 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.
56
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
TRUNCATED DISTRIBUTIONS OF TRAVEL DISTANCES contd.
0.50
0.40
t* - 0.055
Frequency
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.
0.0644
0.0503
0.0497
0.0495
0.0366
0.0359
0.0343
0.0294
0.0211
0.0209
0.30
0.20
0.10
0.00
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 (t* - 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).
0.25
0.75
1.25
1.75
2.25
2.75
3.25
3.75
4.25
4.75
Dimensionless virtual particle velocity
Probability density function Pv of
normalized virtual travel velocity rv for
runs of shortest duration. The values of
excess Shields number (t* - 0.055) are
given in the legend. Note that Pv appears
to be independent of (t* - 0.055)!
57
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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
1.0
Probability of non-exceedance
0.9
0.8
0.7
0.6
rv
Pne (rv , trun )   Pv (~
rv , trun )d~
rv
0.5
0.4
0
0.3
0.2
0.1
0.0
0.0
3.0
6.0
9.0
12.0
Virtual particle velocity (cm/s)
Shortest duration
Double the shortest duration
The probability of nonexceedance 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.
15.0
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:
Pne (rv , trun1)  Pne (rv , trun 2 )
The results to date do indeed indicate this
invariance, as illustrated by the plot to the
left.
58
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
PREDICTOR FOR vv
Estimates of virtual velocity vv can be obtained in the following simple form:
vv 
Ltd t 2   Ltd t1 
t 2  t1
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:


0.90
vv

vˆ v 
 1.34 t  0.055
RgD50
59
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
IS THIS REALLY A GOOD PREDICTOR?
From Slide 52:
q  Lˆ a vˆ v
Furthermore, from Slides 51 and 21, respectively:


1.50
qb

q 
 2.66 t  0.055
RgD50 D50

and,
sˆ y 
sy
D50


 3.09 t  0.055
0.56
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
La  sy
or thus Lˆ a  sˆ y
Substituting the above relation for Lˆ a into the relation above it for sˆ y , and the
predictor for vˆ v presented in the previous slide, a new predictor for q* is found:


q  4.14 t  0.055
1.46
60
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
IS THIS REALLY A GOOD PREDICTOR? contd.
The two bedload equations


q  2.66 t  0.055
and

q
 4.14 t

1.50
 0.055
1.46
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
  0.642 or La  0.642sy  1.99(t  0.055)0.56
The above equation provides the first objective evaluation of active layer
thickness that specifically indicates that it should increase with increasing Shields
number!
61
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
TRACER DISPERSAL FOR EQUILIBRIUM BEDLOAD TRANSPORT
Under morphodynamic equilibrium transport conditions:
h
0
t
then,
qb
0
x
and
Db  Eb
The conservation equation from Slide 47 for the tracer stones in the bed deposit, fb,
becomes:
1    P y f
b
p
S
t
 Eb ftr  fb  pEnt y 
And the conservation equation from Slide 49 for the tracer stones in bedload
transport, ftr, reduces to the following:
 

ftr
ftr

 qb
 Eb ftr  Eb   fb pEnt y  dy
t
x
 -

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).
62
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
TRACER DISPERSAL FOR EQUILIBRIUM BEDLOAD TRANSPORT contd.
The initial conditions on fb and ftr are
fb (x, y, t) t0  fbI (x, y) , ftr (x, t) t0  ftrI(x)
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
ftr (x, t) x 0  ftrF(t)
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 x t (t) and
the standard deviation of travel distance t (t) are given as,

x t (t) 

 
 
0

 
0
xfb (x, y, t)dydx

fb (x, y, t)dydx
, 2t (t) 


0

 
(x  x t (t))2 fb (x, y, t)dydx


0

 
fb (x, y, t)dydx
63
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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.
Let’s 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,
Lna
 p y  dy  1
e
or,
pe y  Lna   0 , PS y  Lna   1

In an analogous way,
Lna
Lna
 p y  dy  1
Ent

and
 p y  dy  1
Dep

64
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
REVISITING THE ACTIVE LAYER FORMULATION contd.
Recalling the conservation equation for tracer stones in the bed deposit presented in
Slide 47:
 fb h fb 
1 p PS y  
  Db ftr  fb  pDep y
 t t y 
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:
Lna
 Lna f

 

h
b
1  p    PS y dy  PS y fb yLna  PS y fb y   fb pe y dy 
t 
  t
 

Lna
 Lna

 Db ftr  pDep y  dy   fb pDep y  dy
 


65
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
REVISITING THE ACTIVE LAYER FORMULATION contd.
Recalling the conservation equation for tracer stones in the bed deposit presented in
Slide 47:
 fb h fb 
1 p PS y  
  Db ftr  fb  pDep y
 t t y 
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:
Lna
 Lna f

 

h
b
1  p    PS y dy  PS y fb yLna  PS y fb y   fb pe y dy 
t 
  t
 


 Db ftr

1

pDep y dy  fb pDep y dy

Lna
Lna
0
Then,
L
Lna


h 
 na fb
1  p    PS y dy  PS y fb yLna   fb pe y dy  Db
t 

 

  t

Lna


ftr   fb pDep y  dy



66
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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:
fb (x, y, t)  fa (x, t) ,    y  Lna
Substituting this into
L
Lna


h 
 na fb
1  p    PS y dy  PS y fb yLna   fb pe y dy  Db
t 


 
  t

Lna


ftr   fb pDep y  dy



yields the form

1  p   fa

 t
Lna
Lna

h 

PS y dy  t PS y fb yLna  fa pe y dy   Db ftr  fa  pDep y dy


Lna
67
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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:
fb (x, y, t)  fa (x, t) ,    y  Lna
Substituting this into
L
Lna


h 
 na fb
1  p    PS y dy  PS y fb yLna   fb pe y dy  Db
t 


 
  t

yields the form

1  p   fa

 t
Lna


ftr   fb pDep y  dy



1
1
Lna
Lna



h







P
y
dy

P
y
f

f
p
y
dy
 S
   Db ftr  fa   pDep y dy
b y L
a  e
 S
na
t 
 



Lna
1
to finally obtain:

 fa
1  p  

 t

h
PS y dy  t fb yLna  fa
Lna



  Db ftr  fa 


68
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
HERE IS THE EQUIVALENT FORMULATION
Let’s 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):
L
La 
na
 P y  dy
S

Let’s also define the variable fI [1] to represent the fraction of tracer stones at the
maximum depth of scour Lna:
fI  fb y L
na
Replacing these two definitions in the last equation of the previous slide:
fa
h




f

f
a
I
a   Db ftr  fa 
t
 t

1    L
p
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. 69
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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
L
f
f
 tr  qb tr  Eb
t
x
 na

fa  pEnt y  dy  ftr 
 

Hence,

ftr
f
 qb tr  Eb ftr  Eb fa
t
x
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):
fa
1  p  La  Eb fa  Db ftr  Db  Eb  fI
t
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.
70
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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, doi:10.1029/2001WR000637.
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.
71
1D SEDIMENT TRANSPORT MORPHODYNAMICS
with applications to
RIVERS AND TURBIDITY CURRENTS
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.
72