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CHAPTER 28: TRACER STONES MOVING AS BEDLOAD IN GRAVEL-BED STREAMS

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Let’s integrate the equation for conservation of bed sediment presented in Slide 35 over the whole range of possible relative levels y :

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)

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.

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.

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.)

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.

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.

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.

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.

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.

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 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.

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.

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?

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).

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,

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.

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.

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.

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:

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):

All water

All sediment + pores

and that PS(y) is a monotonically increasing function ranging from 0 to 1.

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:

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.

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.

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).

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:

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

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.

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!).

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.

“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.

“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.

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

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 > 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).

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.

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:

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.

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

–

=

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.

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:

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:

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.

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 :

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 :

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:

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 :

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, assuming

“thin” tails for pe(y)

0, because mean

of y is 0 by definition

THE ENTRAINMENT FORMULATION FOR SEDIMENT CONTINUITY OF SLIDE 9

CAN BE RECOVERED FROM THE PROBABILISTIC FORMULATION

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, assuming

“thin” tails for pe(y)

0, because mean

of y is 0 by definition

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.

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.

PROBABILISTIC FORMULATION OF MASS CONTINUITY

FOR TRACER STONES IN BED DEPOSIT

Making use again of a boundary-attached 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):

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,

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:

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

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.

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:

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:

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.

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.

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

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.

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

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

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.

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.

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)!

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.

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:

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:

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!

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).

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,

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,

or,

In an analogous way,

and

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:

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,

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

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:

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):

Let’s 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.

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.

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.

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.

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