Bed load transport and streambed structure in gravel streams
P. Diplas
Department of Civil and Environmental Engineering, Virginia Tech
Blacksburg, VA 24060, USA
Abstract
Phenomena related to gravel bed rivers continue to receive considerable attention.
This effort has been facilitated by the collection of extensive field data under
relatively wide range of flow, bed load transport and channel slope conditions over
the last decade. In some cases, these data have provided the opportunity to go
beyond bed load transport and boundary shear stress correlations and have enabled
researchers to examine the behavior of gravel streams within the context of
watershed processes and characteristics. Several of the gravel bed stream studies
published during the past five years have been analyzed and some of the
information has been synthesized in an effort to attain an improved understanding
about the interplay between sediment supply, armour layer development and bed
load transport. Although the development of the “general equation” of bed load
transport remains elusive, better comprehension about the complexity of the
problem, given the remarkable variability of prevailing field conditions, has been
obtained and improved understanding about overall trends and interaction of the
various processes has been attained. The present state of the subject provides a
sound basis for significant future developments.
Key words: perennial streams, ephemeral streams, armour layer, fractional
transport rate
1
Introduction
During the last three decades issues related to sediment transport and the structure
of channel beds in gravel streams have received considerable attention. Terms like
equal mobility and size-selective transport, pavement and armour layers, fractional
transport and single-diameter based bed load transport, surface and subsurface
based analyses have become commonplace in the gravel-bed rivers literature. The
interest generated in this subject has resulted in improved methods for collecting
bed load data and sampling bed material, which in turn provide much needed
reliable data from field and laboratory settings. These developments have been
instrumental in the evolution of ideas in this area, which are heavily based on the
interpretation of the trends exhibited by available data. As one might expect, the
limited range of conditions represented by each data set reveals part of the puzzle.
However, by now, there are a sufficient number of data sets to allow for the
consideration of the phenomena for a relatively wide range of flow, bed load
transport, and other important parameters. This enables us to approach the subject
from a more global perspective by putting together some of the available pieces and
speculate about some of the missing ones.
At first, the present study provides a brief account of some key contributions to the
sediment transport field that have had an impact in our understanding of gravel
bead stream phenomena made during the past two centuries and even earlier than
that. Next, a more detailed review of some of the recent developments in bed load
transport and composition of the streambed is reported. An effort to synthesize
existing information on gravel streams follows. Some trends of bed load transport
rates based on available field and laboratory data covering a wide range of Shields
stresses are examined and the behavior of the armour layer under changing flow
and sediment supply conditions is discussed.
2
Brief Historical Account
Some of the first insightful descriptions regarding gravel-bed river processes were
made about five hundred years by Leonardo, a keen observer of natural phenomena
and possibly the initiator of the use of experiments as a tool in scientific inquiry. In
his famous Codex Leicester he writes: “A river that flows from the mountains
deposits a great quantity of large stones in its bed … as it proceeds on its course it
carries down with it lesser stones with the angles more worn away, and so the large
stones make smaller ones; and farther on it deposits first coarse and then fine
gravel, and after this follows sand at first coarse and then more fine …” (Richter,
1998). Leonardo attributes downstream fining to abrasion and selective transport,
not much different than the explanations provided by contemporary researchers.
Another observation stated in the same codex, is with regard to sediment eroded in
the mountains, supplied to the river, and subsequently transported to the sea, an
indication about the linkage between rivers and their drainage basins.
The more systematic study of rivers started during the second half of the 19th
century. Until the 1970’s the emphasis, for good reason, was on sandy streams.
Gravel bed streams were not perceived as behaving sufficiently differently from
their sand-bed counterparts to deserve special attention. This perception was
reflected in the ASCE Manual 54 (Vanoni, 1975). Nevertheless, this period was
punctuated by a good number of important studies dealing with various aspects of
gravel-bed streams or which indirectly benefited the study of such streams. An
incomplete list of some of these contributions includes: the development of a bed
load transport model based on the use of tractive force (DuBoys, 1879), description
of downstream fining based on observations from the Alpine Rhine River
(Sternberg, 1875), Gilbert’s (1914) extensive data set based on flume experiments,
2
identification of imbricated structures (Johnston, 1922; Lane and Carlson, 1953),
use of the dimensionless boundary shear stress as a criterion for characterizing the
threshold of particle motion condition (Shields, 1936), development of a bed load
transport expression based on flume experiments with coarse and poorly-sorted
sediments (Meyer-Peter and Muller, 1948), recognition of the fluctuating nature of
turbulent forces near a rough boundary and its importance in particle entrainment
(Einstein and El-Samni, 1949), development of a fractional bed load transport
equation that accounts for hiding effects and is not based on the notion of excess
boundary shear stress (Einstein, 1950). Development of a static armour layer under
starved sediment input conditions in laboratory experiments carried out by Harrison
(1950), the grid-by- number method proposed by Wolman (1953) for sampling
sediment deposits, especially suitable for sampling armour layers, detailed
observations from Klaralven River regarding behavior of poorly sorted and
bimodal sediments, among other things, (Sundborg, 1956), documentation of the
absence of certain grain sizes, in the pea and pebble range, from gravel bed
sediment deposits (Yatsu, 1957), refinement of the threshold of motion criterion,
specifically obtained for poorly sorted sediments (Pantelopulos 1955 and 1957,
Neil 1968) (middle-size fractions move at dimensionless shear stress values
comparable to those required from uniform sediments), a new expression for
describing hiding effects in a sediment mixture (Egiazaroff, 1965), introduction of
the stream power concept in sediment transport (Bagnold, 1966), development of
the Helley-Smith bed load sampler (Helley and Smith, 1971), equivalency criteria
among the various methods employed for sampling bed material proposed by
Kellerhals and Bray (1971). New bed load transport formulae proposed by Ashida
and Michiue (1972) and Ackers and White (1973), and a high quality bed load
transport data set collected by Milhous (1973) from Oak Creek, a gravel-bed stream
in Oregon, USA, using a vortex bed load sampler.
During the 1980’s the number of studies focusing exclusively on gravel bedded
streams grew steadily. This is partly due to the interest generated by the
publications resulting from the Gravel-Bed Rivers Symposia and several key
papers, including those by Parker and Klingeman (1982) and Parker et al. (1982).
Extensive flume and field studies were undertaken to augment the available data
sets. The interest continues unabated to the present time, to the extent that currently
it has possibly surpassed the efforts devoted to the study of sandy streams. A recent
search of the Web of Science data base identified several hundred journal
publications dealing with gravel-bed stream issues. The subject area has matured.
The new ASCE Sedimentation Manual, No. 110, due to be published in 2006, will
include several chapters on gravel bottomed streams, reflecting the explosion of
knowledge acquired in this area over the last 30 years.
3
Recent Developments
Since the Gravel-Bed Rivers volumes provide extensive accounts of the
developments in this area, the present brief review will focus on the highlights of
the many works that were published during the last five years or so. Some of the
3
topics addressed over this time period deal with bed load transport issues,
headwater (high slope) streams, threshold, behavior of ephemeral streams, stream
dependence on drainage basin characteristics, partial transport, and evolution of
armor layer during floods. Most of these issues are intertwined.
Several new bed load formulae were proposed, while some older and well known
equations were tested against new and existing field and laboratory data. Almedeij
and Diplas (2003) used the Oak Creek (Milhous, 1973) and Nahal Yatir (Reid et al,
1995) data, representing the two extremes, very low and very high, respectively, of
bed load transport behavior to develop a two parameter equation, one based on
surface and the other on subsurface material characteristics. The mode of each
(surface and subsurface) population was chosen as a suitable parameter for
representing the corresponding sediment characteristics. This is intended for bed
load transport calculations of unimodal sediments over a wide range of boundary
shear stresses. Field data from three other streams were used to examine the
validity of the proposed equation.
Barry et al. (2004, 2005) proposed a simple power relation for bed load transport
based on total water discharge. They hypothesized that the exponent of this relation
depends on the degree of channel bed armouring, while the coefficient depends on
basin characteristics and the amount of sediment supplied to the stream. They used
a large number of field data, collected from 24 gravel-bed rivers in Idaho, to
parameterize the exponent and coefficient in terms of channel and watershed
characteristics, respectively. They suggested that the expression for the former
parameter should be generally valid from stream to stream while the latter will be
region specific, dependent upon local basin land use and physiography.
Subsequently, they compared their formula, together with four other well known
bed load equations, with data obtained from 17 stream sites located in Colorado,
Oregon, and Wyoming. The results are typical of similar comparisons; the
predicted values are within 2 to 3 orders of magnitude of the observed values.
Though the results regarding the predictive ability of the new and existing
equations are not very encouraging, the approach followed by the authors, to
consider river processes within the context of the characteristics of their basins,
appears to be in the right direction (e.g. Ryan et al, 2005). The role of sediment
supply to stream behavior, in terms of degree of armoring and bed load transport,
has been emphasized by many researchers (e.g. Kellerhals, 1967, Reid et al, 1985,
Kuhnle and Southard, 1988, Dietrich et al., 1989, Buffington and Montgomery,
1999, Lisle et al., 2000, Emmett and Wolman, 2001, Gomi and Sidle, 2003,
Whiting and King, 2003). This is demonstrated by considering the modeling results
represented by sediment feed and recirculating flume experiments. It is well known
that these two flume types provide substantially different results (e.g. Parker and
Wilcock, 1993, Wilcock, 2001). The latter operates as a closed system and the
results depend on the initial conditions, while the former operates as an open
system and the outcome of the experiments depends on the boundary conditions
(sediment feed rate) as well.
4
Ryan et al. (2005) analyzed bed load measurements collected by the US Forest
Service from steep, coarse-grained streams in Colorado and Wyoming in an effort
to identify patterns exhibited by transported sediments. For their work, they
selected nineteen sites representing a variety of channel types, including step-pool,
plane-bed and pool riffle channels. Their results corroborated the two-phase bed
load transport model that has been advocated in the past for gravel bed streams (e.g.
Jackson and Beschta, 1982). Phase I, dominated by low rates of sand transport, is
followed by a rapid increase in both sediment size and transport rate during phase
II, typically triggered by the breakup of the armour layer at near-bankfull
conditions. Furthermore, the authors found that piecewise linear functions, one for
each phase, provided good representation of the bed load transport data.
Whiting and King (2003) used an extensive set of bed load transport and other
channel related data collected by the US Geological Survey and the US Forest
Service (King et al., 2004) at sites along 12 streams located within the Snake River
Basin in Idaho over long periods to demonstrate the influence of sediment supply
on the make up of the armor layer. The drainage area of the sites varied 50-fold and
the channel slope ranged from 0.0005 to 0.0268. The median size of the subsurface
material, D50s, ranged from 14 to 26 mm, while the corresponding values for the
armour layer, D50, exhibited much wider variation, from 31 to 173 mm. The
resulting degree of coarseness (D50/ D50s) ranged from 1.9 to 7.2, reflecting
substantially different rates of sediment supply from the corresponding drainage
areas. Typically, the median size of the armour layer varies inversely with sediment
supply, while bed load transport increases substantially with sediment supply (e.g.
Kuhnle and Southard, 1988, Whiting and King, 2003). The correlation between
(D50/ D50s) and D* (the ratio of D50s to the median size of transport- and frequencyweighted bed load) (Lisle, 1995) based on the Whiting and King data is shown in
Fig. 1. The degree of armouring increases as the grain size distribution of the bed
load material compared with that of the subsurface decreases (r2 = 0.56). It is
therefore evident that decoupling the stream from its coarse sediment supply,
especially for its mountainous sections, imposes a limitation on the validity of a
bed load transport formula outside its data base. The effects of sediment supply on
stream behavior probably account for the observation reported in the literature that
bed load transport formulae based on laboratory experiments (with sediment fed at
the upstream end or recirculated), which do not experience sediment supply
problems, often overestimate bed load transport rates collected from the field.
However, consideration of the sources supplying sediment to the stream remains a
difficult task. Coarse sediment supplying events, such as landslides and bank
collapses, tend to be episodic and difficult to model. Furthermore, they introduce
considerable variability/fluctuations on bed load transport for identical flow
conditions as they affect the availability of sediment.
A surface-based transport equation, suitable for bimodal sediments, was proposed
by Wilcock and Crowe (2003). The development of the sediment transport model
was based on forty-eight flume runs, performed using five different experimental
sediments. The flume operated in the sediment recirculating mode. Several aspects
5
of the Parker (1990) surface-based bed load formula have been adopted in the new
formulation. Further testing with additional field and laboratory data will be
necessary to examine the validity of this formulation. The same experiments were
used to examine the role of the sand content on the mobility of a sediment mixture
and the evolution of the armour layer with flow strength (Wilcock et al., 2001). A
substantial increase of gravel transport rate, more than an order of magnitude, with
sand content was observed. However, for a given sediment mixture, no significant
change in the grain size of the surface layer with changing flow strength was
detected. This led them to question the traditionally held belief that the armour
layer weakens with increasing flow strength and transport rate.
Several researchers have pointed out that high gradient streams (S > 0.05, S is the
channel bed slope), such as many headwater streams are, have certain features not
commonly encountered in gravel streams with milder slopes. They include a wider
range of particle sizes, sometimes extended all the way to boulders, stepped-bed
morphology, and shallow flows compared with the coarsest sizes present on the
channel bed, resulting in substantial form drag and distortion of the logarithmic
velocity profile. Rickenmann (2001), among many others (e.g. Gomi and Sidle,
2003, Mueller et al., 2005), pointed out that for many of these streams, flow depth
measurements, which can be used to determine boundary shear stress, are not
usually available because they involve significant errors. To overcome this
shortcoming, he proposed a bed load transport formula based on water discharge.
The empirical formula, derived from flume experiments, is a linear function of the
unit discharge in excess of a critical value, a result reminiscent of the formula
proposed by Schoklitsch (1962). Field data from nineteen mountain streams used in
this study support, on average, such trends. However, the variability from stream to
stream is substantial.
Mueller et al. (2005) examined coupled measurements of flow and bed load
transport characteristics from 45 gravel streams exhibiting a wide range of channel
slopes, to determine the variation of the reference bed shear stress, a stress
associated with a very small but measurable bed load transport rate (e.g. Parker et
al., 1982). They concluded that the dimensionless reference shear stress increases
systematically with channel gradient, ranging from 0.025 at low slopes to values
greater than 0.10 for slopes steeper than 0.02. Furthermore, they found that for
nearly all the cases, the dimensionless bed shear stress at bankfull conditions
modestly exceeded the corresponding reference value. This is consistent with
conclusions reached by Parker et al. (1982) for the case of perennial gravel streams
with milder slopes. The high stresses typically encountered in streams with steep
slopes are moderated by excessive resistance, while smaller particles experience
greater hiding effects due to the very wide range of particle sizes available on the
channel bed. The need for using an appropriate skin friction value, that is the total
boundary shear stress adjusted for form drag contributed by large and exposed
particles, in bed load transport calculations has been emphasized by several
researchers (e.g. Andrews, 2000).
6
Behavior of gravel streams at the two ends of the spectrum, near threshold
conditions and at very high shear stresses, has received considerable attention
lately. Several field studies have verified the persistence of the partial transport
phenomenon in natural streams (e.g. Lisle et al, 2000, Church and Hassan, 2002,
Haschenburger and Wilcock, 2003), which was initially documented in a detailed
way in the laboratory by Wilcock and McArdell (1993). It was also reiterated that
an armour layer of given grain size composition has an additional degree of
freedom in enhancing the stability of its particles, when necessary. This can be
accomplished by rearranging the orientation of the particles (e.g. Proffitt, 1980),
and developing clusters and other surface structures that maximize particle
resistance (e.g. Buffington and Montgomery, 1997 and 1999, Church and Hassan,
2002). As a result, critical shear stress for incipient motion can vary by as much as
an order of magnitude (Church and Hassan, 2002), an important consideration
when bed load transport calculations are based on excess shear stress expressions.
An area of fruitful research during the last decade or so has been based on the
analysis and interpretation of data collected under conditions of very high bed load
transport rates, typically encountered in some ephemeral streams and canyon rivers
(e.g. Laronne and Reid, 1993, Reid and Laronne, 1995, Reid et al, 1998, MartinVide et al., 1999, Powell et al, 2001, Cohen and Laronne, 2005). These streams
exhibit neutral stratification of the bed material in terms of grain size (e.g. Powell
et al, 2001, Cohen and Laronne, 2005). In the case of Nahal Yatir, Israel, though,
where extreme floods generated Shields stresses as high as 13 times the critical
value, a layer of finer particle (with a median size of 6 mm) overlying coarser
material (with a median size of 10 mm) was observed (Laronne et al., 1993). In all
cases, the bed load material became progressively coarser with increasing shear
stress, and reached equal mobility conditions at high shear stress values (> 4.5 the
critical value; Powell et al, 2001). Though massive amounts are in transport in
these streams, the bed load behavior seems to be more straightforward with fewer
fluctuations (Kuhnle and Southard, 1988). However, even under such conditions,
there is disagreement about the validity of a suitable bed load transport formula.
For example, Reid et al. (1996) determined that the Meyer-Peter and Muller (M-P&
M) equation matched very well the bed load transport trends exhibited by Nahal
Yatir, while Martin-Vide et al. (1999) estimated that the bed load transport rates
measured from Riera de les Arenes, a steep ephemeral stream near Barcelona,
Spain, were at least four times as high as those calculated from M-P& M.
4
A bed load model that accounts for fractional sediment transport
The brief review presented in the previous section, as well as the broader literature
on the subject, suggests that the development of a reliable bed load transport
formula, valid for a wide range of gravel bed conditions remains an elusive goal.
Though the physical principles governing bed load transport phenomena are
universal, the variability exhibited by natural streams is daunting. Furthermore,
these physical principles are inadequately represented in the rather simplistic
formulations that have been pursued given the complexity of the problem. As a
7
result we end up with multi-valued bed load functions of a given variable.
Predictions within an order of magnitude of measured values have been accepted as
satisfactory results. Emmett and Wolman (2001), both researchers who have
devoted a lifetime of inquiry on the subject, state: “Given this variety and plethora
of controls of channel form, it is not surprising that no all-encompassing
relationship between morphology and transport has been constructed.” Later on
though they add that absence of a universal bed load transport equation does not
imply “uniqueness and disorder.” Similarly, based on the examination of a large
number of bed load measurements in steep, coarse-grained channels in Colorado
and Wyoming, Ryan et al. (2005) conclude: “while there were many similarities in
observed patterns of bed load transport at the 19 studied sites, each had its own
‘bed load signal’ in that the rate and size of materials transported largely reflected
the nature of flow and sediment particular to that system.” One way to improve the
predictability of a certain equation is to calibrate it by using bed load data from the
stream of interest, and thus provide a site specific adjustment (e.g. Bakke et al.,
1999, Barry et al., 2004).
Several bed load data sets have been plotted in Fig. 2, in terms of a dimensionless
bed load transport parameter, W*, and the Shields stress, τ*. They include field data
from Oak Creek (Milhous, 1973), Jacoby River (Lisle, 1989), Sagehen Creek
(Andrews, 1994), Nahal Yatir (Reid et al., 1995) and experimental data obtained by
Proffitt (1980) (initial and final conditions) and Wilcock and Crowe (2003) for the
sediment mixture containing 6% sand. The terms are defined as,
W 
*
* 
q B* 
q B*
 *1.5
0
 g R Dm
qB
Dm ( R g Dm ) 0.5
0  gH S
(1)
(2)
(3)
(4)
where qB is the volumetric bed load transport rate, τ0 is the boundary shear stress, H
is the flow depth, R = (ρ – ρs)/ ρ, ρ is the fluid density, ρs is the sediment density, g
is the acceleration of gravity, Dm is the mode size of the surface material, and S is
the channel bed slope. On average, the data follow a trend that has been expressed
many times before through the following power formula,
W *   * m
(5)
8
Equation (5) is fitted to each data set separately, and the corresponding exponents
and coefficients are shown in Table 1. By representing the range of τ* values for
each data set by the corresponding mean value, τ*mean, the usual reduction in the
exponent m with increasing τ*mean is observed. These results are plotted in Fig. 3, to
obtain the following expression,
*
m  0.05  mean
1.39
(6)
with r2 = 0.94. This result is consistent with the expectation that m will approach
zero for high τ*mean values.
Extensive field and laboratory observations indicate that some of the consistent
trends exhibited by gravel streams include the increase in the degree of coarseness
of the bed load material with shear stress and the concomitant improvement in the
mobility of the coarser particles, approaching a condition of equal mobility for a
shear stress values several times that of the corresponding critical value (e.g. Parker
et al., 1982, Diplas, 1987, Powell et al., 2001). Such information is important in
determining downstream fining and possible evolution of the armor layer with
changing shear stress. To capture these trends, it is necessary to analyze the data in
terms of fractional transport rates. This approach is followed here for a small subset
of the data shown in Fig. 2, namely the Nahal Yatir and the Proffitt data.
Proffitt (1980) conducted experiments in a non-feeding, non-recirculating sediment
flume to study the development of an armour layer in the presence of poorly sorted
bed material. He used four different sediment mixtures and carried out four
experiments with each mixture. He identified three phases during each experiment.
The initial phase lasted for about one hour after the commencement of the
experiment and was characterized by intense and relatively constant bed load
transport rate in the absence of an armour layer. The final phase was characterized
by the highest degree of bed surface armouring, which was distinct for each of the
sixteen experiments, a condition reached after 20 to 95 hours of run time and a bed
load transport rate of 2.5% or lower of that measured during the initial phase of the
corresponding experiment. During the intermediate phase, the channel bottom and
bed load transitioned between the initial and final phases.
The data collected by Proffitt during the initial and final phases of his experiments
are analyzed here separately in terms of fractional transport rates. The original
sediment mixture, used as the composition of the bed surface layer for the initial
phase, and that of the final armour layer, used for the final phase, are divided into
10 size ranges starting with Di = 0.70 mm and ending with Di = 15.55 mm (Table
2), where Di is the geometric mean diameter of the ith grain size range. Material
finer than 0.5 mm has been assumed to be transported, at least in part, in
suspension and thus has been excluded from further consideration. This material
comprised between 0.2 and 1.7% of the total sediment transported during a run.
Equations (1), (2), (3), and (5) are adjusted for fractional transport as follows
9
Wi 
*
 i* 
*
q Bi
 i*
1.5
0
 g R Di
*
q Bi

q Bi
f i Di ( R g Di ) 0.5
Wi*   i  i* mi
(7)
(8)
(9)
(10)
where fi is the percent of the reference material, original mixture or armour layer,
contained within Di; qBi is the volumetric bed load transport rate of size fraction
represented by the diameter Di. Plots of Wi* vs τi*, for both the initial and the final
phases are shown in Fig. 4. The exponents mi and coefficients αi, obtained by fitting
eq. (10) to the bed load data of each size range, are shown in Table 2. For the initial
phase data analysis, the mi values for the first eight size ranges do not vary much
and do not show any specific trend. It is reasonable to assume that they are
transported under equal mobility conditions. No information is lost by combining
them into a single size range. The same is true for the last two size ranges, which
are similarly combined into one.
The final phase exponents, except for Di = 0.72, show consistent but very gradual
changes, increasing with Di. This suggests that the difference in mobility of
consecutive size ranges is very modest and it is not necessary to maintain 10
ranges. These therefore are reduced to 4, as shown in Table 4.
Equation (10) is fitted once again to the fractional bed load data of the initial and
final phases, based on the fewer size ranges. The corresponding mi and αi values are
included in Tables 3 and 4, for the initial and final phases, respectively. In both
cases, the mi values exhibit consistent behavior, increasing with size range. More
specifically, these trends are described by regression equations (11) and (12) for the
initial and final phases respectively,
mi  2.08 (
Di 0.4
)
D50
(11)
mi  2.69 (
Di 0.4
)
D50
(12)
10
Following Diplas (1987), a new independent variable is used, that takes into
account equations (11) and (12), and the data are plotted in terms of Wi* vs

D
( i ) 0.4
* D50
i
(Fig. 5). The resulting regression equations are described by
 * ( DDi ) 0.4  m'
Wi   i  i 50  i


*
(13)
The exponents mi’, shown in Tables 3 and 4 for the two phases, exhibit very small
variability within each phase, suggesting that the new variable has rendered the curves
fitting the fractional data geometrically similar. This in turn, indicates that the new
variable accounts for the differences in mobility among the particles in the various size
ranges.
The bed load transport rates measured in Nahal Yatir, represent some of the highest
rates ever reported in the literature (Laronne et al., 1994). For the very high shear
stress values considered here (at least 4.5 times the critical Shields stress), they reflect
conditions of equal mobility with respect to the surface material. As such, there is no
need to consider fractional transport analysis for this stream. The corresponding
regression equation for this case is,
W *  7.59 * 0.35
(14)
The analysis of this limited number of data sets suggests that as the Shields stress
value increases (see Table 1) fewer sediment fractions are necessary to capture the
trends exhibited by the bed load transport data. Four and two fractions for the final and
initial phases of Proffitt’s experiments respectively, while a fractional approach was
not deemed necessary for the field data from Nahal Yatir. This is consistent with a
large number of observations indicating that the equal mobility condition becomes a
closer representation of reality with increasing Shields stress values. Because of the
improved similarity obtained with the new variable, all the data analyzed here can be
collapsed on a single curve by considering the reference shear stress value, τri, that
corresponds to Wi* = Wr* = 0.0025 for each size fraction; where Wr* = 0.0025
represents a very low transport rate condition (Parker et al., 1982) (Fig. 6). In this
case, the τri values for Proffitt’s data are obtained from eq. (13) and for Nahal Yatir
from eq. (14). Although additional data for intermediate values of bed load transport
and shear stress are required to verify the trends obtained here, Fig. 6 suggests that
fractional transport rates, like the total transport rates shown in Fig. 2, follow
consistent trends, when proper transformations are employed that account for the
gradual adjustments exhibited by the data, from selective transport dominated events
to conditions where equal mobility prevails.
11
5
Streambed Structure
The conventional thinking about the evolution of the armour layer has been that it
achieves its coarsest state under sediment starved conditions, which also represent the
threshold of motion condition. As the shear stress increases, the armour layer weakens
and at reasonably high shear stresses it disappears, rendering the bed material uniform
in the vertical direction (e.g. Parker and Klingeman, 1982; Dietrich et al., 1989,
Parker, 1990). This thinking is well supported by field data of streams operating at
near threshold and very high shear stresses (e.g. for ephemeral streams). There is a
paucity of data though at active bed conditions, given the difficulty of collecting such
data. Andrews and Erman (1986) provide one of the few, if not the only, widely
quoted measurements of an armor layer during a low flow and a flood event in
Sagehen Creek, a perennial stream in the Sierra Nevada of California. The median size
of the armor layer, D50, at low flow was 58 mm and during the peak of the flood it was
46 mm. The corresponding value reported for the subsurface was 30 mm. Proponents
and opponents of the conventional thinking interpret this result differently.
Similarly, results from sediment feed flumes have been used to support the
conventional thinking (e.g. Kuhnle and Southard, 1988) but they have been countered
by evidence from experiments in sediment recirulating flumes that show insensitivity
of the armor layer composition to varying shear stresses (e.g Wilcock and DeTemple,
2005). However, it is well known that, in general, neither flume type accurately
represents the complex field conditions. In the first flume type, the constant feed rate,
in amount and composition, dictates, to a great extent the outcome. In the second
flume type, the initial conditions (the composition of the bed material placed in the
flume) dictate the outcome. Conservation of mass (bed material) requires that after the
finer particles have infiltrated below the surface layer, there is very little room for
further adjustments in the make up of the armour layer except, possibly, at very high
shear stresses when multiple layers (at least three) of bed material are getting entrained
by the flow. There are cases, though, when one flume type is better suited to model a
specific phenomenon. A case in point is the development of a static armour layer
under starved sediment conditions, e.g. simulating the response of a stream bed due to
the construction of a dam. For the feeding case, this will be approached by cutting off
the sediment input at the upstream end, while the transported sediment is allowed to
escape at the downstream end of the flume. For the recirculating case, this
phenomenon can be modeled by gradual reduction in water discharge, or channel bed
slope, or both in an effort to approach conditions at or below threshold. The former is
closer to reality in this case. In general though, a long flume having a test section at its
downstream end, without sediment feed, will provide conditions that are closer to a
natural setting. Cost and space considerations, however, preclude the wide availability
of such facilities.
The US Geological Survey and US Forest Service data (King et al., 2004) employed
by Whiting and King (2003) and plotted in Fig. 1 indicate that an increase in bed load
coarseness is associated with a weaker armour layer. Although it is difficult to
separate the influence of sediment supply on this trend, the large range of D50/D50s
12
values (from 1.9 to 7.2) is worth pointing out. Given the relative constancy of the
subsurface material composition, this result hardly points to a degree of armoring that
remains constant regardless of the flow and other conditions. Finally, the results from
a large number of ephemeral streams, operating under very high shear stresses and
sediment supply, and the absence of an armour layer thereof, further support the
notion that the armor layer weakens with flood stage, assuming that a critical value is
exceeded. In many perennial streams, this value will represent an infrequent flood
event.
6
Conclusions
The significant contributions, in number and quality, that have been made recently in
gravel bed river mechanics, have provided some new insights, reinforced prior results,
and emphasized the complexity of these stream types. While the development of the
“general equation” of bed load transport might not be imminent, sufficient information
exists to identify well defined trends, even at the fractional transport level. Guidance
also exists on how to deal with engineering aspects of the problem. The subject is
approached in a very systematic way, with both extensive field work and laboratory
experimentation. Various issues are tackled at different scales, from the individual
grain to the entire watershed. More data, and of better quality, continue to be gathered.
The subject is ripe for some significant developments.
Acknowledgements
I am grateful to Hafez Shaheen for his help with this paper. This work was supported
by the National Science Foundation (EAR-0439663).
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17
Table 1. Regression results using eq. (5) for seven bed load data sets
τ* range
τ*mean
Expon. m
Coef. α
r2
Nahal Yatir
0.092 – 0.39
0.241
0.35
7.59
0.91
Proffitt Initial
0.046 – 0.126
0.086
1.54
47.40
0.70
Proffitt Final
0.033 – 0.051
0.042
2.63
48.10
0.20
Jacoby River
0.018 – 0.081
0.050
4.64
1.6 x 104
0.69
Sagehen Creek
0.027 – 0.039
0.033
4.83
3.4 x 104
0.26
Wilcock & Crowe
0.015 – 0.050
0.032
7.65
1.0 x 109
0.85
Oak Creek
0.010 – 0.042
0.026
7.95
3.4 x 109
0.89
Data Source
Table 2. Regression results of W*i = αi τ *imi for initial and final phases of Proffitt’s
bed load data
Size Range
Di
(mm)
(mm)
mi
αi
r2
%fiav
mi
αi
r2
%fiav
0.60 - 0.85
0.72
1.78
0.09
0.20
2.07
2.21
6.65
0.57
3.91
0.85 - 1.20
1.01
1.28
0.09
0.14
4.29
1.97
14.66
0.66
7.13
1.20 - 1.68
1.42
1.35
0.18
0.22
4.82
2.02
33.72
0.75
10.01
1.68 - 2.41
2.01
1.67
0.61
0.38
6.58
1.96
64.83
0.77
13.67
2.41 - 3.35
2.84
1.86
1.63
0.45
5.23
1.93
121.86
0.78
11.05
3.35 - 4.76
3.99
2.28
7.81
0.51
13.94
1.90
208.74
0.84
16.00
4.76 - 6.35
5.50
2.65
33.05
0.48
17.64
2.06
528.51
0.84
16.50
6.35 - 9.52
7.78
2.83
39.17
0.55
7.89
2.14
975.28
0.77
5.41
9.52 - 12.7
11.00
3.81 1.3x103
0.39
12.68
3.87
5.14x105
0.55
4.54
12.7 - 19.0
15.55
5.05 5.5x104
0.38
16.27
3.69
2.10x105
0.32
4.61
2.46
0.37
91.41
2.38
0.69
92.83
Ave./Total
Final phase data
Initial phase data
18
Table 3. Regression results using eqs (10) and (13) for Proffitt’s initial phase
data using two grain size ranges
Size Range
Di (mm)
mi
αi
r2
mi ’
(mm)
(1)
(2)
(3)
(4)
(5)
0.85 - 9.52
2.85
1.95
117.56
0.82
2.08
9.52 - 19.0
13.45
3.58
2.37x 105
0.52
2.08
Table 4. Regression results using eqs (10) and (13) for Proffitt’s final phase
data using four grain size ranges
Size Range
Di (mm)
mi
αi
r2
mi ’
(mm)
(1)
(2)
(3)
(4)
(5)
0.85 - 1.68
1.20
1.29
0.13
0.18
2.64
1.68 - 3.35
2.37
1.77
0.98
0.43
2.75
3.35 - 9.52
5.65
2.44
17.65
0.45
2.69
9.52 - 19.0
13.45
3.41
315.55
0.40
2.66
19
D50/D50s
8
6
4
2
0
0
5
10
D
15
*
Figure 1. Variation of the degree of armouring with change in degree of bed load
coarseness
20
1.E+02
Nahal Yatir
1.E+01
1.E+00
Proffitt Initial
1.E-01
W*
1.E-02
Proffitt
Final
Jacoby River
Sagehen Creek
1.E-03
1.E-04
Proffitt Final Phase
Proffitt Initial Phase
Wilcock and
Crowe
Wilcock and Crowe
Oak Creek
1.E-05
Oak Creek
Nahal Yatir
1.E-06
Sagehen Creek
Jacoby River
1.E-07
1.E-03
1.E-02
*
1.E-01
1.E+00
Figure 2. Dimensionless bed load transport rate, W*, versus Shields stress, τ*, for several
field and laboratory data sets.
21
m
9
8
7
6
5
4
3
2
1
0
0
0.05
0.1
0.15
0.2
0.25
0.3
* mean
Figure 3. Variation of the exponent m for the data sets shown in Fig. 2
22
1.E+01
5.5
3.99
7.78
11
2.84
2.0
1.42
1.0
0.72mm
1.E+00
15.6
W*r = 0.35
Intial phase bedload data
1.E-01
3.99
5.5
2.84
2.0
1.42
1.0
0.72mm
7.78
11
W*i
1.E-02
15.6
W*r = 0.0025
1.E-03
Final phase bedload data
1.E-04
Di mm
1.E-05
1.E-06
1.E-02
1.E-01
1.E+00
*i
Figure 4. Plot of W*i versus τ*i for Proffitt’s fractional bed load transport data
during initial and final phases. Ten size ranges for each case are considered.
23
10
W*i = W*r (x)2.08
r2 = 0.76
W*r = 0.0025
1
0.1
2.67
W*i = W*r (x)
2
r = 0.81
W*r = 0.0025
W*i
0.01
0.001
0.0001
Final Phase (Proffitt Data)
Initial Phase (Proffitt Data)
0.00001
0.000001
0.1
1
10
100
x = (T*i/T*r)^(Di/D50)^0.4
Figure 5. Wi* versus τ* (Di/D50) ^0.40 for the initial (two size ranges) and final (four size
ranges) phases of Proffitt’s bed load data.
24
1.E+04
1.E+03
1.E+02
1.E+01
W*i
1.E+00
1.E-01
1.E-02
1.E-03
Final Phase (Proffitt)
1.E-04
Initial Phase (Proffitt)
1.E-05
Nahal Yatir
1.E-06
1.E-01
1.E+00
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1.E+07
1.E+08
1.E+09
1.E+10
(*i/*r) (Di/D50)^0.4'
Figure 6. Plot of the modified similarity collapse for Proffitt’s (initial and final
phases) and Nahal Yatir data.
25